Probing the early universe: a review of primordial nucleosynthesis beyond the standard big bang

PROBING THE EARLY UNIVERSE: A REVIEW OF PRIMORDIAL NUCLEOSYNTHESIS BEYOND THE STANDARD BIG BANG

Robert A. MALANEY Institute of Geophysics and Planetary Physics, University of Caljfornia, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA and Canadian Institute .for Theoretical Astrophysics, University of Toronto, Toronto, ON, Canada M5S 1A7 and Grant J. MATHEWS P-Division, Physics Department, University of California, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA

NORTH-HOLLAND

PHYSICS REPORTS (Review Section of Physics Letters) 229, No. 4 (1993) 145—219. North-Holland

PHYSICS REPORTS

Probing the early universe: a review of primordial nucleosynthesis beyond the standard big bang Robert A. Malaney a,b and Grant J. Mathews C a b C

Institute of Geophysics and Planetary Physics, University of California, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA Canadian Institute for Theoretical Astrophysics, University of Toronto, Toronto, ON, Canada M5S JA 7 P-Division, Physics Department, University of California, Lawrence Livermore National Laboratory, Livermore, CA 94550, USA

Received January 1993; editor: D.N. Schramm Contents: 1. Introduction 147 2. Standard big bang nucleosynthesis 148 2.1. The standard model 148 2.2. Recent developments 150 3. Inhomogeneous and anisotropic models 154 3.1. Homogeneous anisotropic models 155 3.2. Adiabatic fluctuations 156 3.3. Isothermal or isocurvature baryon number inhomogeneities 156 3.4. The QCD phase transition 157 3.5. Other means to generate baryon number density fluctuations 162 3.6. Nucleosynthesis and baryon diffusion 164 4. Late-time decay or annihilation 171 5. Neutrinos 175 5.1. Neutrino degeneracy 175 5.2. Sterile neutrinos 177 5.3. Neutrino mass 178 5.4. Neutrino oscillations 179 5.5. Electromagpetic properties 181 5.6. Weak interaction properties 184 6. Cosmic strings 184

6.1. Ordinary cosmic strings 6.2. Superconducting cosmic strings 7. Time variation of fundamental constants 8. Other variants 8.1. Shadow matter 8.2. Baryon oscillations 8.3. Massive charged particles 9. Observations and galactic evolution effects relevant to non-standard models 4He 9.1. 9.2. 2H and 3He 9.3. 7Li 9.4. 6Li 9.5. Beryllium and boron 10. Nuclear uncertainties 10.1. Standard big bang reactions 10.2. Inhomogeneous models 10.3. Late-decaying or annihilating particle models 10.4. Other models 11. Conclusions References

184 186 188 191 191 192 192 192 193 195 195 200 201 203 203 205 207 208 208 209

Abstract: The scientific literature is rich with studies of non-standard primordial nucleosynthesis. We review a number of these variants on the simplest standard big bang (SBB) nucleosynthesis model and discuss some of the impact these studies have had on cosmology and particle physics. Primordial nucleosynthesis is one of the earliest probes of the universe, and non-standard nucleosynthesis models allow for an exploration of the range of conditions which might have prevailed during the first few minutes. In particular, nonstandard models often allow for a larger range of conditions to be present in the early universe than those allowed by the SBB while still satisfying observational constraints such as the inferred primordial isotopic abundances and the number of neutrino species derived from recent e+ e collider experiments. By considering alternatives to the SBB we can also determine the model dependence of the constraints imposed on particle physics and cosmology from primordial nucleosynthesis.

0370-1573/93/s 24.00

© 1993 Elsevier Science Publishers B.V. All rights reserved

1. Introduction The Standard Big Bang (SBB) model is a cornerstone of modern cosmology. One of the most outstanding successes of this model has been the quantitative prediction of primordial abundances for light elements [Wagoner et al. 1967; Wagoner 1973]. Based on a simple set of assumptions (isotropy and homogeneity; validity of general relativity; principle of equivalence; no particle degeneracy; thermodynamic equilibrium) SBB nucleosynthesis is able to reproduce the observationally inferred primordial abundances of D, 3He, 4He and 7Li using only one free parameter, the baryon-to-photon ratio i7. Since the abundances of these primordial isotopes span some ten orders of magnitude, this is indeed an impressive success for SSB nucleosynthesis. Furthermore, recent results from the LEP and SLC collider studies on the width of the Z°particle [Aarnio et al. 1989; Abrams et al. 1989; Adeva et al. 1989; 1991, 1992; Akrawy et al. 1989; Decamp et al. 1989, 1990; Dorfan et al. 1989] are in excellent agreement with the complimentary constraint on the number of relativistic neutrino families, N~< 3.3, predicted by SBB theory [e.g. Steigman et al. 1977, 1986; Olive et al. 1990; Walker et al. 19911 even before these measurements were carried out. At the time when the first big bang predictions were made the upper limit to the number of neutrino families from particle physics was in the thousands. In view of these remarkable successes one may question the motivation for investigating variants on the SBB theory. There are at least three solid motivations: (i) The first of these stems from the crucial implications of SBB nucleosynthesis on our understanding of the universe most notably on the baryon-to-photon ratio, i~.The constraints on ~ imposed by SBB nucleosynthesis forces the conclusion that the majority of matter in the universe is in some unknown non-baryonic form. This is the main reason for the widely held notion that the universe consists largely of “exotic” dark matter, and is the major motivation for all non-baryonic “dark” matter searches. One should therefore consider seriously any variants of the SBB model which might influence the range of allowed values for the baryon-to-photon ratio from primordial nucleosynthesis. (ii) In the SBB it is assumed that the universe has expanded from an initial state of very high temperature and density. At present, the thermodynamic history of the universe can be discussed (though not necessarily understood) back to times just after the Planck time t iO~~ s, at which point the temperature and density are T 1019 GeV and p 1078 GeV fm3, respectively. Although recent observations [e.g. Smoot et al. 19921 of the cosmic microwave background anisotropy may provide some glimpse of conditions in the universe near the Planck time when the universe was in the inflationary epoch [Davis et al. 19921, we still have no direct proof that the universe ever was at such extreme conditions. Indeed, although it only extrapolates back to times of t 10—2 where T 10 MeV and p 100 g cm3, primordial nucleosynthesis is still, perhaps, the best direct probe of the young universe. Careful and detailed studies of variant nucleosynthesis scenarios are therefore necessary to quantify the broadest range of conditions of the early universe consistent with the presently derived primordial abundances. Similarly, the yields from primordial nucleosynthesis are very sensitive to conditions in the early universe. Therefore, the observed primordial isotopes can probe the physics of non-standard models which may have been determined at times much earlier than those addressed in the standard big bang model. (iii) Finally, there is the particle—astrophysics connection. There has been a growing trend in —

“~

“~

147

~,

148

R.A. Malaney and G.J. Mathews, Probing the early universe

the past decade or so to use cosmological considerations as a means of investigating the validity of extensions to the standard model of high-energy physics (SU (3 )~ ® SU (2) L ® U (1) y). In many instances primordial nucleosynthesis is the only probe of such extensions. The nucleosynthesis constraint on the number of neutrino generations (now experimentally verified) was only one such constraint. For example, QCD, grand unified theories, the properties of new particles, topological defects, and the time dependence of fundamental constants can all be constrained using primordial nucleosynthesis. It is important to quantify the nucleosynthesis constraints on particle-physics models which often imply significant variants of the SBB. The scientific literature in the past few years has been rich with work which describes possible variants on SBB nucleosynthesis. Some of the most widelydiscussed of these variants have been those related to late-decaying massive particles, the QCD phase transition in the early universe and its resulting prediction of inhomogeneity, unusual neutrino properties, and cosmic strings. In addition, some less widely discussed variants on the standard model such as time-dependent fundamental constants and shadow matter have also been studied. Due to the lack of any recent review article on these matters, this avalanche of new ideas regarding non-standard primordial nucleosynthesis has become quite bewildering. We recognize that with so many significant contributions it is not possible to give an adequate discussion of all of them. Nevertheless, in these pages we attempt to summarize in one article the span of models related to non-standard nucleosynthesis, with particular emphasis on some of the most recent ideas. We begin with a short updated review of the SBB nucleosynthesis model and then review various departures from the standard model. We also summarize the implications for non-standard models of some new observations of the isotopic abundances, and highlight the improvements required from both astronomical observation and experimental nuclear physics as well as studies of galactic chemical evolution effects required for discrimination between the competing scenarios for standard and non-standard primordial nucleosynthesis.

2. Standard big bang nucleosynthesis For detailed discussions of SBB nucleosynthesis and the large amount of effort involved in inferring the primordial abundances, the reader is referred to one of the many fine reviews written on the subject [Schramm and Wagoner 1977; Boesgaard and Steigman 1985; Matzner 1986; Kolb and Turner 1990], or to some of the original papers [Gamow 1946; Alpher et al. 1948, 1950, 1953; Hoyle and Taylor 1964; Peebles 1966; Wagoner et at. 1967] and major studies [Wagoner 1973; Yang et at. 1984; Olive et al. 1990; Walker et al. 1991; Smith et al. 1993]. Here, we just introduce the basic ideas and relations of SBB nucleosynthesis. These relations will be useful for our later discussions of non-standard nucleosynthesis models. In all of our discussions we have adopted Ii = c = 1. 2.1. The standard model By considering the constraints imposed on Einstein’s field equations by an isotropic and homogeneous Robertson—Walker metric, one can derive the Friedmann models of the universe. The evolution of the scale factor R of these models is described by a simple first order differential equation which, ignoring the cosmological constant, is just, 2+k=!.!~PR2,

(~)

(2.1)

R.A. Malaney and G.J. Mathews, Probing the early universe

149

where p is the energy density of the universe and the curvature index, k = 0,±l,is a dimensionless constant which measures the curvature of space. We can see from this equation that the critical density required for a flat (k = 0) universe is given by 3H2 Pc~~ (2.2) where H = (1/R) (dR/dt) is the Hubble parameter. The energy density is commonly expressed in terms of a dimensionless density parameter, Q = P/Pc, such that Q = 1 describes a flat Einstein—DeSitter universe. For all times relevant for primordial nucleosynthesis, k<
(2.3)

and eq. (2.1) can be rewritten H2 (t)

(ldR\2

=

8irG —~—-p,

(2.4)

giving the direct dependence of the universal expansion rate on the energy density. For an ideal gas in thermal equilibrium, the number density of particles of type i with momenta between p and p + dp is given by a Fermi—Dirac or Bose—Einstein distribution, nt(p)dp

=

_Lgjp2

[e*~ (Ei(P)_ iii) ±i]

dp,

(2.5)

where E, (p) is the energy of the particle, the ±sign is for bosons and + for fermions, and g is the number of spin states of the particle (g = 1 for neutral massless leptons, g = 2 for charged leptons and photons). Integrating E (p )n (p) over all momenta gives the energy density for particle i. In the standard big bang, with all chemical potentials, /t~,set to zero, the total energy density of the universe at the epoch of nucleosynthesis is given by —

p=py+pv,+pi,

(2.6)

where p~,p,.,~and p 1 are the energy densities due to neutrinos, photons and charged leptons, respectively (including antiparticles). At the temperatures of primordial nucleosynthesis only electrons (and positrons) contribute to the charged leptons. The contribution from neutrinos is given by (assuming only relativistic particles) p~=

[n~(p)+ nv,(p)] dp,

(2.7)

which to reduces to 2 2/15)(kT~)4, (2.8) Pu, ~ 8(,r where T~is the neutrino temperature. The energy densities from photons p~and relativistic charged leptons p are given by P1 = ~Pl = 2(kT 4, 1~ir 1)

(2.9)

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R.A. Malaney and G.J. Mathews, Probing the early universe

where T1 is the photon temperature (henceforth we absorb the Boltzmann constant k into the temperature). For temperatures above 1 MeV the neutrinos and photons are in thermal equilibrium (T~= T1 = T). Equations (2.4), (2.6), (2.8), and (2.9), plus the fact that temperature decreases inversely with the scale factor, can be used to derive the relation between temperature and time in the early universe. —1/2

4(.j.~—)

MeV, T~l.6g~ where g~r( T) represents the effective number of relativistic species, g~ff(T) =

~

(2.10)

(2.11)

+ ~ ~IJ~mi gFe~i.

Bose

We briefly consider the three important time epochs of importance to nucleosynthesis: (a) At a time oft 10—2 s (T 10 MeV) the weak interaction rates are faster than the universal expansion rate. The energy density of the universe is dominated by relativistic particles y, e~,e, v~,i.., in thermal equilibrium (from eq. (2.11) we have g~r 10.75), and the neutron-to-proton ratio is given by its equilibrium value, n/p = exp(—AM/T), where AM is the neutron—proton mass difference. Due to the high ambient temperature, the nuclear statistical equilibrium is shifted to a photodissociated nucleon gas and no nucleosynthesis occurs. (b) At t 1 s (T 1 MeY) the weak interaction rates can no longer maintain thermal equilibrium and the n/p ratio is “frozen in” at a value of 1/6. After this time the n/p ratio is only modified by free neutron decay. A crucial point to note here is that any increase in p will lead to an increase in the expansion rate and a higher freeze-out temperature. This implies a higher equilibrium n/p ratio when weak reactions freezeout which will lead to a higher 4He abundance when nucleosynthesis begins. At T 0.5 MeV the e~e pairs annihilate heating the photon gas, but not the neutrino gas which decoupled from the plasma at T 1 MeV. The temperature is still too high, however, for nucleons to assemble into nuclei. (c) At t 100 s (T 0.1 MeV) the photodissociation rate of deuterium is low enough for significant D production to occur along with the production of the other primordial isotopes 3He, 4He and 7Li. However, the nuclear reaction rates diminish too rapidly to maintain nuclear statistical equilibrium. To calculate the nucleosynthesis yields exactly the time dependence of the temperature must be coupled to a network of nuclear reaction rates [Wagoner et al. 1967; Wagoner 1973]. The coupled differential equations which describe the time evolution of the isotopic abundances must then be integrated until the density and temperature fall to the point where nuclear reactions become ineffective. ~

2.2. Recent developments

In recent years a number of developments have impacted SBB nucleosynthesis. The most exciting of these developments is the measurement of the width of the Z°particle derived from the results of the LEP and SLC e~e colliders [Aarnio et a!. 1989; Abrams et al. 1989; Adeva et a!. 1989, 1991, 1992; Akrawy et al. 1989; Decamp et at. 1989, 1990; Dorfan et al. 1989]. Since the width of the Z°is directly related to the number of light (m~< Mz/2) particles which can couple to the Z° it is possible to determine an independent limit on the number of light neutrinos. From the results so far reported [e.g. Adeva et at. 1991] one finds an upper limit to the number of light neutrino species of N,, ~ 3.25 at the 2a level (see fig. 1).

R.A. Malaney and G.J. Mathews, Probing the early universe

0.26

36

—

87

I 88

I 89

I 90

I

I

I 91

I 92

i

-

151

iii,

ii

I

I 93

II 94

—

95

96

Ecm (GeV)

Fig. 1. The measured cross section for e+ e —~ hadrons as a function of the center-of-mass energy [Adeva et al. 1989]. The dotted, solid, and dashed curves correspond to two, three, and four neutrino families, respectively.

—

I 10~0

I 100

Fig. 2. Light-element abundance versus baryon-to-photon 4He mass fraction, (D + 3He)/H, and 7Li/H. ratio ,~for the The boxes indicate observationally acceptable regions for 17: the horizontal lines give upper and lower bounds to the deduced primordial abundances; the vertical lines show the resulting bounds on ~. The dashed curves show the 2a abundance curves as determined by Monte Carlo simulation. The hatched regions show the values of ~ for which there is concordance between computed abundances and observational constraints. Taken from [Smith et al. 1993].

New techniques have been developed to determine the neutron half-life. In particular, those based on the storage of ultra-cold neutrons [Mampe et al. 1989; Afimenkov et at. 1990] have led to a considerable reduction of the uncertainty in this important input parameter. By combining this latest result with other recent results since 1986, Smith et al. [1993] infer the neutron half-life to be 10.26 ±0.03 mm. This is to be compared with the value of 10.6 ±0.2 mm employed in previous studies only a few years ago [e.g. Yang et al. 1984]. The change in the neutron half-life slightly reduces the 4He abundance and considerably reduces its uncertainty [Olive et al. 1991; Walker et al. 1991; Smith et al. 1993]. New experimental studies and analyses of key reactions in the SBB nuclear reaction network have resulted in modified production rates. The most significant of these modified rates are those concerned with the production of 7Li and to a lesser extent the D + 3He abundance. Reference to, and compilations of many updated rates can be found in [Caughlan and Fowler 19881 and [Smith et al. 1993]. Much work has also been carried out to inferr the primordial abundances of light isotopes from

152

R.A. Malaney and G.J. Mathews, Probing the early universe

observation. In many metal-poor halo stars a very narrow plateau of stars is observed [Spite and Spite 1982, 1986; Spite et at. 1984; Hobbs and Duncan 1987; Rebolo et at. l988b]. In the effective temperature range 5500 < T~.< 6350 K, there is an approximately uniform abundance of Li relative to hydrogen. An average of 35 stars with Teff> 5500 weighted by the quoted uncertainties for [Li] defined as log(Li/H) + 12 2.08 ±0.07(2u) [Walker et at. 19911. A systematic study of the evolution of Li in the atmospheres of old halo stars has been carried out [Deliyannis et al. l990b] which may allow for refined limits on the primordial 7Li abundance to be deduced from observations of these plateau halo stars. By fitting the data to standard non-rotating stellar evolution models the primordial Li abundance was inferred to be tog(Li/H) + 12 ~ 2.21 (at the 2a level). With allowance for stellar diffusion processes the limit was increased to log(Li/H) + 12 < 2.36. Allowing for rotational mixing increases this limit to 3.1. It is important, therefore, to appreciate that the surface lithium lies in a very thin region ~ 0.05M®) of the stellar surface and a number of subtle processes (e.g. magnetic bubbles, changes in opacity, turbulent mixing) could affect the inferred primordial abundance (see section 9.3). An improved determination of the 4He abundance has also been pursued. In order to determine the primordial 4He abundance one must correct for stellar processing. Such a correction is usually attempted by correlating the 4He abundance with metallicity and extrapolating to zero metallicity. Usually observations are taken of highly-ionized H II regions in low-metallicity galaxies with oxygen as the metallicity tracer [Peimbert and Torres-Peimbert 1974]. It has been pointed out [Kunth and Sargent 19851, however, that oxygen could actually be a poor diagnostic of 4He production since 4He is produced in stars of mass M ~ 2M® whereas oxygen is produced only in stars of mass M > 12M®. Since it is expected that nitrogen and carbon are also produced in M ~ 2M® stars, it has been argued that these elements might be an improved diagnostic of the primordial 4He abundance. By compiling data sets of 4He as a function of C, N and 0 new determinations [Pagel 1988; 1989; 1991; Pagel and Simonson 1989; Steigman et a!. 1989; Torres-Peimbert et at. 1989; Fuller et a!. 1991; Pagel et at. 1992; Mathews et al. 1993b] of the 4He abundance have found Y~= 0.23 ±0.01. It has been noted, however, that if the early contributions to nitrogen production have significant secondary sources, as seem required by the data, then the mean primordial helium abundance may be as low as 0.22 [Fuller et a!. 1991; Mathews et at. 1993a; Batbes et at. 1993]. It has also been recently noted [Skillman 1993] that new emissivity corrections may increase the optimum value for Y~above 0.23. In all of these analyses there are large possible systematic uncertainties (-...~ 0.005 or more [Pagel 1991]) which are comparable to the statistical errors (~-~0.005) in the extrapolations of the data to zero metallicity. So a reasonable upper limit to the observed helium abundance is probably Y~<0.24 (a fuller discussion of these issues is given in section 9.1) The effects of these new developments can be seen from the Monte Carlo study [from Smith et at. 1993] shown in fig. 2, where the D, 3He, 4He and 7Li abundances are shown as a function of ~ io, the baryon-to-photon ratio in units of 10—10. The inferred primordial abundances assumed, and the constraints they impose on ~10 are given in table 1. The region of consistency of i~,is bounded from below by the lower limit inferred on D + 3He and bounded from above by the upper limit inferred on 4He, is 2.86 < < 3.77. Previous analyses [e.g. Walker et al. 1991], found that the upper bound was constrained by 7Li and not 4He. However, the new reaction rates employed in [Smith et at. 1993] result in a 20% reduction of 7Li at high The range of ~ can be converted into a permitted range of the baryonic contribution to the closure density ~b using the relation ~.

Qbh2

=

3.77 x l03~

3, 1o(T1/2.76)

(2.12)

R.A. Malaney and G.J. Mathews, Probing the early universe

153

Table 1 Table 1. The inferred primordial abundances assumed in the study of [Smith etal. 1993]. Element Constraint deuterium deuterium helium-4 and helium-3

D/H 1.8 x i0~ 3He)/H < 9.0 x l0~ (D +
lithium

1.1 x 10—10 < 7Li/H

2.3 x 10’° (Pop H) 1.3 x iO~ (Pop I)

where h is the Hubble parameter in units of 100 km s1 Mpc’, and T 1 is the present microwave background temperature. T1 = 2.76 ±0.02 (2a) K is a weighted mean of results from COBE [Matheret a!. 1990] and other recent recent measurements at wavelengths greater that 1 mm [Gushet at. 1990; Kaiser and Wright 1990; Kogut et a!. 1990; Meyer et at. 1990; Palazzi et at. 1990 and refs. 2therein; Amiccietetal.at.1993]. 1991]. From eq. (2.12) the allowed range of Qb is given 0.015De[Smith byThe 0.011
154

R.A. Malaney and G.J. Mathews, Probing the early universe

Reviews of the dark matter problem are given in [Trimble 1987; Turner 1987; Tremaine 1992; Ashman 1992]. Another potential source of discrepancy with the SBB model is that 4He can only be consistent with other light-element constraints if Y~, > 0.237 [Smith et at. 1993]. This is to be compared with the upper limit inferred from observation, Y~, < 0.24 (although new emissivity corrections [Skitlman 19931 may increase this upper limit). At this level of comparison, small effects, hitherto neglected, may be important. For example, it has been recently argued [Rana and Mitra 1991] that neutrino re-heating could lead to a 0.3% reduction in Yr,. Finally, it is useful for our subsequent discussion to construct some semi-empirical formulae. For 2.5
0.228 + 0.01 ln~ 10+ 0.0!2(N,, —3) + 0.185 (T8~.8),

(2.13)

where r~is the neutron mean life measured in seconds. This can be rewritten in terms of the number of neutrino species N,,

=

3.0—0.8!n,~10+ 19

fY~—0.228\ fr~—889.8\ ~,. 0.228 —15 889.8

)

~.

)

(2.14)

Adopting the lower bound of n~o 2.8 [Walker et at. 1991], the lower 2a bound on r~,and the upper bound Y~~ 0.24,0 experimental the number of neutrino species is constrained to be N,,
3. Inhomogeneous and anisotropic models The idea that there could be deviations from the homogeneous isotropic Friedmann—Robertson— Walker (FRW) conditions during the big bang is not new [e.g. Misner et al. 1973; Ellis 1984 and refs. therein]. There are essentially three deviations from the FRW conditions to consider [Misner et at. 1973; Schramm and Wagoner 1977; Ellis 1984]. They are: (i) Homogeneous but anisotropic models [Hawkingand Taylor 1966; Thorne 1967; Barrow 1976; 1984; Olsen 1978; Rothman and Matzner 1984; Kurki-Suonio and Matzner 19851; (ii) adiabatic fluctuations in the radiation energy density [Gisler et al. 1974; Olsen and Silk 1978]; and (iii) isothermal (constant temperature) or isocurvature (constant total energy density) fluctuations in the baryon number density [Harrison 1968; Zeldovich 1975; Epstein and Petrosian 1975; Barrow and Morgan 1983; Rees 1984; Sale and Mathews 1986]. Models have also been considered which involve a combination [Centretla et at. 1986; Kurki-Suonio and Centrella 1990] of the above effects. For (i) and (ii) the effect on primordial nucleosynthesis is primarily through variations in the expansion rate. For (iii) the nucteosynthesis is also influenced by variations in the baryon-to-photon and neutron-to-proton ratios as discussed below.

R.A. Malaney and G.J. Mathews, Probing the early universe

155

3.1. Homogeneous anisotropic models When the three curvature of the universe is very close to isotropic [Eardley et a!. 19721, the lowest order effect of anisotropic regions on the FRW expansion rate is to add two terms to eq. (2.4); (3.la) H2 = ~irGp+ a2 +R*, where a is the scalar shear measuring the magnitude of the shearing anisotropic flow and R5 is the scalar curvature of the comoving 3-spaces orthogonal to the fluid-flow 4-vector. Because the shear enters quadratically it can only increase the expansion rate relative to the SBB model. Hence, anisotropic models tend to increase the primordial helium abundance by causing the n/p ratio to freeze out at an earlier time~and higher temperature. More generally, the shear can be written as a tensor [Barrow 1984] which to first order evolves as ~

+ 3(R/R)a, 1

=

—(R,~ ~5~~R*) + 8irG(T~~ ~.ô,i77), —

(3.lb)

—

where R,~is the Ricci 3-curvature and ~ is the stress energy tensor. The evolution of the shear term has been studied in homogeneous but anisotropic models [Hawking and Taylor 1966; Thorne 1967; Barrow 1976; Olsen 1978; Rothman and Matzner 1984; Kurki-Suonio and Matzner 1985]; in the context of Kasner or Bianchi type I models [Landau and Lifshitz 1975] for local anisotropic islands of flat-space curvature (i.e. R* = R,~= ~ = 0 in (3.la)). For such models a and the results can be characterized by a temperature, T5, above which the shear scalar dominates the dynamics. By appropriately choosing the volume associated with the anisotropic islands and T5, the deuterium and helium abundance constraints can be satisfied even in a universe with ~b = 1 for T5 l0~K [e.g. Olsen 1978; Olsen and Silk 1978]. The shear also decays away sufficiently rapidly [Rothmanand Matzner 1984] to damp the fluctuations to a level of AT/T 4 x 10~ and thus satisfy observed limits to variations of the cosmic microwave background temperature 2 from primordial nucleosynthesis, [Smoot Ct al.constraint 19921. Asis far as the limits on allowed values forfraction a the strongest from primordial helium mass which increases by a factor of 1.03 [Kurki-Suonio and Matzner 1985] for an energy density in anisotropy comparable to the radiation energy density at T = 5 x 1010 K. This restricts the anisotropy energy density to be less than the radiation density before nucleosynthesis (at T 1010 K when the helium abundance is fixed by the weak reaction freeze out), and implies an exceedingly small anisotropy at the present time [Olsen 1978]. Helium production in anisotropic models which do not contain the Friedman model as a special case have also been studied [Barrow 19841. In that paper three exact non-Friedmanian cosmological models were explored. These were a Bianchi 1V 0 tilted anisotropic model [Rosquist 1983], a Bianchi II radiation solution [Collins and Stewart 1971; Wainright 1984], and a radiation filled Bianchi Type IV model [Collins 1971]. In these models the previously simple evolution of the scalar shear is complicated by the anisotropic part of the stress energy tensor and/or in eq. density (3.1 b). 2 time dependence as the curvature radiation energy The shear energy term can then have the same t so that the shear energy does not dominate over the radiation energy density at early times as in models with flat or isotropic curvature. Therefore, even substantial anisotropies only increase the helium abundance by 0.01 to 0.05 [Barrow 1984]. Of course, the metrics associated with such non-Friedman anisotropic models cannot be a viable description of the universe since they imply quadrupole anisotropies in the cosmic microwave background, AT/ T 0.1—1, which are ~

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R.A. Malaney and G.J. Mathews, Probing the early universe

ruled out by observation. On the other hand, as with the Friedman models discussed above, the dynamics of the early universe might have been dominated by local regions with large curvature anisotropies even if the mean anisotropy was zero. These models are thus yet another example of how nucteosynthesis-motivated constraints derived from the most straightforward modifications to the standard big bang may be circumvented by non-standard models. 3.2. Adiabatic fluctuations Adiabatic fluctuations with AT/T> 6.3 x tO~in the cosmic microwave background have been ruled out at the 95% confidence limit down to an angular size of? 0.0015 deg [Hogan and Partridge 1989]. However, the angular size of a region of the microwave background corresponding to the horizon at the time of primordial nucteosynthesis is exceedingly small V 10—6 deg), and even if formed, fluctuations this small could have damped out by the time of recombination [Liang 19771. So it is at least possible that there could have been significant adiabatic fluctuations on the scale of the horizon during primordial nucleosynthesis. Adiabatic fluctuations with wavelengths smaller than the horizon during nucleosynthesis would appear as oscillating acoustic waves, while those larger than the horizon during nucleosynthesis will act like separate universes each evolving as a separate standard big bang. Thus, one would expect fluctuations larger than the horizon to produce primordial nucleosynthesis yields similar to the standard homogeneous big bang, i.e. each region would evolve through the nucleosynthesis epoch identically (for the same baryon-to-photon ratio) albeit at a different time. However, it has been demonstrated [Gisler et at. 1974; Olsen and Silk 1978] that large wavelength adiabatic fluctuations can lead to variations in the local curvature such that the expansion rate through the epoch of nucleosynthesis is different for different regions. This can lead to substantial changes in the nucleosynthetic yields. For small wavelength fluctuations there can be many oscillations during the epoch of nucteosynthesis so that the yields only reflect the average. On the other hand, the expansion rate can be affected due to the additional energy density associated with the acoustic waves. The maximum effects [Gisler et a!. 1974; Olsen and Silk 1978] on the yields from primordial nucleosynthesis are from models with moderate-sized fluctuations with wavelengths just comparable to the size of the horizon during primordial nucleosynthesis. In this case there can also be an effect due to the oscillating energy density. For example deuterium could be produced preferentially in the rarefaction part of an acoustic wave where the temperature is lower while helium would be only modestly affected. Relativistic time dilation could also affect nucleosynthesis [Gister et at. 19741 as matter moves with the acoustic waves. In addition, the specific entropy at the time of nucleosynthesis can be reduced relative to the standard model. The reason for this is that the entropy is later increased as the acoustic waves are damped and thermatized. Thus, the baryon-to-photon ratio at the time of nucteosynthesis could be greater than that inferred from the present microwave background temperature. 3.3. Isothermal or isocurvature baryon number inhomogeneities Because only a small fraction of the total energy density is in baryons during the epoch of primordial nucleosynthesis, there is little difference between isothermal and isocurvature fluctuations in the baryon density. The effects of an inhomogeneous distribution of baryon-to-photon ratio was considered even in some of the earliest work on big bang nucleosynthesis [e.g. Wagoner 1973]. The simplest effects of baryon inhomogeneities can be derived in the limit of larger than horizon sized

R.A. Malaney and G.J. Mathews, Probing the early universe

157

fluctuations so that regions of different baryon density could be treated as separate big-bang models which could be averaged after the end of nucleosynthesis. A number of such studies have been made [e.g. Harrison 1968; Wagoner 1973; Zeldovich 1975; Epstein and Petrosian 1975; Barrow and Morgan 1983; Wagoner 1973; Yang et al. 1984]. These have concluded that it might be possible to produce deuterium in the low-density regions and 4He in the high-density regions, and thus allow for a larger value of q or ~b than in baryon inhomogeneous models. If the high-density regions could somehow be collapsed into baryonic dark-matter remnants then all of the light-element abundance constraints could even be satisfied for an Qb = 1 universe [Rees1984; Sale and Mathews 1986]. However, in all of these early studies it was assumed that the distance scale of the fluctuations was much longer than the baryon diffusion scale. Possible effects of baryon diffusion were, therefore, not considered. There has, however, more recently been a great deal of renewed interest in primordial nucleosynthesis in a universe with an inhomogeneous distribution of baryon number. There are two reasons for this. One is the recognition that besides the effects of varying baryon-to-photon ratios there is an important modification of the nucteosynthesis yields due to the diffusion of neutrons into tow-baryon density regions [Applegate et at. 1987]. This causes a variation of the neutron-to-proton ratio as welt. The second reason for the renewed interest is the emergence of plausible scenarios for the development of such baryon-number inhomogeneities before the epoch of nucleosynthesis [Witten 1984]. The most popular mechanism for the generation of baryon-number inhomogeneities has been a first order QCD phase transition, although a number of other processes could also contribute (see section 3.5). We therefore begin with a review of possible sources for baryon-number inhomogeneities and then present a general discussion of how such fluctuations (independent of the means for their generation) could affect the yields from primordial nucleosynthesis. 3.4. The QCD phase transition There have been a number of studies of the QCD transition from quark—gluon plasma to a hadronic gas [e.g. Crawford and Schramm 1982; Kolb and Turner 1982; Suhonen 1982; Bonometto 1983; Dixit and Suhonen 1983; Hogan 1983; Lodenquai and Dixit 1983; Bonometto and Sakeltariadou 1984; Gyutassy Ct al. 1984; Kurki-Suonio 1985; Kajantie and Kurki-Suonio 1986; Fuller et a!. 1 988a,b; Murugesan et at. 1990]. In some of this early work [e.g. Crawford and Schramm 1982] it was speculated that primordial fluctuations from this transition might lead to the production of planetary-sized black holes. This possibility has been also considered in more recent work [Halt and Hsu 19901. However, the first suggestion that the QCD transition may lead to the production of isothermal baryon-number fluctuations was based upon a discussion [Witten 1984] of how the thermodynamics of the phase transition could produce a concentration of baryon number. The basic ideas have been discussed and amplified an a number of subsequent works [Apptegate and Hogan 1985; Iso et at. 1986; Applegate et at. 1987; Alcock et al. 1987; Fuller et al. 1988b; Kurki-Suonio 1988]. The idea can be summarized as follows. If the transition from unconfined quark—gluon plasma to normal hadronic matter is first order then the temperature history of the universe through the phase transition should proceed as depicted schematically in fig. 3 [from Cleymans et at. 1983]. As the temperature reaches the coexistence temperature, T~,of the phase transition the universe will supercool until bubbles of hadronic phase spontaneously nucleate as illustrated at the top of fig. 4 [from Kurki-Suonio 1988] for three different nucleation site densities. The latent heat released from the formation and growth of these hadron bubbles will then reheat the universe to the coexistence temperature producing the plateau visible on fig. 3. The continued release of latent heat

158

R.A. Malaney and G.J. Mathews, Probing the early universe

r0< lOi.tm

intermediate

ç3> 10cm

I {MeVJ

300~\

I

_______

I

t [~s~

Fig. 3. A schematic view of the temperature history through the QCD phase transition. The bold and dashed curves correspond to phase transitions which occur at temperatures of 170 and 120 MeY, respectively [Cleymans et al. 1983].

Fig. 4. A schematic picture of the evolution of baryon density for three different initial separation scales, r0 [KurkiSuonio 1988]. The bottom shows sketches of the final density profiles. The first frame depicts the bubble formation stage. The second frame shows the contracting droplet stage after bubble percolation. The third frame depicts the baryon density profiles just after the end of the phase transition.

associated with the growth of hadron bubbles will then balance the cooling due to expansion so that the universe stays at the coexistence temperature. About half way through the phase transition the hadron bubbles should collide and percolate. If the bubble surface tension is large enough then the shrinking regions of quark—gluon plasma may become spherical bubbles as depicted in the middle of fig. 4. During this process of bubble growth the baryon number is expected to preferentially reside in the quark—gluon plasma phase. This increased solubitity of baryon number in the high-temperature phase can be understood on the basis of some very simple thermodynamic arguments [Witten 1984; Alcock et at. 1987; Fuller et at. l988b; Kurki-Suonio 1988] described below. Eventually, alt of the quark—gluon plasma is converted to hadron phase and the universe can again continue to coo!. However, there can remain local fluctuations in the baryon density as shown at the bottom of fig. 4. A critical component of the above scenario is that the QCD transition is first order. Regarding the order of the transition, the first results of finite-temperature Monte Carlo lattice QCD calculations seemed to clearly indicate a first order deconfining phase transition in the pure gauge theory without quarks [Fukugita et a!. 1 989a,b 1. There were also preliminary indications that the chiral-symmetry breaking transition in the theory with quarks occurs at the same temperature and was weakly first order [Fukugita and Ukawa 1986; Gupta et a!. 1986, 1988; Kovacs et at. 1987; Gottlieb et al. l987a; Gavai et at. 1987; Fukugita et al. 1987; Fukugita 1988; Kogut and Sinclair 1988]. However, with the most recent results from dedicated special purpose machines, the possibility of a first order

R.A. Malaney and G.J. Mathews, Probing the early universe

159

QCD transition seems less likely [Brown et al. 1990; Vaccarino 1991; Fukugita and Hogan 1991]. Nevertheless, calculations with realistic quark masses stilt require a number of approximations and the possibility of a weakly first-order transition may not yet be entirety ruled out. 3.4.1. Equilibrium ratio of baryon number densities Ifthe QCD phase transition is first order the quark—gluon plasma and hadron-gas phase can coexist during the transition. To a first approximation, in the high-temperature quark—gluon plasma phase, baryon number is carried by nearly massless relativistic quarks, whereas in the low-temperature hadron phase, baryon number is carried by massive m>> T~nonrelativistic baryons. A Bottzmann factor, em/T, therefore suppresses the concentration of baryon number in the low-temperature phase. To be more precise, the net baryon number density, ~b, in either phase is determined from the derivative of the thermodynamic potential, Q, with respect to the chemical potential, 1u, 1 fdQ\ 8/~JVT (3.2) V\ where V is the volume. Q is derived from the partition function Z through Q = T ln Z. Note that Q is also used to denote the closure parameter. However, to be consistent with the literature we use the same symbol here for the thermodynamic potential. It should be clear from the context which meaning is implied. For the quark—gluon plasma we can approximate Q with —

Qqg

=

_T~j~NCNfvT4[l +

+

-

~NgVT’~ + By,

(3.3)

where N~is the number of colors, Nf is the number of quark flavors (2 or 3), Ng = 8 is the number of gluons, and B is the QCD vacuum energy. In the limit of small chemical potential, p~,then the baryon number in the quark—gluon plasma can be approximated =

(3.4)

~

For baryons in the hadron phase we have (for ,u << m) =

2gT4V ~

cosh(i~ub/T)K 2(im/T),

(3.5)

where K2 is a modified Bessel function of second order. For T << m eq. (3.5) implies 36

-mIT

~)

-

~T)

~j7)

e

.

(.)

In the limit of only nucleons contributing to baryon number the ratio of baryon density in the two phases can therefore be approximated: 2em/Tc n~2(~r~’\~ n~ 9 8 ) (m/T~)3I2 ‘~‘

~.

When the sum over all hadronic species [Hernández et al. 1990; 1991] is included [Atcock et al. 1987; Turner 1988] the equilibrium ratio of baryon densities in the two phases varies from 50 at D 100 MeV, to 10 at T 150 MeV. ‘—j

~

160

R.A. Malaney and G.J. Mathews, Probing the early universe

I

I

I

I V

o

2

—

—

-

~<

Quark-gluon plasma

Y

continuum

0.1—

—

0.0

—

—

T~

_________________________________ 5.2

5.3

g~~3 Hadron

q q

~qg=37

1.5xT,

—0.1 ~ 5.1

Hadron gas

5.4

q

5.5

6/g2 Fig. 5. Baryon number susceptibilities, Xs, as a function of coupling parameter, 6/g2, from the Monte Carlo lattice calculations of [Gottlieb et al. 1 987b I. Horizontal arrows label the value of Xs for two flavors of quarks on an 8~x 4 lattice and the continuum. Vertical arrows indicate values of the coupling corresponding to the phase transition temperature and 1.5 times the transition temperature.

Fig. 6. A schematic view [Fuller et al. 1 988b] of baryon transport during the phase transition.

It is worth pointing out that the simple Fermi-gas assumption in the above derivation has been made at least plausible by lattice QCD calculations of the baryon number susceptibility, Xs aflb/aPb, in the two phases [Gottlieb et al. 1987b]. As can be seen in eqs. (3.4) and (3.6) the baryon density varies linearly with the baryon chemical potential in both phases (Pg = Pb in equilibrium). Therefore, the chemical potentials cancel from eq. (3.5) and the ratio of susceptibilities is equivalent to the ratio of baryon number densities. On the lattice this quantity can only be easily calculated for Pb = 0 (which is a good approximation for the big bang). The lattice calculations [Gotttieb et at. 1 987b I shown in fig. 5 indicate that the ideal gas model represented in eqs. (3.4) and (3.6) gives reasonable estimates of the susceptibility in the two phases, i.e. the calculations show a rapid change in the baryon susceptibility at the phase transition. However, the statistical uncertainty in the susceptibility for the hadron phase is too large to determine the equilibrium susceptibility ratio. One can only conclude that the equilibrium ratio is greater than tO. There have been other attempts to estimate corrections to the ideal gas model besides lattice QCD. For example Olive [1988] has estimated the effects of hadron interactions on the equilibrium ratio of baryon number densities and concludes that the ratio is greater than 10 even at very high temperature. Also, McLerran [1987] has considered a derivation based upon the large-N limit of QCD and Murugesan et at. [1990] have considered effects of relativistic quantum statistics in both phases and corrections for finite hadron size. 3.4.2. Baryon number transport Of course an equilibrium ratio which is greater than unity does not by itself necessarily lead to the production of baryon number inhomogeneities at the time of nucleosynthesis. The final density

R.A. Malaney and G.J. Mathews, Probing the early universe

161

contrast will depend upon the dynamics of baryon number transport in both phases during the transition [Witten 1984; Fuller et at. 1 988a,b; Kurki-Suonio 1988; Alcock et al. 1989]. If diffusion of baryon number in the hadron phase were efficient then any density fluctuation would vanish after its formation. However, the transport of baryon number in the hadron phase is restricted due to scattering from the background relativistic particles and hadrons [Fuller et al. t988b]. It has also been shown [Kurki-Suonio 1988] that unless the effective diffusion length of quarks in the high-temperature phase is at the same time relatively large, there will not be significant density fluctuations. This is due to a kind of snow-plow effect where only a small total baryon number excess builds up next to the moving phase boundary. On the other hand it has been argued [Alcock et at. 1989] that development of turbulence in the quark—gluon plasma may substantially increase the transport of baryon number away from the phase boundary in the quark—gluon plasma allowing for the generation of significant baryon-number fluctuations. Another important effect to consider is the actual transport of baryon number from the quark— gluon plasma to the hadron phase. Since there are no baryon-number violating processes occurring during the QCD phase transition, baryon number has to physically transport across the moving phase boundary as nucleated bubbles of hadron phase grow. This process of baryon transport corresponds to three color-singlet quarks moving toward and through the phase boundary as depicted in fig. 6 from [Fuller et at. l988b]. There has been at least one attempt to estimate this transport in the context of the chromoelectric flux tube model [Sumiyoshi et at. 1989, 1990] which has been used to describe baryon production in hadron showers at high energy. In this picture the baryon number transport occurs when a flux tube appears outside the phase boundary. The formation of baryons is significantly hindered which can lead to the production of large baryon number density fluctuations in the regions of shrinking quark—gluon plasma. 3.4.3. Nucleation Another important consideration is the actual density of hadron bubbles produced after the supercooling phase. This relates directly to the mean separation between fluctuation sites after the phase transition. If the separation between fluctuations is too short then the diffusion of baryons before nucleosynthesis will restore the homogeneous conditions of SBB nucleosynthesis [Terasawa and Sato 1 989a,b,c; Kurki-Suonio and Matzner 1989, 1990; Mathews et at. 1 99Ob]. Ifthe separation distances are too distant then there will not be enough nucleon diffusion to significantly alter the nucleosynthesis yields from those of two separate SBB models averaged together. The density of nucleation sites can be estimated from classical nucleation theory from the free energy associated with the formation of bubbles [Kajantie and Kurki-Suonio 1986; Alcock et at. 1987; Fuller et at. 1 988b]. Unfortunately, such estimates require knowledge of the surface energy of the hadron bubbles and the latent heat of the transition. At present, neither of these quantities is well enough known to significantly limit the range of the density of fluctuation sites. However, there has been some recent progress in lattice gauge theory calculations [Kajantie et at. 1990, 1992; Huang et al. 1990]. An inferred ratio of surface tension to the cube of the critical temperature, a/ 7~= 0.24 ±0.06, from [Kajantie et at. 1990] implies, for T~ tOO MeV, that the separation distance between fluctuations is only 4.7 ±0.7 cm which is too small to be interesting for nucleosynthesis (see below). Another uncertainty comes from the fact that in a realistic nucleation scenario there should be a distribution of different nucleation site separation distances. An attempt has been made [Meyer et at. 1991] to estimate the effect of such a distribution on the nucleosynthesis yields. Including such a spectrum of fluctuation separation distances slightly increases the 4He and 7Li abundances. There is also a question of the transport of the latent heat released by the nucleation and growth

162

R.A. Malaney and G.J. Mathews, Probing the early universe

~tl’ir

Fig. 7. An example of the nonlinear evolution of a hadron bubble in cartesian coordinates from the geometric model described in [Freese and Adams 19901.

of hadron bubbles. If the heat is only transported by neutrinos it has been shown [Freese and Adams 19901 that the system becomes unstable to the growth of dendritic structure as new bubbles of hadron phase continue to nucleate throughout the transition. This would lead to a close arrangement of fluctuations (as shown in fig. 7) which would preclude interesting nucleosynthesis. However, it is expected that hydrodynamic shock waves [Witten 1984; Kajantie and Kurki-Suonio 19861 should also be important in heating up the universe to the coexistence temperature which would shut off the dendritic growth process. 3.5. Other means to generate baryon number density fluctuations In addition to the mechanisms specified above there have been other speculations as to how baryon density fluctuations might arise and influence primordial nucleosynthesis during the big bang. One possibility is from superconducting cosmic strings as discussed in section 6.2. Another speculated possible outcome of the QCD transition would be the formation [Witten 19841 and subsequent evaporation [Alcock and Farhi 1985] of quark—matter agglomerates. As long as the agglomerates contained a sufficiently high baryon number they could leave behind large baryon density fluctuations after evaporation [Applegate and Hogan 19851. It has, however, been argued that quark—matter agglomerates may have a larger survivability than derived in [A!cockand Farhi 1985] both due to reabsorption of hadrons [Madsen 1986] and the low estimated tunneling rate for baryons to be ejected from the nuggets based upon the chromoelectric flux tube model [Sumiyoshi et al. 1989, 1990]. Both of these effects would slow the net evaporation rate. If such quark—matter agglomerates were present and undergoing evaporation during the epoch of nucleosynthesis they might also have an effçct on primordial abundances by acting as a sink and/or source of baryons [Madsen and Riisager 1985; Schaffer et at. 1985]. One way this might happen is that they could develop a local positive charge distribution near their surface. They would thus repel protons yet devour neutrons. This would decreases the n/p ratio thereby reducing the helium abundance produced. This process would also change the baryon-to-photon ratio. A similar effect

R.A. Malaney and G.J. Mathews, Probing the early universe

163

could be achieved with baryonic non-topological solitons, or Q-balls [Madsen l990a]. In [Madsenand Riisager 1985; Schaffer et al. 1985] it was concluded that a universe closed by baryonic matter was possible when quark-matter agglomerates were present. It has been argued, however [Madsen 1986], that primordial nucleosynthesis is sensitive to the estimated hadron emission rates, and that this needs reinvestigation with better rates. Another possible QCD-related phase change which could produce baryon inhomogeneities is the transition from a kaon condensate to an ordinary hadron gas [Nelson 1990]. This transition could be first order and could occur after the deconfining/chiral-symmetry restoring phase transition. Shrinking bubbles of kaon phase would concentrate baryon number both because of chemical baryon-number equilibrium (analogous to the quark—hadron transition) and because neutrino cooling can shrink the bubbles without losing baryons. The net effect could be to produce baryon inhomogeneities on a potentially interesting scale for primordial nucleosynthesis as discussed below. Another suggestion [Fukugita and Rubakov 1986] is that a late epoch of baryogenesis associated with supersymmetric GUTs would produce large baryon number-density fluctuations from small temperature fluctuations. In this scenario [Linde 1985] it is assumed that a scalar field composed of a combination of scatar-quark and scalar-lepton fields takes a value far from the minimum of its potential. The oscillation of this scalar field in the potential leads to particle creation with nonzero baryon number and a reheating of the universe. This process, however, generates too much baryon number, i.e. a baryon-to-photon ratio 1. A reduction in the baryon-to-photon ratio is achieved via instanton-like effects during the later etectroweak transition. It is during this tatter step that temperature fluctuations could lead to large baryon-number fluctuations [Fukugita and Rubakov 1986]. Similarly, it may be possible that the entire baryon asymmetry was erased just before the electroweak transition and then regenerated by anomalous baryon-number violating processes. [Kuzminet al. 1985; McLerran 1989; Dine et at. 1990, 1991, 1992a,b,c; Cohen et a!. 1990, 1991 a,b; Nelson et al. 1992]. Temperature fluctuations might in this case produce baryon-number fluctuations [McLerran 1989]. Another possibility is a restriction of baryon transport through the phase boundary which is a part of the baryogenesis process [Dine et at. 1990, 1991, 1 992a,b,c; Cohen et at. 1990; 1991 a,b; Nelson et a!. 1992]. Such restriction of the flow of baryon number across the phase boundary might produce baryon number density fluctuations by an analogous mechanism to that which can operate in a first-order QCD phase transition as described above. Alternatively, we speculate that isothermal baryon-number fluctuations might result from baryon number violating processes associated with normal cosmic strings [Alford et at. 1989] or the generation of axion strings during the electroweak transition [Snyderman 1988]. Regarding the generation of baryon number fluctuations at the electroweak scale, it is amusing [Jedamzik et at. l993b] that the size of the horizon (‘~0.2 cm) at the time of the electroweak transition, when expanded to the time of nucleosynthesis, becomes just comparable to the optimum scale for effects of inhomogeneities on primordial nucleosynthesis as we show below. Thus, even though the electroweak transition occurs at a high temperature it could impact primordial nucleosynthesis. As one final speculation on the origin of isothermal baryon number fluctuations it has also been pointed out [Ignatius et al. 1992] that a QCD Z(3) phase transition may occur at a temperature of 10 TeV. Such a phase transition would also produce large baryon number fluctuations. In this case, however, the distance scale is probably too small even after Hubble expansion for the baryon inhomogeneities to have an appreciable effect on the yields from primordial nucleosynthesis. As we shall see, small scale baryon inhomogeneities are destroyed by baryon diffusion prior to primordial nucleosynthesis. A TeV-scale phase transition is, therefore, not likely to affect primordial

164

R.A. Malanev and G.J. Mathews, Probing the early universe

nucleosynthesis unless there are baryon inhomogeneities over super-horizon distances at that scale. 3.6. Nucleosynthesis and baryon diffusion 3.6.1. Light element nucleosynthesis In the previous sections we discussed how various mechanisms may have produced primordial baryon number-density fluctuations large enough to survive to the epoch of primordial nucleosynthesis. Now we review the changes in the nucleosynthetic yields produced by the existence of these inhomogeneities. The essential features of the evolution of baryon-number fluctuations prior to and during the epoch of nucteosynthesis have been explored in a number of works [Atcock et at. 1987, 1989, 1990; Applegate et at. 1987, 1988; Applegate 1988; Audouze et a!. 1988; Fuller et al. t988a,b; Kurki-Suonio 1988; Kurki-Suonio et al. 1988, 1990; Malaney and Fowler l988a,b; Kurki-Suonio and Matzner 1989, 1990; Terasawa and Sato l989a,b,c; Mathews et at. 1988, l99Ob, l993b; Reeves et al. 1990; Reeves 1991; Jedamzik et at. 1993a] We briefly outline the evolution here. From the time of their formation until the freeze out of weak reactions at T 1 MeV the baryon fluctuations should change little. At high temperatures (T> 50 MeV) the hadron density is high enough to slow the diffusion of the baryons. Until the temperature drops below 1 MeV the neutron-to-proton ratio will be maintained at the equilibrium ratio. Nevertheless, there will be some small diffusion of nucleons during the fraction of the time that they appear as neutrons [Applegate and Hogan 1985; Applegate et al. 1987, 1988] As the’ temperature drops below 1 MeV the neutrons and protons obtain separate identities and thus different interaction rates with the background relativistic plasma. The diffusion of protons is inhibited due to the fact that they are charged and therefore couple strongly to the background electron—photon plasma. The diffusion of neutrons on the other hand is only weakly inhibited due to the interaction of their magnetic moment with the background plasma and neutron—proton scattering. Diffusion constants for protons and neutrons have been discussed by [Applegate and Hogan 1985; Applegate et al. 1987, 1988; Banerjee and Chitre 1991; Kurki-Suonio et al. 1992]. For protons the most important scattering is with electrons and positrons. A nonrelativistic nondegenerate estimate for the proton diffusion coefficient is “~

cm2 s~, (3.8) D~= 8cv2A ~ (J...~ xehlx 2 2.56 Ax lO~1 + xe2x1t-”~ \~m + 2x2 0J 1 + 2x + 2x where x = T/m~and A 5 is a dimensionless Coulomb logarithm. For neutrons the dominant contributions are from both electron scattering and proton scattering, -

-.~

D;1=D~j+D~

(3.9)

where ~(M\(l~

=

x tø~

1

e x(l+3x+3x2)

eh/x

cm2 s~,

(3.10)

where Ic = 1.91 nuclear magnetons is the anomolus magnetic moment of the neutron and M is the neutron mass. Equation (3.10) assumes zero chemical potential and Maxwell—Boltzmann statistics.

R.A. Malaney and G.J. Mathews, Probing the early universe

165

Recently, an alternative form for neutron—electron scattering (eq. (3.10)) has been suggested [Banerjee and Chitre 1991] based upon the lowest-order Chapman—Enskog approximation to relativistic kinetic theory. Although eq. (3.10) does not go to the correct limit at low temperature, Kurki-Suonio et al. [19921 have shown that the corrections make little difference in the calculated nucleosynthesis yields because the electron-scattering diffusion coefficient only dominates at high temperature. Therefore, eq. (3.10) is adequate for most purposes. The diffusion coefficient for neutron—proton scattering is written [Applegate Ct al. 1987], ~

=

T 1/2 1 (~-~)

(3.11)

,

where cy~~ is the neutron—proton s-wave scattering cross section given by

=

ira~ 3ira? 2 + (1 ~rsask2)2 + (k)2 + (1 ~rtatk2)2’ (a~k) —

—

(3.12)

where as = —23.71 fm and at = 5.432 fm are the singlet and triplet scattering lengths, respectively, and r~ = 2.73 fm and rt = 1.749 fm are the effective ranges. Equation (3.11) is based upon a simple mean-free-path estimate for the diffusion coefficient. In the lowest order Chapman—Enskog approximation [Banerjee and Chitre 1991; Kurki-Suonio et al. 1992] eq. (3.11) is multiplied by an additional factor of (3/8) ~ where ad is the classical hard-sphere scattering cross section. For a~i= o~,this factor is approximately 1.15. Including this additional factor increases the distance scale for the optimum nucleosynthesis yields by the square root of this factor. Otherwise, the yields are largely independent of which diffusion coefficient is used [Kurki-Suonio et al. 19921. Figure 8 shows the calculated diffusion lengths for neutrons and protons as a function of temperature before and during the epoch of primordial nucleosynthesis [Applegate 1988]. Once the temperature has dropped below the photodissociation threshold for deuterium (T 100 key) the epoch of nucleosynthesis can begin. However, because of neutron diffusion the conditions for nucleosynthesis will involve regions of varying n/p ratio. In the neutron-rich regions in particular, the available protons can quickly be absorbed into deuterium and subsequently 4He. Further nucleosynthesis will then await the decay of neutrons into protons. This delayed nucleosynthesis can occur at relatively low baryon density compared to the standard big bang model with the same average baryon to photon ratio. This low baryon density is reduced even further relative to the standard big bang due to the fact that nucleosynthesis is occurring at later times when there has been more universal expansion. These effects combine to produce a local increase of deuterium and 4He in the neutron-rich regions. The 7Li abundance is also high there. The possibility for A > 7 isotopic production in these regions has also been discussed considerably [Boyd and Kajino 1989; Malaney and Fowler 1989; Kajino and Boyd 1990; Kajino et al. 1990; Terasawa and Sato 1990; Coc et al. 1993; Kawano et al. 1991; Applegate et al. 19931 (see section 9). Figure 9 (bottom) shows an example [Malaney and Fowler 1 988b] of computed light-isotope abundances as a function of time in a neutron-rich region for a calculation in which neutron diffusion after the onset of nucleosynthesis is not included. In the proton-rich high-density regions the nucleosynthesis is limited (as in the SBB) by the availability of neutrons. Since the neutrons may be depleted by diffusion into the low-density regions, the produced helium mass fraction is commensurately reduced along with heavier elements relative to the SBB at the same average baryon density. Nevertheless, the baryon density can be so high that significant light and intermediate-mass production can occur [Kajino et al. 1990] in the

166

R.A. Malaney and G.J. Mathews, Probing the early universe

/

100

10

;~o6

proton scattering)

0.5

-

/

‘He

/

‘He

—0.5 log Te (MeV)

—~

-~

10

50

100 Time (s)

500

1000

/

102

//

/

0.0

‘Be

10 ~

,/~~ofls

~i0~

2

i

neutrons (without

10~

‘He

—1.0

Fig. 8. Comoving diffusion lengths against the electron temperature Te [Applegate 1988]. Curves show the rms comoving diffusion distances traveled by a baryon up to the time the universe cools to a temperature Te. The curves are labeled by the baryon-to-photon ratio in units of 10—8.

l0~

io~

io~ T~ee,e(s)

Fig. 9. Computed nucleosynthesis yields [Malaney and Fowler 1988b] as a function of time in the neutron-rich regions (bottom) and proton-rich regions (top) ofa two-zone inhomogeneous big bang model with ~b ~ fv = 0.11 and R = 50.

proton-rich regions. Figure 9 (top) shows an example of elemental nucleosynthesis as a function of time in a proton-rich region. The first calculations attempted to describe baryon diffusion by a schematic mixing of neutrons before primordial nucleosynthesis [Alcock et at. 1987; 19889; 1990; Applegate et al. 1987, 1988; Audouze et at. 1988; Fuller et at. l988a,b; Rana et a!. 1990]. However, once nucleosynthesis begins neutrons are quickly depleted in the proton-rich zones. The gradient for neutron diffusion then reverses and neutrons can flow back into the high density regions. As they flow into the high density regions they can destroy 7Be by the 7Be(n, p)7Li(p, a)4He reaction [Malaney and Fowler 1988b]. Figure 10 shows the comoving neutron abundances in various regions as a function of time for a calculation in which nucleosynthesis and diffusion are followed in a number of zones [Mathews et at. 1 990b]. The low-density zones come into neutron-diffusion equilibrium by t 50 s, which is about the time that nucleosynthesis begins in the high-density zones. By t 100 s the loss of

R.A. Malaney and G.J. Mathews, Probing the early universe

‘I

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167

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flb= 1.0 R ~10~ 030

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Fig. 10. Comoving neutron densities as a function of time for different zones during an inhomogeneous big bang [Mathews et al. 19901,1.

-

.1

II~

0.1

I

1.0

10.0

100.0

I 1000.0

10000.0

Radius (I,,)

Fig. 11. Computed nucleosynthesis yields [Mathews et al. 1 990b] as a function of separation distance in a baryonnumber inhomogeneous big bang model which includes neutron diffusion before, during, and after the nucleosynthesis epoch.

neutrons by nuclear reactions in the high-density regions causes the neutron density in all of the zones to be nearly equal. However, further nuclear reactions in the high-density regions depletes neutrons by up to 18 orders of magnitude. Neutrons then begin to diffuse back into the high-density zones. The rate of this back diffusion is inhibited, however, due to the universal expansion, which means a larger proper distance. The short mean free path of the neutrons in the high-baryon density regions also limits the rate of diffusion back into the high density regions. Between 1000 and 2000 s neutron-diffusion equilibrium is again established among the zones. Beyond 2000 s most of the original neutrons would have decayed. However, some new neutrons are produced from the D (D, n ) 3He reaction. In all such calculations it has been found that it is necessary to have good zoning particularly in the vicinity of the boundary between high-density and low-density regions [Mathews et al. l990b]. Unless the grid size is less than the neutron diffusion length during a time step, the amount of diffusion can be overestimated. As far as parameters characterizing such inhomogeneous big bang calculations are concerned it has been found [Mathews et al. 1988, 1990b; Kurki-Suonio and Matzner 1989, 1990; Kurki-Suonio et al. 1990; Terasawa and Sato 1989a,b,c] that a large density ratio, R (not to be confused with the scale factor), of the high to low-baryon density regions (R > 1 O~),and a relatively small volume fraction, f~ ~ 0.1, for the high density zones gives the best agreement with primordial abundances (R and f~defined in section 3.4). By minimizing the volume and maximizing the density in

168

R.A. Malaney and G.J. Mathews, Probing the early universe

the high-density regions the neutrons are more efficiently depleted by diffusion away from the high-density regions so that less 4He is produced. At present, little can be said about what values R, f~,and the separation distance should take. An upper limit to R can be placed from the effects of neutrino induced heating of the high density zones of about 106 [Hogan 1988; Heckler and Hogan 1992; Jedamzik and Fuller 1993; Jedamzik et al. l993b]. QCD lattice gauge theory calculations [e.g. Kajantie et al. 1990] suggest that the separation distances are small, ~ 10 cm (comoving at 100 MeV) for a QCD generated fluctuation. For other fluctuations generated at higher temperatures, an upper limit on the separation distance can be imposed by the horizon at a given temperature. In comoving coordinates at 100 MeV this implies a separation distance of < 1 o~[100 MeV/ T] m. The value of f~ depends upon the dynamics of the process generating the fluctuation. It is difficult to quantify constraints on this quantity. Small values for f~ can arise from any mechanism in which baryon number is concentrated into shrinking bubbles. The volume of the resulting fluctuation is probably determined by the internal pressure of baryons. However, the baryon pressure is not likely to be significant compared to the radiation pressure until baryons are compressed to quite small volumes and high baryon densities. Figure 11 shows an example [Mathews et al. 1 990b I of nucleosynthesis yields as a function of separation distance between fluctuations in an = 1 universe. There is an optimum distance which gives a closest agreement with observed abundances, although the abundances can not be brought into agreement with observations for Qb = 1 even for the somewhat liberal observational constraints suggested on this figure. For more realistic constraints [e.g. Walker et al. 1991] the agreement with observations is much worse. As has been pointed out [Schaeffer 1989] there can also be problems with the microwave background isotropy if one wishes to close the universe with baryons. So in all it would seem that the early speculation that an Qb = 1 universe might be allowed in baryon inhomogeneous models is ruled out. The main discrepancies are overproduction of 4He and 7Li. By allowing for R to be as large as 106 a value of Qb as large as 0.4 (for Y~< 0.24 and h = 0.5) might be allowed [Mathews et a!. 1993b] if a destruction mechanism such as late time expansion is invoked to destroy lithium (see below). Figure 12 [Kurki-Suonio et al. 1990] summarizes the allowed regions of ‘i and separation distance, 1, for a calculation in which R was restricted to be < 100. This value for R is too small to allow for significant increases in ~j beyond that allowed by the SBB. Recall that from eq. (2.12) for h 0.5, Q~,= 1 implies ~j = 60 x 10_b. As noted above, somewhat higher values of ~ are allowed [Kurki-Suonio and Matzner 1990; Mathews et al. l990b] when larger fluctuation amplitudes are included. Values of i~> 23 x 10—10 are probably excluded even in the most favorable circumstances [Mathews et a!. 1993b]. The effects of averaging over a random distribution of fluctuation distances was calculated [Meyer et a!. 19911 for the QCD phase transition using classical nucleation theory [Kajantie and Kurki-Suonio 1986; Alcock et at. 1987; Fuller et al. 1988] and related to a single parameter characterizing the average separation distance. The effects on the nucleosynthesis yield of averaging over a distribution of fluctuation separation distances are shown in fig. 13. This decreases the maximum allowed value for Qb based upon the 4He abundance by about 0.1. Besides the back diffusion of neutrons which can occur after nucleosynthesis begins in the highdensity regions there is another effect which can occur at late times. As the e+e pairs annihilate and the photon density decreases, the hydrodynamic pressure in the high-density regions can cause the protons to rapidly expand into the low-density regions [Alcock et al. 1990; Jedamzik et al. 1 993a]. If this process occurs at the right temperature (T 30 keY) then significant destruction of lithium and other light elements could ensue. This would improve the agreement between the lithium production and the abundance constraints. The most recent evaluation of this process,

R.A. Malaney and G.J. Mathews, Probing the early universe

1000

I

100

I

3He,

~lc,

169

4He,7Li

D+

1

—

D

I

I

I

0

I

-

Li1

10

7L1

-

D

-

~

910

~

20

10

30

40

50 600 70 80 90100

x 10 Fig. 12. Allowed regions in the ,~and I plane from light-element abundance constraints [Kurki-Suonio et al. 1988]. The area outlined by bold lines are the only regions consistent with all constraints when the density contrast, R, is less than 100.

however [Jedamzik et al. 1 993a], indicates that when the detailed hydrodynamic evolution of the fluctuations is properly analyzed, this mechanism is not an efficient destroyer of lithium.

I

.001

I

.003 I

IIIII~

I

~I

II I

x

I

IIIII~

I

I

I

1111

(2.7/T 3 (H 2 0) 0/50)

111111

-~

I

I

III

Fig. 13. An example [Meyer et al. 1990] of the minimum value of Yp as a function of ~b after averaging over a statistical distribution of fluctuation separations (solid curve) compared with unaveraged [Mathews et al. l990b] results (short-dashed curve) and standard big bang (long-dashed curve).

170

R.A. Malaney and G.J. Mathews, Probing the early universe

3.6.2. CNO isotopes and heavy elements As mentioned earlier a possible prediction of the inhomogeneous models is the production of significant intermediate mass (e.g. CNO ) isotopes. Since the SBB scenario produces essentially zero heavy elements, the presence of a “cosmic floor” of heavy elements in very metal-poor stars could be a strong indicator of inhomogeneity at the time of nucleosynthesis. It is worth keeping in mind that substantial production of heavy elements in baryon-inhomogeneous models is possible even in models with small ~b. By simply making f~ small (~10~)and R large (~l0~)substantial heavy element production is possible in the high-density proton-rich regions without significantly affecting the other light element constraints [Jedamzik et a!. 1993b] or 7Li. This is because, for i~ sufficiently large in the high density regions, 7Be is destroyed by alpha capture to “C [Wagoner et a!. 1967]. The most likely reaction sequence leading to the production of CNO isotopes in the neutron rich regions is believed to be [Malaney and Fowler 1988a,b; Applegate et al. 1988] ‘H(n, y)2H(n, y)3H(d, n)4He(3H, y)7Li(n, y)8Li(ct, n) “B ‘‘B(n,y)’2B($,v)’2C(n,y)’3C(n,y)’4C.

(3.13)

In the proton-rich regions, a capture chain analogous to stellar high-temperature hydrogen burning can lead to heavy elements [Kajino et at. 1990]. In [Kajino et a!. 1990] a network extending to A = 28 was employed in order to determine possible observational signatures of inhomogeneity from the intermediate-mass elements. They found possible signatures both from the neutron capture productions in the low density regions and hotCNO and rp-process [Wallace and Woosley 1981] production in the proton-rich zones. We caution that these calculations are based on two-zone models, whereas multizone calculations [Terasawa and Sato 1990] indicate that in some of the parameter space, two-zone models overestimate the level of CNO production. There have also been subsequent reaction rate studies [e.g. Wiescher et a!. 1990; Koehler and Graff 1991] indicating that the production of elements heavier than carbon may be more difficult than in earlier calculations. Nevertheless, [Jedamzik et a!. 1993b] have shown that even with diffusion treated carefully, substantial heavy element production is possible in inhomogeneous models, without violating any of the light-element constraints. In [Applegate et al. 1993] the network of heavy nuclei was extended through the actinide nuclei so that the actual development of the neutron capture network up to the heavy nuclei could be studied. It is amusing to note that the very first work on primordial nucleosynthesis was concerned with the possibility of heavy-element production during the big bang [Alpher et a!. 1948, 1953; Alpher and Hermann 1950]. In that work, it was merely assumed that a high n/p ratio could be present. Now some 40 years later we at last have a physical motivation for this in the neutron rich regions formed in baryon-inhomogeneous big bang models. In [App!egate et a!. 1993] it was found that under optimum conditions some significant neutron capture nucleosynthesis to very heavy nuclei was possible. For most of the parameter space, however, the neutron density diminishes due to the universal expansion and neutron capture, before there can be significant fission cycling. In any event, the last neutron captures occur relatively close to the line of beta stability so that the final abundances appear more like an s-process distribution (i.e. abundance peaks at A = 88,138,208) than an r-process (i.e. abundance peaks at A = 80,130,195). Figure 14 shows computed abundances for heavy elements [Applegate et al. 1993] for an optimum inhomogeneous big bang.

R.A. Malaney and G.J. Mathews, Probing the early universe

I

I

I

I

I

I

I

171

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100

—

x I

I I

150

I

I

I

200

I

I

I

I 250

I

I

I

I

300

MASS NUMBER

Fig. 14. Heavy element abundances (crosses) computed in an optimum r-process scenario for a baryon-inhomogeneous big bang [Applegate et al. 1993]. Solar r-process abundances are indicated by triangles. The final result contains l0~times the solar abundance in heavy nuclei in the low-density region. The peaks, however, more closely resemble an s-process than an r-process abundance distribution.

4. Late-time decay or annihilation Particles which contribute significantly to the energy density at early times and decay or annihilate on a time scale of 1 to 108 s can affect primordial nucleosynthesis. The energy released in such processes could alter the abundances from those of the standard big bang by changing the energy density and entropy during nucleosynthesis from that inferred by the standard scaling of the present microwave background temperature. Such decays or annihilations could also lead to photodissociation and energetic hadron interactions with the light nuclei produced during primordial nucleosynthesis. Such processes can be important even if the particles decay or annihilate long after the nucleosynthesis epoch. There have been a number of candidate particles proposed for late-time decay such as massive neutrinos [Dicus et al. 1978; Miyama and Sato 1978; Lindley 1979,1985; Hut and White 1984; Scherrer 1984; Ellis et at. 1985; Scherrer and Turner l988a,b; Terasawa et al. 1988], gravitinos predicted by supersymmetric theories [Fayet 1984; Scherrer and Turner l988a,b; Scherrer et al. 1991; Delbourgo-Salvador et at. 1992] (mG ‘—s 20 GeV to 1 TeY), or decaying primordial black holes [Lindley 1980; Carlson et at. 19901 possibly produced during the quark—hadron phase transition [Hall and Hsu 1990]. There are also a number of cold weakly-interacting massive dark-matter candidate particles (WIMPS) for late time annihilations [Dotgov and Kirilova 1988; Hagelin and Parker 1990]. These include a Dirac or Majorana neutrino which annihilate through coupling with the Z°,or a light supersymmetnc particle such as a higgsino or photino. The idea is that the annihilation rate of

172

R.A. Malaney and G.J. Mathews, Probing the early universe

such particles freezes out at some point such that these particles remain as the present cold dark matter. In general the results of late time annihilation models are quite similar to the results of particle decay models, the main difference being the replacement of particle decay rates with particle annihilation rates. For both models the most important result is that much larger values of the present baryon-to-photon ratio are permitted than in the standard big bang. This is mostly due to effects of alteration of the synthesized abundances by photon- and hadron-induced dissociation. We note here that other astrophysical processes, such as accreting massive black holes [Gnedin and Ostriker 1992], could also lead to a late time photon-induced dissociation, in this case after the epoch of recombination. Much of the work on this subject [e.g. Audouze et al. 1985; Dominguez-Tenreiro 1987; Yepes and Dominguez-Tenreiro 19881 has considered the effects of radiative decay. High-energy photons from decaying particles can induce photodissociation and increase the entropy and energy density thereby affecting the baryon-to-photon ratio and the expansion rate. These high-energy photons include both the primary radiation and secondary scattered radiation. It was found [e.g. Audouze et al. 1985] that it might be possible for an = 1 universe to satisfy the light-element abundance constraints due to the destruction of 4He and production of D from photodissociation. There is, however, a tendency [e.g. Audouze et a!. 1985; Dominguez-Tenreiro 1987; Yepes and Dominguez-Tenreiro 1988; Dimopoulos et a!. 1988a,b] to overproduce 3He in such radiatively decaying particle scenarios. Limits to the decaying particle mass, lifetime, and abundance relative to photons have been derived from the light-element abundance constraints in a number of works [Dicus et al. 1978; Miyama and Sato 1978; Lindley 1979,1985; Hut and White 1984; Scherrer 1984; Ellis 1985; Scherrer and Turner l988a,b; Terasawa et al. 1988; DominguezTenreiro 1987; Yepes and Dominguez-Tenreiro 1988; Dimopoulos et a!. l988a,b] An example of the limits on the product of abundance times mass and limits on the lifetime for massive neutrinos or gravitinos are summarized in fig. 15 from [Scherrer and Turner l988a]. A detailed study of the late decaying particle scenario has been made [Dimopoulos et al. 198 8a,b] which included not only the photodissociation of the heavier elements, but also nucleosynthesis in hadronic showers. This latter effect then becomes the main source of primordial light elements. At kinetic energies greater than 1 GeY the hadrons from the decay predominantly lose their energy by strong interactions. Scattering from background 4He nuclei produces energetic D, 3H, 3He, and 4He. These “heated” nuclei then produce a new phase of energetic nucleosynthesis which resets the results of earlier nuclear reactions. Below 1 GeY the nuclei eventually thermalize by electromagnetic interactions to become a part of the background plasma. The photon spectrum per X decay is ~

(Mx/2E

=

(4.la)

1~x)1/E3h/2, E
where M~ is the particle mass, and where the cut-off Em~ due to yy Emax

(m~/25T)ln(~/5 x lO_tO).

=

2H, 3He, 6Li and 7Li abundances are given by

The equations for the df,/dt

=

—p

e+ e is given by (4.lb)

E~

f~9Fxe_T~~r~j —n

f f~.

(E)o~ 1,1(E)dE,

(4.2)

where the subscript i refers to nuclei of type i, of which çej are produced per X decay, n1 is the number density, f, is the number density divided by the thermal photon density, Q, is the binding

R.A. Malaney and G.J. Mathews, Probing the early universe

173

5Gev ii

el 0)

0

-

0

0MeV 3 MeV

A 0

ci

ne)

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5

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n n 200 (1-0)5 2015 (1-0)5 200 Me~’ Gay May GeV May ,,

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n 5

,,

a

n~4s-wt 3He ~i

2(X)

Gay Gay

-

-

-~

11-0)5 200 200 Gay 1.4ev May 00 2 25 25 52 100 63 63 100 100 lix) I00~ no CX) ~ 5

00 0eV m 312: 5GeV

TOTALS:

—

niE’SGeVI

: ~

nE .(I-.)SGaV] .

.0.22

+

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~ 0eV

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~nl°~ ~

\0

152 _x 0

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I

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I

—

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.

~

•

-

1304

:

L.

+ +

4He : 3.

-

.006

06

‘0.05

r (sec)

d

Fig. 15. The permitted region of the product of abundance relative to photons, r, times the mass m 1, and the permitted region of the lifetime, TX, allowed for a fourth generation massive neutrino, and two scenarios for gravitino production in a late-decaying particle scenario [Scherrer and Turner 1 988a,b] in which only radiative decays occur. Solid curves near the bottom give constraintsfor a massive neutrino with the indicated mixing angles. Horizontal segments give constraints for a gravitino with indicated masses generated after inflation. The solid line near the top shows the constraint for primordial gravitinos with an abundance comparable to the photon abundance near the Planck time. The heavy line demarcates the forbidden region.

019

-

~

.0.0

Fig. 16. Contribution [Dimopoulos et al. 1988] of a 5 GeV neutron to a hadron shower. On each branch the probability of taking that branch from the parent node is indicated. The final yields of suprathermal light elements are indicated below.

~,

energy, a2~is the photodissociation cross section, r~is the effectiveThe branching forisXrelated decaying 4 (E) is the non-thermal photon spectrum. quantityratio f 0r,~ to into baryons, and f2 1 the fraction of the 4He produced by the SBB model which is destroyed by the particle decay. An example of the contributions, from a hadron shower induced by a 5 GeV neutron is given in fig. 16 from [Dimopoulos et al. l988b]. From eqs. (4.1a) and (4.2), there are essentially four parameters in the calculation of the nudeosyiIthetic yields, namely, the decay lifetime, rx (= nj’), the particle mass, M~,the abundance of

R.A. Malaney and G.J. Mathews, Probing the early universe

174

100

II

II

I_0.~r71~~~77

4He

~~7.0X104G0VT

_

T

5sec

5.~45x10

I~ [x/b] = 1 .27 x 1 0~

7 Li~

I

io3

1o4

io5

io6

I—. i~7

iü8

TIME (sec)

Fig. 17. Abundances of light elements as a function of time for typical allowed parameters [Dimopoulos et al. 19881 in the late-decaying particle scenario with hadron showers.

particle X, f~9,and the baryonic branching ratio in the decay, rb. There is only a weak dependence on the effective statistical weight g~r,and on the baryon-to-photon ratio ij. Figure 17 shows the evolution of abundances with time in one of the allowed models. At t 1 o~—106 s the abundances drop due to the onset of photodissociation. The abundances after this drop are largely dependent upon the energetic nuclear reaction rates but do not depend strongly upon the baryon density as in the SBB model. Although there is some uncertainty in the nuclear reactions (see section 10) the authors deduced [Dimopoulos et al. 1988a,b] that the energetic reactions would produce primordial abundances in the ratio: D : 3He : 6Li : 7Li

‘S-’

1: 1: iO~: 10—6

The striking prediction of a primordial 6Li abundance an order of magnitude greater than the primordial 7Li abundance is in strong contrast to the SBB model. The comparable production of D and 3He is also in potential conflict with observations [Walker et al. 1991]. The possibility that this much primordial 6Li could be destroyed in stars is not yet ruled out [Brownand Schramm 1988; Deliyannis et at. 1990], although searches for primordial 6Li have been conducted. These observations may ultimately provide a definitive test of this mode! (see section 9.4). There is one other possible effect of late-decaying particles the late generation of the baryon to photon ratio. Cline and Raby [1991]have shown that in the minimal low energy model of supergravity, decaying gravitinos could violate R-parity. As a consequence baryon number can be generated from the decay of squarks (the supersymmetric partners of quarks) into two anti-quarks. The gravitino would most naturally decouple at temperatures slightly below the Planck temperature and decay at a potentially interesting temperature for nucleosynthesis (T 1 MeY). This has recently been studied [Scherrer et al. 1991; Delbourgo-Salvador et al. 1992]. The standard big bang model is modified in this picture because the universe passes through a phase in which the energy density is dominated by gravitinos. The photons, baryons, and neutrino densities then evolve differently than in the SBB, and the n/p ratio is shifted by the injection of hadrons. Roughly speaking, it is necessary for the gravitino decay epoch to have been completed by a temperature of —

R.A. Malaney and G.J. Mathews, Probing the early universe

175

2 MeY to avoid a conflict with primordial nucleosynthesis. This places a constraint on the decay lifetime and therefore the mass of the gravitino. The overabundance of light elements, particularly 4He and 7Li, implies a gravitino mass ~ 50 TeV and a mean decay lifetime ~ 0.04 s [Scherrer et al. 1991]. In the range of 50 to 80 TeY it is even possible to lower the helium abundance below that of the standard big bang by as much as 0.005. 5. Neutrinos

As can be seen from the discussions of section 2, neutrinos play a very important role in the early evolution of the universe. The results from the LEP and SLC et e colliders [Aarnio et al. 1989; Abrams et al. 1989; Adeva et at. 1989, 1991, 1992; Akrawy et al. 1989; Decamp et al. 1989, 1990; Dorfan et at. 1989] have effectively ruled out an extra generation of light neutrinos beyond the three we presently know to exist. This eliminates some of the uncertainty from the SBB model. However, there are several extensions to the standard cosmological and particle physics models which would allow the known neutrinos to perturb primordial nucleosynthesis. We now discuss these effects and how they constrain the new hypothesized neutrino properties. 5.1. Neutrino degeneracy

The universe appears charge neutral to very high precision [Lyttleton and Bondi 1959]. Therefore, any universal net lepton number beyond the net baryon number must reside entirely in the neutrino sector. Since the present relic neutrino asymmetry is not directly observable there is no firm experimental basis for postulating that the lepton number for each species is zero. In fact, the best constraints on the lepton numbers arise from studies of primordial nucleosynthesis [Yahil and Beaudet 1976; Beaudet and Yahil 1977; David and Reeves 1980a,b; Steigman 1985; Fry and Hogan 1982; Terasawa and Sato 1985, 1988; Bianconi et al. 1991; Olive et a!. l99!a; Kang and Steigman 1992; Starkman 1992]. The lepton number for each neutrino generation i is defined to be L 1 = (n~— n0, )/n7, where n01 is the neutrino number density. The total lepton number of the universe is then the sum over the three generations of neutrino species L = ~ L~1,i = e, 0u, r. It is normally assumed in studies of the early universe that the individual lepton numbers L are very small. By very small we mean comparable to the baryon number B, where B = (flb — ~b ) /n~,.A small lepton number for the universe is somewhat “natural” in many grand unified models [Dimopoulos and Feinberg 1979; Schramm and Steigman 1979; Nanopoulos et al. 1980; Turner 1981; Olive et a!. 199la]. For instance, in an SU(5) model [Georgi and Glashow 1974] where it is assumed that B — L = 0, the predicted value of B is much smaller than the observed value of B 1 o~.In other grand unified models where B — L is not conserved, a nonzero value for B — L can be generated prior to the electroweak phase transition. However, B L conserving, but B + L violating, non-perturbative physics in the standard electroweak model SU (2 )L ® U (1) y [Glashow 196!; Weinberg 1967; Salam 1968] may erase any initial net asymmetries between IBI and LI [Kolb and Turner 1983; Kuzmin et al. 1985; McLerran 1989; Bochkarev et at. 1990; Dine et at. 1990, 1991, 1992a,b,c; Cohen et al. 1990, l99ta,b; Nelson et al. 1992; Kuzmin et at. 1985]. Observations of B (see [Trimble 19871 for a review) would then indicate that the initially generated value of L was 1 O~. These arguments apply only to the total lepton number of the universe and not to the individual lepton numbers. The difference between individual lepton numbers is conserved by the sphalerons of the electroweak model and hence an initially large asymmetry between different neutrino species —

R.A. Malaney and G.J. Mathews, Probing the early universe

176

will remain even at temperatures far below that of the weak phase transition [Kotb and Turner 1987; Kuzmin et a!. 19871. It is therefore possible for the total lepton number of the universe to be small (ILl l0~),while the individual lepton numbers are large (IL~I>> l0~). Assuming Fermi—Dirac distributions, the lepton numbers can be characterized by a set of degeneracy parameters, = 1a ‘/T, where ~t is the chemical potential of each neutrino species. In terms ‘-.-~

~‘

of these degeneracy parameters the individual lepton numbers are given by 24’~(~t2 + ~). L, = (n1 — fl7)/~7 6.8 X l0 Also, the neutrino energy density of a degenerate universe is given by =

~

~T4(~+~=.i~+

(5.!)

(5.2)

which in the limit of large degeneracy reduces to Pu,

(5.3)

(t/8ir2)T4~.

It can be seen from eqs. (2.6), (2.9) and (5.3) that in the limit of large degeneracies, the neutrinos dominate the energy density of the universe. Non-zero lepton numbers of the universe can effect nuc!eosynthesis primarily in two ways [Wagoner et al. 1967; Yahil and Beaudet 1976; Beaudet and Yahil 1977; David and Reeves 1980a,b; Steigman 1985; Fry and Hogan 1982; Terasawa and Sato 1985, 1988; Bianconi et al. 199!; Olive et a!. 1991 a; Kang and Steigman 1992; Starkman 1992]. First, the excess energy density in a neutrino-degenerate sea leads to an increased expansion rate for the universe. Since the weak interactions will decouple earlier, this has the net result of increased 4He production relative to a SBB model (which assumes L, = 0). Secondly, the non-zero electron neutrino degeneracy can directly effect the equilibrium n/p ratio at weak freeze out. The altered neutrino chemical potential shifts the equilibrium of the reactions e+p4—*vc+n,

lJe+p~e~+fl.

(5.4)

The equilibrium n/p ratio is related to the electron neutrino degeneracy by n/p

=

exp(—AM/T 4

—

ce),

(5.5)

where T~ is the weak freeze out temperature for this reaction (which now also depends on the degeneracy parameters). Figure 18 from the study of [David and Reeves l980a] shows the compatibility regions of the ~ versus ~ plane for a wide range of Q,3. The most stringent constraint placed on any particular degeneracy parameter is that placed on the electron degeneracy, ~e, from considerations of primordial nucleosynthesis. In spite of the since revised limits on primordial lithium, consistency with the presently inferred primordial abundances are not much different from those derived by [David and Reeves l980a] viz., —0.5 ~$i~ ~ 1.5 (~ 0) [e.g. Starkman 1992]. This constraint is more stringent than that derived from the age of the universe (the mass-energy limit), which is (~~~)h/4~ 140 [Freese et al. 1983] The role played by the neutrino mass on this constraint is discussed in section 5. [Freese et a!. 1983] also considered the more stringent constraint arising from considerations of the, still unresolved, formation of galactic structure. The constraints placed by nucleosynthesis studies on other neutrino degeneracies, which do not directly influence the n—p interactions, is less clear. However, the limit imposed by nucleosynthesis,

R.A. Malaney and G.i Mathews, Probing the early universe

0.7 2.6 4.3 14 I -0.3

27

46

I -

.~

-0.07

~

0 1 2

4

6

8

10

12

177

14

16

Le

18

Fig. 18. The compatibility regions of the ~e—~ plane for Q,~ranging from 0.04 to 0.7 [David and Reeves 1980a 1. The 1b~ superscript ‘ signifies that any extra energy density is folded into ~ and the Hubble parameter h = 0.75 is adopted in ~ The solid and dashed curves represent 1a and 2a compatibility with the then inferred primordial abundances, respectively.

which again depends on observational uncertainties, is generally somewhat better than that imposed by the mass-energy limit. The nucleosynthesis constraint is found to be ~ 20 [e.g. Starkman 1992]. (It should be noted that if only one neutrino species is assumed degenerate then the above limits are more tightly bound.) Using the additional freedom allowed by invoking degeneracy of more than one neutrino species, it is possible [Beaudet and Yahil 1977; David and Reeves l980a; Olive et al. 1991a; Starkman 1992] to reconcile the observed primordial abundances with those predicted by an = 1 universe. Due to the increase in the universal expansion rate, however, the range of degeneracies compatible with the Pop II lithium abundance for ~b = I may be incompatible with the timescale required for galactic structure formation [Steigman 1985; Kang and Steigman 1992]. If one uses the constraints imposed by the structure formation argument, the limits on the degeneracies become —0.06 ~ ~ 1.1 and ~

K,L,

Tl ~ 7 [Kang and Steigman 1992]. However, if the primordial lithium abundance is actually greater than the Pop II value (see section 9.3) this constraint can be relaxed. Also, it has been suggested [Starkman 1992], that if the neutrinos have a mass it may be possible for the universe to become matter dominated earlier, thereby allowing galactic structure to form within the right 7Li timescale. although theproblems implications may be dependent on the adopted abundanceNonetheless, and the still unresolved of galactic structure formation, it is clearprimordial that neutrino degeneracy could play a crucial role during the nucleosynthesis epoch. Finally, it is interesting to note that neutrino decay may play an important role in setting up neutrino degeneracies [Enqvist et al. l992b; Madsen 1992]. If the decay takes p!ace after chemical decoupling but prior to thermal decoupling, interesting results for nucleosynthesis follow. One should also be aware how neutrino oscillations in a degenerate universe can influence nucleosynthesis (see below). 5.2. Sterile neutrinos If the presence of right-handed (RH) “sterile” neutrinos, yR (SU (2) L ® U (1) y singlets) are allowed, then interesting scenarios can arise when mechanisms for their conversion into left-handed (LH) “active” neutrinos, zn., are introduced. Hereafter when we refer to RH neutrinos we mean

178

R.A. Malaney and G.J. Mathews, Probing the early universe

both RH neutrinos and LH anti-neutrinos. By “sterile” here we mean that they do not transform under the standard-model gauge group. In the SBB model RH neutrinos play no role. Even if they exist they are assumed to have decoupled at T > 100 GeY. The neglect of RH neutrinos in nucleosynthesis calculations is to a large extentjustified from consideration of the statistical weight, g, of the relativistic particles present before and after the QCD phase transition. Since the evolution of the universe through the phasetransition and particle-annihilation epochs is characterized by constant comoving entropy density, the amount of heating of the LH sea relative to the RH sea is given by TLH/ TRH = (g, /g2 ) 1/3 where the subscripts 1 and 2 refer to the times before and after the annihilation epoch, respectively, and g1 and g2 are the effective statistical weights of relativistic particles at these times. Prior to confinement, the statistical weight in relativistic particles is g1 56.5; whereas, after the quarks are confined, the only nearly relativistic strongly interacting particles will be pions, so that g2 17.25 and TLH/TRH 1.5. This implies that at the time of nucleosynthesis the relative energy densities in a LH and RH neutrino species will be PLH/PRH 5, so that a sterile neutrino species counts as less than 1/5 of an additional neutrino flavor. Muons and pions should annihilate or drop out of equilibrium at roughly T 100 MeY, and if we add their statistical weight to the differential heating of the LH and RH seas then a RH neutrino species would count for only about 0.! of an additional flavor. On the other hand, if the decoupling of the RH sea is after muon and pion annihilation then the RH components count for almost a full extra neutrino flavor, which would be incompatible with the limit N~< 3.3 discussed in section 2. Thus, the RH components of the neutrino would have to decouple prior to the QCD epoch, which we define to occur at a temperature T = TQCD. It is important to note here that, because this argument is based on added degrees of freedom contributed by the RH neutrinos, we must assume that neutrinos are purely Dirac particles. Any Majorana contribution to the neutrino mass term will invalidate the argument. A particle species, x, is decoupled from the background plasma when >nja~H,

(5.6)

where a is the cross section of the interaction x with background particle i, and H is the Hubble parameter as given by eq. (2.4) which can be rewritten as 2mj1g~2T2,

H

=

where m~

(5.7)

(4ir3/45)h/

1is the P!anck mass. Several studies have used the condition that any RH neutrino species must be decoupled prior to the QCD phase transition to impose constraints on the mass, electromagnetic oscillation, and weak-interaction properties of neutrinos. We now discuss these studies in more detail. 5.3. Neutrino mass Consider a neutrino of mass m~and energy E0. One can show that the cross section for helicity-flip in any scattering event is given by [Shapiro et a!. 1980; Perez and Gandhi 1990] (5.8)

2FT2 is the cross section for the normal non-flip scattering event (GF is the Fermi where aL G coupling constant). Demanding that eq. (5.6) be valid at T = TQCD and adopting statistical weights

R.A. Malaney and G.J. Mathews, Probing the early universe

179

relevant to the epoch prior to quark confinement, leads to the following constraint on the neutrino mass [Fuller and Malaney 1991] m~<<300 keY (100 MeV/TQCD)”2. Assuming TQCD

(5.9)

100 MeY results in a cosmological limit of m

0 << 300 keY. The present experimental upper limits for the muon and tau neutrino are 270 keY and 35 MeV, respectively [Hernándezetal. 1990, 1991]. Finally, we note that the constraint in eq. (5.9) can be altered for the tau neutrino if it were massive enough (m0 ~ 1 MeY), as then it would be non-relativistic during nucleosynthesis [Kolb and Scherrer 1982; Kolb. et at. 1991]. In such circumstances the universe is matter dominated as it passes through nucleosynthesis and the isotopic yields can be perturbed from their SBB values. This effect would modify somewhat the limit of eq. (5.9) for the other neutrino species. Introduction of additional new properties for such massive neutrinos, for example large magnetic dipole moments [Kawano et al. 1992], could obviously influence this argument. 5.4. Neutrino oscillations

If at least one generation of neutrino has a non-zero mass, and if the weak interaction states are not mass eigenstates ofthe free Hamiltonian, then the possibility exists for flavor changing neutrino oscillations to take place. To see this consider the most general case where the mass eigenstates can be different from the weak eigenstates, with a unitary transformation connecting them. Considering only two neutrino flavors Ve and v15, we can write the mass eigenstates v1 and v2, with masses m, and m2, respectively, as (v,~ = (cosO —sinO~(ve~ Y’2)

\stnO

cosO

,/

(5.10)

\v,1j

parameterizing the unitary transform via a mixing angle 0. Ignoring terms which contribute only an overall phase, we can write the equation of motion in the weak basis as 2cos2O öm2sin20’\ (ye (5 11) .0 (~e’\ 1 (2EV~ — 2öm ‘at~v44Ek, öm2sin20 0 )~v,L —

where E is the neutrino energy, and ôm2 = m~— mf. V~represents a potential energy term present only in matter (see below). For the vacuum 1’~ = 0. Clearly then, for ~m2 ~ 0, oscillations from one flavor to another can occur. The probability, F, that an electron neutrino will remain an electron neutrino in the vacuum can be written as P

=

1 —sin2 (20)sin2 (irr/Lo),

(5.12)

where r is the distance traversed by the neutrino in time 1, and L 0 is the neutrino oscillation length. This latter quantity2.is given by L0 = 4irE/ôm Neutrino interactions in matter contribute a non zero term for Ve, viz., Vev~GFne,

(5.13)

(5.14)

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R.A. Malaney and G.J. Mathews, Probing the early universe

where ~e is the electron number density. From the form of eq. (5.11) then, it is evident that a resonant condition exists for \/~GFfle =

(ôm2/2E)cos2O.

(5.15)

This describes the so-called MSW mechanism [Wolfenstein 1978, 1979; Mikheyev and Smirnov 1986]. It is widely believed that this effect may resolve the apparent discrepancy between the neutrino flux predicted by the standard solar model and the value measured by the underground neutrino experiments (for a review of the solar neutrino problem see [Bahcall 1989; Bahcall and Pinsonneault 1992]). In the standard cosmological mode! in which only LH neutrinos, 1L, and RH antineutrinos, 11R, exist, oscillations would not be of any significance. This is because the neutrino flavor (and antineutrino) number densities are nearly equal. However, in the presence of RH sterile neutrinos, ~R, an interesting scenario may arise when allowance is made for their conversion into LH active neutrinos. Original calculations of this effect in the early universe [Dolgov 1981; Khlopov and Petcov 1981; Fragion and Shepkin 1984; Langacker et al. 19871 imposed stringent constraints on the mixing parameters. However, it was subsequently pointed out [Nötzold and Raffelt 1988] that in the context of the early universe, finite temperature corrections are important. For example, the self energy due to this correction of an active electron neutrino, in the primordial plasma is described by Ve

=

~

(5.16)

where n~,is the photon density, and A 55. The term L~contains terms proportional to both the lepton and baryon asymmetries, all of which are anticipated to be of order 10—10. A description of the evolution of the neutrino system in the primordial plasma can readily be obtained by use of eq. (5.11), with V~as given by eq. (5.16) and the replacement v,~—~ ~ Several re-analyses of the phenomenon of oscillations into sterile states subsequently appeared, with the result that the constraints on the oscillation parameters became much less stringent [Barbieri and Dolgov 1990, 1991; Kainulainen 1990; Enqvist Ct al. 1990, l992a]. Two classes of solutions are in fact allowed, depending on the sign of öm2. To see this consider the resonance condition for the primeval plasma: 2V’~(3)GFT3(±Lp— AT2/M~,J)= (ôm2/3.15T) cos2O.

(5.17)

The +L~ (—Lv) term in this relation corresponds to neutrino (antineutrino) transitions into inert states. Although L 0 is no longer a constant due to variations in the neutrino and antineutrino number densities, it has been shown [Enqvist et al. 1991] that it dynamically evolves to zero. In this2 case seen from resonance condition only constraints. be satisfied The for <0. itIn can the be scenario ôrn2 >eq.0, (5.17) the lackthat of athe resonance leads to muchcan weaker ~m allowed region of the oscillation parameters are shown in fig. 19 [Enqvist et al. 1990]. Here the space relevant to neutrino oscillations in the solar interior is entirely within the allowed region. However, we do note that non-equilibrium effects [Barbieri and Dolgov 1991] may slightly extend the disallowed region shown in fig. 19, to the extent that it marginally impacts the solar neutrino problem. Finally, it is perhaps not too surprising to find that relaxation of the bounds shown in fig. 19 are possible in the presence of additional new physics. For example, in the presence of large Majorana couplings, the bounds on the mixing parameters can be weakened by several orders of magnitude [Babu and Rothstein 1992].

R.A. Malaney and G.J. Mathews, Probing the early universe

181

100

~eN~

1o6

ioi

_3 2 2B,

~-l

Sin

Fig. 19. The nucleosynthesis bound on electron neutrino mixing parameters [Enqvist et al. 1990].

As an aside, it is interesting to note that the arguments given above can be used to impose stringent constraints on the pseudo-Dirac mass splitting [e.g. Dixon and Nir 1991; Cline and Walker 1992; Enqvist et al. t992c]. This is an important point in regard to a wide class of neutrino models which purport to account for the widely discussed 17 keY neutrino. In the presence of large initial neutrino degeneracies (L~>> l0~),the first term of eq. (5.17) dominates the neutrino self-energy. In this case, oscillations between the different (active) neutrino flavors become important [Savage et al. 1991]. Here the neutral currents make the dominant contributions to the effective mass differences between the neutrino species. For this scenario, the effect of resonant oscillations on the SBB limits for the leptonic charge of the universe have been investigated, and it was found that for a large range of öm2 and 0 significant flavor oscillations can occur, subsequently altering the degeneracy of individual neutrino species [Savage et al. 1991]. In the presence of these new oscillations the best limits on the degeneracy of the universe are not in fact those derived from the nucleosynthesis studies discussed above, but from the age of the universe. 5.5. Electromagnetic properties 5.5.1. Magnetic dipole moment The possibility that the neutrino possesses a large magnetic dipole moment, Pu, is of considerable interest in astrophysics. For example, in evolved stages of stellar evolution a large 4u~has crucial implications for cooling rates via neutrino emission [Bernstein et al. 1963; Domogatskii and Nadezhin 1971; Sutherland et al. 1976; Marciano and Parsa 1986; Fukugita and Yazaki 1987; Raffelt et al. 1989; Raffelt 1990]. A large neutrino magnetic moment has also been proposed as an explanation for the low solar neutrino flux and its apparent correlation with the solar cycle [Cicernos 1971; Okun 1986; Yoloshin and Yysotskii 1986; Yoloshin et al. 1986a,b]. In the standard electroweak model the Dirac neutrino obtains, through radiative corrections (fig. 20a), a magnetic moment of 9p~ (5.18) Pu ~ 3 X t0~ where PB is the Bohr magneton. In extensions to the standard etectroweak theory, however, it is possible to generate much larger neutrino magnetic moments. For example, in a left—right

,

(~~)

182

R.A. Malaney and G.J. Mathews, Probing the early universe

a)

)b)

~

WL+WR

m~ 1’L

1L

VR

7

)c)

11R

~‘L

~<(JJf(

m~) ~“L \.,‘t.~)

~‘R

~‘L

1L

1R

m~ 1R

1L

Fig. 20. (a) Diagrams giving rise to the magnetic dipole moment of the neutrino in the standard model. (b) Diagram giving rise to the magnetic dipole moment of the neutrino in a left—right symmetric model. (c) A possible diagram giving rise to a large magnetic dipole moment of the neutrino, where ~+ is a charged scalar particle.

symmetric model [Kim 1976; Beg et al. 1978; Schrock 1982] Pu 10’4pB (fig. 20b). If there exists an appropriate charged scalar particle with the correct coupling (fig. 20c) then Pu could even be as large as 10’°pB [Fukugita and Yanagida 1987]. Astrophysical constraints on neutrino magnetic moments come from consideration of stellar cooling rates, Pu ~ lO’0—lO’2p~[Bernstein et at. 1963; Domogatskii and Nadezhin 1971; Sutherland et a!. 1976; Marciano and Parsa 1986; Fukugita and Yazaki 1987; Raffe!t et al. 1989; Raffelt 1990], or from analysis of the neutrino burst detected from supernova 1 987A Pu ~ 10— ‘2PB [Barbieri and Mohapatra 1988; Goldman et al. 1988; Lattimer and Cooperstein 1988]. Some caution is perhaps warranted in regard to this latter constraint, however, since it assumes an understanding of stellar collapse and the galactic magnetic field. Currently, the best experimental limits on neutrino magnetic monit~.tsare [Kyuldjiev 1984] Pu, <

1.5 x t010p~

(5.19a)

and [Abeet al. 1987] p~<9 x l0’°p~.

(5.19b)

In the early universe a large magnetic moment for the neutrino wilt have important consequences for two reasons. First, the neutrinos could be kept in equilibrium with the photons to much later times than predicted by the standard cosmological model. In this situation the neutrinos could share in the entropy production by the e~e~annihilations at T 0.5 MeY, consequently altering conditions prior to nucteosynthesis. Secondly, a neutrino magnetic moment breaks the chiral invariance of the neutrino sector in the Lagrangian. That is, the vertex coupling the neutrino to the electromagnetic current only connects states of opposite helicity. This means that electron—neutrino scattering may result in a heticity change (e±+ v 1 ~ e±+ v,), and e+e annihilation always produces both helicity states. From the above discussion it is clear that, as with a neutrino mass or oscillations, a neutrino magnetic moment gives rise to a mechanism whereby the RH neutrino sea can equilibrate with the LH neutrino sea. The RH helicity states remain coupled to the primordial plasma if the

R.A. Malaney and G.J. Mathews, Probing the early universe

183

rate of exciting interactions remains larger than the expansion rate. This constraint reduces to >J, n,a (Pu,) < H, where a (Pu,) is the scattering cross section of neutrino species i. By requiring that the RH neutrinos decouple from the primordial plasma early enough, so as not to overproduce 4He, the neutrino magnetic moment is constrained to be Pu < 2 x l0”PB [Morgan 198la,b]. A numerical study of all the nucleosynthesis yields as a function of Pu could be useful in providing other constraints. The situation becomes more involved if large primordial magnetic fields exist. A simple scaling law coupled with the present day intergalactic magnetic field [Zeldovich et at. 1983], suggests that a field of 1 0~G could have been present at the epoch of nucleosynthesis. Apart from its direct consequences [Thorne 1967], a magnetic field would enhance the spin-flip mechanism just described [Lynn 1981; Shapiro and Wasserman 1981]. Neutrinos in the early universe, and in the presence of an external magnetic field, could flip helicity states at a rate [Shapiro and Wasserman 1981] (5.20)

TB = 2PuBp/1E,

where B~is the component of the primordial magnetic field perpendicular to the neutrino velocity. In order that rapid flipping of helicity states not result in a substantial population of RH neutrinos, and a resulting overabundance of 4He, the neutrino magnetic dipole moment is constrained by the following limit (assuming B~is proportional to the square of the cosmic distance scale), Pu

~ 2

~<

!0’8pB/Bo,

(5.21)

where B 0 is the strength of the present intergalactic magnetic field in units of tO~G. Taking into account magnetically induced neutrino oscillations and effects of neutrino refraction [Fukugita et al. 19881, a weaker limit to the neutrino magnetic dipole moment is obtained, 6pu/Bo. (5.22) Pu < l0~ Recently, [Enqvist et at. 1 992d] have used similar arguments in the context of the galactic dynamo mechanism. Finally, in closing this section we note that studies regarding the solar neutrino problem have considered the combined effects of flavor mixing, magnetic spin flip and matter interactions [Akhmedov and Khlopov 1988; Lim and Marciano 1988]. It would perhaps be interesting to carry out a similar study for the early universe. Neutrino charge radius Even though the neutrino is neutral, radiative corrections give rise to an isotropic separation of charge. The size, r, of this neutrino charge radius is defined by 5.5.2.

(r2)

6 ~

(5.23)

where F (q2) is the electromagnetic form factor and q is the momentum transfer. From arguments similar to those given above for a neutrino magnetic moment, a cosmological limit on (r2) can be derived. Adopting a(e~e

—~

vi)

2(r2)2q2,

=

3~irc~

(5.24)

184

R.A. Malaney and G.J. Mathews, Probing the early universe

and demanding decoupling prior to TQCD leads to the constraint (r2) ~ l032 cm2 [Grifols and Massó 1987]. This upper limit is compatible with that determined from terrestrial accelerator experiments [Bergsma et a!. 1984]. 5.6.

Weak interaction properties

The requirement that any RH neutrino species decouple prior to TQCD also leads to interesting constraints on the weak interaction properties of neutrinos. For example, consider the RH counterpart, WR, of the intermediate vector boson, WL, which in the standard etectroweak model mediates interactions between neutrinos and charged particles. The cross section for such reactions vary as OL cx G2FT2, where GF is the usual Fermi coupling constant for LH neutrinos. If we assume that the cross section for RH neutrinos varies in a similar manner, namely cJR cx G~T2,where GR is the coupling constant for RH neutrinos, then the decoupling temperature for RH neutrinos varies as Td cx G~213.Since the LH neutrinos decouple at 1 MeV, this leads to a constraint on the RH-coupling constant ‘—i

GR~GFT~,

(5.25)

which for TQCD 100 MeY results in GR ~ 1O3GF [Bond et a!. 1980; Olive et a!. 1981]. Similarly, if GR cx M~,where MWR is the mass of WR, then this leads to tie constraint MwR > MWL (TQcD)314

(5.26)

which for TQCD 100 MeY results in MWR > 2.5 TeY [Olive et al. 1981]. If GR or MWR are in violation of the above limits, 4He will again be overproduced.

6. Cosmic strings Stable topological defects in the early universe have received considerable attention in recent years. In particular, cosmic strings have been intensively studied. This is largely motivated by early numerical studies [e.g. Albrecht and Turok 1985; Turok and Brandenberger 1986; Brandenberger et a!. 1986] which seemed to indicate that cosmic strings could provide seed density fluctuations for the formation of large scale structure. However, subsequent studies [e.g. Bennett and Bouchet 1988, t989, l990b; Allen and Shellard 1990] have shown that too few massive 1oops are formed for this idea to work. Superconducting cosmic strings have also been proposed as a galaxy and large-scale-structure formation scenario [Ostriker and Cowie 1981; Ostriker et al. 1986]. Although this idea also appears ruled out by the lack of observed [Mather et al. t 9901 distortion in the cosmic microwave background spectrum. Nevertheless, although this scenario for galaxy and largescale-structure formation seems no longer viable, there are other potentially interesting effects which the presence of ordinary or superconducting cosmic strings may have on the early universe, namely to change the yields from primordial nucteosynthesis. We discuss these below. 6.1. Ordinary cosmic strings In the simplest scenario, ordinary cosmic strings are formed from the breaking of a U (1) symmetry by a complex scalar field. The most important cosmic-string parameter is its mass per

185

R.A. Malaney and G.J. Mathews, Probing the early universe

I

I

1o—• MPS—Bound~

-

G~

t~hI

CosmicTime

a

Fig. 21. Evolution of the background radiation energy density and the gravitational radiation energy density released from ordinary cosmic strings as a function of time [Bennett 1986].

Fig. 22. Upper bounds on G~ufrom nucleosynthesis (solid lines) plotted as a function of a. For comparison the millisecond pulsar bounds (MPS) and the microwave background bounds (MBR) are shown. The lower bound on G~irequired for galaxy formation, and the upper limit on a from the numerical simulations are also shown (from [Quirôs 1991]).

2.The unit length, 4u. This is related to the mass scale at symmetry breaking, p~= Aq~review mass per unit length required for galaxy formation is believed to be given bythrough Gu -~ 10—6. of cosmic strings and their role in galaxy formation can be found in [Vilenkin 1985; Accetta and Krauss 1990]. The production of cosmic strings in the early universe may also have profound implications for nucleosynthesis. Cosmic strings can undergo relativistic oscillations which subsequently result in the emission of substantial gravitational radiation, the rate of which is given by L -~ YgGaU2 where ~ is a dimensionless number which depends upon the shape of the loop (typically, Yg = 50—100 [Vachaspati and Vilenkin 1985]). If the rate of such radiation is large enough, a significant fraction of the universal energy density during the radiation dominated era could be in the form of gravitons. The energy density in gravitons is given by ~,

,

p(t)

y~’12[2~i’12G/3t2] ln(t/t*)

,

(6.1)

where t~ 10—30 s is the time at which frictional forces become unimportant. The important point to note here is that the gravitational energy density is not red-shifted (as is the case for the background radiation) while it remains stored in the cosmic string loops. This can be seen from fig. 21 where the time evolution of the radiation energy density and the gravitational energy density produced by the decay of cosmic strings are shown. It is the logarithmic term in eq. (6.1) which allows the gravitational radiation to eventually overcome the radiation produced by the background relativistic particles. This additional energy density could increase the expansion rate and change the nucleosynthesis yields [Davis 19851. The contribution to the energy density from the loops themselves, however, will be small relative to the gravitational energy density during the radiation dominated era.

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R.A. Malaney and G.J. Mathews, Probing the early universe

Using the fact that any new radiation source must not contribute more than 7% of the background radiation at the time of nucleosynthesis, allows constraints to be placed on the cosmic string parameters. By calculating the energy density in gravitation at any particular time, and coupling this directly into a nuclear reaction network, a constraint on the value of 1~can be derived [Davis 1985; Bennett 1986; Kiokawa 1991; Quiràs 1991; Bennett and Bouchet 1991; Caldwell and Allen 1992] (assuming all strings are characterized by a single value for u). Using the high resolution numerical simulations of [Bennett and Bouchet 1990b; Allen and Shellard 19901, fig. 22 displays the calculations of [Quirôs 19911 which shows the nucleosynthesis bounds on Gu as a function of a, the initial loop size relative to the to the horizon scale. The smallest loop scale that the numerical simulations presently resolve is a -~ 10—2. Shown for comparison are the bounds from the millisecond-pulsar stability [Stinebring et a!. 19901 and microwave background fluctuations [Bennett and Bouchet 1991]. In the limit a —f 0, the bound on Gu is found to be [Quirôs 1991] Gu< [2.6+ 12.9(3—N0)] x 10—6,

(6.2)

where N0 is the effective number of neutrino families. This bound is similar to that determined by [Bennett and Bouchet 19911, who find this limit to be stronger than their bound derived from the millisecond pulsar [Bennett and Bouchet 1 990a], although slightly weaker than that imposed by the microwave background. We also note that the above results are similar to those of [Caidwell and Allen 19921 except for a tighter pulsar bound relative to the work of [Bennett and Bouchet l990a]. This can be traced to differing models of gravitational wave emission. 6.2. Superconducting cosmic strings There exists the possibility that cosmic strings may, in some circumstances, be superconducting [Witten 1985]. Such superconducting cosmic strings could be produced in the simplest scenario through the breaking of a U (1) 0 U (1)’ symmetry by two complex scalar fields. There are also several proposed mechanisms for inducing large currents in superconducting strings [Copeland et al. 1987; Mijié 1989], most of which require the presence of a primordial magnetic field. If they exist, superconducting strings can have implications for cosmology. It has been proposed, for example, that they could influence galaxy formation via an explosion (rather than implosion) mechanism [Ostriker and Cowie 1981; Ostriker et a!. 1986]. At the same time they could be copious producers of radio waves, y-rays, and high-energy cosmic rays [Chudnovsky et al. 1986; Vilenkin and Field 1987; Aryal et al. 1987; Vilenkin 1988]. The cosmological contribution from exploding cosmic strings, however, is severely constrained by the observed lack of distortion in the cosmic microwave background spectrum [Mather et al. 1990]. Superconducting strings could, however, affect primordial nucleosynthesis and do so in a way which is more dramatic than for ordinary strings. One reason for this is that superconducting strings emit electromagnetic radiation in addition to2Yem~ gravitational Thestructure rate of emission where radiation. a is the fine constant, of I electromagnetic radiation is given by Lem = (j/j~) is the current carried by the string and Ic is the critical current at which growth is terminated by particle production. The magnitude of this critical current for bosonic charge carriers is given by J~= e~(a similar expression related to the mass of the charge carriers can be found for fermionic strings). For Gu -~ 10—6, I~ 1020 A. At a distance r from the string (assume r is much less than ioop size) a magnetic field is generated by the current j the strength of which is given by B~(r)—~2j/r. As was the case for gravitational radiation, the additional contribution to the energy density in the form of electromagnetic radiation can perturb the nucleosynthesis yields. Assuming Yem = Yg, ,

,

R.A. Malane’y and G.J. Mathews, Probing the early universe

187

the ratio of energy emitted in electromagnetic radiation relative to gravitational radiation is given by (6.3) (I/Ic)2a(Gp)~. By including the extra electromagnetic energy density in the calculations new constraints on the mass per unit length of the superconducting strings can be obtained [Hodges and Turner 1988]. For small values of f most of the radiation losses from the string are in the form of gravitational and not electromagnetic radiation. The constraint on p is therefore the same as that found for ordinary strings. For intermediate values of f less gravitational radiation is produced and the constraint on p becomes somewhat weakened. For large values of f, however, the magnetic energy density produced by the string can become very large, and in order to satisfy the 4He abundance constraint, p must be reduced so as to decrease the electromagnetic energy produced by the string. If in fact the superconducting string can carry a very large current, a more profound effect may take place [Malaney and Butler 1989]. The moving string and its magnetic field can deflect charged particles away from the immediate vicinity of the string [Hurley 1961], somewhat like the deflection of the solar wind around the earth. In the rest frame of the string, a region completely free of charged particles can develop. The distance rf at which the charge-free region ends can be estimated by considering the balance between the magnetic pressure and radiation pressure at time t, viz.,

f

=

B~(r~)/8m~ (l0t~’2)4,

(6.4)

When the mean-free path of neutral particles is small relative to rf then all particles are swept from the vicinity of the string. By considering the neutron and proton diffusion coefficients, it can be seen that such a particle-free region would not last very long. The ratio of the neutron-to-proton diffusion coefficients at the epoch of nucleosynthesis is given by D~/D~ -‘~ 1 05 (section 3.6), and consequently the timescale for neutron diffusion will be much faster than that for protons. Indeed, the timescale for neutron diffusion (—s 10 s) is much less than than the timescale for nucleosynthesis (—~ 1000 s), whereas the opposite is true for proton diffusion. As such, neutrons rapidly diffuse back into the wake, thereby forming regions in the universe which are neutron-rich. In such a situation, the universe at the onset of nucleosynthesis can consist of proton-rich areas (regions in which a passing string has pushed in additional charge particles), and neutron-rich areas (the wakes of strings into which the neutrons have rapidly diffused). Such a situation resembles that of the inhomogeneous models discussed in section 3. The volume fraction f~,of the universe which is in the form of neutron-rich regions was estimated by [Malaney and Butler 1989] to be

-~

0.lve~/(yema)’12,

(6.5)

The situation is complicated by the uncertainty regarding the value of v to be adopted for a superconducting string network. The evolution of such a network may be substantially different to that of an ordinary string network [Butler et al. 1991], and unfortunately the only values of ii presently available are those derived from the numerical simulations of ordinary networks [Albrecht and Turok 1985; Turok and Brandenberger 1986; Brandenberger et al. 1986; Bennett and Bouchet 1988, 1989, 1990b]. [Malaney and Butler 1989] find significant departures from the SBB isotopic yields for reasonable values of the string parameters. However, these calculations are not dynamic in the sense that they do not take into account successive passages of the string at any point in space, nor do they

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include diffusion effects. Both these processes can actually occur during the nucleosynthesis epoch. Further, it has been pointed out that at relatively late times ~ 106 s) catastrophic “quenching” of superconducting strings could result in a significant flux of high energy y-rays capable of photodissociating the light primordial isotopes produced earlier [Hodges et al. 1988]. This clearly is yet another potential source for alteration of the nucleosynthesis yields. Finally, we note that a small effect on the 4He abundance could also arise due to the high magnetic fields in the vicinity of the strings which could affect the weak interaction rates [Amsterdamski 1988] (see section 5.3), or cause a precession of neutrinos into RH sterile states (see section 5.2). Detailed numerical simulations of dynamical primordial nucleosynthesis in the presence of superconducting strings which also takes into account other string properties, such as electromagnetic radiation, would be very useful. Although it is hoped that such studies will enable a more quantitative statement to be made, it is already evident that superconducting cosmic strings can have important consequences for primordial nucleosynthesis. 7. Time variation of fundamental constants Because the results of primordial nucleosynthesis depend upon the strength of the gravitational interaction (through the expansion rate), the weak interaction rates (through the n/p ratio freezeout temperature), the strength of electromagnetic interactions (through the n p mass difference) and the strong interactions (through the nuclear reaction rates) it is possible to use primordial nucleosynthesis to place stringent constraints [Yang et a!. 1979; Rothman and Matzner 1982; Canuto and Goldman 1983; Khare 1986; Koib et al. 1986; Dixit and Sher 1988; Vayonakis 1988; Accetta et al. 1990; Coley 1990; Damour and Gundlach 1991; Casas et al. 1 992a,b; Serna and Dominguez-Tenreiro 1992; Serna et al. 1992] on any time variation of the fundamental constants, particularly for any change which has decreased monotonically since the big bang. There are some physical motivations for such time dependences . One example is in Kaluza— Klein and super-symmetric theories with extra dimensions [e.g. Koib et al. 1986; Vayonakis 1988; Co!ey et al. 1990]. In such Kaluza—Klein-like theories, the fundamental constants are actually defined in D + 4 dimensions and the presently observed four-dimensional values are the result of compactification of the extra D dimensions. The fundamental constants become time dependent since their observed value depends upon the radius (volume) of the compact D-dimensional space. It is difficult to find a model in which the four dimensions vary and the D dimensions do not. It is therefore most natural to suppose that the compactified dimensions contract, expand or oscillate with time [Kolb et al. 1986]. These changes in the volume of the D space change the values of the fundamental constants during primordial nucleosynthesis compared to their present day values. In [Kolb et al. 1986] the limits on the size of extra dimensions during nucleosynthesis were investigated. For simplicity the volume VD of the D-dimensiona! space was represented by a single mean radius, VD RD. The effects of variations on the gravitational constant, G, the Fermi constant, GF, and the fine structure constant, a, which depend upon the radius of the compactified dimensions as given in table 2, were considered. The different RD dependences relate to the different origins for gauge symmetries in the two theories. In addition, variations of GF due to possible changes in the mass of the W boson were ignored. It was concluded that at the time of nucleosynthesis there could not have been more than a 0.5% deviation of the compactified dimension radius from its present size in D = 6 supersymmetry or greater than a 1% deviation for Kaluza—Klein models. It was pointed out by Khare [1986] however that neutrino degeneracy could cancel the effects of the extra dimensions so that a reduction of the —

R.A. Malaney and G.J. Mathews, Probing the early universe

189

Table 2 Variation of fundamental constants with changes in compactified geometry [KoIb et al. 1986]. Theory Kaluza—Klein(D superstrings

=

a/ao 6)

2 (R/R0)—6 (R/R 0)

G/G

(R/R0 (R/R 0)_D 6 0)—

2 (R/R0) 6 (R/R 0)—

compactified dimension radius by a factor 4 might be allowed. To obtain present limits on the present variation rate of fundamental constants, a power law time dependence, e.g. G r-~[Barrow 1978; Yang et a!. 1979; Koib et al. 1986; Accetta et al. 1990; Serna and Dominguez-Tenreiro 1992] is usually considered. These typically imply exceedingly small present rates of change, e.g. O/G ~ ±9x l0—’~y’ [Accetta et a!. 1990]. Another theory which incorporates a time variation in the gravitational constant is the Brans— Dicke scalar—tensor theory of gravity [Brans and Dicke 1961]. This simplest extension to general relativity has recently received renewed interest [e.g. Accetta et a!. 1990; Damour and Gundlach 1991; Casas et a!. 1 992a,b; Serna et a!. 1992] because it has been proposed as a means to restore the original ideas of inflation while avoiding the cosmological difficulties. In such models the inflation of the scale factor can be a power law in time rather than exponential. This is the so-called extended inflation scenario [La and Steinhardt 1989; Steinhardt 1990; Steinhardt and Accetta 1990]. Further motivation for such scalar—tensor gravity theories is that they may also appear as a low-energy limit of superstring theories. In all but the flat space version of this model the explicit time variation of G must be followed numerically [Greenstein 1968; Dicke 1968; Serna et al. 1992]. In extended inflation [Lo and Steinhardt 1989] G can even oscillate with time [Steinhardt 1990]. For the flat-space matterdominated version of this model [Yang et a!. 1979] G/G 0 ~ (T/T0)h/~~+U

(7.1)

where T is the temperature and cv is the scalar—tensor coupling constant. Requiring that the inferred value of G at the time the universe becomes matter dominated is less than the value during nucleosynthesis, and from the primordial helium constraint, it can be concluded that, for N,. = 3, G cannot have increased by more that 1% from its present value. Thus, cv > 380 [Casas et a!. l992a] from eq. (9.1). However, for N,. = 2, G can vary by as much as 20—40% and cv > 50 [Yang et a!. 1979; Casas et a!. 1992a]. For such large values of cv the Brans—Dicke theory is indistinguishable from general relativity at the present time. For more thorough calculations of the constraints on Brans—Dicke—Jordan theories see [Damour and Gundlach 1991; Serna et al. 1992]. Another possibility is that of scale-covariant Friedman—Robertson—Walker cosmological in which the notion of absolute units is abandoned [Rothman and Matzner 1982; Canuto and Goldman 1983; Serna and Dominguez-Tenreiro 1992]. Since units are associated with interactions, abandoning absolute units implies that the ratio of proper length intervals can become a function of space and time, a violation of the strong equivalence principle. Mathematically this is described by a gauge function, fi which can vary in space and, in particular, time and accounts for the change of units. [Serna and Dominguez-Tenreiro 1992] have studied such models considering not only the effect on nucleosynthesis from a scaling of fundamental constants, but the modifications to standard kinetic

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R.A. Malaney and G.J. Mathews, Probing the early universe

J

I

I

I

Y~~O.26

0

0.02

I

I

N~3

0.04

0.06

0.08

0.10

It

Fig. 23. Allowed fractionof vacuum energy, x, to total energy density during primordial nucleosynthesis based upon indicated constraints [Freese et al. 1987].

theory which modify the nuclear reaction rates as well. As in the above studies they found that /3 could at most vary from its present value by 1%. Another possibility is that of a time varying cosmological constant. One of the outstanding puzzles in particle astrophysics is why the cosmological constant (or vacuum energy) is so small [Schwarzschild 1989]. Although there has been some interesting discussion of this point [Coleman 1988; Banks 1988], it is not presently understood why the upper limit to the present value of the cosmological constant is only < 10b0 eV cm3 [Peebles 1984] compared to the relevant particle physics energy scale (the Planck scale, 10126 eV cm3). Resolving this embarassing discrepancy is the primary motivation for supersymmetric theories. They avoid this problem by canceling the vacuum energy with supersymmetric partners. Another possible explanation of this dilemma is that the vacuum energy has decayed over time to its present small value. Freese et a!. [1987] investigated the constraints imposed by primordial nucleosynthesis in a universe with such decaying vacuum energy. They postulated a vacuum energy coupled to radiation such that the vacuum energy is always a constant fraction (0 < x < 1) of the total vacuum-plus-radiation energy density. For such a cosmological model the vacuum energy redshifts away with time along with the radiation. The main effects of a nonzero vacuum energy during the radiation dominated epoch are to speed up the expansion rate (due to increased energy density), and maintain the universe at a higher temperature than the standard big bang. The latter effect dominates so that the net result is to decrease the helium abundance because the universe cools below the photodisintegration threshold for deuterium at a later time than for the SBB. A longer timescale for the formation of deuterium implies fewer neutrons have survived decay and a lower proper baryon density (and hence less 4He produced) during primordial nucleosynthesis. It was concluded that in such a model the vacuum energy must remain significantly less than the radiation energy density during primordial nucleosynthesis to satisfy the abundance constraints. Figure 23 [Freese et al. 1987] summarizes the

-~

R.A.

Malaney and G.J. Mathews, Probing the early universe

191

allowed fraction of the total energy density, x, in vacuum energy as a function of baryon-to-photon ratio for three neutrino types. At most only 8% of the energy density could be in vacuum energy or 4He would be underproduced. Another possible quantity which could have a time dependence during the nuc!eosynthesis epoch is the baryon-to-photon ratio due to a late baryogenesis epoch [e.g. Scherrer et al. 1991; DelbourgoSalvador et a!. 1992]. The overabundance of light elements, particularly 4He, requires that the temperature at which baryogenesis ends be greater than 2 MeV (see section 4). —‘

8. Other variants We conclude our presentation of non-standard primordial nuc!eosynthesis models by discussing some other particle physics models which have been suggested as possible sources of perturbation to the SBB model. The effect of these perturbations in the early universe has not yet been fully explored and it may be that they have only a minor bearing on the nucleosynthesis yields. 8.1. Shadow matter

Compensation for the asymmetry of left and right chirality states of ordinary particles in superstring models is provided by the presence of shadow matter particles. These particles will have the mirror properties of ordinary particles if the compactification of additional dimensions retains the symmetry of left and right states. The forces between such mirror particles are identical to those between ordinary particles. However, the mirror and ordinary particles are linked to each other only through the gravitational interaction. A mirror universe with the same temperature history as the ordinary universe would lead to an increase in the effective degrees of freedom from g~cc = 10.75 to g~r= 21.5, corresponding to roughly six neutrino families. From our discussion of section 2 it is clear that such a mirror universe is ruled out and the total contribution to the energy density from shadow matter must be less than a few percent [e.g. Dubrovich and Khlopov 1989]. However, it might be possible to compensate the increased helium production by shifting the time of freeze-out of the weak interactions to an earlier epoch [Kramarovskii et a!. 1992]. It is also possible [e.g. Bartlett and Hall 1991] that the mirror universe could have a temperature history substantially different from the ordinary universe. This arises, for example, if the mirror universe had a different initial temperature or if re-heating processes occurring within it took place at different times from those occurring in the ordinary universe. If the mirror universe had a much lower temperature than the ordinary universe at the epoch of nucleosynthesis, then since p cx T4 the energy density in the form of mirror particles will be diluted relative to the energy density in the form of ordinary particles. In these circumstances the mirror universe could essentially play no role in the nucleosynthesis occurring in the ordinary universe. This argument is very similar to that discussed for sterile neutrinos in section 5.2. If the existence of mixed particles which experience both mirror and ordinary forces is presumed, however, then temperature equilibration between the two different universes could be obtained if the coupling between them is large enough. Due to radiative corrections mixed-mirror particles could carry conventional electric charges of order ee where e 1 O~—I~_6~ Values of e in this range, however, would lead to a large re-heating of the mirror universe and a subsequent increase in g~.trwhich would be incompatible with the inferred primordial 4He abundance. In order for this

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R.A. Malaney and G.J. Mathews, Probing the early universe

bound not to be violated c ~ 3 x 10—8 is required [Carlson and Glashow 1987; Khlopov et a!. 1991]. It has been suggested [Bartlett and Hall 19911 that at some point after nucleosynthesis a coupling between the ordinary universe and a cooler shadow universe, similar to that just described, would lead to a cooling of the ordinary photons and therefore a reduction in their comoving photon density. This idea attacks the notion that the baryon-to-photon ratio, ~j, is the same now as it was at the time of nucleosynthesis. SBB nucleosynthesis is allowed to take place with a low and then photon cooling subsequently increases ~ to accommodate a baryon critically-closed universe. In this scenario the shadow particles are not mirror particles. ~,

8.2. Baryon oscillations Oscillations requiring a change in the baryon number zXB = 2 are not allowed in the simplest GUTs such as SU (5) (although they naturally account for ~B = 1 oscillations). Other models such as SO (10), SU (16) or superstrings, for example, do allow for AB = 2 oscillations with periods r of order l05_108 s [Mohapatra and Marshak 1980; Panda 1983; Lazarides et al. 1986; Mohapatra and Vafle 1987]. ~B = 2 oscillations would enable neutron—antineutron transformations to take place, giving rise to potentia!!y interesting nucleosynthesis effects. Unless t > iO~s, the nucleosynthesis yields are perturbed from their SBB values [Halm 1989]. This nucleosynthesis limit provides an alternative to the experimental limit of r ~ 108 s, which is based on several theoretical and experimental assumptions [Fidecaro 1983; Takita 1986] 8.3. Massive charged particles Finally, we mention the possible role played by stable massive charged particles (CHAMPS) [Cahn and Glashow 1981] during nucleosynthesis. It has been proposed that such heavy particles may compose the dark matter [Chivukula and Walker 1989; DeRüjula et al. 1990; Dimopoulos et al. 1990]. Since they could bind to the primordial isotopes with binding energies comparable in some cases to nuclear binding energies, CHAMPS could significantly alter some important nuclear reaction rates. The details of such changes and their effect on SBB nucleosynthesis is difficult to calculate and has not been fully worked out. However, the presently inferred terrestrial flux limits on CHAMPS (although dependent on several assumptions about galactic dynamics and the solar wind) would seem to imply that their abundance in the early universe was low enough so as not to significantly affect nucleosynthesis [Chivukula and Walker 1989; DeRtIjula et al. 1990; Dimopoulos etal. 1990]. 9. Observations and galactic evolution effects relevant to non-standard models Most of the non-standard models we have discussed are either speculative or based upon theoretical foundations which are not fully developed. In some circumstances, such as the QCD phase transition, the theoretical difficulties are complex and not likely to be overcome in the near future. In these circumstances it is important to investigate what role observation may play in discriminating between the different non-standard models and in constraining possible extensions to the cosmological and particle physics standard models. In this way observational studies impact future theoretical development. Here we will discuss some recent observational studies which directly impact the study of primordial nucleosynthesis models.

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We also discuss how observations pertaining to the primordial isotopes and non-standard nudeosynthesis models should be interpreted in light Of the uncertainties arising from chemical evolution. A review as to how galactic chemical evolution might constrain baryonic dark matter limits is given by [Pagel 1990]. The galactic evolution of light-element abundances and its effect on the constraints from primordial nucleosynthesis have been studied in a number of papers over the past two decades [Reeves et al. 1970; Truran and Cameron 1971; Meneguzzi et al. 1971; Mitler 1972; Reeves et a!. 1973; Audouze and Tins!ey 1974; Reeves 1974; Mathews 1977; Reeves and Meyer 1978; Walker et a!. 1985; Arnould 1986; Delbourgo-Sa!vador et al. 1985; Mathews et al. l990a; Brown 1992]. In addition to any big bang contribution to the light e!ement abundances, there is most certainly at least some stellar contribution to 3’4He [Roodet a!. 1976, 1992; Hartoog 1979] as well as contributions to 6Li, 7Li, 9Be, ‘°Band “B from cosmic-ray induced spallation reactions [Walker et al. 1985]. There may also be contributions from supernovae to 7Li and “B [Epstein et a!. 1988; Woosley and Haxton 1988; Dearborn et a!. 1989; Woosley et a!. 1990; Brown et a!. 1991] as well as 9Be [Ma!aney 1992], and contributions to the abundance of 7Li from red giant stars [Cameron and Fowler 1971; Smith and Lambert 1985] or compact objects [Canal 1970; un 1990]. The uncertainties surrounding the possible galactic contributions to the light elements therefore introduce some uncertainty into the primordial abundances. Even the Pop II lithium abundance could be contaminated by a significant fraction (> 1/2) from cosmic-ray lithium produced by the a + a reaction [e.g. Steigman and Walker 1992]. Another uncertainty comes from the fact that deuterium, lithium,’ beryllium and boron are extremely fragile. Thus, even though a thin layer of surface abundance may remain intact during the main sequence evolution of warm Pop II stars [Spite and Spite 1982, 1986], essentially all of the initial interior stellar abundance of these elements is destroyed before the end of the star’s lifetime. The destruction factors for these elements have recently been studied as a function of progenitor mass and metallicity over the entire stellar initial mass function [cf. Dearborn 1992]. For all models the initial mass fraction of these elements is destroyed by at least 99% before re-ejection into the interste!!ar medium independent of mass or metallicity. Thus, another source of uncertainty in constraints on the big bang comes from the uncertainty in the degree of stellar processing of primordia! material. Yet a third uncertainty comes from the degree to which stellar surface abundances on main sequence stars reflect the initial abundance or are the resu!t of gradual depletion. This is particularly important for 7Li as discussed below. 9.1. 4He

From the previous sections it should be clear that the primordial 4He abundance is often the most sensitive constraint on nonstandard big bang models. Although the yield of 4He is not very sensitive to i~,it does depend strongly on the expansion rate and the n/p ratio at the time of weakreaction freeze-out without much nuclear-physics uncertainty. However, the inferred primordial helium abundance involves an important correction for galactic production which is usually derived from an assumed linear correlation with metallicity, which can be derived from the closed box model of galactic chemical evolution with instantaneous recycling of stellar material [e.g. Mathews et a!. l993a], Y

=

Y~+ (E~Y/i~Z)Z,

(9.1)

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194

where Z should be the heavy-element mass fraction, but is usually taken to be the number abundance of a heavy element relative to hydrogen. The maximum likelyhood regressions from the observed [Peimbert and Torres-Peimbert 1974; Kunth and Sargent 1985; Pagel 1988, 1 989a, 1991; Page! and Simonson 1989; Torres-Peimbert et al. 1989; Steigman et al. 1989; Olive et a!. 1991; Walker et al. 1991; Fuller et a!. 1991; Pagel et a!. 1992; Mathews et a!. 1993b] correlation of metallicity with helium abundance depends upon the subset of the available data used. For example, [Pagel 1991] found that V,, = 0.225 ±0.005 when the fit is restricted to Y versus 0/H data with meta!!icity less than 25% solar, whereas a fit including higher metallicity points including the Sun and Orion gives Y,, 0.24 ±0.01. Using similar restrictions on the data set, [Fuller et al. 1991] argue for V~= 0.22 ±0.01, thereby allowing for V~as low as 0.21. However, [Page! et a!. 1992] argue against the [Fuller et al. 1991] analysis due to deficiences in the data set. One also obtains different values for V,, depending upon whether one fits 0/H, N/H, or C/H. There is also a slight change if points aze removed which might be contaminated from the presence of nearby Wo!f—Rayet stars [Pagel et al. 1992]. It has also recently been pointed out [Skillman 1993] that emissivity corrections increase the helium abundances for the low metallicity points of Pagel et al.’s [1992] data set. Chemical evolution corrections are also important [Mathews et al. !993a], particularly for nitrogen, which may have a significant contribution from secondary stellar sources and therefore be delayed relative to the formation of helium. A best linear fit to the corrected and Wolf—Rayet excluded data of [Page! et al. 1992] gives, V

=

0.228(±0.005)+ !l5(±40)0/H

(9.2)

V

=

0.233(±0.004)+ 14l5(±510)N/H,

(9.3)

or

whereas, introducing corrections for chemical evolution effects [Mathewset al. l993b] and fitting the correlations of Y, 0/H, and N/H simultaneously [Balbes et a!. 1993] reduces the optimum primordial helium abundance from this data set to Y~= 0.227 ±0.006. Of course, there are still a large number of other potential systematic uncertainties [Davidson and Kinman 1985; Pagel 1991; Peimbert et a!. 1992] associated with the observed helium abundances such as corrections for neutral helium, uncertainties in the ionizing UV flux and in effective recombination coefficients, processes involving grains, interstellar reddening, absorption lines of metals, detector nonlinearity, collisional excitation, and temperature fluctuations. There also are a number of possible sources of uncertainty in the extrapolation even when chemical evolution models are used due to possible effects of mixing times and outflow of metal-rich material. In addition, there remains the caveat of some source of 4He, such as hypothetical Pop III stars [Bond, Arnett and Carr 1984; Carr 19901, in which the concomitant production of metals is very low. Thus, although the statistical errors in the above analyses are rather small, the potential systematic errors probably increase the uncertainties to 0.01 so that it is not yet possible to conclude that the observed primordial helium abundance is inconsistent with the allowed lower limit obtainable from the standard big bang with 3 relativistic neutrino species, 1’,, > 0.23 7 [Smith et a!. 1993]. However, if the upper limit to the primordial helium abundance is ever established to be less than 23.7%, then the SBB can not reproduce such a low helium abundance without overproducing (2H + 3He) and 7Li. Then some of the nonstandard effects discussed in this review may be required to resolve the discrepancy such as baryon inhomogeneity [e.g. Mathews et al. !993a], neutrino degeneracy [e.g. Kang and Steigman 1992; Starkman 1992], time-dependent physical constants

R.A. Malaney and G.J. Mathews, Probing the early universe

195

[e.g. Ko!b et al. 1986], or late baryogenesis [Scherrer et a!. 1991; Delbourgo-Salvador et a!. 1992]. It might also be possible within the standard big bang to reduce the primordial helium abundance lower limit by reducing ~ and imposing an unconventional galactic chemical evolution scenario to circumvent the constraint on the upper limit to (2H +~He)/H. There is also an upper limit to the V~value which can be consistent with the standard big bang and 3 light neutrino species. The observed Pop II lithium abundance (7Li/H < 2.3 x b_b) requires ~ < 5.87 which implies V~< 0.247 [Smith et al. 1993]. This limit differs slightly from some previous analyses [e.g. Olive et al. 1991; Walker et a!. 199!] due to an overall 20% decrease in reactions producing lithium at high baryon-to-photon ratio. 9.2. 2Hand3He Since deuterium is mostly converted to 3He in stars it is useful [Dearborn et a!. 1986] to consider these two nuclides together when determining constraints from primordial nucleosynthesis. The evolution of the sum of deuterium plus 3He is not completely mode! independent but it is less mode! dependent than to consider either species separately. Accounting for material which has been cycled through stars [Olive et a!. 1991; Walker et al. 1991], the upper limit can be written (2H + 3He)/H < Ar” (X®/X~)y 23® where A® deuterium factor, g3 islifetime, the fraction 3He which survives to isbethere-ejected at astration the end of a star’s and of initial deuterium plus Y23® is the presolar sum of deuterium and 3He obtained primarily from an analysis of gas rich meteorites. Taking A® ~ 1/3 and g 2H + 3He)/H < l0~ [Olive 3 1/4 gives ana upper limit of ( understanding of astration et a!. 199!; Walker et al. 1991]. Clearly however, more quantitative factors and survival fractions would help to strengthen this constraint. It is also important to determine the primordial deuterium abundance. Deuterium is the most fragile of the light elements. Essentially all of the initial deuterium is destroyed in the inner convective region before the star contracts to the main sequence [Bodenheimer 1966; Mazzitelli and Moretti 1980]. Therefore, deuterium is the best indicator of the degree to which light-elements have been destroyed. Present estimates are that deuterium destruction could have been as little as a factor of 2 or as large as a factor of 50 [Gry et al. 1984; Delbourgo-Salvador et a!. 19871. However, it has been argued that destruction by more than a factor of 15 might lead to an overproduction of 3He [Delbourgo-Salvador et al. 1985] since the destruction of deuterium returns as produced 3He. Figure 24 shows the helium and deuterium abundances as a function of Galactic age from a galactic evolution calculation of [Delbourgo-Salvador et a!. 1985]. Deuterium destruction in the range, 1/2 ~3Dp/Dpresent ~ 1 / 15, is consistent with a presolar mass fraction of 3He of 4 ±2 x 1 O~, although we note that higher destruction factors are not completely ruled out. The fact that the astration factor for deuterium is about the same as that for lithium allows the ratio of 7Li to D to be used as a constraint on big bang models almost independently of the astration factors [Mathews and Viola 1979]. However, the application of this ratio is limited due to the uncertainties in the galactic contribution to 7Li. 9.3. 7Li

It should be quite clear by this point that the determination of the primordial abundance of both lithium isotopes is of crucial importance to the study of non-standard nucleosynthesis. Most observational studies of lithium abundances, however, do not have the resolution necessary ~ 0.08 A) to determine the 6Li/7Li ratio. Since a relatively small amount of 6Li is produced during SBB

196

R.A. Malaney and G.J. Mathews, Probing the early universe

>< CD)

2.510’

‘He

3He)

X(

: . : ~.—‘T

~

i.:::

.28

1.10~’

-

012

31,

0.20 0.22

56789101112 Time

Fig. 24. Mass fractions of D, 3He and 4He as a function of time (in Gyr) from the chemical evolution studies of [Delbourgo-Salvador et al. 19851.

nucleosynthesis, and 6Li/7Li, ratios of < 0.1 are observed in several stars [Cohen 1972; Spite et al. 1984; Anderson et al. 1984; Hobbs 1985; Pilachowski et a!. 1989], abundance determinations of elemental lithium are usually taken to represent the 7Li abundance (see however section 9.4). Of the primordial isotopes, determination of the primordial abundance of 7Li remains the most controversial. The problem has been that observations of the lithium abundance in the atmospheres of Pop I stars [e.g. Cayre! et al. 1984; Hobbs and Pilachowski 1988] and of the interstellar medium in the galactic disk [Ferlet and Dennefeld 1984] result in a number fraction Li/H l0~, whereas observations of Pop II stars [Spite and Spite 1982, 1986; Spite et al. 1984; Hobbs and Duncan 1987; Rebolo et a!. 1988b] result in Li/H b_b. The reason for the order of magnitude difference between the different sets of observations, and the question of which measurement more closely represents the actual primordial abundance of lithium, has been the subject of some debate, although the consensus has been that the lower Pop II abundance represents the true primordial value [e.g. Boesgaard and Steigman 1985]. As already stated, however, the situation is confused by galactic and stellar evolution effects. In addition, interpretation of interstellar lithium observations requires complicated corrections for ionization, cosmic ray spallation, and grain depletion processes. The main argument that the Pop II lithium abundance represents the true primordial lithium value is motivated by the fact that Pop II stars are the oldest stars in the galaxy, and should therefore be representative of the abundances in the primordial material out of which these stars formed. Evidence for this view comes from the lithium abundance plateau determined in a wide spectral class of Pop II stars [Spite and Spite 1982, 1986; Spite et al. 1984; Hobbs and Duncan 1987; Rebolo et a!. 1988b]. In a sophisticated analysis of the problem [Deliyannis et a!. 1990] constructed a series of Li isochrones for the parameter range 10 ~ Age ~ 20 Gyr, 1.1 ~ a ~ 1.5, 0.0001 ~ Z ~ 0.001, and compared them with their selected plateau stars using a x2 fitting procedure. By adopting a conservative error on each of the observed abundance determinations of 0.2 and 0.4 for the plateau and coo! stars, respectively, they derive a primordial Li value of [Li] ,~= 2.17 ~ (2 a). Since this value accounts for any small scale depletion that can occur in the plateau stars, it should be considered as a better estimate than that obtained simply from a statistical mean of the plateau star abundances. By including diffusion into their standard stellar evolution codes [Deliyannis et a!. 1990] again obtained some acceptable x2 fits to the available data. In this situation some additional, albeit small, depletion of the primordial Li had occurred and the revised value for the primordial Li abundance became [Li],, = 2.31 ~ (2a). Although additional studies by Deliyannis -‘~

~

R.A. Malaney and G.J. Mathews, Probing the early universe

197

Abundance relative to H Li7

/

.10

(—Li6

i~1.20

.25 Mr/Me

.30

Fig. 25. Lithium and beryllium abundance [Schramm et al. 1990] as a function of interior mass near the surface of a 1.3 M® star.

and Demarque [1990]have suggested that the above range may be an over-estimate of the Li depletion due to diffusion processes, [Pinsonneault et al. 1992] have found that by allowing for rotation a primordial lithium abundance as high as [Li ],, < 3.1 is possible. Thus, conservatively, 2.04 < [Li]~ < 3.1(2cr) can be adopted as the primordial Li value as obtained from analysis of the Pop II halo stars. Note that this upper limit is consistent with the observed Pop I value. The stellar models are thus marginally consistent with a view [e.g. Mathews et al. 1 990a; Brown 1992] that 7Li is gradually removed from stellar atmospheres. Possible mechanisms for the transport of 7Li to hotter regions where it can be destroyed include rotationally induced mixing [Vauclair 1988; Pinsonneault et a!. 1992], magnetic bubbles [Hubbardand Dearborn 1980], mass loss [Wilson Ct a!. 1987], uncertainties in opacities [Stringfellow et a!. 1989], or diffusion [Michaud 1986; Michaud et al. 1984]. Since any surviving primordial lithium should reside in a thin surface layer (-..~0.05M®, see fig. 25) even a very subtle correction to the standard stellar models may gradually deplete lithium and other light-element abundances [Mathewset al. 1 990a 1. In this scenario, the older stars would have destroyed more of their original lithium, thereby explaining the lower lithium abundance seen in the Pop II stars. The Pop I lithium abundance might then be more representative of the primordial lithium abundance. Any such depletion mechanism, however, must lead to a uniform depletion in all plateau halo stars. It has been argued [eg. Spite and Spite 1982, 1986] that this is unlikely since the plateau halo stars represent a range of stellar parameters. Any depletion mechanism would, therefore, have to be independent of such parameters. Indeed, Deliyannis et al. [1990] defined a number of conditions that such depletion models must satisfy. Further, Pinsonneault et al. [1992] have found models with rotation which satisfy these conditions; where lithium can be depleted by factors of 5 to 10 and still reproduce a flat lithium plateau. There is, in fact, a small observed dispersion in the plateau for metal-poor stars compared to that for Pop I stars can be traced to differences in the structure and evolution of these stars. For example, in both metal-poor and Pop I stars, differences in their total angular momentum are removed during the pre-main sequence. However, because of the thin, rapidly retreating convective zones, in metal-poor stars, the angular momentum transport is ineffective in depleting lithium in the pre-main sequence. Therefore, most of these stars should exhibit only slight differences in lithium depletion during their lifetime.

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We note, however, that this description of events is challenged by the recent non-detection of lithium in otherwise ordinary halo stars. Hobbs et a!. [1991] find no detectable lithium feature in the metal-poor subdwarf G186-26, and likewise Thorburn [1992] observes no lithium in the metal-poor dwarfs Gl22-69 and G139-8. The upper limits to the lithium abundance in these stars is roughly one order of magnitude below the Pop II lithium plateau. Other evidence for dispersion in the lithium plateau exists. By analyzing the available data in the color-equivalent width plane (which removes many of the uncertainties associated with transformation to the Tea-abundance plane), Deliyannis et a!. [199!] claim a small dispersion of 20 percent exists in the data. Such a dispersion is, in fact, consistent with the rotational depletion hypothesis. In addition, rotational depletion predicts that there should be the rare plateau star with a Li abundance we!! above the mean plateau value. Such a star would correspond to one with an initial angular momentum which was unusually low. De!iyannis et al. [1991] argue that if interstellar reddening is taken into account then one of the plateau stars, BD 23°3912,does in fact possess a Li abundance 50% above the mean plateau abundance. The important implications of such a conclusion are clear the present abundance of Li in the plateau stars may not represent the primordial value. Further observations which would impact the question of intrinsic dispersion within the halo plateau stars are clearly desirable. An important constraint on lithium evolution in the disk has been obtained by Hobbs and Pilachowski [1988]who determined lithium abundance as a function of age for stars in five Pop I open clusters with ages ranging from l08_ 1010 years. Even though the measured lithium abundances for these clusters decrease with age, when extrapolated to the plateau region, the lithium abundances are consistent with no change for the past 10 Gyr [Hobbs and Pilachowski 1988]. We do note, however, that there is some uncertainty in the age of the oldest clusters. This apparent consistency of the Li/H abundance with time has been shown [Mathews et al. 1 990a; Brown 1992] to be consistent with chemical evolution models for either the destruction or galactic production of lithium. Figure 26 shows several possible scenarios [Brown 1992] for the galactic evolution of 7Li compared with the observations [Hobbs and Pilachowski 1988]. In the lower set of curves, the 7Li abundance is presumed to begin with a low primordial abundance (7Li/H -~10~0)as given by the SBB mode! with a baryon-to-photon ratio of ~ -~ 3 x 10_b. This value is then enriched to the present Pop I value (Li/H -~l0~)by different stellar sources as labeled. The other scenario begins with a high primordial 7Li abundance (Li/H 7 x iO~)with no stellar sources but astration of interstellar lithium with an exponentially decreasing star formation rate. The presently observed Pop II value is obtained by gradual main-sequence depletion of lithium. The depletion of interstellar 7Li from a high primordial value to the present Pop I value is achievable within the constraints on these models as is growth from a low Pop II value to the Pop I value. Thus, based only upon the stellar and solar-system observations of Li/H, it is difficult to distinguish empirically between high or low primordial values. The discrimination between models with high or low primordial lithium may require a measurement of Li/H for material which has not experienced much stellar processing. It would be useful, ther~fore,to observe the lithium abundance directly in the halo or intergalactic medium. In this regard, observation of interstellar lithium toward SN 1 987A seemed, in principle, a promising way to determine the primordial lithium abundance. The fact that the supernova occurred in the LMC has several advantages. Due to the high galactic latitude of the LMC, the lithium abundance in the galactic halo, and not the galactic disk, can be measured. Since the galactic halo is less influenced by stellar activity, the lithium abundance there should more closely represent the primordial value. Similarly, the low star formation rate in the LMC [Rocca-Volmerange 1983], as well as its low ~

—

-..~

-~

R.A. Malaney and G.J. Mathews, Probing the early universe

199

1e—08

le-09

rgbig

—

ibbn 1e—10

j:::.::!:~:.j:::.

0

2

4

I

6

8 10 Time (Gyears)

I

12

14

16

18

Fig. 26. InterstellarLi/H as a function oftime [Brown 1992] for chemical evolution models with an exponentially decreasing star formation rate. These are compared with observed abundances, mostly from [Hobbs and Pilachowski 1988]. The curves labeled mg, sn, and rgbig are models in which lithium is enriched from the Pop II value by galactic production in low-mass red giants, supernovae, or intermediate-mass red giants respectively. The curve labeled ibbn corresponds to a model with a high primordial lithium abundance which is astrated by star formation to the present Pop I value.

dust content [Schwering and Israel 1989, 1991], may allow for reduced stellar activity, and reduced grain depletion corrections, respectively. Four searches for interstellar lithium toward SN 1 987A have been reported [Vidal-Madjar et al. 1987; Sahu et al. 1988; Baade and Magain 1988; Malaney and Alcock 1990]. The early detection of interstellar lithium found in [Vidal-Madjar et al. 1987] was not verified by the more detailed investigations of [Baadeand Magain 1988; Malaney and Alcock 1990]. The much larger upper limit to the primordial Li/H ratio given by Malaney and Alcock [1990]arises from a more conservative estimate of the present uncertainties involved in determining a lithium abundance from an interstellar Li I line toward SN 198 7A. These uncertainties primarily concern ionization corrections and depletion processes onto stellar grains. These issues are discussed more fully in [Baade et al. 1991] in which all of the data concerning interstellar lithium toward SN 1 987A are combined. Since the more conservative estimate of Li/H ~ 4 x 1 O~encompasses the Li/H ratio observed in both Pop I and Pop II stars, this is somewhat disappointing. The observational data of interstellar lithium cannot presently resolve the issue of the true primordial lithium abundance unless one is willing to believe that the ionization and grain depletion processes are currently much better understood than as discussed in [Malaney and Alcock 1990]. Future studies of the gas structure and other elemental abundances in the LMC should allow for a better understanding of depletion processes which have occurred there, which in turn may allow for a more accurate determination of the primordial lithium abundance from the absorption spectra of the interstellar medium in the direction of SN 1 987A. For halo stars with a surface temperature ~5500 K, the observed surface lithium abundance

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remains constant as the iron abundance increases [Rebolo et al. 1 988b] from [Fe/H] = —4 to —2. This has been taken as evidence [e.g. Reeves et al. 1990] that the Pop II stellar value is the true primordial abundance. This certainly is an important indicator of the fact that the observed lithium is indeed primordial. However, this is not a proof that there has been no main sequence destruction of the original primordial Li in these stars [Mathews et a!. 1990a; Vauclair 1988]. The reason is that the growth of the iron abundance in the early Galaxy and halo probably occurs over a very short time interval [Twarog1980; Carlberg et al. 1985; Matteucci and Greggio 1986; Schuster and Nissen 1989; Mathews et al. l990a,1992]. The variation of [Fe/H] from —4 to —2 probably occurs in less than the first 1 O~yr. Therefore, any gradual destruction of lithium, say from the Pop I value to the apparent Pop II value, would not be discernible over the time interval sampled by most of the [Li/H] versus [Fe/H] correlation. It is also true that if the Pop I 7Li abundance is built up by galactic sources from an initial Pop II value, any galactic source of 7Li would not affect the correlation until the accumulated galactic yield exceeded the primordial contribution (-..~10% of the Pop I value). This would happen at [Fe/H] -~—1.0 if the stellar source of 7Li tracks the stellar source of iron. Thus, one expects a flattening of the [Li/H] versus [Fe/H] correlation at low metallicity in either scenario and the correlation of lithium with iron does not by itself distinguish between models with low or high initial primordial lithium. 9.4. 6Li

An estimate of the 6Li/7Li isotopic ratio can be obtained from the measured wavelength of the composite lithium feature and from an analysis of the line profile using spectrum synthesis techniques. Of the few stars so far analyzed for the presence of 6Li most are of no use [Cohen 1972; Spite and Spite 1984; Anderson et a!. 1984; Hobbs 1985; Pilachowski et al. 1989] in obtaining reliable information on the primordial value due to their high metallicity and/or low effective temperatures, T~.High-metallicity stars may have been contaminated by the production of lithium isotopes during the chemical evolution of the galaxy. Stars with low values of Tefi possess deep convective zones in their atmospheres, in which the fragile lithium isotopes can be transported from the stellar surface to the hot inner regions where they can be destroyed. To circumvent these problems, one must observe hot (Teff ~ 6300 K) low-metallicity ([Fe/H] < —1.4) halo subdwarf main sequence stars (evolved stars can be contaminated with products of their own internal nucleosynthesis). The main destruction reactions for 6Li and 7Li are 6Li (p, 3He )4He and 7Li (p, a )4He, respectively. The rate for the destruction of 6Li is roughly a factor of 100 larger than the 7Li destruction rate over the temperature range relevant to the inner regions of a stellar atmosphere [Schramm et a!. 1990] (see fig. 25). Using this fact, coupled with the dependence of atmospheric convection on surface temperature, it is found that for Teff ~ 6300 K, 6Li should be sufficiently preserved [Brown and Schramm 1988; Deliyannis et al. 1989]. Consequently, the observed lithium lines for stars with this temperature should provide an upper limit to the primordial 6Li/7Li ratio. Brown and Schramm [1988]list some halo subdwarfs which satisfy this required criteria. A careful analysis of the Li I )~6707profile in one of these stars, HD 84937, was carried out by [Pilachowski et al. 1989] who concluded 6Li/7Li < 0.1 in this star. More recent observations of HD 84937 [V. Smith et al. 1993], claim detection of 6Li at the level 6Li/7Li = 0.05 ±0.02 (fig. 27), in addition to an upper limit of 6Li/7Li < 0.02 in HD 19445. This detection of 6Li in HD 84937 would be consistent with production of 6Li by cosmic rays coupled with the mild depletion predicted by non-rotating stellar models [Deliyannis et al. 1990]. The upper limit in the lower mass star HD 19445 is also consistent with the depletion factors anticipated from non-

R.A. Malaney and G.J. Mathews, Probing the early universe

1.02

I

~5 —

I

I

—

~

I

-

p

km/s

Log e(Li)—2.12

.‘

w..+0.892A r~1.0O16 .92

I

I

HD84937

-

.94

I

201

I

I

6707.4

f(’U)oOO

,

i

I

I

I

6707.6

—

f(’Li)0.05

~“,

S=0.153A

-

6707.8 Wavelength (A)

fCLi)’—O.lO

I

I

I

6708

I

6708.2

Fig. 27. The Li I line in HD84937: the filled circles are the observed points, and the curves are theoretical spectra with 6Li fractions f(6Li). The curve corresponding to f(6Li) = 0.05 appears to give the best fit to the data. The differing adopted microturbulence ~, the total lithium abundance a, the velocity shift w, the continuum shift r, and the smoothing parameter S are also shown (taken from [Smith et al. 1993]).

rotating models. However, [V.Smith et al. 1993] argue that their detection of 6Li in HD 84937 would be inconsistent with the large depletion factors anticipated from rotating subdwarf models [Pinsonneault et al. 19921. Therefore, if rotational depletion is applicable to HD 84937, 6Li production from some other source would be required. As increased production of 6Li by cosmic rays would likely conflict with observations of Be and B in halo dwarfs (see below), an alternative source could be stellar flare production. Failing this, it could mean that some primordial production of 6Li is required. If, in fact, the 6Li observed in HD 84937 was primordial it may be tentative evidence in support of the late-decaying particle scenario discussed earlier. Clearly though, in order to reconcile the observations with the large primordial abundance of 6Li predicted by the studies of [Dimopoubos et al. 1988a,b], one must conclude that some physical effect such as rotationally induced mixing [Pinsonneau!t et al. 1992] is applicable to a large sample of dwarfs. Due to its key importance in testing late-decaying particle models, and the crucial role of stellar depletion, similar studies of 6Li in other halo subdwarfs would obviously be important. As far as galactic chemical evolution is concerned, it is expected that the evolution of 6 Li should be very similar to that of 9Be described below unless there is indeed a very large primordial contribution. 9.5. Beryllium and boron As mentioned in section 3 it has been speculated that baryon-inhomogeneous models might produce large quantities 9Be [Boyd and Kajino 1989; Malaney and Fowler 1989; Kajino and Boyd

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R.A. Malaney and G.J. Mathews, Probing the early universe

1990] relative to that produced by the SBB model, although such high beryllium yields have not been achieved in more detailed studies [Terasawa and Sato 1990]. Nevertheless, a determination of the primordial Be/H ratio might enable constraints to be placed on the inhomogeneous models. Observations of Be in stars is made difficult by the predicted low abundance for primordial 9Be. Even in the most favorable calculations (two-zoned), the maximum abundance of 9Be produced in the inhomogeneous models is Be/H l0’~, and in more realistic (multi-zoned) inhomogeneous models [e.g. Terasawa and Sato 1990] the abundance is reduced to less than -.~l0’~.However, current observational techniques are at the point where Be/H as small as 10—13 is detectable even for relatively faint halo subdwarfs. Using spectral synthesis techniques of the Be II lines at 3130 A, Be abundances and upper limits have been obtained for a number of metal-deficient dwarf stars [Rebolo et a!. 1 988a; Ryan et al. 1990, 1992; Gilmore et al. 1991, 1992]. Since 9Be is more stable against destruction than the lithium isotopes (see fig. 25), stellar astration of 9Be may not be as important if the stars selected for study have the same characteristics as those chosen for determination of primordial lithium. Furthermore, there are presently no proposed galactic sources for beryllium other than cosmic ray spallation (except perhaps neutrino nucleosynthesis [Malaney 1992]). In fact, the recent literature is rich with cosmic ray spallation calculations which purport to account for the beryllium seen in the low-metallicity dwarfs [Gilmore et a!. 1991; Ryan et a!. 1992; Steigman and Walker 1992; Duncan et al. 1992; Walker et a!. 1992; Prantzos et a!. 1993]. Another piece of evidence in favor of a galactic source for beryllium is the observation [Duncanet al. 19921 of boron in halo dwarfs, one of which (HD 140283) also has a beryllium detection. The observed ratio of boron to beryllium (-..~ 10) is consistent with a cosmic ray origin. Unlike the case for 7Li, however, the 9Be abundance does not maintain a uniform abundance with decreasing metallicity, but instead tracks to some extent the iron abundance. This can be seen from fig. 28 where the data showing the Be abundances in stars with a wide range of metallicity is shown. Also shown are the recent boron detections [Duncanet a!. 1992]. This trend cannot be related to differences in the degree of stellar depletion [Deliyannis and Pinsonneault 1990] and must be due to differences in the initial stellar abundances. This is evidence that beryllium is being produced along with iron in the early galaxy, and at the current levels of sensitivity we have not detected primordial 9Be. However, there is a problem in that the observed linear growth of Be with Fe is not what one would expect from the simplest cosmic ray nucleosynthesis mode! in which beryllium would grow as a secondary element, and therefore increase quadratically with Fe. It would not seem, however, that this linear behavior is related to a primordial Be component [Page! 1991]. Rather, the resolution of this dilemma seems to require cosmic ray activity which is more efficient relative to the enrichment rate of the C, N, and 0 spallation target nuclei (or else a hitherto unknown stellar source). The most sophisticated analysis of the problem to date has been the work of [Prantzos et al. 1993], where an attempt to couple halo evolution, evolved GCR spectra, nuclear spallation reactions, and galactic chemical evolution is made. A principal conclusion of their study is that compatibility with the observed linear Be versus Fe behavior (and also the observed Li/Be ratio) is only attainable when time-evolution of both the total GCR flux and spectra are allowed for. They argue that such behavior may be naturally accounted for by more efficient confinement of cosmic rays arising from dynamical evolution of the galactic halo. In an extension of their calculation, however, [Malaney and Butler 1993] have argued that consideration of the nuclear mean free path limits the usefulness of this effect, and that linearity can only be maintained over a limited range of metallicity. Although it seems unlikely that current observations of beryllium and boron in metal-poor halo -~

R.A. Malaney and G.J. Mathews, Probing the early universe

~

~

203

B

Duncan e~aI~992

1Fe/HI

Fig. 28. Be and B detections in metal-poor halo stars. Observations of HD 140283 by different groups are joined by a short line. The solid lines are linear fits. Typical errors are indicated on the right hand side of the plot.

dwarfs seriously constrain inhomogeneous big bang models [Terasawa and Sato 1990], nevertheless, the importance of determining their abundances remains. If future observations can show that the Be or B abundance in metal-poor stars ceases at some point to track the metallicity and levels off at a level higher than that predicted by the SBB model, this may provide evidence of baryon inhomogeneity during the epoch of nucleosynthesis. If not, these data still provide valuable clues to the astration and galactic production of light elements during the early history of the Galaxy.

10. Nuclear uncertainties The use of observation to constrain extensions to the SBB model requires accurate predictions of the nucleosynthesis yields produced by the different non-standard models. Such accurate predictions rely to a large extent on accurate input nuclear data. Here we discuss some recent experimental advances, and outline the impact that uncertainties in the nuclear data have on our ability to determine the validity of some models. We commence with uncertainties in the nuclear input data of the SBB mode!. 10.1. Standard big bang reactions For a list of reactions included in the SBB network see [Wagoner et a!. 1967; Wagoner 1973; Schramm and Wagoner 1977; Yang et al. 1984; Smith et al. 1993]. Given the current uncertainties in the nuclear reaction rates, [Smith et al. .1993] concluded from their Monte Carlo study that there are 12 key nuclear reactions that need to be included in a SBB network. These are listed in table 3. Any other reaction not included in this table was found to alter the predicted abundance yields by an amount less than that arising from the intrinsic uncertainties of the reaction rates

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R.A. Malaney and G.J. Mathews, Probing the early universe

Table 3 Uncertainties of the key SBB nuclear reaction rates [Smith et al. 1993]. 1. 2. 3. 4. 5. 6. 7.

Reaction

1-a uncertainty

neutron lifetime 2H p(n, y) ~H(p, y)3He ~H(d, n)3He 2Jj(~Jp)3H 3H(d, n)4He ~H(ce,y)7Li

0.42% (3.73 s) 7% 10% 10% 10% 8% T 2— 7.2T 2— 0.56Tb)%, 9 T< 10: (29— 5.9T~ 9,, + 4.0T~ 9,, = T9 + 0.0419 T9 > 10: 8.1% 10% 8% T 2+ 4.OT 2— 0.02T~b)%, 9 T9b < 10: 9,, — 0.25T9~ = T(27 — 15T~ 9 + 0.783 8% T9> 10: 9.7% 9%

3He(n, p)3H 9. 3He(d, p)4He 8. 10. 3He(~,y)7Be

7Li(p, s)4He 11. 12. 7Be(n, p)TLi

listed. [Riley and Irvine 1991] did a similar variational study of reactions rates to see effects on the nuclide abundances, resulting in the same 12 reactions. The effect of reaction rate uncertainties have been investigated in the primordial nucleosynthesis studies of [Yang et a!. 1984; Walker et a!. 1991; Beaudet and Reeves 1984; Delbourgo-Salvador et al. 1985; and Riley and Irvine 1991]. However, [Krauss and Romanelli 1990] made the first quantitative study of the uncertainties of reaction rates utilizing all available data in a systematic fashion, by means of Monte Carlo methods. [Krauss and Romanelli 1990] used 10 ofthe reactions listed in table 3 (they did not consider 3He(n, p)3H and 3He(d, p)4He). Including the twelve reactions of table 3 [Smith et a!. 1993] extended the previous Monte Carlo analysis, and also incorporated temperature-dependent reaction uncertainties. Table 3 lists the new reaction rate uncertainties determined by [Smith et al. 1993]. Additionally, errors that arise from the numerical computation itself, which were neglected by previous SBB studies, were included. These errors, although relatively small, are not insignificant: for example, the previously used nucleosynthesis code of [Kawano 1988; Kawano et a!. 1988] shows computational errors as large as 6% (for 7Li at low ,~values). The study of [Smith et a!. 1993] employed a new version of the program [Kawano 1992] which included corrections for these errors. It is worthwhile to note, the striking advance in recent years in determinations of the neutron halflife, tn, which have greatly reduced the uncertainty of rn as an issue for primordial nucleosynthesis. [e.g. Mampe et a!. 19891. As a result of this the corresponding uncertainty in the value of V~ (-..~0.001) is now less than the observational uncertainties associated with Y~(section 9.1). At the present time the nuclear reactions leading to the production of 7Li have the largest uncertainties of any of the important SBB reactions. The consequences of all the uncertainties listed in table 3 can be seen in fig. 2, and a brief description of their implications is given in section 2.2. used, For a more detailed discussion the reader is referred to [Smith et al. 1993].

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10.2. Inhomogeneous models The nuclear reaction networks required for investigation of primordial nucleosynthesis yields from baryon inhomogeneous models require the addition of several new reactions to the standard networks. This is particularly so in regard to production of isotopes up to and through the CNO region. This has stimulated many experimental and theoretical studies of the new reactions and their associated rates [Boyd and Kajino 1989; Malaney and Fowler 1989; Kajino and Boyd 1990; Wiescher et a!. 1989, 1990; Kawano et al. 1991; Paradellis et a!. 1990; Wang et al. 1991; Brune et a!. 1991; Kubono et al. 1991, 1992; Barhoumi et al. 1991; Boyd et al. 1992a,b; Rauscher et al. 1992; Czerki et al. 1992; Knape et al. 1992; Cecil et al. 1992]. We discuss below in more detail some recent advances in the experimental situation. The uncertainties in the production of 7Li in the inhomogeneous models are compounded by the inclusion of new reactions such as 7Li(3H, n)9Be [Boyd and Kajino 1989]. However, the dominant uncertainty with regard to 7Li production still arises mainly from the SBB reactions listed in table 3. This is simply a consequence of the lower abundance of isotopes such as 3H relative to protons, neutrons, and alpha particles. It is clear that the large overabundance of 7Li predicted by some inhomogeneous calculations (see section 3) cannot be accounted for by uncertainties in the nuclear data. As stated earlier it may be possible to produce CNO isotopes in inhomogeneous models through the reaction sequence described in (3.13). There does exist a smaller branch through 7Li(a,y)’’B(n,y)’2B(fl,v)’2C.

(10.1)

With Qb = 1 and a density contrast, R = 50, less than 15% of the flow was believed to pass through this weaker chain [Malaney and Fowler 1 988a 1. However, measurements [Wiescher et al. 1989] have determined a somewhat lower 7Li (n, y ) 8Li cross section. This has the consequence of a larger flow (~factor 2) through the 7Li (a, y)” channel. Other important reactions with regard to heavy isotope formation may be the “leak” reaction sequences 8Li(n, y)9Li(fl, u)9Be(p, a)6Li, or 8Li(p, n)8Be 2a. A systematic investigation of the importance of these reactions over a wide range of the parameter space (0.1 <~b < 1, 10 < R < l0~)has been carried out [Kawano et al. 1991]. This study confirms that in those regions of the parameter space where significant heavy element production occurs, the 7Li (n, y ) 8Li (a, n)” reaction sequence dominates. The same study also found that ~ (p, n)8Be 2a was the most important leak reaction, destroying up to 50% of the 8Li. A new reduced rate for 8Li (d, n ) 9Be [Paradellis et al. 1990] was adopted by Kawano et al. [19911 causing this reaction to become unimportant. Due to the presence of the radioactive isotope 8Li (t 112 1 s) in the heavy-isotope producing chain, the is difficult to determine. 8Limagnitude (a, ~ of hasCNO nowproduction become available [Paradellis et a!. However, direct on point out, though, that the reliability of the extrapolation 1990; Boyd et al.experimental 1992a]. We data should of the experimental data to energies relevant to primordial nucleosynthesis has been questioned [Rauscher and Oberhummer 1992]. As discussed in sections 3.6 and 9.5, a potentially important indicator of inhomogeneity at the time of nucleosynthesis might be the presence of a high level of 9Be in very metal-poor stars [Boyd and Kajino 1989; Malaney and Fowler 1989; Kajino and Boyd 1990; Malaney 19931 produced via the 7Li(3H, n)9Be reaction. Direct experimental information on the rate for this reaction is now available from different groups [Brune et al. 1991; Barhoumi et a!. 1991] which show good agreement. ‘°Bmay also be produced in significant quantities in the inhomogeneous models [Kajino and Boyd 1990]. It can be produced via 9Be(n, y)’0Be(fl)’°B, but ‘°Be(p,a)7Li can lead to —~

206

R.A. Malaney and G.J. Mathews, Probing the early universe

N,utronR~eh Region

io—~

-

/

-

10-12

-

-

~//~

Temperature

9K]

[10

Fig. 29. The destruction rate of ‘4C for each of the dominant destruction reactions as a function of temperature for an = 0.02, R = 500 and ~b = inhomogeneous model [Kawano et al. 19911.

rapid destruction of the ‘0Be prior to its decay to ‘°B. An experimental determination of the rate of this latter reaction has recently become available [Knape et a!. 1992]. The isotope 14C is pivota! in determining the further build up of heavy isotopes up through the A 20 region and beyond. The dominant destruction reaction of ‘4C however, is currently the subject of some uncertainty. In the earlier studies it was believed that the dominant reaction was ‘4C (a, y) 180 [Applegate Ct al. 19881. Recent studies of reaction rates on 14C have shown, however, that the ‘4C(n, y)15C [Kajino et al. 1990], ‘4C(p, y)’5N [Wiescher et al. 1990] and ‘4C(d, n)’5N [Kawano et a!. 1991] reaction rates may exceed the ‘4C(ct, y)’80 rate [Funk and Langanke 1989]. An experimental study [Wang et a!. 199!] of the 11B(a, p)’4C rate has shown this reaction to be unimportant. Figure 29 shows the destruction rate of ‘4C for each of the four primary destruction channels as well as its beta-decay rate in the neutron-rich region of a typical inhomogeneous mode! [Kawano et a!. 1991]. From this, the importance of each reaction can be seen. We caution, however, that some of the reactions are uncertain by about a factor of three and further analyses of the above reaction rates are required before any more definitive conclusions can be drawn. As has been pointed out [e.g. Kurki-Suonio and Matzner 1988, 1990; Mathews et al. 1 990b; Terasawa and Sato 19901 it is important to use an accurate treatment of diffusion during nuc!eosynthesis (i.e. the use of multi-zone models instead of two-zone mode!s). When diffusion is treated properly A > 7 isotope production, as described above, becomes seriously inhibited in a large region of the parameter space. One reason for this can be a dearth of neutrons in the low-density regions. That is, if the dep!etion of neutrons from the low-density regions by diffusion back into the high-density zones is fast relative to nuclear reaction rates in the low-density zones, then the build up to heavier nuc!ei in the low-density regions is halted. The timescale of the late-time diffusion of neutrons back into the high-density regions is therefore important. Build up of heavy nuclei in the high-density regions can be inhibited by (p, a) reactions unless the baryon density is high enough for alpha captures to bypass Be and B nuclei and produce A > 12 nuclei as in [Wagoner et al. 1967]. This subject has been discussed in section 3.6.1. The main uncertainty in the diffusion rates comes from the use of approximate treatments rather than an exact solution of the Bo!tzmann equation [Kurki-Suonio et a!. 19921. The dominant diffusion coefficient in the high-density zones is that due to neutron—proton scattering. Although there might be some uncertainty in the nuclear ,

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cross section, the main uncertainty (possibly as much as a factor of two [Kurki-Suonio et a!. 1992]) in the diffusion coefficients comes from the appfoximations in their derivation. On the other hand, uncertainties in the diffusion coefficients can be compensated to some extent by other parameters, e.g. the separation distance between fluctuations. Moreover, it should be noted that current observations of Be in metal-poor halo dwarfs (section 9.5) currently offer no evidence of primordial Be production. For a more detailed discussion of these issues the reader is referred to [Malaney 1993]. Finally, we note that a new measurement of the 2H(2H, y)4He reaction rate [Barnes et al. 1987] has !itt!e effect on primordia! nucleosynthesis ca!cu!ations [Madsen 199Db] even though there exist significant quantities of deuterium in the inhomogeneous models, and the new reaction rate is a factor of 30 higher than the previous value (at the relevant temperature). This is because deuterium is destroyed faster by the neutron and proton induced reactions than by deuterium capture. Also, 4He is produced dominantly by the 3He + n and 3He + 3He reactions rather than by + 2H. 10.3. Late-decaying or annihilating particle models As discussed in section 4 several studies have investigated the effects of late-decaying particles, all of which conc!ude that the re-synthesis of the light nuclei can lead to dramatic departures from the predictions of standard primordial nucleosynthesis. One of the most recent and extensive studies in this regard has been the work of [Dimopoulos et a!. 1988a,b]. This study includes the consequences of both the electromagnetic and the baryonic decay of a massive particle, X. The proper study of the effects of such particle decays on primordial nucleosynthesis requires a very large amount of nuclear-cross-section data over a wide energy range. It is clear from tab!e 3 of [Dimopou!os et a!. l988bJ that the available data is sparse. For some reactions there is no data at all, while for others the available data is contradictory or has large experimental errors. Such a lack of experimenta! information has important consequences for the late-decaying particle scenario in that it substantial!y limits its predictive powers. The high 6Li/7Li ratio is the most important result of [Dimopoulos et a!. l988a,b] since it provides a possible test of the late-decaying particle scenario. The principa! reason for the high 6Li/7Li ratio can be readily understood. 7Li is produced by the reactions a~+ a p + 7Li and a* + a —i n + 7Be, whereas 6Li is produced by the 3H* + a n + 6Li reaction (the asterisk denotes an energetic particle produced by the decay of particle X). Due to a Z2 (Z being the nuclear charge) dependence in the Cou!omb energy-loss formula, the energetic a* particles lose energy, as they Coulomb scatter through the plasma, four times faster than 3H* partic!es. Also, since the cross section used for the 3H* + a n + 6Li reaction is 5 times larger than the cross sections used for the a* + a p + 7Li and a* + a n + 7Be reactions, then a ratio of 6Li/7Li 20 can be expected (a* and 3H* are produced in comparable quantities by the X decay). There exists no information on dU/dEa for nuc!eon(N)—a scattering above EIab 2 GeV. This is unfortunate since the production of 7Li is very sensitive to the a* spectrum. There is also little information on the photodestruction cross section for 7Be. Ca!culation of the 6Li abundance is also hampered by a lack of sufficient experimental data. The energetic 3H* are produced by ine!astic p—a scattering. The only experimental data on dapa...t/dQ exist at 90 GeV. In addition, data on the 3H + a n + 6Li cross section is limited, with no data available for E~b> 18 MeV. Other uncertainties which affect the predictions of the latedecaying particle scenario are the lack of data on the ine!asticity of baryon collisions, the energy distribution of neutrons from N—a scattering, and the neutron yield in nuc!eon—nucleon collisions. -~

-~

—~

-~

—~

—÷

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Although difficu!t to estimate, this lack of nuc!ear data could easily produce a factor of 2 uncertainty in the predicted 6Li/7Li ratio. Even such a large error, however, may not account for the discrepancy between the observed and predicted 6Li/7Li ratio. It would seem that extensive ga!actic-chemical-evolution effects must be introduced in order to salvage the late-decaying-particle model. 10.4. Other models The uncertainty of the input nuclear data discussed in section 10.1 for the SBB mode! plays a similarly important role in deviants from the standard picture. As discussed above the most important uncertainties in the SBB mode! are those associated with the reactions of table 3. None of the other non-standard models !isted above predict the production of significant heavy-e!ement flow through the radioactive isotope 8Li, or the presence of high-energy cascades. Therefore, the nuc!ear reaction uncertainties associated with inhomogeneous and late-decaying particle models will have little relevance to these other models.

11. Conclusions We have seen how the SBB model of primordial nucleosynthesis is at the present time in good agreement with both the inferred primordial abundances and with the number of neutrino fami!ies as determined from the recent e+ e collider experiments. The SBB model of nucleosynthesis is the simp!est model of isotope production in the early universe in that on!y one free parameter (17) determines the isotopic yields. However, we have argued that the important conclusions inferred from SBB nuc!eosynthesis warrant, and perhaps demand, detailed investigations as to how robust the SBB model is with respect to alterations of the standard cosmological scenario and to extensions of the SU (3 )~ ® SU (2 )L ® U(1) y particle physics model. By demanding that such perturbations and extensions also lead to agreement with the inferred abundances, we can turn the argument around and determine information and constraints on any a!ternative cosmological and particle physics model. In this review we have attempted to look at primordial nucleosynthesis from both these angles. We have determined the robustness of the standard model by detailing how far one must perturb different aspects of the cosmo!ogical model in order to spoil agreement with observation, and we have also highlighted what constraints this in turn imposes on any new physics which might exist. In table 4 we summarize the main aspects of non-standard primordial nucleosynthesis we have covered in this review. In column one the type of non-standard model under investigation is listed, and in column two the main effects of this particular model on the nucleosynthesis yields relative to the standard model predictions are given. In column three we list the type of information and constraints which can be determined from the limits imposed on the particular non-standard model by the inferred primordial abundances. We have not listed the individual non-standard models in any particular preference or re!evance to reality. That is left as an exercise for the reader! Table 4 represents the wealth of information that can be obtained from investigations of primordial nucleosynthesis beyond that of the standani model. We hope that by this point, the reader, even if he strongly believes in the validity of the standard mode!, has become convinced that such studies are indeed interesting, worthwhile, and fruitful. We close with a word of caution. The information listed in table 4 is based on the premise that on!y one particu!ar non-standard big bang model is relevant at any one time. That is, we have not

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Table 4 Effect on abundances and main constraints imposed by the different types of non-standard models (SCS means superconducting cosmic string). Effect baryon inhomogeneities particle decay neutrino degeneracy neutrino oscillations neutrino electromagnetics massive neutrinos RH weak interactions cosmic strings anisotropy time varying constants primordial black holes mirror matter

Abundances relative to SBB 4He, I7Li, ICNO ~1 I3He, ~4He,T6Li jI4He jI4He I4He I4He I4He I4He (normal); j4He, I7Li (SCS) I4He ~14He T3He, ~4He,I6Li T4He

Constraints QCD physics properties of new particles lepton charge of universe create/alter degeneracy dipole moment, charge radius neutrino mass mass of M~,coupling constants mass/length, currents shear forces, magnetic fields superstrings, relativity number density, Hawking radiation viability, coupling constants

considered the possibility that the early universe should be represented by some aspects of two or more non-standard models. The neglect of such considerations represents the unwritten ru!e that one is not allowed to invoke two “tooth fairies” in any one avenue of research. It is c!ear, however, that if one does invoke two non-standard models at the same time then the information as listed in table 4 would have to be revised. This worry is increased if one considers that one or more of the models listed in table 4 may at some time be confirmed by laboratory measurements or astronomical observations. If so, then clearly much of the information presented in tab!e 4 would require renewed investigation. We live in exciting times. Observations currently planned or already underway on the next generation of telescopes should allow better discrimination between the different nucleosynthesis mode!s, and verify or ru!ë out extensions to the standard cosmological and particle physics models. Searches for both baryonic and exotic dark matter are at the planning phase or already underway. Detection of either would clearly have an important impact on our understanding of primordial nucleosynthesis. Such studies are full of promise for resolving fundamental questions about the physics and conditions prevailing in the early universe. Acknowledgment This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under contract number W-7405-ENG-48 and Nuclear Theory Grant SF-ENG—48. We are grateful for discussions with C.R. Alcock, J. Applegate, M.B. Aufderheide, M. Butler, D.D. Clayton, D.S.P. Dearborn, C. Deliyannis, G.M. Fuller, L. Kawano, H. Kurki-Suonio, J. Madsen, M. Savage, D.N. Schramm, M. Smith, and N.J. Snyderman. References Aarnio, P. et al., 1989, (DELPHI Collaboration), Phys. Lett. B 231, 539. Abe, K. et al., 1987, Phys. Rev. Lett. 58, 636.

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