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Neutron diffusion, primordial nucleosynthesis, and the r-process
PHYSICS REPORTS (Review Section of Physics Letters) 163, Nos. 1-3 (1988) 141-154. North-Holland, Amsterdam
NEUTRON DIFFUSION, PRIMORDIAL NUCLEOSYNTHESIS, AND THE r-PROCESS
James H. APPLEGATE Department of Astronomy, Columbia University, New York, NY 10027, USA
Abstract: The application of standard particle physics to big bang cosmology predicts that the temperature interval between roughly 1 TeV and the onset of primordial nucleosynthesis at 100 keV should be rich in physical phenomena; electroweak symmetry breaking, chiral symmetry breaking, and quark confinement should all occur in this interval. First-order phase transitions can produce entropy inhomogeneities because of the uneven release of latent heat, and by shocks and detonations if significant supercooling occurs. In addition, stable objects such as black holes or soliton stars can be produced, and these objects can generate entropy by accretion. These possibilities suggest that the assumption of homogeneity in the standard model of big bang nucleosynthesis could be seriously wrong. In this article I analyse nucleosynthesis in inhomogeneous cosmologies, and I conclude that if baryon density contrasts of order 10 can be created on the proper scales, then the success of the standard model in predicting light-element abundances can be matched by a model containing a closure density of baryons. I argue that the prediction of the cosmological production of r-process elements by the inhomogeneous models is the most promising means of discriminating between these models and the standard model.
1. Introduction
Primordial nucleosynthesis is the best probe of physical conditions in the early universe available today. In the standard model [1], the big bang is responsible for the production of 90% of the 4He in the universe, all of the deuterium, and it is a significant contributor to the production of 3He and 7Li. Within the context of the standard model, powerful conclusions can be drawn from the observed abundances of deuterium and 4He. Primordial 4He production is sensitive to the number of relativistic particle species at the time of nucleosynthesis, and the observations will allow at most one more light two-component neutrino species. The deuterium production is very sensitive to the baryon density during nucleosynthesis, and the requirement that the big bang produce at least the current deuterium abundance implies that baryons must fail to provide enough mass to close the universe by at least a factor of five. The application of standard particle physics to cosmology predicts that the temperature interval between roughly 1 TeV and the onset of primordial nucleosynthesis at 100 keV should be rich in physical phenomena; electroweak symmetry breaking, chiral symmetry breaking, and quark confinement all occur in this interval. This richness of phenomena, any of which can disturb an initially uniform universe, suggests that of the assumptions made in the standard model, the assumption of homogeneity is the one most likely to be seriously in error. In this article I discuss the consequences that baryon density inhomogeneities produced by the electroweak or QCD transitions can have for primordial nucleosynthesis. I conclude that if density inhomogeneities of sufficient amplitude (density contrasts ~>10 are needed) are produced on the right comoving distance scales, then a universe containing a closure density of baryons can produce the observed abundances of the light elements because a fraction of the universe undergoes nucleosynthesis 0370-1573/88/$4.90 ~ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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in environments in which neutrons outnumber protons. I argue that the most sensitive test of the inhomogeneous model is an observational search for the cosmological abundance of r-process elements that will be produced in neutron-rich environments. The conventional wisdom that there is absolutely no evidence that O = 1 is beginning to be questioned. (The parameter 12 = P/Pc, where p is the mean mass density in the universe and Pc = 3H2o/87rG = 1.9 x ]0-29h2g/cm3, where H 0 is the Hubble constant and h is H 0 in units of 100km/s Mpc, is the mean mass density needed to close the universe.) Two independent studies, sensitive to the global value of/2, which prefer O = 1 have recently been presented. Fowler [2] derives an age of to = 11.0 - 1.6 Gyr for the universe from uranium-thorium nuclear cosmochronology, and he obtains a Hubble constant of H 0 = 58 -+ 5 km/s Mpc from the maximum luminosity of type Ia supernovae [3] with the assumption that 0.6 M o of 56Ni is produced in the explosion. The product of these numbers is Hot o = 0.65 -+ 0.11, which corresponds to 0.7 -100 MeV will leave an observable record. The most important physical process affecting density perturbations on the relatively small scales I consider is neutron diffusion. Once produced, a baryon density perturbation will try to diffuse back to uniformity. Neutrons, being neutral, have a much longer mean free path than protons. As long as the weak interactions are in equilibrium the diffusion lengths of neutrons and protons are equal because protons diffuse by converting into neutrons, diffusing, and changing back. The proton distribution is frozen when the weak interactions fall out of equilibrium, but the neutrons continue to diffuse. Since there are five protons for every neutron when the weak interactions decouple, any region with less than 1/5 the mean baryon density will be neutron rich once the neutrons diffuse back to uniformity. Neutron diffusion and nucleosynthesis in homogeneous neutron-rich regions was studied in detail by Applegate, Hogan, and Scherrer [9] (hereafter AHS), who concluded that neutron-proton diffusive segregation could occur on a wide range of scales of interest, and that the nucleosynthetic consequences of the neutron-rich regions were drastic enough to bring an 12 = 1 universe composed entirely of baryons into agreement with the light-element abundances. In the following sections I describe and extend the AHS calculation, with particular attention given to the possibility that the neutron-rich regions proposed by AHS (see also ref. [8]) can produce r-process elements.
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143
2. Neutron diffusion Upper and lower limits to the distance scales on which a perturbation in baryon density can affect nucleosynthesis through neutron-proton diffusive segregation are determined by the diffusion lengths of neutrons and protons at the onset of nucleosynthesis. At high temperatures neutrons and protons intratransmute rapidly through weak interactions, so their diffusion lengths are equal. The distribution of protons is frozen when the weak interactions decouple, the neutrons continue to diffuse, so diffusive segregation can occur. The magnetic moment of the neutron scatters electrons and positrons with a cross section =10 -30 cm 2, and the neutron-proton cross section "--10-23 cm 2. Coulomb collisions with electrons and positrons give the proton a transport cross section =10 -24 cm 2. With the baryon to photon number ratio r/= nb/n ~ in the range 10 -8 to 10 -1°, these cross sections show that the mean free path of the neutron is roughly 106 times that of the proton. In section 2.1, I outline the theory of neutron diffusion prior to nucleosynthesis presented by AHS. In section 2.2, I extend this theory to describe a process not considered by AHS: the diffusion of neutrons back into proton-rich regions after the start of nucleosynthesis.
2.1. Neutron diffusion prior to nucleosynthesis If the decay of the neutron and expansion of the universe are neglected, the neutron density n is described by
On/Ot = D. VZn,
(2.1)
where D, is the diffusion coefficient. After a time t the rms distance a neutron has diffused is given by d = (6Dnt) 1/2 .
(2.2)
Since the weak interactions decouple at t = 1 s and nucleosynthesis begins at t = 200 s, neutron decay is a minor perturbation to the diffusion and can be neglected. The expansion of the universe makes D n a function of time. This complication is treated in detail by AHS, but it is a good approximation to use eq. (2.2) and evaluate D n at the time of interest. I normalize the scale factor a(t) of the universe to a = 1 when the neutrino temperature is 1 MeV (T e = T v = T at this time). Since T v evolves as a T = constant through e+e - annihilation, the comoving diffusion length d/a is given by
d/a = Tv(MeV ) [6Dn(t)t] 1/2 .
(2.3)
The diffusion of nucleons while the weak interactions are in equilibrium can be described simply because a nucleon does all of its diffusing while it is a neutron. The fraction of time X n that a nucleon spends as a neutron in thermal equilibrium is given by X n = (1 +eQ/r) -1 ,
(2.4)
where Q = 1.29MeV is the neutron-proton mass difference. Thus, eqs. (2.2) and (2.3) should be multiplied by XI,/2 to give the nucleon diffusion length while the weak interactions are in equilibrium.
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144
The proton diffusion length is the nucleon diffusion length when the weak interactions decouple. I assume that the weak interactions remain in equilibrium until an effective weak decoupling temperature Tweak is reached, and decouple instantaneously once the electron temperature falls below Tweak. I determine Tweak by equating the equilibrium neutron fraction, given by eq. (2.4), to the asymptotic value of X, = 0.164 found in a detailed computation by Peebles [10] when he neglects neutron decay. This method gives Tweak "= 794 keV. Neutrons are scattered by electrons and positrons, and by protons. The diffusion coefficients Dne , appropriate for neutron-electron scattering, and Dnp , appropriate for neutron-proton scattering, were calculated by AHS. They found 1/x
D.e = 2.0 x 101° e
cm2/s,
(2.5)
where x = T/mec 2 and f(x) = 1 + 3x + 3x 2, and 6.5
x
- -
Dnp
Xp
101°
T 1/2
e 3 cm2/s 178 O'np T ~
(2.6)
'
where Xp is the mass fraction in protons, Te and T~ are the electron and neutrino temperatures in MeV, O'np is the neutron-proton scattering cross section in fm 2, and r/8 = 108 ~7, where -q is the baryon to photon ratio at the current epoch. The value of r/used here should be appropriate for the region into
l0 T
~ i0 6 i
E o
10 4
/~'/ ' / ~ . 'r/8:1
protons
J
~
10 3
0.5
I
I
I
I
J
I
I
I
0.0
I
J
I
I
- 0.5 log
I
I
I
I
I
I
I
-I.0
Te (MeV)
Fig. 1. Comoving diffusion distance as a function of Hubble time or temperature. The solid curve shows the rms comoving distance d(T)/a (normalized to a = 1 at T = 1 MeV) travelled by a baryon up to time T. After weak decoupling, neutron and proton transport are shown separately.
J.H. Applegate, Neutron diffusion, primordial nucleosynthesis, and the r-process
145
which the neutrons are diffusing, not the global value of 77. The neutron diffusion coefficient D n is given by Dn I =
D,-~
+ O n-1 p .
(2.7)
Near the onset of nucleosynthesis at T = 100 keV neutron-proton scattering dominates if r/>-- 10- ~0 and neutron-electron scattering dominates if rt ~<10 -1°. Comoving diffusion lengths for neutrons and protons for the cases r/8 = 1, which is the global 77 appropriate for a closure density of baryons 078 = 3IZoh2), ~78= 1/30, which is the global r/implied by the deuterium abundance in the standard model, and for the case D, = D,e are given in fig. 1. Density perturbations on comoving scales less than the proton diffusion length are destroyed by diffusion while T >-- Tweak, so the proton diffusion length is a lower limit to the scales on which density perturbations can affect nucleosynthesis. (A more stringent lower limit is discussed in section 2.2.) The neutrons can diffuse back to uniformity on all scales smaller than the neutron diffusion length. Regions of baryon density less than 1/5 of the mean baryon density, which corresponds to specific entropy 5 times the mean entropy, will be neutron rich once the neutrons diffuse back to uniformity. Density perturbations on scales larger than the neutron diffusion length can affect nucleosynthesis, but they cannot be filled to uniformity with neutrons. 2.2. Neutron diffusion during nucleosynthesis
At high temperatures the equilibrium of the reaction n + p ~ d + ~/strongly favors free neutrons and protons. Once the temperature falls to 100 keV the equilibrium starts to favor deuterium; this event marks the onset of primordial nucleosynthesis. Once deuterons can appear in quantity, subsequent reactions produce 4He rapidly until one of the primary reactants, neutrons or protons, is exhausted. Nucleosynthesis occurs in a proton-rich environment in the standard model, so neutrons are quickly exhausted. In neutron-rich regions protons will be exhausted by the production of 4He, so these regions will have late-time supply of neutrons. The dramatic implications this late-time exposure to neutrons has for nucleosynthesis is the topic of section 3. Here, I discuss another of its implications: Once nucleosynthesis begins, the sense of the neutron density gradient reverses, and neutrons diffuse back into proton-rich environments. In this section I estimate the time required for regions of various sizes to lose their neutrons, and show that only the largest regions that can be uniformly filled with neutrons prior to nucleosynthesis can retain their neutrons long enough to significantly affect nucleosynthesis. The neutron diffusion time out of neutron-rich regions can be estimated with a simple onedimensional model. Let the region from x = 0 to x = L be filled with neutrons, and let the region from x = L to x = oo represent the proton-rich region. Note that L is the physical, not the comoving, size of the neutron-rich region. Inside the neutron-rich region (region 1) the neutron density is described by On I 82n~ 8t = D1 8x 2 "
(2.8)
In the proton-rich region the neutrons are rapidly consumed by the n(p, ~/)d reaction. In the proton-rich region (region 2) the neutron density is described by o~n2
0~2n2
Ot = D 2 0 x z
A2n z ,
(2.9)
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146
where a 2 = n p ( O r o ) is the rate of the n(p, y)d reaction. Free neutron decay is usually slower than diffusion out of the neutron-rich regions, so I have neglected it. It can be incorporated in a straightforward manner. The diffusion equations (2.8) and (2.9) have solutions n l ( x , t) = N 1 e -Kit c o s ( a × )
r/2(X, t) = N 2 e -~2' e -~{x-L) ,
,
(2.10)
where the decay rates K1 and r 2 are given by K 1 = D1 a 2
(2.11)
K2 ~- a 2 -- D 2 ~ 2
The boundary conditions demand that the densities n(x, t) and the diffusion fluxes F = - D d n / a x be continuous at x = L for all t. These conditions give K1 = r2,
N 1cos(aL) = N 2 ,
D l a N I sin(aL) = D 2 f l N 2 .
(2.12)
I assume that the global 7/= 10 -8, which corresponds to a universe with O b = 1 078 = 3Obh2) • To sufficient accuracy the proton density in the proton-rich region is ~Tp= r/nv = 2 × 10 20 T39 cm -3 ,
(2.13)
where T 9 is the temperature in units of 10 9 K. The diffusion coefficient in region 2 is D.p, given by eq. (2.6). The e+e - pairs have annihilated by the time of nucleosynthesis, so T v = Te/1.401. Near T 9 = 1, O'np-----10b, so Dnp is given by D,p = 1011 T95/2 cmZ/s. At
T9 =
By
T9 =
(2.14)
0.7, characteristic of the later stages of nucleosynthesis, D 2 = D n p ( T 9 = 0.7) = 2 × 1011 c m 2 s -1. 0.7 most of the baryons in the neutron-rich regions are in 4 H e (see figs. 2a and b). The diffusion coefficient for n-et scattering is D.~ = v T / 3 n , o'.,, where v x = 5 × 10 8 T19/2 cm/s is the velocity of a thermal neutron, and o-,~ = 0.76 b [11] is the low-energy n-or scattering cross section. I assume that the neutron-rich region was devoid of baryons until diffusion filled it with neutrons; thus the baryon density in region 1 is n b = ~r/nv = 3 which gives n~ Dn~ =
3
=
× 1019
T39 c m
-3 ,
nb/4 = 8
× 1018
T39 c m
x 1013
T9 5/2 cm 2 s -1 ,
(2.15) -3,
assuming that X, = 1. The diffusion coefficient is (2.16)
and D 1 -- D n ~ ( T 9 = 0.7). Note that the r a t i o D.,JDnp = 300 is independent of temperature. The neutron capture rate for the n(p, y)d reaction using eq. (2.13) and the FCZ [12] value for (try) is given as a function of temperature in table 1. At T 9 = 0.7, /~2 = /~np = 3 s -1, neutron capture is very fast.
J.H. Applegate, Neutron diffusion, primordial nucleosynthesis, and the r-process t (sec)
t (sec)
102
105
I
I
104
1.c ~
0 8
0.6
--
~//v\\\
0.4--
~/~.. X
i o.~-
64 '_,,o -
147
\\f'~" decay
/'"" X",
/ / //L----_. //..
_
O.~ -
.
o,
102
105
104
I
I
I
f.
x~\\ 4
-
~
\/,
-
-
./~
o.~-
-,t
.\
L,-..-----
=
¼ ""i',,
~,.k~ ]()
10-1
II
I
10-2
IN
I
I
I
I
10-1
T (MeV)
I
I
l
10-2 T(MeV)
Fig. 2. Evolution of cosmic abundances in a medium initially dominated by neutrons with (a) "%, = 10-s, (b) -q,, = 10-1°. Mass fraction is shown as a function of temperature/time for various species. The top portion is a linear plot, the bottom portion logarithmic. The dashed line gives the neutron abundance in the absence of all reactions other than neutron decay. Note the secondary peak in D production at about r,, the neutron half-life.
Table 1 Rate of the n(p, ~,)d reaction
L 1.0 0.7 0.5 0.3 0.1
,% (s-')
t (s)
8.4 3.0 1.1 0.26 0.011
177 361 708 1970 17 700
I combine the second and third boundary condition in eq. (2.12), assume that a L ~ 1, and find D2
(aL)2=-~
(flL),
(2.17)
which, when combined with eq. (2.11), gives f i t = ½(1 + 4,t2L2/D2) u2 - ~1 .
(2.18)
If L'~(Dz/4A2) u 2 = 105cm, then flL = A2L2/D2 and the decay rate K = D l O t 2 = '~2 = 3s-~; s m a l l regions lose their neutrons as fast as the protons can capture them. If L >> (D2/4)1.2)1/2 then fl =
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148
(h2/O2) 1/2 = (3 km) -x, and the decay rate is K --
(AzD2) 1/2 L
8 X 105 cm -1 L s
(2.19)
This formula breaks down when L is large enough for aL =1. As long as a L ~ l , a L = (AaD2/D2x)I/4L x/2 = 10 - 4 LX/2(cm). Thus, significant gradients appear in region 1 for L = 10 8 cm. For L > 108 cm the product aL = zr/2 and the decay rate is 7r2Dx K-
4L 2 -
2 x 1014cm 2 -x L2 s
(2.20)
At T9 = 0.7 the neutrino temperature is 43 keV, so the comoving distances in fig. 1 must be multiplied by 23 to give physical distances. The largest d/a can be 107 t o 107.5 cm MeV, for T 9 = 0 . 7 , if np scattering is unimportant down to low temperatures. This gives L < 7 x 108 cm. Free neutron decay, which gives K = 1.09 x 10 -3 s -1, dominates for L > 4 x 108 cm. Thus K = 10 -2 s -x to 10 -3 s -1 is representative of the effective decay rate in the larger neutron-rich regions. Proton-rich regions will begin nucleosynthesis before neutron-rich regions because of the higher baryon density in the proton-rich regions. Thus, small neutron-rich regions can lose their neutrons before they even begin nucleosynthesis. After the e+e - pairs have annihilated, the time and temperature are related by T9 = 13.3/t 1/2, with t in seconds. If proton-rich regions begin nucleosynthesis at T 9 = 1, neutron-rich regions begin at T 9 = 0.9, and both finish at T 9 = 0.7, the time interval between the onset of nucleosynthesis in the two regions is 35 s and the total duration of nucleosynthesis is 200 s. Regions that lose their neutrons faster than K = 0.03 s -x will probably not affect nucleosynthesis (note that the loss of neutrons affects both the n/p ratio and -q), so the lower limit on L is roughly 2 x 107 cm, which produces a lower limit on d/a of (d/a),l,in ~-- 10 6 c m M e V .
(2.21)
This lower limit is slightly larger than the comoving horizon scale at T = 100GeV, so entropy inhomogeneities produced in the electroweak transition will not affect nucleosynthesis. The electroweak transition can affect nucleosynthesis if it produces stable objects, such as soliton stars [13] or black holes [141.
3. Nucleosynthesis Neutron diffusion in the inhomogeneous cosmology discussed in this article can affect primordial nucleosynthesis through three mechanisms: light-element nucleosynthesis in homogeneous neutron-rich regions, light-element nucleosynthesis by neutrons diffusing back into proton-rich regions at late times, and heavy-element nucleosynthesis by rapid neutron capture in neutron-rich regions.
3.1. Light-element nucleosynthesis The nucleosynthesis of elements up to 160 in homogeneous neutron-rich environments was computed in detail by AHS. Their results are summarized in tables 2 and 3, taken from AHS, which give
J.H. Applegate, Neutron diffusion, primordial nucleosynthesis, and the r-process
149
Table 2 Products of neutron-rich cosmological nucleosyntbesis (F, = 1) "q.~
X~,('He)
log X.,(2H)
log X~,(3He)
log X~(7Li)
log X.~(A >-12)
log(~/XD)
10T M 10-1°
28.5% 81% %%
-1.% -2.5 --3.3
-3.14 -3.7 --4.5
-7.47 -6.7 --6.6
-13.42 -10.3 --7.7
10 -8
~%
--4.I
-5.3
--6,7
-5.5
10 -7
I~% 1~% ~% ~%
-5.0 --5.9 --6.7 --7.2
--6.1 --7,0 --7.9 --8.5
-6.7 --6.8 --6.9 --7.3
-12.8 -12.5 --12.3 -12.1 -12.0 --11.9 --11,7 --11.2
10 -9
10-6 10-5 10-4
--4.0 --2.8 --1.7=2% (18%)
Table 3 Sensitivity of element abundances to initial neutron fraction F. for r/.s = 10-~ log X,~(TLi)
log Xn~(A >-12)
F,
X~('He)
log X~,(2H)
log Xn,(3He)
1
~%
-4.1
-5.3
-6.7
-5.5
0.9 0.8 0.7 0.6 0.5 0.4 0.3
@% ~% ~% ~% ~% 67% 50%
-4.1 -4.2 -4.2 -5.2 -6.4 -7.6 -8.9
-5.3 -5.3 -5.4 -6.0 -5,5 -5.3 -5.2
-6.6 -6.6 -6.6 -8.4 -7.2 -7.1 -7.1
-5.5 -5.6 -5,8 -9.1 -10.4 -10.8 -11.1
the predicted mass fractions of 2H, 3He, *He, 7Li, and the sum of the mass fractions for A -> 12 for an initially pure neutron region as a function of 77 in the region, and the light-element mass fractions as a function of the initial neutron fraction for 77= 10-8. Abundances as a function of time during nucleosynthesis for the cases 77= 10 -8 and r/= 10 -1° are given in figs. 2a and b, also taken from AHS. The most important conclusion from the AHS nucleosynthesis calculation is that an inhomogeneous universe with ag~ = 1 (closure density of baryons), in which a fraction fn = 0.2 of the neutrons undergo nucleosynthesis in neutron-rich regions which occupy a fraction fv = 0.2 of the volume, can explain the abundances of deuterium, aHe, and the population I value of 7Li. The conversion factors needed to relate the baryon to photon ratio and the mass fractions in the neutron-rich regions to the global averages are
1L
*/"~- 6 f~ 770
(3.1)
X0 = ~f.X,s,
(3.2)
where the subscript '0' refers to the global average value of a quantity and the subscript 'ns' refers to the value in the neutron-rich region. The factor 1/6 is the neutron to baryon ratio at the onset of nucleosynthesis. The product Xns(2H)Bns is nearly constant. If this is converted to the global average of the deuterium mass fraction, and the requirement Xo(2H) = 3 x 10 -5 [1] is used, then 7/0 = 3 x 10 -8 fv. Thus O b = 1 (r/0 ---3 x 10 -8 Dq,h 2) is consistent with deuterium production if fv =0.2. Inhomogeneous nucleosyn-
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150
thesis produces less 4He than a homogeneous model with the same global value of O b because it takes two neutrons to make one 4He nucleus in a proton-rich environment and four neutrons to make one 4He nucleus in a neutron-rich environment. The same number of neutrons are used less efficiently in inhomogeneous nucleosynthesis. The reduction is by a factor AXo(4He)/Xo(4He) = _ l f~.
(3.3)
Thus, fn = 0.2 reduces the/2 b = 1 4He production by the right amount to agree with the primordial 4He abundance. The situation concerning the other two isotopes, 3He and 7Li, associated with the primordial nucleosynthesis is less clear. Both the observational and theoretical aspects of the cosmological role of 3He are ambiguous [1]. The observations suggest that X(3He) = X(ZH) at the present epoch. The AHS model produces X(3He)= 0.06X(ZH), so AHS underproduces 3He. However, 3He can be both produced and destroyed in stars, so 3He is not a sensitive cosmological probe. Lithium offers a potentially more sensitive test because of the quality of the observations. The standard model predicts X0(7Li)-~ 10 -9, which is in good agreement with the VLi abundance found in metal-deficient (population II) dwarf stars [15, 16]. The AHS model predicts X0(7Li) --- 10 -8 with f, = 0.2, and this is in good agreement with the 7Li abundance found in a wide variety of population I sites [1]. The standard model interprets the pop II 7Li mass fraction as the primordial value. However, this interpretation requires the uncomfortable assumption that an unknown physical process, which is sufficiently robust to produce a factor of 10 more 7Li than is produced in the standard model, turn on in pop I sites and produce the pop 1 7Li abundance. Furthermore, this process must be assumed to not contaminate the pop II 7Li abundance. The AHS picture is favored if the pop I 7Li abundance is primordial, but this interpretation requires that an unknown destruction mechanism uniformly deplete the pop II stars by a factor of 10 and leave pop I untouched. Spite and Spite [15] (see also ref. [16]) give highly suggestive arguments that the pop II abundance is primordial, but neither of the interpretations of 7Li given above is as convincing as the interpretations of deuterium or 4He within the context of the standard model, and neither will be until the key missing ingredient in the analysis, the nature of the process responsible for the factor of 10 difference in the pop I and pop II 7Li abundances, is well understood. The effect of neutrons diffusing back into proton rich-environments at late times will be to increase the production of ZH and 3He in these regions. The effect should be qualitatively similar to the effect of the decay of heavy antineutrinos (gnome+anything, followed by ~ e + p ~ e ÷ +n) analysed by Scherrer [17], who found that significant amounts of 2H and 3He could be produced by a late-time source of neutrons. The most striking difference between the AHS calculation and the standard model is the production of nuclei with mass numbers A -> 12. The predictions of X , s ( A >- 12), given in table 2, are several orders of magnitude larger than the predictions of the standard model with the same value of r/. Most of the mass in A -> 12 is in 14C. (The reaction network used by AHS was truncated at 160, so the value of X ( A >-12) and the isotopic distribution may not be accurate.) The most likely reaction path for ~4C production is t + ot"--~7Li , 7Li + n ~ 8Li,
J.H. Applegate, Neutron diffusion, primordial nucleosynthesis, and the r-process
151
8Li + o t ~ liB + n , liB + n ~ 12B ,
12B---) 12C 71-e-
"~])e ,
]2C + n ~ 13C, 13C -[- n ~ 14C.
The bottleneck in this sequence is most likely 8Li because it has a half-life of 0.85 s. For r/0 = 10 -8 and r/. s = 1.7 x 10 -9 the mixed mass fraction Xo(A >- 12) is -----10-9, most of which will be 14N, produced by the [3-decay of ~4C. This is about 10 -3 Of the heavy-element mass fraction of the lowest-metaUicity star known [18]. Note that if r/ns is increased by a factor of 6 the value of X(A >- 12) is increased by 100. It is possible that the cosmological floor to the abundance of CNO nuclei predicted by AHS can be detected or ruled out, but the level of the floor is very sensitive to r/ns and can be very low. A potentially much more sensitive test is provided by the fact that the CNO nuclei are produced in a neutron-rich environment where they can seed an r-process, which allows the abundance of r-process elements to be used as a cosmological probe.
3.2. Heavy element nucleosynthesis The production of r-process elements from 14Chas three distinct steps: the match, the fuse, and the bomb. The match is a sequence of reactions that produces 2°Ne from 14C. The most likely sequence is 14C(ot, ~/)lSO(n, ~)190([3-)19F(n, ~/)2°F([3-)2°Ne ;
(3.4)
of these the key is ~4C(a, -/)180. The fuse is a series of (n, ~/) reactions and B-decays that convert 2°Ne to neutron-rich nuclei in the A = 60 region. The fuse is essentially a light-element n-process [19], but because the (n, ~/) cross sections are rather low, t r - 0.1 mb to 10 mb, the reaction path leads through mostly known [3-decays [most (n, ~/) cross sections are, however, not measured]. The bomb explodes once "real r-process nuclei", with neutron capture cross sections of 100 mb to 1 b, have been produced. The heavy-element r-process is characteristic of the long-time solution [20], which is dominated by fission cycling. The number of fission cycles is the largest uncertainty in the prediction of the final mass fraction of r-process elements. In this section I will give an outline of the theory of the primordial r-process; the details will be presented elsewhere [21]. The ~4C produced in the AHS calculation starts to be produced at T 9 = 1 and it is close to its final abundance by T9 = 0.7. I assume that the production of 2°Ne occurs in the same temperature interval as the production of 14C, and that the (n,-/) reactions and [3-decays that take 2°Ne into the r-process region occur at T 9 < 0.7. I characterize the strength of the various reactions by the product try" of the thermally averaged cross sectidn o- and the exposure r [22], where z is defined by d~" = on dt, where n is the number density of irradiating particles and v is their mean thermal velocity. With this definition the number of reactions dN per target nucleus caused by exposure dr is dN = tr dr. I assume that rl, s = 1.7 x 10 -9, appropriate to 7/0 = 10 -8, and I take X n = 0.6, X~ = 0.4, and Xp = 10 -2 as representative of rough averages of these quantities in the interval 1 > T 9 > 0.7. With these assumptions the exposures are rp = 2 X 10 28 cm -2 ,
r~ = 1029 cm -2 ,
r n = 7 x 10 29 cm -2 .
(3.5)
Theory ofsupernovae
152
The proton mass fraction changes from Xp = 10 -3 for ?~ns= 10-8 to Xp = 10 -1 for 7/, S = 10 -1°, while X, and X~ are relatively insensitive to 77,s in the interval 0.7 < T 9 < 1. I compute the cross section for 14C(a,-/)180 from the published rate [23]. The cross section varies from 2.4 × 10 -34 c m 2 at T 9 = 1 to 6 × 10 -36 c m 2 at T 9 = 0.7. I adopt or = 3 × 10 -35 c m 2, appropriate for T 9 = 0.8, as a suitable average; this gives t r r = 3 × 10 -6. The neutron capture 180(n, ~/)190 must compete with 180(p, ot)15N. The cross section for 180(p, ct)15N is about 2 × 10 -27 c m 2 [24], which gives trrp =40. I estimate or = 6 × 10 -30 c m 2 for 180(n,~/)190 from known resonance properties [11]; this gives o-zn = 4. About 90% of the 180 produced by laG(or, ~')180 ends up as 15N and about 10% continues on the path (3.4). The neutron capture 19F(n, 302°F dominates 19F(p, 0 0 1 6 0 by about a factor of 200 [11, 24]. The half-lives of the two 13-decays a r e t l / 2 ( 1 9 0 ) = 26.9s and t112(Z°F)= l l . 0 s [25]. Since it takes 200 s to go from T 9 = 1 to T 9 = 0.7, there is plenty of time for these 13-decays. I estimate the yield of the match to be X(2°Ne) = 3 x 10 -7 X(14C)~---10 -14 ,
(3.6)
where both estimates assume rhs = 1.7 × 10 -9. The number density of neutrons is given by n n = 3.4 × 1019 f(t)T39 cm -3 ,
(3.7)
where f(t) is the fraction of neutrons present at the onset of nucleosynthesis which are still free at time t. At T 9 = 0.7 1 t a k e f = 0.4. At later times I assume that the neutrons disappear with the decay constant r computed in section 2.2. Thus
f(t) = 0.4 e x p [ - r(t - to)],
(3.8)
where t o = 372 s is the time at which T 9 = 0.7. This gives a total neutron exposure for T 9 < 0.7 of 6 × 1029 -2 r n -- - cm Kt 0
(3.9)
If K < 10 -2 S -1 the neutron exposure is ~ 1 0 29 cm -2. I estimate the exposure needed to convert Z°Ne to the A = 60 region by assigning a mean cross section to all neutron captures, and then setting trz, = 60. If only neutron captures were considered, then Orn = 40 would be required. However, I assume that neutrons can be captured until a ~-decay as fast as the (n, 3,) reactions can be found. Roughly 20 13-decays are needed, so I use trr n = 60. I estimate o- = 4.2 mb by taking the geometric mean of the 43 nuclei between 2°Ne and 58Fe whose (n, ~/) cross section was computed by Woosely et al. [26] using Hauser-Feshbach theory. The exposure required to get to A = 60 is zn ---2 x 1028 cm -2
(3.10)
This is 10% of the total exposure for x = 0.01 S -1. The mean neutron capture time for tr = 4 mb is = 0.1 s. Thus [3-decays with half-lives of 0.2 s are needed for the [3-decays to keep up. Known B-decays [25] with half-lives ~<1 s are common for the elements in question. The fuse probably bums in <60 s. Heavy nuclei have neutron capture cross sections tr ~> 100 mb. If I take tr = 100 mb, r = 1029 cm-2,
J.H. Applegate, Neutron diffusion, primordial nucleosynthesis, and the r-process
153
neglect fission cycling, and assume that fast 13-decays can always be found, then every r-process seed captures 104 neutrons. Since fission cycling occurs about every 140 neutron captures, this corresponds to 70 fission cycles. However, the neutron capture time for or = 100 mb is =0.01 s, so the time needed to 13-decay cannot be neglected. The time needed to complete the fission cycle once is at least the sum of the [3-decay half-lives of the nuclei in the nucleosynthetic path [20]. The total 13-decay waiting time is estimated to lie in the range [27] 0.1 s < r~ < 3 0 s ;
(3.11)
the factor of 300 uncertainty in r~ is the major uncertainty in the computation of the yield of the primordial r-process. If the exposure lasts 100 s, the number of fission cycles is n = 100/r~,
(3.12)
and 3 < n < 1000 is consistent with the uncertainty in r~. The mass fraction in r-process elements produced in the neutron-rich regions is Xn~(r) = -~00" 2"X(Z°Ne) - 10 -13 2",
(3.13)
which gives a mixed mass fraction of
Xo(r) = 3 x 10-1s2"
.
(3.14)
If n > 40 the neutrons will be exhausted and X,s(r)~ 0.1, which corresponds to Xo(r ) = 3 × 10 -3, will be produced. This is an overproduction by about 10 relative to the sun. This brings out an important feature of the primordial r-process: A very small mass fraction of seeds can produce a very large mass fraction of r-process elements because, relative to stellar r-process sites, the neutron exposure in the big bang lasts a very long time. Stars with r-process depletions of 1000 relative to the sun have been observed [28, 29]. The elements observed are in the rare earth region between the r-process peaks at A = 130 and A = 195. Since A < 120 is not produced in the long-time solution [20], I determine the relevant r-process mass fraction for the sun by summing the r-process contributions to A > 120 in the abundance table of Cameron [30]. I obtain Xo(r ) = 10 -7, which gives a limit of 10 -1° for cosmological production. This gives 1 200.2,X(20Ne), 10-1°> 3--0 " 20
(3.15)
which gives n < 15. The observations are already a significant constraint. The fact that the limit on the cosmological production is already near the low side of what is possible suggests that the cosmological r-process abundance, if it exists, may not be too far below the current observational limit.
4. Conclusion
I have presented an inhomogeneous cosmological model which can match the success of the standard model in predicting light-element abundances with O = 1 in baryons. The inhomogeneities can be
154
Theory of supernovae
produced in the quark confinement transition. The model predicts a cosmological production of r-process elements by rapid neutron capture in neutron-rich environments. If detected, this cosmological production of r-process elements would be a relic of the time when quarks first condensed into hadrons when our universe was 10-4s old.
Acknowledgements This work was done in collaboration with Craig Hogan and Bob Scherrer, and I gratefully acknowledge their contributions to the results presented here. I would like to thank Bob Malaney for pointing out the correct reaction sequence involved in the production of t4c.
References [1] A.M. Boesgaard and G. Steigman, Annu. Rev. Astron. Astrophys. 23 (1985) 319. [2] W.A. Fowler, Tenth Edward Arthur Milne Lecture, Oxford Univ., Quart. J. R. Astron. Soc. 28 (1987) 87. [3] W.D. Arnett, D. Branch and J.C. Wheeler, Nature 314 (1985) 377. [4] E.D. Loh and E.J. Spillar, Astrophys. J. (Lett.) 307 (1986) L1. [5] E. Shaya, in: Galaxy Distances and Deviations from Universal Expansion, eds B.F. Madore and R.B. Tully (Reidel, Dordrecht, 1986) p. 229. [6] A. Yahil, in: The Virgo Cluster of Galaxies, eds O,G. Richter and B. Binggeli (ESO, Garching, 1985) p. 359. [7] P.J.E. Peebles, Astrophys. J. 284 (1984) 439. [8] J.H. Applegate and C.J. Hogan, Phys. Rev. D 31 (1985) 3037. [9] J.H. Applegate, C.J. Hogan and R.J. Scherrer, Phys. Rev. D 35 (1987) 1151. [10] P.J.E. Peebles, Astrophys. J. 146 (1966) 542. [11] Neutron Cross Sections, Vol. 1, BNL #325 (1973), available from the National Neutron Cross Section Center, Brookhaven National Laboratory. [12] W.A. Fowler, G.R. Caughlan and B.A. Zimmerman, Annu. Rev. Astron. Astrophys. 5 (1967) 525. [13] T.D. Lee, Phys. Rev. D 35 (1987) 3637; see also Columbia Theoretical Physics preprints #355, 356, 357. [14] K. Freese, R. Price and D.N. Schramm, Astrophys. J. 275 (1983) 405. [15] F. Spite and M. Spite, Astron. Astrophys. 115 (1982) 357. [16] M. Spite, J.P. Maillard and F. Spite, Astron. Astrophys. 141 (1984) 56. [17] R.J. Scherrer, Mon. Not. R. Astron. Soc. 210 (1984) 359. [18] T.C. Beers, G.W. Preston and S.A. Shectman, Astron. J. 90 (1985) 2089. [19] J.B. Blake and D.N. Schramm, Astrophys. J. 209 (1976) 846. [20] P.A. Seeger, W.A. Fowler and D.D. Clayton, Astrophys. J. Suppl. 11 (1965) 121. [21] J.H. Applegate, C.J. Hogan and R.J. Scherrer, Astrophys. J. (1988), to appear. [22] D.D. Clayton, Principles of Stellar Evolution and Nucleosynthesis (McGraw-Hill, New York, 1968). [23] R.V. Wagoner, Astrophys. J. (Suppl.) 18 (1969) 247. [24] M.J. Harris, W.A. Fowler, G.R. Caughlan and B.A. Zimmerman, Annu. Rev. Astron. Astrophys. 21 (1983) 165. [25] C.M. Lederer and V.S. Shirley, eds, Table of Isotopes, 7th Ed. (Wiley, New York, 1978). [26] S.E. Woosley, W.A. Fowler, J.A. Holmes and B.A. Zimmerman, At. Data Nucl. Data Tables 22 (1978) 371. [27] D.N. Schramm, in: Essays in Nuclear Astrophysics, eds C.A. Barnes, D.D. Clayton and D.N. Schramm (Cambridge Univ. Press, Cambridge, 1982) p. 325. [28] C. Sneeden and M. Parthasarathy, Astrophy. ~ 267 (1983) 757. [29] C. Sneeden and C.A. Pilachowski, Astrophy. J. (Lett.) 288 (1985) L55. [30] A.G.W. Cameron, in: Essays in Nuclear Astrophysics, eds C.A. Barnes, D.D. Clayton and D.N. Schramm (Cambridge Univ. Press, Cambridge, 1982) p. 23.
NEUTRON DIFFUSION, PRIMORDIAL NUCLEOSYNTHESIS, AND THE r-PROCESS
James H. APPLEGATE Department of Astronomy, Columbia University, New York, NY 10027, USA
Abstract: The application of standard particle physics to big bang cosmology predicts that the temperature interval between roughly 1 TeV and the onset of primordial nucleosynthesis at 100 keV should be rich in physical phenomena; electroweak symmetry breaking, chiral symmetry breaking, and quark confinement should all occur in this interval. First-order phase transitions can produce entropy inhomogeneities because of the uneven release of latent heat, and by shocks and detonations if significant supercooling occurs. In addition, stable objects such as black holes or soliton stars can be produced, and these objects can generate entropy by accretion. These possibilities suggest that the assumption of homogeneity in the standard model of big bang nucleosynthesis could be seriously wrong. In this article I analyse nucleosynthesis in inhomogeneous cosmologies, and I conclude that if baryon density contrasts of order 10 can be created on the proper scales, then the success of the standard model in predicting light-element abundances can be matched by a model containing a closure density of baryons. I argue that the prediction of the cosmological production of r-process elements by the inhomogeneous models is the most promising means of discriminating between these models and the standard model.
1. Introduction
Primordial nucleosynthesis is the best probe of physical conditions in the early universe available today. In the standard model [1], the big bang is responsible for the production of 90% of the 4He in the universe, all of the deuterium, and it is a significant contributor to the production of 3He and 7Li. Within the context of the standard model, powerful conclusions can be drawn from the observed abundances of deuterium and 4He. Primordial 4He production is sensitive to the number of relativistic particle species at the time of nucleosynthesis, and the observations will allow at most one more light two-component neutrino species. The deuterium production is very sensitive to the baryon density during nucleosynthesis, and the requirement that the big bang produce at least the current deuterium abundance implies that baryons must fail to provide enough mass to close the universe by at least a factor of five. The application of standard particle physics to cosmology predicts that the temperature interval between roughly 1 TeV and the onset of primordial nucleosynthesis at 100 keV should be rich in physical phenomena; electroweak symmetry breaking, chiral symmetry breaking, and quark confinement all occur in this interval. This richness of phenomena, any of which can disturb an initially uniform universe, suggests that of the assumptions made in the standard model, the assumption of homogeneity is the one most likely to be seriously in error. In this article I discuss the consequences that baryon density inhomogeneities produced by the electroweak or QCD transitions can have for primordial nucleosynthesis. I conclude that if density inhomogeneities of sufficient amplitude (density contrasts ~>10 are needed) are produced on the right comoving distance scales, then a universe containing a closure density of baryons can produce the observed abundances of the light elements because a fraction of the universe undergoes nucleosynthesis 0370-1573/88/$4.90 ~ Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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Theory of supernovae
in environments in which neutrons outnumber protons. I argue that the most sensitive test of the inhomogeneous model is an observational search for the cosmological abundance of r-process elements that will be produced in neutron-rich environments. The conventional wisdom that there is absolutely no evidence that O = 1 is beginning to be questioned. (The parameter 12 = P/Pc, where p is the mean mass density in the universe and Pc = 3H2o/87rG = 1.9 x ]0-29h2g/cm3, where H 0 is the Hubble constant and h is H 0 in units of 100km/s Mpc, is the mean mass density needed to close the universe.) Two independent studies, sensitive to the global value of/2, which prefer O = 1 have recently been presented. Fowler [2] derives an age of to = 11.0 - 1.6 Gyr for the universe from uranium-thorium nuclear cosmochronology, and he obtains a Hubble constant of H 0 = 58 -+ 5 km/s Mpc from the maximum luminosity of type Ia supernovae [3] with the assumption that 0.6 M o of 56Ni is produced in the explosion. The product of these numbers is Hot o = 0.65 -+ 0.11, which corresponds to 0.7 -100 MeV will leave an observable record. The most important physical process affecting density perturbations on the relatively small scales I consider is neutron diffusion. Once produced, a baryon density perturbation will try to diffuse back to uniformity. Neutrons, being neutral, have a much longer mean free path than protons. As long as the weak interactions are in equilibrium the diffusion lengths of neutrons and protons are equal because protons diffuse by converting into neutrons, diffusing, and changing back. The proton distribution is frozen when the weak interactions fall out of equilibrium, but the neutrons continue to diffuse. Since there are five protons for every neutron when the weak interactions decouple, any region with less than 1/5 the mean baryon density will be neutron rich once the neutrons diffuse back to uniformity. Neutron diffusion and nucleosynthesis in homogeneous neutron-rich regions was studied in detail by Applegate, Hogan, and Scherrer [9] (hereafter AHS), who concluded that neutron-proton diffusive segregation could occur on a wide range of scales of interest, and that the nucleosynthetic consequences of the neutron-rich regions were drastic enough to bring an 12 = 1 universe composed entirely of baryons into agreement with the light-element abundances. In the following sections I describe and extend the AHS calculation, with particular attention given to the possibility that the neutron-rich regions proposed by AHS (see also ref. [8]) can produce r-process elements.
J.H. Applegate, Neutron diffusion, primordial nucleosynthesis, and the r-process
143
2. Neutron diffusion Upper and lower limits to the distance scales on which a perturbation in baryon density can affect nucleosynthesis through neutron-proton diffusive segregation are determined by the diffusion lengths of neutrons and protons at the onset of nucleosynthesis. At high temperatures neutrons and protons intratransmute rapidly through weak interactions, so their diffusion lengths are equal. The distribution of protons is frozen when the weak interactions decouple, the neutrons continue to diffuse, so diffusive segregation can occur. The magnetic moment of the neutron scatters electrons and positrons with a cross section =10 -30 cm 2, and the neutron-proton cross section "--10-23 cm 2. Coulomb collisions with electrons and positrons give the proton a transport cross section =10 -24 cm 2. With the baryon to photon number ratio r/= nb/n ~ in the range 10 -8 to 10 -1°, these cross sections show that the mean free path of the neutron is roughly 106 times that of the proton. In section 2.1, I outline the theory of neutron diffusion prior to nucleosynthesis presented by AHS. In section 2.2, I extend this theory to describe a process not considered by AHS: the diffusion of neutrons back into proton-rich regions after the start of nucleosynthesis.
2.1. Neutron diffusion prior to nucleosynthesis If the decay of the neutron and expansion of the universe are neglected, the neutron density n is described by
On/Ot = D. VZn,
(2.1)
where D, is the diffusion coefficient. After a time t the rms distance a neutron has diffused is given by d = (6Dnt) 1/2 .
(2.2)
Since the weak interactions decouple at t = 1 s and nucleosynthesis begins at t = 200 s, neutron decay is a minor perturbation to the diffusion and can be neglected. The expansion of the universe makes D n a function of time. This complication is treated in detail by AHS, but it is a good approximation to use eq. (2.2) and evaluate D n at the time of interest. I normalize the scale factor a(t) of the universe to a = 1 when the neutrino temperature is 1 MeV (T e = T v = T at this time). Since T v evolves as a T = constant through e+e - annihilation, the comoving diffusion length d/a is given by
d/a = Tv(MeV ) [6Dn(t)t] 1/2 .
(2.3)
The diffusion of nucleons while the weak interactions are in equilibrium can be described simply because a nucleon does all of its diffusing while it is a neutron. The fraction of time X n that a nucleon spends as a neutron in thermal equilibrium is given by X n = (1 +eQ/r) -1 ,
(2.4)
where Q = 1.29MeV is the neutron-proton mass difference. Thus, eqs. (2.2) and (2.3) should be multiplied by XI,/2 to give the nucleon diffusion length while the weak interactions are in equilibrium.
Theory o[ supernovae
144
The proton diffusion length is the nucleon diffusion length when the weak interactions decouple. I assume that the weak interactions remain in equilibrium until an effective weak decoupling temperature Tweak is reached, and decouple instantaneously once the electron temperature falls below Tweak. I determine Tweak by equating the equilibrium neutron fraction, given by eq. (2.4), to the asymptotic value of X, = 0.164 found in a detailed computation by Peebles [10] when he neglects neutron decay. This method gives Tweak "= 794 keV. Neutrons are scattered by electrons and positrons, and by protons. The diffusion coefficients Dne , appropriate for neutron-electron scattering, and Dnp , appropriate for neutron-proton scattering, were calculated by AHS. They found 1/x
D.e = 2.0 x 101° e
cm2/s,
(2.5)
where x = T/mec 2 and f(x) = 1 + 3x + 3x 2, and 6.5
x
- -
Dnp
Xp
101°
T 1/2
e 3 cm2/s 178 O'np T ~
(2.6)
'
where Xp is the mass fraction in protons, Te and T~ are the electron and neutrino temperatures in MeV, O'np is the neutron-proton scattering cross section in fm 2, and r/8 = 108 ~7, where -q is the baryon to photon ratio at the current epoch. The value of r/used here should be appropriate for the region into
l0 T
~ i0 6 i
E o
10 4
/~'/ ' / ~ . 'r/8:1
protons
J
~
10 3
0.5
I
I
I
I
J
I
I
I
0.0
I
J
I
I
- 0.5 log
I
I
I
I
I
I
I
-I.0
Te (MeV)
Fig. 1. Comoving diffusion distance as a function of Hubble time or temperature. The solid curve shows the rms comoving distance d(T)/a (normalized to a = 1 at T = 1 MeV) travelled by a baryon up to time T. After weak decoupling, neutron and proton transport are shown separately.
J.H. Applegate, Neutron diffusion, primordial nucleosynthesis, and the r-process
145
which the neutrons are diffusing, not the global value of 77. The neutron diffusion coefficient D n is given by Dn I =
D,-~
+ O n-1 p .
(2.7)
Near the onset of nucleosynthesis at T = 100 keV neutron-proton scattering dominates if r/>-- 10- ~0 and neutron-electron scattering dominates if rt ~<10 -1°. Comoving diffusion lengths for neutrons and protons for the cases r/8 = 1, which is the global 77 appropriate for a closure density of baryons 078 = 3IZoh2), ~78= 1/30, which is the global r/implied by the deuterium abundance in the standard model, and for the case D, = D,e are given in fig. 1. Density perturbations on comoving scales less than the proton diffusion length are destroyed by diffusion while T >-- Tweak, so the proton diffusion length is a lower limit to the scales on which density perturbations can affect nucleosynthesis. (A more stringent lower limit is discussed in section 2.2.) The neutrons can diffuse back to uniformity on all scales smaller than the neutron diffusion length. Regions of baryon density less than 1/5 of the mean baryon density, which corresponds to specific entropy 5 times the mean entropy, will be neutron rich once the neutrons diffuse back to uniformity. Density perturbations on scales larger than the neutron diffusion length can affect nucleosynthesis, but they cannot be filled to uniformity with neutrons. 2.2. Neutron diffusion during nucleosynthesis
At high temperatures the equilibrium of the reaction n + p ~ d + ~/strongly favors free neutrons and protons. Once the temperature falls to 100 keV the equilibrium starts to favor deuterium; this event marks the onset of primordial nucleosynthesis. Once deuterons can appear in quantity, subsequent reactions produce 4He rapidly until one of the primary reactants, neutrons or protons, is exhausted. Nucleosynthesis occurs in a proton-rich environment in the standard model, so neutrons are quickly exhausted. In neutron-rich regions protons will be exhausted by the production of 4He, so these regions will have late-time supply of neutrons. The dramatic implications this late-time exposure to neutrons has for nucleosynthesis is the topic of section 3. Here, I discuss another of its implications: Once nucleosynthesis begins, the sense of the neutron density gradient reverses, and neutrons diffuse back into proton-rich environments. In this section I estimate the time required for regions of various sizes to lose their neutrons, and show that only the largest regions that can be uniformly filled with neutrons prior to nucleosynthesis can retain their neutrons long enough to significantly affect nucleosynthesis. The neutron diffusion time out of neutron-rich regions can be estimated with a simple onedimensional model. Let the region from x = 0 to x = L be filled with neutrons, and let the region from x = L to x = oo represent the proton-rich region. Note that L is the physical, not the comoving, size of the neutron-rich region. Inside the neutron-rich region (region 1) the neutron density is described by On I 82n~ 8t = D1 8x 2 "
(2.8)
In the proton-rich region the neutrons are rapidly consumed by the n(p, ~/)d reaction. In the proton-rich region (region 2) the neutron density is described by o~n2
0~2n2
Ot = D 2 0 x z
A2n z ,
(2.9)
Theory of supernovae
146
where a 2 = n p ( O r o ) is the rate of the n(p, y)d reaction. Free neutron decay is usually slower than diffusion out of the neutron-rich regions, so I have neglected it. It can be incorporated in a straightforward manner. The diffusion equations (2.8) and (2.9) have solutions n l ( x , t) = N 1 e -Kit c o s ( a × )
r/2(X, t) = N 2 e -~2' e -~{x-L) ,
,
(2.10)
where the decay rates K1 and r 2 are given by K 1 = D1 a 2
(2.11)
K2 ~- a 2 -- D 2 ~ 2
The boundary conditions demand that the densities n(x, t) and the diffusion fluxes F = - D d n / a x be continuous at x = L for all t. These conditions give K1 = r2,
N 1cos(aL) = N 2 ,
D l a N I sin(aL) = D 2 f l N 2 .
(2.12)
I assume that the global 7/= 10 -8, which corresponds to a universe with O b = 1 078 = 3Obh2) • To sufficient accuracy the proton density in the proton-rich region is ~Tp= r/nv = 2 × 10 20 T39 cm -3 ,
(2.13)
where T 9 is the temperature in units of 10 9 K. The diffusion coefficient in region 2 is D.p, given by eq. (2.6). The e+e - pairs have annihilated by the time of nucleosynthesis, so T v = Te/1.401. Near T 9 = 1, O'np-----10b, so Dnp is given by D,p = 1011 T95/2 cmZ/s. At
T9 =
By
T9 =
(2.14)
0.7, characteristic of the later stages of nucleosynthesis, D 2 = D n p ( T 9 = 0.7) = 2 × 1011 c m 2 s -1. 0.7 most of the baryons in the neutron-rich regions are in 4 H e (see figs. 2a and b). The diffusion coefficient for n-et scattering is D.~ = v T / 3 n , o'.,, where v x = 5 × 10 8 T19/2 cm/s is the velocity of a thermal neutron, and o-,~ = 0.76 b [11] is the low-energy n-or scattering cross section. I assume that the neutron-rich region was devoid of baryons until diffusion filled it with neutrons; thus the baryon density in region 1 is n b = ~r/nv = 3 which gives n~ Dn~ =
3
=
× 1019
T39 c m
-3 ,
nb/4 = 8
× 1018
T39 c m
x 1013
T9 5/2 cm 2 s -1 ,
(2.15) -3,
assuming that X, = 1. The diffusion coefficient is (2.16)
and D 1 -- D n ~ ( T 9 = 0.7). Note that the r a t i o D.,JDnp = 300 is independent of temperature. The neutron capture rate for the n(p, y)d reaction using eq. (2.13) and the FCZ [12] value for (try) is given as a function of temperature in table 1. At T 9 = 0.7, /~2 = /~np = 3 s -1, neutron capture is very fast.
J.H. Applegate, Neutron diffusion, primordial nucleosynthesis, and the r-process t (sec)
t (sec)
102
105
I
I
104
1.c ~
0 8
0.6
--
~//v\\\
0.4--
~/~.. X
i o.~-
64 '_,,o -
147
\\f'~" decay
/'"" X",
/ / //L----_. //..
_
O.~ -
.
o,
102
105
104
I
I
I
f.
x~\\ 4
-
~
\/,
-
-
./~
o.~-
-,t
.\
L,-..-----
=
¼ ""i',,
~,.k~ ]()
10-1
II
I
10-2
IN
I
I
I
I
10-1
T (MeV)
I
I
l
10-2 T(MeV)
Fig. 2. Evolution of cosmic abundances in a medium initially dominated by neutrons with (a) "%, = 10-s, (b) -q,, = 10-1°. Mass fraction is shown as a function of temperature/time for various species. The top portion is a linear plot, the bottom portion logarithmic. The dashed line gives the neutron abundance in the absence of all reactions other than neutron decay. Note the secondary peak in D production at about r,, the neutron half-life.
Table 1 Rate of the n(p, ~,)d reaction
L 1.0 0.7 0.5 0.3 0.1
,% (s-')
t (s)
8.4 3.0 1.1 0.26 0.011
177 361 708 1970 17 700
I combine the second and third boundary condition in eq. (2.12), assume that a L ~ 1, and find D2
(aL)2=-~
(flL),
(2.17)
which, when combined with eq. (2.11), gives f i t = ½(1 + 4,t2L2/D2) u2 - ~1 .
(2.18)
If L'~(Dz/4A2) u 2 = 105cm, then flL = A2L2/D2 and the decay rate K = D l O t 2 = '~2 = 3s-~; s m a l l regions lose their neutrons as fast as the protons can capture them. If L >> (D2/4)1.2)1/2 then fl =
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148
(h2/O2) 1/2 = (3 km) -x, and the decay rate is K --
(AzD2) 1/2 L
8 X 105 cm -1 L s
(2.19)
This formula breaks down when L is large enough for aL =1. As long as a L ~ l , a L = (AaD2/D2x)I/4L x/2 = 10 - 4 LX/2(cm). Thus, significant gradients appear in region 1 for L = 10 8 cm. For L > 108 cm the product aL = zr/2 and the decay rate is 7r2Dx K-
4L 2 -
2 x 1014cm 2 -x L2 s
(2.20)
At T9 = 0.7 the neutrino temperature is 43 keV, so the comoving distances in fig. 1 must be multiplied by 23 to give physical distances. The largest d/a can be 107 t o 107.5 cm MeV, for T 9 = 0 . 7 , if np scattering is unimportant down to low temperatures. This gives L < 7 x 108 cm. Free neutron decay, which gives K = 1.09 x 10 -3 s -1, dominates for L > 4 x 108 cm. Thus K = 10 -2 s -x to 10 -3 s -1 is representative of the effective decay rate in the larger neutron-rich regions. Proton-rich regions will begin nucleosynthesis before neutron-rich regions because of the higher baryon density in the proton-rich regions. Thus, small neutron-rich regions can lose their neutrons before they even begin nucleosynthesis. After the e+e - pairs have annihilated, the time and temperature are related by T9 = 13.3/t 1/2, with t in seconds. If proton-rich regions begin nucleosynthesis at T 9 = 1, neutron-rich regions begin at T 9 = 0.9, and both finish at T 9 = 0.7, the time interval between the onset of nucleosynthesis in the two regions is 35 s and the total duration of nucleosynthesis is 200 s. Regions that lose their neutrons faster than K = 0.03 s -x will probably not affect nucleosynthesis (note that the loss of neutrons affects both the n/p ratio and -q), so the lower limit on L is roughly 2 x 107 cm, which produces a lower limit on d/a of (d/a),l,in ~-- 10 6 c m M e V .
(2.21)
This lower limit is slightly larger than the comoving horizon scale at T = 100GeV, so entropy inhomogeneities produced in the electroweak transition will not affect nucleosynthesis. The electroweak transition can affect nucleosynthesis if it produces stable objects, such as soliton stars [13] or black holes [141.
3. Nucleosynthesis Neutron diffusion in the inhomogeneous cosmology discussed in this article can affect primordial nucleosynthesis through three mechanisms: light-element nucleosynthesis in homogeneous neutron-rich regions, light-element nucleosynthesis by neutrons diffusing back into proton-rich regions at late times, and heavy-element nucleosynthesis by rapid neutron capture in neutron-rich regions.
3.1. Light-element nucleosynthesis The nucleosynthesis of elements up to 160 in homogeneous neutron-rich environments was computed in detail by AHS. Their results are summarized in tables 2 and 3, taken from AHS, which give
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149
Table 2 Products of neutron-rich cosmological nucleosyntbesis (F, = 1) "q.~
X~,('He)
log X.,(2H)
log X~,(3He)
log X~(7Li)
log X.~(A >-12)
log(~/XD)
10T M 10-1°
28.5% 81% %%
-1.% -2.5 --3.3
-3.14 -3.7 --4.5
-7.47 -6.7 --6.6
-13.42 -10.3 --7.7
10 -8
~%
--4.I
-5.3
--6,7
-5.5
10 -7
I~% 1~% ~% ~%
-5.0 --5.9 --6.7 --7.2
--6.1 --7,0 --7.9 --8.5
-6.7 --6.8 --6.9 --7.3
-12.8 -12.5 --12.3 -12.1 -12.0 --11.9 --11,7 --11.2
10 -9
10-6 10-5 10-4
--4.0 --2.8 --1.7=2% (18%)
Table 3 Sensitivity of element abundances to initial neutron fraction F. for r/.s = 10-~ log X,~(TLi)
log Xn~(A >-12)
F,
X~('He)
log X~,(2H)
log Xn,(3He)
1
~%
-4.1
-5.3
-6.7
-5.5
0.9 0.8 0.7 0.6 0.5 0.4 0.3
@% ~% ~% ~% ~% 67% 50%
-4.1 -4.2 -4.2 -5.2 -6.4 -7.6 -8.9
-5.3 -5.3 -5.4 -6.0 -5,5 -5.3 -5.2
-6.6 -6.6 -6.6 -8.4 -7.2 -7.1 -7.1
-5.5 -5.6 -5,8 -9.1 -10.4 -10.8 -11.1
the predicted mass fractions of 2H, 3He, *He, 7Li, and the sum of the mass fractions for A -> 12 for an initially pure neutron region as a function of 77 in the region, and the light-element mass fractions as a function of the initial neutron fraction for 77= 10-8. Abundances as a function of time during nucleosynthesis for the cases 77= 10 -8 and r/= 10 -1° are given in figs. 2a and b, also taken from AHS. The most important conclusion from the AHS nucleosynthesis calculation is that an inhomogeneous universe with ag~ = 1 (closure density of baryons), in which a fraction fn = 0.2 of the neutrons undergo nucleosynthesis in neutron-rich regions which occupy a fraction fv = 0.2 of the volume, can explain the abundances of deuterium, aHe, and the population I value of 7Li. The conversion factors needed to relate the baryon to photon ratio and the mass fractions in the neutron-rich regions to the global averages are
1L
*/"~- 6 f~ 770
(3.1)
X0 = ~f.X,s,
(3.2)
where the subscript '0' refers to the global average value of a quantity and the subscript 'ns' refers to the value in the neutron-rich region. The factor 1/6 is the neutron to baryon ratio at the onset of nucleosynthesis. The product Xns(2H)Bns is nearly constant. If this is converted to the global average of the deuterium mass fraction, and the requirement Xo(2H) = 3 x 10 -5 [1] is used, then 7/0 = 3 x 10 -8 fv. Thus O b = 1 (r/0 ---3 x 10 -8 Dq,h 2) is consistent with deuterium production if fv =0.2. Inhomogeneous nucleosyn-
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150
thesis produces less 4He than a homogeneous model with the same global value of O b because it takes two neutrons to make one 4He nucleus in a proton-rich environment and four neutrons to make one 4He nucleus in a neutron-rich environment. The same number of neutrons are used less efficiently in inhomogeneous nucleosynthesis. The reduction is by a factor AXo(4He)/Xo(4He) = _ l f~.
(3.3)
Thus, fn = 0.2 reduces the/2 b = 1 4He production by the right amount to agree with the primordial 4He abundance. The situation concerning the other two isotopes, 3He and 7Li, associated with the primordial nucleosynthesis is less clear. Both the observational and theoretical aspects of the cosmological role of 3He are ambiguous [1]. The observations suggest that X(3He) = X(ZH) at the present epoch. The AHS model produces X(3He)= 0.06X(ZH), so AHS underproduces 3He. However, 3He can be both produced and destroyed in stars, so 3He is not a sensitive cosmological probe. Lithium offers a potentially more sensitive test because of the quality of the observations. The standard model predicts X0(7Li)-~ 10 -9, which is in good agreement with the VLi abundance found in metal-deficient (population II) dwarf stars [15, 16]. The AHS model predicts X0(7Li) --- 10 -8 with f, = 0.2, and this is in good agreement with the 7Li abundance found in a wide variety of population I sites [1]. The standard model interprets the pop II 7Li mass fraction as the primordial value. However, this interpretation requires the uncomfortable assumption that an unknown physical process, which is sufficiently robust to produce a factor of 10 more 7Li than is produced in the standard model, turn on in pop I sites and produce the pop 1 7Li abundance. Furthermore, this process must be assumed to not contaminate the pop II 7Li abundance. The AHS picture is favored if the pop I 7Li abundance is primordial, but this interpretation requires that an unknown destruction mechanism uniformly deplete the pop II stars by a factor of 10 and leave pop I untouched. Spite and Spite [15] (see also ref. [16]) give highly suggestive arguments that the pop II abundance is primordial, but neither of the interpretations of 7Li given above is as convincing as the interpretations of deuterium or 4He within the context of the standard model, and neither will be until the key missing ingredient in the analysis, the nature of the process responsible for the factor of 10 difference in the pop I and pop II 7Li abundances, is well understood. The effect of neutrons diffusing back into proton rich-environments at late times will be to increase the production of ZH and 3He in these regions. The effect should be qualitatively similar to the effect of the decay of heavy antineutrinos (gnome+anything, followed by ~ e + p ~ e ÷ +n) analysed by Scherrer [17], who found that significant amounts of 2H and 3He could be produced by a late-time source of neutrons. The most striking difference between the AHS calculation and the standard model is the production of nuclei with mass numbers A -> 12. The predictions of X , s ( A >- 12), given in table 2, are several orders of magnitude larger than the predictions of the standard model with the same value of r/. Most of the mass in A -> 12 is in 14C. (The reaction network used by AHS was truncated at 160, so the value of X ( A >-12) and the isotopic distribution may not be accurate.) The most likely reaction path for ~4C production is t + ot"--~7Li , 7Li + n ~ 8Li,
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151
8Li + o t ~ liB + n , liB + n ~ 12B ,
12B---) 12C 71-e-
"~])e ,
]2C + n ~ 13C, 13C -[- n ~ 14C.
The bottleneck in this sequence is most likely 8Li because it has a half-life of 0.85 s. For r/0 = 10 -8 and r/. s = 1.7 x 10 -9 the mixed mass fraction Xo(A >- 12) is -----10-9, most of which will be 14N, produced by the [3-decay of ~4C. This is about 10 -3 Of the heavy-element mass fraction of the lowest-metaUicity star known [18]. Note that if r/ns is increased by a factor of 6 the value of X(A >- 12) is increased by 100. It is possible that the cosmological floor to the abundance of CNO nuclei predicted by AHS can be detected or ruled out, but the level of the floor is very sensitive to r/ns and can be very low. A potentially much more sensitive test is provided by the fact that the CNO nuclei are produced in a neutron-rich environment where they can seed an r-process, which allows the abundance of r-process elements to be used as a cosmological probe.
3.2. Heavy element nucleosynthesis The production of r-process elements from 14Chas three distinct steps: the match, the fuse, and the bomb. The match is a sequence of reactions that produces 2°Ne from 14C. The most likely sequence is 14C(ot, ~/)lSO(n, ~)190([3-)19F(n, ~/)2°F([3-)2°Ne ;
(3.4)
of these the key is ~4C(a, -/)180. The fuse is a series of (n, ~/) reactions and B-decays that convert 2°Ne to neutron-rich nuclei in the A = 60 region. The fuse is essentially a light-element n-process [19], but because the (n, ~/) cross sections are rather low, t r - 0.1 mb to 10 mb, the reaction path leads through mostly known [3-decays [most (n, ~/) cross sections are, however, not measured]. The bomb explodes once "real r-process nuclei", with neutron capture cross sections of 100 mb to 1 b, have been produced. The heavy-element r-process is characteristic of the long-time solution [20], which is dominated by fission cycling. The number of fission cycles is the largest uncertainty in the prediction of the final mass fraction of r-process elements. In this section I will give an outline of the theory of the primordial r-process; the details will be presented elsewhere [21]. The ~4C produced in the AHS calculation starts to be produced at T 9 = 1 and it is close to its final abundance by T9 = 0.7. I assume that the production of 2°Ne occurs in the same temperature interval as the production of 14C, and that the (n,-/) reactions and [3-decays that take 2°Ne into the r-process region occur at T 9 < 0.7. I characterize the strength of the various reactions by the product try" of the thermally averaged cross sectidn o- and the exposure r [22], where z is defined by d~" = on dt, where n is the number density of irradiating particles and v is their mean thermal velocity. With this definition the number of reactions dN per target nucleus caused by exposure dr is dN = tr dr. I assume that rl, s = 1.7 x 10 -9, appropriate to 7/0 = 10 -8, and I take X n = 0.6, X~ = 0.4, and Xp = 10 -2 as representative of rough averages of these quantities in the interval 1 > T 9 > 0.7. With these assumptions the exposures are rp = 2 X 10 28 cm -2 ,
r~ = 1029 cm -2 ,
r n = 7 x 10 29 cm -2 .
(3.5)
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152
The proton mass fraction changes from Xp = 10 -3 for ?~ns= 10-8 to Xp = 10 -1 for 7/, S = 10 -1°, while X, and X~ are relatively insensitive to 77,s in the interval 0.7 < T 9 < 1. I compute the cross section for 14C(a,-/)180 from the published rate [23]. The cross section varies from 2.4 × 10 -34 c m 2 at T 9 = 1 to 6 × 10 -36 c m 2 at T 9 = 0.7. I adopt or = 3 × 10 -35 c m 2, appropriate for T 9 = 0.8, as a suitable average; this gives t r r = 3 × 10 -6. The neutron capture 180(n, ~/)190 must compete with 180(p, ot)15N. The cross section for 180(p, ct)15N is about 2 × 10 -27 c m 2 [24], which gives trrp =40. I estimate or = 6 × 10 -30 c m 2 for 180(n,~/)190 from known resonance properties [11]; this gives o-zn = 4. About 90% of the 180 produced by laG(or, ~')180 ends up as 15N and about 10% continues on the path (3.4). The neutron capture 19F(n, 302°F dominates 19F(p, 0 0 1 6 0 by about a factor of 200 [11, 24]. The half-lives of the two 13-decays a r e t l / 2 ( 1 9 0 ) = 26.9s and t112(Z°F)= l l . 0 s [25]. Since it takes 200 s to go from T 9 = 1 to T 9 = 0.7, there is plenty of time for these 13-decays. I estimate the yield of the match to be X(2°Ne) = 3 x 10 -7 X(14C)~---10 -14 ,
(3.6)
where both estimates assume rhs = 1.7 × 10 -9. The number density of neutrons is given by n n = 3.4 × 1019 f(t)T39 cm -3 ,
(3.7)
where f(t) is the fraction of neutrons present at the onset of nucleosynthesis which are still free at time t. At T 9 = 0.7 1 t a k e f = 0.4. At later times I assume that the neutrons disappear with the decay constant r computed in section 2.2. Thus
f(t) = 0.4 e x p [ - r(t - to)],
(3.8)
where t o = 372 s is the time at which T 9 = 0.7. This gives a total neutron exposure for T 9 < 0.7 of 6 × 1029 -2 r n -- - cm Kt 0
(3.9)
If K < 10 -2 S -1 the neutron exposure is ~ 1 0 29 cm -2. I estimate the exposure needed to convert Z°Ne to the A = 60 region by assigning a mean cross section to all neutron captures, and then setting trz, = 60. If only neutron captures were considered, then Orn = 40 would be required. However, I assume that neutrons can be captured until a ~-decay as fast as the (n, 3,) reactions can be found. Roughly 20 13-decays are needed, so I use trr n = 60. I estimate o- = 4.2 mb by taking the geometric mean of the 43 nuclei between 2°Ne and 58Fe whose (n, ~/) cross section was computed by Woosely et al. [26] using Hauser-Feshbach theory. The exposure required to get to A = 60 is zn ---2 x 1028 cm -2
(3.10)
This is 10% of the total exposure for x = 0.01 S -1. The mean neutron capture time for tr = 4 mb is = 0.1 s. Thus [3-decays with half-lives of 0.2 s are needed for the [3-decays to keep up. Known B-decays [25] with half-lives ~<1 s are common for the elements in question. The fuse probably bums in <60 s. Heavy nuclei have neutron capture cross sections tr ~> 100 mb. If I take tr = 100 mb, r = 1029 cm-2,
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153
neglect fission cycling, and assume that fast 13-decays can always be found, then every r-process seed captures 104 neutrons. Since fission cycling occurs about every 140 neutron captures, this corresponds to 70 fission cycles. However, the neutron capture time for or = 100 mb is =0.01 s, so the time needed to 13-decay cannot be neglected. The time needed to complete the fission cycle once is at least the sum of the [3-decay half-lives of the nuclei in the nucleosynthetic path [20]. The total 13-decay waiting time is estimated to lie in the range [27] 0.1 s < r~ < 3 0 s ;
(3.11)
the factor of 300 uncertainty in r~ is the major uncertainty in the computation of the yield of the primordial r-process. If the exposure lasts 100 s, the number of fission cycles is n = 100/r~,
(3.12)
and 3 < n < 1000 is consistent with the uncertainty in r~. The mass fraction in r-process elements produced in the neutron-rich regions is Xn~(r) = -~00" 2"X(Z°Ne) - 10 -13 2",
(3.13)
which gives a mixed mass fraction of
Xo(r) = 3 x 10-1s2"
.
(3.14)
If n > 40 the neutrons will be exhausted and X,s(r)~ 0.1, which corresponds to Xo(r ) = 3 × 10 -3, will be produced. This is an overproduction by about 10 relative to the sun. This brings out an important feature of the primordial r-process: A very small mass fraction of seeds can produce a very large mass fraction of r-process elements because, relative to stellar r-process sites, the neutron exposure in the big bang lasts a very long time. Stars with r-process depletions of 1000 relative to the sun have been observed [28, 29]. The elements observed are in the rare earth region between the r-process peaks at A = 130 and A = 195. Since A < 120 is not produced in the long-time solution [20], I determine the relevant r-process mass fraction for the sun by summing the r-process contributions to A > 120 in the abundance table of Cameron [30]. I obtain Xo(r ) = 10 -7, which gives a limit of 10 -1° for cosmological production. This gives 1 200.2,X(20Ne), 10-1°> 3--0 " 20
(3.15)
which gives n < 15. The observations are already a significant constraint. The fact that the limit on the cosmological production is already near the low side of what is possible suggests that the cosmological r-process abundance, if it exists, may not be too far below the current observational limit.
4. Conclusion
I have presented an inhomogeneous cosmological model which can match the success of the standard model in predicting light-element abundances with O = 1 in baryons. The inhomogeneities can be
154
Theory of supernovae
produced in the quark confinement transition. The model predicts a cosmological production of r-process elements by rapid neutron capture in neutron-rich environments. If detected, this cosmological production of r-process elements would be a relic of the time when quarks first condensed into hadrons when our universe was 10-4s old.
Acknowledgements This work was done in collaboration with Craig Hogan and Bob Scherrer, and I gratefully acknowledge their contributions to the results presented here. I would like to thank Bob Malaney for pointing out the correct reaction sequence involved in the production of t4c.
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