Neutrinos in the Universe

Neutrinos in the Universe

T. D~ GRAAF Institute of Theoretical Astronomy, Cambridge, on leave of absence from the Institute for Theoretical Physics, University of Groningen, Netherlands

SUMMARY

In this article we consider the relations which exist between the properties of the neutrino, a manifestation of Nature on its smallest scale, and on its largest scale the behaviour of the Universe as a whole. The Hot Big Bang theory describes the thermal history of the Universe, during which the lepton era was particularly important with regard to weak interaction physics and the predicted neutrino processes. Possible modifications of the theory of weak interactions change these processes and the cosmological rSle of the neutrinos. We give several possible estimates and upper limit values for the neutrino energy density, which can be obtained from the cosmological model and experimental data.

1. INTRODUCTION The s t u d y of p h y s i c a l processes in stellar interiors i n d i c a t e s t h a t n e u t r i n o s m a y p l a y a v e r y i m p o r t a n t role in a s t r o p h y s i c s , where t h e y p r o v i d e m e c h a n i s m s for emission of e n e r g y a n d d e t e r m i n e t h e t i m e scale of e v o l u t i o n for o b j e c t s like s u p e r n o v a e a n d p l a n e t a r y nebulae. A s t r o p h y s i c a l n e u t r i n o processes h a v e been considered b y several authors, e.g. Chin (1968), a n d i t has been p o i n t e d o u t t h a t several a s t r o p h y s i c a l p h e n o m e n a d e p e n d s t r o n g l y on t h e p r o p e r t i e s of t h e n e u t r i n o processes, which are d e s c r i b e d b y t h e t h e o r y of w e a k i n t e r a c t i o n s . Because of difficulties c o n n e c t e d w i t h t h e o b s e r v a t i o n of t h e i n t e r e s t i n g a s t r o p h y s i c a l o b j e c t s a n d w i t h t h e e x p e r i m e n t a l d e t e c t i o n of specific w e a k i n t e r a c t i o n processes in t h e l a b o r a t o r y , several aspects of these considerations are r a t h e r u n c e r t a i n . I t is still m o r e u n c e r t a i n t o d r a w conclusions a b o u t cosmological neutrinos, t h e n e u t r i n o s which are n o t p r o d u c e d in one of t h e local sources, b u t originate in some w a y f r o m earlier stages of t h e universe. One m a y assume t h a t t h e y h a v e a homogeneous d i s t r i b u t i o n , for which t h e p r o p e r t i e s are d e t e r m i n e d b y t h e processes t h a t t o o k place d u r i n g t h e cosmic evolution. These processes are d e s c r i b e d b y t h e laws of t h e m a c r o c o s m o s : t h e b e h a v i o u r of t h e U n i v e r s e as a whole, which is s t u d i e d in cosmology, a n d b y t h e laws of t h e microcosmos: e l e m e n t a r y p a r t i c l e physics, especially t h e t h e o r y of w e a k interactions. Good surveys of m o d e r n cosmology h a v e been given b y Zeldovich a n d N o v i k o v (1964, 1965), whose b o o k on R e l a t i v i s t i c A s t r o p h y s i c s (Zeldovich a n d N o v i k o v , 1967) describes in d e t a i l t h e i m p o r t a n t d e v e l o p m e n t s which h a v e t a k e n place in this new field of research where general r e l a t i v i t y , e l e m e n t a r y p a r t i c l e physics a n d a s t r o p h y s i c s are l i n k e d together. 6,~ vA

161

162

Neutrinos in the Universe

I n the last decade the most important discovery in observational cosmology has been the detection of the microwave background radiation by Pcnzias and Wilson (1965) (Dautcourt and Wallis, 1968 ; Field, 1969). This isotropic microwave radiation has been interpreted as a black body radiation, originating from a thermal distribution of photons with a temperature of about 3°K and can be considered as a relic of an earlier "primeval fireball" stage of the Universe. I n this stage of the universe the temperature and density would have been very high according to the " h o t big b a n g " model, which was originally put forward by Gamow (1946, 1949, 1956) and later modified by Hayashi (1950) and others (e.g. Alpher, Follin and Herman, 1953). I t will be considered in more detail in Section II. After Einstein introduced the general theory of relativity, Fricdmann (1922, 1924) obtained a solution of the Einstein equations for a homogeneous, isotropic matter distribution which undergoes expansion. This non-static model of the universe was afterwards confirmed by Hubble (1929), who detected the red shift of remote nebulae. Further considerations specify the state of matter during the earlier periods: I n the primeval fireball the temperature was very high and all elementary particles with rest mass m, for which mc2
T. DE GRAAF

163

stages of the universe which disappeared during the further evolution. I n the anisotropic models, described by Thorne (1967), Misner (1967) and Doroshkevich et al. (I968, 1969), the neutrinos play an important rSle, as they provide a mechanism which can smooth out the anisotropy. We do not consider the anisotropic models, but only state that the indicated mechanism (neutrino viscosity) depends on the weak interaction processes, e.g. the way in which the neutrinos decouple during the evolution of the universe. For more details we refer to the quoted literature on this subject and to the chapter on anisotropic models inthe book of Zeldovich and Novikov (1967). Another topic which will not be discussed in this lecture concerns the gravitational interaction of neutrinos, e.g. the gravitational effect of a cosmic neutrino sea (Marx, 1967). A review on the neutrino and the theory of gravity has been given by Kuchowicz (1969). For an extensive compilation of references on neutrino astrophysics and related fields we refer to his review reports (Kuchowicz, 1966, 1968) and to a monograph by Bugayev et al. (1970). II. THE HOT BIG BANG THEORY I n relativistic cosmology a homogeneous isotropic model of the universe is described by the Robertson-Walker metric with line element ds e = c2 dt 2 - R e ( t )

(dr ~ + redO e + r e sin~ O dq) 2)

1 +

(1)

where R(t) is the scale factor, the characteristic size of the Universe, and the curvature index/c determines whether the model is closed and spherical, open and flat or open and pseudo-spherical for k equal to + 1, 0 or - 1 , respectively. Substitution in the Einstein equations with eosmologieal constant A = 0 provides the following relations (el. Tolman, 1943; Adler, Bazin and Schiffer, 1965) +

2~- +

(2)

R2 - ~ - - e ,

+ R--~ =

- -

c2

p.

(3)

With R and )~ we indicate the time derivatives dR~dr and d2R/dt 2. If the scale factor does not depend on time we have /~ -- R = 0 (Static Universe), and the resulting equations can only be solved with non-negative pressure p and density Q if we take p = 9 = 0. I t was the Russian meteorologist Friedmann (1922, 1924), who found a non-static solution of these equations, where R(t) is an increasing function of time which describes the expansion of the Universe. This function can be given in such a way that the expansion time t is equal to zero at the singular initial state of the Universe with R -- 0. Some years after the publication of this non-static model Hubble (1929) detected the red shift of remote nebulae, which confirmed that the matter in the Universe is indeed expanding. At the present time t = t o the expansion can be characterized by the Hubble constant H 0 and the deceleration parameter %, which are defined by

t 4 = - f f ,=,o, %=6a*

(--~-),:,o"

(5)

164

N e u t r i n o s in the Universe

I n his discussion of the present data from which the Hubble constant can be determined + 19 Sandagc (1968a) finds H 0 . . . .~_~.2 . 15 km sec -1 Mpe -1 which corresponds to the following value for the Hubble time Ho I -- (1-3 ± 0"3) x 101° y. (6) He remarks, however, t h a t it seems possible t h a t H e could be as small as 50 k m sec -1 Mpc -1, corresponding to H o 1 ___ 1.9 × 101° y. Substitution of (4) and (5) in Eqs. (2) and (3), taken at the present time t -- t o provides kc 2 Ro~ -

8zG 3 ~' -- H~,

kc ~

(7)

8~G -

Po

-

2qoH

o

-

(s)

F r o m relation (7) we find t h a t the geometric structure of the universe as a whole depends also on the ratio/20 = ~0/Qc of the present density ~0 to the critical density ~c (Zeldovich and Novikov, 1967). This critical density is defined b y the following expression in which the given value corresponds to the Hubble constant with H o i = 1.3 × 101° y. ~c=

3Ho2 8~G - 1 . 1

x

10_99

g e m -s.

(9)

The three cases with curvature index k = - 1, 0 or + 1 correspond, respectively, to /20 < 1,/20 = 1 o r / 2 0 > 1. The pressure and the density are related b y the equation of state, which takes a simple form in two specific cases: For the radiation-type Universe with only relativistic particles (with rest mass m and m c 2 < k T ) we have p = 31 ~c ~

(10)

a n d for the m a t t e r - t y p e Universe (with m c 2 > k T ) p = 0.

(11)

I n these cases the deceleration parameter qo can be related to the relative density factor /20: F r o m Eqs. (7)-(11) we conclude t h a t qo = / 2 0 in the radiation-type Universe and q0 = ½£20 in the m a t t e r - t y p e Universe. The geometric structure of the Universe as a whole c a n thus also be determined b y measurement of the deceleration parameter. This is, however, very difficult with the existing extragalactic observations. Figure 1 gives the different possibilities. B y combining Eqs. (2) and (3) the following relation can be f o u n d : d p dt (~R3) ~- c ~

dR a

d---/--- 0.

(12)

This determines the conservation of e n t r o p y for a specific co-moving volume during the expansion. Because of the high temperatures in the early stages of the Universe which are assumed in the hot big bang model (Gamow, 1946, 1949, 1956), the available particles are all relativistic and we can use the radiation-type model, for which (12) becomes with (10): d dt

(QRa) = 0.

(13)

T. DE Gr~Ar

165

(i)

¢Y

(2) 2 c~

I I

*o Expansion time,

t

1~o.1 Model

(1)

Metric

(2)

(3)

pseudo-spherical

euclidean

spherical

indefinite

expansion

contraction after maximum R

Evolution Curvature index k

-- 1

0

A-1

Relative density factor $2o

< 1

= 1

>1

% for radiation-type model

< 1

= 1

> 1

qo for matter-type model

<½

= ½

>½

Thus ~R 4 is a constant during the expansion, which we can call Q. F r o m equation (2) we get after substitution of eR a = Q: (dR)28~GQ 1 = 3 R~ kc2" (14) F o r small values of R we can neglect the t e r m - k c ~ and find after integration the following solution for the expansion time

t =

(

3 )1/2R2. 32~GQ

(15)

This result can also be obtained when we assume t h a t the radiation-type model of the universe is valid until the present time t o (the density m a y be dominated b y relativistic particles, e.g. neutrinos). After integration of Eq. (2), where we use QR4 = ~oR~, the expansion time can be expressed in the following w a y as a p r o d u c t of the t t u b b l e time and a function of the relative density factor ~0 and the expansion parameter R / R o: t ----H~)1~1/2(1 - ~0)-1 {(1 + - -

R2 1 - f 2 o ) 1/2 } Ro2 ~o - 1 .

(16)

166

Neutrinos in the Universe

For f20 ~ 1 or for small values of the expansion parameter (in early stages of the Universe) this can be approximated by

t = . l Hoi[2~l/+ (-~o ) 2

(17)

which is equivalent to (15), for

Q = ~oR~ =

3H~T2oR~ 8zG

Relation (15) is thus valid for all stages of a flat radiation-typeUniverse (with k ----0), but it also holds for the non-euclidean radiation-type models in early stages of the Universe. I n these cases we find from (15) the following expression for the density Q 3 p = R---Y = 32~Gt2 _ 4.5 × 105t-2g cm -a.

(18)

The corresponding total energy density becomes = Qc2 - 4 × 102et-2 erg cm -a _ 2-5 × 10as- eV cm -a.

(19)

During the expansion the energy density decreases and therefore the temperature in the hot big bang model is also a decreasing function of time. This time dependence is determined by the number and type of relativistic elementary particles which are available at each stage. I n the evolution of the Universe several stages can be distinguished (Harrison, 1968 a, 1968 b). At very high temperatures (kT > M:,c 2) the Universe is in its hadron era, where all strongly interacting particles with Mc 2 < k T are relativistic and have energy densities of the same magnitude as the photon energy density. For M~c 2 > k T > mec 2 one has the lepton era, where photons and leptons are the particles which determine the energy density. After annihilation of the electrons (kT < mec 2) the Universe enters its radiation era, which lasts until the temperature becomes smaller than about 3000°K. Thereafter the Universe is in the stellar era, for which it is assumed that matter becomes the main constituent of the total density, protons and electrons recombine and radiation is decoupled from matter. For photons we have the following relations for number density and energy density at temperature T (Chiu, 1968) oo

n~, = - -

- -

7~2

e x-

1

--

20T 3 cm -3,

(20)

0 oo

kT ~kT~a f

x adx

= aTaergcm_3 '

(21)

0

with radiation density constant ~2/¢4 a 15haca - 7"56 x 10-15ergcm - a d e g -4_~ 4"7 x 10 - 3 e V c m -adeg -4.

(22)

For non-degenerate neutrinos these relations become (Landau and Lifshitz, 1958)

l n~=nv=--~j

oo

/

~

eX+ 1 _ 7"5T3cm -a,

(23)

x3dx

(24)

0 o0

kT(kT)af

e ~ = e v =-~-~2 - ~ c

eX+ 1 0

7 aT4 erg era_3 ' 16

T. DE GRAAF

167

and for non-degenerate relativistic electrons (also for muons) oo

1 ~kT~af ne- = ne÷ = ~--~\-~-c ]

x~dx ex + 1 - 15Taem-a'

(25)

0

kr

x3 x

~2 \ h e ]

Je x+ 1

7_ a T 8

4 erg e m -3.

(26)

0

If we assume t h a t there are no other particles present in the Universe, the total energy density in the lepton era is provided by photons, both kinds of neutrinos (which are assumed to be non-degenerate) and electrons (mec2 < k T < m,c~). I n this case the total energy density becomes with (21), (24) and (26), s = 9/2aT ~, whereas for higher temperatures (m,c 2 < k T < M~c 2) muons are also available and we have s = 25[4aT 4. I n general e can be expressed by the relation = uaT 4 (27) in which u is an increasing function of the temperature. I t is possible t h a t the existence of other zero mass particles (e.g. as yet unknown neutrinos which differ from v~ and v~) determines a higher value of u and changes the relation between temperature and expansion time. I t has been shown (Shvartsman, 1969 ; Sunyaev and Zeldovich, 1969) t h a t the number and density of these particles cannot be too high, for this would imply a large He ~ abundance which is very unlikely. This effect might, however, be compensated by neutrino degeneracy (Fowler, 1970) or anisotropy of the Universe in the initial state (Zeldovich and Novikov, 1967). F r o m (18), (19) and (27) we get the following relations between temperature and expansion time :

3c 2 ~1/,

T = \3~Ga]

7¢-1/4t-1/~ ~- 1"5 × 101° ~-l/4t-1/2°K

(28)

and t --- 2.3 × 102 ~-1/2Tff2 sec

(29)

in which the temperature Tg is expressed in units of 109°K. I n the lepton era we can take = 9/2 and find t -----10~Tff2. (30) I n this way we obtain relations between the expansion time, the density (18) and the temperature (28). The stages of evolution of the Universe are summarized in Table I, which can be obtained from these relations (cf. Harrison, 1968b). Particles with zero mass remain relativistic during the expansion of the Universe and form a thermal background which does not annihilate. The only component of this background which has been detected could be the background microwave radiation which m a y be interpreted as a cosmic black body radiation of photons resulting from the big bang (Penzias and Wilson, 1965; Dautcourt and Wallis, 1968). For its present temperature the value Tu ~ 2-7°K has been established. This corresponds to an energy density s 7 = aT~ ~_ 0.25 eV cm -a.

(31)

We indicate the present temperature of the photon gas by Tv, which is different from the present temperature of the neutrino gas, T:, as we shall see in the following sections. I n these sections we consider the background of neutrinos which might also result from the big bang and study the properties of this cosmic neutrino background.

Neutrinos in the Universe

168

TABLE 1. The Thermal History of the Universe according to the Hot Big Bang Theory Expansion time t

Density

Temperature

~

Hadron era

larger than nuclear density

4 x 10-5 see

3 × 10~4g cm-3

Lepton era

3 sec

Events

T

kT ~ Mnc2

annihilation of all hadrons when the temperature decreases

1"7 × 1012°K

ir~c 2 ~ k T ~ me c2

5 × 104g cm-3

annihilation of p at T ~ 1.2 × 1012°K decoupling of vg at T ~ 1.2 × 1011°K decoupling of Ve at T ~ 1"8 × 101°°K annihilation of e at T ~ 5"9 × 10a°K

5.9 × 109°K

R a d i a t i o n era

/cT ~ meg2

~)rad ~ ~matter cosmological

nucleosynthesis (?) 10s years

10-51 g cm-3

3000 °K

Stellar era

end of primeval fireball photons decouple; galaxy and star formation ~)rad ~ ~mattcr (?)

1.3 × 101°years (present)

~ 3 × 10-31 g cm-3 (matter)

T r ~ 3°K

Tv ~ 2°K

I I I . PHYSICAL PROCESSES IN THE LEPTON ERA, DECOUPLING AND ANNIHILATION F i r s t we w a n t to consider the processes between e l e m e n t a r y particles i n the lepton era a n d compare the characteristic times for these processes with the expansion time (30). Strong i n t e r a c t i o n s do n o t play a n i m p o r t a n t rSle, for the h a d r o n s are non-relativistic a n d their d e n s i t y is v e r y small compared to the lepton a n d p h o t o n density. E l e c t r o m a g n e t i c i n t e r a c t i o n s are responsible for scattering a n d a n n i h i l a t i o n processes like y + e - - , y + e;

e- + e+ ~ y + ~,,

(32)

which d e t e r m i n e the t h e r m o d y n a m i c e q u i l i b r i u m between p h o t o n s a n d charged leptons. The cross-sections of these processes are high enough to m a i n t a i n this e q u i l i b r i u m long after the lepton era. W e a k i n t e r a c t i o n processes are u s u a l l y described b y the U n i v e r s a l F e r m i i n t e r a c t i o n , which was proposed b y F e y n m a n a n d Gell-Mann (1958). If this t h e o r y gives a n appropriate description of the physical p h e n o m e n a , the following processes are possible a n d their crosssections can be calculated: (a) Processes with electrons and electron-neutrinos ~e -]- e - -~ ~e -[- e - ; ~e -~" e+ --'~ ~e "~- e + ;

re -{- e~ -+ ~e -I- e +,

(33)

~e -~- e - ~

(34)

e - -~- e + ~-~ ~'e "~ Ve"

Pe -~ e - ,

(35)

T. DE GRAAF

169

(b) Processes with muons and muon-neutrinos

(36) v, + # + - * v~ + #+;

~, + # - -* ~, + # - ,

#- +#+~

~ + ~.

(37) (38)

(c) Processes with both kinds o/leptons ~u- --* e - + ~e + r~;

#+ --* e + + ve + ~ ,

(39)

ve + e- ~ ~ + ~ - ;

ve +/~+ ~ ~p + e+,

(40)

There can also occur processes with photons and neutrino-antineutrino pairs which involve weak and electromagnetic interactions and virtual muons or electrons, like ve + ~e ~ 3~.

(42)

The corresponding cross-sections are, however, very small and therefore these processes need not be taken into account (de Graaf and Tolhoek, 1966). The processes (33)-(38) are described by the "diagonal terms" in the Universal Fermi Interaction Hamiltonian ~%fl = ~ -g~ (J~ + J~)(J~ + Ji) + h.e.

(43)

Here g is the weak interaction coupling constant and J ~ (•h) and i~ (i~-) represent the charged hadron and lepton currents, respectively. The lepton current i f = ~e e + ~ #

(44)

describes e.g. annihilation of an electron-antineutrino or electron and creation of an electron-neutrino or positron and it is charge increasing. I t should be remarked that the given interaction Hamiltonian does not predict scattering processes like v~We-,v~+e

or

VeWIZ~Ve+I~.

With this theory the following total cross-sections for neutrino-electron scattering can be obtained (Feynman and Gell-Mann, 1958; Bahcall, 1964). a(v~ + e- - , ve T e-) ----a0[~o2/(1 + 2co)],

(45)

a(ve + e+ -* ~e + e+) -----(a0w/6) [1 -- (1 + 2o~)-s],

(46)

where co is the neutrino energy in the electron rest system, expressed in terms of the electron rest energy: ~o = E~/mec ~ and a 0 is determined by the weak interaction coupling constant g and the mass m e: (70 =

492m~ ~h 4

= 1-7 x 10 -44 em 2.

(47)

For low energies (with co ~ 1) these cross-sections become

~(~e + e-) = ~(~e + e+) = a0~2

(48)

and for high energies we get (with w >> 1) ~(~e + e-) = 3cr(ve + e+) = ½ ~oW.

(49)

170

Neutrinos in the Universe

I n t h e l e p t o n era t h e neutrinos s c a t t e r with electrons a n d positrons in a r e l a t i v i s t i c F e r m i gas a t t e m p e r a t u r e T, which is described b y t h e n u m b e r d e n s i t y (25) a n d t h e e n e r g y d e n s i t y (26). A v e r a g i n g t h e t o t a l cross-section over this electron d i s t r i b u t i o n p r o v i d e s in t h e relativistic case with eo >~ 1 a n d k T >:>m~c ~ (Bahcall, 1964)

( ~ ( ~ + e - ) ) ~ , r = 3 ( ~ ( ~ + e+))~,r -~ y ~ d o

x 3"2

= 0"27~0o)T 9.

(50)

The m e a n free p a t h '~ve = (~O') he)-1 of a n e u t r i n o in t h e electron gas is d e t e r m i n e d b y this a v e r a g e cross-section a n d t h e electron density. F o r t h e n e u t r i n o e n e r g y we t a k e t h e m e a n e n e r g y in t h e n o n - d e g e n e r a t e F e r m i d i s t r i b u t i o n f r o m (23) a n d (24) : e,

= 3.2--kT

= 0.54T9.

(51)

The c h a r a c t e r i s t i c t i m e for n e u t r i n o s c a t t e r i n g is defined as v~, = 2~,/c a n d we g e t

~;~ = (~(ve + e-) + a ( ~ + e÷)} n~c.

(52)

The relativistic electron d e n s i t y n~ is given b y Eq. (25) a n d can also be expressed as a f u n c t i o n of To: 1 3

'~ = ~-~ ~ hc ]

× T ~(3) = 1-5 × lO~'T~.

(53)

I n the relativistic case we find therefore for t h e c h a r a c t e r i s t i c time, using Eqs. (50)-(53) T:: ~ 6-7 × 105T~ 5 sec.

(54)

The e q u i l i b r i u m is n o t o n l y e s t a b l i s h e d b y s c a t t e r i n g processes like (33) a n d (34), b u t also b y t h e a n n i h i l a t i o n process (35). I t is i n t e r e s t i n g t o show t h a t t h e c h a r a c t e r i s t i c t i m e for t h e t r a n s f e r of e n e r g y f r o m electrons to n e u t r i n o s in t h e process e- + e + + ve + +~ is of t h e same order as (54). I n t h e r e l a t i v i s t i c case ( k T > m~c 2) the r a t e of this e n e r g y t r a n s f e r is according to Chin (1968)

ds~ = 4.45 × 1015T~ erg em -s sec -1. dt

(55)

A t t h e r m o d y n a m i c e q u i l i b r i u m we get from (24) t h e t o t a l e n e r g y d e n s i t y of b o t h n e u t r i n o s and antineutrinos : 7 4 = 6-6 x 10~IT~ erg cm -3. s~, = -~-aT (56) This e n e r g y d e n s i t y can be o b t a i n e d f r o m t h e process e- + e + --* ve + ve in a t i m e

/ d ~ ~-1

v~ ----s ~ ~ - - ~ - )

----- 1"5 × 106T~ 5 sec

(57)

which can be considered as a c h a r a c t e r i s t i c t i m e for this process (Zeldovich a n d N o v i k o v , 1967). W e see t h a t T~v has t h e same o r d e r of m a g n i t u d e a n d t e m p e r a t u r e d e p e n d e n c e as ~e

•

Comparison of these results with (30) i n d i c a t e s t h a t t h e c h a r a c t e r i s t i c times increase m u c h faster with decreasing t e m p e r a t u r e t h a n t h e e x p a n s i o n t i m e of t h e Universe. A t a certain t e m p e r a t u r e t h e s h o r t e s t characteristic t i m e becomes e q u a l to t h e e x p a n s i o n t i m e a n d t h e

T. DE GR~_A~

171

neutrinos cease to interact effectively with the other particles. We can define the decoupling temperature of the neutrinos as the temperature for which vre = t and get re T~e c ~_ 1.8 × 101°°K.

(58)

The decoupling time at which this takes place is obtained from (30) and we get re td~ ¢ -- 0'3 see.

(59)

We see t h a t the temperature (58) is higher t h a n the annihilation temperature of the e e electrons Tann, which we define with kTan n = mec 2 or e Tan n = 5"9 × 109°K.

(60)

This justifies the use of the relativistic expression for the cross-section (50), the density (53) and the rate (55) in the deduction of the characteristic times. I t can be shown (Fowler and Hoyle, 1964) t h a t for T = T~n n the electron density ne reduces to about 80% of the value which would follow from the relativistic expression (53). For T = ½ Ten n this fraction becomes ~ 50 % and for T = ½ T~nnit is ~ 20%. This indicates t h a t the electron-positron pairs annihilate at temperatures around Tan n.e We see t h a t the values of Td¢ ere and Teann differ only b y a factor 3, which might be even less for reasons which will be discussed in Section IV. If we assume t h a t the electron-neutrinos decouple before the electron-positron pairs annihilate, the present value of the neutrino energy density can be established in the following way (Alpher, Follin and Herman, 1953; Zeldovich and Novikov, 1967). Because of the conservation law (12) the total entropy of all relativistic particles in a volume V 4

(61)

S = --~ ~ a T a V

is a constant during the expansion of the Universe. For constant g (e.g. ~ = 9/2 when m~,c 2 > k T :> mec 2) the temperature decreases as R -1, but this does not hold during electron-positron annihilation, when u decreases from 9/2 to 11/4. This means t h a t the temperature Tv of the photon gas which is interacting with the annihilating electrons, is increased relative to the temperature T v of the electron-neutrinos which ceased to interact and for which the entropy is separately conserved. I n this way we get the following relations for the temperatures before and after electron annihilation: Se + S r = - ~

4

-~ + 1 aTaV = -~a

7

Sr = --~ x 8 a T 3 V

4

= ~

:~g = S~,,

(62)

= S'~.

(63)

7

× 8aTa~V'

F r o m these relations we obtain for the present temperature T ~ of the electron-neutrino gas (Alpher, Follin and Herman, 1953), T~. -- - ~ -

T v--- l ' 9 ° K .

(64)

Under the assumption t h a t the muon-neutrinos decouple at a stage before the annihilation of the muons, the same method would provide the following temperature for the muon-neutrino gas (Harrison, 1968a) ~: 116 '~'/3 Tr, = [-~-3-] Tr" (65)

172

Neutrinos in the Universe

I t can, however, be pointed out (de Graaf, 1970b) that this is not the case in the usual picture of weak interactions, in which these neutrinos decouple at a later stage, after muon annihilation. The processes with muon-neutrinos are given in (36)-(41) and their crosssections are established in the same way as for the electron-neutrino processes. The muons are relativistic for k T > m . c 2 or T~ > 1.2 x 10a. (66) I n the relativistic case the averaged cross-section and the characteristic time for the scattering process of muon-neutrinos and muons is found from (50) and (52) after replacing me by m . . According to (47) a0 contains the factor m~ and we see from (47), (50) and (51) that the mass factors cancel in (52). We get therefore the same characteristic time v~. for v~ - # scattering as ~e in (54) for ve - e scattering, if the temperature satisfies (66). The corresponding @coupling temperature (58) takes, however, a lower value and therefore we have to calculate ~ g and the decoupling temperature T ~ c with the non-relativistic quantities in the following way: The non-relativistic cross-section is in the muon case also equal to (48) : a(~

+/~-)

= ~(~

+/~+)

= ~ 0 ~ 2 = ~oCO2

(67)

because

m~2

4gm~ a~ =

~h 4

=

2 ~0 me

and

~

E~

~% =

(m~c2) 2

~ --

2

~.

m~

The thermal motions of the muons do not significantly increase the cross-section as given in the rest frame of the muons. For the neutrino energy we use again the mean value (51) and for the muon density we take the non-relativistic Fermi distribution with (Chiu, 1968) 2 (1226) n f = ha (2~m~kT) a/2 exp ( - m~,c2/kT) ~- 4.5 x 1032 T~ I~ exp T----~ "

(68)

The relation for the characteristic time becomes in this case ~

= 7"5 T~ v2 exp ( 1226 / [ T 9 ]"

(69)

The processes like v~ + e- -+ re ~-/~- have a threshold value for the energy which makes the characteristic time very large for temperatures with )~T > m~,c9=. The characteristic time (69) increases much faster with decreasing temperature than (54). The muon-neutrinos thus become non-interacting in a much shorter period than the electron-neutrinos. They also decouple at a higher temperature, which can be obtained by taking (69) equal to the expansion time (30). This provides the relation [ 1226/~ T~ a/~ exp \ - - ~ 9 ] - 14.6

(70)

from which we find the @coupling temperature Tde¢ _ 1.2 × 10n°K.

(71)

The annihilation temperature of the muons is much higher, namely Ta~.n -~ 1"2 × 1012°K.

(72)

During their annihilation the muons transfer energy and entropy to the electrons and photons and also to the neutrinos. The muon-neutrinos start to decouple when only very few muons are left. This means that the resulting temperature T.. for the muon-neutrinos

T. DE GRAAF

173

becomes approximately equal to the temperature T~, for the electron-neutrinos and these temperatures satisfy the relation 116 ~1/3 ( 4 ~1]3 - ~ - ~ ] T~ < T,~ ~ T~e = \-i-l-] T~.

(73)

I n Fig. 2 we summarize the elementary particle processes which take place in the lepton era of the Universe and the properties of the leptons, according to the U F I theory. Processes which follow from other theories are also indicated and will be further described in the following Section.

Electromagnetic interactions

e

Weak

interactions

7 +e ~7,+e e++e---7 + X f

Annihilation at

5.9 x io 9 °K

re+ e ~ z,e+ e e++e-~-ye+ ~e

/

~' -~ -_

)~ -FjU,--7" + F "~

Y + y~7"+;,'+~" not important

I.L ~ e + ve + u ~+F~e+e

t ~ + /.4.---,.-7" + "f

Annihilation (1I I-2 x 1012°K r"L

u~,+ p. ~ u~,+ I1 F++ /.~=~vF+ ~l~

Ve+F ~ + F

~+e ~+e

e+ +

Decou~ling (~t Z/e 1"8 xlO°°K

e-..-~/j.++i.c

t.~i~-u,~?

Decoupling at 1"2 xlOll°K

FIG. 2

~V. ALTERNATIVE THEORIES OF WEAK INTERACTIONS AND THEIR CONSEQUENCES The foregoing calculations were based on the assumption t h a t the Universal Fermi Interaction describes the weak interaction processes under consideration. I n this theory the local interaction Hamiltonian (43) is given by the product of two charged currents, with one specific value for the coupling constant g, and conservation laws for quantities like the lepton numbers L e and L~. There have been proposals to modify this theory and the influence of such modifications on processes in the lepton era will be considered here. (a) Neutral lepton currents The experimental data (Perkins, 1969) permit rather substantial rates for semileptonie processes like ve + P - * ve + P, (74) v~ + p --* ~ + n + ~+,

(75)

which are only possible if the interaction Itamiltonian contains a contribution from neutral lepton currents (Albright, 1969). I n a non-local theory one can introduce a scheme with a neutral intermediate boson (Glashow, Iliopoulos and Maiani, 1970), which also predicts

174

Neutrinos in the Universe

processes like

(76) ~ ~- e---* ~# ~- e-;

v s ÷ e + ~ ~ ÷ e +,

(77) (78)

e- + e + ~ / ~ - + # %

(79)

I f t h e cross-sections for t h e processes (76)-(78) are t h e same as those for t h e processes (33) to (35), t h e m u o n - n e u t r i n o s b e h a v e in t h e same w a y as t h e electron-neutrinos. This m e a n s t h a t t h e characteristic t i m e s are t h e same for t h e r e l e v a n t p h y s i c a l processes a n d t h u s also t h e decoupling t e m p e r a t u r e s T ~ c a n d Tde ~ c for b o t h kinds of neutrinos. E l e c t r o n - n e u t r i n o s a n d m u o n - n e u t r i n o s would decouple in t h e same p e r i o d a n d w i t h t h e same speed, whereas in t h e c o n v e n t i o n a l t h e o r y t h e m u o n - n e u t r i n o s d e c o u p l e d m u c h faster t h a n the electronneutrinos. The r e l a t i o n {73) holds in this m o d i f i e d t h e o r y w i t h T~e = T ~ . (b) Non-conservation o] lepton number A n o t h e r p o s s i b i l i t y of getting equal t e m p e r a t u r e s is p r o v i d e d b y a t h e o r y in which a n oscillation process Ve ~- vs b e t w e e n b o t h k i n d s of n e u t r i n o s is i n t r o d u c e d (Pontecorvo, 1968; G r i b o v a n d P o n t e c o r v o , 1969). The e x p e r i m e n t a l results are n o t in c o n t r a d i c t i o n w i t h a c e r t a i n a m o u n t of n o n - c o n s e r v a t i o n of l e p t o n n u m b e r , which occurs if there exists a direct i n t e r a c t i o n b e t w e e n electron-neutrinos a n d muon-neutrinos. I n a stage of t h e l e p t o n era where m u o n - n c u t r i n o s no longer r e a c t with electrons a n d muons, t h e y m i g h t still be coupled to these particles t h r o u g h t h e oscillation process ve ~ vs. This m e a n s t h a t we a g a i n g e t similar decoupling t e m p e r a t u r e s a n d times for b o t h kinds of neutrinos. (c) Diagonal interactions with a different coupling constant R e c e n t t h e o r e t i c a l w o r k (Gell-Mann et al., 1969) indicates t h a t a calculation of t h e d i a g o n a l processes like re + e -* ve + e a n d % + / ~ - . % + ~u m i g h t involve different k i n d s of divergencies t h a n n o n - d i a g o n a l processes like fl-decay. This could i m p l y t h a t the corresponding coupling c o n s t a n t gD m i g h t be different from t h e coupling c o n s t a n t gp which is o b t a i n e d from fl-decay e x p e r i m e n t s . A s t r o p h y s i c a l d a t a (Stothcrs, 1970) i n d i c a t e t h a t gp can h a v e a value satisfying

0.lg~ < go ~< 10g~,

(s0)

whereas t h e C E R N n e u t r i n o e x p e r i m e n t (Cundy et al., 1970) p r o v i d e s the u p p e r limit

go < lSg~

(81)

a n d d e t e c t i o n w i t h r e a c t o r n e u t r i n o s gives (Reines a n d Gurr, 1970)

g . < 2g~.

(82)

A difference in t h e coupling c o n s t a n t for d i a g o n a l processes influences t h e b e h a v i o u r of t h e leptons d u r i n g the l e p t o n era of t h e Universe. I f we assume t h a t b o t h coupling c o n s t a n t s differ b y a f a c t o r C(gD = cg~), we find for t h e r e l e v a n t processes cross-sections which differ a f a c t o r c 2 from t h e ones in t h e U F I t h e o r y (a' = c2a). A n increase of t h e cross-section b y a f a c t o r c2 > 1 would i m p l y a decrease of t h e c h a r a c t e r i s t i c t i m e a n d also of the decoupling t e m p e r a t u r e (T'de¢ ---- C-~/aTd¢~). The d e r i v a t i o n of T ~ as given in Section I I I would be diff e r e n t if T~¢¢ became smaller t h a n T~,.. W e saw t h a t t h e electron d e n s i t y n e reduces to 80 % of its relativistic value, ff T --- T~,,e . F o r Td~¢' ~< T ~ , t h e n e u t r i n o s s t a r t to decouple a f t e r t h e y h a v e r e m o v e d p a r t of t h e energy of t h e a n n i h i l a t i n g electrons a n d positrons. I n

T. DE GRAAF

175

that case the decoupling temperature Tre would not be equal to (64), but it would satisfy 11 ]

Tr < T~e < Tr.

(83)

I n t h e U F I theory we found T ~ c - 3T~n ~. We would get T~¢¢ ~ T~nnfOr c -2/3 ~ 1/3 or c >~ 5. I t therefore follows that the results of Section I I I are also true in the modified theory with different coupling constant gD for gD > 5g~.

(84)

From the experimental upper limit (82) we see that the possible values of gD satisfy condition (84) and it can be concluded that the relation (64) is also valid in the modified theory with gD ~e g~.

V. DEGENERATE NEUTRINOS IN THE UNIVERSE

I n the foregoing sections it has been assumed that the neutrino background in the Universe consists of a non-degenerate Fermi gas for which the relations (23) and (24) are valid. I t is also possible to generalize the considerations about the hot big bang theory to a case with degenerate neutrinos. This has been described by Fowler et al. (Wagoner, Fowler and Hoyle, 1967; Fowler, 1970). I n other theories (which are not compatible with the hot big bang theory) it has been supposed that the Universe contains a completely degenerate Fermi gas of neutrinos or antineutrinos with Fermi energy E~. The first work on universal neutrino degeneracy has been done by Weinberg (1962) and Pontecorvo and Smorodinskii (1962), who also indicated experimental methods to find an upper limit for the Fermi energy of the neutrinos. I t can easily be derived that the particle density and the energy density of the neutrinos arc related to the Fermi energy in the following way: 47~

n~ = 3(hc)a E~ cm -a = 2.2 x 1012 E~ cm -a, s~ =

Yg

(he)a E 4 e V c m -a = 1"6 x 1012E~,eVcm -a,

(85) (86)

in which E r is expressed in eV. Weinberg stated that fl-spectra will be cut off at a certain energy below the maximum energy available, if there exists a Fermi gas of electron-antineutrinos. This follows from the fact that in the process n - . p + e- + ~e the final antineutrino cannot have an energy value which is lower than the Fermi energy of the Fermi gas. I n the same way the fl+-emission might be inhibited by a Fermi gas of electron neutrinos. The same effect might, however, also be due to a finite value of the neutrino mass. The most recent determination of the upper end of//-spectra by Bergkvist (1969) gives as a result for the upper limit of the Fermi energy E~ < 60 eV.

(87)

Another possibility for drawing conclusions about the value of E r is provided by the energy spectrum of ultrahigh-energy particles in cosmic rays, and by special assumptions about their origin and the interaction with a possible Fermi background of neutrinos. This has been done by Cowsik et al. (1964), who find E~ ~ 2 eV and by Konstantinov et al.

176

Neutrinos in the Universe

(1964, 1968), who find several possible values, depending on specific assumptions concerning the scattering cross-section and the cosmological behaviour of cosmic rays. The total cross-section for the inelastic scattering process v ~- N -~ l ÷ N* with production of a charged lepton 1 (e or/~) and a baryon resonance N* which m a y decay according to N * --* N + ~ , can be given as (Perkins, 1969) atot(~ + N) :" 0"6 x 10 -38 E (°) cm ~.

(88)

E (°) is the neutrino energy in the rest system of the nucleon, expressed in GeV, and can have values between 1 GeV and 15 GeV. I n the cosmic coordinate system, where the mean m o m e n t u m of the background particles is zero, a high energy proton with m o m e n t u m and energy Ep collides with a neutrino in the Fermi gas with m o m e n t u m ~ and energy E~. The neutrino energy in the rest system of the proton is equal to E~°) -- E~y(1 -- fl cos v~)

(89)

where 7 = Ep/ipc2;

fl = JPl c / E = v/c

and vq is the angle between ~ and ~. The inelastic scattering process is only possible ff E (°) is larger than the threshold value of the process, about 300 MeV. This implies that the high-energy protons can only react with the Fermi sea of neutrinos ff EFy(1 ÷ fl) > 300 MeV.

(90)

This condition holds for proton energies with Ep >~ 1.4 × 1017 E~ 1 eV

(91)

where the Fermi energy E F is given in eV. The experimentally determined energy spectrum of cosmic ray particles shows a rather sharp change in the high-energy region at about 1015 eV. Among other phenomena the spectral index ~ in the relation for the flux N ( E ) d E ~ E -~' d E increases from ~ 2 . 6 to 3"1. One of the possible explanations for this might be the fact t h a t particles of the highest energy are of metagalactic origin, whereas for E < 1015 eV they are created in the Galaxy. According to Ginzburg and Syrowatskii (1964) another possible cause of the described feature might be a change in the nature of the interaction of the ultrahigh-energy particles with nuclei in the atmosphere, for E ~ 1015 eV. Moreover, a change in the primary spect r u m might occur due to a change in the conditions for the disappearance of particles from the Galaxy at an energy of about 1015 cV. Yet another possible reason for the change in the spectrum of high-energy particles could be that for E > 1015 eV the protons lose their energy in inelastic scattering processes with a Fermi gas of neutrinos, whereas these processes cannot take place for E < 1015 eV because of condition (91). We see t h a t this is the case for a Fermi energy E r ~ 140 eV.

(92)

According to (86) this corresponds to a very large value of the neutrino energy density which is incompatible with the usual cosmological models. Therefore it is most probable that the change in the energy spectrum at 1015 eV has a different origin, e.g. one of the mechanisms t h a t was mentioned before. A very crude estimate of the Fermi energy E F can also be given by comparing the mean free p a t h of the protons in the neutrino sea with the distance L which the cosmic rays

T. DE GR~r

177

travel from their source to the Earth. For the mean free path X of protons with high energy E r the following expression can be found (de Graaf, 1970a) ~

1 0 a5

7-1E~4 cm

(93)

in which 7 = Er/Mpc 2 and E~ is given in eV. If we assume extragalactic origin of the ultrahigh-energy cosmic rays, in sources at the borders of the observable Universe, the distance L can be taken as ~ 10~s cm. No protons with Ep > 102° eV and thus y > 10 n are detected. Besides many other possibilities, one of the explanations for this fact might be that for 7 ~ 1011 the value of 2 b e c o m e s equal to ~ 1 0 2 s c m , or 10~4E~ 4 ~ 102s, which leads to a Fermi energy E F ~ 0.1 eV. (94) This can be considered as an upper limit for the Fermi energy, which has also been found by Rekalo (1970). I n the above treatment it has been assumed that cosmic ray particles with an energy larger than 102° eV do not penetrate to the Earth because of the interaction with the cosmic background of low-energy neutrinos. I t is, however, also possible t h a t no protons of energy higher than 1020eV are created in the sources or, if they are created, that annihilation takes place by means of other scattering processes, e.g. with galactic matter, or with photons in the cosmic black body radiation.

VI. TEE DENSITY OF COSMOLOGICALNEUTRINOS I n this section we want to consider which value or upper limit for the neutrino energy density follows from the theories which have been given in the foregoing sections (cf. de GraM, 1970a). (a) The upper limit ]or e~ /tom nucleochronology I n Section I I we found Eq. (16) for the expansion time of a radiation-type Universe. The age t o of the Universe which is defined as the time that elapsed after the singular state with R = 0, is found by substituting R = R 0 in this equation. This provides (Zeldovich and Novikov, 1967) to = H~I( 1 + , / ~ o ) - ' "

(95)

For the density of observable matter in galaxies, averaged over the Universe, Oort (1958) has given ~m ~ 3 × 10 -al g c m -a, which corresponds to an energy density em ~ 1.7 × 102 eV cm -a.

(96)

According to new theories on intergalactic high-velocity clouds (Oort, 1969; 1970) the total matter density might be considerably higher because of a large contribution of the intergalactic gas. I t might be as large as ½ ~c, where ~c is the critical density which was given in (9) and which corresponds to an energy density ec - 6-2 x 10a eV cm -a.

(97)

If we assume that the background of neutrinos has a much larger energy density than e m, the use of the radiation-type model of the Universe is justified and Eq. (95) can be given for its age. This possibility has also been considered by Pontecorvo and Smorodinskii (1962), Weinberg (1962) and Wataghin (1968). We see that for increasing density the age of the Universe becomes shorter. From the abundances of certain radioactive isotopes and their half-lives the age of solid components of the solar system can be established.

178

Neutrinos in the Universe

This method of nucleoehronology provides a time (Clayton, 1968; Sandage, 1968b)

t, ~_ 4.6 x 109y z - ~1- H ~

1

(98)

which is expressed here in units of the Hubble time (6). This time t, determines a minimum value for the age of the Universe. With (95) we conclude that from to ~> t, follows ~o ~< 4. Thus we get an upper limit for the total neutrino energy density : et°t < e t°t ~ 4ec --~ 2"5 × 104 eV cm -a.

(99)

A lower value of the upper limit for the total energy density can be obtained if one compares t o with the age of the Galaxy which can be established with methods of nucleocosmochronology. In a recent determination Fowler (1970b) gives for this age t G = (11.7 _+ 2) × 103 y. The relation (99) should also be modified, if the Hubble time differs significantly from (6). I f the background of low-energy neutrinos indeed had an energy density larger than ec its cosmological consequences would be important; the radiation-type Universe would be closed. I t is, however, impossible to detect such a neutrino background with any existing physical technique, if s~ would satisfy (99). (b) The energy density o/ neutrinos in the hot big bang theory We saw before t h a t under certain assumptions on the model of the Universe and the weak interaction processes it is possible to derive the present temperature T~ for the neutrino background which is given by (64). I t has been shown t h a t this value for the present temperature can be taken for both electron-neutrinos and muon-neutrinos. From (24) we find for the total energy density of both kinds of neutrinos and antineutrinos

etot ~-- 4 × ~ 7 aT4 = -~ 7 × [ - -4~ /~4/aaT¢~ O.4e, -~ O.l eV cm -a.

(100)

This value m a y be higher in the case of neutrino degeneracy. (e) The upper limit for s~ /rom fl.decay I n Section V we gave expression (86) for the energy density of a completely degenerate Fermi gas of neutrinos and the upper limit (87) which has been established from fl-deeay for the Fermi energy. From both relations the following upper limit for the energy density of electron-antineutrinos is found: s~e < 1.9 x 1019 eV cm -a.

(101)

The given value is much higher than the upper limit (99) and such a very high density is not in accordance with the cosmological model. (d) A possible value/or s~ /rom cosmic ray data With a number of rather crude assumptions we found the upper Hmit value (94) for the Fermi energy of the neutrinos. According to (86) this corresponds to the following energy density limit for e, ~< 1-6 x l0 s cV cm -a (102) which is also much higher than the upper limit (99).

(e) The energy density o/ neutrinos /rom main sequence stars The foregoing energy density values of cosmological neutrinos can be compared with the energy density of astrophysical neutrinos, which originate from physical processes in

T. DE GRAA~

179

s t a r s or o t h e r objects. A n u m b e r of possible sources of n e u t r i n o s h a v e b e e n s u m m a r i z e d b y F o w l e r (1966) who e s t i m a t e d t h e n e u t r i n o e n e r g y o u t p u t of stars, quasi-stellar objects, r a d i o g a l a x i e s a n d collapsing stars. H e concluded t h a t t h e n e u t r i n o e n e r g y d e n s i t y in t h e U n i v e r s e can o n l y be of cosmological significance, if its origin lies in cosmological processes. To i l l u s t r a t e t h i s we consider m a i n sequence stars like t h e Sun, where a smaU f r a c t i o n ( ~ 4 % ) of t h e e n e r g y is e m i t t e d in t h e f o r m of n e u t r i n o s which are p r o d u c e d in t h e t h e r m o n u c l e a r processes of t h e p p - a n d CN-ehain. T h e p h o t o n l u m i n o s i t y of t h e S u n is (Chin, 1968) L r = 3-9 × 108a erg sec -1 ~- 2.5 × 1045 eV sec -1 (103) a n d t h e n e u t r i n o l u m i n o s i t y is therefore a p p r o x i m a t e l y L , ~ 1044 eV sec -4.

(104)

W e assume t h a t all stars of t h e G a l a x y (with t o t a l n u m b e r Ns ~ 10 ~x) p r o d u c e this a m o u n t of e n e r g y in t h e f o r m of n e u t r i n o s d u r i n g t h e whole life t i m e of t h e Universe, for which we t a k e H~ 1 ~ 1"3 x 10 ~° y ~ 4 × 10 ~7 see. Because of t h e v e r y small cross-sections for t h e n e u t r i n o processes t h e n e u t r i n o s are n o t a n n i h i l a t e d , b u t a c c u m u l a t e in t h e Universe. T h e o b s e r v a b l e U n i v e r s e contains N o ~ 3 x 109 galaxies a n d its v o l u m e can be t a k e n as V ~ 7.3 x 1084 cm a (Allen, 1964). I n this w a y we g e t t h e following crude e s t i m a t e for t h e e n e r g y d e n s i t y of t h e n e u t r i n o s f r o m m a i n sequence s t a r s : e~ ~" L , N s N c ; H ~ I V

- 1 "~

10 -a eV cm -a.

(105)

T h i s v a l u e is v e r y small in c o m p a r i s o n w i t h t h e densities which were e s t a b l i s h e d for cosmological neutrinos. I t is n o t m u c h smaller t h a n t h e d e n s i t y of p h o t o n s f r o m starlight, for which we can t a k e ( K o n s t a n t i n o v a n d K o c h a r o v , 1964) sy --~ 10 -2 eV cm -a.

(106)

T h e r e s u l t s for t h e different possible e n e r g y v a l u e s are s u m m a r i z e d in T a b l e 2. TABLE 2. The Energy Density of Neutrinos in the Universe and Other Energy Density Values Origin ~Nucleochronology Hot big bang theory Experiments on fl-decay Cosmic ray data Main sequence stars Matter in galaxies Photons from starlight Photons in the "cosmic black body radiation" Critical density

Energy density in eV cm -3 sv ~ 48c~2"5 × 104 sv '~ 0"1 sv ~ 1"9 × 10TM s~ ~ 1-6 × l0 s s, ~ I0 -~ sm "~ 1"7 × 10~ e r ~_ 10-2 s~ ~ 0.25 so --~6.2 × 10~

I wish to a c k n o w l e d g e t h e helpful discussions w i t h W . A. F o w l e r , E . R . H a r r i s o n , M. R e e s a n d G. S t e i g m a n d u r i n g t h e p r e p a r a t i o n of this review, t h e facts of which were also discussed a t t h e Conference on A s t r o p h y s i c a l A s p e c t s of t h e W e a k I n t e r a c t i o n s a t Cortona, I t a l y (June 1970).

180

Neutrinos in the Universe

REFERENCES .ADLER,1~.,BAZIlq,M., and SCHI~'FER,M. (1965) Introduction to General Relativity, McGraw-Hill, New York. ALBRmHT, C. H. (1969) On the Existence o] Weak Neutral Currents, CERN preprint TH 1066. ALL~.~, C. W. (1964) Astrophysical Quantities, The Athlone Press, London. ALPHER, R. A., FOLLIN, J. W., and HERMAN, R. C. (1953) Phys. Rev. 92, 1347. BAHCXLL,J. N. (1964) Phys. Bey. 186, B 1164. BERGKVIST, K. E. (1969) In Proceedings o] the Topical Con/erence on Weak Interactions, Report CERN 69-7, Geneva. BUGAYEv, E. V., KoTov, Yu. D. and ROZENTAL,I. L. (1970) Kosmicheskie myuony i neitrino, Atomizdat, Moskva. C~t~c, H. Y. (1968) Stellar Physics, Blaisdell Publ. Comp., Waltham, Mass. CLAYTON, D. D. (1968) Principles o/Stellar Evolution and Nucleosynthesis, McGraw-Hill, New York. COWSIK,R., P.~L, Y., and TA:NDON,S. N. (1964) Phys. Letters 18, 265. CUNDY, D. C. etal. (1970) Phys. Letters 81B, 478. DAUTCOURT, G. and WALLm, G. (1968) •ortschr. der Phys. 16, 545. DE GRAAY,T. and TOLHO]~K,H. A. (1966) Nucl. Phys. 81,596 and A 99, 695. D~. GRAAE,T. (1970a) Astronomy and Astrophysics 5, 335. DE GRXAI~,T. (1970b) Lettere al Nuovo Cim. 4, 638. DOROSKKEWCH, A. G., ZELI)OVIC~, YA. B., and I~OVIKOV,I. D. (1968) Sov. Phys. J E T P 26, 408. DOROSHKEWC~, A. G., ZELDOWCJZ,YA. B., and NOVIKOV,I. D. (1969) In Problems o/Theoretical Physics, Nauka, Moskva. FEYNMA~, R. P. and GELL-M~N~, M. (1958) Phys. Rev. 109, 193. FIELD, G. B. (1969) Rivista del Nuovo Cim., Serie 1 1, 87. FOWLER, W. A. and HOYLV., :F. (1964) Astrophys. J. Suppl. ~o. 91. I~(~WLER, W.A. (1966) In Proceedings o/ the International School o/ Physics "Enrico Fermi", 1965, Academic Press, New York. FOWLER, W. A. (1970a) Comm. on Astrophys. and Space Physics 2, 134. FOWLER, W. A. (1970b) In George Gamow Memorial Volume, to be published by Columbia University Press, New York. FRIED~A~N, A. (1922) Z./. Phys. 10, 377. FRIED~A~N, A. (1924) Z./. Phys. 21,326. GAMOW, G. (1946) Phys. Rev. 70, 572. GAMOW, G. (1949) Rev. Mod. Phys. 21,367. GxMow, G. (1956) Vistas in Astronomy, ed. A. BEER, 2, 1726. G]~LL-MAN~,M. et al. (1969) Phys. Rev. 179, 1518. GII~ZBURG,V. L. and SYROW,TSKIr, S. I. (1964) The Origin o/Cosmic Rays, Pergamon Press, Oxford. GLASHOW, S. L., ILmPOULOS, J., and MAIANI,L. (1970) Weak Interactions with Lepton-Hadron Symmetry, preprint Lyman Lab. of Physics, Harvard University. GRIBOV, V. and PO~TECORVO, B. (1969) Phys. Letters 28 B, 493. H~RRISON, E. R. (1968a) Phys. Rev. 167, 1170. Ht~RR~SO~, E. R. (1968b) Phys. Today 21, 31. HAYASI~, C. (1950) Progr. Theor. Phys. Japan 5, 224. HURBLE, E. (1929) Proc. Nat. Acad. Sci. Wash. 15, 168. KO~ST~T~OV, B. P. and KoC~ROV, G. E. (1964) Soy. Phys. J E T P 19, 992. KOI~STANTII~OV,B. P., KOCHAROV,G. E., and ST~I~BU~OV,YU. N. (1968) Izvestiya Akademii Nau~ SSSR, Seriya Fizicheskaya 32, 1841. Kvc~ow~cz, B. (1966) The Bibliography o/the Neutrino, l~uclear Information Center, Warsaw; Gordon and Breach, New York, 1968. KUcHowIcZ, B. (1968) Nuclear and Relativistic Astrophysics and Nuclidic Cosmochemistry: 1963-1967, l~uclear Information Center, Warsaw. Kuci~ow~cz, B. (1969) l~ortschr, der Phys. 17, 517. LANDAU, L. D. and L~FSC~TZ, E. M. (1958) Statistical Physics, Pergamon Press Inc., New York. M~RSH~K, R. E., RI~ZUDDn~ and RrA~, C.P. (1969) Theory o/ Weak Interactions in Particle Physics, John Wiley and Sons, New York. MARX, G. (1967) Acta Phys. Hung. 22, 59. M~SNER, C. W. (1968) Astrophys. J. 151,431. OOI~T, J. H. (1958) In Proceedings o] the Solvay Con]erence on the Structure and Evolution o/the Universe, Brussels.

T. D~ G R ~ F

18I

OORT, J. I~. (1969) Nature 224, 1158. OORT, J. H. (1970) Astronomy and Astrophysics 7, 381,405. PE~ZIAS, A. A. and W~so~, R. W. (1965) Astrophys. J. 142, 419. PERKINS, D. H. (1969) In Proceedings o] the Topical Con/erence on Weak Interactions, Report CERN 69-7, Geneva. PONTECORVO, B. and S~ORODINSXII,YA. (1962) Sov. Phys. J E T P 14, 173. PONTECORVO, B. (I968) Sov. Phys. J E T P 26, 984. R~I~ES, F. and GURR, H. S. (1970) Phys. Rev. Letters 24, 1448. R~ALO, M. P. (1970) Ukrainskii Fiz. Zhur. 15, 923. SA~DA(~, A. (1968a) Astrophys. J. 152, L 149. SANDAGE, A. (1968b) The Time Scale ]or Creation; in Galaxies and the Universe (ed. L. WOLTJ~R), Columbia University Press, New York. SHVARTSMA~,V. F. (1969) Soy. Phys. J E T P Letters 9, 184. STCrHERS, R. :B. (1970) Phys. Rev. Letters 24, 538. SU~:¢AEV, R. A. and ZELDOVICg, YA. B. (1969) Comm. on Astrophys. and Space Physics 1, 159. T~OR~E, K. S. (1967) Astrophys. J. 148, 51. TOL~N, R. C. (1934) Relativity, Thermodynamics and Cosmology, Clarendon Press, Oxford. WAaO~R, R. V., FOWLE~, W. A., and HOrLE, F. (1967) Astrophys. J. 148, 3. WATA(~HI~,A. (1968) Nuovo Cim. 55B, 258. W~I~BERO, S. (1962) Phys. Rev. 128, 1457. Z~LDOWC~, YA. B. and Novr~ov, I. D. (1964) Soy. Phys. Uspekhi 7, 763. ZELDOVIC~, YA. B. and NOVLKOV,I. D. (1965) Sov. Phys. Uspelchi 8, 522. ZELDOWC~, YA. B. (1965) Survey o/ Modern Cosmology, in Advances in Astrophysics, Academic Press, New York. Z~LDOVIC~, YA. B. and NOWKOV, I. D. (1967) RelyativistsIcaya AstrofiziIca, Nauka, Moskva.