recombination of the primeval plasma and light -inos

Volume 144B, number 1,2 PHYSICS LETTERS 23 August 1984

RECOMBINATION OF THE PRIMEVAL PLASMA AND LIGHT -INOS

P. SALATI Universit~ de Savoie, Chambkry, France and LAPP, Annecy-le- Vieux, France

and

J.C. WALLET LAPP. Annecy-le-Vieux, France

Received 10 April 1984

The recombination of neutral hydrogen in the early universe is reinvestigated taking into account light neutral fermions, stable or radiatively unstable. When these fermions are stable, their main effect is to increase the expansion rate of the universe, and to increase the fossilized ionization x e of matter. Big bang nucleo-synthesis provides density constraints on the baryonic components and if we assume that the universe is closed with light neutral fermions, we can set limits upon Xe: 4 × 10- 4 < x e < 2 × 10 -3 instead of the previous baxyon-dominated universe result: 3 × 10 -s < x e < 3 X 10 -4. If the light neutral fermions decay radiatively, the emitted photon is the UV-range and reionizes the neutral matter. We point out that matter can be completely reionized at a redshfft Z ~ 100 for radiative lifetimes in the range 102°-1024 s. Supersymmetry provides us with such a light "ino". The reaction photino -o photon + gravitino exhibits the good relation between the photino lifetime and the ionizing photon energy.

If we admit that the standard hot big bang model

(see e.g., refs. [1,2]) is essentially correct, the matter of the very early universe was very hot and dense. When the temperature was over 104 K, it was completely dissociated into electrons and ions like protons and ionized helium for the most part. As the universe expanded, it cooled and when its temperature reached over 4000 K at a time of about 105 y, the baryonic matter combined with electrons to form neutral matter. The recombination of the primeval plasma has

been studied [3] and the details of the transition between ionized and neutral matter are of great interest.

Recombination plays a crucial role in galaxy formation [4] because the perturbations in matter density cannot develop or grow to form galaxies in an ionized medium, due to the Thomson friction of electrons upon the thermal radiation. So the recombination period is determinant for the evolution of perturbations which lead to gravitation-bound systems.

As radiation scatters onto the matter via the quasi- elastic Thomson diffusion the small scale angular in-

0.370-2693/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

homogeneities of the cosmic background radiation (CBR) depend directly on the recombination features.

Electrons and protons catalyse the formation of molecular hydrogen [5].

Finally, an intergalactic medium (IGM) may well be constituted by the relics of recombined matter which have escaped from gravitational collapse. So, observation of the IGM is a good way to test the theory and to go back into the past at the time matter recombined.

Independently, it turns out that two experimental observations are in conflict. The so-called "missing- mass" problem arises from the fact that the observed mean baryonic density is about 3 × 10-31g/cm 3 although the universe appears to have a global mean

density of Ptotal ~ Pcritic = 5 X 10 -30 (g/cm 3) [Ho/ (50 km/s Mpc)] 2 (H 0 is the present value of the Hubble constant), due to the observation of its quasi- flatness. Particle physics may give a solution to this problem. Non minimal conventional GUTs predict the existence of low mass neutrinos [6] whereas super-

61


symmetric models [7] include in their spectrum light weakly interacting "-inos", as photinos or gravitinos. Such weakly interacting particles could have easily escaped detection for the moment and may provide the missing contribution to the total mass density. It is worth noticing that such a light neutral fermion with a mass of about 5 0 - 1 0 0 eV is sufficient to solve the missing mass problem. Another effect of that hypothetical hidden non baryonic matter would be to in- fluence strongly the formation of the galaxies.

In this paper, we examine the recombination of neutral hydrogen in the presence of stable or unstable light neutral fermions (the so-called "inos"). In the first part, we recall the main features o f the recombination and compute numerically the solution of the differential equation governing it if there is an "-inos" population in addition to hydrogen. Previous calcula- tions (Peebles, Zeldovich [3] ) assumed that baryonic matter was alone, and gave, with Ptotal ~ 5 X 10 -31 g/cm 3, a residual ionized fraction of baryons of over 10- 4. The addition o f light-inos increases the expansion rate of the universe, changing significantly the recombination: the residual ionized fraction of baryons is increased by a factor O(10) compared to previous results. We compare afterwards our results with two approximate solutions. In the second part of the paper, we study the recombination in the presence of unstable light "-inos" which radiatively decay into a light- er particle and a photon. It turns out that the emitted photons are in the ultra violet range and can reionize the recombined hydrogen. We point out that matter can be completely reionized at a redshift Z ~ 100 for radiative lifetimes as large as 1022 s. Some astrophysi- cal data suggest that the IGM is ionized [8] ~ . The hy- pothesis of the photoionization of the IGM by radiatively decaying neutrinos or photinos has been previously formulated by Sciama [9] who dealt with lifetimes o f about 1027s owing to a "recent" (Z ~ 3.5) ionization. The study of a possible photoionization of the IGM at the earlier epoch of the recombination is also of interest.

Recombinat ion with stable light-inos. We first deal here with the general scenario of the recombination of neutral hydrogen. We have to study the evolution in time of the chemical reaction

+x See also Sciama [9].

electron + proton ~ neutral hydrogen + pho ton . (1)

When the universe expands, the temperature decreases and all the reagents are diluted. This induces a quenching of the reaction and the ionized fraction x e of baryonic matter

x e = np/n B = ne/n B (2)

is frozen at some fixed value. The main features of the recombination have to

be recalled: (i) When the temperature is very high, reaction (1)

is in thermal and chemical equilibrium. We get what is called the Saha equilibrium. The concentration of the different constituents is given by the constant of the reaction:

nenp /n H = ( 2 ~ m e k T ) 3 / 2 e - B 1/k T /h3 , (3)

where he, np and n H are respectively the densities of electrons, protons and hydrogen atoms. B 1 is the binding energy of the ground state, m e is the electron mass. The ionized fraction x e obeys the relation:

x 2 / (1 - Xe) = ( 27rmek T)3/Ze-Bl /k T /nB h3 = A ( T) , (4)

where n B is the baryon density and decreases like T 3 when the universe expands. It is noticeable that at fixed T, the lower nB, the larger x e. In our case n B is suf- ficiently small to explain the high value o f x e even for a low value of the temperature. For instance:

x e ----- 99% when T = 5000 K , (5)

corresponding to an energy k T ' " 0.5 eV small compared to the binding energy B 1 = 13.6 eV of the ground state.

So, at a high temperature, the different processes that allow the reagents to interact are very fast. It implies that the Saha equilibrium is respected and that x e is given by its equilibrium value:

Xe = Xe Saha = 1 [ -A + (.4 2 + 4A) 1/z ] . (6)

(ii) The recombination proceeds only via the excited states. A direct recombination toward the ground state is largely inhibited because, basically, the UV photon produced in such a reaction is not degenerated into a lower energy one, but keeps its frequency. If such a process occurred, the UV radiation background would grow and the equilibrium would be shifted to-

62


ward the ionization, due to a large increase of the pho- todissociation of the ground state. So, that process would be stopped and inversed.

(iii) The different excited states are in thermal contact with the cosmic background radiation. So, i f N n l refers to the populat ion of the state (n , / ) , one has

Nnl/N2s = (2 /+ 1) exp[(B n - B 2 ) / k T ] . (7)

B n is the binding energy of state n, l. The main conse- quence is that the n = 2 state is the most densely popu- lated among the excited states.

(iv) The n = 1 and n = 2 states are in close kinetic equilibrium. The 2p + 1 s Lyman-a transition has a very high rate. So the ratio N2s/N1 s goes towards its equilibrium value n a with a relaxation time some 20 orders of magnitude smaller than the universe expansion time. n a is the average number of photons per mode in the Lyman-a line,

N2s/Nls = n a = e(Bz BI )/k T + R K ,

K = Xa3/167rH. (8)

R is the net rate of Lyman-c~ photon product ion per unit volume. ~'a = 1215 A is tile wavelength o f a Lyman-a photon and H is the Hubble constant. It is noticeable that N2a is very small compared to N 1 s- Thus, when a net recombinat ion occurs, the final result is the creation of neutral hydrogen in the ground state. Even if the recombinat ion transits via the n = 2 state, not only the populat ion of that state remains negligible compared to the ground state one, but also

the ratio N 2 J N l s is always given by n a. (v) Finally, at each time an e l ec t ron -p ro ton pair

recombines, an n = 2 a tom is formed. That state evolves rapidly for the abovementioned reasons:

it can be photodissociated into a new electron-proton pair;

it can decay into the ground state and a Lyman-~ photon;

it can deexcite via a two-photons transition with a rate given by Als2s = 8.227 s - 1.

The equation describing the evolution o f x e reads:

_ d x e / d t = C[acnBX2e _ 13c( 1 _ Xe)e(B~ - B 1 ) I k T ]

C = (1 + Als2sKnls) / [1 + Knls(/3 c + Als2s)] . (9)

a c is the recombination factor of an e l ec t ron -p ro ton

pair toward the excited states of hydrogen. ~c is the photodissociat ion factor of these excited states, mainly of the n = 2 one. The inhibition factor C reflects the three choices a n = 2 state has to evolve. 1 stands for the Lyman-a transition. KnlsAls2s corresponds to the two-photon transition. Kn ls13 c describes the ionization of the atom. That inhibition factor C simplifies:

C~--- Als2s/(Als2s + t3c) (10)

just only because the two-photon transition alone is enough to drive the recombination and outweights the Lyman-a transition.

Now, we turn on to the discussion of the results of the numerical integration of eq. (9). We have consider- ed a present Hubble constant H 0 equal respectively to 50 km/s Mpc and 100 km/s Mpc, and we have varied the mass fraction g2 B of the baryons

FZB = PB/Pc (11)

as indicated on fig. 1 a and 1 b. The lower value corresponds to the visible baryonic mass density of 3 X 10-3 t g/cm 3 observed. As the universe is quasi-fiat, we have assumed that

Ptotal ~ Ocritic = 2 X 10-29(g/cm 3)

X [H0/(100 km/s Mpc)] 2 (12)

Fig. l a and Ib show the evolution of the ionized fraction x e with the temperature. Peebles found that for ~2 B = 1 and Ptotal = 1.8 X 10-29 g/cm 3 the residual ionization had a frozen value of 2 X 10 - 5 . It is note- worthy that for about the same total mass density, but for a low ~B of 1.5%, the residual ionized fraction x e is equal to 1.3 X 10 -3 , two orders of magnitude larger. The behaviour of the frozen value o f x e versus H 0 and g2 B depends on two properties:

When Ototal increases, the Hubble constant at every time increases. The faster the universe expands, the sooner the chemical reaction freezes and the higher the residual ionization is;

When g2 B decreases, the chemical equilibrium is shift. ed toward the ionization, so x e increases. This explains the fact that x e is multiplied by over 70 when g2 B decreases from 1 to 0.015 w i t h H 0 = 100 km/sMpc. So the residual ionization lies between 10 - 5 and 10 - 3 de- pending on H 0 and ~2 B.

Finally we present here a simple approximation to the recombination of neutral hydrogen and we corn-

63

Volume 144B, number 1,2 PHYSICS LETTERS 23 August 1984 lla I0-I

lO-Z

I0-~ ~ j ~ / / H°=1OOkm/S/Mpc

10 % ~ ~ 110 i i i i 0.5 1.0 1.5 2.0 2.5 3.0 35 4.0 t,.S

Temperature in 10 3 K

1 b i J i ,

10 -1 ' / 10-2

Ho=5Okm/S/Mp C

10-~ ~ 1.0

10_ 5 ~ j i 0._~ 1 ~ . 5 2'.0 2L.5 3.0 3.5 4.0 4.5 Temperature in 10 ) K

Fig. 1. The ionized baryonic matter fraction x e is p lot ted against the universe temperature for several values of the baryonic mass fraction I2 B and two values of the present Hubble constant: Ho = 100 km/s Mpc (a) and H o = 50 km/s Mpc (b).

pare it to a previous one used by Zabotin and Nasel'ski] [10] (see fig. 2a). Basically, we have to compare two characteristic evolution times:

The Saha equilibrium x e evolves with a relaxation time given by

7"equilibriu m = ( - d In x e Saha/dt)- 1

= [(2 -- X e Saha)/(1 -- x e Saha)]

X [H(B1/kT- 3/2)] - 1 , (13)

The recombination process has a characteristic time given by

rproces s = ( - d In Xe/dt )- 1 = (Co~cnBXe)- 1

(1 + ~c/Als2s)l(~cnBXe. (14)

1

10 -1

lo-z

10-3

10-~

lo-s

ximltion a ~ , 1 I~ / "

/Saha equilibrium ;

Ho=SOkm/S/Hpt ./ / a ~B =0'06 /

015 170 1.5 21.0 2.5 310 31.5 /,I.0 ~..5 Temperature in 10 3 K

1 QB=0"06

b 3.9 t,.0 t,.1 4.2 t,.3


Fig. 2. (a) The ionized fraction x e is plot ted against the universe temperature, and four behaviours of Xe are shown: the Saha equilibrium; the numerical integration using a resolu- tion of the differential eq. (9) on a computer; the approximation (a) done by Zabotin and Nasel'skl]; and our approximation (b). (b) The ratio R of ~'process/Zequilibrium is plot ted against the universe temperature. Note that when the age of the universe grows, its temperature decreases and R increases.

At a high temperature, when 7.process < 7"equilibrium, the recombination has plenty of time to follow the evolution of the equilibrium. So x e is given by the Saha value x e equilibrium of formula (6). Then, as the universe expands, it cools down. At a temperature T F ranging from 4000 to 4300 K the two times are equal and the freezing occurs. We remark that the quenching appears at a high value XeF. Fig. 2b shows the increase in time of the r a t i o 7.process/Tequilibrium. The reaction cannot follow the Saha equilibrium, it decouples. The equation o f evolution simplifies to

-dXe/dt ~ C%nBx2e . (15)

64


and the approximate solution after decoupling is given by

Xe(T < TF)

xFOtc(T4 = 1)nB(r 4 = 1) - F 1+ - Xe Hubble(T 4 = 1)

f TF dT 4 ]-1 X j - - - (16)

T4 1 +~T 4 e -B2/T4! '

where a =/3°/Alszs, t3 ° = (/3c/T4)eBz/T4 and T 4 is the temperature expressed in units of 104 K. Formula (16) shows clearly that x e reaches an asymptotic value as the universe cools. We get a residual ionized fraction

Fig. 2a is a plot of the numerical integrated x e, the Saha equilibrium, the approximation (a) used by Zabotin and Nasel'ski'i and our approximation (b) versus the temperature, for f2 B = 0.06 and H 0 = 50 km/s Mpc. Between 2500 and 3500 K, the two ap- proximations (a) and (b) are equal but give a result two times larger than the integrated solution. Below 2500 K, the approximation (a) is smaller than the integrated solution but approximation (b) predicts the good value.

Finally, if we assume that the universe is closed with light neutral fermions that are stable, we can set a lower limit upon x e. Big bang nucleosynthesis provides density constraints [11] on the baryonic components. A lower limit on the amount of baryonic matter in the universe can be derived from combined D and 3He abundances, ~2 B [H0/(100 km/s Mpc)] 2 0.01 and the observed abundances of 4He (mass fraction Y~< 0.25), D and 7Li result in an upper limit to baryonic matter density ~2 B [H0/(100 km/s Mpc)] 2 < 0.034. If light -inos are introduced to close the universe, the ionized fraction x e is always larger than 4 X 10-4;

4 × 10-4 ~Xe ~ 2 X 10 -3 (17)

instead of the previous baryon-dominated universe result

3X 10-5 ~<~Xe ~<3 X 10 -4 . (18)

Recombination in the presence o f unstable light- inos. Now, we assume that, in addition to the hydrogen, there is a population of light unstable -inos with mass m in the range 30-100 eV, whose mass density

would provide the missing contribution to close the universe. That population went out of thermal equilibrium at some high temperature T d larger than 1 MeV, when the rate of interactions with matter fall below the expansion rate of the universe. Its number density is frozen at the value corresponding to a gas of 2 helicity-states light fermions

n(T~)/T 3 = (3/27r 2) ~(3) (k~c) 3 cm -3 K -3. (19)

If T d ~ 1 MeV, which is the case for the neutrino or the supersymmetric partner of the photon, the photino. After the departure from the thermal equilibrium the population begins a free expansion and its number density is diluted by the cosmic expansion and slowed down by the decays. We assume that the neutral fre- talons can decay principally into a photon and another fermion of mass much smaller than m,

f ~ f ' + 7 , (20)

with energy of the photon = m/2. To avoid problems with cosmological data, we assume that the lifetime r of f is larger than 1022 S. With the range of mass under consideration, smaller lifetimes may contribute too much to the various photon backgrounds (especially when 1012 s <~ r ~< 1022 s) or distort the 3 K cosmic background radiation (104 < r < 1012 s) [2,12].

At the epoch of the recombination the population of those nuetral fermions declays slowly at rest. This acts as an extra source of UV photons which can strongly alter the kinetics of the recombination and even induce a complete reionization of the primeval hydrogen at Z ~ 100 if the decay rate I" = 1/~- is not very far from its upper value 10- 22 s- 1. It turns out that the photo-ionization cross-section o I of hydrogen is very large. Even for a 50 eV photon, its value is 10-19 cm 2, some 5 orders of magnitude larger than the Thomson cross section. As soon as a UV photon is produced by a decaying light-ino, it is absorbed by a neutral hydrogen atom. Once the recombination becomes appreciable, the mean free time of an ionizing photon is much smaller than the characteristic evolution time of the ionized fraction x e, as can be seen by evaluating the ratio

tx e/tionizing photon

> tu/aO/(1/olcnls ) = 4.5 X 107~2B(1 -- Xe)T 3/2 >~ 1. (21)

65


We conclude that the ionizing part of the radiation spectrum is in close thermal contact with matter and that a kinetic equilibrium is established among the ionizing photons whose density is n I. They are in- volved in 4 processes:

(i) They are created by the decay of the f particles whose density is N. Their production rate per unit volume is FN.

(ii) They come from the recombination on the ground state of e - p pairs. The rate of the process is ~lsCe 2 .

(iii) They are absorbed in photoionizing a neutral hydrogen atom. The corresponding absorption rate per unit volume is oien 1 sni -

(iv) They are red-shifted with a rate given by (B 1 / kT)n I due to the general expansion of the universe,

When the kinetic equilibrium is achieved, the number density of ionizing photons is given by

n I = (als n2 + FN)/(B1H/kT + olcnls ) . (22)

The new differential equation driving the recombination reads now as follows:

-dXe/dt = right hand term of eq. (9)

+ (1 4 °Icng(1 --Xe)~ - 1

X (alsnB x2 - °IC(1-Ze) I"N) (23) •

To exhibit the effects of the new term on the solution, we present the following rough approximation: the ratio p = CrXCnB(1 - -Xe) / (B1H/kT)~ 108 T 5/2 (1 - x e ) is large once the recombination is appreciable. So we approximate the factorized term into brackets by 1/p. The second term in (23) that describes the reionization can be cast into the form:

(PN/nB) [(ao-25~B/F)X2e/(1 -- Xe) -- 1] . (24)

We remark that for P ~ O(10- 24~B)S- 1, (24) can be approximated by -FN/n n just because 10 25 ~2 B X p - 1 Xe 2 (1 -- X e ) - I fails very quickly below 1. On the other hand, the term containing/3 c is suppressed by the exponential factor e -B 1/k T so that the leading contribution to the reionization is given by -FN/n. At some temperature between 1000 and 3000 K, the reionization goes over the recombination and x e in-

creases. When F < O(10 _24 ~B)S- 1 the behaviour o fx e looks like the classical one and the smaller F, the later the reionization. Finally, when F reaches the ex- treme 10 _30 s - 1, the reionization of the hydrogen becomes negligible.

Particle physics could provide us with such a light radiatively unstable "-ino". We remark that the above considerations cannot generally apply to a hypothetic heavy neutrino because the rate for its radiative decay

Pheavy ~ ~ + Plight (25)

is strongly suppressed by GIM factors. In the consider- ed range of mass, its radiative lifetime would be larger 1033 s, excluding any possible reionization. (However, note that the Zee model [13] by the unnatural choice of a suitable Higgs sector, predicts lifetimes between 1020 and 1023 s.) Supersymmetry seems to us more appealing. The supersymmetric companion of the photon, the photino ~,, might be an example of the "-ino" we have dealt with previously. Its mass can be small and it couples to the photon and to a massless goldstino, or a light gravitino if the goldstino has been eaten up by the +! helicity states of the gravitino via --2 the super-Higgs mechanism [ 14]. Although a class of models predicts a heavy gravitino with a mass of the order of the SU(2) X U(1) breaking scale, the hypo- thesis of a light gravitino with a mass < 1 eV, cannot be presently ruled out. Taking the usual ~-3,-gravitino vertex gives the photino lifetime

r = 1.65 X 1030 s [d/10000GeV 2] 2 [1 eV/m T ]5, (26)

where x/-ffis the supersymmetry breaking scale, d and the mass of the gravitino m3/2 are related by the formula:

m 3 / 2 = K d / x / ~ , w i t h K = 4 X 10 - 1 9 G e V -1 .(27)

The best experimental lower bound on the scale x/-ff comes from the non-observation of the reaction

e + + e - ~ photino + antigravitino + photon (28)

at CELLO. x/dcannot be smaller than 100 GeV [15] and so could lie near the value of the SU(2) X U(1) breaking scale, x/d "= 100 GeV gives a photino lifetime

z = 1.65 × 1020 s [100eV/m~-] 5 (29)

and is of interest for the reionization of the primeval

66


1 s

10-I . S

10-,1°-3 ° a 1 I. I - - I i - - l ~ I

0.5 1.0 1.5 2.0 2.5 3.0 3.5 t~.0


.. Lifetime

x ~ I0-I ~ ~ ~ x I02ZS

IO-= 100,m/S,'Mp,

10-3 b

0t.5 i _115 i 21.5 , L _ _ 0 1.0 2.0 3.0 3.5 /*.0 Temperature in 10 3 K

Fig. 3. The ion ized bay ron i c ma t t e r f r ac t i on x e is p l o t t e d against the universe temperature for several values of the life- t ime of the unstable "- ino", and for two different conditions: (a) maximum baryonic abundance: H o = 50 km/s Mpc and s2 B = 0.14; (b) min imum baryonic abundance: H o = 100 km/s Mpc and ~2 B = 0.01

hydrogen. We have numerically integrated eq. (23) whenH 0 = 50 km/sMpc and ~t3 = 0.14, a n d H 0 = 100 km/s Mpc and ~2 = 0.01, and varied the "-ino" lifetime as indicated on figs. 3a and 3b. A photino mass lying within 30 and 100 eV largely reionizes the primeval hydrogen. Such a process has many conse- quences:

(i) Matter and radiation are in close thermal contact for a much longer time than in the all-baryonic case. Basically, the decoupling occurs when the thermal relaxation time of the matter temperature T m toward the radiation temperature T7 becomes larger than the typical expansion time of the universe, so when the ratio:

Thermal relaxation time/Expansion time

=4 .6X 106 T45/2Xe/(1 +Xe) (30) overreaches 1. Peebles stated that in the baryonic-dominated universe, the decoupling occurred around T7

2100 K. When the universe is closed with stable non-baryonic matter, the decoupling is delayed to T 7

300 K and finally, if reionization occurs the thermal coupling between matter and light could be strong even at the epoch galaxies formed, towards Z = 10. In that later case we point out that even if the thermal decoupling occurs, all the energy released by the light "-ino" radiative decay heats only the baryons whose temperature grows. Eq. (23) is no longer valid but some features of the matter behaviour can be drawn. As the recombination depends on the matter temperature like T m 1/2, that process becomes inhibited and reionization appears to be so important that the thermal contact could well be reestablished. Otherwise, electrons are heated to a very high temperature prob- ably as large as 10 5 K, owing to a very hot plasma in the IGM.

(ii) If the baryonic medium is highly ionized, the Thomson scattering of the CBR upon electrons is important and the resulting Thomson friction of matter upon light is non negligible. The mass density inhomogeneities ~ p B / p B are damped until the temperature

d r o p s b e l o w 7"damping decoupling [I 6] :

Tdamping decoupling/Tthermal decoupling

= (mp/me) 2/3 ~ 100. (31)

So the mass density fluctuations are allowed to grow later in that scenario than in the standard one. It af- fects galaxy formation.

Off) The small-angular scale inhomogeneities of the CBR could be affected by the reionization of hydrogen and comparison with experimental data should be very interesting. However, a more settled conclu- sion requires an accurate analysis which will follow in a forthcoming publication. Finally, light "-inos" could provide a possible explanation of the hypothetical ionization of the IGM. Present data show that there is little neutral matter in the IGM but nothing is clearly settled about the ionized species. If the IGM was found to be neutral, a new cosmological bound would be derived excluding "-ino" radiative lifetimes smaller

67


than 1024 s and constraining the supersymmet ry

breaking scale d.

We would like to thank P. Fayet , G. Girardi and

B.J.T. Jones for valuable discussions and comment s ,

and G. Girardi for a cri t ical reading o f the manuscr ipt .

References

[ 1 ] S. Weinberg, Gravitation and cosmology (Wiley, New York).

[2] A.D. Dolgov and Ya.B. Zeldovich, Rev. Mod. Phys. 53 (1981) 1.

[3] P.J.E. Peebles, Astron. J. 153 (1968) 1; V.G. Kurt, R.A. Sunyaev and Ya.B. Zeldovich, Sov. Phys. JETP 28 (1969) 146.

[4] P.J.E. Peebles and J.T. Yu, Astron. J. 162 (1970) 815. [5 ] J. Silk, Formation and evolution of galaxies and large

structures in the universe (Reidel, Dordrecht) p. 253. [6] For a review and references, see for example: P.

Langacker, Phys. Rep. 72 (1981) 185;

F. Fukugita, T. Yanagida and M. Yoshimura, Phys. Lett. 106B (1981) 183.

[7] For a review and references, see for example: P. Fayet and S. Ferrara, Phys. Rep. 32C (1977) 1.

[8] P.M. Gondhalekar et al., C.C., 1982, to be published. [9] D.W. Sciama, Mon. Not. R. Astron. Soc. 198 (1982) 1;

see also: N. Cabibbo, G.R. Farrar and L. Maiani, Phys. Lett. 105B (1981) 155.

[ 10] N.A. Zabotin and P.D. Nasel'sk~'i, Sov. Astron. 26 (1982) no. 3.

[11] D.N. Schram and R.V. Wagoner, Ann. Rev. Nucl. Sci. 27 (1977) 37.

[12] R. Kimble, S. Bowyer and P. Jakobsen, Phys. Rev. Lett. 46 (1981) 80.

[13] A. Zee, Phys. Lett. 93B (1980) 389. [14] P. Fayet, Phys. Lett. 70B (1977)461; 84B (1979)421;

86B (1979) 272. [15] P. Fayet, Phys. Lett. l17B (1982) 460, and private com-

munication. [ 16 ] M.J. Rees, Formation and evolution of galaxies and large

structures in the universe (Reidel, Dordrecht) p. 239.

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recombination of the primeval plasma and light -inos

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