Volume 189, number 1,2 PHYSICS LETTERS B 30 April 1987

DECAY OF GRAVITINOS AND P H O T O - D E S T R U C T I O N OF L I G H T E L E M E N T S

Masahiro KAWAS AKI and Katsuhiko SATO

Department of Physics, Faculty of Science, The University of Tokyo, Tokyo 113, Japan

Received 15 December 1986

Gravitinos are produced in the reheating phase of the inflationary universe. We consider the destruction of light elements (4He, 3He and D) by the high energy photons produced by the decay of gravitinos. The spectrum of high energy photons, which was roughly treated previously, is obtained more precisely by a numerical integration of the Boltzmann equation and a constraint on the reheating temperature is derived. We conclude that the allowed reheating temperature is as high as 109-1012 GeV for gravitinos with a mass between 100 GeV and 1 TeV and a much higher reheating temperature is allowed for gravitinos with mass greater than 1 TeV. This constraint is weaker than previous work by more than two orders of magnitude in the mass range above - 300 GeV.

1. Introduction. The mass of gravitinos is ex- pected to be O(100 GeV) in many phenomenologi- cal supergravity theories [1]. But gravitinos with a mass between O(1 keV) and O(10 TeV) are forbid- den in standard big bang cosmology [2], since such gravitinos have a lifetime longer than the cosmic time at primordial nucleosynthesis ( = 10 2 s), and disastrously alter the abundances of light ele- ments. Therefore the existence of gravitinos con- flicts with standard cosmology, unless the number density of gravitinos is suppressed. In the infla- tionary universe, primordial gravitinos are diluted by the exponential expansion and their cosmo- logical effects become negligible. But gravitinos are produced secondarily in the reheating phase after the inflation. The number density of sec- ondary gravitinos depends only on the maximum reheating temperature. Thus a stringent constraint can be imposed on the reheating temperature by considering the cosmological effects of the sec- ondary gravitinos [3-9].

If the lightest supersymmetric particles, which are stable, are photinos, and no other SUSY par- ticles are lighter than gravitinos, then gravitinos decay into photons and photinos. Most of the emitted photons are thermalized by pair creation off thermal photons, pair creation off nucleons and Compton scattering off thermal electrons, but some photons with an energy greater than the

threshold for photo-destruction may destroy the light elements (4He, 3He and D). Previous work [7,8] shows that the most stringent constraint comes from requiring that the destruction of 4He does not overproduce 3He and D. In refs. [7,8] a reheating temperature T R higher than 107-101° GeV was forbidden for gravitinos with a mass between 100 GeV and 10 TeV. But such a low reheating temperature makes it difficult to gener- ate the baryon-an t ibaryon asymmetry when one builds a realistic model. Therefore the existence of gravitinos is dangerous even in the inflationary universe.

However, in previous work the spectrum of high energy photons, which determines the de- struction rate of light elements, was calculated by using a simple approximation. Therefore it is im- portant to estimate the destruction rate of light elements by using a precise spectrum. In this paper we evaluate the spectrum by a numerical integration of the Boltzmann equations incorpo- rating various radiative processes, and derive a new constraint on the reheating temperature. As is seen later, our constraint is much weaker than previous work, thus gravitinos do not become so dangerous in the inflationary universe.

2. Radiative processes. High energy photons which are produced by the decay of gravitinos lose

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Volume 189, number 1,2 PHYSICS LETTERS B : 30 April 1987

their energy by pair creation off thermal photons and nucleons and Compton scattering off thermal electrons, while high energy electrons and positrons, which are produced by pair creation, scatter off thermal photons and create high energy photons (inverse Compton scattering). Hereafter we call pair creation off thermal photons the double photon process and pair creation off nucleons the pair creation. We must solve the Boltzmann equations which include the contribu- tion of these radiative processes to obtain the time evolution of high energy photon and electron spectra. We describe the Boltzmann equations as

- - = I C ON~I(EY) D P

ON (e ) ON (e ) + -

Ot Ot Ot

ONv(Ev) e bNv(Ev) c + Ot + Ot

0Nv(Ev) } (1)

+ 0t DE'

aUe(Eo)0t = 0 Vo(eo)0t ic + 0Uo(Ee)0t D P

o&(eo) ,,

+ 0----7---- ' (2)

where Nv(Ev) and Ne(E~) are the energy spectra of high energy photons and electrons, and the subscripts IC, DP, P, C and DE refer to the contribution to ON/Ot due to inverse Compton scattering, double photon pair creation, pair crea- tion, Compton scattering and the photons pro- duced by the decay, respectively.

The number density of gravitinos with mass M3/2 after inflation is calculated in ref. [6] and is given by

n3/2(t) = 2.35 X 10-13nv TR9(1 -- 0.018 In TR9 )

(3) where TR9 - TR/ /10 9 GeV and n v is the number density of thermal photons. When gravitinos de- cay into photons and photinos, the contribution to ( 0 N v ( E v ) / / 0 t ) D E is given by

ONv(Ev) DE 1 0t = 2.35 X 1013nv

X TR9(1 -- 0.018 In TR9 )

X 8 (Ev - M3/2/2 ) . (4) 24

Here r is the lifetime of gravitinos. Since we assume that the mass of photinos is negligible, the emitted photons have the energy ½ M3/2. When the mass of the photinos is negligible (m~ << M3/2), the lifetime of gravitinos is given by [6]

r = 4 X 108 s (100 GeV/M3/2) 3. (5)

Details of the other contribution to ON/Ot are given in refs. [10,11] and we describe only the summary here. Inverse Compton scattering and double photon pair creation proceeds much faster than Compton scattering and pair creation, be- cause the number density of thermal photons is 109-101° times that of thermal electrons and nucleons in our universe. However, double photon pair creation has the threshold (Ev% > ( m e c 2 ) 2,

where % is the energy of the thermal photon and rn e is the mass of electrons), hence only a small part of the thermal photons can partake this pro- cess. For a given temperature Tv(MeV), the criti- cal energy E* above which double photon pair

10 e

10 5

10 4

10 3

10 2

101

1

M3/2 = 100 GeV

T = 4x108 sec 1

T R = 101° GeV -~ J

2x \

t=5x10 7

101 10 2

ENERGY

10 3 MeV

Fig. 1. Time evolution of the spectrum of high energy photons for M3/2 = 100 GeV, T R =101° GeV and ~- = 4 x 108 s.

Volume 189, number 1,2 PHYSICS LETTERS B 30 April 1987

creation dominates over Compton scattering is estimated in ref. [10] as

Ev* = 1 /70 kT~ MeV, (6)

Photons with energy smaller than E* are thermal- ized only by Compton scattering and pair crea- tion, and thermalization proceeds more slowly than by double photon pair creation. Hence such pho- tons have much chance to destroy light elements.

The time evolution of the spectrum o f high energy photons is shown in fig. 1 for M3/2 = 100 GeV, ~- = 4 x 108 s and T R = 101° GeV.

3. Photofission of light elements. The evolution of the total number of the light elements i in a comoving volume is given by

dN4/dt = - 2 ( 4 ~ anything) N4, (7)

dN3/dt = - ~ ( 3 ~ anything)N 3 + 27(4 ~ 3)N4, (8)

dN2/dt = -27(2 ~ anything)N 2 + 27(4 --> 2 ) N 4

+ 2 ( 3 ~ 2) N3, (9)

where N4, N 3, and N 2 denote the total number of 4He, 3He and D in a comoving volume, and ~ ' s are the destruction rates and given by

~,(i ~ j ) + f ~ cNv(E, t)oi_~)(E ) dE, (10) E ~ ,

where % , ~ and E i__,J are the cross section and threshold of the photofission reaction i ~ j ("1", "2" and "3" denote D, 3He and 4He, respectively). We consider the following reactions.

4He + ~ ~ 3 H + p (19.8 MeV),

4He + ~, ~ 3 He + n (20.6 MeV),

4He + .y ~ 2 H + p + n (26.1 MeV),

3He + "t ~ 2H + P (5.5 MeV),

3He + ), ~ p + p + n (7.7 MeV),

2He + ~, ~ p + n (2.2 MeV),

where the numbers in brackets denote the threshold. We use the cross section of these reac- tions in refs. [12-15].

4. Constraint on reheating temperature. We calculate N~ using the high energy photon spec- trum which is obtained by numerical integration of eqs. (1) and (2). The time evolutions of N/ are shown in fig. 2 and for M3/2 = 400 MeV and T R = 1012 GeV and in fig. 3 for M3/2 = 2.5 TeV and T R = 1015 GeV. In this figure we see that when the mass of gravitinos is small, a significant amount of 4He is destroyed into D and 3He and this leads to overproduction of D and 3He. While, when the mass is large, only D is destroyed. This is because from eq. (6) the lifetime is short in this case and thermalization occurs very rapidly. The abundances of light dements are restricted by observation to 4 H e / H > 0.07, (D + 3 H e ) / H < 1 x 10 -4 and D / H > 1 x 10 -5 [16]. However, we perform the numerical calculation with the initial abundances 4 H e / H = 8.8 x 10 -2 and 3 H e / H = D / H - 1 x 10 -5. Therefore we must impose a milder restriction for the abundances of deuterons than .observation requires. Thus we require that the resultant abundances of light elements satisfy the following conditions;

4 H e / H > 0.07, (11)

(O + 3 n e ) / n < 10 -4, (12)

D / H > 4 x 10 -6. (13)

The contour maps of the abundances are shown in figs 4-6. The constraint on the reheating tempera- ture is obtained from our requirements (11)-(13). In the mass range between 100 GeV and 1 TeV the most stringent constraint come from the re- quirement that 3He + D should not be overpro- duced, and in the mass range greater than 1 TeV deuteron destruction imposes the most stringent constraint on the reheating temperature.

5. Discussion. Our constraint on the reheating temperature is not so severe as in previous work [7,8]. The limiting temperature is higher than refs. [7,8] by a factor 2-102 for gravitinos with a mass between 100 GeV and 300 GeV and by more than two orders of magnitude for gravitinos with a mass greater than 300 GeV. The allowed reheating temperature is as high as 1012 GeV for gravitinos with a mass of about 1 TeV. This is because in the previous work radiative processes were treated

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Volume 189, number 1,2 30 April 1987

i0-~

LU (.~ 10-2

Z < ("1 10 -3

Z 03 <~ 10 -4

10 -5

PHYSICS LETTERS B

' t . . . . I ' ' ' I . . . . I

Mass = 400GeV : L i fe = 6 .3E+6sec

Reheat ing Temp. = 1 .0E+12 GeV

I ' i ,

4He

3He+ D

10.5 , , , I , ,,,I , ; , I , , ,Jl , , L i , , , 10 5 10 6 10 7 10 a

TIME (sec)

Fig. 2. Time evolution of light elements (D, 3He and 4He) for M2/3 = 400 GeV, T a = 1012 GeV and ~- = 6.3 x 106 s.

10-1

I 0 -2 LU O Z lO-3 < 63 Z 10--4

03 <:~ 10.5

10 ~

' I . . . . I ' I . . . . I I ' ' '

Mass = 2 .5 TeV : L i fe = 2 .5E+4 sec

Reheat ing Temp, = 1 .0E+15 GeV '*He

3H e

, , , I , , ,,I

10 5

TIME (sec)

D 10-7 I i = I i i i = l , I . . . .

10 3 10 4 10 6

26

Fig. 3. Time evolution of light elements (D, 3He and 4He) for M2/3 = 2.5 TeV, TR = 1015 OeV and ~" = 2.5 × 104 s.

Volume 189, number 1,2 PHYSICS LETTERS B 30 April 1987

T.

1 2 1o

GeV .m13

I ] I

~'- 6 x 1 ~ - - - ' : - ~ 0 -2 /

= 8 ×10 2 101°

10 9

10 8 I [ I I I

IOOGeV 1TeV M 3/2

Fig. 4. Contour map of 4He/H on the M3/2-T R plane. Observation requires 4 He /H > 0.07.

G e V

10 ~s

1014

1013

TR 1012

~/ ~ I0~ 6

1011 i

101° 1 T e V

I I I

M3/2

I

IOTeV

Fig. 6. Contour map of D/H. We require D / H > 4 × 10-6.

GeV 1013 i , J ,

1°12

lO 11

rR lO lo

10 9 -

10 8 ~ i I i i 10 0 GeV 1 T e V

M 3/2

Fig. 5. Contour map of (D+3He)/H. Observation requires (D + 3He)/H < 10-4.

roughly and the estimate of the spectrum of high energy photons was not precise. Especially the critical energy above which double photon pair creation dominates over Compton scattering was overestimated in ref. [7,8], which made the de- struction rate larger and the constraint more severe.

With the low reheating temperature imposed by the previous work it is difficult to generate baryon asymmetry. (but it is not impossible if non-thermal processes are considered [4].) We think that the reheating temperature which is allowed in our calculation is high enough to produce the baryon-ant ibaryon asymmetry in the universe [17]. Therefore we conclude that gravitinos in the infla- tionary universe are not dangerous in the mass range which most of the phenomenological super- gravity models require.

We thank H. Kodama and N. Terasawa for stimulating discussions. The numerical calcula- tions were carried out on a FACOM M-380R Computer at the Institute for Nuclear Study at the

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Volume 189, number 1,2 PHYSICS LETTERS B 30 April 1987

University of Tokyo. This work was supported in part by the Grant-in-Aid for Science Research Fund of the Ministry of Education, Science and Culture No. 60302024.

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