Cascade nucleosynthesis in the early universe

!ll[ll I $_,I ',11~ -"L'/.'] Ill,']

Nuclear Physics B (Proc. Suppl.) 43 (1995) 62~55

EI.SEVIER

PROCEEDINGS SUPPLEMENTS

Cascade nucleosynthesis in the early universe* R.J. Protheroe ~, T. Stanev b and V.S. Berezinskyc aDepartment of Physics, University of Adelaide, SA 5005, Australia. bBartol Research Institute, University of Delaware, Newark, DE 19716, U.S.A. CINFN Laboratori Nazionali del Gran Sasso, 67010 Assergi (AQ) Italy. We describe a c a l c u l a t i o n of e l e c t r o m a g n e t i c cascading in r a d i a t i o n a n d m a t t e r in t h e early universe i n i t i a t e d by t h e decay of massive particles or by some o t h e r process. We have used a c o m b i n a t i o n of M o n t e Carlo a n d n u m e r i c a l t e c h n i q u e s which enables us to use exact cross sections, where known, for all t h e relevant processes. In cascades i n i t i a t e d after t h e e p o c h of big b a n g nucleosynthesis "/-rays in t h e cascades will p h o t o d i s i n t e g r a t e 4He, p r o d u c i n g aHe a n d d e u t e r i u m . Using t h e observed aHe a n d d e u t e r i u m a b u n d a n c e s we are able to place c o n s t r a i n t s on t h e cascade energy deposition as a function of cosmic time. In t h e case of t h e decay of massive p r i m o r d i a l particles, we place limits on t h e i r density as a f u n c t i o n of their m e a n decay time.

1. I N T R O D U C T I O N

radiation:

Electromagnetic cascades in the early universe can be initiated by the decay of massive particles [1,2], or by their annihilation, by cusp radiation of ordinary cosmic strings [3], by super-massive particles "evaporating" from superconducting cosmic strings [4-6], by evaporation of primordial black holes, and probably by some other processes. These cascades result in the production of aHe and D by disintegration of 4He by photons in the cascade, and we shall refer to this as "cascade nucleosynthesis". A cascade is initiated by a high energy photon or electron, and develops rapidly in the radiation field mainly by photon-photon pair production and inverse Compton scattering: e q- ")'bb ---+ el -+- ,yt,

7 nt- "Ybb --+

e+ + e - .

(1)

When cascade photons reach energies too low for pair production on the black body photons, the cascade development is slowed, and further development occurs in the gas by ordinary pair production, but with electrons still losing energy mainly by inverse Compton scattering in the black body *The research of R J P is supported by a grant from the Australian Research Council. The research of TS is supported in part by DOE G r a n t DE-FG-91ER40626. 0920-5632/95/$09.50 ~ 1995 Elsevier Science B.V. All rights reserved. SSDI 0920-5632(95)00452- l

7 + Z --+ Z + e + + e - ,

e q- 7bb --+ e' q- 7 "

(2)

As first pointed out by Lindley [7], since the observed ratios of D/4He and 3He/4He are very small (~ 10-5 - 10-4), cascade nucleosynthesis can put strong constraints on the energy going into the particles initiating cascades (7, e+,e -) in the early universe. Cosmological applications of cascade nucleosynthesis have been discussed in several papers [7,8,1,2]. Reno and Seckel [9] have explored the consequences of massive particle decay into unstable hadrons during the era of primordial nucleosynthesis when particles in the resulting hadronic cascades interact with nucleons affecting the neutron to proton ratio, and hence changing the relative abundances of 4He, aHe, D and other light isotopes. For massive particle decay at somewhat later epochs (~ 10a - 10r s), Dimopoulos et al. [10] have considered the breakup o f 4 l i e by hadronic cascades. For the epoch under consideration in the present paper (> 3 x 107 s) unstable hadrons resulting from decay of massive particles decay into neutrinos and an electromagnetic component (electrons and photons) before they get a chance to interact. Full details of the present work may be found in ref. [11].

R.J Protheroe et al./Nuclear Physics B (Proc. Suppl.) 43 (1995) 62 65

2. T H E P H Y S I C A L

PROCESSES

Electromagnetic cascades in the early universe take place rapidly in the radiation field, and then slowly in the matter. The processes involved in the cascade in the radiation field are photonphoton pair production, inverse Compton scattering and photon-photon scattering. For interactions in the matter, the following processes occur: ordinary (Bethe-tteitler) pair production on hydrogen and helium, Compton scattering of energetic photons by electrons, photoproduction of pions in photon-proton and photon-helium collisions, and bremsstrahlung by energetic electrons on hydrogen and helium. For ordinary pair production we use the full energy dependent cross sections which are significantly different at 10 100 MeV, where photodisintegration takes place, from asymptotic values. The effective cross section for photodisintegration of 4He is the sum over partial cross sections of all channels giving rise to the nucleus in question, weighted by the multiplicity. For example, for photodisintegration of 4He into D we have

~Dff(E) = ~(7, pnD;E)+ 2c~(7, DD;E),

(3)

and for photodisintegration of 4He into 3He we have

a~e(E) = a(7, aHen;E)+a(7,3Hp;E)

(4)

where we have included production of3H because it decays into 3He. For the photodisintegration cross sections, we have used data of Arkatov et al. [12]. Above threshold, the cross section for production of 3He is much higher than for production of D. During pion photoproduction on 4He the nucleus almost always fragments, and we therefore also take account of photodisintegration during pion photoproduction. This is particularly important for calculating the abundance of deuterium, but is less important for 3He production [11].

63

tion of Monte Carlo techniques to cascades dominated by the physical processes described above over cosmological time intervals is impractical. The approach we use here is based on the matrix multiplication method described by Protheroe [13] and subsequently developed by Protheroe & Stanev [14]. We use a Monte Carlo program to calculate the yields of secondary particles due to interactions with the thermal radiation and matter. The yields are then used to build up transfer matrices which describe the change in the spectra of particles produced after propagating through the radiation/matter environment for a time 6t. Manipulation of the transfer matrices as described below enables one to calculate the spectra of particles resulting from propagation over arbitrarily large times. 3.1. M a t r i x m e t h o d We use logarithmic energy bins of width A l o g E -- 0.1 covering the energy range from 10 - 3 GeV to 108 GeV. The energy spectra of electrons and photons in the cascade at time t are represented by by vectors Fie(t) and F;(t) which give respectively the total number of electrons, and photons, in the j t h energy bin at time t. The numbers of nuclei produced by photodisintegration of 4He nuclei by photons in the cascade are also represented by vectors, r?(t) and F~(t), which give respectively the total number of 3He nuclei and D (2H) nuclei produced by interactions of photons having energy in the j t h energy bin at time t. Du We define transfer matrices, Tij (60, which give the number of particles of type u = e (electron), 7 (photon), 3 (3He) or 2 (deuterium) in the bin j which result at a time 6t after a particle of type # =e or 7 and energy in the bin i initiates a cascade. Then, given the spectra of particles at time t we can obtain the spectra at time (t + 6t)

(5)

[F(t + St)] = [T(ht)][F(t)] where

3. M E T H O D

OF CALCULATION

To take account of the exact energy dependences of all the cross sections we can use the Monte Carlo method. However, direct applica-

iF]=

Fe F~ F2

,

iT]=

T ~ T ~¢ T ~ T "rr T ~3 T "r3 T ~2 T "~2

0 0 I 0

0 0 0 I

.(6)

64

R..I Protheroe et al./Nuclear Physics B (Proc. Suppl.) 43 (1995) 62 65

The transfer matrices depend on particle yields, Y/j , which we define as the probability of producing a particle of type /3 in the energy bin j when a primary particle with energy in bin i undergoes an interaction of type a. To calculate Y/~Z we use a Monte Carlo simulation. For inverse Compton scattering and photon-photon pair production we have used the computer code described by Protheroe [13,15], updated to model interactions with a thermal photon distribution of arbitrary temperature, except that we replace any electron produced with energy in the Thomson regime by all the photons from inverse Compton scattering that would subsequently be produced while the electron cools. For photon-photon scattering, we have used the cross sections given by Berestetskii et al. [16]. We require 1 / 6 t be much larger than the largest interaction rate in the problem, inverse Compton scattering in the Thomson regime FT 10-11(1 + z) s -1, and hence typically we use 5t ~ 101°/(1 + z) s. The cascade is followed for a time tm~× which must be much longer than the largest interaction time for interactions with matter. To complete the calculation of the cascade over time tm~x using repeated application of the transfer matrices would therefore require t m ~ x / 6 t ~ 5 X 1015 steps. This is clearly impractical, and we must use the more sophisticated approach described below. Once the transfer matrices have been calculated for a time 6t, the transfer matrix for a time 26t is simply given by applying the transfer matrices twice, i.e. [T(25t)] = [T(6t)] 2.

(7)

In practice, it is necessary to use double precision and to ensure that energy conservation is preserved after each doubling. The new matrices may then be used to calculate the transfer matrices for a time interval 4at, and so on. A time interval 2" 6t only requires the application of this 'matrix doubling' n times. As a test we have run the program over such large time intervals and switched off all processes except photon-photon pair production, inverse Compton scattering and photon-photon scattering so that our results could be compared directly

with those of Svensson and Zdziarski [17] and found satisfactory agreement. Finally, we take account of redshifting because the expansion rate of the universe, H, can be comparable to the interaction rates in matter. 4. R E S U L T S

AND

DISCUSSION

We have performed the cascade calculation to find the number of deuterium and 3He nuclei produced by a cascade initiated at the epoch of redshift z~. We give the number of nuclei per 1 GeV of total cascade energy, so that the total number of nuclei is obtained by multiplying by the total energy of the cascade in GeV. The results are given in Fig. 1 for redshifts at which cascades are initiated, zc, in the range 102 - l07. We obtain a limit on the density of long-lived particles X which can decay into a cascadeproducing particle c (X --+ c + anything) with energy fraction f¢ and branching ratio be. First, N(D +3 He, vx) is obtained from N(D +3 He,t(z~)), given in Fig. 1 by convolving with an exponential distribution of decay times. We then use these results together with the upper limit inferred from measurements of 3He in meteorites and the solar wind making assumptions about stellar processing and galactic chemical evolution[18] (3He -4- D)/H < 1.1 × 10 -4, to obtain an upper limit to $

~*X -- n x m x pc

(8)

plotted in Fig. 2. We also plot the result of Ellis et al. [1] and the result we would have obtained if we had used the asymptotic cross sections for ordinary pair production. REFERENCES

1. J. Ellis, G.B. Gelmini, J.P. Lopez, D.V. Nanopoulos, S. Sarkar, Nucl. Phys. B373, 399 (1992). 2. V.S. Berezinsky, Nucl. Phys. B380, 478 (1992). 3. R.H. Brandenberger, Nucl. Phys., B 2 9 3 , 812 (1987). 4. E. Witten, Nucl. Phys., B249, 557 (1986).

65

R..Z Protheroe et al./Nuclear Physics B (Proc. Suppl.) 43 (1995) 62-65

-1

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1

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-2

j

A

S .....

":~.,\

o.oo,.... 0.008

/

.o=

.

",•....~.~,,,,f

¢=0 0

°-°°41: 0.002 ~ I.~--~- ~

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J

0,006

0.000

.

3

4 5 log(1 + z~)

6

Figure 1. The number of (a) 3He nuclei and (b) D nuclei produced per GeV of cascade energy at redshift zc. Results are shown for various h (and ~tb): dotted curves - 0.4 (0.125); full curves - 0.7 (0.025); and dashed curves - 1.0 (0.01).

5. J.P. Ostriker, C. Thompson, and E. Witten, Phys. Lett., 180B, 231 (1986). 6. C.T. Hill, D.N. Schramm, and T.K. Walker, Phys. Rev., D36, 1007 (1987). 7. D. Lindley, Mon. Not. R. Astr. Soc., 193,593 (1980). 8. J. Ellis, D.V. Nanopoulos, S. Sarkar, Nucl. Phys. B259, 478 (1985). 9. M.H. Reno and D. Seckel, Phys. Rev. D37, 3441 (1988). 10. S. Dimopoulos, R. Ezmailzadeh, L.J. Hall and G.D. Starkman, Nucl. Phys. B311, 699 (1989). 11. R.J. Protheroe, T. Stanev and V.S. Berezinsky, University of Adelaide preprint ADP-AT94-5 (revised), submitted to Phys. Rev. D (1994). 12. Yu.M. Arkatov, P.I. Vatset, V.I. Voloshchuck, V.A. Zolenko, I.M. Prokhorets, and V.I. Chmil', Soy. J. Nucl. Phys., 19, 598 (1974). 13. R.J. Protheroe, 1986, Mon. Not. R. Astr.

-4 -2

0

,

,

,

2

4

6

log[~x (y)]

,

Figure 2. Upper limit, gt~, to the fraction of the present closure density that massive particles would contribute if they had not decayed with mean decay time ~'x, multiplied by fcbc/~b. Results are shown for various h (and ~tb): dotted curves - 0.4 (0.125); full curves - 0.7 (0.025); and dashed curves - 1.0 (0.01). Also shown are results we would obtain if we used asymptotic pair production cross sections ( . . . . . . . . . ), and results of Ellis et al. [1] ( ).

Sot., 221,769 (1986). 14. R.J. Protheroe R.J., T. Stanev, Mon. Not. R. Astr. Soc., 264, 191 (1993). 15. R.J. Protheroe, Mon. Not. R. Astr. Soc., 246, 628 (1990). 16. V.B. Berestetskii, E.M. Lifshitz, and L.P. Pitaevskii, in Quantum Electrodynamics (L.D. Landau and E.M. Lifshitz, Course in Theoretical Physics, vol. 4, p. 566) (Pergamon Press, Oxford, 1980). 17. R. Svensson, and A.A. Zdziarski, Astrophys. J., 349,415 (1990). 18. C.J. Copi, D.J. Schramm, and M.S. Turner, Science, submitted (1994).