Table Of ContentADP-AT-94-5 (Revised)
ELECTROMAGNETIC CASCADES AND CASCADE NUCLEOSYNTHESIS
IN THE EARLY UNIVERSE
1 2 3
R.J. Protheroe , T. Stanev and V.S. Berezinsky
1
Department of Physics, University of Adelaide, SA 5005, Australia.
2
5 Bartol Research Institute, University of Delaware, Newark, DE 19716, U.S.A.
9 3
INFN Laboratori Nazionali del Gran Sasso, 67010 Assergi (AQ) Italy.
9
1
n
a
J
6
1 We describe a calculation of electromagnetic cascading in radiation and matter in the
early universe initiated by the decay of massive particles or by some other process. We
4 have used a combination of Monte Carlo and numerical techniques which enables us to
0
use exact cross sections, where known, for all the relevant processes. In cascades initiated
0
9 after the epoch of big bang nucleosynthesis (cid:13)-rays in the cascades will photodisintegrate
0 4 3 3
He, producing He and deuterium. Using the observed He and deuterium abundances
4
9 we are able to place constraints on the cascade energy deposition as a function of cosmic
/
h time. In the case of the decay of massive primordial particles, we place limits on the
p
density of massiveprimordialparticles as a function of their mean decay time,and on the
-
o expected intensity of decay neutrinos.
r
t
s
a
98.80.-k Cosmology
98.80.Ft Origin and formation of the elements
25.40.Sc Spallation reactions
23.70.+j Heavy-particle decay
1
1 Introduction
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, 5, 6],
by evaporation of primordial black holes, and probably by some other processes. These
3 4
cascades result in the production of He and D by disintegration of He by photons
in the cascade, and we shall refer to this as \cascade nucleosynthesis". The epoch of
(cid:24) 6
interest for cascade nucleosynthesis has redshift z <10 , by which time production of
the light elements D, He and Li by big bang nucleosynthesis has taken place. At this
epoch, electron-positron pairs are no longer in equilibrium,and black body photons, (cid:13)bb,
constitute the densest target for electromagnetic cascade development.
A cascade is initiated by a high energy photon or electron, and develops rapidly in the
radiation (cid:12)eld mainlyby photon-photon pair production and inverseCompton scattering:
0 0 + (cid:0)
e+(cid:13)bb ! e +(cid:13) ; (cid:13) +(cid:13)bb ! e +e : (1)
Such electron{photon cascading through radiation (cid:12)elds involving photon{photon pair
production and inverseComptonscattering governsthespectrumof high-energyradiation
in a variety of astrophysical problems [7, 8, 9, 10, 11, 12, 13, 14, 15].
When cascade photons reach energies too low for pair production on the black body
photons, the cascade developmentis slowed, and further developmentoccurs in the gas by
ordinarypairproduction, but withelectronsstilllosingenergymainlybyinverseCompton
scattering in the black body radiation:
+ (cid:0) 0 0
(cid:13) +Z ! Z +e +e ; e+(cid:13)bb ! e +(cid:13) : (2)
4 3 4
As (cid:12)rst pointed out by Lindley [16], since the observed ratios of D/ He and He/ He
(cid:0)5 (cid:0)4
are very small ((cid:24) 10 (cid:0) 10 ), cascade nucleosynthesis can put strong constraints on
+ (cid:0)
the energy going into the particles initiating cascades ((cid:13);e ;e ) in the early universe.
Cosmologicalapplicationsofcascadenucleosynthesishavebeendiscussedinseveralpapers
3 6
[16, 17, 1, 2]. The strongest constraints will be placed for redshifts between 10 and 10 .
6 3
At z > 3 (cid:2) 10 energies of cascade photons are below the threshold for D and He
3
production, while at z < 10 , direct observation of isotropic X-rays and (cid:13)-rays places
more severe limits on the cascade energy deposition.
2 The Processes
Electromagnetic cascades in the early universe take place rapidly in the radiation (cid:12)eld,
and then slowly in the matter. The processes involved in the cascade in the radiation
(cid:12)eld are photon-photon pair production, inverse Compton scattering and photon-photon
scattering. For interactions in the radiation (cid:12)eld, the mean interaction rates for all three
2
2
processes in black body radiation with temperature T are given by (assuming E (cid:29) mc )
Z Z
1 (cid:22)max (cid:27)int(s)(1(cid:0)(cid:12)(cid:22))
(cid:0)int(E;z) = c n[";T(z)] d(cid:22)d"; (3)
"min (cid:0)1 2
where n(";T) is the di(cid:11)erential photon number density of the radiation (cid:12)eld, (cid:27)int(s) is
the cross-section for the process, s is the square of the total center of momentum frame
energy, (cid:22) = cos(cid:18) is the cosine of the interaction angle, and (cid:12)c is the velocity of the
primary particle ((cid:12) = 1 for photons). Here E is the energy of the primary particle
and " is the target photon energy. For photon-photon scattering and inverse Compton
scattering there is no threshold energy, and hence "min = 0 and (cid:22)max = 1. The threshold
2 4 2 4
condition for photon-photon pair production is s > 4m c , giving "min = m c =E and
2 4
(cid:22)max = 1 (cid:0) 2m c =("E). We assume that the present temperature of the microwave
background radiation is T0 = 2:735 K.
Reno and Seckel [18] have explored the consequences of massive particle decay into
unstable hadrons during the era of primordial nucleosynthesis. Here, particles in the re-
sulting hadronic cascades interactwith nucleonsa(cid:11)ecting theneutron to proton ratio, and
4 3
hence changing the relativeabundances of He, He, D and other light isotopes. For mas-
3 7
sive particle decay at somewhat later epochs ((cid:24) 10 { 10 s), Dimopoulos et al. [19] have
4
considered the breakup of He by hadronic cascades. For the epoch under consideration
7
in the presentpaper (> 3(cid:2)10 s), unstable hadrons resultingfromdecay of massiveparti-
cles will decay into neutrinos and an electromagnetic component (electrons and photons)
before interacting. We therefore only consider the cascade due to the electromagnetic
component but include in our later discussion the fraction of the massive particle's rest
mass energy carried away by neutrinos.
For interactions in the matter, the following processes are included: ordinary (Bethe-
Heitler)pairproductiononhydrogenandhelium,Comptonscatteringofenergeticphotons
byelectrons,photoproduction ofpionsinphoton-proton andphoton-heliumcollisions,and
bremsstrahlung by energetic electrons on hydrogen and helium. Since the matter density
(cid:0)3
depends on epoch as (cid:26) / (1+z) , for interactions with matterthe interactionrates scale
with z as
(M) 3 (M)
(cid:0)int (E;z) = (1+z) (cid:0)int (E;z = 0): (4)
R
In the case of radiation, the number density of black body photons, nbb = n(";T)d",
(cid:0)3
also depends on epoch as nbb / (1+z) . However, because the target photon energies
also depend on epoch, " / (1+z), for interactions with radiation,
(R) 3 (R)
(cid:0)int(E;z) = (1+z) (cid:0)intf(1+z)E;z = 0g: (5)
Mean interaction rates are illustratedin Fig. 1 for all of the processes discussed above.
Inthecaseofinteractionswithradiation, thesearegivenforthepresentepoch, (1+z) = 1,
but they may be scaled to any epoch shifting the corresponding curve by a factor (1+z)
towards lower energies as described in Eq. (5).
3
3 Qualitative Treatment of Electromagnetic Cas-
cades at Large Redshift
7
The black body radiation at the epochs of interest (z < 10 ) is characterized by temper-
ature, T(z) = 2:735(1+z) K, and photon number density,
2 3 3 (cid:0)3
nbb = 4:22(cid:2)10 T2:75(1+z) cm ; (6)
where T2:75 = 0:995 is the temperature in units of 2.75 K.
The density of baryonic gas is
(cid:0)29 3 2 (cid:0)3
(cid:26)b(z) = 1:88(cid:2)10 (1+z) (cid:10)bh g cm (7)
where(cid:10)b = (cid:26)b=(cid:26)c isthemassfractionofbaryonsintheuniverseandhisHubble'sconstant
(cid:0)1 (cid:0)1
in units of 100 km s Mpc . From the recent review by Copi, Schramm and Turner
[20], for standard nucleosynthesis one has
2
0:009 (cid:20) (cid:10)bh (cid:20) 0:02; (8)
and
0:4 (cid:20) h (cid:20) 1:0: (9)
The baryonic gas consists mainly of hydrogen ((cid:24) 77% by mass) and helium ((cid:24) 23% by
(cid:0)2
mass). The radiation length for gas of this composition is X0 (cid:25) 66:6 g cm .
The characteristic interaction rates for photon-photon pair production (PP) and or-
dinary pair production, i.e. Bethe-Heitler (BH) process, are
s
(cid:0)12 3 Ea(z) (cid:0)1
(cid:0)PP(E;z) (cid:25) 2:2(cid:2)10 (1+z) exp((cid:0)Ea(z)=E) s ; (10)
E
at E (cid:28) Ea(z), where
2 6
mec 1:12(cid:2)10
Ea(z) = = GeV; (11)
kT (1+z)
and, using the asymptotic pair production cross sections,
(cid:0)22 3 (cid:0)1
(cid:0)BH(E;z) (cid:25) 1:7(cid:2)10 (1+z) s : (12)
We must compare these with the expansion rate
(
(cid:0)18 3=2 (cid:0)1 4 2 (cid:0)4
3:24(cid:2)10 h(1+z) s for z < 2:3(cid:2)10 (cid:10)h T2:75
H(z) = (cid:0)20 (cid:0)2 (cid:0)2 (cid:0)1 4 2 (cid:0)4 (13)
1:65(cid:2)10 T2:75(1+z) s for z > 2:3(cid:2)10 (cid:10)h T2:75
Strictly, we should use the correct energy-dependent cross sections (described later), but
this is su(cid:14)cientfor the present discussion. At high energies, the cascade develops entirely
4
on the black body photons by photon-photon pair production and inverse Compton scat-
tering. The characteristic interaction rates for these processes are much higher than the
expansion rate, H(z), and thus one can assume the cascade spectrum is formed instantly.
We shall refer to this spectrum as \the zero-generation spectrum".
The zero-generation spectrum extends up to a maximum energy, EC, which is deter-
3
mined at low redshifts (z < 10 ) by
(cid:0)PP(E;z) (cid:25) H(z); (14)
3
and at high redshifts (z > 10 ) by
(cid:0)PP(E;z) (cid:25) (cid:0)BH(E;z): (15)
The maximumenergy obtained in this way is given approximately by
(
4 (cid:0)1 3
4:5(cid:2)10 (1+z) GeV for z < 10
EC(z) (cid:25) 4 (cid:0)1 3 (16)
4:7(cid:2)10 (1+z) GeV for z > 10 :
For small z, the zero-generation spectrum for a cascade of primary energy E0 can
be approximately calculated analytically as described by Berezinsky et al. [11] with the
result 8
(cid:0)1:5
>< K0(E=EX) for E < EX
(0) (cid:0)2
n(cid:13) (E) (cid:25) K0(E=EX) for EX < E < EC (17)
>:
0 for EC < E
(0) 3 (cid:0)1
Here n(cid:13) (E) gives the di(cid:11)erential photon spectrum, EX = 1:78(cid:2)10 (1+z) GeV is the
power-law break-energy appropriate for a black body target photon spectrum, and
E0
K0 (cid:25) 2 (18)
EX[2+ln(EC=EX)]
is the normalization constant. This spectrum is con(cid:12)rmed by Monte Carlo simulation at
z = 0.
When the cut-o(cid:11) energy, EC given by Eq. (16), is less than the threshold for pho-
4
todisintegration of He nuclei ((cid:24) 20 MeV), cascade nucleosynthesis is ine(cid:14)cient. This
(cid:24) 6
condition restricts the epoch of cascade nucleosynthesis to z <2(cid:2)10 . At large redshifts
the spectrum is considerably distorted by (cid:13)(cid:13) ! (cid:13)(cid:13) scattering, as was (cid:12)rst demonstrated
by Svensson and Zdziarski[21], and wetake this intoaccount in our accurate calculations.
3 6
At redshifts between 10 and 2 (cid:2)10 , where the subsequent cascade development is
viaordinary pairproduction and inverseComptonscattering, thezero-generation photons
survive for a time determined by either the energy loss rate for Compton scattering or
the interaction rate for ordinary pair production in the gas. During this time they can
produce light nuclei by photodisintegration. The electrons and positrons give rise to (cid:12)rst
generation photons as a result of inverse Compton scattering. These (cid:12)rst generation
photons then produce the second generation photons, etc. Each generation of photons is
5
strongly shifted to low energiesbecause the inverseCompton scattering isin the Thomson
regime, and only 1 { 2 generations of photons are su(cid:14)ciently energetic to induce cascade
nucleosynthesis.
(cid:24) 3
At redshifts z <10 , when interaction timesbecomelarger than the Hubble time,only
thezero-generation photons areproduced, andthe e(cid:11)ectivenessof cascadenucleosynthesis
diminishes as z decreases. At these redshifts, however, cascade photons can be observed
directlyandsuchdirectobservations morestronglyconstraintheenergydensityofmassive
primordial particles.
3.1 Approximate calculation of cascade nucleosynthesis
3 6
For the redshift range 10 { 2 (cid:2) 10 the zero generation photons survive ordinary pair
(cid:0)1
production and Compton scattering for a time [(cid:0)BH(E)+k(E)(cid:0)CS(E)] where
4=3
k(E) (cid:25) 1(cid:0) (19)
2
ln(2E=mec )+1=2
is the average fraction of energy lost in Compton scattering. Neglecting subsequent gen-
erations, the number of D nuclei produced by the cascade is given approximately by
Z
(cid:0)1 (0) e(cid:11)
ND (cid:25) nHec [(cid:0)BH(E)+k(E)(cid:0)CS(E)] n(cid:13) (E)(cid:27)D (E)dE (20)
where nHe is the number density of He nuclei, (cid:27)D is the e(cid:11)ective cross section for photo-
disintegration of He into D.
The e(cid:11)ective cross section is the sum over partial cross sections of all channels giving
rise to the nucleus in question, weighted by the multiplicity. For example, for photodis-
4
integration of He into D we have
e(cid:11)
(cid:27) (E) = (cid:27)((cid:13);pnD;E)+2(cid:27)((cid:13);DD;E); (21)
D
4 3
and for photodisintegration of He into He we have
e(cid:11) 3 3
(cid:27)3He(E) = (cid:27)((cid:13); Hen;E)+(cid:27)((cid:13); Hp;E) (22)
3 3
where we have included production of H because it decays into He. For the photodis-
integration cross sections, we have used data of Arkatov et al. [22]. The e(cid:11)ective cross
sections used for photodisintegration are plotted in Fig. 2(a). We see that the important
photon energy range is between 25 MeV and 100 MeV and that, above threshold, the
3
cross section for production of He is much higher than for production of D.
4
Intheenergyrange wherephotodisintegration of Heisimportant,thepairproduction
cross sections have not yet reached their asymptotic values, and are strongly energy
dependent. We show in Fig. 3 the pair production cross sections for hydrogen and helium
for the two cases: fully ionized matter, and neutral matter. For E < 100 MeV, the
6
cross sections are almost independent of ionization state, but are quite di(cid:11)erent from the
asymptotic values. The interaction rate for pair production is given by
(cid:0)BH(E) = [nH(cid:27)(cid:13)H(E)+nHe(cid:27)(cid:13)He(E)]c: (23)
Hence, from Eq. (20) we obtain
Z e(cid:11) (0)
(Y=4)(cid:27)D (E)n(cid:13) (E)
ND (cid:25) dE (24)
(1(cid:0)Y)(cid:27)(cid:13)H(E)+(Y=4)(cid:27)(cid:13)He(E)+(1(cid:0)Y=2)k(E)(cid:27)CS(E)
where Y is the fraction of helium in the early universe by mass, and (cid:27)CS(E) is the Klein-
3
Nishina cross section. A similar equation governs the number of He nuclei produced.
4 Accurate Calculation
To take account of the exact energy dependences of all the cross sections we use the
Monte Carlo method. However, direct application of Monte Carlo techniques to cascades
dominated by the physical processes described above over cosmological time intervals
presents some di(cid:14)culties, which we will try to address in the following sections.
The approach we use here is based on the matrix multiplication method described
by Protheroe [9] 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 (cid:14)t. Manipulation of the transfer
matrices as described below enables one to calculate the spectra of particles resulting
from propagation over arbitrarily large times.
4.1 Matrix method
Weuse 110 (cid:12)xedlogarithmicenergybins of width(cid:1)logE = 0:1 coveringtheenergyrange
(cid:0)3 8
from10 GeVto 10 GeV.For example,the energyrange of the jth energybin runs from
(j(cid:0)31)=10 (j(cid:0)30)=10
10 GeV to 10 GeV. The energy spectra of electrons and photons in the
e (cid:13)
cascade at time t are represented by by vectors Fj(t) and Fj (t) which give respectively
the total number of electrons, and photons, in the jth energy bin at time t.
4
The numbers of nuclei produced by photodisintegration of He nuclei by photons in
3 2
the cascade are also represented by vectors, Fj(t) and Fj(t), which give respectively the
3 2
total number of He nuclei and D ( H) nuclei produced by interactions of photons having
energy in the jth energy bin at timet. That is, in this case the energy bin index refers to
the energy of the photon responsible for the photodisintegration, and not to the energy
of the produced nucleus, which is negligible.
(cid:22)(cid:23)
We de(cid:12)ne transfer matrices, Tij ((cid:14)t), which give the number of particles of type (cid:23) = e
3
(electron), (cid:13) (photon), 3 ( He) or 2 (deuterium) in the bin j which result at a time (cid:14)t
7
after a particle of type (cid:22) =e or (cid:13) 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+(cid:14)t)
X110h i
e ee e (cid:13)e (cid:13)
Fj(t+(cid:14)t) = Tij((cid:14)t)Fi(t)+Tij((cid:14)t)Fi (t) ; (25)
i=j
X110h i
(cid:13) e(cid:13) e (cid:13)(cid:13) (cid:13)
Fj (t+(cid:14)t) = Tij((cid:14)t)Fi(t)+Tij ((cid:14)t)Fi (t) ; (26)
i=j
X110h i
3 e3 e (cid:13)3 (cid:13)
Fj(t+(cid:14)t) = Tij((cid:14)t)Fi(t)+Tij ((cid:14)t)Fi (t) ; (27)
i=j
X110h i
2 e2 e (cid:13)2 (cid:13)
Fj(t+(cid:14)t) = Tij((cid:14)t)Fi(t)+Tij ((cid:14)t)Fi (t) ; (28)
i=j
e (cid:13)
where Fi(t) and Fi (t) are the input electron and photon spectra (number of electrons or
photons in the ith energy bin).
We could also write this as
[F(t+(cid:14)t)] = [T((cid:14)t)][F(t)] (29)
where
2 3 2 3
e ee (cid:13)e
F T T 0 0
6 (cid:13) 7 6 e(cid:13) (cid:13)(cid:13) 7
6 F 7 6 T T 0 0 7
[F] = 6 3 7; [T] = 6 e3 (cid:13)3 7: (30)
4 F 5 4 T T I 0 5
2 e2 (cid:13)2
F T T 0 I
4.2 Transfer matrix calculation
(cid:11)(cid:12)
The transfer matrices depend on particle yields, Yij , which we de(cid:12)ne as the probability
of producing a particle of type (cid:12) in the energy bin j when a primary particle with energy
(cid:11)(cid:12)
in bin i undergoes an interaction of type (cid:11). To calculate Yij we use a Monte Carlo
simulation. For inverse Compton scattering and photon-photon pair production we have
usedthecomputercodedescribedbyProtheroe[9,12],updatedtomodelinteractionswith
a thermal photon distribution of arbitrary temperature. For photon-photon scattering,
we have used the cross sections given by Berestetskii et al. [23].
In the case of inverse Compton scattering in the Thomson regime, the basic matrix
method fails to predict correctly the electron spectrum, and hence the emitted photon
spectrum. Thisisbecausethefractionof energylost perinterctionissmall,ande(cid:11)ectively
the electrons su(cid:11)er continuous energy losses,
dE
2
= (cid:0)bE : (31)
dt
Injection of one electron with energy E0 at t = 0 should result in one electron with energy
(cid:0)1 (cid:0)1
E(t) (cid:25) (bt+E0 ) (32)
8
at time t (i.e. there is very little spread in the (cid:12)nal energy). However, the basic matrix
method would give rise to a broad energy distribution with mean energy equal to E(t).
In the present problem, electrons in the Thomson regime lose energy by inverse Compton
scattering at a rate very much higher than by the competing process, bremsstrahlung.
Therefore we immediatly replace any electron produced with energy in the Thomson
regime, or in the transition region between Thomson and Klein-Nishina regimes, by all
the photons from inverse Compton scattering that would be produced while the electron
IC
subsequently cools. We de(cid:12)ne a matrix G which gives the number of photons produced
ij
in energy bin j by an electron injected with energy in bin i (cid:20) m. We have chosen the
maximumenergy of electrons treated in this way to be just below the photon-photon pair
production threshold, i.e. m = 77 at the present epoch. For 61 (cid:20) i (cid:20) 77 (at the present
IC
epoch) the Monte Carlo method is used to calculate G for j (cid:21) 61, and the remaining
ij
array elements, for which the electron is well inside the Thomson regime, are obtained
from the distribution function which we obtain by numerical integration,
R R
2 (cid:13) (cid:0)4 (cid:0)1
dN 9mec 0 d(cid:13)(cid:13) d"" n(")f("^1)
= R ; (33)
dE 16 d""n(")
where f("^1) is given in ref. [24].
For interactions with matter (ordinary pair production, bremsstrahlung, Compton
scattering, photoproduction) we assume a composition 77% hydrogen and 23% helium
(cid:6) (cid:6)
by mass. Photon and electron yields from photoproduction follow from the (cid:25) ! (cid:22) !
(cid:6) 0
e and (cid:25) ! 2(cid:13) decays. In the case of ordinary pair production and bremsstrahlung,
yields are calculated for both neutral and fully ionized matter, and mixed in a ratio
appropriate to the fractional ionization at the epoch for which the transfer matrices are
to be calculated. For the fractional ionization we use Eq. (3.95) of Kolb and Turner [25].
It is important to emphasize that in the most important energy range (20 { 100 MeV)
the cross{sections for bremsstrahlung and pair production are very strongly energy de-
pendent. For pair production we use the directcalculations of Hubbel, Gimmand Overbo
[26] for hydrogen and helium, in which the screening correction (for neutral matter) is
explicit. For bremsstrahlung we use the expressions of Koch and Motz [27] with form
factors for hydrogen and helium adjusted to represent the more precise values of Tsai
[28]. The cross-sections for fully ionized matter are calculated with the bremsstrahlung
formulae valid in the absence of screening by the atomic electrons.
3
In the case of production of He and deuteriumnuclei, the yieldis simplythe e(cid:11)ective
cross section for production of the nucleus in question divided by the total cross section.
4
There is, however, a further complication. During pion photoproduction on He the nu-
cleus almostalways fragments,and we musttherefore take account of photodisintegration
during pion photoproduction. We know of no experimental data giving branching ratios
for the various possible (cid:12)nal state nuclei, and new measurements are urgently required.
4
In the mean time, we assume this process to be similar to the break-up of He during
collisions with nucleons in which pions are produced. Following Meyer [29] we assume
3 3
(cid:12)nal states ( Hen) : ( Hp) : (DD) : (pnD) are produced in the ratio 2:2:1:2 with neg-
9
ligible production of the (cid:12)nal state (2p2n). Hence the e(cid:11)ective cross sections used for
photodisintegration during pion photoproduction are
4
3
(cid:27)e(cid:11)(D;pions) = (cid:27)e(cid:11)( He;pions) = (cid:27)(photoproduction): (34)
7
The e(cid:11)ective cross sections used for photodisintegration during pion photoproduction
have been added to Fig. 2(a). In Fig. 2(b) we have plotted the e(cid:11)ective cross sections
for both photodisintegration and disintegration during photoproduction multiplied by
(cid:0)0:7 3
E . For this log{linear plot, the areas then show the relative contributions to He
(cid:0)1:7
and D production for an E photon spectrum (a single power-law approximation to
the photon spectrum in the cascade). Clearly, disintegration during photoproduction is
particularly important for calculating abundances of deuterium, but is less important for
3 3 4 (cid:0)4
He production. Since the inferred ratio of ( He + D) to He is of the order of 10 ,
3
this implies that photodisintegration of He and D is unimportant in determining their
abundances. We therefore neglect this process.
To calculate the transfer matrices we have used a modi(cid:12)cation of the semi-analytical
technique described by Protheroe & Stanev [14]. From Fig. 1 we see that at all epochs
the highest interaction rate is that for inverse Compton scattering by electrons at low
energies where the scattering is in the Thomson regime, (cid:0)IC ! (cid:0)T = nbb(cid:27)Tc. If (cid:14)t is
much shorter than the shortest interaction time in the cascade, i.e. (cid:14)t (cid:28) 1=(cid:0)T, then
ee ICe breme
Tij((cid:14)t) (cid:25) (cid:14)ij[1(cid:0)(cid:14)t(cid:0)e(Ei)]+(cid:14)t[(cid:0)IC(Ei)Yij +(cid:0)brem(Ei)Yij ]; (35)
e(cid:13) IC(cid:13) brem(cid:13)
Tij ((cid:14)t) (cid:25) (cid:14)t[(cid:0)IC(Ei)Yij +(cid:0)brem(Ei)Yij ]; (36)
(cid:13)e PPe CSe BHe photoe
Tij ((cid:14)t) (cid:25) (cid:14)t[(cid:0)PP(Ei)Yij +(cid:0)CS(Ei)Yij +(cid:0)BH(Ei)Yij +(cid:0)photo(Ei)Yij ]; (37)
(cid:13)(cid:13)
Tij ((cid:14)t) (cid:25) (cid:14)ij[1(cid:0)(cid:14)t(cid:0)(cid:13)(Ei)]+
scat(cid:13) CS(cid:13) photo(cid:13)
+(cid:14)t[(cid:0)scat(Ei)Yij +(cid:0)CS(Ei)Yij +(cid:0)photo(Ei)Yij ]; (38)
e3
T ((cid:14)t) (cid:25) 0; (39)
ij
e2
Tij((cid:14)t) (cid:25) 0;; (40)
(cid:13)3 PD3 (He) photo3
Tij ((cid:14)t) (cid:25) (cid:14)t[(cid:0)PD(Ei)Yij +(cid:0)photo(Ei)Yij ]; (41)
(cid:13)2 PD2 (He) photo2
Tij ((cid:14)t) (cid:25) (cid:14)t[(cid:0)PD(Ei)Yij +(cid:0)photo(Ei)Yij ]; (42)
33
Tij((cid:14)t) = (cid:14)ij; (43)
22
Tij((cid:14)t) = (cid:14)ij (44)
where
(cid:0)e(Ei) = (cid:0)IC(Ei)+(cid:0)brem(Ei); (45)
and
(cid:0)(cid:13)(Ei) = (cid:0)PP(Ei)+(cid:0)scat(Ei)+(cid:0)CS(Ei)+(cid:0)photo(Ei)+(cid:0)BH(Ei)+(cid:0)PD(Ei) (46)
give the total interaction rates of electrons and photons. In the equations above, we
have used the following abbreviations: IC (inverse Compton), brem (bremsstrahlung),
10
