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〉 PostScript processed by the SLAC/DESY Libraries on 27 Mar 1995. ASTRO-PH-9503094 t e e h v wn a w He/D tegra- tation y non- 3 ys with t from the -ra y of Grand
5 atory 60637-1433 or He-photodisin e apply this fact to kground. It is sho 4 y the deca ab W TION 60510-0500 y bac tegration initiated b FERMILAB-Pub-95/051-A tegration are also discussed. ago, IL YS, AND y (HECR) pro duction. Suc -ra ed from
ophysics an Sasso, and V. S. Berezinsky e National L e nd that this pro cess cannot b e 3 ; ermi Institute 2 ; W ago, Chic 1 oF ago, IL 60637-1433 ould result in anomalously high ed di use He photo disin olution assuming standard fragmen tegration b ound. The constrain hs. 4 Q), Italy CT CK HOLES ophysics Center y atory, Batavia, IL t pro duction of HECR b hramm e Livermor olution assumptions. gi (A or onomy & Astr Hb provided byCERNDocumentServer e, CA 94550 2 enc ago, Chic ab brought toyouby awr H since it w 2 k hole induced photodisin ABSTRA ts on these mo dels deriv ,D.N.Sc He-photodisin atori Nazionali del Gr ator L ermilab Astr 4 4 ; He and Livermor hemical ev 1 3 or GY COSMIC RA OR TOPOLOGICAL DEFECTS, eler e blac 67010 Asser ab c c CORE artment of Astr NASA/F 2 artment of Physics, Enric Dep INFN L 1 5 Dep MASSIVE BLA 3 AND NUCLEOSYNTHESIS: , K. Jedamzik ts on massiv The University of Chic 2 eV. The constrain ; TIONS F 1 20 kground rules out the dominan ermi Institute, The University of Chic t source of primordial ermi National A F e10 HELIUM PHOTODISINTEGRA University of California, L oF v 4 HIGH ENER G. Sigl e b een prop osed as p ossible sources of ultrahigh energy particles and y bac v -ra Enric
e consider the pro duction of h without violating at least the W kground radiation (CMBR) and from the observ IMPLICA functions. Constrain Uni cation particles in mo dels with cosmological ev that for reasonable primary particleordinary injection strings sp ectra supor erconducting cosmic annihilatingepoc strings, monop oles, unlik cannot pro ducedi use the HECR ux at the presen bac energies ab o tion are compared to corresp onding limits from sp ectral distortions of the cosmic micro mo dels ha constrain top ological defect mo dels of highest energy cosmic ra ratios in con ict with standard c the predominan thermal energy releases during early cosmic ep o c
1 Intro duction
In this pap er we consider various constraints inferred from the p ossible photodisintegra-
4
tion of He in the early universe. Following Prothero e, Stanev, and Berezinsky [1] we note
that the photodisintegration of this isotop e can b e employed to place stringent limits on early
cosmic energy injections asso ciated with, for example, decaying particles [2, 3], evap orating
black holes [4], or annihilating top ological defects [5, 6 , 7, 8, 9, 10 ]. Our fo cus here will
4
b e particularly on constraining the latter scenario. It has also b een suggested that He-
photodisintegration in the early universe could b e a pro duction mechanism for the observed
3
light-element abundances of deuterium and He [11]. In this work we will study the feasibility
3 2
of such a scenario and show that the ( He/ H) ratio p oses a problem to it. We will show
3 2 2 3
that photodisintegration yields ( He= H) >> 1 and since H is destroyed and He increases
3 2
with evolution, measures of ( He/ H) place severe constraints on photo disintegration.
Nonthermal energy releases at high redshifts may leavevarious observable signatures.
The cosmic microwave background radiation (hereafter, CMBR) has b een measured to have
a blackb o dy sp ectrum to very high accuracy [12 ]. Any injection of energy b etween redshifts
3 6
of z ' 10 and z ' 3 10 may pro duce observable sp ectral distortions of the blackbody spec-
trum [13]. Here the lower redshift represents the approximate ep o ch of decoupling (assuming
no re-ionization), whereas the higher redshift represents the ep o ch at which double-Compton
scattering is still ecient enough to completely thermalize signi cant energy releases [14 ].
The di use -ray background observed at the present epoch can also b e used to constrain
<
300 1000 pair pro duction by -rays early cosmic energy injections [15]. For redshifts z
4
on protons and He is rare so that the universe b ecomes transparentto -rays with energies
b elow E . Here the energy E is
max max
2
1
m T
e
E ' ' 17GeV ; (1)
max
15T 1eV
where T is the CMBR temp erature and m is the electron mass. E is related to the
e max
+
threshold energy for e e -pair creation by high-energy -rays scattering o CMBR-photons.
Any radiation with energies ab ove this threshold is e ectively instantaneously \recycled" by
+
e ) and inverse Compton scattering of the created electrons pair pro duction ( ! e
CM BR
and p ositrons (e ! e ). These pro cesses yield a degraded -ray sp ectrum with generic
CM BR
1:5
energy dep endence / E considerably b elow E b efore steep ening and nally cutting o
max
at E [3]. Signi cant energy releases in form of high-energy -rays and charged particles
max
at ep o chs with redshifts b elow z ' 300 1000 may therefore pro duce a presentday -ray
background and are sub ject to constraint.
6
For redshifts smaller than z ' 10 stringent constraints on various forms of injected
4
energy can also b e derived from the p ossible photo disintegration of He and the concomitant
3
pro duction of deuterium and He. The injection of high-energy particles and -rays ab ove
the energy threshold E will initiate an ep o ch of cascade nucleosynthesis subsequentto
max
the ep o ch of standard primordial nucleosynthesis at T 100 keV . The abundance yields of
2 3 4
H and He pro duced by He-photodisintegration during cascade nucleosynthesis are quite 1
indep endent from the primary -ray and charged particle energy sp ectra. Deuterium and
3
He abundance yields dep end only on the amount of injected energy and the injection ep o ch.
For the detailed calculations leading to these conclusions the reader is referred to the work
by Prothero e, Stanev, and Berezinsky [1]. The nucleosynthesis limits on the release of energy
into the primordial gas can b e up to a factor of 100 more stringent than equivalent limits
on energy releases derived from distortions of the CMBR-blackb o dy sp ectrum.
6
> >
For redshifts z 10 , corresp onding to CMBR-temp eratures of T 200 eV, the photo-
4
disintegration of He is inecient. This is b ecause the energy threshold for pair pro duction
4
He 4
<
E . The b est nu- falls b elow the energy threshold for He-photodisintegration , E
max
th
cleosynthesis limits on decaying particles and annihilating top ological defects in the cosmic
< <
temp erature range 1 keV T 10 keV come from the p ossible photodisintegration of deu-
terium [3, 16]. These limits are stronger than analogous limits from distortions of the CMBR
blackb o dy sp ectrum.
In this narrow temp erature range limits on decaying particles and top ological defects
may, in fact, b e more stringent due to e ects of injecting antinucleons. Antinucleons may
5
>
10 GeV or when there is a b e pro duced during pair pro duction for -energies E
CM BR
signi cant hadronic decaychannel for a massive decaying particle or top ological defect. These
4
antinucleons can then annihilate on He thereby pro ducing approximately equal amounts of
2 3
H and He [17]. We will, however, not further pursue this idea here.
For temp eratures ab ove T ' 1 keV there are virtually no constraints on decaying parti-
cles and top ological defects from distortions of the CMBR blackb o dy sp ectrum. However,
stringent limits on decaying particles and top ological defects may obtain from the injection
of hadrons (for a review see [3]). An injection of mesons and baryons generally increases the
4
> >
100 keV ) T neutron-to-proton ratio and results in increased He-mass fractions (1 MeV
2 3
> >
10 keV ; [18 ]). It has b een sug- T and/or increased H and He-abundances (100 keV
4 2 3
gested that a combination of He-hadro destruction and H, He-photo destruction induced
by a late-decaying particle (T 3 keV )may bring big-bang-pro duced light-element abun-
dances close to observationally inferred abundance constraints for a wide range of fractional
contributions of baryons to the closure density, [19 ].
b
3 2
The observational signatures of such scenarios are primordial isotop e ratios of ( He/ H)'
6 7
2 3 and Li= Li 1, contrasting the predictions of a standard, or inhomogeneous, big-bang
freeze-out from nuclear statistical equilibrium.For a wide range of parameters, such as de-
2
caying particle life times and hadronic branching ratios, these mo dels would overpro duce H
3
and He and therefore the calculations by Dimop oulos et al. [19 ] do also serve as constraints
3 2
on particle parameters and abundances. We note here that the high ( He/ H) ratio mayin
fact b e a severe problem for such scenarios.
In this pap er we restrict ourselves to constraints derived from the e ects of nonthermal
6
<
energy injections at ep o chs with redshifts z 10 . The outline of the pap er is as follows.
2
In Section 2 we brie y review the observationally inferred light-element abundances of H
3 4
and He. We then consider He-photodisintegration scenarios and their compatibility with
the observations. In Section 3 we study the e ects of p ossible energy injection by sup er-
2 3
conducting strings, ordinary strings, and magnetic monop oles on the primordial H and He 2
abundances, the distortions of the CMBR-blackbody, and the di use -ray background. In
these scenarios we assume that such top ological defects would radiate on a level such that
they could pro duce the observed highest energy cosmic rays at the present epoch. Conclu-
sions are drawn in Section 4. Throughout this pap er we will mostly use c =h =1.
4
2 Constraints on He-Photo disintegration as the pre-
dominant Source of Primordial Deuterium
4
In this section weinvestigate scenarios whichhave He-photodisintegration as an ecient
3
pro duction mechanism of the light-element abundances of deuterium and He. In this study
3 2
we are naturally led to consider the primordial ratio of ( He/ H) . This is b ecause the ratio
p
3 2
of these light isotop es emerging from the big bang nucleosynthesis pro cess, ( He/ H) ,
BBN
4 3 2
is quite di erent from that emerging from the He-photodisintegration , ( He/ H) . In
photo
3 2 3 2
<
particular, we exp ect generic isotop e ratios of ( He/ H) 1, and ( He/ H) 1.
BBN photo
4
We will show that this fact can b e used to severely constrain the photodisintegration of He
as the principal source of primordial deuterium. We will also show that the observationally
2 3
inferred abundances of H and He may imply a factor 2-3 more stringent constraints on the
primordial numb er densities of decaying particles and on the energy injected by top ological
defects than previous work has assumed.
3 2
The most accurate determination of a ( He/ H)-ratio is thought to come from solar
3
system observations of He abundances. Geiss [20] reanalyzed the existing data and inferred
3
for the abundances of deuterium and He at the time of solar system formation
3
He
5 5
< <
1:2 10 1:8 10 ;
H
2
H
5 5
< <
3:3 10 ; (2) 1:6 10
H
3
He
< <
0:34 1:13 :
2
H
2 3
A determination of the interstellar medium abundances of H and He is less precise due
2 6
<
to observational diculties [21]. The observed ( H/H)-ratios ranges b etween 5 10
2 5 3
<
( H=H ) 2 10 [22 ]. Interstellar ( He/H)-ratios are observed in the range 1:1
ISM
5 3 5 3 2
< <
10 ( H e=H ) 4:510 [23]. These abundances imply a present( He/ H)-isotop e
ISM
3 2
< <
ratio of 0:55 ( He= H) 9.
ISM
Deuterium is the most fragile of the light isotop es. It is easily destroyed during the
2 3
pre-main sequence evolutionary stage of stars via H(p, ) He. Furthermore, there are no
plausible galactic pro duction sites for deuterium. Epstein, Lattimer, and Schramm [24]
summarize the arguments against a galactic origin of deuterium. The chemical evolution
3 3
He is less clear. It is known that He is destroyed to some extent in massive stars of
3
> <
(M 5 8M ), whereas low-mass stars (M 1 2M )may b e net pro ducers of He.
3
3
This theory is supp orted by the observations of He abundances in planetary nebulae. It
is certainly very reasonable to assume that standard chemical evolution mo dels can only
3 2
increase the primordial ( He/ H) -ratio,
p
3 3
He He
>
: (3)
2 2
H H
t p
3 2
In this expression ( He/ H) denotes the isotop e ratio at some cosmic time t and the pri-
t
3 2
mordial isotop e ratio ( He/ H) includes any pre-galactic pro duction mechanism, suchas
p
4
big bang nucleosynthesis and He-photodisintegration in the early universe. Note that the
3 2
inferred ( He/ H) ratios at the time of solar system formation and the present epoch are
3 2
consistent with the assumption of monotonically increasing ( He/ H) ratios with time.
3 2
The ( He/ H)-ratio in a standard homogeneous big bang nucleosynthesis (hereafter,
10 3 2
is ( He/ H) ' 0:2. An upp er SBBN) scenario at baryon-to-photon ratio =310
SBBN
3 2 4
limit on the ( He/ H)-ratio in SBBN can b e obtained by requiring the He-mass fraction
<
0:25, whereas a lower limit on this isotop e ratio can b e estimated from the to satisfy Y
p
4 2
<
3 10 . This yields the SBBN range conservative b ound ( H=H )
3
He
< <
0:09 0:55 : (4)
2
H
SBBN
3 2
Typical ( He/ H)-isotop e ratios resulting in inhomogeneous big bang scenarios are not very
di erent from those in Eq. (4).
The detailed calculations by Prothero e, Stanev, and Berezinsky [1] show that the abun-
3 2
dance ratios of ( He/ H) pro duced during cascade nucleosynthesis in the early universe
exceed
3
He
>
8 ; (5)
2
H
photo
for a wide range of fractional contributions of baryons to the closure density, , Hubble
b
1
1
parameters H in units of 100 k m sec Mpc , h, and ep o chs of energy injection. This
0
4
is b ecause in He-photodisintegration the e ective cross sections for the two-nucleon pho-
4 2 4 2 2
He( ,pn) H and He( , H) H] are roughly ten times smaller than toabsorption pro cesses [
4 3
the e ective cross sections for the single-nucleon photoabsorption pro cesses [ He( ,p) H and
4 3
He( ,n) He] [25 ].
Note that Eq. (5) applies strictly only under the following assumption. In cascade nu-
cleosynthesis it is assumed that the main fraction of radiation is injected ab ove the energy
+
threshold Eq. (1) for ! e e pair creation. Pair creation and inverse Compton
CM BR
1:5
scattering will then yield a generic -ray sp ectrum with energy dep endence / E be-
5
low E =2 and / E ab ove b efore cutting o at E . These -rays can b e e ective
max max
4
in photo disintegrating He where the comp eting pro cess is the consumption of -rays by
Bethe-Heitler pair pro duction on hydrogen and helium.
When radiation is injected b elow E the -rays mayhave a sp ectrum quite di erent
max
1:5
from the b ehavior / E dep ending on the actual -ray source. In principle, it is then
4
4 3 2
conceivable to photo disintegrate He in suchaway that isotop e ratios of ( He/ H)' 1 result.
This could b e accomplished bya -ray source which preferentially radiates ab ove energies
of E ' 100 MeV but b elow E . This is b ecause only in the energy range b etween the
max
4
4 He
He-photodisintegration threshold E =19:8MeV and E ' 100 MeV the e ective cross
th
3 4
section for He pro duction in He-photodisintegration is roughly ten times larger than the
2
>
e ective cross section for H pro duction in this pro cess. For -ray energies E 100MeV these
cross sections are roughly equal. In practice, any such scenario has to o ccur at relatively low
3
<
redshifts z 10 so that there will not develop a \softer" second generation -ray sp ectrum
pro duced by Bethe-Heitler pair pro duction and inverse Compton scattering. In this case,
however, signi cant deuterium pro duction would require -ray uxes whichwould exceed
the presentday di use -ray background.
3
It should b e noted that -rays could also b e e ective in photo disintegrating He and
2 3 2
H and thereby in resetting any initial ( He/ H) -isotop e ratio pro duced during cascade
photo
4 3
nucleosynthesis to smaller values. However, the relative abundances of He-targets to He-
3 4
targets is approximately 10 10 to 1, so that for roughly equal photo disintegration cross
3
sections the numb er densities of -rays in the energy range b etween the He-photodisintegra-
3 4
He He 3 4
tion threshold E =5:4MeV and E =19:8MeV should b e 10 10 times larger than
th th
4
He
the numb er densities of -rays with energies ab ove E . Such a scenario would require an
th
extremely \soft" -ray sp ectrum.
4
We can derive limits on the allowed contributions of He-photodisintegration to the pri-
2 3 3 2
mordial H and He abundances. This can b e done by employing the solar system ( He/ H)-
isotop e ratio from Eq. (2) and assuming that this ratio represents a conservative upp er limit
3 2
on the primordial ( He/ H)-isotop e ratio [refer to Eq. (3)]. Note that when either one of
Eqs. (2) or (3) do es not apply one of the widely used standard assumptions of galactic chem-
ical evolution has to break down. We can derive an upp er limit on the fraction of deuterium
photo
4
contributed to the primordial deuterium abundance by He-photo disintegration. A f
2
H
simple calculation of the abundance average then yields
3 3
He He
( ) ( )
2 2
photo
BBN
H H
<
f : (6)
2
3 3
H
He He
( ) ( )
2 2
photo BBN
H H
3 2
By using the upp er limit for ( He/ H) from Eq. (2), the lower limit in Eq. (4) for the
3 2 3 2
( He/ H) -ratio, and Eq. (5) for the ( He/ H) -ratio we derive
SBBN photo
photo
<
f 13% : (7)
2
H
It is evident that the contribution of deuterium pro duced during cascade nucleosynthesis to
3
the total primordial deuterium abundance has to b e small in order to not overpro duce He.
4
This also implies that generic He-photodisintegration scenarios can not be the predominant
2 3
production mechanism of the primordial H and He light-element abundances. The stringent
limit of Eq. (7) can only b e evaded when either there existed an extremely \soft" -ray source
3
in the early universe or when generic features of the galactic destruction/pro duction of He
2
H are for some yet unknown reason not understo o d. and 5
Gnedin and Ostriker [11 ] have prop osed the interesting scenario of a very early formation
(z ' 800) of massive black holes. If these black holes do accret material which emits a
4
quasar-like X -ray and -ray sp ectrum they may induce the photodisintegration of He and
the reionization of the universe. The reionization of the universe would cause primordial
CMBR uctuations to b e erased, whereas the pro cessed -ray sp ectrum could constitute
the di use -ray background at the present epoch. They concluded that this selfconsistent
mo del could evade the upp er limit on given by the observed deuterium abundance and a
b
3
SBBN scenario since deuterium and He would have b een pro duced, at least in part, in the
photo
4
He-photodisintegration pro cess. For typical mo dels they pro duce a fraction f ' 50% of
2
H
4
the total primordial deuterium abundance by He-photo disintegration. Clearly, this fraction
3 2
is in con ict with the limit of Eq. (7) and would result in to o high ( He= H) ratios [26 ].
p
photo
We can also constrain the fraction f which can b e contributed to the total sum of
2 3
( H + He)
3 4
the primordial deuterium- and He-abundances by He-photo disintegration. This parameter
is limited by
photo
<
f 35% 55% : (8)
2 3
( H + He)
Any annihilating top ological defects or decaying particles abundant enough to initiate an
epoch of cascade nucleosynthesis such that more than 35% of the presently observed abun-
2 3
dance sum of ( H+ He) is contributed by this cascade nucleosynthesis are sub ject to con-
straint. The limits given in Eq. (8) are a factor 2-3 b etter than equivalent limits assumed in
previous work.
2 3
These limits can b e put into context by the upp er limit on the sum of H and He inferred
from the solar system data and chemical evolution mo dels by Geiss [20 ]
2 3
H + He
4
<
1:1 10 : (9)
H
4
In Figure 1 we show constraints from He-photodisintegration on the maximum allowed en-
ergy release as a function of redshift. To pro duce this gure wehave used Eq. (9) and
the upp er range given in Eq. (8). For comparison we show analogous limits from p ossible
distortions of the CMBR background. These are taken from reference [12 ]. It is seen that
4
over a wide range of redshifts the limits from He photo disintegration are more stringent
than the limits from CMBR distortions. Also shown are constraints from the di use -ray
background which result from the generic cascade sp ectrum (see section 3.4).
3 Energy Injection from Top ological Defects
and Highest Energy Cosmic Rays
3.1 History of Energy Injection in Defect Mo dels
It is commonly b elieved that cosmic rays are pro duced mostly by rst order Fermi accel-
eration (see e.g. [27 , 28 ]) at astrophysical sho cks in the presence of magnetic elds. The 6
highest energies seem to b e reached in relativistic sho cks contained in radiogalaxies and ac-
tive galactic nuclei (see e.g. [29, 30 , 31 , 32 ]). The recent observation of cosmic rays ab ove
20
10 eV by the Fly's Eye [33, 34] and AGASA [36, 37 ] exp eriments, and the exp eriment
at Yakutsk [38, 39 ] may,however, not b e easily explained by this mechanism [40 , 41 , 42 ].
Therefore, it has b een suggested that such sup erhigh energetic cosmic rays could havea
non-acceleration origin [5, 8 , 41 , 43 , 44 , 45, 46 ] as, for example, the decay of sup ermassive
elementary \X" particles asso ciated with Gand Uni ed Theories (GUTs). These particles
could b e radiated from top ological defects (TDs) formed in the early universe during phase
transitions caused byspontaneous breaking of symmetries implemented in these GUTs. This
is b ecause TDs, like ordinary or sup erconducting cosmic strings and magnetic monop oles,
on whichwe will fo cus in this pap er, are top ologically stable but nevertheless can release
part of their energy in form of these X-particles due to physical pro cesses like string collapse
15
or monop ole annihilation. The X-particles with typical GUT scale masses ( 10 GeV )
decay subsequently into leptons and quarks. The strongly interacting quarks fragmentinto
4 5
a jet of hadrons which results in typically of the order of 10 10 mesons and baryons.
It is assumed that these hadrons then give rise to a substantial fraction of the HECR ux,
whereas the contribution from the lepton primary is often approximated to b e negligible.
It also causes a more or less uniform global energy injection whose sp ectrum is determined
by the cascades pro duced by the interactions of the primary decay pro ducts with various
4
background radiation elds. This energy injection is sub ject to the constraints from He-
photodisintegration discussed in the previous section as well as to constraints from sp ectral
CMBR distortions and the observed -ray background.
The X-particle injection rate dn =dt as a function of time t or redshift z usually is
X
parametrized as [44 ]
dn
X
4+p
/ t : (10)
dt
It is imp ortant to note that the e ectivevalue of p may dep end on the ep o ch. Given that and
using standard cosmological relations for t(z ) [47] one can describ e the X-particle injection
history byintro ducing the dimensionless function
(dn =dz )(z )
X
f (z ) ; (11)
t (dn =dt)(t )
0 X 0
1
where t =2H =3 is the age of the universe (we assume a at universe, = 1, throughout
0 0
0
this pap er).
For example, for annihilating magnetic monop oles it can b e shown [46 ] that p = 1 for
t> t and p =1:5 for t< t , where t is the time of matter-radiation equality.
eq eq eq
As a second example, let us lo ok at collapsing cosmic string lo ops. These may include
ordinary as well as sup erconducting strings. Let us assume that the history of lo ops consists
of two distinct evolutionary stages. We will see b elow that sucha schematic representation
can b e used for b oth sup erconducting strings and ordinary strings. In the rst stage the
2
lo op slowly radiates gravitational radiation with a p ower 100G . Here, G is Newton's
2
is the energy p er unit length of the string in terms of the GUT symmetry constant and ' v 7
breaking scale v . This will decrease the lo op length, L(t), at an e ective rate v 100G,
g
L(t)= L v (t t ): (12)
b g b
In this expression t and L denote the birth time and the lo op length at birth, resp ectively.
b b
Numerical string simulations [48 , 49 , 50 ] suggest that lo ops are b orn with a typical length
L = t with b eing a dimensionless constant which can b e as small as a few times v .To
b b g
simplify the calculation we will assume that all lo ops are b orn with the same length L .
b
Note that the gravitational radiation asso ciated with this rst stage of string lo op evo-
lution should not haveany e ects on CMBR distortions, the di use -ray background, or
4
result in photo disso ciation of He. The existence of gravitational radiation during the ep o ch
of primordial nucleosynthesis, however, can e ect abundance yields bychanging the cosmic
16
<
expansion rate [51 ]. For symmetry breaking scales v 10 GeV as discussed in this pap er
this e ect is negligible.
Once the lo op enters the second evolutionary stage gravitational radiation b ecomes a
sub dominant energy loss mechanism. The lo op starts to collapse at a rate which grows
considerably b eyond the gravitational rate v by radiating other forms of energy, one of
g
them b eing X-particles. The decay pro ducts of these X-particles may then contribute to the
HECR ux observed at the present epoch. Weschematically assume here that during this
second evolutionary phase a fraction f of the total energy in lo ops smaller than a certain
critical length scale, L (t), is instantaneously released in form of X-particles. This is a go o d
c
approximation as long as the time which lo ops sp end in their second evolutionary phase is
short compared to the cosmic time t. Denoting the birth rate of closed string lo ops p er unit
volume b eing chopp ed o of the string network at birth time t by(dn =dt) we can then
b b t
b
write the rate of X-particle pro duction p er unit volume as
" #
3
dn dn dt R(t ) L (t)
X b b b c
(t)= f : (13)
dt dt dt R(t) m
X
t
b
Here R(t) is the cosmic scale factor and m = gv is the X-particle mass in terms of the
X
<
1). Furthermore, (dt =dt) takes symmetry breaking scale v and the Yukawa-coupling g (g
b
account of the time delaybetween the birth of a string lo op at time t and the nal phase of
b
3
X-particle evap oration at later time t. Finally, the factor [R(t )=R(t)] accounts for dilution
b
due to the cosmic expansion b etween t and t. If the string network exhibits scaling b ehavior
b
the birth rate of closed string lo ops can b e written as [44]
dn
b
= ; (14)
4
dt t
b
t
b
where is a dimensionless constant which is approximately related to by the relation
0:1 [52].
The p ossible existence of sup erconducting cosmic strings within certain GUTs was rst
prop osed by Witten [6]. Ostriker, Thomson and Witten [7] (hereafter, OTW) discussed quite 8
severe p otential cosmological consequences and also suggested that these ob jects might con-
tribute to the ultrahigh energy cosmic ray ux. This was further pursued by Hill, Schramm,
and Walker [8] who mainly investigated fermionic sup erconducting string lo ops which could
pro duce HECR by ejecting sup erheavy fermion pairs towards the end of their evolution.
With resp ect to the schematic scenario describ ed ab ovetwo cosmic ep o chs have then to
<
b e considered for sup erconducting cosmic strings of this typ e. For cosmic time t t , all
tr
existing lo ops are radiating dominantly in electromagnetic and/or X-particle radiation. In
terms of the (in general time dep endent) saturation length L (t) the relevant condition which
s
compares electromagnetic and gravitational energy loss rates reads [8]
" #
2
gL (t)
s
>v : (15)
g
t
In this case the lo ops have not exp erienced a gravitational radiation dominated energy loss
phase, but rather have directly entered the phase of comparatively fast collapse at birth. We
can therefore approximate t ' t and the critical length L (t) is the minimum of the birth
b c
length, L (t), and the saturation length, L (t). In contrast, for cosmic times t>t the
b s tr
epoch at which a string lo op reaches its second evolutionary stage is primarily determined
by gravitational energy loss, t ' (v = )t, and L (t)= L (t). Using Eqs. (13) and (14) this
b g c s
leads to the following time dep endence of the X-particle injection rate:
8
h i
3
< R(v t= )
g
4
dn
X t L (t) if t> t
s tr
R(t)
/ (16)
4
:
dt
t Min(L (t); t) if t< t
s tr
The saturation length for sup erconducting strings dep ends on the intergalactic magnetic
eld history [8] and is therefore strongly mo del dep endent. OTW originally considered an
intergalactic eld whose energy density scales like the CMBR energy density. In this scenario
the saturation length is roughly constant in time and can b e written as
!
2
1=3
B v
0 0
1 2=3
L (t) const: 10 g pc ; (17)
s
9 15
10 G 1 Mpc 10 GeV
where B and are strength and coherence length of the intergalactic eld to day. Using
0 0
Eq (15), the transition time t which separate the two string evolution ep o chs is in terms of
tr
redshift z given by
tr
!
1
1=2
2=3
v B
0 0
1=6 3
: (18) z =4:78 10
tr
9 15
10 G 1 Mpc 10 GeV
3
For the calculations p erformed in the following we will use z =210 . Since we will match
tr
the two functional time dep endences in Eq. (16) at t = t and since Min(L (t); t)= t
tr s
3
for t t we will use dn =dt / t for all t< t for a lower b ound on energy injection.
tr x tr
3
Furthermore, we will neglect the time dep endence coming from the factor [R(v t= )=R(t)]
g
in Eq. (16) in case t = v t= < t and t>t . A more detailed treatmentwould haveto
b g eq eq 9
takeinto account the chronological order of t , t and t as well as the nite collapse time
b eq
of a string lo op in its second evolutionary stage. This would b e mo del dep endent via the
parameters from Eqs. (17) and (18). It is, however, easily seen that such e ects lead to
X-particle injection rates which can only b e larger at early times than the injection rates of
our simpli ed treatment Eq. (16). Our calculations will therefore give us conservatively low
estimates for the total energy release in X-particles. Within these approximations Eq. (16)
> <
is of the form of Eq. (10) with p = 0 for t t and p = 1 for t t .
tr tr
In principle, for sup erconducting strings the energy radiated in form of X-particles is
determined by the mo del. In Ref. [53 ] it was shown that the ultrahigh energy particles are
absorb ed in the strong magnetic eld pro duced by the electric current in the string lo ops.
Instead, it was suggested that most of the string energy would b e lib erated in the form
of neutrinos [54 ]. We shall demonstrate here that even without these e ects in the OTW
scenario, where L (t) is approximately constant in time, it is barely p ossible to pro duce
s
the observed HECR ux for reasonable mo del parameters. It has b een shown [8] that in
scenarios where L (t) grows with time the saturation length at the present epoch, L (t ),
s s 0
has necessarily to b e smaller than the L (t ) in the OTW scenario. Such scenarios would, for
s 0
example, b e given when intergalactic magnetic elds are increased by dynamo e ects. In this
case it follows from Eq.(16) that the HECR ux at the present epoch can not b e pro duced
by sup erconducting cosmic strings even when f = 1. In the opp osite case (L (t) growing
s
<
1 can repro duce the observed HECR ux. in time) scenarios are conceivable where an f
However, for a given universal HECR ux more energy would have b een injected outside of
the strong magnetic eld region in the past compared to the OTW scenario. Therefore, if
to o much energy tends to b e injected within the OTW scenario, as will b e shown to b e the
case b elow, the other scenarios are also unlikely to b e able to explain the observed HECR
ux.
In the case of ordinary strings it has b een shown that well known physical pro cesses like
cusp evap oration are not capable of pro ducing detectable cosmic ray uxes [10 , 55 ]. It has,
however, b een suggested that a small fraction f of all lo ops could b e formed in states which
would lead to their total collapse within one oscillation p erio d after formation [45 ]. The
total energy in these kinds of lo ops would b e released in form of X-particles. Then, t t
b
and L (t)=L t so that
c b
dn
X
1 3
= f m t ; (19)
X
dt
which is of the form of Eq. (10) with p = 1. Recently, there has b een a claim [56 ] that lo ops
in high-harmonic states are likely to self-intersect and decayinto smaller and smaller lo ops,
nally releasing their energy in relativistic particles. Eq. (19) would b e a reasonable go o d
approximation also in this case.
Up to nowwehave only considered the functional form of the X-particle injection rate
dn =dt up to an absolute normalization. If we assume that HECR are pro duced by decaying
X
X-particles radiated from top ological defects we can normalize to the di erential HECR
ux j (E ) observed to day(t=t) at a xed energy E = E . In these mo dels one
H ECR 0 obs
20
>
exp ects to observe mainly -rays at energies E 10 eV [57 ]. We de ne the e ectiveX-
10
particle fragmentation function into -rays, (dN =dx)(x) where x =2E=m , as the e ective
X
di erential primary -raymultiplicity p er injected X-particle multiplied by2=m [10]. Then
X
the normalization dep ends on the -ray attenuation length (E ) and on (dN =dx)(x)at
x=2E =m :
obs X
# "
1
dn 2m dN 2E
X X obs
j (E ) (20) (t ) =
H ECR obs 0
dt (E ) dx m
obs X
! !
1
2
m (E ) j (E ) GeV cm sec sr
X obs H ECR obs
40
' 8:16 10
16 31
10 GeV 10 Mpc 4 10
" #
1
dN 2E
obs
3 1
cm sec :
dx m
X
In the last expression of Eq. (20) and in the following wehave used the numb ers for E =
obs
20
2 10 eV .
Using the parametrization of X-particle injection history, Eq. (11), and the normalization
Eq. (20) we are now in a p osition to derivevarious constraints on TD mo dels for HECR from
limits on energy injection into the universe.
3.2 Limits from Cascade Nucleosynthesis
3 3 4
In Ref. [1] the number N ( He;D;z)of He and D nuclei pro duced via He-photodisintegra-
tion p er GeV electromagnetic cascade energy injected into the universe was calculated as a
function of redshift z . These functions dep end only weakly on h and . Therefore, using
b
Eqs. (11) and (20) and assuming that a fraction f of the total energy release in high energy
c
particles go es into the cascade one gets
! ! ! !
1 1 1
3 2
2
He h h m (E )
b X obs
' 9:7f (21)
c
16
H 0:02 0:75 10 GeV 10 Mpc
photo
!" #
1
Z
2
dN 2E f (z) j (E ) GeV cm sec sr
obs H ECR obs
3
dz ; N ( He;z)
31 3
4 10 dx m (1 + z )
X
where the integral is p erformed over the range in Fig. 4 of Ref. [1]. An analogous formula
2
applies for the pro duced deuterium fraction ( H=H ) . Using Eq. (8) and the b ound
photo
3 2 4
( He + H)=H 1:1 10 we can imp ose the constraint
!
3
He + D
5
<
5 10 : (22)
H
photo
This leads to lower limits on the fragmentation function taken at x =2E =m whichin
obs X
the three cases discussed in the previous section read
8 9
5
# ! ! "
> >
1 1
for monop ole annihilation 1:4 10
2
< =
h h dN 2E
b obs
6
>
2:0 10 for ordinary strings f
c
> >
dx m 0:02 0:75
: ;
11 X
1:8 10 for the OTW scenario 11
! !
1
2
2
m (E ) j (E ) GeV cm sec sr
X obs H ECR obs
: (23)
16 31
10 GeV 10 Mpc 4 10
This has to b e compared with exp ected fragmentation functions in the di erent defect
scenarios. In case of monop oles and ordinary strings this function is mainly determined
by the hadronization of the fundamental quarks created in X-particle decays. At HECR
energies it is reasonable to assume a p ower law b ehavior [45, 46 ]. In sup erconducting string
scenarios the e ective sp ectrum of HECR, which if at all able to leave the high magnetic eld
region around these strings, could well b e altered byinteractions with these strong elds.
Nevertheless it is still reasonable to assume that at least at HECR energies this sp ectrum
hasapower law form.
It can easily b e shown that a prop erly normalized p ower law fragmentation function
q 2
(dN =dx)(x) / x (q>0) ob eys (dN =dx)(x) 2x for all q>0. Thus, b ecause of
Eq. (23) the OTW scenario is inconsistent with these p ower law fragmentation functions
3
>
6:9 10 . In contrast, the monop ole annihilation and indep endentof m as long as f
X c
ordinary cosmic string scenarios are compatible with reasonable fragmentation functions.
3.3 Limits from Cosmic Microwave Background Distortions
Early non-thermal electromagnetic energy injection can also lead to a distortion of the cos-
mic microwave background. We fo cus here on energy injection during the ep o ch prior to
recombination. A comprehensive discussion of this sub ject was recently given in Ref. [58].
Regarding the character of the resulting sp ectral CMBR distortions there are basically two
6 5
p erio ds to distinguish: First, in the range 3 10 ' z >z > z '10 between the ther-
th y
malization redshift z and the Comptonization redshift z , a fractional energy release u=u
th y
leads to a pseudo-equilibrium Bose-Einstein sp ectrum with a chemical p otential given by
' 0:71u=u. This relation is valid for negligible changes in photon numb er whichisa
go o d approximation for the Klein-Nishina cascades pro duced by the GUT particle decays we
3
are interested in [58]. Second, in the range z >z >z ' 10 between z and the recom-
y rec y
bination redshift z the resulting sp ectral distortion is of the Sunyaev-Zel'dovichtyp e [59]
rec
with a Compton y parameter given by4y=u=u. The most recent limits on b oth and y
were given in Ref. [12]. The resulting b ounds on u=u for instantaneous energy release as a
function of injection redshift [13] are shown as the dashed curve in Fig. 1.
Since energy injection by top ological defects would b e a continuous pro cess it is convenient
to de ne an e ective fractional energy release into the CMBR in the following way:
Z
z
th
u f m dn (z )
b X X
dz : (24)
4
u u dz (1 + z )
z
ef f rec
0
Here f is the fraction of the total energy release in high energy particles which contributes
b
4
to the CMBR distortion, u is the CMBR energy densitytoday, and (z ) is given by10
0
divided by the function shown as the dashed curve in Fig. 1. This e ective energy release is 12
4
constrained to b e smaller than 10 [13]. Similar to Eq. (23) this leads to the lower limits
8 9
5
" # !
> >
1
1:2 10 for monop ole annihilation
< =
h dN 2E
obs
5
>
1:5 10 for ordinary strings f (25)
b
> >
dx m 0:75
: ;
X
10
1:1 10 for the OTW scenario
! !
1
2
2
m (E ) j (E ) GeV cm sec sr
X obs H ECR obs
:
16 31
10 GeV 10 Mpc 4 10
These constraints are less stringent than the constraints Eq. (23) from cascade nucleosyn-
thesis. For the OTW scenario e ectivepower law fragmentation functions are inconsistent
>
0:11. with CMBR distortions for f
b
It should b e noted that in the sup erconducting string scenario there is an additional con-
tribution to the CMBR distortions even if HECR are not pro duced at all. This contribution
comes from the Sunyaev-Zeldovich e ect caused by the hot gas pro duced around the string
by emission of electromagnetic radiation b efore it reaches saturation length and p otentially
starts to emit HECR. This was discussed in Ref. [7]. Our restriction to distortions caused
by HECR alone therefore renders our constraints conservative.
3.4 Limits from the -ray Background
Electromagnetic cascades which are started at relatively low redshifts z pro duce an isotropic
-radiation in the observable energy range. The ux in this radiation puts an upp er limit
on the p ossible ux of ultrahigh energy particles. The most stringent constraint comes from
the upp er limit to the observed isotropic ux at E ' 200 MeV , whichwas rep orted to b e
8 2 1
7 10 ( MeV cm sec sr ) [61 ].
The limits derived b elow crucially dep end on the assumptions ab out fragmentation of
X-particles into the usual particles like protons, pions, photons, electrons etc., and on the
assumption ab out cosmological evolution of X-particle pro duction [see Eq. (10)].
We shall assume that the fragmentation function for the decay of X-particles with mass
m into particles i (i =p, ,e) has the form
X
(q 1)
2 dN dN E 1
i i i
(x)= (E ;m )= A ; (26)
i X i
m dx dE m E
X i X i
where E is the energy of particle i and A is a normalization constant. For q we shall fo cus
i i
on the values b etween q = 1 inspired by scaling distribution in inelastic pp-scattering and
q =1:32 according to QCD calculations [5].
As far as evolution is concerned we shall consider two cases: (i) absence of evolution and
(ii) the \weak" evolution, as given by Eq. (19)and inspired by the development of a network
of cosmic string lo ops [62 ]. The strong evolution with p<1 [see Eq. (10)] results in more
stringent limits and we shall skip it in this pap er.
Let us rst turn to the non-evolution case (i). Let the HECR ux observed at E =
obs
20 31 2 1
2 10 eV , j (E ) ' 4 10 ( GeV cm sec sr ) , b e caused by protons or -rays.
H ECR obs
1
3 1
The generation function for these particles in GeV cm sec can then b e found as
4
(E )= j (E ) ; (27)
i obs H ECR obs
(E )
i obs
which also leads to Eq. (20). This can b e extrap olated to other energies by using the fragmen-
tation function Eq. (26). In Eq. (27) i =p or , and (E ) is again the attenuation length
i obs
for these particles in the CMBR eld. From Eqs. (26) and (27) one can then nd the total
3 1
energy pro duction q in form of protons, -rays and electrons (i =p, ,e) in GeV cm sec .
i
The energy released in electrons and -rays (pro duced directly or through the decayof
other particles) go es into electromagnetic cascades (the cascade energy pro duction due to
protons is considerably less). Using the usual quark counting one can estimate that ab out
10% of the total energy release go es into electrons and thus into the cascades. The ux of
the cascade photons can then b e found as [63]
1
c (2=3)H q
cas
0
cas 3=2
j (E )= E ; (28)
obs
1=2
4
[2 + ln(E =E )]E
x
a x
where E and E are characteristic cascade energies which for z = 0 are given by E '
a x a
4 3
8 10 GeV and E ' 5:1 10 GeV , and q is equal to the energy release in the form of
x cas
electrons and -rays.
8
From Eqs. (27) and (28) we nd the cascade ux at E ' 200 MeV to b e 8 10 ,
8 9 2 1
3 10 , and 9 10 ( MeV cm sec sr ) for q =1:1, 1.2 and 1.32, resp ectively, assuming
16
m =10 GeV . These numb ers should b e compared with the observational upp er limit
X
8 2 1
7 10 ( MeV cm sec sr ) [61]. For q =1:32 the predicted ux is one order of magnitude
less.
Let us nowgoover to the case of evolution (ii). The cascade limit b ecomes more stringent
in this case b ecause the cosmological ep o chs with large z give no contribution to the presently
20
observed HECR ux at E ' 10 eV , while they contribute strongly to the cascade energy
density due to the enhanced energy release at earlier times. We shall restrict ourselves to
the case of weak evolution here where integration over redshifts results only in a logarithmic
factor.
It is easy to understand the existence of a \critical" ep o ch (with redshift z )inour
c
problem. It is de ned as E (1 + z )=E (z ), where E is a photon energy at z = 0 and
c x c
E (z ) is the turn-over energy of the cascade sp ectrum at redshift z .For E ' 200 MeV
x c c
one nds z ' 100. If weintegrate the evolution function Eq. (19) over the redshift interval
c
between z = 0 and z = z we obtain
c
1
c H q ln(z )
cas c
0
cas 3=2
(E )= j ; (29) E
obs
1=2
4 2 + ln[E (z )=E (z )] [E (0)]
a c x c x
where q is found with the help of Eq. (27) and the fragmentation function Eq. (26) using
cas
the energy transfer into p, and e at large redshifts.
6
For q =1:1, 1.2, and 1.32, the ux Eq. (29) at E ' 200 MeV is numerically 8 10 ,
6 7 2 1
, and 9 10 ( MeV cm sec sr ) , resp ectively.For X-particle masses di erent from 3 10 14
16 16 2 q
m =10 GeV these uxes havetobemultiplied by(m =10 GeV ) . These numb ers
X X
8 2 1
are considerably higher than the upp er limit 7 10 ( MeV cm sec sr ) as long as m is
X
16
not much smaller than 10 GeV . These considerations can b e translated into the lower limit
>
q 1:6 for the index of an assumed p ower law injection.
Note that Chi et al. [64] derived similar limits by considering cascade development in the
CMBR and in the infrared and starlight elds. These limits dep end to some extent on the
history and intensity of these less well known backgrounds. However, in the case of \weak
evolution" of TDs considered here the comparatively strong injection at high redshifts leads
to cascading probably mostly in the CMBR, whereas the authors of Ref. [64 ] were more
concerned with low redshift injection where these other backgrounds are more imp ortant.
As a conclusion we claim that for a fragmentation function of the form of Eq. (26) with
20
< < >
reasonable values for q ,1 q 1:32, the explanation of observed HECR at E 10 eV as
16
protons or -rays from the decay of GUT scale X-particles with m ' m ' 10 GeV
X GU T
is incompatible even with the \weak" cosmological evolution of their pro duction. The non-
evolution case is not severely constrained by these arguments.
4 Conclusions
4
Wehave discussed limits on cosmic high energy particle injection derived from He photo-
disintegration, CMBR distortions and the di use -ray background. Wehave found that the
3
>
5 10 nucleosynthesis limits give the most stringent constraints for ep o chs with redshift z
whereas at lower redshifts particle injection is predominantly limited by its contribution to
the di use -ray background (see Fig. 1). These constraints were applied to top ological
defects p otentially radiating sup ermassive GUT scale (\X") particles which subsequently
decayinto high energy leptons and hadrons. The history of high energy particle injection
is more or less determined within these defect mo dels. The mo del dep endent parameters to
b e xed are the numb er density of X-particles radiated within unit time and the e ective
fragmentation function for the decay pro ducts of these X-particles. Wehave assumed that
the ux of these decay pro ducts contributes signi cantly to the presentday observed HECR
ux. This allowed us to formulate our constraints as lower limits on the fractional energy re-
20
lease at HECR energies (' 10 eV ) which is mainly determined by the -ray fragmentation
function. Wehave found that for reasonable -ray fragmentation functions sup erconducting
strings can not explain the HECR ux without violating at least the b ound coming from
4
He-photo disintegration. In contrast, magnetic monop ole and ordinary cosmic string mo dels
pro ducing observable HECR uxes are most severely constrained, but not yet ruled out, by
their contribution to the di use -ray background.
In the second part of the pap er wehave studied the p ossibility that the presently ob-
4
served deuterium has b een pro duced by an epochof He-photodisintegration subsequentto
a standard nucleosynthesis scenario. Such an epochmayhave b een initiated by the decayof
particles, the annihilation of top ological defects, or, in general, the pro duction of energetic
<
-rays byany source. Wehave found that only a small fraction ( 10%) of the observed
4
deuterium mayhave its origin in the pro cess of He-photodisintegration since, otherwise, 15
3 2
anomalously large primordial ( He/ H)-ratios would result. A larger fraction of the pri-
mordial deuterium contributed by this pro cess would require either standard assumptions of
chemical evolution to break down or the existence of -ray sources in the early universe which
radiate with extremely \soft" -ray energy sp ectra. Wehave shown that a scenario which
employs massive black holes to repro cess the light element abundances from a standard big
2 3
bang nucleosynthesis pro cess [11] is in con ict with H and He observations. Wehave also
3 2 4
used the anomaly in the ( He/ H)-ratios pro duced during He-photodisintegration to slightly
tighten constraints on the abundances and parameters of decaying particles and top ological
defects.
Acknowledgments
This work was supp orted by the DoE, NSF and NASA at the University of Chicago, by
the DoE and by NASA through grantNAGS-2788 at Fermilab, and by the Alexander-von-
Humb oldt Foundation. This work was also p erformed under the auspices of the U.S. De-
partment of Energy by the Lawrence Livermore National Lab oratory under contract number
W-7405-ENG-48 and DoE Nuclear Theory Grant SF-ENG-48.
References
[1] R. J. Prothero e, T. Stanev, and V. S. Berezinsky, Preprint LNGS-94/106 (1994).
[2] D. Lindley, MNRAS 188, 15p (1979); V. S. Berezinsky, Nucl. Phys. B380, 478 (1992).
[3] J. Ellis, G. B. Gelmini, J. L. Lop ez, D. V. Nanop oulos, and S. Sarkar, Nucl. Phys.
B373, 399 (1992); see also references therein.
[4] S. Miyama and K. Sato, Prog. Theor. Phys. 59, 1012 (1978); D. Lindley, MNRAS 193,
593 (1980); T. Rothman and R. Matzner, Astrophys. Space Sci. 75, 229 (1981).
[5] C. T. Hill, Nucl. Phys. B224, 469 (1983).
[6] E. Witten, Nucl. Phys. B249, 557 (1985).
[7] J. P. Ostriker, C. Thompson, and E. Witten, Phys. Lett. 180, 231 (1986).
[8] C. T. Hill, D. N. Schramm, and T. P.Walker, Phys. Rev. D36, 1007 (1987).
[9] R. H. Brandenb erger, Nucl. Phys. B293, 812 (1987).
[10] P. Bhattacharjee, Phys. Rev. D40, 3968 (1989).
[11] N. Yu. Gnedin and J. P. Ostriker, Astrophys. J. 400, 1 (1992).
[12] J. C. Mather et al., Astrophys. J. 420, 439 (1994). 16
[13] E. L. Wright et al., Astrophys. J. 420, 450 (1994).
[14] P.J.E.Peebles, Physical Cosmology (Princeton: Princeton Univ. Press, 1971);
R. A. Sunyaev and Ya. B. Zel'dovich, ARA&A 18, 537 (1980); A. P. Lightman, Astro-
phys. J. 244, 392 (1981).
[15] J. I. Trombka and C. E. Fichtel, Phys. Rep. 97, 173 (1983).
[16] S. Dimop oulos, R. Esmailzadeh, L. J. Hall, and G. D. Starkman, Nucl. Phys. B311,
699 (1989).
[17] F. Balestra et al., Il Nuovo Cimento 100A N.3, 323 (1988); F. Balestra et al., Phys.
Lett. B230, 36 (1989).
[18] M. H. Reno and D. Seckel, Phys. Rev. D37, 3441 (1988).
[19] S. Dimop oulos, R. Esmailzadeh, L. J. Hall, and G. D. Starkman, Astrophys. J. 330,
545 (1988).
[20] J. Geiss, in Origin and Evolution of the Elements ed. N. Prantzos, E. Vangioni-Flam,
and M. Casse (Cambridge: Cambridge Univ. Press), 89 (1993).
[21] K. A. Olive, R. T. Ro o d, D. N. Schramm, J. W. Truran, and E. Vangioni-Flam, Preprint
FERMILAB-Pub-338-A (Septemb er 1994), submitted to Astrophys. J.
[22] A. Vidal-Ma jar, Adv. SpaceRes. 11, 97 (1991); R. Ferlet, in Astrochemistry of Cos-
mic Phenomena, ed. P. D. Singh (Dordrecht: Kluwer), 85 (1992); J. L. Linsky et al.,
Astrophys. J. 402, 694 (1992).
[23] T. M. Bania, R. T. Ro o d, and T. L. Wilson, in Origin and Evolution of the Elements, ed.
N. Prantzos, E. Vangioni-Vlam, and M. Casse (Cambridge: Cambridge Univ. Press),
107 (1993).
[24] R. I. Epstein, J. M. Lattimer, and D. N. Schramm, Nature 263, 198 (1976).
[25] A. N. Gorbunov and V. M. Spiridonov, Sov. Phys. JETP 34(7), N.4, 600 (1958);
Yu. M. Arkatov et al., Sov. J. Nucl. Phys. 19, 598 (1974).
[26] Note that this is not evident from the gures and tables given in reference [11],
N. Yu. Gnedin, private communication.
[27] for a review see R. Blandford and D. Eichler, Phys. Rep. 154, 1 (1987).
[28] T. K. Gaisser, Cosmic Rays and Particle Physics (Cambridge: Cambridge University
Press, 1990).
[29] A. M. Hillas, Ann. Rev. Astron. Astrophys. 22, 425 (1984). 17
[30] P. L. Biermann and P. A. Strittmatter, Astrophys. J. 322, 643 (1987).
[31] Astrophysical Aspects of the Most Energetic Cosmic Rays, eds.: M. Nagano and
F. Takahara (World Scienti c, Singap ore, 1991).
[32] C. T. Cesarsky, Nucl. Phys. B (Pro c. Suppl.) 28B, 51 (1992).
[33] D. J. Bird et al., Phys. Rev. Lett. 71, 3401 (1993); Astrophys. J. 424, 491 (1994).
[34] D. J. Bird et al., Astrophys. J. 441, 144 (1995).
[35] Proc. of Tokyo Workshop on Techniques for the Study of Extremely High Energy Cosmic
Rays, (ICRR, Univ. of Tokyo, 1993).
[36] S. Yoshida et al., Institute of Cosmic Ray Research (Univ. of Tokyo) Rep ort ICRR-
Report-325-94-20, to app ear in Astropart. Phys. (1995).
[37] N. Hayashida et al., Phys. Rev. Lett. 73, 3491 (1994).
[38] N. N. E mov et al. in Ref. [31 ], p. 20.
[39] T. A. Egorov in Ref. [35 ], p. 35.
[40] P. Sommers in Ref. [35], p. 23.
[41] G. Sigl, D. N. Schramm, and P. Bhattacharjee, Astropart. Phys. 2, 401 (1994).
[42] J. W. Elb ert and P. Sommers, Astrophys. J. 441, 151 (1995).
[43] P. Bhattacharjee in Ref. [31].
[44] P. Bhattacharjee, C. T. Hill, and D. N. Schramm, Phys. Rev. Lett. 69, 567 (1992).
[45] P. Bhattacharjee and N. C. Rana, Phys. Lett. B246, 365 (1990).
[46] P. Bhattacha jee and G. Sigl, Preprint FERMILAB-Pub-94/387-A, to app ear in Phys.
Rev. D.
[47] E. W. Kolb and M. S. Turner, The Early Universe (Addison-Wesley, New York, 1990).
[48] B. Allen and E. P. S. Shellard, Phys. Rev. Lett. 64, 119 (1990).
[49] A. Albrecht and N. Turok, Phys. Rev. D40, 973 (1989).
[50] D. P. Bennet and F. R. Bouchet, Phys. Rev. Lett. 60, 257 (1988); 63, 2776 (1989).
[51] B. Allen and R. R. Caldwell, Phys. Rev. D45, 3447 (1992).
[52] D. Austin, E. J. Cop eland, and T. W. B. Kibble, Phys. Rev. D48, 5594 (1993). 18
[53] V. S. Berezinsky and H. R. Rubinstein, Nucl. Phys. B323, 95 (1989).
[54] R. Plaga, Astrophys. J. 424, L9 (1994).
[55] A. J. Gill and T. W. B. Kibble, Phys. Rev. D50, 3660 (1994).
[56] X. A. Siemens and T. W. B. Kibble, Preprint hep-ph/9412216.
[57] F. A. Aharonian, P. Bhattacharjee, and D. N. Schramm, Phys. Rev. D46, 4188 (1992).
[58] W. Hu and J. S. Silk, Phys. Rev. D48, 485 (1993); see also references therein.
[59] Ya. B. Zel'dovich and R. A. Sunyaev, Astrophys. Space Sci. 4, 301 (1969).
[60] R. Svensson and A. A. Zdziarski, Astrophys. J. 349, 415 (1990).
[61] C. E. Fichtel et al., Astrophys. J. 217, L9 (1977); 222, 833 (1978); D. J. Thompson,
and C. E. Fichtel, Astron. Astrophys. 109, 352 (1982).
[62] A. Vilenkin, Phys. Rep. 121, 263 (1985).
[63] V. S. Berezinsky, S. V. Bulanov, V. A. Dogiel, V. L. Ginzburg, and V. S. Ptuskin,
Astrophysics of Cosmic Rays (North-Holland, 1990).
[64] X. Chi, C. Dahanayake, J. Wdowczyk, and A. W. Wolfendale, Astropart. Phys. 1, 239
(1993).
Figure Captions
Figure 1: Maximal energy release in units of the CMBR energy density allowed by the
constraints from the observed -ray background at 200MeV (dotted curve), CMBR distor-
4
tions (dashed curve, from Ref. [13]), and He-photodisintegration as a function of redshift z .
These b ounds apply for instantaneous energy release at the sp eci ed redshift ep o ch. The
logarithm is to the basis 10. 19