SLAC - PUB
June 1987
T/AS
Is the Universe
Nucleosynthesis
-
with
Closed by Baryons?
a Late Decaying
SAVAS
University,
RAHIM
Stanford
i..
Stanford
-
Particle*
of Physics
Stanford,
California,
94305
ESMAILZADEH
Linear
University,
Massive
DIMOPOULOS
Department
Stanford
- 4356
Accelerator
Stanford,
LAWRENCE
J.
Department
Center
California,
94305
HALL
of Physics
-- -
University
of California,
GLENN
Berkeley,
University,
Qa 720
D. STARKMAN
Department
Stanford
California,
of Physics
Stanford,
California,
94305
-- -
Submitted
_. .-
*
to The Astrophysical
Work supprted in part by the Department
Journal
of Energy, contract DEAC03-76F00515.
4
._
ABSTRACT
:A late decaying
(7 > 105sec), massive (M
new phase of nucleosynthesis
-
place when the hadronic
’ hadronic
4 He causing
in the keV era.
decay products
in which hadroproduction
independent
role is to provide
are in- agreement with
..
- .
element
with
initiates
production
the a m b ient
protons
The primordial
balances photodestruction.
a
takes
and
abun-
Since fixed
abundances of these elements are
of the physics before the keV era, whose only important
at least the observed abundance of 4He.
of 4He is subsequently
particle
given by the fixed points of the corresponding
points erase all previous memory, the primordial
completely
Light
showers and 4 He hadrodestruction.
dances of D, 3He, 6Li and ‘Li-are
rate equations
interact
2 10 GeV)
hadrodissociated.
observations;
The primordial
furthermore
Any overabundance
element
they are independent
for a very broad range of ClBhz which
includes the previously
0.03 -5 flBhz
for the late decaying particle
5 1.1. An ideal candidate
gravitino.
-- -
-- -
2
abundances
of RBhz
forbidden
range
may be the
4
._
- 1. Introduction
One of the most fascinating
is that
at least 9 0 % of the matter
“dark
-
matter”
in the universe is invisible.
or
could consist of the usual baryons and leptons in states that
do
or absorb with
ternatively,
it could consist of new particles
sufficient
The most compelling
ternative
comes from primordial
intensity
to be observable at present.
that
reason in favor of the latter
nucleosynthesis
ful implementation
requires that
0.014 < RBhz
0.03 then 4He and 7Li are overproduced
non-baryonic
al-
( Peebles 1966; Wagoner 1967;
1984); its success-
< 0.03. If RBhz
and D is underproduced
is larger than
(eg. Wagoner
-
O ther arguments
m icrowave
-
Al-
do not e m it electromagnetic
Wagoner 1968; Wagoner et al 1967; Yang et al 1984; Primack
1967).
cosmology
This invisible
not radiate
radiation.
_-
consequences of recent observational
background
and the separation
In fact,
against baryonic dark matter
recent
radiation
(where adiabatic
of visible from invisible
observations
arising from distortions
matter
perturbations
led Peebles (Peebles 1987a,b) to consider purely
are assumed)
are less solid (Primack
of the large scale structure
baryonic
of the
1984).
of the universe
universes;
have
similarly
-- Blumenthal,
thal,
Faber, F lores and Primack
Dekel and Primack
!-lBh;
(Blumenthal
(Blumenthal
et al 1987) have considered theories with
> 0.03.
W e are aware of two classes of attempts
RB = 1 using particle
al 1987) and Alcock,
barybn
segregation
to distinct
_. .-
et al 1986) and Blumen-
physics.
Applegate,
Fuller and Mathews
and neutron
diffusion
regions of nucleosynthesis.
nucleosynthesis
Hogan and Scherrer
(Alcock
with
(Applegate
et
et al 1987) have shown that
after the QCD
W ith
3
to reconcile
suitable
phase transition
choice of unknown
lead
QCD
4
._
the net yields of the light elements in an DB = 1 universe could be
parameters
m a d e to agree with
observation,
three orders of m a g n itude.
except that
The importance
new physics to the standard
In practice
thesis operates.
calculations
of Audouze
Tenreiro
no
QCD parameters
are indeed
this could be the way standard
nucleosyn-
it is very difficult
to make thoroughly
convincing
using these ideas.
A second class of attempts
_-
by two or
of this work is that. it introduces
m o d e l. If the unknown
in the range chosen by these authors,
-
7Li was overproduced
et al (Audouze
to reconcile RB = 1 with
et al 1983) and Dominguez-Tenreiro
1987). Audouze et al try to solve the deuterium
lem by introducing
a late radiatively
to photodissociation
nucleosynthesis
of 4He into D.
decaying particle
Unfortunately,
account for the 4He, D and 3He abundances.
such scenarios with purely radiatively
is that
(Dominguez-
underproduction
prob-
whose decay products
lead
they cannot simultaneously
In particular,
we will show that
decaying X’s neccessarily lead to either too
much 3He or too much 4He.
Dominguez-Tenreiro
sociation
-- -
claims .to have a positive
and anti-nucleons;
scenario using only photodis-
however, he ignores several d o m inant
processes, in-
&ding:
a)photodissociation
cess that can sufficiently
b) hadronic
of the overabundant
reduce their abundance without
getting
rid of 4He ;
showers, the m a in source of D, 3H, 3He, 6Li, 7Li, and 7Be.
Our general framework
standard
mass seven elements, the only pro-
is much broader than the case nB = 1, or even the
big bang. It applies to any theory which, in the 10 keV era (t N 104sec),
has-an overabundance
of the remaining
of 4He and is independent
of the pre 10 keV abundances
elements . In such a theory, we add a long lived (7 2 105sec)
4
massive
(M
X 1OGeV)
general properties
X interact
with
reactions
that
reduction
-
particle
the ambient
and photon
3He,
parameters.
6.Li, 7Li
the abundances
of the light
triggered
induced
of D, 3He, 6Li,
out to’ be independent
previously
ratios
-- -
forbidden
with
absolute
nuclei
and destruction
fix ed points
for a very
erases all memory
of initial
magnitude
of the primordial
depend on one combination
by the requisite
the ratios
range 0.03 < StBhz 5 1.1.
observations
amount
Thus,
abundances
of hadrodestruction
of
conditions,
of the physics
of the X for a
of these abundances
is a nontrivial
of the properties
range
The fixed points’ are essentially
of RBhz for a very large range of RBh:
the relevant
7Li and
are in turn
wide
7Li are camp letely independent
Furthermore,
chain
of the light
by nuclear physics and do not depend on the properties
wide range of its parameters.
-
6Li,
of
by the X decay. The rate equations
that went on before the decay of the X particle.
determined
D, 3He,
These light
production
behavior
decay products
and cause nuclear
nuclei
4He.
reach their
Since fixed point
The baryonic
and alpha particles
overabundant
by the radiation
hadron
D,
protons
lead to the production
photodissociated
elements
decays in the few keV era and whose
are stated in the next section.
of the previously
describing
X that
that
includes
the agreement
test
turns
the
of these
of these ideas.
The
of D, 3He, 6Li and 7Li does
of X. Another
combination
of the initially
overabundant
is fixed
4He.1
1
The
necessary
If RBhg
4He.
=
If nBhz
amount
1 then
of
4He
hadrodestruction
we need to hadrodissociate
< 1 we n eed to hadrodestroy
approximately
less. Our fractional
-- -of 3He, 6Li and 7Li are given by the fixed point condition
on RBh:.
5
depends
on
RBh:.
15% of the
abundances
and do not depend
4
._
These two combinations
completely
determine
the primordial
abundances of D,
3He, 4He, 6Li and 7Li in our framework.
In section 2 we introduce
the general requisite
In section 3 we describe the observations
-predictions.
-
against which
In section 4 we set up the formalism
showers, electromagnetic
by the decay products
properties
we must compare our
for the study of the hadronic
showers, and the nuclear processes that
of X in the hot a m b ient plasma.
solutions
of the rate equations that demonstrate
fixed point
behavior,
flBhz
estimates.
simulations
etc.
which confirm
In section 7, we contrast
Bang Nucleosynthesis,
F inally,
independence
and identify
in section 8 we summarize
are triggered
In section 5 we present
the analytic
sults of numerical
of the X particle.
the aforementioned
In section 6 we present the re-
and refine our analytic
our predictions
with
fixed point
those of Standard
Big
ways in which their differences may be tested.
our results and present our conclusions.
-.
-- -
6
,- -
4
._
2. Properties
of the Decaying
In this section we describe the properties
responsible
for altering
the light
depends on the lifetime
-
mass Mx,
decay.
unimportant,
of the particle
abundances.
7-x, the baryonic
and the abundance relative
W e will
simplicity
element
branching
the X particle
lifetime
In order not to distort
synthesis
in X decay r~,
the
later.
W h ile
the spin of X is
our framework
numbers.
neutral.
is independ.ent
However,
For
of physics
for simplicity,
we take
to be > lo4 seconds in order to ensure that X decays are
late enough not to substantially
2-
ratio
we will assume that it is a color singlet and electrically
before lOkeV, so long as 4He is overproduced.
(Silk’and
The element
must be m a d e about the other quantum
As stressed in the introduction,
-
X whose decays are
to photons f$ which X’s have before they
discuss each of these parameters
assumptions
Particles
disturb
the conventional
the cosmic background radiation,
era of nucleosynthesis,
we take 7-x < 10’ seconds
Stebb’ms, 1983). In fact our results show that
rx lies in the range of
9 X lo5 sec.
-- -
In the analysis of this paper we assume that X does not carry baryon number
and does not have a large scattering
This
assumption
is necessary if fg
cross-section
2
f~
=
with
nB/n,B
protons
but
and neutrons.
is unnecessary
for
2
If X carries baryon number
and
fx
2
fB,
then its decays will
-- -By more than a factor of two. If X has a large scattering
nucleons, then our hadronic
shower calculations
7
dilute
4He
cross section with
are altered.
,-. -
4
._
Our results show that rg should be in the approximate
:;
range of 0 (10e5) to
0 (1). It is reasonable to be anywhere in this range. X could decay directly
final state containing
-
involve
baryons,
loops or virtual
in which case r~ = 0 (I),
.. _-.
Our results
also. show that
order lo6 GeV; hence, the X particle
there is an upper
at the weak scale. Rather this long lifetime
can arise naturally
scale or Grand-Unified
results
particle
Successful gravitino
show that
f$/ f B
to Mx
of
quarks and leptons
from a dimension
5 operator
with
at the Planck
its mass and lifetime
in
cosmologies with 0.1 < nBh2 2 I
et al 1987).
are discussed elsewhere (Dimopoulos
dur
with
scale, or a dimension
scale. A familiar
this range is the gravitino.
bound
is very long lived for its mass. It is most
interactions
at the intermediate
,m
mass so that it can decay
reasonable that X not have renormalizable
6 operator
-
or such a decay m ight
states, rB = 0 (10e5 - 10m3).
intermediate
The X mass must be larger than twice the proton
baryonically.
to a
should be in the range 0 (lo-‘)
to 0 (102).
Such an abundance may have an origin related to the cosmic baryon asymmetry,
-- -
in which case
annihilation
f-$
-
fB
is reasonable.
Alternatively
rate would give acceptable values for
X has only Planck scale interactions.
scale after inflation
would give
f;k
if Mx
fg.
< Mw,
Another
In this case, reheating
then a weak
possibility
is that
to the intermediate
in the desired range.
3
Our analysis does not apply to the case that the X carries negative baryon
-- Yirumber
and is light enough that the average anti-nucleon
of the decay products
is an order of m a g n itude
8
or more.
to nucleon ratio
-
The allowed
fg
ranges of Mx-and
xf”
A
particles
-
never dominate
radiation
during
with
dominated
such that
Mx
fB
When this is combined
are correlated
1OOGeV - 0(1)’
the allowed values for TX , it is apparent
the energy density
of the universe.
that the X
The universe
at the era of X decay. Hence the time-temperature
is still
relation
this era is given by:
and X decays occur in the keV era.
We ignore the possibility
-neutrinos.
Decays to neutrinos
of interest,
extremely
-
that X has totally
the -probability
small?
are invisible
invisible
in the sense that,
of a decay neutrino
If such totally
invisible
decays, for example
interacting
into
for the rx and MX
in the hot plasma
is
decays occur, our results still apply but
.
4
The few neutrinos
abundances.
-- -
For neutrinos
neutrino-nucleon
are Dp--+e+n
become
parameters
via np+dy,
present
of interest
nB
cannot
of sufficiently
pions,
produces
and the latter
low energy that
the dominant
neutrons,
processes
some of which
depletes
4He, since the vast
at the keV era are in 4He.
For the ranges of X
to us, these abundance
= 1 scenarios
-- --dances are substantially
produce
The former
and un+ep.
of neutrons
in the plasma can alter the light element
and anti-neutrinos
scattering
deuterium
majority
However,
which do. interact
can be constructed
adjusted
by neutrino
these here, since the 7Li overabundance
9
modifications
are negligible.
where
D and 4He abun-
reactions.
We do not pursue
is not addressed by this mechanism.
with
f$
appropriately
in the electromagnetic
plasma,
with
for example
proper
resealing
increased.
We also assume that there is no missing
and baryonic
in neutrinos,
of Mx,
decays.
If some energy is lost from
then our analysis
rg and
and results
still
energy
the hot
apply,
but
f$.
-
,a.-
10
-
4
._
-3. Observations
Unlike
2;
many other fields in modern cosmology, there exists a wealth
with which to test our theoretical
light elements.
Since the early qualitative
have attempted
to determine
the very small fractional
dial abundances.
observational
difficulties
light element abunin measuring
precisely
the great challenge of this pro-
relate the measured abundances to actual primor-
In this chapter we briefly
field as it pertains
the primordial
abundances involved,
gramme has been to correctly
summarize
the current
status of the
to this work.
DEUTEkIUM
Deuterium
-
synthesis of the
successes of Big Bang nucleosynthesis
more accurately
dances. Besides the considerable
..
of the primordial
and Hoyle 1967: Wagoner 1967; Wagoner 1968), astronomers
-(Wagoner,Fowler
-3.1.
understanding
of data
is detected mostly
I and deuterated
m o lecules.
in the interstellar
through
the isotopic shift of lines of atomic H
This approach has been used in stellar atmospheres,
gas and in planetary
atmospheres.
Stellar
D has not been
detected, though upper lim its have been obtained.
A conservative
lim it is D/H
10m6 (Boesgaard and Steigman 1985). Deuterium
in the interstellar
<
-- been detected
in the Lyman
stars and to more distant
interstellar
profile, m u ltiple
stellar H I complicate
m e d ium has
lines of H I along the line of sight to nearby cool
0 and B stars; however, stellar
contamination
clouds in the line of sight, and high velocity
the derivation
of the interstellar
D/H
of the
“puffs” of
values (Vidal-Madjar
et al 1983; Gry et al 1983; Gry et al 1984). Boesgaard and Steigman take the most
probable
interstellar
et al claim
that
D/H
value to be 0.8 - 2.0 x 10m5, although
5 x 10m6 - 5 x 1 0 m 5 is consistent
11
with
Vidal-Madjar
observations
(Vidal-
Madjar
et al 1983) and some-authors
along the line of sight toward
I observations
nearby
of deuterated
-and MacFarlane
Jupiter
and Combes
fractionation
affected
its abundance.
To determine
and other
the primordial
rate determination
enhancement
abundance
of the current
of evolution
(Dupree
abundance
of the abundance
et al 1977).
troposphere
1982).
Voyager
yield
Some authors
a D/H
(Hubbard
of D present in the atmospheres
will be close to the presolar
chemical
values as low as 2.4 x 10e6
in the Jovian
1980) argue that the amount
and Saturn
standing
cool stars
molecules
range of 1.2 - 3.1 x 10m5 (Encrenaz
-
have quoted
value, while
others
of
believe that
processes may have significantly
of deuterium
requires
and a complete
to the present
both
an accu-
theoretical
under-
day. During
the chemical
evolution
of the galaxy any primordial
D which is processed through stars is
completely
destroyed.
The efficiency of processing depends on the birth rate
:\
..
function,
the initial
mass function,
mass loss rates,
infall
to and mixing
in the
- .
galactic
ficient,
disk, etc.
Various
so the present
value (for example
models
abundance
suggest that
astration
of D could be l/2-
is 50% to > 90% ef-
< l/20
of its primordial
Page1 1982).
-- -
We take as our range for the post-astration
abundance
of deuterium
5.0 x 10m6
to 5.0 x 10-5.
3.2.
HELIUM-
As stated
mordial
by Boesgaard
nucleosynthesis
munication,
destroyed
3
A.M.
is very unclear
Boesgaard
and produced
and Steigman
private
the value of 3He as a probe
at present
(also R. Wagoner
communication).
(by conversion
private
com-
Since 3He can be both
of D and by incomplete
12
of pri-
H burning)
in
4
.-
stars, the present abundance ceuld be much greater or much less than the primordial abundance.
abundance.
The situation
Presolar
is exacerbated
abundances inferred from measurements
the solar wind are approximately
while observations
-
1 - 2 x 1 0 m 5 (Boesgaard
nia et al, as communicated
with any physical parameters
by Boesgaard)
to account for by stellar
The pre-Pop
on meteorites
and Steigman
and
1985),
of H II regions yield a wide range of values as high as 1.5 x 10m4,
with no apparent correlation
be difficult
by the wide range in the observed
(Rood et al 1984; Ba-
Such a range of 3He abundances may
processing of the solar system abundances.
I average 3He abundance
could be significantly
greater
than the
solar system value.
W e adopt as our maximum
range for the pre-Pop I average 3He abundance
1 x 1o-5 - 1.5 x 10-d.
:
3.3.m
HELIUM-4
-.
According
to Boesgaard and Steigman the best value for the primordial
lium mass fraction
Yp comes from isolated extragalactic
results from various authors give Yp = 0.245f0.003
-. -
Steigman,
G a llagher
and Schramm
(1987) reported
H II regions.
(Kunth
Combined
1983). More recently
0.235 f 0.012, while
(1987) has quoted 0.22. The actual value will not be of great significance
work,
except to affect, in an easily computed
13
Page1
for our
fashion, the allowed branching
tio and abundance of the X, and the range of permitted
scenarios.
he-
ra-
pre-keV nucleosynthesis
4
._
LITHIUM
3.4.
The 7Li abundance observed in stars varies over a wide range. 7Li is produced
A
by spallation
2.5 x lo6
-
destroyed
and can be destroyed in stellar regions which are hotter
OK by the reaction
7Li(p,a)4He.
in a region of a star is a strong function
lo6 years for T = 3.5 X lo6
region:
The timescale
et al 1984).
(Maurice
hence different
on which
of the temperature
of that
7Li is circulated
from
depletions of surface 7Li.
I stars,
such as the sun, this has led to significant
lifetime
of the star.
T,, and
In many population
depletion
of 7Li
over the
It has been argued that the 7Li abundance seen in many low metallicity,
dwarf, population
..
II stars reflects the true primordial
abundance
little
is constant
7Li depletion
(at N lo- ” the hydrogen
has occurred
halo
abundance (Spite and Spite,
1982). The evidence for this claim is that, for 5500° < Teff
-.
OK
the surface
of the star, such as its effective temperature
stars have different
7Li is
OK and 1011 years for T = 2.0 x lo6
The depths to which
depends on various properties
than about
< 6300°, the 7Li
abundance),
in any of these stars. For Teff
suggesting
that
< 5500” the 7Li
abundance drops sharply.
-- It is known from studies of the spectral
these old stars is predominantly
lineshape that
7Li and not 6Li (Maurice
the lithium
seen in
et al 1984). Although
6Li has not been s een in these stars, it could be there at a 1 0 % level compared
with
7Li.
6Li is destroyed in stars by the process 6Li(p, 3He)4He.
ature, this occurs with
Henie
a star with
a cross-section
even m ild
depending on the fraction
_- .-
At any given temper-
almost 100 times that of 7Li destruction.
7Li depletion
of surface material
14
can have very large 6Li depletion,
convected to regions of sufficiently
In the dwarf
high temperature.
where
little
7Li depletion
is probable.
According
is argued
-a temperature
to one theoretical
of T = 2.3 x lo6
this case essentially
Another
analysis
possibility
The observed
the 10-l’
(1982),
a large depletion
of 6Li
must be convected
and diffused
OK for all these stars (independent
is fully convective
down to
of Te). In
be destroyed.
is that 6Li will be efficiently
D.Vandenberg
and Spite
of the 7Li data of these stars
all the 6Li of these stars would
phase, if the protostar
L.Nelson,
by Spite
to have occurred,
et al 1986) th e surface material
(Rebolo
-
halo stars studied
destroyed
in the protostellar
at a high enough temperature
(D.Bond,
in preparation).
7Li abundance
seen in Population
in Population
II. This
I is 0 (lo-‘);
suggests that
this is higher
the 7Li
abundance
than
in the
disk has been enriched.
There
is also a reported
of sight toward
-.
primordial
ratio
evolution
sonable
limit
of 6Li/7Li
< l/10
two stars (Snell et al, 1981).
depending
on the history
of the disk depends
assumption
that
This may or may not represent
of the intervening
on the amount
6Li is destroyed
in the ISM along the lines
of galactic
in the astrated
the
gas. The chemical
astration.
matter
It is a rea-
(Audouze
et al
-- 1983).
With
abundance
this in mind
to be 5 x lo-l1
we consider
to 5 x lo-‘.
a maximum
range for the primordial
7Li
4
._
4. Physics of the KeV Era
:,
W h e n a particle
of mass greater than a few GeV decays during the keV era,
some of the energy goes into elecromagnetic
hadrons.
-
Both
classes of decay products
the background
plasma.
showers and write
In this chapter
down the resulting
decay products,
and some goes into
form showers as they thermalize
we examine the development
dynamical
with
of these
equations for the abundances of
the light elements.
4.1.
ELECTROMAGNETIC
SHOWERS
W h e n a massive X decays electromagnetically
_ energetic
photons,
kev) background
particles
-.
electrons
plasma.
In thermalizing
into the relatively
with
They found that
matic photons
cool (7’ = 1 - 30
the background,
relativistic
electron-photon
these energetic
when energetic
photons
are injected
of energy E,, the d o m inant
mechanism
This has a kinematic
resulting
lose energy chiefly through
shower electrons,
ing off background
photons.
After
in a plasma.
into a gas of monochrofor photon
threshold
energy loss is
Eth = mz/T
inverse Compton
several pair-production
photons are below the pair-production
and Varda;
shower production
e+ - e- pair production.
remaining
it injects very
create showers. Burns and Lovelace (1982) and Aharonian
nian. (1985) studied
-. -
and positrons
or hadronically,
threshold
. The
scatter-
m e a n free paths, the
and are found to have
the energy spectrum
-- .
These post pair-production
photons
are called “breakout”
16
photons.
In the ab-
sence of ambient
ton spectrum
charged
would
-
with
photons
photons
temperature
T. Energetic
plasma,
photons
background
ing becomes
- photons
comparable
the now dominant
Note that
E,,,
-- -
bound
have a thermal
can pair produce
This
en-
and fP = fe by charge neutrality
( T < m,)
. The
lose energy
scattering
is time dependent
by Compton
is continued
scattering,
until
Compton
shower
and photo-dissociation,
and monotonically
between
it depends on flgh z. This dependence
scatter-
scattering
photons
have
using
as described
later:
As E,,,
elements,
to photo-destruction
Compton
off
distribution
increasing.
of the various
species become susceptible
mostly
process for the .most energetic
the most energetic
thresholds
E maz depends on the interplay
off ambient
lowers the threshold
From then on, we evolve the photon
Compton
dis-
, with
At this point
passes the photodissociation
tightly
the pho-
and protons
to the pair production
--mz/25T.
they
also contains
The shower evolution
of the shower.
energy Ema,
scaterring,
electrons
can therefore
electrons.
photons
distribution.
however,
fp E rap/n7 = 2.8 10m8RBhz,
energetic
Delbruck
are not monoenergetic,
in the tail of the thermal
ergy. The primordial
-.
and neglecting
not evolve further.
The background
tribution
particles,
sur-
more and more
(cf. fig.1).
Since
and pair production,
is weak and has only a slight effect on our
results.
We now discuss the normalization
carry
baryon
number,
or if M x
of the photon
>> lGeV,
- 10% of its energy in the form of baryons.
a.sma.11 fraction
of Mx.
the break out spectrum
Ignoring
spectrum.
the average
to have total
energy Mx.
17
hadronic
Hence, decay baryons
energy loss to neutrinos,
If X does not
jet
carry
we therefore
Defining
has only
&,(E)dE
off only
normalize
to be the
4
._
number of breakout
photons of energy E produced
per X decay, we take5
0 I E F Emax
P-2)
E max I E ,
which is plotted
-
in fig. 2. These photons are responsible for the photo-dissociation
of the light elements.
4.2.
SHOWERS
CONTAINING
PRIMARY
BARYONS
W h e n an X decays, it has some branching
ratio r~ to produce baryons . The
average number of baryons produced in a hadronic
e+ -e-
jet data (Schwitters
1983) at E,,
- average about five nucleon-antinucleon
= Mx
pendent on E,,.
-
.
. For Mx
pairs, with
and even smaller values as Mx-+lGeV.
X decay can be deduced from
= lTeV,
there are on
- 5 GeV energy per nucleon,
The m u ltiplicity
is logarithmically
W e take the average numbers of jet protons
de-
and jet neutrons
to be equal.
As a proton
bremsstrahlung
-- -
verse Compton
electrons
moves through
off background
scattering
and plasmon
the background
protons
off thermal
excitations
greater than - 1 GeV the d o m inant
below 1 GeV Coulomb
scattering
plasma,
or electrons,
it can lose energy by
synchrotron
photons Coulomb scattering
and strong interactions.
radiation,
in-
off background
At kinetic
energies
energy loss mechanism is strong interactions;
and excitation
of plasma oscillations
d o m inate
5
Although
this power law may be altered
-- %y the averaging over the thermal
tively
insensitive
by the inclusion
photon distribution,
to the exact shape of the spectrum.
18
of Delbruck
or
our results are rela-
._
.
(though
small).
the rate for inverse Compton
Neutrons
lose energy only through
W h ile strong interactions
d o m inate
scatter many times off background
energetic
-
d,3H,3He
can collide with
light
background
their energy loss, the energetic
protons and alpha particles,
is
alphas producing
baryons
producing
several
some of the background
nuclei lose energy by Coulomb
energetic baryons thermalize
W e proceed to calculate
is large, the energy transfer
strong interactions.
and 4He, and destroying
Even as these energetic
remaining
scattering
6Li,7Li
alphas.
scattering,
they
and 7Be. Eventually
the
and join the sea.
&, the number of nuclear species i produced
per X
decay. Let
rp’a(Er)
= &
n,& (E,.)
xkb ngrkb(Er)
(4.3)
!!
be the probability
that a baryon r (r = n,p) of energy E,, scatters off a nuclear
specie8 a (a = p, CZ) via reaction
-
(Table 1). Let tik(E,-,Em)
Em per reaction
j,,
where j,
be th e number
runs over the reactions
of species m produced
in table’
with
energy
ja at baryon energy E. Then
-- -
is the average number
of species m of energy Em produced
by a baryon
r of
energy E, in one baryon collision.
For example,
alpha particles
consider a 5 GeV neutron
is one-tenth
the number
(cf F ig. 3). The number
density of protons.
alpha cross section at 5 GeV is 130 mb. The total
is-4Pmb.
Therefore
alpha is 25%.
the probability
The inelastic
of scattering
neutron-alpha
19
The total
neutron-proton
off a proton
density of
neutron-
cross section
is 75%, and off an
cross section at 5 GeV is 100 mb;
,m -
4
of that 30 m b is na -+
nP;ff-dpnn
dpnn,- and 11 m b is na! d
ddn. Therefore
= 0.25 x +j$ x $& = 0.06
and
nPzff-ddn
= 0.25 x +k$ x $$ = 0.02.
,-
-Thus
-
*gd = 0.06 + 2
X
0.02 = 0.10.
It is essential to include secondary baryons in the shower calculations.
For
baryon energy (X 1 GeV) th ere are three classes of secondary baryons:
large initial
a) baryons which
have elastically
energy;
b) baryons
of their
initial
which
energy;
parts of the target
Unfortunately
scattered,
retaining
have inelastically
c) debris baryons
almost
scattered,
which
losing
all of their
a fraction
are the old target
alphas; these have energies of approximately
i is not well determined
taken -to be (Bucza,
private
1 - E
protons
or
50 to 200 MeV.
in the energy range of interest.
communication)
initial
between 4 0 % and 75%.
It is
At lower
-.
Neutrons
energies these classes become less distinct.
-
25 MeV
processes.
-- -
where the elastic neutron-proton
Protons
energy by Coulomb
,I.
To obtain
cross section d o m inates
1 GeV below which
must be traced to scattering
down to
all other
they rapidly
lose
and plasmon excitations.
the (i’s, we must sum the contributions
etc. baryons,
must be traced
average over the neutron
and proton
of all secondary, tertiary,
contributions,
and m u ltiply
by the baryon m u ltiplicity:
cm
=
VB
X :xX
r=n,p
E,
c
E,t,E,tt,...>lGeV
c
E,I,E,tI,...>25MeV
-- -
{rgm(Ein,Em)
+r gnr (Ein, E,,)[,,gm(E,,,Em)+
.
20
(4-5)
-
4
._
+nrgnrt(Ent,
En,,)
W e have taken the initial
-
the baryon m u ltiplicity,
(nrrPrn(Enlr,
Em)
energies, Ei,,
+
* * *) +nt
gprr(pt/grn
of the primary
branching
Using the procedure
_-
vg, to be 5, which are their values at MX
outlined
above, we calculate
spectively
the average number of deuterons,
neutrons,
produced
scattered
out of the thermal
particles
destroyed.
A posteori,
-
per X decay; (*Hi,
scatter
tritons,
&j,&H,
is partly
into
EsHe, and en, re-
3He nuclei, and - 25 MeV
the average number
sea to MeV energies; and (E,
of alpha particles
the number of alpha
The results are listed in Table 2. All the cross-sections
and
of the e’s are found in Table 3.
the values of these e’s can be easily understood.
a 5 GeV nucleon takes approximately
producing
on Mx
= 1TeV. As
ratio rB.
other data necessary for the determination
..
* . *)I+
baryons to be 5 GeV, and
discussed in section 4.3, the dependence of these quantities
the baryonic
+
eight scatterings
two or three 200 MeV neutrons
en route.
two or three more times before falling
For E = 0.25,
to fall below 500 MeV,
These 200 MeV neutrons
below the threshold
for helium-4
-- destruction.
Thus, about fifteen scatterings
are off background
Q ’S, and three quarters
GeV nucleons we therefore
destroyed.
are approximately
of these
For five 5
x 3/4 x 5 = 15 ‘He’s
to be
the cross sections for single 3He, 3H and
equal.
channel and, for low energy neutrons,
One quarter
of those break up 4He.
expect about 15 x l/4
In case of a 4He breakup,
D production
occur in all.
However,
there is also a two deutron
a 3Hd channel.
and more tritium
W e therefore
deuterium
than tritium,
calculated
values, (2 = 7, J3H = 6 and &He = 4. One quarter
21
than helium-3,
expect more
in agreement with
the
of nucleon alpha
4
._
scatterings
are elastic, we therefore expect 15
close to the calculated
value of seven.
The calculation
&~i
l/4
x
x l/4
x
5 = 5 energetic Q ’S,
A
of &~i,
and
involves the spectra of energetic 3H,
(7~~
3He, and 4He p reduced in the baryon shower.
-
duction
mechanism for 6Li is tcr--+n6Li.
E” interact
with
so the probability
electromagnetically
For example, the d o m inant
The probability
that a tritium
an c1!and produce a 6Li while traveling
that a tritium
pro-
of energy
.- -
a distance dx is
of energy E’ produce a 6Li while losing energy
to the background
plasma is
E’
n~~~t-,~~i(E”)(~)/EadEl’.
J
0
dE/dx
is the Coulomb
using formula
and plasmon energy loss of a triton.
This is calculated
5w
(13.88) of Jackson (Jackson 1975)
dE
-XX
dx
Z2~
-wp”
AU
In (
212
Wpbmin
P-6)
13
with
b m in = max(-
-- and A = 0 (1). The probability
1
L),
ymv ’ ymv2
47rncr
wp2 = m
that a triton
22
produced
in a N - (Y collision
have
4
._
energy E’ is
1
dUiva-t+...
(
oNcx+t+...
This is taken to be independent
range of interest
-
of the incoming
(Meyer 1972). F inally,
= (it + $,)
hi
x /meNa
>E”
dEt
+
t+
...
0
nucleon energy over the energy
the 6Li yield is
_m -
(duN;;t+.)E’
t
E’
nd,t-+n6
X
Li (E”) (2)
(4.7)
1E,,dE”dE’,
s
0
-where we have assumed that a(Ncr-+t
n6Li)
= a(a3Hk
Coulomb
+ p’Li).
loss formula.
+ . . .) = a(Nat3He
The factor of l/4
This gives
+ . . a) and a(&
for ~~~~ is due to the l/Z2
-+
in the
M 3 - ‘7 10w5.
(6,5
- .
The calculation
production
.
incident
and
reactions are a*cr+p’Li
elastic scattering.
-- -
of &L;
is similar
t7j36
and a*cx+n7Be
However, the spectrum
nucleon energy.
to that
W e therefore
of
of the energetic alphas depends on the
calculate
the probability
=
with
that
a background
E’
n~~~~-p7~~(Ett)(~)~E~~dE”dE’,
s
0
ax&sum
P”‘(EN)
of an 7Li (or 7Be ),
w
P”‘(EN)
The d o m inant
, with o* coming from N--Q!
nucleon of energy EN , having undergone an elastic collision
Q , result in production
(6~i.
over all such elastic collisions
[7Li M 1 - 2.5 10e6 and
t7B6
in the shower. This yields
m 1 - 2.5 1 0 m 6 .
_.. .
23
(44
a
lithium-7
to &~i
of E6,5i
The ratio
is easily understood.
Energetic
lose energy a factor of Z2 = 4 faster than the tritons
o’s which
produce
producing
lithium-
:,
6. Also the average production
that of 7Li.
Since the yield of energetic
of energetic
tritons,
Equipped
abundances
4.3.
EFFECTIVE
(Schwitters
depends
baryon
for Mx
than lithium-7.
is stronger.
ti, we took the baryon
= 1 TeV;
the dependence
and the energy
of the primary
baryons
values of the X mass,
actually
vg depends
threshold
can be absorbed
branching
ratio,
and 3 is a factor
t’s, on the energy of the primary
consists
of five 1 GeV baryons,
the last section would
give us new values for the e’s, &(lGeV)
0.35EE(5).
3 - 0.4.
Therefore
= 5 GeV
where
into an
ratio,
of the yields,
if a shower
the
multiplicity
vg = 5 and Ei,
The value of vg, however,
branching
governing
of X decay, and solve them.
on the energy, except near the baryon production
baryonic
For instance,
equal to the number
RATIO
show.er yields,
energy
Here rB is the true baryonic
_. .-
BRANCHING
the baryon
of five larger than
down the equations
the period
on the mass of the X. For other
the dependence
;;
:
times more lithium-6
during
BARYONIC
1983) The number
logarithmically
effective
(Y’S is approximately
these yields we can write
of the light elements
and the primary
-
we expect twenty
with
In calculating
..
cross section of 6Li is a factor
24
which
represents
shower baryons.
a similar
analysis
= 0.35&(5GeV),
to
Working
with
P& instead of ?-g allows us to consistently
To compute
for the e’s as given for a 1 TeV X particle.
use the same values
the true value of the
A
baryonic
-
branching
ratio
It is worth
mentioning
a function
of the primary
&He(lGeV)
e 7Li scales slightly
that
baryon
= 0.4[3He(5GeV),while
ignored this difference.
lTeV,
4.4.
;:
as a function
differently
energy for low Mx.
&Li(lGeV)
This means that for Mx
may be higher than our result
_-.
-
rB, one needs vg x F/5
by a factor
the 7Li abundance may be slightly
of the X mass.
than the other t’s as
For example,
= 0.5ETLi(5GeV).
we have
W e have
< 1 TeV, the true 7Li abundance
of -
0.5/0.4.
Similarly,
for Mx
>
lower than indicated.
REACTIONS-
.
Let fi be the reduced number density of species i, the number density divided
by the number
density of thermal
photons,
n7,,,.
The equations
governing
the
light element abundances are:
f7,(E)
= ~7f(E)f;;I’xe-t~Tx
P-9)
dE
7
-
f7*
wn,,,
C(feac
+
fdCUz7(E))
25
-
+ %thC
Tai* (E)T
gA(E)fAufAdE
0
(4.10)
-
.fzHe((ov)
n3 He+3Hp
+
(ov)n3He+a7)+
+ f6Li((ov)n6Li+a3H +
+ f7Li(ov)n7Li-y8Li +
(01/)n6&,77~i)+
f7B6(02])n7Be+p7Li)
- fnrn
fd
=
+ fpfnn7,h(~V)np+d7
ritdf$rXe-t’rx
E maz
(4.11)
+
fr*(E)(faar~Y-d+...(E)
%hc
+
f3Ho+H+d+n(E)
/
0 -
f f3Heb+He+d+p(E))dE
.. E
-
m’;7*(E)codD,(E)dE)
J
+
fd%h(fn(ov)nd+3Hy
0
+
fnn7,~(f3He(6v)n3He-3Hp
+
f6Li(~v)n6Li-+3Ha
E maz
+
fanTthC
f-y* (E)G7+3H+...
(E)dE
s
0
E tnaz
-
-
f,*(E)o&,(E)dE
f3HnvthC
-
J
0
26
?-$~~6Lif~l?xt?-t’rx
+
fd(av)nd+3Hy)(4.12)
- ,.
E maz
+
_
n-w&c
+
fnfTBe?&,
-
f7L&y,h
J
fr*
(E)(f7L;~r7Li-n6Li(E)
(au>
n7 Be+p7
Li
+
+
fyBer7
( ~.f7*(E)E~~Liw~
Be+
+
0
.
27
f’Beoy7Be+L@))dE
f?+J)n’Li+Li)
(4-15)
(4.16)
E naaz
I-
-
f7B,nr,h(fn(4v),7Be-p’Li
(4.17)
fr*
+
(E)ca,D7Be(E)dE
s
0
-
f7Ber7Be,
cross section, oD ’s are photodissociation
where UC is the Compton
-
gA(E)
is the mean number of neutrons
A by a photon of energy E, and
number
density
of thermal
In the photon equation,
fr*
are represented
Many
of the terms
equation,
..
photons,
and A and A’ represent
the number
of electrons,
of X’s divided
equilibrium.
density of photons
any light nuclei.
can be ignored.
is much greater
is direct
by the
These
in Fig. 4.
in the above equations
since 2, = sooo J,(E)&!3
of nucleide
after the x’s fall out of thermal
diagrammatically
photons
density
re p resents the reduced number
(E)
source- of energetic
by photodissociation
is the number
fg
with energy between E and E+dE,
equations
produced
cross sections,
production
than
from
In the photon
tn, the only important
X decays.
Also,
because
-
-- -
density
fe,
is much greater than the number
the light elements,
and because the Compton
photo-dissociation
cross-sections,
Compton
scattering
iy(E)
The photon
lytic
equation
and numerical
off electrons.
=
the only important
The photon
Er@)ff;hC++x
-
solutions.
Note that,
photo-reaction,
to zero because of the threshold
an effective
f-l*(E)
theta function
when E,,,
photons
to
is
is now
(4.18)
fenrthCQC(E).
simplifying
both the ana-
is less than the threshold
then poem”” fr*
(E)a(E)dE
in the source term
in time in all photo-reaction
28
is large compared
sink of energetic
equation
decouples from the rest, greatly
energy, Q, for a particular
isequal
cross-section
densities of
terms.
&
; thus, there is
In the neutron
tion
is brought
equation,
about
the neutrons
were taken to be thermal.
by n-p elastic scattering,
which
dominates
Thermalizaall other
reac-
L,
tions, including
neutrons
per X decay, was therefore
As a neutron
-
decay. t,, which had been calculated
sink, neutron
taken to be the yield of thermal
capture
because of these elements’ low abundnaces.
terium,
and n3He--wy
are electromagnetic,
the neutron
neutrons.
by the mass six and seven elements,
ignored
modifications,
to be the yield of - 25MeV
equation
Neutron
capture
and hence suppressed.
by deu-
With
..
-
Unfortunately,
-tion remains
light elements
-- -
n3He+p3H
coupled
cannot be ignored at early times so the neutron
to the 3He equation.
as a source of neutrons.
to an effective
value.
However,
dances are only noticeably
-
(4.19)
-
ble, they contribute
the tabulated
these
is
- fn%h (fpCo2))np-+d7
+ f3He(((Tv)n3He+3Hp)
- fJn.
can be
We did drop photodissociation
As the various
elements
en which is up to a factor
equaof the
become vulnera-
of three larger than
ex post facto we find that the final element
affected if En is increased by a factor
abun-
of approximately
100.
Several other
glected
a3&n6Li,
terms were dropped
as a source of 3H because
a3He+p6Li
for 3H, 3He and *He
from
6Li
abundances
and aa+p7Li
respectively
the equations.
were ignored
since so little
29
n6Li-m3H
are very small.
as depletion
was neSimilarly,
mechanisms
6Li and 7Li were produced,
ie
(cLi <
(sH, &He and J.IL~ <
(2.
The new 3H equation
is
+ fn%th f3He(423He--rWp
(4.20)
E maz
+ fcn7,hC
-
fr*
/
(JqdE+
(qJa7--*3H+...
,-
0
E maz
-
fr* (E)o&,(E)dE.
f3HnythC
s
0
with
similar
modifications
to the 3He and CI:equations.
We have now reached the point
numerical
-
solutions
scenarios
which
elements
inferred
4.5.
where we can discuss both the analytic
to these equations
are not in conflict
and demonstrate
with
the existence
the primordial
abundances
and
of no = 1
of the light
from observations.
UNCERTAINTIES
Although
we were as careful
as possible,
there are several sources of uncer-
-- tainty
in our calculation
data was available;
which we were unable to avoid.
for others,
or had large experimental
the paramters
Mx,r&,
we cared to precisely
of the uncertainties
crucial
widen
errors.
and
Many
(ff;/fp) ;
determine
parameter
data was incomplete,
of these translate
th us, they would
the properties
in the data imply
to keep this in mind
the allowed
the existing
For certain
since a favourable
space for the X, while
30
contradictory,
in
only be of relevance
if
in the ratios
alteration
no
into uncertainties
of the X particle.
uncertainties
quantities
However,
of the t’s.
in the & ratios
an unfavourable
some
It is
would
alteration
-
4
could close it entirely.
data followed
by Monte
From our results
-
Improved
values for the &‘s will require new experimental
Carlo studies of the baryonic
it will
become clear that
uncertainty
are in the determinations
-production
mechanism
is a*a-+p7Li;
energetic alpha particle
produced
the most important
of 67 and &Li.
as a function
of E,,
sources of
the d o m inant
for 7Be, it is o*o--+n7Be.
of the CX*‘s. Unfortunately,
for N - cx elastic scattering
For 7Li,
in elastic N - CYscattering.
section for acr-+7 falls exponentially
the energy distribution
showers.
Here cz* is an
Because the crossis very sensitive to
J7
we could not find da/dE,
above Elab - 2GeV (Klem
et al 1977; Meyer 1972).
t o b e constant for nucleon energies above 1.78 GeV, since
W e took a-l(da/dE,)
it is the same for 800 MeV to 1.78 GeV. W e were also unable to find any data
on the photo-dissociation
of 7Be. W e assumed that the photo-dissociation
crosssections of 7Be is equal to the cross-section for 7Li because they are image nuclei.
:
one m ight
-
Coulomb
expect a lower threshold
for 7Be than for 7Li because of the greater
energy. This would m e a n a very slightly
elements than we report,
For ‘Li,$he
lower abundance of mass seven
since more 7Be would be destroyed.
d o m inant
production
mechanism
is 3H*cx-m6Li.
The energetic
-- 3H’s are produced in p-a
inelastic scattering.
only for EFb = 90,300MeV
(Meyer
W e could find data on daP”‘t/dCl
1972). S ince it was identical
energies, we assumed it was the same for all the EP of interest.
for this is that the reaction
remaining
3H as a spec tator.
threshold
to Et = 14MeV
n!Li--+ta
and n6Li*--+ta
for Et > 18MeV.
_
.
is essentially
a nucleon-nucleon
The data available
obtained
by detailed
scattering,
balance from
with
the
was 0 from
measurments
of
No data was available
For higher energies we used the cross-section
31
The justification
for 3Ha-m6Li
(G. Hale private communication).
for those two
for 3Hecx+6Lip
and 3Hea+6Li*p
42MeV.
at E,
= 42MeV
Unless there is some unexpected
these energies, the linear
The 3H spectrum
contribute
-
uncertainties;
to determine
however,
14MeV
in the cross-section
to within
energy, so higher
and
between
approximately
25%.
energies do not
the magnitude
of the errors
introduced
they could account for a 70% error in &Li/Js
uncertainties
inelasticity,
range of interest
-
with
between
by these
and in &Li.
apply to all the t’s; we list them in order of our
of -their importance:
a)The
..
structure
is accurate
decreases exponetially
The remaining
estimate
interpolation
interpolated
to EsL~.
It is difficult
tween
and linearly
E, of inelastic
(W. Bucza,
baryon
private
0.4 and 0.75 , the extreme
collisions
is unknown
communication)
and was taken
values communicated
to us.
The error
in E
the number
of scatterings
tion cannot
be precisely
determined.
This leads to a large ( approximately
50%)
uncertainty
in all the shower yields.
We have checked that the uncertainty
in the
of the yields
b) There
&/&
baryon
to be be-
means that
ratios
of an energetic
in the energy
before thermaliza-
is smaller.
is apparently
no data on the energy distribution
of the debris neu-
-- trons from inelastic
N-o.
to the shower yields.
neutrons
Their
are effective
neutrons
are more effective
than-protons.
yield
The debris neutrons
so uncertainty
in the &/es
ratio
are not thermalized
in destroying
An important
in nucleon-nucleon
contribute
significantly
energies were taken to be 50 - 200MeV.
3H producers,
itself in an uncertainty
c)Since
scattering.
*He
of approximately
by Coulomb
and synthesizing
source of neutrons
collisions
number
manifests
20%.
scattering
the other
is pp-+pnh.
is not well known
32
in their
Low energy
below
at lGeV,
light
they
elements
Since the neutron
Egb
-
3GeV,
the
number
of energetic
neutrons
d) The multiplicities
gies. At lTeV,
included
and energies of jet baryons
the theoretical
discussed in section
-
can also be included
the shower for Ei,
extrapolations
of r;j.
Presumably,
in P; through
> 10GeV.
are not known
are uncertain
3.4 the effect of vg variation
in the definition
elements
is uncertain.
by a factor of two. As
on the t’s, has been explicitly
the effect of Ein variation
the factor
7. However
we have not calculated
of two smaller
in high &ix
scenarios.
e) There are large error bars in many of the experiments,
report
different
(Meyer
1972).
cross sections.
on the E’s
It is possible that the ratio of 7Li to the other light
could be up to a factor
N - c1!scattering
at high ener-
Different
experiments
especially
at nearby
those on
energies often
The fits to the data are not always compelling.
f) The cross-over between strong interactions
and Coulomb scattering
dominant
energy loss mechanism for protons is not a sharp boundary.
..
somewhere
-
between
500MeV
g) In the Compton
mic enhancement
the photon
-- -
strength
h) The energy
thermalization
in (d), the dependence
probably
be embedded
have a unique
(En >> 25MeV,
a O(lO%)
However,
the destruction
effect,
it will
might
slightly
baryons
alter
alter
the effective
was not considered.
of the e’s on the primary
energy when actually
i) We ignored
ergetic
in ri.
the logarith-
term.
of the primary
discussed
we ignored
since it does not significantly
treatment
scattering
distribution
of the photons,
scattering,
A more careful
of the Compton
It lies
and 1GeV.
due to forward
energy.
as the
this assumes that
baryon
the primary
they have some distribution
of light
Ep 2 1GeV)
elements
baryons.
have approximately
33
other
energy
than
As
can
baryons
of energies.
*He
Since hadrodestruction
the same percentage
by very enof *He
is
effect on the
other
light elements.
j) We have also neglected
cross-section
for anti-nucleons
cross section.
not produce
-
to destroy
significant
baryonic
*He by annihilating
produced
showers.
10e3
light element
production,
the results
of the baryonic
showers.
Delbruck
of Burns
(77-+77)
is less clear.
fp.
and consequently
and Lovelace
scattering
the total number
rBYg
the *He
and Aharonian
scattering
be the subject
fg.
We shall
depletion,
is negligible
was ignored.
do
for an anti-nucleon
nucleons,
Therefore,
The
the annihilation
However,
by anti-nucleons
This should
l/3
scatter
nucleon.
for RBhz = 1, because Compton
the situation
is -
It is also possible
a constituent
hence other
Delbruck
..
do not multiple
is 2. 5
k) In the treatments
of the light elements.
off nucleons
is the same as primary
this combination
Vardanian
destruction
to scatter
Hence, anti-nucleons
of anti-nucleons
see that
anti-nucleon
and
compared
to
and
It is possible
to ignore
dominates.
For n~hz
of further
investigation.
< 1,
1) Our
baryon
-- -
formula
velocities
for
greater
(D.Seckel,
private
freezeout,
the relevant
with
coming
than the thermal
velocity
velocity
baryons
are -
would
1keV.
have a kinetic
is 8MeV
triton
A triton
energy with
for
in abundances
at
nucleus
energy of only - 6MeV.
The
the bulk of production
of > 20MeV,
so this should
in 6s. The threshold
for aa-+7Lip
is 36Mev in the lab frame
- m)
only
in the plasma
or helium-3
tritons
no effect on
is valid
of the electrons
Since we are interested
temperatures
for 6Li production
from
losses by energetic
communication).
a corresponding
threshold
Coulomb
not be a major
source of error
and hence this has
E7.
As the X’s decay, they
range of interest,
this dilution
add entropy
to the system.
is at the very most a factor
34
In the parameter
of two and the time-
4
temperature
-
relation
responding
initial
7 is higher,
abundance
of *He
coming
necessitate
a small adjustment
estimates
largely
unaffected.
so there
from
For the sake of clarity,
discussions,
is a very
Standard
specified.
of the errors on the t’s:
energy distribution,
3Ha-+6Lin;
70% error in J7, beyond
large (factor
of 2) d iscrepancies
500MeV,
(e/e3
and the sensitivity
We emphasize
that
from
in the initial
This will
values of the 6’s in all further
50% error in the overall
70% error
increase
When needed, we will adopt the following
experimental
in
slight
B 1, the cor-
5.12 and 8.2.
we will use the central
unless otherwise
for RBh:
Big Bang Nucleosynthesis.
in equations
calculation;
data.
However,
from the error in E; 40% error in &/[s
neutron
..
is essentialy
normalization
from the uncertainty
error, and roundoff
the lack of data
arising
in the debris
error in the shower
on pa~--+~H + .. m and
the 50% in the overall normalization,
in the data on pa elastic
of the calculation
our final abundances
scattering
to uncertainties
at Elab 2
in reading
for the light elements,
from
the
and hence
the size of the allowed
crucially
parameter
space which
on the actual values of the &‘s.
35
we will
derive for the X, depend
4
._
5. Nuclear
Abundances
From Fixed Points
The -equations for the light nuclear abundances
plicated
to solve analytically.
keep only the d o m inant
-
However,
production
(4.10)- (4.17)
are too com-
in each case a good approximation
and destruction
is to
terms, where the equations
_a
take the form:
f’= J(t) - fryi).
The behaviour
of the solutions
to
(5.1)
(5-l)
is discussed in detail in Appendix
both for general J and l? and for the three particular
of interest.
A,
forms of which prove to be
There we show that if
(5.2)
..
then equation
-
(5.1)
order- t r M r-l.
reaches its fixed point
equilibrium
either condition
is violated
tf
< t,, then
= J(t)/I’(t),
on tim e scales of
Beta use the fixed point is tim e dependent we will refer to it as
the state of quasi-static
-- -
f(t)
f
The photon
(5.2)
(QSE). Equation
(5.1) remains in QSE until
at a tim e t,, or it freezes out at a tim e tf.
If
freezes out at its QSE value.
equation
(4.18)
is of the form (5.1), and has QSE solution:
f (E) = 6-,4E)f;7ck.-t/Tx
r*
f enrthcaC(E)
(5.3)
The response time, t,, is much less than r, at early times, so the photons rapidly
approach QSE. Although
t, < tf for the photons
tim e the photon equation
leaves QSE, J is exponentially
36
(cf appendix),
t, >> 7,. By the
small and
fr* m 0.
=--
-
4
._
W e now begin the arguments
production
these can easily be approximated
point is that n3Hej3Hp
-
identify
the d o m inant
destruction
and
(4.10) - (4.17). Except for the coupling between 3He and
terms in
the neutrons,
which
is important
by the form
(5.1). The crucial
only at early times, before the onset of 3He
photo-dissociation,
when both the proton density and the ratio of 3He to protons
is high.
we are ultimately
However,
Since these are determined
the temperature
interested
only
in the final
abundances.
during the period of quasi- static equilibrium,
when
has already fallen to 2 lOkev, and 3He photodissociation
begun, we can ignore, for the purposes of these estimates,
3He. The simp lified neutron
neutron
has
depletion
by
equation,
(5.4
.fn = GEnfTTxe-t/Tx - fn+,, fp(O?l)np--td7
- fnrn,
is decoupled, and is of the form (5.1). The QSE solution
is therefore
(5.5)
W e can now bring the remaining
equations into the form
(5.1). W e add the
-- 3He and 3H eq uations,
and 7Li and 7Be equations, taking the photo-dissociation
cross-sections of equal mass species to be equal, to obtain equations for the mass
three
and mass seven elements.
important
Photodissociation
of 7Li and 7Be is not an
source of 6Li, because, in X decays, more (jLi is produced
7Li and 7Be combined.
W e ignore the thermal
these elements as they are small.
of-qIiasi-static
equilibrium
neutron
destruction
than are
terms for all
The equations to which we apply the method
are therefore:
_- .37
4
._
F-6)
+ fPfd+h
f3 =
-
(4
np+dy
-t’Tx
r;E3f;;rxe
-
fdfy*ny,,,C~dDy~(Emaz
- Qa7+3)
(5.7)
- Q4)
(5.8)
+ fa f7*n7,hca,7,30(E,,,
- f3f7*n7,,ca~7e(Emaz
fol = -r&i~f~rxe-t/Tx
-
Qd)
-
93)
- f~n7,hcf7*o&~(E,,,
(5-g)
fsLi = rT;&LifJ;lYXe-il~x
- f6Ljnrth(fy*CDrDsLie(Emo,z
i7
-where
reaction
..
-
= &t7f$irxe-t’Tx
threshold
Q6) + fn(o.l/)n6Li-+ta)
natural
m e a n ings,
and the Q ’s are
(5.6)- (5.10) lead t o acceptable abundances if rg = O ? In
B we show that this is impossible:
too much 3He is produced.
either too little
This is why the program
*He is destroyed or
of Audouze et al (Audouze
et al 1985) cannot succeed. Instead we rely on a destruction
-- -
(5.10)
- Q7),
energies.
Can the equations
appendix
-
7 f7f7*n7,,,co~B(Emoz
of7 and uf7 have their
f3,f&&,
in hadronic
showers
to lower the- *He abundance.
Dropping
the photo-dissociation
term from the a equation
tent so long as 7-x < t4, the threshold
exactly
to obtain
(which is consis-
for *He photo-dissociation),
we integrate
:
f&4
Taking
-
.- -
= for(O)- r&gf;;.
(5.11)
EE = 10, we find
?-;(f;;/fb)
=
1.15 lo-3
38
-
(Y - .24)
4.
(5.12)
=e
._
.
where Y is the final primordial-mass
fraction
density
after primordial
nucleosynthes.
production
is twice the number
of protons
fp is the
of *He and
fp
reduced number
is taken to be a constant
:,
since the net proton
hence
Af,/f,
Consider
-
terms
were
next
equations
discarded
In the deuterium
of deuterium,
it is smaller
dissociation-will
Neutron
dominate
final 6Li abundance
The equations
Since
we
abundances.
do
not
we can drop this as a source term
capture
is because the 7~‘s that
-- -
element
the ratio
of deuterium
from neutron
permit
the
in all the equa-
production
capture
Several
by protons
in hadron
is
q3,.
nrth (ov;lp+Q fp) ’ f” + 25(
neutron
these- estimates.
light
analysis.
production
n
-
this
equation,
$1+
Thus although
for the other
in
of *He,
showers to deuterium
..
hadro-destroyed,
=54%.
photo-dissociation
tions.
of *He’s
by protons
at freezeout
capture
is comparable
and can be ignored
can be disregarded
we will be forced to consider
neutron
to shower production.
capture
during
for the purpose
as a sink for 6Li.
will be such that
the period
of
This
photo-
of QSE, when the
is determined.
for ii (i = D, 3He, 6Li, 7Li) then take the form:
f; = (r;Ji - e,f) f;;rxe--rxt
(5.13)
P
The ci are defined by
E maz (t)
-- Q(t)
=
-$
e7* (JmpW
s
0
39
(5.14)
where we have ignored
tering.
The times
for nuclei
form
i. For
are defined by Emaz(ti)
ti
t
<
it is clear that
ti
of (5.1), and since for rx > 2
-with these fixed point
-
the energy and angular
dependence
= Qi the photodissociation
ci(t)
vanishes.
105.sec,
x
of the Compton
t,
>
threshold
(5.13)
Equation
the abundances
ti,
scat-
has the
freeze-out
values:
.a
(5.15)
This is quite
dependence
millibarn.
a remarkable
result.
The freeze-out
on i because the photodissociation
Hence, we predict
are approximately
that
just the ratio
: f3:
values of ci have only a weak
cross-sections
the ratios
of these light
are all of order
element
abundances
of the &
f6Li
F=:1:1:10-90-y
-
compared
with
the values inferred
fd:f3:
-- -
f6Li:f7
The success of these estimates
abundances
is entirely
yields approximately
scenario
(5.16)
from observation
M
:
1:1:(?):10-(5t06).
is remarkable.
different
The physics which determines
from the physics of the standard
the same ratios of light element
does make certain
testable
predictions.
abundances.
In particular,
7Li and the 3He to D ratios
are higher
discUss this further
6, where we present our numerical
in chapter
show how it is reconciled
a
with
than in the standard
observations.
40
these
scenario;
yet it
However,
both
our
the 6Li to
scenario.
We will
results,
and will
-
It is important
nearly
independent
trivial
test of these ideas.
The light
-
to note that
.
of the properties
element
abundances
1. The *He abundance
2. The abundances
Mx/r$
in (5.14)
of these fixed point
of the X particle,
abundances
and are therefore
depend on the properties
are
a non-
of X as follows:
.a -
depends only on rif3(f,$-/fp),
of D, 3He, 6Li and 7Li depend
as we now show.
(4.2)
the ratios
Taking
~$7 N lmb
on the single parameter
and 0~ N 30mb,
and using
gives
MX
(5.17)
ci N 0.3GeV
Using this in (5.15)gives
t,m= fX4
..
Successful
-
abundances
are thus obtained
0.3GeV
(5.18)
&jx >
for
Mx N_ 105GeV
(5.19)
-
G
-. -
The numerical
results
of the next section confirm
these results for a broad
range of parameters.
We now consider
is now different;
the case RBhz # 1. The amount
equation
(5.12)is replaced
of requisite
*He depletion
by
rjf3-f;; N 1.5 x 1o-3 - O.O35(Y - 0.24) + 3.1 x lo-*
ln(nBhz)
(5.20)
fP
The-abundances
of the remaining
light elements are still given by equation
(5.15).
‘.
rs does not depend
on ClBh:;
e’s have only a logarithmic
41
dependence
on nBhz
=-
4
._
arising from the electromagnetic
Coulomb losses of equation
tion
on OBh:
(5.14)has a m ild sensitivity
from E,,,.
(4.6); ci(tf)
of equa-
This arises from the de-
pendence on the electron density of the crossover from pair creation to Compton
scattering.
Hence we obtain successful light element abundances for a wide range
of nBh2 including
the previously
excluded region of 0.03 5 nBhz
-
42
5 1.1.
6. -Numerical
For -the numerical
A
(2.3).
in section
tion
(4.18)
analysis,
Our program
analytically
Results
we return
to the full set of equations
is as follows
. We first
in the quasi-static
limit,
developed
solve the photon
since the response
equatime
-
7 -’
is much less than
greater
..
-
feCOc)-l
M
we use the method
stability
the coupled
neutron
.
3& equation
(4.12)
;Li
(4.16)
scale in the problem.
t&(f$/fp).
A
a) At t f
(4.14)
of backward
(Wagoner,
private
and 3He equations
and the d equation
of features
numerically.
differencing
because it provides
communication).
(4.15).
We then solve
th en successively
the
(4.17), the
In Fig. 5, we plot the abun-
of t for representative
are readily
to
Throughout
(4.11), and the 7Be equation
and the 6Li equation
number
Using the solution
((4.19),(4.13)),
dances of the light elements as a function
7x3
(6-l)
2(F)3sec
we solve the Q equation
numerical
equation
(n7,,
any other time
the photon-equation,
the program,
=
values of Mx/rL,
apparent:
lo5 - lo6 seconds, the abundances of d, 3H, 3He, 6Li , and 7Li drop
-- significantly.
This represents
the various
elements begin to drop off is consistent
as illustrated
the post-threshold
of the tQ’s
photodissociation
decline.
except that of 7Be
This is the onset of QSE. 7Be of course
to decay into 7Li . This also explains
danze after the initial
tritium
the ordering
drop, all the abundances,
out and do not change further.
continues
with
The order in which
in figure 1.
b) Following
flatten
the onset of photodestruction.
the slight rise in the 7Li abun-
One can also note the importance
as a source of deuterium.
43
of 3He and
c) Except
dances.
for 4He the final abundances
All information
about
are independent
the initial
conditions
of the initial
is quickly
abun-
erased by shower
production.
As expected,
-
-r$(f$/fp)
= 1.5
and for 7-x 5 6
plot
depletion
x
elements
105sec, when photodissociation
of the light
elements
, and r&(&l&).
vulnerable
down to their
Our
-
while
there
l/20.
is still
and A4x,
on. In Fig. 6, we
is increased
the various
by the drop in the final
too far, 4He becomes
some photon
power.
and the much smaller
This
rise in the D
Li is of course unaffected.
next
D,3He,6Li
O(lO),
as rx
If rx is increased
causes the large rise in the 3He abundance,
abundance.
(5.12),
of rZ for representative
This is evidenced
QSE values.
to photodestruction
of 4He turns
as a function
W e see that
reach QSE before freeze-out.
abundances
as per equation
10m3 - O.O33(Y, - .24). This is valid for all Mx/r$
x
the abundances
values of Mx/r&
of 4He fixes ri(&-/fp)
goal is to constrain
and 7Li.
while
We first
observations
Mx/rlf3
using
note that
indicate
the observed
our predicted
that
presently
If we take the Pop II value of 7Li
abundances
ratio
6Li/7Li
as representing
of 6Li
5
l/10
of
to 7Li
is
or possibly
its primordial
abun-
-- dance, then,
as in the standard
Pop I stars is non-primordial.
expect
to produce
0 (lOeg)
scenario,
Since the 7Li abundance
6Li.
Although
to the Pop I value of < 10-l’
stars),
the observed
6Li/7Li
al 1981) suggests that
exp-ected
lo-’
This results
within
ratio
pre-Pop
to 5 10-l’
we must assume that
of < l/10
I stellar
requires
in a 50 - 90% reduction
that
approximately
in deuterium,
in
we
6Li is depleted
(as discussed
in the interstellar
processing
44
in Pop II is 0 (10-l’)
it is conceivable
the Pop I star
the lithium-7
for Pop II
medium
is necessary.
To deplete
50 - 90% stellar
and a smaller
(Sell
et
the
processing.
reduction
or
even an increase 6 in 3He and 7Li.
:,
likely to produce
abundances
stellar
required
processing
large uncertainties,
-
Such a scenario
consistent
with
as well as on the values of M-x/r&
expect the primordial
their
while
6Li or stellar
However,
a sizable fraction
abundance
attractive
scenario
-
and
The exact amount
of
to be primordial,
to be 0 (lo-*).
deuterium
then we
In order to reduce this 6Li
either that Pop I stars uniformly
of the pre-Pop
I material
is destroyed
of 3He may remain.
and a small deuterium
and within
and r,.
of O(lOeg)
we would require
processing
helium-3
6
6Li abundance
to 5 10-l’,
99% efficiency.
observations.
plausible
depends on the actual values of &, and (7, which have
If we take the Pop I 7Li abundance
abundance
seems both
with
deplete
approximately
90 -
to the same extent
We expect
abundance.
the result
as 6Li,
to be a large
This is not a particularly
our errors is excluded.
-
-
Let A be the fraction
lar processing,
processing
-- -
of primordial
material
and gi be the fraction
in the
“average”
star,
which
of element
then
Dt
=
does not undergo
i which
AD,
survives
stelstellar
+ (1 - A)gDD,
and
3Het L~A3He,+(1-A)g3(3Hep+Dp)
(Y an g et al 1984).
Here the subscript
t indicates
and the subscript
p denotes
mordial
values.
of deuterium
ing.
I abundances
The D term in the 3He equation
to helium-3;
go is generally
l/4 - l/2,
- Xnd
the pre-Pop
although
7Li obey similar
taken
comes from the conversion
3He can also be formed
to be zero, while
this number
equations,
H burn-
to be in the range
and could be zero. 6Li
6Li is not a source of 7Li; g6 B 0,
while gr could be of the order of gs. 7Li might
45
by incomplete
gs is taken
is model dependent
although
the pri-
also be produced
in stars.
._
.
Returning
we constrain
astration
-
to the scenario-in
f7 to lie in the interval
factors
8 x lo-l1
- 2 x lo-lo.
from approximate1
the astration
and helium-3,
factor to deuterium
1.5 x 10-4.
We allow
40% errors
in
Uncertainties
between
1
abundances
[z/(3,
5 x 10m6 <
for 50% errors
x
constrained
104GeV
and 2
to be between
in
Mx/r$
x
with
105sec.
vs. rZ plane for various
upper limit
on rx corresponds
dissociation;
the lower limit
out happens
after photo-dissociation
the astrated
D and 3He abundances
f3 <
in the
is constrained
to be
our predictions.
rx
In Fig. 7, we plot
amounts
the condition
of the e’s,
as discussed
to the onset of 3He production
comes from
10-l’.
and requiring
normalization
and &/[s
&j/t3
and 9
x
< 5 x 10e5 and 1 x low5 <
consistent
2 x ld5sec
allowed region in the Mx/ri
assuming ga = l/4,
We find that
105GeV,
x
to (0.25 - 1)
the
15% to over 90%. We apply
in the overall
and 70% errors
section of this paper.
fd
is primordial,
We then calculate
necessary to reduce the 6Li abundance
These are found to be anywhere
for the astrated
_-
which the Pop II 7Li abundance
t, >
of astration.
is
the
The
from 4He phototQ,
so that
freeze
- .
consistently
-- -
both the primordial
is the primordial
of 3He and 3H has begun.
that we consistently
require
-- Yin figure
ratio is greater
but indicates
x
range for .Mx/r1;3
10Y5 and helium-3
vs. rx,
than 1, as
constraining
models.
This is within
that an improved
9, we plot the primordial
the estimated
fd, f3, fsLi
values of the t’s.
46
central value
errors for
value would be of interest.
values of
for several values of rx, using the central
deu-
in 1 - 6 x 10B5. We find
[7 to be 70% higher than its calculated
low (< 75%) astration
this quantity,
3He/D
range,but
ratio of 6Li to 7Li.
to lie in the range 0.8 - 2
to obtain
vary over most of the allowed
and astrated
In Fig. 8, we plot our preferred
terium
In this region
and
f7 vs. Mx/rL
We see, once again ,
that
As remarked
is determined
fshi
-
..
f3
consistently
and
f7
>
f2
and
-f6 2 f7,
in the previous
exclusively
. Mx
chapter,
primordial
it is important
M x / rg determines
by 4He.
is entirely
for the predicted
free (so long as Mx
values.
to note that
the overall
2 1OGeV).
r&
(f,$/ fp)
scale of
fd, f3,
The approximate
.- -
-value of rz is a predictionP
-
7
The
analytic
techniques
dicting
the light element
applied
to other
simply
- Xdering
believe
in the previous
abundances.
nucleosynthesis
the output
the dominant
understanding
applied
did well
in pre-
If, as is likely, these techniques
can be
calculations,
of complicated
We thank
47
it should
numerical
sink and source terms,
of the results.
chapter
not be necessary
programmes.
By con-
one can gain a quantitative
Bob Wagoner
for suggesting
to
this.
7. Testable
Our- theory
A
Differences
of nucleosynthesis
those of the standard
model.
the two theories, improved
-
and processed
predictions
(eg.
a “Li
standard
7Li.
-
model.
that
cannot
In table
(unprocessed)
the
model with those of our theory for both primordal
and
depend on the amount
in primordial
abundances
which is more than thirty
We also predict
(Table
distinguish
should do so. The
of abundances in both primordial
matter.
differ from
4), we contrast
difference
abundance
abundances
of abundances
of astration.
occurs for 6Li.
We
times bigger than that of the
higher D, 3He, and 3He/D
A subset o f these differences could potentially
spectral
..
disk)
Model
present observations
Note that the disk abundances
The most striking
predict
nuclear
future measurements
of the standard
disk matter.
predicts
Although
‘theories differ in their predictions
matter
From the Standard
and slightly
lower
be tested by looking at quasar
lines.
As far as disk matter,
we predict
more 6Li,3He,
and 3He/D
than the stan-
dard- model.
Finally,
-- -
that
our theory
of the -standard
or equal to that
can accomocdate
model,
against the standard
that
as well as any 4He abundance
of the standard
the 4He or 7Li abundances
an 7Li abundance
big bang.
Any observation
are lower than previously
model, but would be consistent
48
thought
with
is smaller
which
that
than
is less than
suggests that
would be evidence
our theory.
-) -
8. Discussion
of Results
and Conclusions
In this paper we have analyzed the synthesis and destruction
;,
in the keV era of the hot big bang via hadronic
induced by the decays of heavy particles.
of light elements
and electromagnetic
We have found primordial
showers
abundances
.m -
-
‘of d, 3He, 4He 6Li and 7Li for RBhz equal to or less than unity,
our uncertainties,
are in agreement
We have shown that
primordial
with
cosmologies
nucleosynthesis
x
within
observations.
0.03 < stg 5 1.1 are consistent
if there is a particle
2
r;--f% = 1.15
with
which,
X with
properties:
105sec < rx < 9 X 105sec
X
_ (Y -4. o-24) +
10-3
(8-l)
2.2 X loA
(8.2)
ln hfRB
fB
..
1
1OOOGeV
1
< r;3
200
<lo
Mx
-
We have commented
-- -
on how these parameters
is outside
the range of (8.1)
condition
for obtaining
to obtain
acceptable
A conventional
primordial
results
do not depend
irrelevant,
value.
(8.2)
Y, and (8.3)
nucleosynthesis
The initial
is the
is required
occurs in two stages.
30 - 100 keV and a new era at T -keV.
on wether
since no memory
Equation
II. If rx
d, 3He, 6Li and 7Li abundances.
there ever was such a conventional
require only that above a few keV, the 4He abundance
primordial
arise in section
4He mass fraction
a scheme in which
era at 2” -
(8.3)
th e scheme does not work.
an acceptable
We have described
might
with
conditions
for d, 3He,
of them survives
49
In fact our
era.
We
is larger than its observed
and 7Li
once the evolution
are completely
equations
reach
=--
4
._
their fixed point.
Therefore
the only memory
a keV is the existence of a substantial
The standard
osynthesis with
which
-
R = 1 cosmology of the hot big bang has conventional
0.014 < RBhi
< 0.03. The dark matter
to $2 is lo-100 times the baryonic
W e have proposed an alternative
with
a keV era of unconventional
This
requires
correct
an exotic
unstable
includes exotic particles
chosen so that the exotic
contribution.
scheme for accommodating
synthesis and destruction
particle
nucle-
with
R = 1 : nB = 1,
of the light elements.
parameters
chosen to give the
4He and d abundances.
This alternative
completely
scheme provides motivation
for exploring
the possibility
of a
consistent cosmology with R B = 1. This would require understanding
the formation
of large scale structure
1987a,b) the present form
observe this dark matter
In addition
in such universes with
and location
of the dark matter,
dark matter
nB
= 1 (Peebles
and how we could
to be baryonic.
our scheme for nucleosynthesis
can accommodate
tween 0.014 .and 1.1 . Hence study of the formation
-- -
above
4He abundance.
are stable and which have masses and couplings
contribution
-
of physics at temperatures
in these universes is also of interest
any RBhz be-
of large scale structure
and
(Peebles 1987a,b and Blumenthal
et al 1987)
The m o d e l does have definite,
guish it from standard
W e predict
the primordial
7LiF
that
standard
m o d e l.
testable,
predictions,
which
distin-
Big Bang nucleosynthesis.
the primordial
abundance
W e predi ct that
hopefully
3He/D
is greater
than one and that
of 6Li is of the same order as or larger than that of
in the disk 3He,6Li
F inally,
ratio
and 3He/D
are higher than in the
it seems possible, in light of recent observations,
50
that
the primordial
4He abundance
is lower than
this proves to be so, then standard
to explain
both the observed
in our model
-
elements,
fUnCtiOn
..
-
is essentially
we would
Of RBhE.
regard
Big Bang nucleosynthesis
deuterium
and helium-4
independent
it merely
had been previously
If
may not be able
Since the 4He abundance
of the abundances
as a revised
believed.
prediction
of the other
for r&(f$-fp)
light
as a
.- -
4
._
- APPENDIX
F ixed point
:-
-
analysis for source-sink
In this appendix
A
equations
we derive the analytic
solution
f’ = J(t) - frp)
The general solution
-St:
f(t) = e
to equation
.-
to the equation
J,l? 2 0.
(A.1)
(A-1)
is
l?(t’)dt’
,
(A4
I
where tz is the initial
this can be integrated
time, which we take to be t = 0. If J and I are constants
exactly
to obtain
f(t)
= evr,(,
-
O n tim e scales of order I’-l,
-- -
- f,
this approaches the equilibrium
general J and I’ are tim e dependent.
Making
(A-3)
+ g.
value J/I’.
Now in
the ansatz
f =;+s,
(A-4)
we find that
i=-rs-J [ !--C I
rJ
r’
If
(A-5)
-
(A-6)
52
-
then 6 becomes much less than
J/I’ in a response time,
t,,
such that
t+tr
r(t’)dt~ - 1.
,s
Therefore,
-
f
so long as condition
= J/I’,
(QSE).
If condition
(A.6)
t
(A.6)
which we previously
(A-7)
f
holds,
referred
approaches
its fixed point
to as the state of quasi-static
is violated
at a time
then
t,,
f
value,
equilibrium
leaves quasi-static
equilibrium.
If, at some time
t; if also St” I’dt <
c
t, St” Jdt’ <<
1, then
f
If tf<
- .
-- -
d oes not grow appreciably
appreciably.
are met; we say that
f
after
We define tf as the
freezes out,
f(t = 00) = f(Q).
t, then f freezes out at its QSE value
There
are
Case I: J = leet/‘;
three
particular
-7, I?
cases
which
are
of
interest
to
equation
((5.4))
when only the decay term is
as a sink (this is valid well before tf ) . In this case, the solution
can be expressed
us:
constant
This is the form of the neutron
included
f
then
does not diminish
time when both these conditions
-
f(t),
in terms of known
(A.2)
functions.
- -
f(t)
= ---((I
!py;
- eert)
T
53
+ emrtf (0).
(A-8)
_
_
4
If I-l
.K r, then, for t z+ I’-ls
f(t) = +.
Case II: J = T[t)emt/‘,
-
The photon
r(t)
equation
= ct+
((4.18))
-t/r
(A.9)
; .7, c
constant,
is of this form.
n > 0
In this case, condition
(A.6)
becomes-
This is satisfied
if t <
(57) 1/m and t <
(s/n) &.
tc M rnin(($r)‘/“,
..
n-l
= (-)
tf
Thus
(i) --‘)
(A.ll)
1
1--n.
(A.12)
5
-
Case III:
J = .~(tle-~/‘,
The equations
-- -
((5.10))
I’(t)
f y(t)emtl’
for d ((5.6)),
are brought
;
.I constant
3H + 3He ((5.7) ), 6Li ((5.9)),
and 7Li + 7Be
into this form.
If 7 is a constant,
then the solution
(A.2)
can again be integrated
exactly to
obtain
f(t)
= e7T(e-t”-1)
F=A
If rr >
1, then the correction
f. + $1
_
L + ee7?-(fo - z).
7
7
to J/Y is small.
54
e7r(e-tl’-l))
(A.13)
-- -
4
._
If 7 is tim e dependent,
the QSE solution
is
f(t) = g.
Condition
(A.6)
(A.14)
becomes
-
,m
(A.15)
I 1 I < ye-‘/‘.
7
For any particular
tf
< t,.
form of -7, one can compute
In the equations of interest,
r(t)
with
i;
=
70(1-
1=
root, t,,
t > t,
($)-“)
otherwise
y)““(l_
(3-y”
any roots.
e-t/‘.
, is greater than t, then it represents a critical
r0 = rc such that
In all the equations of interest r0 > rc and t,,,
abundances are therefore
(A.17)
5
the onset of .QSE. Note that there is also a critical
--. -
(A.16)
,
and t, is the larger of the two roots of
5
If the smaller
and t, and check whether
7 is of the form
0
0 < n < 1. Then tf 5 r1n7,r
-
-
tf
approximately
t, < tf
< t,.
tim e for
(A.17)has
The final
the QSE abundances at freeze-out.
55
i -
-
4
._
- APPENDIX
The Necessity for Baryonic
B
Decays of X
-
As RB is increased to unity,
progressively
the conventional
era of nucleosynthesis
more 4He and 7Li and less d and 3He.
that a condition
In this appendix
for X decays to give acceptable adjustments
is that fg must be non-zero.
duced by photons,
leptons
That
we show
to these abundances
is, we show that electromagnetic
or mesons cannot
produces
photodestroy
showers in-
4He and 7Li and
synthesize D and 3He by the desired amounts.
At first
sight a scheme with only electromagnetic
showers from X decays
appears promising. A large 7Li photodestruction
and a much smaller 4He photodestruction
fraction
sociation
will
-
can be arranged
e-t41rx
t7,
since in this case only a
lead to n, D and other remnants.
by protons
also has a mechanism
to form
for deuterium
production.
The photodis-
Some of the neutrons
D before they decay.
which is m a d e will be photodissociated.
will
-
of X’s decay after t4 when 4He is vulnerable.
of 4He will
be captured
by choosing rx
Hence this scenario
The m a jority
of the deuterium
However, the tail of the 4He destruction
lead to D which have large survival
probabilities,
since they are m a d e late
enough that few X’s remain to decay.
This simple and appealing
D abundances
3a~
picture
by electromagnetic
for the adjustment
showers is ruined
This can be seen by just considering
of the 7Li, 4He and
by the overproduction
of
the coupled 3He and 4He equations.
Suppose that most of the 4He is destroyed after t”4, where t”4 is the tim e at which
56
._
.
ay4 has reached a sizable fraction
f4 =
of its maximum
Then for t > f4
-rxt
yc4f4f;;rxe
f3 = (c4f4 -
value.
(B.1)
-rxt
c3f3)f;;k
P.2)
Here
can be taken to be the sum of the scaled 3H and 3He number
f3
and cs will involve
an appropriate
average of the photodestruction
and 3He. For times late enough that 4He can be photodestroyed
(B.2)is
slightly
smaller than the c4 in (B.l).
densities,
ratios of 3H
to D, the c4 in
This small correction
will not affect
our argument.
Since t > t”4 and since
approximation
equations.
..
cd f4 >>
-
is only to be reduced
f4
to take c 3, c4 and
.
f4
as constants
It is then straightforward
by 0 (10%))
it is a good
in the right hand sides of these
to integrate
(B.2)and
use the fact that,
to obtain
c3f3(t”4)
$$$
= c4(l
c3
(B-3)
- ezp(-C3e-rxi44)J
-- -
This ratio
reaching
(B.2).
of abundances
quasi-static
If cse -rxG
From the solution
is never small.
equilibrium
is satisfied
< 1, the exponential
to (B.l),
If cseerxi4
unimportant.
this quantity
for each 4He destroyed;
We conclude
f4, then either too little
that
g
can be expanded
of 4He, which must be taken to be 0 (&).
being produced
so that
> 1, the condition
N c4e-rxt4.
is seen to be the fractional
destruction
so that
Th is case corresponds
if 4He destruction
57
the fixed point
2
photodissociation
4He is destroyed
N 2,
for
f3
of
to a 3H or 3He‘
of 3H or 3He being
occurs predominantly
or too much 3He is produced.
after
This
leaves open the possibility
and f4. If the destruction
optimistic
where
freezout
proportional
must
during
over-depleted
t
6,
=
c4(t4
This
+
6)
-
t4
+ 6 and 6 <
fi/f4
c3
reach QSE and then
t4.
b/t4 5 10m3. For significant
at t4 + 6. This
requires
the fraction
case is therefore
more X decays
rx
< 6 2 5 x lo3
of X’s remaining
not only fine tuned,
large value for
-
Since of4 is approximately
6 after t4, significantly
were this small,
of an absurdly
.-
f$.
but
Of course,
by t4
also leads
7Li would
be
by a very large factor.
RB = 1 and X having
decays does not work:
too large a value for the primordial
is produced.
< mproton is completely
Thus Mx
scenario
which
showers.
Furthermore,
abundance
t4
.
’
5 10m3 requires
We have shown that a scenario with
-- -
be for (B.2)to
f3
-N
_
14
the time interval
if rx
2 e-looo.
to the requirement
-
at
to 6 for.small
However,
-is e -UTx
between
close to t4, then c4 < cg. The most
ratio would
occur- at, say, t4 + 6/2 than
seconds.
..
f3/f4
4He is destroyed
so that
occurred
4He destruction
sufficient
occurs sufficiently
case for a small
freeze out rapidly
that
utilizes
excluded.
only non-baryonic
3He/4He
Any successful RB = 1
late decays must consider
the complications
this analysis
it is better
by baryodestruction
suggests that
rather
of baryonic
to reduce the 4He
than by photodissociation.
58
abundance
=P
Acknowledgements
It is a pleasure to thank
:,
especially
Ann Boesgaard,
Bob Wagoner for very valuable
also like to thank
David
Schramm,
Mike Turner
discussions and suggestions.
Bob Wagoner for the use of the output
and
We would
from his code.
S.D. and G.D.S. acknowledge
der Grant
NSF-PHY-86-12280
ence and Engineering
knowledges
at SLAC.
support
L.J.H.
to Stanford
University.
by the Department
acknowledges
Science Foundation
G.D.S.
of Energy, contract
support
by the Director,
and Nuclear
of Energy under contract
under Grant PHY-85-15857.
Foundation
by the National
Research Council of Canada Postgraduate
sources, Office of High Energy
. the U. S. Dept.
support
Fellowship
L.J.H.
Physics,
and a Presidential
59
Scholar.
Sci-
R.E. ac-
DE-AC03-76SF00515
Office of Energy
Division
DE-AC03-7600098
also acknowledges
is a National
un-
of High Energy
Reof
and in part by NSF
partial
support
Young Investigator
Award.
by a Sloan
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Yang,J.,
Turner,M.S.,
Steigman,G.,
Schramm,D.N.
281,493.
64
and Olive,K.A.
1984, Ap.J.,
FIGURE
:-
1) -%mzas a function
susceptible
of time.
The times when the various
to photodestruction
2) The post pair production
CAPTIONS
elements become
are indicated.
“breakout”
spectrum
of energetic photons at 106s,
arising from the decay of an X with Mx
3) The contribution
probability
of one 5GeV n to a baryon
of taking
that
Q’S destroyed,
4) The reactions
5) Abundances
..
4He,3He,3H,d,
are indicated
which determine
times its central
= 1.27
node is indicated.
the
The
of
below
as a function
x 10m3,
On each branch,
and n’s, and the number
the abundances
of the light elements
104GeV , ;$( f2/fp)
shower.
branch from the parent
final yields of suprathermal
background
= 100GeV.
of the light elements
of time for Mx/r&
r, = 4.5 x 105s , with
= 7 x
17 equal to 1.7
value and EsLi = ezLi.
6) Abundances
7
x 104GeV
of the light elements as a function
, r&(f$/fp)
and so is included
-- -
x 10m3.
= 1.27
of X lifetime
3H eventually
for A4x/r&
=
decays into 3He
in the final 3He abundance.
7) The allowed region in the M x /r* B - rx plane for 30%,40%,50%,
60%,75%
and 90% astration.
8) Preferred
range of Mx/ri
9) Abundances
r;(f;;/fp)
and rx
of the light elements as a function
= 1.27 1O-3 , and rx
= 3,4,5
and 7Li.
-
65
of lWx/ri
105s. a)D,
for
b) 3He and c) 6Li
Table 1. Reactions
contributing
to baryon
shower development
PP-+PP
PP+PP+s
5,
6,
7,
8,
9,
10,
11,
1%
13,
14,
15,
1%
17,
18,
1%
..
-
2%
pp--+prm’s
w-v
np-mpr’s
p4He+p4He
p4He--+p4Hen’s
p4He--+n4Her’s
p4He+np3He
p4He+2p3H
p4 He+p2d
p4He+2pnd
p4He+2n3p
n4He+n4He
n4He--m4He#s
n4He-mp3H
n4He+2n3He
n4 He+n2d
n4He+p2nd
n4He-+3n2p
.a
-
._
.
Table- 2., Hadron
Yields
E = 0.25
E = 0.60
I:
-15
-7
ed
7
3
t 3H
6
2.5
E 3He
4
1.5
t 4He*
7
3.5
-
_- -
t 6Li
7 x 1o-5
3 x1o-5
&Li-
2.5 x lo-
1 x lo+
I 7Be
2.5 x lo-
1 x 1o-6
en
..
slho wer yield
18
9
._
4
Table
.
Reaction
:r
..YU1”
Energy
R
A rnmnilat.inn
V.
AL
nf all
-~LLy”..a”IV’L
Range
VI
WII
the
“IL”
data
~ranrl
Ux.u”xA
UUbU
Reference
Useful Data
Mitler
30 - 120 MeV
ototal
30 - 110 MeV
ototal
4He(a,p)7Li
60 - 140 MeV
ototal
4He(a,p)7Li
40 - 180 MeV
Qtotal
Glagola
4He(a,p)7Li
40 - 400 MeV
gtotal
Meneguzzi
4He(cx, n)7Be
40 - 180 MeV
ototal
Glagola
4He(a,p)7Li
4He(a,p)7Li,
7Li*
Kozlovsky
1972
and Ramaty
1974
King et al 1975
-
_a
4He(3He,p)6Li
4He(a,
z)“&i
4He(a,
z)‘Li
10,28,40
MeV
* cm , ototal
40 - 180 MeV
60-
MeV
Ototal,
utotal
m3H(~, n)6Li
0 - 14 MeV
Utotal
x)
“Li(n,p)‘He
et al 1978
and Brown
Meneguzzi
G.Hale,
1977
et al 1978
Mitler
up2aytial
40 - 1000 MeV
et al 1971
Glagola
utotal
4He(cr, z)6Li
6Li(n,
Koepke
et al 1978
1972
et al 1971
private
communication
4-7MeV
u;artial
3-9MeV
Utotal
Bresser et al 1969
10 - 150 keV
Qtotal
Sowerby et al 1970
964 keV
Utotal
Rossorio-Garcia
and Beason 1977
..
-’
GLi(n, a)t
-6Li(ti,
CE)~H
6Li(n,
t)4He
100 - 600 MeV
a)t
lOOeV-20 MeV
Utotal
t)4He
27 MeV
d:Tm
c~)~H
10 - 1000 keV
z)
200eV-2 MeV
.‘Li(n,
d)5He
14.4 MeV
df:,
“Li(n,
y)7Li
lOeV- 10 MeV
Utotal
3 - 10 MeV
Utotal
6Li(n,
- ‘Li(n,
‘Li(n,
‘Li(n,
6Li(n,p)6He
Stephany
Brown
g (0”)
Utotal
uabsorption
and Knoll
et al 1977
Nuclear
Cross Sections
Proceedings,
Neutron
Standard
Lowell
1976
Ref. Data
Sowerby et al 1970
Valkovic
Nuclear
et al 1965
Cross Sections
Proceedings,
Lowell
1976
1972
-
._
Energy
Reaction
6Li(n,
gtotal
Proceedings,
Lowell
1976
6Li(n, 2np)4He
0 - 20 MeV
ototal
Proceedings,
Lowell
1976
6Li(P,
>
7Li(n,
)
7L+,
5
M~V
3 - 15 MeV
5
lo-2
7Be(n,p)7Li
1 MeV
_
104
MeV
=5 1 MeV
3He(n, d3H
3ke(n,p)3H
“He(n;
104
14.4 MeV
y)*Li
>
_
14.8 MeV
t), (n, d)
7Li(P,
1 MeV
lo-2
t)“He
7Li(n,
da
dt’,rn
14 MeV
7)7Li
7Li(n,
7)4He
3He(n, d)D
-.
Reference
0 - 20 MeV
‘Li(n,
-
Useful Data
n’a!)4He
‘Li(n,p)‘He
..
Range
_
0.1 - 10 MeV
et al 1972
(oV)capture
Fowler et al 1967
adestruction
Reeves 1974
Lindsay
-d:zm
-
Merchez
Proceedings,
-d;fiR
da
dCl,,dE
Miljanic
et al 1973
Lowell
et al 1970
(ou)capture
Fowler et al 1967
Ddeetruction
Reeves I974
(gu)capture
Fowler et al 1967
ototal,
dc
dR
Nuclear
Cross Sections
2
1 MeV
(ou)capture
Fowler et al 1967
2
1 MeV
(ou)capture
Fowler et al 1967
0.1 - 10 MeV
1976
ototal
Nuclear
Cross Sections
3He(7, P)D
5.5-
MeV
aphotodise
Gorbunov
and Varfolomeev
1964
3He(7, n)2p
7.7-
MeV
aphotodias
Gorbunov
and Varfolomeev
1964
3H(7, n)D
5.5-
MeV
aphotodiea
Faul et al 1981
3H(7, p)2n
7.7-
MeV
aphotodiss
Faul et al 1981
4He(7, pn)D
25-
MeV
aphotodiss
Gorbunov
and Spiridonov
1958a
4He(7, n)3He
20.6-
MeV
aphotodiss
Gorbunov
and Spiridonov
195813
19.8-
MeV
aphotodiss
Gorbunov
and Spiridonov
195813
Gorbunov
and Spiridonov
1958b
4He(w)3H
4He(7,
qr,
)
n)p
20-
MeV
photodiss
Ototal
2.2-
MeV
photodiss
utotal
Govarets
et al 1981
React ion
Energy
Range
Useful Data
Reference
1975
6L47,
)n
5-
MeV
,,photodiss
6Lq7,
)P
5-
MeV
aphotodisa
)3He, 3H
18-
MeV
aphotodiss
lo-
MeV
aphotodiss
Berman
1975
7-
MeV
aphotodias
Berman
1975
aphotodise
Berman
1975
6Li(7,
7Li(7,p)6He
‘L. 7Li(7,
)n,2n
7h(y,
t)4He
7Be(7,
>
2.5-
MeV
aphotodiss
3-.MeV
Berman
assumed equal to ‘Li(7,
Wong et al 1970
assumed same as 7Li
nb, 7)d
2 1 MeV
(4
capture
Fowler
et al 1967
n(d, 7)3H-
2 1 MeV
(4
capture
Fowler
et al 1967
v, PP
p-4He
)(4He,
)d,3He,3H
pi4He
p--He
1 MeV-10
GeV
1 MeV-10
GeV
1 -MeV-10
GeV
3 MeV-1 GeV
0.6-1.8 GeV
oelastic, Uinelastic, gelastic
oelastic,oinelastic
cpartial
do
dR
elastic
elastic
4.E
dR
Meyer 1972
Meyer 1972
Meyer 1972
Meyer
Klem
1972
et al 1977
P-G
1 MeV-10
GeV
oelastic,
uinelastic
Meyer
1972
p-3He
1 MeV-IO
GeV
belastic,
uinelastic
Meyer
1972
n(4He,pn)3H
90,300 MeV
dv
dEa ,q
Meyer 1972
)r
._
ICAVIL
-I.
Primordial
Primordial
SBBN
a~“I.zb”~;u
u.llU
hJ.LJUI,
aauuILuaIIL,c;u
7Li
6Li
D
3He
6.4 x 1O-5
1.2 x 1o-4
2.6 x lo-lo
8.4 x lo-l1
few 10m5
5 few 10m5
< lo-l1
lo-lo
50%
3.2 x 1O-5
7.5 x 1o-5
1.3 x lo-lo
(10-9)
Disk:
75%
1.6 x 1O-5
5.2 x 1O-5
6.5 x lo-l1
(10-9)
Disk:
90%
0.6 x 1O-5
3.9 x 1o-5
2.6 x lo-l1
(1O-g)
few 10v5
5 few 10e5
< lo-l1
(10-9)
< lo-l0
1o-g
Observation
Primordial
particle
-compared
8 x 1O-5 -4
and astrated
scenario
with
cleosynthesis.
-
IUll”lUIc&I,
Disk:
SBBN
..
I
with
M,/rg
observations
-
x 1O-4
1O-5 -4
abundances
of D, 3He,6Li
= 7-x 104GeV,
and with
x 1O-4
and 7Li in a decaying
TX = 4.5 x 105s and (7 = 1.7J:,
the predictions
of Standard
Big Bang Nu-
-
IO2
IO’
-
+Li6
-
10-*
+ ‘7 I
U
IO0
-- 9-87
IO2
IO4
t
(set)
Fig. 1
‘He3
,+(H3
IO6
IO8
5848A 1
._
IO8
L-
I
I
I
’
I
I
I
I
’
IO7
I
>
r”
W
-L
‘. w
I
I
I
’
4
Mx = 100 GeV
-i
I
’
t=106s
IO6
IO5
IO4
IQ3
0
9-87
I
I
10*
2
.
4
E
I
6
(MeV)
-- -
Fig. 2
I I
I
8
IO
5848A2
._
.
5GeVn
I---\
-
-
-
I
pnp
nnna
;
200 (145 xx) (I-E)5 200
5 200
GeV MN
GeV MeV GcV MeV GeV GeV
-
j,
-
n
nk 4Hi* 3He 3H d (I-c)5
GeV
loo
2
---25
25
52
loo
-loo IO0 100 100 100 100
.n
p’
200 200
MeV MeV
63
63
looloo
TOTALS:
.
-- -
n(E =fjGeV)
9 3
+ 301:(-)22
1304
.* 414
n[E=(I-c)5teV]
: g
$
+
n(E=2OOMeV)
1 5 $25 3
. 41 7
+
p(E= 200MeV)
K
“a
.-B--8 7
.
*
g
f
*
5
= 0.65
631*04l
1304
*
63 I
+ 557
= 0.58
100 I = 0.19
Ix,4
4He :
32L
1304
‘3He :
g
3H
:
251
1304
d
:
22 I D 0.10
I30 4
Fig. 3
~(306
*
+ = 0.05
ZOO5
5848A3
-
I
.
A
70e
-
..
-.
-
9-87
584aA4
n
Fig. 4
-
IO4
-. 12-87
105
TIME (set)
Fig. 5
106
107
IO8
584aii5
4
100
"
1 "
"
1 "
"
1 "I'
) "'I
-
4He
‘I’
h
= 7.0~10~
M,/rg*
-
IO+-
rB*[x/b]
= 1.27
X
GeV
'o-3
10-g
- .
6Li
I
I
I
I
I
I
.I
I
I
I
2
-- 12-87
T,(lO%ec)
Fig. 6
-- -
I
I
I
I
, 7Li
I I
I
I )
10
5848A6
t-l
-- -
12-87
0
l
I
I
I
I
I
0.5
I
I
I
1
M ,/Q*
(IO5 GeV)
Fig. 7
1
I
I
I
I
I
1.5
2
584aA7
6.0,,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
,
5.5 -
5.0 -
4.5 TV
-r”
4.0 -
3.5 3.0-l
-- -12-87
'1
""I
0.4
""I
""I
0.6
M,/rg'(lO
Fig. 8
0.8
4 GeV)
"'I
1
1.2
504aAa
--
A=x=3x105sec
- 10-2
m Tx =4x105sec
. =x =5x105sec
10-3
10-4
-
-
.
10-5
-- 12-87
-
Mx/rg*(104GeV)
Fig. 9a
5848A9
._
I
_m
A zx
=3x105sec
=X
=4x105sec
TX
=5x105sec
IO'
42-87
Mx/rg*(104GeV)
Fig. 9b
5848AlO
-
._
10-g
c
100
-- - 12-87
M,/rg*
(lo4 GeV)
Fig. 9c
5848All