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\begin{center}
\today     \hfill    LBNL-41826 \\
%~{} \hfill ICRR-Report-??-98-?? \\
~{} \hfill \\


\vskip 0.2in

{\large \bf Radiative Decay of a Long-Lived Particle and Big-Bang
Nucleosynthesis}%
\footnote{
This work was supported by the Director, Office of Energy
Research, Office of Basic Energy Services, of the U.S.
Department of Energy under Contract DE-AC03-76SF00098. K.K. is
supported by JSPS Research Fellowship for Young Scientists.
}

\vskip 0.3in

Erich Holtmann$^{a}$, M. Kawasaki$^{b}$, K. Kohri$^{b}$, and  
Takeo Moroi$^{a}$

\vskip 0.2in

{\em $^{a}$Theoretical Physics Group,
     Earnest Orlando Lawrence Berkeley National Laboratory,\\
     University of California, Berkeley, California 94720}

\vskip 0.2in

{\em $^{b}$ Institute for Cosmic Ray Research, The University of Tokyo,
     Tanashi 188-8502, Japan}
 
\vskip 0.4in

\end{center}

\begin{abstract}

The effects of radiatively decaying, long-lived particles on big-bang
nucleosynthesis (BBN) are discussed. If high energy photons are
emitted after BBN, they may change the abundances of the light
elements through photodissociation processes, which may result in a
significant discrepancy between the BBN theory and observation.  We
calculate the abundances of the light elements, including the effects
of photodissociation induced by a radiatively decaying particle, and
we derive a constraint on such particles by comparing our theoretical
results with observations.  Taking into account the recent
controversies regarding the observations of the primordial D and
$^4$He abundances, we derive constraints for various combinations of
the measurements.  We also discuss several models which predict such
radiatively decaying particles, and we derive constraints on such
models.

\end{abstract}

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\section{Introduction}
\label{sec:intro}

Big-bang nucleosynthesis (BBN) has been used to impose constraints 
on neutrinos and other hypothetical particles predicted by particle 
physics, because BBN is very sensitive to the thermal history of the 
early universe at temperatures $T \lesssim 1$ MeV~\cite{BBN-review}.

Weakly interacting massive particles appear often in particle physics.
In this paper, we consider particles which have masses of
$\sim O(100{\rm ~GeV})$ and which
interact with other particle only very weakly ({\it e.g.}, through
gravitation).  These particles have
lifetimes so long that they decay after the BBN of the
light elements (D, $^3$He, $^4$He, etc.), so
they and their decay products may affect the
thermal history of the universe.  In particular, if the long-lived
particles decay into photons, then the emitted high energy
photons induce electro-magnetic cascades and produce many soft
photons.  If the energy of these photons exceeds the binding energies
of the light nuclides, then photodissociation may profoundly alter
the light element abundances.
Thus, we can impose constraints on the abundance and lifetime of
long-lived particles, by considering the photodissociation
processes induced by its decay.
There are many works on this subject, such as the
constraints on massive neutrinos and gravitinos obtained by the
comparison between the theoretical predictions and
observations~\cite{Lindley,Raddec,KM1,KM2,hadron}.

A couple of years ago, Hata {\it et al.}~\cite{HSSTWBL} claimed that
light-element observations seemed
to conflict with the theoretical predictions of
standard BBN.  Their point was that standard BBN predicts too
much $^{4}$He, if the baryon number density is determined
by the D abundance inferred from observations;
equivalently, standard BBN
predicts too much D, if the baryon number density is
determined by the $^{4}$He observations.  Inspired by this ``crisis
in BBN,'' many people re-examined standard and non-standard BBN by
including systematic errors in the observations, or by introducing
some non-standard properties of neutrinos~\cite{xinu,mstau}.  In a
previous paper~\cite{PRLHKM}, we investigated the effect upon BBN
of radiatively-decaying massive particles. These particles induce
an electro-magnetic cascade.  We found that in a certain
parameter region, the photons in this cascade destroy only D,
so that the predicted abundances of D, $^{3}$He, and $^{4}$He fit the
observations.

However, since the ``BBN crisis'' was claimed, the situation
concerning the observations of the light elements has changed.  First,
the D abundances in highly red-shifted quasar absorption systems (QAS)
have been observed by two groups. The abundance of D in high-$z$ QAS
is considered to be the primordial value. Thanks to these direct new
observations, we no longer need to use poorly-understood models of
chemical evolution to infer the primordial abundance from the material
in solar neighborhood.  Unfortunately, however, the D abundance
measured by the first group~\cite{songaila, higp, no-interloper} is not
consistent with the abundance measured by the other~\cite{BurTyt}.  We
may have to wait for more data before we can decide the primordial
abundance of D.

Second, the observations of $^{4}$He have also changed.  For a long
time, the relatively low $^{4}$He abundance (viz., $Y\simeq 0.234$,
where $Y$ is the primordial mass fraction of
$^{4}$He)~\cite{pagel,OliSkiSte} was believed.  Recently, however, a
higher $^{4}$He abundance ($Y \simeq 0.244$) has been
reported~\cite{ThuIzo,IzoThuLip}.  The typical errors in $^{4}$He
observations are less than $\simeq 0.005$, so we have discordant
data for $^{4}$He as well as for D.

Since we have new observations for D and $^{4}$He, the previous
constraint on the radiative decay of long-lived particles must
be revised.  In addition, the statistical analyses on radiatively
decaying particles are insufficient in the previous works.  Therefore,
in our present paper, we perform a better statistical analysis of
a long-lived, radiatively-decaying particle, and of the resultant
photodissociations, in order to constrain the abundances and lifetimes
of long-lived particles.  In deriving the
constraint, we use all four combinations of the observed abundances of D
and $^{4}$He, because it is premature to decide which data are correct.
As a result, it will be shown that for a certain combination of the
observed data, we have a discrepancy between observations and standard
BBN theory.  Moreover, we show in that case that a long-lived particle
with appropriate abundance and lifetime can solve the discrepancy.  In
the other cases, standard BBN fits the observations, so we derive
stringent constraints on the properties of long-lived particles.

In this paper, we also include the photodissociations of
$^7$Li and $^6$Li for the first time. As we will show later, the
destruction of $^7$Li does not dramatically affect the predicted D and
$^{4}$He, in the region where the 
observed D and $^{4}$He values are best fit.
However, the $^6$Li produced by the destruction of $^7$Li can be
two orders of magnitude more abundant than the standard BBN
prediction of $^6$Li/H $\sim O(10^{-12})$. We discuss the possibility
that this process may be the origin of
the $^6$Li which is observed in some low-metallicity halo stars.

In Sec.~\ref{sec:SBBN} we study how consistent the theoretically
predicted abundances and observations are, in the case of standard
BBN.  The radiative decay of long-lived particles is considered in
Sec.~\ref{sec:BBNX}, and the particle physics models which predict such
long-lived particles are presented in Sec.~\ref{sec:model}.  Finally,
Sec.~\ref{sec:summary} is devoted to discussion and the conclusion.

\section{Standard Big-Bang Nucleosynthesis}
\label{sec:SBBN}

We begin by reviewing standard big-bang nucleosynthesis (SBBN).
We are interested in the light elements, since their
primordial abundances can be estimated from observations. In
particular, we check the consistency between the theoretical
predictions and the observations for the following quantities:
 \beq
  y_2 &=& n_{\rm D} / n_{{\rm H}},\\
  Y &=& \rho_{^4{\rm He}} / \rho_{{\rm B}}, \\
  y_6 &=& n_{^6{\rm Li}} / n_{{\rm H}}, \\
  y_7 &=& n_{^7{\rm Li}} / n_{{\rm H}}.
 \eeq
where  $\rho_{{\rm B}}$ is the total baryon energy density.
Notice that we do not discuss $^3$He, since we would have to
use a poorly-understood model of chemical evolution if we
were to estimate its primordial abundance.
However, even without $^3$He, we obtain a
non-trivial constraint on the BBN model, as we will show below.

In this section, we first review the observations of the light
elements, and the extrapolations back to the primordial abundances.
Next, we describe our theoretical calculations of these abundances, by
using standard big-bang theory as an example.  Finally, we compare the
theoretical and observed light-element abundances to determine how
well the SBBN theory works.

\subsection{Review of Observation}
\label{sect-obs}

Let us start with a review of the observations of the light element
abundances. Two factors complicate the interpretation of the
observations of the light-element abundances.  First, there are
several observational results (both for D/H and for $^4$He)
which are not consistent with each other, within the quoted errors.
This fact suggests that
some groups have underestimated their systematic error.\footnote
 {We do not believe that the discordant measured abundances
 are evidence of inhomogeneity, because such a large-scale
 primordial inhomogeneity is ruled out by the observed smoothness
 of the cosmic microwave background~\cite{inhomo}.}
We believe it is premature to judge which measurements are reliable;
hence, we consider all possible combinations of
the observations when we test the consistency between theory and
observation.  Second, some guesswork is involved in the extrapolation
back from the observed values to the primordial values, as we shall
discuss below. Keeping these factors in mind, we review the
estimations of the primordial abundances of D, $^4$He, $^6$Li, and $^7$Li.

D/H has been measured in the absorption lines of highly red-shifted
(and therefore presumably primordial) H$_{\rm I}$ (neutral hydrogen)
clouds which are backlit by quasars.  However, the D/H measurements
from these QAS generally fall into two classes, viz., high and low,
which differ by almost an order of magnitude.

The first three measurements (all in the direction of QSO 0014+813)
were high~\cite{songaila, earlyQASH, no-interloper}, in the range $y_2=(1.9-2.5)\times
10^{-4}$.  Since these original observations, there have been
additional measurements~\cite{higp, recentQASH} of high D/H in this and
other QAS.  However, Carswell {\it et al.} state that there is a
significant probability that their ``deuterium'' may actually be
Doppler-shifted hydrogen~\cite{earlyQASH} in an interloping H$_{\rm
I}$ cloud.  Steigman~\cite{Doppler} claims that this may be the case
in other measurements, as well, although Rugers and
Hogan~\cite{no-interloper} say that an interloper is very unlikely.
Finally, Tytler, Burles, and Kirkman~\cite{TytBurKir} reobserved
QSO 0014+813 and found that their higher-quality data yield a very
large uncertainty in D/H.

On the other hand, Tytler {\it et al.}~\cite{earlyQASL}
have found much smaller values of D/H, viz.
$y_2 \sim 2.4 \pm 0.4$, in the directions of QAS 1937-1009
and QAS 1009-2956.
However, a reanalysis~\cite{reanQASL} of Tytler's QAS 1937-1009
data yields a much higher D/H value.  Similarly, new data
for QSO 1937-1009~\cite{reobQASL} also yields higher D/H.

Because of these conflicting measurements, we will perform
several analyses in our
paper.  For our low values, we use the recent determination of
Burles and Tytler~\cite{BurTyt}.
This value is slightly higher than their original measurement,
because they use an improved model of the cloud and have a better
measurement of the neutral hydrogen:
%
\beq
%
\mbox{\rm Low: }
y^{obs}_2 = (3.39 \pm 0.25) \times 10^{-5}.  \label{lowD}
%
\eeq
%
We take our high value from
Rugers and Hogan~\cite{no-interloper}:
%
\beq
%
\mbox{\rm High: }
y^{obs}_2 = (1.9 \pm 0.5) \times 10^{-4}.  \label{highD}
%
\eeq
%

In this paper, we do not rely upon the presolar and
interstellar-medium measurements of D and $^3$He, because of the
uncertainty involved in extrapolating back to the primordial
abundance of
D/H.  An analysis based upon these measurements will appear in a
separate paper by one of the authors (E.H.).  This analysis generally
agrees with the low QAS D/H in this paper.

The primordial $^4$He abundance is deduced from observations of
extragalactic H$_{\rm II}$ regions (clouds of ionized hydrogen).
Currently, there are two classes of $Y^{obs}$, reported by several
independent groups of observers.  Hence, we consider two cases:
one low, and one high.

We take our low $^4$He abundance from Olive, Skillman,
and Steigman~\cite{OliSkiSte}.
They used measurements of $^4$He and O/H
in 62 extragalactic H$_{\rm II}$ regions, and linearly extrapolated
back to O/H$= 0$ to deduce the primordial value
%
\beq
%
\mbox{\rm Low: }
Y^{obs} = 0.234 \pm (0.002)_{stat} \pm (0.005)_{syst}   \label{lowHe}.
%
\eeq
%
(When they restrict their data set to only the lowest metallicity
data, they obtain $Y^{obs}= 0.230 \pm 0.003$.)  Their
systematic error comes from numerous sources, but they claim that no
source expected to be much more than 2\%.  In particular, they
estimate that stellar absorption is of order 1\% or less.

We take our high $^4$He abundance from Thuan and
Izotov~\cite{ThuIzo}.
They used measurements of $^4$He and O/H in a new sample of 45 blue
compact dwarf galaxies to obtain
%
\beq
%
\mbox{\rm High: }
Y^{obs} = 0.244 \pm (0.002)_{stat} \pm (0.005)_{syst}   \label{highHe}.
%
\eeq
%
The last error is an estimate of the systematic error, taken from
Izotov, Thuan, and Lipovetsky~\cite{IzoThuLip}.
Thuan and Izotov claim that He$_{\rm I}$ stellar absorption is an
important effect; this explains some of the difference between their
result and that of Olive, Skillman, and Steigman.

Rather than attempting to judge which group has done a better job of
choosing their sample and correcting for systematic errors, we
prefer to remain open-minded.  Hence, we shall use both the high and
low $^4$He abundances, without expressing a preference for one over
the other.

The $^7$Li/H abundance is taken from observations of the surfaces of
Pop II (old) halo stars.  In general, Li/H decreases with decreasing
stellar surface temperature, since cooler ({\it i.e.}, lower mass) stars
have deeper convection zones, and $^7$Li is destroyed in the warm
interior of a star.  However, Spite and Spite~\cite{SpiSpi} found that
at high surface temperatures, $^7$Li/H levels off into a ``plateau.''
This is interpreted as the primordial value of $y_7$.  Similarly, it
was found that $^7$Li decreases with decreasing Fe/H (iron indicates
non-primordial matter), but $^7$Li levels off at very low
metallicities ([Fe/H] $\leq -1.5$).\footnote
 { [Fe/H] $\equiv
   \log_{10} (n_{\rm Fe}/n_{\rm H})
 - \log_{10} (n_{\rm Fe}/n_{\rm H})_{presolar}.
 $}
Using data
from 41 plateau stars, Bonifacio and Molaro~\cite{BonMol} determine
the primordial value $\log_{10}(y^{obs}_7) = -9.762 \pm (0.012)_{stat}
\pm (0.05)_{syst}$.  Bonifacio and Molaro
argue that the data provides no evidence for $^7$Li/H depletion in the
stellar atmospheres (caused by, {\it e.g.}, stellar winds, rotational
mixing, or diffusion).  However, for our analysis, we shall adopt the
more cautious estimate of Hogan~\cite{Hog} that $^7$Li may have
been supplemented (by production in cosmic-ray interactions) or
depleted (in stars) by a factor of two:~\cite{factor-of-two}
 \beq
  \log_{10}(y^{obs}_7) = 
  -9.76 \pm (0.012)_{stat} 
  \pm (0.05)_{syst}
  \pm (0.3)_{factor~of~2}.
 \label{Li7}
 \eeq

Because $^6$Li is so much rarer than $^7$Li, it is much more difficult
to observe.  Currently, there is insufficient data to find the ``Spite
plateau'' of $^6$Li.  However, we can set an upper bound on
$^6$Li/$^7$Li, since it is generally agreed that the evolution of
$^6$Li is dominated by production by spallation (reactions
of cosmic rays with the interstellar medium). The upper bounds
on $^6$Li/$^7$Li observed in low-metallicity
([Fe/H] $\leq -2.0$) halo stars range from~\cite{HobTho}
$y_6/y_7 \lesssim 0.045$ to $y_6/y_7 \lesssim 0.13$. (Note that
the primordial $^6$Li and $^7$Li have both been destroyed in material
which has been processed by stars.)

Rotational mixing models~\cite{SmiLamNis} yield a survival
factor for $^7$Li of order 0.05 and a survival factor for $^6$Li
of order 0.005.  Therefore, the upper bound for primordial
$^6$Li/$^7$Li ranges approximately from
 \beq
  y^{obs}_6/y^{obs}_7 \lesssim 0.5 \ \ {\rm to} \ \ 1.3.
 \label{Li6}
 \eeq
Since we have only a rough range of upper bounds on $^6$Li, and no
lower bound, we will not use $^6$Li in our statistical analysis
to test the concordance between observation and theory.
Instead, we will just check the consistency of our theoretical
results with the above constraint.

\subsection{Theoretical Calculations}

Theoretically, the primordial abundances of the light elements
in SBBN
depend only upon a single parameter: the baryon-to-photon ratio
$\eta$. In our analysis, we modified Kawano's nucleosynthesis
code~\cite{Kaw} to calculate the primordial light-element
abundances and uncertainties.

In our calculation, we included the uncertainty in the neutron
lifetime~\cite{PDG} and in the 11 most important nuclear reaction
rates~\cite{SmiKawMal}.  We treated the neutron lifetime and the
nuclear reaction rates as independent random variables with Gaussian
probability density functions (p.d.f.'s).  We performed a Monte-Carlo
over the neutron lifetime and the 11 nuclear reaction rates, and we
found that the light-element abundances were distributed approximately
according to independent, Gaussian p.d.f.'s. Therefore, the
p.d.f. $p_{tot}^{th}$ for the theoretical abundances is given by the
product of the Gaussian p.d.f's
\beq
p^{Gauss}(x;\bar{x},\sigma) &=&
 \frac{1}{\sqrt{2 \pi}\sigma}
 \exp \left[ -\frac{1}{2}
            \left( \frac{ x-\bar{x} }{\sigma} \right)^2
      \right]
\label{p_i^th}
\eeq
for the individual
light elements:
%
\beq
p_{tot}^{th} (y^{th}_2,Y^{th}, \log_{10}y^{th}_7)
 = p^{Gauss}_2(         y^{th}_2) \times
   p^{Gauss}_4(         Y^{th}  ) \times
   p^{Gauss}_7(\log_{10}y^{th}_7).        \label{p^th}
\eeq

In Fig.~1,  we have plotted the theoretical predictions for the
light-element abundances (solid lines) with their one-sigma errors
(dashed lines), as functions of $\eta$.

The dependences of the abundances on $\eta$ can been seen
intuitively\cite{BBN-review, SBBN}. The $^4$He abundance is a gentle,
monotonically increasing function of $\eta$. As $\eta$ increases,
$^4$He is produced earlier because the ``deuterium bottleneck'' is
overcome at a higher temperature due to the higher baryon
density. Fewer neutrons have had time to decay, so more $^4$He is
synthesized.  Since $^4$He is the most tightly bound of the light
nuclei, D and $^3$He are fused into $^4$He. The surviving abundances
of D and $^3$He are determined by the competition between their
destruction rates and the expansion rate.  The destruction rates are
proportional to $\eta$, so the larger $\eta$ is, the longer the
destruction reactions continue.  Therefore, D and $^3$He are
monotonically decreasing functions of $\eta$. Moreover, the slope of D
is steeper, because the binding energy of D is smaller than $^3$He.
 
The graph of $^7$Li has a ``trough'' near $\eta \sim 3 \times
10^{-10}$.  For a low baryon density $\eta \lesssim 3 \times
10^{-10}$, $^7$Li is produced by $^4$He(T, $\gamma$)$^7$Li and is
destroyed by $^7$Li(p, $\alpha$)$^4$He. As $\eta$ increases, the
destruction reaction become more efficient and the produced $^7$Li
tends to decrease. On the other hand for a high baryon density $\eta
\gtrsim 3 \times 10^{-10}$, $^7$Li is mainly produced through the
electron capture of $^7$Be, which is produced by $^3$He($\alpha$,
$\gamma$)$^7$Be.  Because $^7$Be
production becomes more effective as $\eta$ increases, the synthesized
$^7$Li increases. The ``trough'' results from the overlap of these two
components. The dominant source of $^6$Li in SBBN is D($\alpha$,
$\gamma$)$^6$Li. Thus, the $\eta$ dependence of $^6$Li resembles that
of D.

We have also plotted the 1-$\sigma$ observational constraints. The
amount of overlap of the boxes is a rough measure of the consistency
between theory and observations.  We can also see the favored ranges
of $\eta$. However, we will discuss the details of our analysis more
carefully in the following section.

\subsection{Statistical Analysis and Results}
\label{sec-bbnResults}

Next, let us briefly explain how we quantify the consistency
between theory and observation. For this purpose, we define the
variable $\chi^2$ as
%
\beq
 \chi^2 = \sum_i \frac{(a_i^{th} - a_i^{obs})^2}
                      {(\sigma_i^{th})^2 + (\sigma_i^{obs})^2} \label{chi2}
\eeq
%
where $a_i = (y_2, Y, \log_{10}y_7)$, and we add the systematic errors
in quadrature:
$(\sigma_i^{obs})^2=(\sigma_i^{syst})^2+(\sigma_i^{stat})^2$.  (See
Appendix~\ref{app:analysis} for a detailed explanation of our use of
$\chi^2$.)  $\chi^2$ depends upon the parameters of our theory
(viz. $\eta$ in SBBN) through $a_i^{th}$ and $\sigma_i^{th}$.

Notice that we do not include $^6$Li in the calculation of $\chi^2$,
since the $^6$Li abundance has not been measured well. Instead, we
check that $y^{th}_6/y^{th}_7$ satisfies the bound (\ref{Li6}).  In
the case of SBBN, we found that the $^6$Li abundance is small enough
over the entire range of $\eta$ from $8.0 \times 10^{-11}$ to $1.0
\times 10^{-9}$.  (Numerically, $y^{th}_6/y^{th}_7 < 5\times 10^{-4}$,
which is well below the bound (\ref{Li6}).)

With this $\chi^2$ variable, we discuss how well the theoretical
prediction agrees with observation. More precisely, we calculate from
$\chi^2$ the confidence level (C.L.)  with which the SBBN theory is
excluded, at a given point in the parameter space of our theory
(for three degrees of freedom):
%
 \beq
  {\rm C.L.} &=& \int_0^{\chi^2}
  \frac{1}{2^{3/2}\Gamma(\frac32)}y^{\frac12}e^{- \frac{y}{2}} d y
  \label{chidis}\\
&=&
  -\sqrt{\frac{2 \chi^2}{\pi}}
  \exp \left( -\frac{\chi^2}{2} \right)
  + {\rm erf} \left( \sqrt{\frac{\chi^2}{2}} \right),
 \label{CL}
 \eeq

In Fig.~2, we have plotted the $\chi^2$ and
confidence level at which SBBN theory is excluded by the observations,
as a function of $\eta$.  We find that high D is allowed at better
than the 68\% C.L. at $\eta \sim 2 \times 10^{-10}$, while low D and
high $^4$He is allowed at better than the 68\% C.L. at $\eta \sim 5
\times 10^{-10}$.  However, for low D and low $^4$He, no value of eta
works at the 91.5\% C.L.

The case of low $^4$He and low D suggests a discrepancy with standard
BBN.  Some people believe that this casts doubt on the low D or low
$^4$He measurements\cite{CopSchTur}.  However, we do not want to
assume SBBN theory and use it to judge the validity of the
observations; rather, we use the observations to test BBN theory.
Therefore, we give equal consideration to all four combinations of the
observed abundances.

Before closing this section, we apply our analysis to constrain the
number of neutrino species. Here, we vary $\eta$ and the number
$N_\nu$ of neutrino species, and we calculate the
confidence level as a function of these variables. The results are
shown in Fig.~3a,b for four combinations of the
observations. We can see that the standard scenario ($N_\nu=3$)
results in a good fit, except for the case of low D and low $^4$He. In
fact, low D and low $^4$He prefers $N_\nu \sim 2$, as pointed out by
\cite{HSSTWBL, mstau}. However, for the other combinations,
$N_\nu = 3$ is 
completely consistent with observation. In Table~\ref{table:nueta}, we show
the 95$\%$ C.L. bounds for the number of neutrino species $N_\nu$ and
$\eta$ in the four cases.

\begin{table}[t]
\begin{center}
\begin{tabular}{lcc}
\hline\hline
  & $N_\nu$ (95 $\%$ C.L.) & $\eta \times 10^{10}$ (95 $\%$ C.L.)\\
\hline
Low $^4$He $\&$ Low D &
   2.1$^{+1.0}_{-0.8}$ &
   4.7$^{+1.0}_{-0.8}$ \\
Low $^4$He $\&$ High D &
    3.0$^{+1.2}_{-1.1}$ & 
    1.8$^{+1.9}_{-0.5}$\\
High $^4$He $\&$ Low D &
     2.8$^{+1.0}_{-1.0}$ &
     5.0$^{+1.0}_{-0.8}$ \\
High $^4$He $\&$ High D &
     3.7$^{+1.3}_{-1.2}$ & 
     1.9$^{+2.1}_{-0.6}$\\
\hline\hline
\end{tabular}
\caption{Observational constraints on $\eta$ and $N_\nu$ in SBBN}
\label{table:nueta}
\end{center}
\end{table}

\section{BBN + $X$}
\label{sec:BBNX}

In this section, we discuss the implications of a radiatively
decaying particle $X$ for BBN. For this purpose, we first
discuss the behavior of the photon spectrum induced by $X$. Then we
show the abundances of the light elements, including the effects of the
photodissociation induced by $X$. Comparing these abundances with
observations, we constrain the parameter space for $\eta$ and $X$.

\subsection{Photon spectrum}

In order to discuss the effect of high-energy photons on BBN, 
we need to know the shape of the photon spectrum induced by the
primary high-energy photons from $X$ decay.

In the background thermal bath (which, in our case, is a mixture of
photons $\gamma_{\rm BG}$,
electrons $e^{-}_{\rm BG}$, and
nucleons $N_{\rm BG}$), high energy photons lose their
energy by various cascade processes. In the cascade, the photon
spectrum is induced, as discussed in various literature\cite{spectrum}
. The important
processes in our case are:
 \begin{itemize}
  \item Double-photon pair creation
($\gamma +\gamma_{\rm BG} \rightarrow e^{+} +e^{-}$)
  \item Photon-photon scattering
($\gamma +\gamma_{\rm BG} \rightarrow \gamma +\gamma$)
  \item Pair creation in nuclei
($\gamma  +N_{\rm BG} \rightarrow e^{+}  +e^{-} + N$)
  \item Compton scattering
($\gamma  +e^{-}_{\rm BG} \rightarrow \gamma  +e^{-}$)
  \item Inverse Compton scattering
($e^{\pm} +\gamma_{\rm BG} \rightarrow e^{\pm}  +\gamma$)
 \end{itemize}
(We may neglect double Compton scattering
$ \gamma + e^{-}_{\rm BG} \rightleftharpoons \gamma + \gamma + e^- $,
because Compton scattering is more important for thermalizing
high-energy photons.)
In our analysis, we numerically solved the Boltzmann equation
including the above effects, and obtained the distribution function of
photons, $f_\gamma (E_\gamma)$. (For details, see
Refs.~\cite{KM1,KM2}.)

In Fig.~4, we show the photon spectrum for several
temperatures $T$. Roughly speaking, we can see a large dropoff at
$E_\gamma\sim m_e^2/22T$ for each temperature.  Above this threshold,
the photon spectrum is extremely suppressed.

The qualitative behavior of the photon spectrum can be understood in
the following way. If the photon energy is high enough, then
double-photon pair creation is so efficient that this process
dominates the cascade. However, once the photon energy becomes much
smaller than $O(m_e^2/T)$, this process is kinematically
blocked. Numerically, this threshold is about $m_e^2/22T$, as we
mentioned. Then, photon-photon scattering dominates.  However, since
the scattering rate due to this process is proportional to
$E_\gamma^3$, photon-photon scattering becomes unimportant in the
limit $E_\gamma\rightarrow 0$.  Therefore, for $E_\gamma\ll
O(m_e^2/T)$, the remaining processes (pair creation in nuclei and
inverse Compton scattering) are the most important.

The crucial point is that the scattering rate for $E_\gamma\gtrsim
m_e^2/22T$ is much larger than that for $E_\gamma\ll m_e^2/22T$, since
the number of targets in the former case is several orders of
magnitude larger than in the latter. This is why the photon spectrum
is extremely suppressed for $E_\gamma\gtrsim m_e^2/22T$. As a result,
if the $X$ particle decays in a thermal bath with temperature
$T\gtrsim m_e^2/22Q$ (where $Q$ is the binding energy of a nuclide)
then photodissociation is not effective.
 
\subsection{Abundance of light elements with $X$}

Once the photon spectrum is formed, it induces the photodissociation
of the light nuclei, which modifies the result of SBBN. This process is
governed by the following Boltzmann equation:
 \begin{eqnarray}
  \frac{dn_N}{dt} + 3Hn_N &=& \left[\frac{dn_N}{dt}\right]_{\rm SBBN}
  - n_N \sum_{N'}\int dE_\gamma 
  \sigma_{N\gamma\rightarrow N'}(E_\gamma) f_\gamma (E_\gamma)
 \nonumber \\ &&
  + \sum_{N''}n_{N''} \int dE_\gamma 
  \sigma_{N''\gamma\rightarrow N} (E_\gamma) f_\gamma (E_\gamma),
 \end{eqnarray}
 where $n_N$ is the number density of the nuclei $N$, and
$[dn_N/dt]_{\rm SBBN}$ denotes the SBBN contribution to the Boltzmann
equation. To take account of the photodissociation processes, we
modified the Kawano code~\cite{Kaw}, and calculated the abundances of
the light elements. The photodissociation processes we included in our
calculation are listed in Table~\ref{table:pf}.

\begin{table}[t]
\begin{center}
\begin{tabular}{rlrrr}
\hline\hline
& {Photofission Reactions} & 1$\sigma$ Uncertainty & Threshold Energy & Ref.\\
\hline
 1. &   ${\rm D} + \gamma \rightarrow p + n$
                      &  6\% &  2.2 MeV & \cite{Evans}\\
 2. &   ${\rm T} + \gamma \rightarrow n + {\rm D}$
                      & 14\% &  6.3 MeV & \cite{ZP208-129,PRL44-129}\\
 3. &   ${\rm T} + \gamma \rightarrow p + 2n$
                      &  7\% &  8.5 MeV & \cite{PRL44-129} \\
 4. &$^3{\rm He} + \gamma \rightarrow p + {\rm D}$
                      & 10\% &  5.5 MeV & \cite{PL11-137} \\
 5. &$^3{\rm He} + \gamma \rightarrow n + 2p
                    $ & 15\% &  7.7 MeV & \cite{PL11-137} \\
 6. &$^4{\rm He} + \gamma \rightarrow p + {\rm T}$
                      &  4\% & 19.8 MeV & \cite{PL11-137} \\
 7. &$^4{\rm He} + \gamma \rightarrow n +~^3{\rm He}$
                      &  5\% & 20.6 MeV & \cite{CJP53-802,PLB47-433} \\
 8. &$^4{\rm He} + \gamma \rightarrow p + n + {\rm D}$
                      & 14\% & 26.1 MeV & \cite{SJNP19-598} \\
 9. &$^6{\rm Li} + \gamma \rightarrow {\rm anything}$
                      &  4\% &  5.7 MeV & \cite{Berman} \\
10. &$^7{\rm Li} + \gamma \rightarrow 2n + {\rm anything}$
                      &  9\% & 10.9 MeV & \cite{Berman} \\
11. &$^7{\rm Li} + \gamma \rightarrow n +~^6{\rm Li}$
                      &  4\% &  7.2 MeV & \cite{Berman} \\
12. &$^7{\rm Li} + \gamma \rightarrow~^4{\rm He} + {\rm anything}$
                      &  9\% &  2.5 MeV & \cite{Berman} \\
13. &$^7{\rm Be} + \gamma \rightarrow p +~^6{\rm Li}$
                      &      &          & \\
14. &$^7{\rm Be} + \gamma \rightarrow~{\rm anything~except}~^6{\rm Li}$
                      &      &          & \\
\hline\hline
\end{tabular}
\caption{Photodissociation processes, and the 1-$\sigma$ uncertainty
         in the cross sections.
         Since there is no experimental data on photodissociation
         of $^7$Be, we assume in this paper that the rate for Reaction 13
         is the same as for Reaction 11, and the rate for Reaction 14 is
         the sum of the rates for Reactions 10 and 12.}
\label{table:pf}
\end{center}
\end{table}

The abundances of light nuclides will be functions of the lifetime of
$X$ ($\tau_X$), the mass of $X$ ($m_X$), the abundance of $X$ before
electron-positron annihilation 
%
\begin{equation}
    \label{yx}
 Y_X = n_X/n_\gamma,
\end{equation}
%
and the baryon-to-photon ratio ($\eta$).  In our numerical BBN
simulations, we found that the nuclide abundances depend only on the
mass abundance $m_X Y_X$, not on $m_X$ and $Y_X$ separately.  In
Figs.~5 -- 8, we show the abundances
of light nuclei in the $m_X Y_X$ vs. $\tau_X$ plane, at fixed $\eta$.

We can understand the
qualitative behaviors of the abundances in the following way.
First of all, if the mass density
of $X$ is small enough, then the effects of $X$ are negligible, and hence we
reproduce the result of SBBN. Once the mass density gets larger, the SBBN
results are modified. The effects of $X$ strongly depend on $\tau_X$,
the lifetime of $X$. As we mentioned in the previous section, photons with
energy greater than $\sim m_e^2/22T$ participate in pair creation
before they can induce photofission. Therefore, if the
above threshold energy is smaller than the nuclear binding energy,
then photodissociation is not effective.

If $\tau_X\lesssim 10^4$~sec, then $m_e^2/22T \lesssim 2$MeV at the decay time of $X$, and photodissociation is negligible for all elements.
In this case, the main effect of $X$ is on the $^4$He
abundances: if the abundance of $X$ is large, its energy density
speeds up the expansion rate of the universe, so the neutron
freeze-out temperature becomes higher. As a result, $^4$He abundance is
enhanced relative to SBBN.

If $10^4$~sec $\lesssim\tau_X\lesssim 10^6$~sec, then 2~MeV $\lesssim
m_e^2/22T\lesssim$ 20~MeV.  In this case, $^4$He remains intact,
but D is effectively photodissociated through the process
${\rm D}+\gamma\rightarrow p+n$.

If the lifetime is long enough ($\tau_X\gtrsim 10^6$~sec), $^4$He can
be also effectively destroyed. In this case, the destruction of even a
small fraction of the $^4$He can result in significant production of
D, since the $^4$He abundance is originally much larger than that of
D. This can be seen in Figs.~5 and 6:
for $\tau_X\gtrsim 10^6$~sec and $10^{-10}$~GeV $\lesssim
m_XY_X\lesssim 10^{-9}$~GeV, the abundance of D changes drastically
due to the photodissociation of $^4$He. If $m_XY_X$ is large enough,
all the light elements are destroyed efficiently, resulting in very
small abundances.

So far, we have discussed the theoretical calculation of the light element
abundances in a model with $X$ decay. In the next section, we compare the
theoretical calculations with observations, and derive constraints on
the properties of $X$.

\subsection{Comparison with observation}

As we mentioned in Section~\ref{sect-obs}, we have two $^4$He and two D
values which are inferred from various observed data to be the primordial
components.  In this section we compare the theoretical
calculations with these observed abundances and show how we can
constrain the model parameters in each of the four cases.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{Low $^4$He 
($Y^{obs}=0.234\pm (0.002)_{stat}\pm (0.005)_{syst}$)}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Recalling that the low observed $^4$He value [Eq.~(\ref{lowHe})]
is consistent with the theoretical calculation at low $\eta$ in the
case of SBBN, we expect that we can obtain rigid constraints on the
model parameters for the high observed D value
[Eq.~(\ref{highD})] . On the other hand, for the low observed D
value [Eq.~(\ref{lowD})] we search the parameter space for regions of
better fit than we can obtain with SBBN.

\subsubsection*{Low D
($y^{obs}_2 = (3.39 \pm 0.25) \times 10^{-5}$)
} \label{l2l4}

In Fig.~9 we show the contours of the confidence
level computed using three elements (D, $^4$He, and $^7$Li), for some
representative $\eta$ values ($\eta_{10}=2,4,5,6$),  where
 \begin{eqnarray}
  \eta_{10}\equiv \eta \times 10^{10}.
 \end{eqnarray}

The region of parameter space which is allowed at the 68\%
C.L. extends down to low $\eta$ (see
Fig.~9a).  Near $\eta_{10}=2$, deuterium is
destroyed by an order of magnitude (without net destruction of $^4$He),
so that the remaining deuterium agrees with the calculated low
$^4$He. We also plotted the regions excluded by the observational
upper bounds on $^6$Li/$^7$Li. The shaded regions are $y_6/y_7
\gtrsim 0.5$, and the darker shaded regions are  $y_6/y_7 \gtrsim
1.3$. Even if we adopt the stronger bound $y_6/y_7 \lesssim 0.5$, our
theoretical results 
are consistent with the observed $^6$Li value.

In Fig.~10, we show the contours of the confidence
levels for various lifetimes, $\tau_X = 10^4, 10^5, 10^6$ sec. As the
lifetime decreases, the background temperature at the time of decay
increases, so the threshold energy of double-photon pair creation
decreases. Then for a fixed $m_X Y_X$, the number of photons
contributing to D destruction decreases. Thus, for shorter lifetimes,
we need larger $m_X Y_X$ in order to destroy sufficient amounts of
D. The observed abundances prefer nonvanishing $m_X Y_X$.

In Fig.~11, we show contours of $\chi^2$ which have
been projected along the $\eta$ axis into the
$\tau_X$ - $m_X Y_X$ plane.  By projection, we mean taking the lowest
C.L. value along the $\eta$ axis for a fixed point ($\tau_X$, $m_X Y_X$).

The lower $m_X Y_X$ region, {\it i.e.} $m_X
Y_X\sim 10^{-14}$ GeV, corresponds to SBBN, since there are not enough
photons to affect the light element abundances. It is notable that
these regions are outside of the 68\% C.L. This fact may suggest the
existence of a long-lived massive particle $X$ and may be regarded as
a hint of physics beyond the standard model or
standard big bang cosmology. 

For example, in Fig.~12 we show the predicted abundances of 
$^4$He, D, $^7$Li and $^6$Li adopting the model parameters $\tau_X =
10^6$ sec and $m_X Y_X = 5 \times 10^{10}$ GeV. The predicted
abundances of $^4$He and $^7$Li are nearly the same as in SBBN.
Only D is destroyed; its abundance decreases by about 80\%.
At low $\eta \sim (1.7-2.3) \times 10^{-10}$ in this model,
the predicted abundances of these
three elements agree with the observed values.
It is interesting that the produced
$^6$Li abundance can be two orders of magnitude larger than the SBBN
prediction in this parameter region. The origin of the
observed $^6$Li abundance, $^6$Li/H~$\sim O(10^{-12})$ is usually
explained by cosmic ray spallation; however, our model demonstrates
the possibility that $^6$Li may have been produced by the
photodissociation of $^7$Li at an early epoch. Our
$^6$Li prediction is consistent with the upper bound Eq.~(\ref{Li6}).

Despite this, it is worth noting that SBBN lies within the 95\% C.L.
agreement between theory and observation.
In Fig.~11 the 95\% bound for $\tau_X \lesssim 10^6$
sec comes from the constraint that not much more than 90\% of the
deuterium should be destroyed; for $\tau_X \gtrsim 10^6$ sec the
constraint is that deuterium should not be produced from $^4$He
photofission.  In Table~\ref{table:ll} we show the representative
values of $m_X Y_X$ which correspond to 68\% and 95\% confidence
levels respectively, for $\tau_X = 10^4 - 10^9$ sec.

\begin{table}[t]
\begin{center}
\begin{tabular}{ccccccc}
\hline\hline
 & $\tau_X=10^4$ sec & 10$^5$ sec & 10$^6$ sec & 10$^7$  sec &
 10$^8$ sec & 10$^9$ sec \\ \hline
 95\% C.L.&
 9$\times$10$^{-6}$&
 9$\times$10$^{-9}$&
 1$\times$10$^{-9}$&
 7$\times$10$^{-11}$&
 2$\times$10$^{-12}$&
 7$\times$10$^{-13}$\\
 68\% C.L.&
 (1 -- 9)$\times$10$^{-6}$&
 (1 -- 7)$\times$10$^{-9}$&
 (2 -- 9)$\times$10$^{-10}$&
 & & \\
 \hline\hline
 \end{tabular}
 \caption{Upper or (upper -- lower) bound on $m_XY_X$ in units of GeV
  for the case of low $^4$He and low D.  Note that the C.L. is for 3
  degrees of freedom, and $\eta$ is varied to give the extreme values
  for $m_XY_X$.}
 \label{table:ll}
 \end{center}
 \end{table}

\subsubsection*{High D
($y^{obs}_2 = (1.9 \pm 0.5) \times 10^{-4}$)
} 

In the case of low $^4$He and high D, SBBN ({\it i.e.}, low $m_X Y_X$) works
quite well for $\eta\sim 2\times 10^{-10}$.  Thus, we expect that we
can strongly bound the parameter space of the $X$-decay model.  In
Fig.~13, we show the 68\% and 95\% C.L. contours for
some representative values of $\eta$.  In order to fit to low $^4$He,
we can place an upper bound on $m_X Y_X$ at a low $\eta$
(Fig.~13a).

There are also small allowed (at better than the 68\% C.L.)  regions
of parameter space at higher values of eta (see
Figs.~13b -- 13d).  These allowed
regions lie at $\tau_X \gtrsim 10^6$ sec, where a small amount of
$^4$He is broken down into D.  However, these allowed regions are
small, because the parameters must be finely tuned to target the D
abundance to $\sim O(10^{-4})$.

In Fig.~14 we show the contour plots for some
representative $\tau_X$ in the same manner as
Fig.~10.

In Fig.~15 we plot the contours projected along the
$\eta$ axis, in a fashion similar to Fig.~11.
Comparing the constraints on $\tau_X$ and $m_X Y_X$ with the case of
the low D (Fig.~11), we find that the 95\% boundary is
moved to higher $m_X Y_X$, for $\tau_X \gtrsim 10^6$
sec. This is because D (produced by $^4$He destruction) is permitted
to be an order of magnitude more abundant than in the case of the low
D value. We show the 68\% and 95\% C.L. upper bounds on $m_X Y_X$ in
Table~\ref{table:hl} at various lifetimes $\tau_X$.

\begin{table}[t]
\begin{center}
\begin{tabular}{ccccccc} 
\hline\hline
 & $\tau_X=10^4$ sec & 10$^5$ sec & 10$^6$ sec & 10$^7$  sec &
10$^8$ sec & 10$^9$ sec \\
\hline
95\% C.L.&
5$\times$10$^{-6}$&
5$\times$10$^{-9}$&
6$\times$10$^{-10}$&
5$\times$10$^{-10}$&
7$\times$10$^{-11}$&
4$\times$10$^{-11}$ \\
68\% C.L.&
3$\times$10$^{-6}$&
3$\times$10$^{-9}$&
3$\times$10$^{-10}$&
4$\times$10$^{-10}$&
5$\times$10$^{-11}$&
3$\times$10$^{-11}$  \\ 
\hline\hline
\end{tabular}
\caption{Same as Table~\protect\ref{table:ll}, except for low $^4$He
and high D.}
\label{table:hl}
\end{center}
\end{table}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsubsection{High $^4$He 
($Y^{obs} = 0.244 \pm (0.002)_{stat} \pm (0.005)_{syst}$)
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The high observed $^4$He abundance [Eq.~(\ref{highHe})] is consistent
with the SBBN theoretical calculations, for both the low and high
observed D abundances [Eqs.~(\ref{lowD}) and (\ref{highD})].
Therefore, we expect to be able to constrain the model parameters in
both cases.

\subsubsection*{Low D
($y^{obs}_2 = (3.39 \pm 0.25) \times 10^{-5}$)
}

For four representative $\eta$ values ($\eta_{10}=2, 4, 5, 6$), we plot
the contours of the confidence level in Fig.~16. In Fig.~2,
we see that the SBBN calculations agree with
the observed abundances for mid-range values of the baryon to
photon ratio ($\eta \sim 5 \times 10^{-10}$).  Thus, the upper
bound for $m_X Y_X$ is plotted in Fig.~16c.
Even at a
low $\eta$ (where the SBBN calculation disagrees with the low observed
D value), the theoretical calculations can match observed data in
the region $10^4 {\rm sec} \lesssim \tau_X \lesssim 10^6 {\rm sec}$
and $m_X Y_X \gtrsim 10^{-10}$, because of the significant destruction
of D.  In Fig.~17 we show the C.L. plots for three
typical lifetimes, $\tau_X = 10^4, 10^5, 10^6$ sec.
Finally, we show the C.L.  contours projected along the $\eta$ axis
into the $\tau_X$ - $m_X Y_X$ plane (Fig.~18).
Table~\ref{table:lh} gives the upper bounds
on $m_X Y_X$ (GeV) which correspond to 68\% and 95\% C.L., for some
typical values of the lifetime.

\begin{table}[t]
\begin{center}
\begin{tabular}{ccccccc} 
\hline\hline
 & $\tau_X=10^4$ sec & 10$^5$ sec & 10$^6$ sec & 10$^7$  sec &
10$^8$ sec & 10$^9$ sec \\ \hline
95\% C.L.&
7$\times$ 10$^{-6}$&
7$\times$ 10$^{-9}$&
8$\times$ 10$^{-10}$&
1$\times$ 10$^{-10}$&
8$\times$ 10$^{-12}$&
3$\times$ 10$^{-12}$ \\ 
68\% C.L.&
5$\times$ 10$^{-6}$&
5$\times$ 10$^{-9}$&
6$\times$ 10$^{-10}$&
8$\times$ 10$^{-11}$&
4$\times$ 10$^{-12}$&
2$\times$ 10$^{-12}$ \\ \hline\hline
\end{tabular}
\caption{Same as Table~\protect\ref{table:ll}, 
except for high $^4$He and low
D.}
\label{table:lh}
\end{center}
\end{table}


\subsubsection*{High D
($y^{obs}_2 = (1.9 \pm 0.5) \times 10^{-4}$)
}

As in the low D case, we now plot C.L. contours for high D for four
typical values of $\eta$ in Fig.~19.  Since the
adoption of the high $^4$He and high D observed values is
consistent with SBBN calculations for low $\eta$, we expect to obtain
bounds on $\tau_X$ and $m_X Y_X$ ({\it e.g.}, Fig.~19a).
In Figs.~19b -- 19d, 
we see that we also have allowed regions for
$\tau_X \gtrsim 10^6$ sec. The reason is same as the case of low
$^4$He and high D; the D final abundances are well-balanced between
production and destruction.

In Fig.~20, we plot the confidence level for $\tau_X =
10^4, 10^5$, and $10^6$ sec.
The range of preferred $\eta$ is relatively narrow. This is because
the case of high D and high $^4$He is only consistent in SBBN for low
value of $\eta$, and in the lifetime range $\tau_X \sim 10^4-10^6$,
the $^4$He abundance is not affected by the radiative decay of $X$.

Next, we show the 68\% and 95\% C.L. contours projected along the
$\eta$ axis (Fig.~21). There is a large region between
the 68\% C.L. and the 95\% (for a fixed $\tau_X$) for two reasons.
First, the uncertainty in the high observed D value is
large.  Second, the $\eta$ predicted from the high observed
$^4$He value has a wide spread.  The overall shape of the 95\%
C.L. line is very similar to the case of low $^4$He and high D.  This
is because the constraint for $\tau_X \gtrsim 10^6$ sec is
particularly sensitive only to the observed D value.

Just as in the case of low $^4$He, the 95\% C.L. line for the
high D value goes to higher $m_X Y_X$ than for the low D value,
because the new D component produced by $^4$He destruction is
allowed to be one order of magnitude larger than in the case of low D. In
Table~\ref{table:hh}, we list the upper bounds on $m_X Y_X$ at the
68\% and 95\% confidence levels, for various values of $\tau_X$.

\begin{table}[t]
\begin{center}
\begin{tabular}{ccccccc} 
\hline\hline
 & $\tau_X=10^4$ sec & 10$^5$ sec & 10$^6$ sec & 10$^7$  sec &
10$^8$ sec & 10$^9$ sec \\ \hline
95\% C.L.&
2$\times$ 10$^{-6}$&
3$\times$ 10$^{-9}$&
3$\times$ 10$^{-10}$&
4$\times$ 10$^{-10}$&
5$\times$ 10$^{-11}$&
3$\times$ 10$^{-11}$ \\
68\% C.L.&
5$\times$ 10$^{-7}$&
6$\times$ 10$^{-10}$&
7$\times$ 10$^{-11}$&
2$\times$ 10$^{-11}$&
1$\times$ 10$^{-11}$&
2$\times$ 10$^{-11}$  \\ 
\hline\hline
\end{tabular}
\caption{Same as Table~\protect\ref{table:ll}, 
except for high $^4$He and high
D.}
\label{table:hh}
\end{center}
\end{table}

\subsection{Additional constraints}

We now mention additional constraints on our model. First, the
the cosmic microwave background radiation (CMBR) was observed
by COBE~\cite{fixsen} to very closely follow a blackbody spectrum.
This gives us a severe constraint on particles with lifetime longer
than $\sim 10^6$ sec~\cite{PRL70-2661}, which is when the
double Compton process
($ \gamma + e^- \rightleftharpoons \gamma + \gamma + e^- $)
freezes out~\cite{lightman}.  After this time, photon number
is conserved, so photon injection from a radiatively decaying
particle would cause the spectrum of the CMBR to become a
Bose-Einstein distribution with a finite chemical potential $\mu$.
COBE~\cite{fixsen} observations give us the constraint
$|\mu| \lesssim 9.0 \times 10^{-5}$. For small $\mu$,
the ratio of the injected to total photon energy density is
given by $\delta \rho_{\gamma}/\rho_{\gamma} \sim 0.71 \mu$.
Thus, we have the constraint
%
\begin{equation}
    \label{COBE}
    m_X Y_X \lesssim 1.7 \times 10^{-10} {\rm GeV}
          \left(\frac{\tau_X}{10^6{\rm sec}}\right)^{-\frac12} \quad
          {\rm for} \ 
     10^6 {\rm sec} \lesssim \tau_X \lesssim 4 \times 10^{10}
     {\rm sec}. 
\end{equation}
%
Note that for lifetimes $\tau_X$ longer than $10^6 {\rm sec}$,
the CMBR constraint is stricter than the bounds from the BBN
which we have discussed above.

In this paper, we have considered only radiative decays; {\it i.e.}, decays
to photons and invisible particles.  If $X$ decays to charged leptons,
it is similar in effect to decay to photons because the charged
leptons also generate the electro-magnetic cascade shower. On the
other hand if $X$ decays only to neutrinos, the constraints becomes much
weaker. In the content of MSSM, the $X$ particle decays to neutrino and 
sneutrino and the emitted neutrinos scatter off the background
neutrinos. Then electron positron pairs are produced and they
subsequently produce the many soft photons through the
electro-magnetic cascade. Because the interaction between the emitted
neutrino and the background neutrino is week, the destruction of the
light elements does not occur very efficiently~\cite{chglep}.
However, if $X$ decayed to hadrons,
we expect that our bounds would tighten, because hadronic
showers could be a significant source of D, $^3$He,
$^6$Li, $^7$Li, and $^7$Be~\cite{hadron}.

\section{Model}
\label{sec:model}

So far, we have discussed general constraints from BBN on radiatively
decaying particles. In the minimal standard model, there is no
such particle. However, some extensions of the standard model
naturally result in such exotic particles, and SBBN may be
significantly affected in these cases. In this section, we present
several examples of such radiatively decaying particles, and discuss
the constraints.

Our first example is the gravitino $\psi$, which appears in all the
supergravity models. The gravitino is the superpartner of the graviton,
and its interactions are suppressed by inverse powers of the reduced
Planck scale $M_*\simeq 2.4\times 10^{18}$ GeV.  Because of this
suppression, the lifetime of the gravitino is very long. Assuming that
the gravitino's dominant decay mode is to a photon and its
superpartner (the photino), the gravitino's
lifetime is given by
 \begin{eqnarray}
  \tau_{3/2}\simeq 4\times 10^5{\rm ~sec} \times
  (m_{3/2}/1 {\rm ~TeV})^{-3},
 \end{eqnarray}
 where $m_{3/2}$ is the gravitino mass. Notice that the gravitino mass
is $O(100{\rm ~GeV}-1{\rm ~TeV})$ in a model with gravity-mediated
supersymmetry (SUSY) breaking, resulting in a lifetime which may
affect BBN.

If the gravitino is thermally produced in the early universe, and
decays without being diluted, it completely spoils the success of
SBBN. Usually, we solve this problem by introducing inflation which
dilutes away the primordial gravitinos. However, even with inflation,
gravitinos are produced through scattering processes of thermal
particles after reheating. The abundance
$Y_{3/2} = n_{3/2}/n_\gamma$
of the gravitino depends
on the reheating temperature $T_R$, and is given by~\cite{KM1}
 \begin{eqnarray}
  Y_{3/2} \simeq 2\times 10^{-11} \times (T_R/10^{10}{\rm GeV}).
 \end{eqnarray}
Therefore, if the reheating temperature is too high, then gravitinos
are overproduced, and too many light nuclei are photodissociated.

We
can transform our constraints on $(\tau_X, m_X Y_X)$ to constraints on
$(m_{3/2}, T_R)$. In particular, we use the projected 95\% C.L. boundaries
from Figs.~11,~15,~18, and~21.  For several values of
the gravitino mass, we read off the most conservative upper bound on
the reheating temperature from Fig.~22, and the results are given by
 \begin{eqnarray}
  m_{3/2}=100{\rm ~GeV} ~~~ 
  (\tau_{3/2}\simeq 4\times 10^{8}{\rm ~sec}) &:& 
  T_R \lesssim 3\times 10^{8} {\rm ~GeV},
 \nonumber \\
  m_{3/2}=1{\rm ~TeV} ~~~ 
  (\tau_{3/2}\simeq 4\times 10^{5}{\rm ~sec}) &:& 
  T_R \lesssim 1\times 10^{9} {\rm ~GeV},
 \nonumber \\
  m_{3/2}=3{\rm ~TeV} ~~~ 
  (\tau_{3/2}\simeq 1\times 10^{4}{\rm ~sec}) &:& 
  T_R \lesssim 3\times 10^{11} {\rm ~GeV}.
 \nonumber
 \end{eqnarray}
 If the gravitino is heavy enough ($m_{3/2}\gtrsim 5{\rm ~TeV}$), then
its lifetime is too short to destroy even D. In this case, our only
constraint is from the overproduction of $^4$He. If the gravitino mass
is lighter, then the lifetime is long enough to destroy D or even
$^4$He. In this case, our constraint on the reheating temperature is
more severe.

Another example of our decaying particle is the lightest superparticle
in the MSSM sector, if it is heavier than the gravitino.  In
particular, if the lightest neutralino is the lightest superparticle
in the MSSM sector, then it can be a source of high energy photons,
since it will decay into a photon and a gravitino. In this case, we
may use BBN to constrain the MSSM.

The abundance of the lightest neutralino is determined when it
freezes out of the thermal bath. The abundance
is a function of the masses of the superparticles, and it becomes
larger as the superparticles gets heavier. Thus, the upper bound on
$m_X Y_X$ can be translated into an upper bound on the mass scale of the
superparticles.

In order to investigate this scenario, we consider the simplest case
where the lightest neutralino is (almost) purely bino $\tilde{B}$.
In this case,
the lightest neutralino pair-annihilates through squark and slepton
exchange. In particular, if the right-handed sleptons are the lightest
sfermions, then the dominant annihilation is
$\tilde{B} + \tilde{B}\rightarrow l^+ + l^-$. The annihilation cross section
though this process is given by~\cite{PLB230-78}
 \begin{eqnarray}
  \langle\sigma v_{\rm rel}\rangle
  = 8\pi\alpha_1^2 \langle v^2\rangle
  \left\{
  \frac{m_{\tilde{B}}^2}{(m_{\tilde{B}}^2+m_{\tilde{l}_R}^2)^2}
  - \frac{2m_{\tilde{B}}^4}{(m_{\tilde{B}}^2+m_{\tilde{l}_R}^2)^3}
  + \frac{2m_{\tilde{B}}^6}{(m_{\tilde{B}}^2+m_{\tilde{l}_R}^2)^4}
  \right\},
 \end{eqnarray}
 where $\langle v^2\rangle$ is the average velocity squared of bino,
and we added the contributions from all three generations by assuming the
right-handed sleptons are degenerate.\footnote
 {If the bino is heavier than the top quark, then the $s$-wave contribution
annihilating into top quarks becomes important. In this paper, we do
not consider this case.}
 With this annihilation cross section, the Boltzmann equation for the
number density of binos is given by
 \begin{eqnarray}
  \dot{n}_{\tilde{B}} + 3H n_{\tilde{B}} = - 2 \langle\sigma v_{\rm
  rel}\rangle (n_{\tilde{B}}^2 - (n_{\tilde{B}}^{{\rm EQ}})^2),
 \end{eqnarray}
 where $n_{\tilde{B}}^{{\rm EQ}}$ is the equilibrium number density of
bino.  The factor 2 is present because two binos annihilate into leptons
in one collision. We solved this equation and obtained the mass
density of the bino as a function of the bino mass and the
right-handed slepton mass. (For details, see {\it e.g.}
Ref.~\cite{Kolb-Turner}). Numerically, for $m_{\tilde{B}}=100$~GeV,
$m_XY_X$ ranges from $\sim 10^{-9}$~GeV to $\sim 10^{-5}$~GeV as we
vary $m_{\tilde{l}_R}$ from 100~GeV to 1~TeV. If $m_XY_X$ is in this
range, BBN is significantly affected unless the lifetime of the bino
is shorter than $10^4$ -- $10^5$~sec (see Tables~\ref{table:ll} --
\ref{table:hh}). The lifetime of the bino is given by
 \begin{eqnarray}
  \tau_{\tilde{B}} = 
  \left[\frac{1}{48\pi}
  \frac{m_{\tilde{B}}^5\cos^2\theta_{\rm W}}
  {m_{3/2}^2M_*^2}\right]^{-1}
  \simeq 7\times 10^4{\rm ~sec}\times
  \left(\frac{m_{\tilde{B}}}{100{\rm ~GeV}}\right)^{-5}
  \left(\frac{m_{3/2}}{1{\rm ~GeV}}\right)^2.
 \end{eqnarray}
Notice that the lifetime becomes shorter as the gravitino mass
decreases; hence, too much D and $^7$Li are destroyed if the gravitino mass
is too large. Therefore, we can convert the constraints given in
Figs.~11,~15,~18, and~21 into upper bounds on the gravitino mass. Since the
abundance of the bino is an increasing function of the slepton mass
$m_{\tilde{l}_R}$, the upper bound on the gravitino mass is more severe
for larger slepton masses. For example, for $m_{\tilde{B}}=100{\rm
~GeV}$, the upper bound on the gravitino mass is shown in
Fig.~23.  At some representative values of the slepton
mass the constraint is given by
 \begin{eqnarray}
  m_{\tilde{l}_R}=100{\rm ~GeV} &:& 
  m_{3/2}\lesssim 1{\rm ~GeV},
 \nonumber \\
  m_{\tilde{l}_R}=300{\rm ~GeV} &:& 
  m_{3/2}\lesssim 500{\rm ~MeV},
 \nonumber \\
  m_{\tilde{l}_R}=1{\rm ~TeV} &:& 
  m_{3/2}\lesssim 400{\rm ~MeV}.
 \nonumber
 \end{eqnarray}
 As expected, for a larger value of the slepton mass, the primordial
abundance of bino gets larger, and the upper bound on the gravitino
mass becomes smaller.

Another interesting source of high energy photons is a modulus field
$\phi$. Such fields are predicted in string-inspired supergravity
theories.  A modulus field acquires
mass from SUSY breaking, so we estimate its mass $m_\phi$
to be of the same order as the gravitino mass
(see for example~\cite{Carlos}).

In the early universe, the mass of the modulus field is negligible
compared to the expansion rate of the universe, so the modulus field
may sit far from the minimum of its potential. Since the only scale
parameter in supergravity is the Planck scale $M_*$, the initial
amplitude $\phi_0$ is naively expected to be of $O(M_*)$.  However,
this initial amplitude is too large; it leads to well-known problems
such as matter domination and distortion of the CMBR.
Here, we regard $\phi_0$ as a free parameter and
derive an upper bound on it.

Once the expansion rate becomes smaller than the mass of the modulus
field, the modulus field starts oscillating. During this period,
the energy density
of $\phi$ is proportional to $R^{-3}$ (where $R$ is the scale factor);
hence, its energy density behaves like that of non-relativistic
matter. The modulus eventually decays, when the expansion rate
becomes comparable to its decay rate. Without entropy production
from another source, the modulus density at the decay time is approximately
 \begin{eqnarray}
  m_\phi Y_{\phi} = 
  \frac{\rho_\phi}{n_\gamma} \sim 2\times 10^{10} {\rm ~GeV}
  \times (m_\phi/1{\rm ~TeV})^{1/2} (\phi_0/M_*)^2,
 \end{eqnarray}
 where $\rho_\phi$ is the energy density of the modulus field. As in our
other models, we can convert our constraints on $(\tau_X, m_X Y_X)$
(Figs.~11,~15,~18, and~21) into constraints on $(m_{\phi}, \phi_0)$.
Using the most conservative of these constraints, we still obtain
very stringent bounds on the initial amplitude of the modulus
field $\phi_0$:
 \begin{eqnarray}
  m_{\phi}=100{\rm ~GeV} ~~~ (\tau_{\phi}\sim 4 \times 10^8{\rm ~sec}) &:&
  \phi_0 \lesssim 2 \times 10^{8} {\rm ~GeV},
 \nonumber \\
  m_{\phi}=1{\rm ~TeV} ~~~ (\tau_{\phi}\sim 4 \times 10^5{\rm ~sec}) &:&
  \phi_0 \lesssim 7 \times 10^{8} {\rm ~GeV},
 \nonumber \\
  m_{\phi}=3 {\rm ~TeV} ~~~ (\tau_{\phi}\sim 1\times 10^4{\rm ~sec}) &:&
  \phi_0 \lesssim 2 \times 10^{10} {\rm ~GeV}.
 \nonumber
 \end{eqnarray}
 Clearly, our upper bound from BBN rules out our naive expectation
that $\phi_0 \sim M_*$. It is important to notice that (conventional)
inflation cannot solve this difficulty by diluting the coherent mode
of the modulus field. This is because the expansion rate of the
universe is usually much larger than the mass of the modulus field,
and hence the modulus field does not start oscillation. One attractive
solution is a thermal inflation model proposed by Lyth and
Stewart~\cite{Lyth-Stewart}. In the thermal inflation model, a
mini-inflation with $e$-fold number $\sim$ 10 reduces the modulus
density. Even if thermal inflation occurs, there may remain a
significant modulus energy density, which decays to high energy
photons. Thus, BBN gives a stringent constraint on the thermal
inflation model.

\section{Discussion and Conclusions}
\label{sec:summary}

We have discussed the photodissociation of light elements due to the
radiative decay of a massive particle, and we have shown how we can
constrain our model parameters from the observed light-element
abundances.  We adopted two D values and two $^4$He values in this
paper, and we obtained constraints on the properties of the
radiatively decaying particle in each case.

When we adopt the low $^4$He and low D values, we find that
% at the 68\% C.L.we need
a non-vanishing amount of such a long-lived, massive particle
is preferred:
$m_X Y_X \gtrsim 10 ^ {-10} {\rm GeV}$ for $10^4 {\rm sec} \lesssim
\tau_X \lesssim 10^6 {\rm sec}$. On the other hand,
% at the 95\% C.L., we have
consistency with the observations imposes
upper bounds on $m_X Y_X$ in each of the four cases.

We have also studied the photodissociation of $^7$Li and
$^6$Li in this paper. These processes do not affect the D and $^{4}$He
abundances, because $^7$Li and $^6$Li are many orders of magnitude less
abundant than D and
$^{4}$He. When we examine the region of parameter space where the
predicted abundances agree well with the observed $^7$Li and the
low $^{4}$He and low D observations, we find that the
produced $^6$Li/H may be of order $10^{-12}$, which is
two orders of magnitude larger than the prediction of SBBN (see
Figs.~7 and~12). The predicted $^6$Li is
consistent with the observed upper bound Eq.~(\ref{Li6}) throughout
the region of
parameter space we are interested in. Although presently it is believed
that the observed $^6$Li abundance is produced by
spallation, our model suggests another origin: the observed
$^6$Li may be produced by the photodissociation of $^7$Li.

We have also discussed candidates for our radiatively decaying
particle. Our first example is the gravitino. In this case, we can
constrain the reheating temperature after inflation, because it determines
the abundance of the gravitino. We obtained the stringent bounds
$T_R \lesssim
10^8 {\rm GeV} - 10^9 {\rm GeV}
$ 
for $100{\rm ~GeV}
\lesssim  m_{3/2} 
\lesssim 1{\rm ~TeV}$.
Our second example is the lightest neutralino which is heavier than
the gravitino. When the neutralino is the lightest superparticle in
the MSSM sector, it can decay into a photon and a gravitino. If we
assume the lightest neutralino is pure bino, and its mass is about 100
GeV, the relic number density of binos is related to the right-handed
slepton mass, because they annihilate mainly through right-handed
slepton exchange. For this case, we obtained the upper bound of the
gravitino mass, $m_{3/2} \lesssim 400 {\rm ~MeV} - 1 {\rm ~GeV}$ for
$100 {\rm ~GeV} \lesssim m_{\tilde{l}_R} \lesssim 1 {\rm ~TeV}$.  Our
third example is a modulus field. We obtained a severe constraint on
its initial amplitude, $ \phi_0 \lesssim 10^8 {\rm ~GeV} - 10^{10}
{\rm ~GeV}$ for $100{\rm ~GeV} \lesssim m_{3/2} \lesssim 1{\rm
~TeV}$. This bound is well below the Planck scale, so it suggests the
need for a dilution mechanism, such as thermal inflation.

\section*{Acknowledgement}

This work was supported by the Director, Office of Energy
Research, Office of Basic Energy Services, of the U.S.
Department of Energy under Contract DE-AC03-76SF00098. K.K. is
supported by JSPS Research Fellowship for Young Scientists.

\appendix

\section{Appendix}
\label{app:analysis}

In this appendix, we explain how we answer the question,
``How well does our simulation of BBN agree with the observed
light-element abundances?'' To be more precise, we rephrase
the question as, ``At what confidence level is our simulation
of BBN excluded by the observed light-element abundances?''

From our Monte-Carlo BBN simulation, we obtain the theoretical
probability density function (p.d.f.)  $p^{th}({\bf a}^{th})$ of our
simulated light-element abundances ${\bf a}^{th} = (y_2^{th}, Y^{th},
\log_{10}y_7^{th})$.  We find that $p^{th}({\bf a}^{th})$ is
well-approximated by a multivariate Gaussian.  (See
Eqs.~(\ref{p_i^th})and~(\ref{p^th}).)  Note that $p^{th}({\bf
a}^{th})$ depends upon the parameters {\bf p} of our theory, {\it e.g.}
{\bf p} $ = (\eta, ...)$.  (The ellipses refer to parameters in
non-standard BBN, {\it e.g.}, $m_X$, $Y_X$, $\tau_X$.)  In particular,
the means and standard deviations of $p^{th}({\bf a}^{th})$
are functions of {\bf p}.

We also construct the p.d.f. $p^{obs}({\bf a}^{obs})$
for the observed light-element abundances, viz.
${\bf a}^{obs} = (y^{obs}_2, Y^{obs}, \log_{10}y^{obs}_7)$.
Since the observations of the light element abundances
are independent, we can factor
%
\beq
%
p^{obs}({\bf a}^{obs})
  &=& p^{obs}_2(y^{obs}_2) \times
      p^{obs}_4(Y^{obs}) \times
      p^{obs}_7(\log_{10}y^{obs}_7)
%
\eeq
%
We assume Gaussian p.d.f.'s for $y^{obs}_2, Y^{obs}$, and
$\log_{10}y^{obs}_7$.  We use the mean abundances and standard deviations
given in Equations~(\ref{lowD})--(\ref{Li7}).
Since we have two discordant values of D/H
and two discordant values of $^4$He,
we considered four cases (each with its own
p.d.f. $p^{obs}({\bf a}^{obs})$):
(i) low $^4$He, low D/H; (ii) low $^4$He, high D/H;
(iii) high $^4$He, low D/H; and (iv) high $^4$He, high D/H.

Then $\Delta{\bf a} = {\bf a}^{th} - {\bf a}^{obs}$ has a
p.d.f. given by
%
\beq
%
p^{\Delta}(\Delta{\bf a})
  &=& \int d{\bf a}^{obs}~p^{obs}({\bf a}^{obs})
        ~\int~d{\bf a}^{th}~p^{th}({\bf a}^{th})
         \delta(\Delta{\bf a} - ({\bf a}^{th} - {\bf a}^{obs}))
      \nonumber \\
  &=& \int d{\bf a}~p^{th}({\bf a})
                  p^{obs}({\bf a} - \Delta{\bf a}),
 \label{pdf}
%
\eeq
%
where we have suppressed the dependence of
$p^{\Delta}(\Delta{\bf a})$ and
$p^{th}({\bf a}^{th})$ on the theory parameters {\bf p}.
Note that when all $p^{th}_i$ and $p^{obs}_i$ are Gaussian,
Eq.~(\ref{pdf}) is easily integrated to yield a product of three
Gaussian p.d.f.'s.
%
\begin{equation}
    \label{gpdf}
    p^{\Delta}(\Delta{\bf a}) = \prod_i \frac{1}{\sqrt{2 \pi}
    \sigma_i} \exp\left[-\frac{({\Delta}a_i-{\Delta}\bar{a}_i)^2}{2
    \sigma_i^2}\right],
\end{equation}
%
where $\Delta\bar{a}_i = \bar{a}^{th}_i-\bar{a}^{obs}_i$, $\sigma^2_i =
(\sigma^{th}_i)^2 + (\sigma^{obs}_i)^2$, and $i$ runs over $y_2$, $Y$ and 
log$_{10}y_7$.

Our question can now be rephrased as, ``At what confidence level is
$\Delta{\bf a} = 0$ excluded?''  The answer,
%
\beq
%
{\rm C.L.}({\bf p})
 = \int_{
         \{ \Delta{\bf a}:
            p^{\Delta}(\Delta{\bf a}; {\bf p})
            > p^{\Delta}(0; {\bf p})
         \}
        }
          d(\Delta{\bf a})~
          p^{\Delta}(\Delta{\bf a}; {\bf p}),
%
\eeq
%
is used in this paper to constrain various scenarios of BBN.
Since we have assumed Gaussian p.d.f.'s, we can easily evaluate this
integral.  The result is conveniently expressed in terms of $\chi^2$.
(See Eqs.~(\ref{chi2}) and~(\ref{CL}).)

Our confidence level is calculated for three degrees of freedom
$\Delta a_i$.  It denotes our certainty that a given point {\bf p} in
the parameter space of the theory is excluded by the observed
abundances.  If the abundances $a_i$ were linear in the theory
parameters {\bf p}, then we could integrate out a theory parameter
such as $\eta$ and set a C.L. exclusion limit (with a reduced number
of degrees of freedom) on the remaining parameters.  However, the
$a_i$ turn out to be highly non-linear, so such a procedure turns out
to have little meaning.  Instead, we shall project out various theory
parameters (as explained in Section~\ref{l2l4}) to present our results
as graphs.



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%
\end{thebibliography}

\newpage

\noindent{{\large\textbf{Figure captions}}}\\

\vspace{0.5cm}
\noindent{Figure 1: SBBN prediction of the abundances of the light
elements. The solid lines are the central values of the predictions, and the
dotted lines represents the 1-$\sigma$ uncertainties. The boxes denote 
the 1-$\sigma$ observational constraints.}\\
% \label{fig:sbbn}

\vspace{.5cm}
\noindent{Figure 2: $\chi^2$ as a function of $\eta$,
          for SBBN with three degrees of freedom. We used four
          combinations of the D and $^4$He abundances
          deduced from observation:
          low  $^4$He and low  D (short-dashed),
          low  $^4$He and high D (dotted-dashed),
          high $^4$He and low  D (solid),
          high $^4$He and high D (long-dashed).}\\
% \label{fig:sbbnchi2}

\vspace{.5cm}
\noindent{Figure 3: C.L. for BBN as a function of $\eta$ and $N_\nu$, with
(a) low value of $Y$, and (b) high value of $Y$. The filled square
denotes the best fit point.}\\
% \label{fig:nnu}

\vspace{.5cm}
\noindent{Figure 4: Photon spectrum $f_\gamma = dn_\gamma/dE_\gamma$
for several
background temperatures $T_{\gamma}^{\rm BG}$.}\\
% \label{fig:photon}

\vspace{.5cm}
\noindent{Figure 5: Abundance of D in the $m_X Y_X$ vs. $\tau_X$ plane with (a)
$\eta=2\times 10^{-10}$, (b) $\eta=4\times 10^{-10}$, (c)
$\eta=5\times 10^{-10}$, and (d) $\eta=6\times 10^{-10}$.}\\
%\label{fig:bbnx_d}

\vspace{.5cm}
\noindent{Figure 6: Same as Fig.~5, except for $^4$He.}
 \label{fig:bbnx_4he}\\
% \label{fig:bbnx_4he}

\vspace{.5cm}
\noindent{Figure 7: Same as Fig.~5, except for $^6$Li.}\\
% \label{fig:bbnx_6li}


\vspace{.5cm}
\noindent{Figure 8: Same as Fig.~5, except for $^7$Li.}\\
% \label{fig:bbnx_7li}


\vspace{.5cm}
\noindent{Figure 9: C.L. in the $m_X Y_X$ vs. $\tau_X$ plane, for low value of D
abundance and low value of $Y$. We take (a) $\eta=2\times 10^{-10}$,
(b) $\eta=4\times 10^{-10}$, (c) $\eta=5\times 10^{-10}$, and (d)
$\eta=6\times 10^{-10}$. The shaded regions are $y_6/y_7
\gtrsim 0.5$ and the darker shaded regions are  $y_6/y_7 \gtrsim
1.3$.}\\
% \label{fig:xchi2_ll}


\vspace{.5cm}
\noindent{Figure 10: C.L. in the $\eta$ vs. $m_X Y_X$ plane for
various values of 
$\tau_X$, for low value of D abundance and low value of $Y$.}\\
% \label{fig:xtau_ll}


\vspace{.5cm}
\noindent{Figure 11: Contours of C.L. projected on $\eta$ axis
          for low value of D abundance and low value of $Y$.}\\
% \label{fig:xtot_ll}


\vspace{.5cm}
\noindent{Figure 12: Predicted light element abundances $^4$He, D, $^7$Li and
 $^6$Li at $\tau_X = 10^6$ sec and $m_X Y_X = 5 \times 10^{10}$
 GeV. The contours which are favored by observation are
 plotted, adopting the low $^4$He and low D values. The dotted line denotes
 the 95\% C.L. and the shaded region 
 denotes the 68\% C.L.. The predicted $^6$Li abundance is two
 orders of magnitude larger  
 than the case of SBBN.}\\
% \label{fig:bbnxll}


\vspace{.5cm}
\noindent{Figure 13: Same as Fig.~9, except for high value of
D abundance and low value of $Y$.}\\
% \label{fig:xchi2_hl}


\vspace{.5cm}
\noindent{Figure 14: Same as Fig.~10, except for high value of
D abundance and low value of $Y$.}\\
% \label{fig:xtau_hl}

\vspace{.5cm}
\noindent{Figure 15: Same as Fig.~12, except for high value of
D abundance and low value of $Y$.}\\
% \label{fig:xtot_hl}


\vspace{.5cm}
\noindent{Figure 16: Same as Fig.~9, except for low value of
D abundance and high value of $Y$.}\\
% \label{fig:xchi2_lh}


\vspace{.5cm}
\noindent{Figure 17: Same as Fig.~10, except for low value of
D abundance and high value of $Y$.}\\
% \label{fig:xtau2_lh}


\vspace{.5cm}
\noindent{Figure 18: Same as Fig.~12, except for low value of
D abundance and high value of $Y$.}\\
% \label{fig:xtot_lh}


\vspace{.5cm}
\noindent{Figure 19: Same as Fig.~9, except for high value of
D abundance and high value of $Y$.}\\
% \label{fig:xchi2_hh}


\vspace{.5cm}
\noindent{Figure 20: Same as Fig.~10, except for high value of
D abundance and high value of $Y$.}\\
% \label{fig:xtau_hh}


\vspace{.5cm}
\noindent{Figure 21: Same as Fig.~12, except for high value of
D abundance and high value of $Y$.}\\
% \label{fig:xtot_hh}


\vspace{.5cm}
\noindent{Figure 22: Contours of 95\% C.L., yielding an upper bound
on the reheating temperature, as a function of the gravitino mass.}\\
% \label{fig:massTr}

\vspace{.5cm}
\noindent{Figure 23: Contours of 95\% C.L., yielding an upper bound
on the gravitino mass, as a function of the right-handed slepton mass.}\\
%  \label{fig:m32_mslR}


\end{document}

