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Possible origins of 'Li discrepancy between SBBN and CMB
through 3He(a,y)7Be while 7Be destruction by 7Be(n,p)7Li is inefficient
because of the lower neutron abundance a t high density (7Be later decays
t o 7Li). Since the WMAP results point toward the high region, a peculiar attention should be paid to 7Be synthesis. In particular, the 7Be+d
reactions could be an alternative t o 7Be(n,p)7Li for the destruction of 7Be,
by compensating the scarcity of neutrons at high Q. Fig. 1 shows (dashdotted lines) that an increase of the 7Be(d,p)24He reaction rate by factors
of 100 t o 300 would remove the discrepancy. The rate for this reactionQcan
be traced t o an estimate by Parker4' who assumed for the astrophysical
S-factor a constant value of lo5 kev.barn. based on the single experimental data available41. To derive this S-factor, Parker used this measured
differential cross section at 90" and assumed isotropy of the cross section.
Since Kavanagh measured only the po and p1 protons (i.e. feeding the *Be
ground and first excited levels), Parker introduced an additional but arbitrary factor of 3 t o take into account the possible population of higher
lying levels. Indeed, a level at 11.35 MeV is also reported42. This factor
should also include the contribution of another open channel in 7Be+d:
7Be(d,a)5Li for which no data exist. In addition, one should note that no
experimental data for this reaction is available a t energies relevant to 7Be
Big Bang nucleosynthesis (Fig. 3), taking place when the temperature has
dropped below lo9 K. A seducing possibility3 to reconciliate, SBBN, 7Li
and CMB observations would then be that new experimental data below
E d = 700 keV (Ecm~ 0 . MeV)
for 7Be(d,p)24He [and 7Be(d,a)5Li]would
lead to a sudden increase in the S-factor as in 1oB(p,a)7Be43~5.
not supported by known data, but considering the cosmological or astrophysical consequences, this is definitely an issue t o be investigated and an
experiment is planned in 2004 a t the Cyclotron Research Centre in Louvainla-Neuve.
Recent theories that could affect BBN include the variation of the fine structure constant44,the modification of the expansion rate during BBN induced
by q u i n t e ~ s e n c e ~modified
gravity46, or leptons asymmetry47. However,
their effect is in general more significant on 4 H e than on 7Li.
It may not be excluded that some bias exists in the analysis of CMB
anisotropies. For instance, it has been argued4* that a contamination of
CMB map by blazars could affect the second peak of the power spectrum
on which the CMB Rbh2 values are based.
Figure 3. The only experimental data available for the 7Be(d,p)24H reaction from Kavanagh (1960). The displayed S-factor is calculated as in Parker (1972) from the differential cross section at 90° (x47r) leading t o the ground and first 8Be excited states. Note
that no data is available at SBBN energies as shown by the Gamow peak for a typical
temperature of T9 = 0.8
4.4. Pregalactic evolution
We note that between the BBN epoch and the birth of the now observed
halo stars, el Gyr have passed. Primordial abundances could have been
altered during this period. For instance, cosmological cosmic rays, assumed
t o have been born in a burst at some high redshift, could have modified
these primordial abundances in the intergalactic medium49. This would
increase the primordial 7Li and D abundances trough spallative reactions,
increasing in the same time the discrepancy between SBBN calculations
and observations instead to reconciliate them.
Another source of alteration of the primordial abundances could be the
contribution of the first generation stars (Population 111). However, it seems
difficult that they could reduce the 7L2 abundance without affecting the D
one, consistent with CMB Clbh2.
The baryonic density of the Universe as determined by the analysis of the
Cosmic Microwave Background anisotropies is in very good agreement with
Standard Big-Bang Nucleosynthesis compared to D primordial abundance
deduced from cosmological cloud observations. However, it strongly disagrees with lithium observations in halo stars (Spite plateau) and possibly
4He new observations. The origin of this discrepancy, if not nuclear, is a
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DECAYING COLD DARK MATTER COSMOLOGY AND
K. ICHIKI *
lfniversity of Tokyo, Department of Astronomy,
7-3-1 Hongo, Bunkyo-ku,
Tokyo, 113-0033, Japan
National Astronomical Observatqy,
2-21-1 Osawa, Mitaka,
Tokyo, 181-8588, Japan
K. TAKAHASHI ? M. OGURI AND K. KOTAKE
University of Tokyo, Departmelit of Pkysics,
7-3-1 Ho,ngo, Bzmkyo-ku,
Tokyo, 113-0033, Japan
We reformulate the theory of cosmological perturbations in decaying cold dark
matter cosmology and calculate cosmological microwave background anisotropies.
By comparing with recent observation we derive a new bound on the life time of
such decaying particles. We show that the d a t a of CMB anisotropies alone do
not prefer the decaying cold dark matter models t o the standard cold dark matter
ones, and constrained to r-' 2 43 Gyr at 3u.
The existence of cold dark matter (CDM) is now strongly believed from
numerous kinds of astronomical phenomena, such as rotation curves in
gabaxies, anisotropies in cosmic microwave background (CMB), and x-ray
emitting clusters of galaxies, combined with the low cosmic baryon density
predicted by big bang nucleosynthesis. The nature of CDM, however, is
still one of the biggest mysteries in cosmology. Moreover, some discrepan*Work partially supported by the Sasakawa Scientific Research Grant from the JSS
TWork is supported by Grant-in-Aid for JSPS Fellows
cies on galactic and sub-galactic scales in standard CDM cosmology have
stimulated numerous proposals for the alternative CDM candidate 2 .
Phenomenologically, decaying cold dark matter (DCDM) cosinology has
been extensively studied in various contexts. Among them, DCDM model
which Cell proposed could solve both over-concentration problem of the
dark matter halos and overproduction problem of small dwarf galaxies in
the standard CDM models '. The authors showed in previous papers 4,5
that introducing DCDM can improve the fits of observational data sets
of Type Ia SN, mass-to-light ratios and X-ray gas fraction of clusters of
galaxies, and evolution of cluster abundance. It is also suggested that
the decay of super-heavy dark matter particle can be the origin of ultxahigh energy cosmic rays above the Greisen-Zatsepin-Kuzmin cut-off of the
Theoretical candidates for DCDM have been proposed by inany authors and their predictions for lifetime of such particles cover a large range
of values 10W2 < 7 < 10" Gyr 7,8,9. Although DCDM attracts strong
phenomenological interests, such particles with lifetime around the age of
the universe may have dangerous observational signals and should thus be
tightly constrained if it decays into visible particles. The most stringent
constraint on such particles may come from the diffuse ganiina ray background observations
However, realistic simulation which takes all energy
dissipation processes into account showed that even the particles with lifetime as short as a few times of the age of the universe still are not ruled
out by recent observations ll. Moreover, as long as we have not identified
what dark matter is, the decay channel cannot escape some uncertainties.
On the other hand, cosmological constraint derived froin CMB in the
present paper is different froin those from diffuse gamma ray background in
the point that it does not depend on the details of the decay products. Here
we assume only that dark matter part.icles decay into relativistic particles
and put a new bound on the life time of such decaying particles.
2. CMB and DCDM
To make theoretical predictions of CMB anisotropies in DCDM cosmology,
we review and formulate cosmological perturbation theory.
2.1. Background Equations
The equations of background energy densities for DCDM particles and its
relativistic daughter particles are given by
= - ~ H ~ D aRr p D M .
Here dot denotes conformal time derivative and r is the decay width of the
dark matter. The equation of state parameters w = P / p are
respectively. We also define effective equation of state,
which are defined by the evolution of energy density, p i = - 3 H ( 1 + w e f f ) p i .
2.2. Perturbation Equations
In the conformal Newtonian gauge, the line element is given by
+ 2 $ ~ ) +d (~1~+ 2 @ ) 6 , j d ~ ” d ~ ’.]
The perturbed Einstein equations in this metric are described as
where M p is Planck mass, 6 , w and ll are perturbations of density, velocity,
and anisotropic stress, respectively.
The conservation of energy-momentum is a consequence of the Einstein
equations. The perturbed part of energy-momentum conservation equations TYL = 0 yields the following equations of motion,
Here gauge invariant density perturbation is defined by
g - 6 Newton
3(1 + , ~ , ! f f ) ~ i. @
bpi - pi
Note that we subtracted and added the decaying term ar6p$, from the
equation of dark matter t o that of daughter radiation. As for the velocity
perturbation, we subtracted ~ H , w $ Y v D M
from the equation of dark matter
to that of daughter radiation to express
the momentum flux between them. These terms are analogous to baryon photon momentum transfer by Compton scattering in the standard CMB
2.3. Boltzmann Hierarchy f o r Daughter Radiation
Further, we need Boltzmann hierarchy for 1 2 2 moments of daughter
radiation, where I stands for multipole monient in a Legendre expansion of
perturbed distribution function. We begin with the Boltzmann equation in
the Newtonian gauge 1 2 ,
where 4' = yA is the comoving 3-momentum with ,nini= 1, and we
write the distribution function of daughter radiation as f D R ( x iy,nJ,
+ Q D R ( x iq;n,,T)).
We should note that, unlike the standard
CDM, f t R is now time dependent in DCDM. To describe the decay process,
let us consider the collision term in the form,
where f D M = f,""(l+
Q D M ) and DM are the distribution function and
mass of DCDM, respectively. Then we get
in linear perturbation theory. The 3rd and 4th terms in 1.h.s of Ey.(18)
lead standard stream and gravitational scattering terms, respectively. We
then Eq.(18) can be written as
The two t e r m in r.1i.s of Eq.(20) are new oiies in DCDM cosmology. The
second term of them corresponds to the flow from the dark matter to daughter radiation in first order perturbation. Unfortunately, the calculation of
this term is coinplicated in general 14, but in a fluid approximation, it
can be easily described. (see Eq. (12) aiid Eq.(13) for 1 = 0 and 1 = 1,
respectively.) For higher inultipoles ( I 2 2) this term vanishes since the
perturbation in the dark matter ( Q D M ) does have only 1 = 0 , l perturbations which correspond to density and velocity perturbation, respectively.
Finally, we have hierarchy for daughter radiation,
where A41 is the coefficient in a Legelidre expansion,
3. CMB Constraint and Discussions
Recent high-resolution measureinelit of CMB aiikotropies by the Wilkinson Microwave Anisot,ropy Probe (WMAP) has become one of the most
stringent, test for cosinology 15,16,17. The standard procedure based on a
Bayesian statistics gives us the posterior probability distribution from which
we obtain the optimal set of cosmological parameters aiid their confidence
Figure 1. CMB angular power spectrum with and without decay of cold dark matter
particles. For both theoretical lines, all the other standard parameters are fixed t o
WMAP optimal values to demonstrate how decay of the dark matter particles modify
the spectrum. Observational d a t a from WMAP are also shown in the figure.
The likelihood functions we calculate are given by the combination of
Gaussian and lognormal distribution of Verde et a1.l'. To include the decay
of dark matter particles described above, we calculate theoretical angular
using a modified
power spectrum of CMB fluctuation (1(1 + l)C~/(27r))
Boltzniann code of CAMB 19, which is based on a line of sight integration
The decay of the CDM particles modify the CMB spectrum in some
ways. The modified evolution of dark niatter will change the expansion
history of the universe and generally cause the shorter look back time to
the photon last scattering surfaces than that of standard CDM cosmology. This will push the acoustic peaks in the CMB spectrum toward the
larger scales. There will be also a decay of gravitational potential at later
epochs due to that of dark matter particles leading t o a larger integrated
Sachs-Wolfe effect (LISW) and thus larger aiiisotropies at low multipoles.
Thus, the amplitudes and locations of the peaks in the power spectrum
of microwave background fluctuations 21 can in principle be used to constrain this cosmology. An Illustration of CMB power spectrum in DCDM
cosmology is shown in Fig. 1.
It is well known, however, that there are another cosmological parameters which also modify the CMB spectrum. Therefore, we have to generate
the full probability distribution function and marginalize over nuisance parameters to obtain the constraint on paranieter(s) which we are interested
in. To realize this, we followed the Markov Chain Monte Carlo (MCMC) approach " and explore the likelihood in seven dimensional parameter space
(six WMAP standard parameters, Rbh', CLh', h, z,,, 'n,, A , , and one
DCDM parameter r). Our results are showii in Fig. 2.
An interesting point is that the parameter of DCDM cosmology, r, dose
not degenerate with other parameters very much as one can see in the left
side of Fig. 2. This means that the change in the CMB spectrum from r
can not be mimicked by other standard parameters. This is the reason why
CMB can put constraint on the life time of DCDM particle effectively. The
right side of Fig. 2 shows marginalized likelihood of t,he life time of DCDM
particle and it is limited by
2 43 Gyr at 3a. One does not find any
sigiial to prefer DCDM to standard CDM cosmology.
We showed that even the current CMB data alone put constraint on the
2 43 Gyr at 3a. Although this result
life time of cold dark matter to rP1
is coinparable to that from the diffuse gamma ray background of Zeaeepour 11, it is completely different in a sense that CMB constraint is purely
gravitational, i.e, it is independent from the details of decay channels.
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