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Possible origins of 'Li discrepancy between SBBN and CMB

Possible origins of 'Li discrepancy between SBBN and CMB

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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.

This is

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.

4.3. Cosrnology

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.


5. Conclusions

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

challenging issue.


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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



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.

1. Introduction

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

spectrum '.

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,




=3H ’

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

dS2 =


+ 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

and added

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


7) =

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

approach 20.

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.

4. Conclusion

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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