Solution to Big-Bang Nucleosynthesis in Hybrid Axion Dark Matter Model
ABSTRACT Following a recent suggestion of axion cooling of photons between the
nucleosynthesis and recombination epochs in the Early Universe, we investigate
a hybrid model with both axions and relic supersymmetric particles. In this
model we demonstrate that the 7Li abundance can be consistent with observations
without destroying the important concordance of deuterium abundance.
arXiv:1202.5603v1 [astro-ph.CO] 25 Feb 2012
Solution to Big-Bang Nucleosynthesis in Hybrid Axion Dark Matter Model
Motohiko Kusakabe1,∗A.B. Balantekin2,3,†Toshitaka Kajino3,4,‡and Y. Pehlivan3,5§
1Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan
2Department of Physics, University of Wisconsin, Madison, WI 53706, USA
3National Astronomical Observatory of Japan 2-21-1 Osawa, Mitaka, Tokyo, 181-8588, Japan
4Department of Astronomy, Graduate School of Science,
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan and
5Mimar Sinan Fine Arts University, Besiktas, Istanbul 34349, Turkey
(Dated: February 28, 2012)
Following a recent suggestion of axion cooling of photons between the nucleosynthesis and recom-
bination epochs in the Early Universe, we investigate a hybrid model with both axions and relic
supersymmetric particles. In this model we demonstrate that the7Li abundance can be consistent
with observations without destroying the important concordance of deuterium abundance.
PACS numbers: 26.35.+c, 98.80.Cq, 98.80.Es, 98.80.Ft
Introduction – Low-metallicity halo stars exhibit a
plateau of7Li abundance, indicating the primordial ori-
gin of7Li . However, the amount of7Li needed to be
consistent with the cosmic microwave background obser-
vations  is significantly more than7Li observed in old
halo stars . (Even though7Li can be both produced
and destroyed in stars, old halo dwarf stars are expected
to have gone through little nuclear processing). Recent
improvements in the observational and experimental data
seem to make the discrepancy worse [4, 5]. One possible
solution is to invoke either nuclear physics hitherto ex-
cluded from the Big-Bang Nucleosynthesis (BBN) [6, 7]
or new physics such as variations of fundamental cou-
plings [8, 9], and particles not included in the Standard
Model [10–42]. Effects of massive neutral relic particles
on BBN were extensively studied [10–28].
More recently an alternative solution to the lithium
puzzle was proposed. Since the last photon scattering oc-
curs after the end of the nucleosynthesis, one can search
for a mechanism for the cooling of photons before they
decouple. It was suggested that dark matter axions could
form a Bose-Einstein condensate (BEC) [43, 44]. Such
a condensate would cool the photons between the end
of BBN and epoch of photon decoupling, reducing the
baryon-to-photon ratio WMAP infers, as compared to
its BBN value . An alternative mechanism for such a
cooling is resonant oscillations between photons and light
abelian gauge bosons in the hidden sector . There are
two prima facie problems with the axion BEC-photon
cooling hypothesis: it overpredicts primordial deuterium
(D) abundance as well as the effective number of neu-
trinos. Even though D is easy to destroy, one does not
expect the sum of abundances of D and3He to change
significantly in the course of cosmic evolution . Hence
it is important to find a parameter region in which pre-
dicted abundances of D and7Li are consistent with ob-
servations. In this letter we demonstrate the existence of
such a parameter region using a model with axions and
massive relic particles.
The Hybrid Model – We carried out BBN net-
work calculations using Kawano’s code [50, 51] by in-
cluding Sarkar’s correction for
JINA REACLIB Database V1.0  is used for light nu-
clear (A ≤ 10) reaction rates including uncertainties to-
gether with data [54–56]. Adopted neutron lifetime is
878.5 ± 0.7stat± 0.3sys s  based on improved mea-
surements . Taking into account the uncertainties in
the rates of twelve important reactions in BBN , we
employ regions of 95% C. L. in our calculations.
We compare our results with the abundance con-
straints from observations. For the primordial D abun-
dance, the mean value estimated from Lyman-α absorp-
tion systems in the foreground of high redshift quasi-
stellar objects is log(D/H) = −4.55 ± 0.03 .
adopt this value together with a 2σ uncertainty, i.e.,
2.45 × 10−5
< D/H < 3.24 × 10−5.
dance measurements in Galactic HII regions through the
8.665 GHz hyperfine transition of3He+yield a value of
3He/H=(1.9 ± 0.6) × 10−5. Although the constraint
should be rather weak considering its uncertainty, we
take a 2σ upper limit and adopt3He/H < 3.1 × 10−5.
We also utilize a limit on the sum of primordial abun-
dances of D and3He taken from an abundance for the
protosolar cloud determined from observations of solar
wind, i.e., (D+3He)/H=(3.6 ± 0.5) × 10−5.
abundance can be regarded as constant at least within
the standard cosmology since it is not affected by stellar
activities significantly despite an effect of D burning into
3He via2H(p,γ)3He would exist .
For the primordial4He abundance, we adopt two dif-
ferent constraints from recent reports: Yp = 0.2565 ±
0.0051  and Yp= 0.2561 ± 0.0108  both of which
are derived from observations of metal-poor extragalactic
HII regions. Adding 2σ uncertainties leads to 0.2463 <
Yp< 0.2667  and 0.2345 < Yp< 0.2777 .
6Li plateau of metal-poor halo stars (MPHSs), yields
the upper limit of6Li/H= (7.1±0.7)×10−12. Adding
a 2σ uncertainty, we adopt6Li/H < 8.5 × 10−12.
4He abundances .
FIG. 1. Abundances of4He (mass fraction), D,3He,7Li and
6Li (number ratio relative to H) as a function of the baryon-to-
photon ratio η or the baryon energy density parameter ΩBh2
of the universe. The thick dashed curves are for SBBN. The
thin dashed curves around them show the regions of 95% C.
L. in accordance with the nuclear reaction rate uncertainties.
The boxes correspond to the adopted abundance constraints
on the SBBN model. The vertical stripes represent the 2σ
limits on ΩBh2or η for the SBBN model (taken from the
constraint by WMAP  and labeled as WMAP7) and for
the axion BEC model (labeled as axion). The solid curves
are the results obtained with the long-lived decaying particle
model with parameters fixed to (τX, ζX)=(106s, 2 × 10−10
GeV) (see text).
7Li abundance,we adopt thelimits
log(7Li/H)= −12+(2.199±0.086) (with 95% C. L.) de-
rived from recent observations of MPHSs in the 3D non-
local thermal equilibrium model , i.e. 1.06×10−10<
7Li/H < 2.35 × 10−10.
Figure 1 shows the abundances of4He (Yp; mass frac-
tion), D,3He,7Li and6Li (number ratio relative to H) as
a function of the baryon-to-photon ratio η or the baryon
energy density parameter ΩBh2of the universe, where h
is the Hubble constant in units of 100 km/s/Mpc. The
thick dashed curves are the results of the standard BBN
(SBBN) with a neutron lifetime of 878.5 ± 0.8 s. Thin
dashed curves around them show regions of 95% C. L.
from uncertainties in the nuclear reaction rates.
boxes represent adopted abundance constraints as sum-
marized above. The vertical stripes correspond to the 2σ
limits on ΩBh2or η. The values provided by WMAP 
(labeled WMAP7) are
η = (6.225+0.314
Values predicted by the BEC model (labeled axion) are
smaller by a factor of (2/3)3/4at the BBN epoch :
η = (4.593+0.232
It can be seen that the adoption of the η value from
WMAP leads to a7Li abundance calculated in the ax-
ion BEC model, which is in reasonable agreement with
the observations. However, we lose the important consis-
tency in D abundance. Ref.  noted that astronom-
ical measurements of primordial D abundance can have
a significant uncertainty as well as a possibility that D
is burned by nonstandard stellar processes. Even if their
assumption were true, stellar processes are not expected
to change the sum of D and3He abundances . As
seen in Fig. 1, the constraint on (D+3He)/H abundance
seems to exclude the original axion BEC model. Ulti-
mately, this model is viable only when the abundance of
(D+3He)/H is reduced through some exotic processes.
It is known that nonthermal photons can be generated
through electromagnetic energy injections by the radia-
tive decay of long-lived particles after the BBN epoch [11,
16]. These nonthermal photons can photodisintegrate
background light elements [10, 11, 19, 20, 24, 25]. We
adopt the method of Ref.  to calculate the nonther-
mal nucleosynthesis, where we incorporated new thermal
reaction rates as described above. In addition, we adopt
updated reaction rates of4He photodisintegration 
derived from the cross section data using precise mea-
surements with laser-Compton photons [64, 65]. Effects
of electromagnetic energy injection depend on two pa-
rameters. One is ζX = (n0
the number ratio of the decaying particle X and the back-
ground radiation before the decay of X, and Eγ0is the
energy of photon emitted at the radiative decay. The
other is τX, the lifetime of the X particle.
Figure 2 shows the parameter space in our hybrid
model. For4He, we adopt the conservative constraint
with larger uncertainty . We also show the contour
for6Li production at the observed level, i.e.,
7.1 × 10−12. This figure shows the result of nonthermal
nucleosynthesis induced by the radiative decay of long-
lived particles with the η value of the axion BEC model
[Eq. (2)]. Except for D and7Li, contours are similar to
those represented in Ref  where BBN epoch η value
is assumed to be the same as WMAP η value [Eq. (1)].
γ)Eγ0 where (n0
FIG. 2. Parameter space of the hybrid model (τX,ζX) for the
value of η = 4.6 × 10−10provided by the axion BEC model.
The contours identify the regions where the nuclei are over-
produced or underproduced (“over” and “low”, respectively)
with respect to the adopted abundance constraints.
fraction (red line) and3He/H (black lines), D/H (green solid
and dashed lines for upper and lower limits, respectively),
and7Li/H (blue line) number ratios are shown. The orange
line is the contour of6Li/H=7.1 ×10−12. In the gray-colored
region all abundances are within the limits of observational
In the very small colored region, calculated primordial
abundances of all nuclides including D and7Li are simul-
taneously in ranges of adopted observational constraints.
We conclude that the present model eliminates the main
drawback of the original axion BEC model by reducing
primordial D abundance via2H(γ,n)1H reaction, where
γ’s are nonthermal photons. We note that the decaying
particle model with the WMAP η value cannot resolve
the7Li problem by itself [23, 25].
The effect of the radiative decay on other elemental
abundances is not significant except for7Li. Since ener-
getic photons produced quickly collide with background
photons and create e+e−pairs, nonthermal photon spec-
tra developed by the decay has a cutoff energy EC =
e/22T where meis the electron mass . The decay
at earlier (hotter) universe then triggers nonthermal pho-
tons with lower cutoff energies. The threshold energies
of7Be and D photodisintegration,7Be+γ →3He+4He
and D+γ → n + p are 1.5866 MeV and 2.2246 MeV,
respectively. These two nuclei are very fragile against
photodisintegration. When the decay occurs early at rel-
atively high T9≡ T/(109K)>
∼106s, nonthermal photon spectra contain pho-
tons to dissociate7Be and D, while keeping other nuclides
intact. The gray region indicates parameters which re-
sult in moderate destruction of D which is overproduced
in the original BEC model because of low η. Above that
region, D is destroyed too much by photodisintegration,
tively) and number ratios of other nuclides relative to H as
a function of T9. Solid lines show the abundances calculated
in the hybrid model with the parameters (τX, ζX)=(106s,
2×10−10GeV) which correspond to the point indicated with
a star in Fig. 2. The dashed lines show the SBBN prediction.
Mass fractions of H and4He (Xp and Yp, respec-
while below it D abundance is too high.
We next present the results of a BBN calculation in our
hybrid model with a fixed set of parameters given by (τX,
ζX)=(106s, 2×10−10GeV) and the η value given in Eq.
(2). This corresponds to the point indicated with a star in
Fig. (2) and yields the required D and7Li abundances as
seen in Figure 3. This figure shows H and4He mass frac-
tions (denoted by Xpand Yp), and n, D, T,3He,6Li,7Li
and7Be number ratios relative to H as a function of the
temperature. The abundances calculated using the axion
BEC + long-lived decaying particle model with the pa-
rameters (τX, ζX)=(106s, 2×10−10GeV) and the η value
provided by BEC model [Eq. (2)] are shown in solid lines
whereas the SBBN prediction is plotted by dashed lines.
The small difference at T9>
∼0.06 observed between solid
and dashed lines is caused by difference between initial
η values. At 0.06>
∼7×10−3(corresponding to the
cosmic time of t ∼ 5×104–4×106s), effects of2H(γ,n)1H
are seen in the decrease of D and the increase of n abun-
dances.We find a slight decrease in
This is caused through reactions7Be(γ,3He)4He (thresh-
old energy: 1.5866MeV),7Be(γ,p)6Li (5.6858 MeV), and
7Be(γ,2pn)4He (9.3047 MeV). The second reaction in-
creases the6Li abundance. Finally, at T9<
where the abundance of long-lived X particle is already
less than 3 % of the initial abundance, effect of4He pho-
todisintegration is to increase3H and n abundances.
In Fig. 1, solid lines show the results for the SBBN
+ long-lived decaying particle model with the same pa-
rameter values, i.e., (τX, ζX)=(106s, 2 × 10−10GeV).
Obviously the abundances of D and7Li (produced partly
as7Be) are reduced, while that of6Li is increased from
those of SBBN.
∼7 × 10−3,
Conclusions – We used a hybrid axion and massive
relic particle model in which axions cool the photons and
relic particles produce non-thermal photons to eliminate
the high D abundance in the original axion BEC model.
Our hybrid model also produces6Li keeping7Li abun-
dance at the level of Population II Spite plateau. Our
work thus demonstrates that the7Li abundance can be
consistent with observations without destroying the im-
portant concordance of deuterium abundance.
This work was supported in part by Grants-in-Aid
for Scientific Research of the JSPS (200244035) and
for Scientific Research on Innovative Area of MEXT
(20105004), in part by JSPS Grant No.21.6817, in part by
the U.S. National Science Foundation Grant No. PHY-
0855082, in part by the Council of Higher Education of
Turkey, and in part by the University of Wisconsin Re-
search Committee with funds granted by the Wisconsin
Alumni Research Foundation.
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