We improve the estimate of the axion CDM energy density by considering the
new values of current quark masses, the QCD phase transition effect and a
possible anharmonic effect.Comment: 7 pages, 6 figures. References are added. A factor is correcte

Ji-Haeng Huh

Jihn E Kim

Komatsu E (WMAP Collaboration)

Kyu Jung Bae

Landau L D

Linde A D

Yao W-M

Zhitnitskii A P

English

arXiv

Pyungwon Ko

Deog Ki Hong

Crossref

Update of axion CDM energy density

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Updated axion CDM energy density
Ji-Haeng Huh
Department of Physics and Astronomy and Center for Theoretical Physics, Seoul National University, Seoul
151-747, Korea
Abstract. We update cosmological bound on axion model. The contribution from the anharmonic effect and the newly
introduced initial overshoot correction are considered. We present an explicit formula for the axion relic density in terms
of the QCD scaleΛQCD, the current quark massesmq’s and the Peccei-Quinn scaleFa, including firstly introduced 1.85 factor
which is from the initial overshoot.
Keywords: axion, cosmology, cold dark matter
PACS: 14.80.Mz, 12.38.Aw, 95.35.+d
INTRODUCTION
The standard model suffers several fine tuning problems.
They are all related to the symmetry of the nature and its
breaking. The gauge hierarchy problem which is one of
the main motivation of the upcoming experiment, LHC
has its origin in the gauge symmetry breaking sector.
Another problem, strong CP problem is deeply related to
a discrete symmetry, CP. The fact that the almost of the
suggestions to solve them are more or less related to the
invisible missing matter in the Universe is interesting.
The strong CP problem can be stated as that a CP-odd
operatorθ̄ tr{GG̃} which cannot be forbidden is unmea-
surably small∼ 10−11 [13]. The introduction of the ax-
ion can solve this problem beautifully [4, 7]. The axionic
solution introduces an irrelevant operatoraFa tr{GG̃}. It
can be thought of as makinḡθ a dynamical field. If then,
the Vafa-Witten theorem tells that CP conserving point
a+ θ̄ = 0 becomes a minimum of the free energy by the
strong interaction [1].
However, the introduction of the axion coupling solely
cannot solve the strong CP problem. One needs one more
ingredient, the relaxation mechanism for axion. The stan-
dard big bang cosmology provides it naturally, so the ax-
ionic solution to the strong CP problem is intrinsically
cosmological one. This point was recognized soon after
the suggestion of the axion [8]. The axion is one of the
possible candidate of the cold dark matter.
After the early studies of cosmological implication
of the axion, Turner [9] made a quantitative, numerical
estimate of the axion energy density with a anharmonic
correction and more precise axion mass formula.
All of their work depends on the assumption of the adi-
abatic process. However, the assumption can be broken
in the two phases of the evolution, at the time of start-
ing to roll down and at the time of QCD phase transition
[16, 10].
We considered the effects of these two phases on the
axion energy density and reanalyzed anharmonic effect.
In addition, many parameters of QCD physics and the
cosmology are changed [17, 12]. We finally represent the
explicit formula in terms of these parameters.
RELIC DENSITY OF AXION
Just after the inflation, a very light scalar field like ax-
ion tends to be frozen at a certain value [6],〈Φ〉 = φ0 =
const., which is determined stochastically. The dynam-
ics of a scalar field in the expanding universe with the
Hubble parameterH(t) is governed by the equation of
motion Φ̈ + 3H(t)Φ̇+V ′(Φ) = 0. The second term in
the left hand side of it prevents roll down of the scalar
field, while second term forces it to move to the mini-
mum. Therefore, after the inflation, the scalar fields stays
at their initial valueφ0 until the forcesV ′′ to roll down
overcome the Hubble friction 3HΦ̇.
If the scalar potential is harmonic one, it starts rolling
down and oscillates when 3H becomes small as the mass
m. In the hydrodynamic description, such a coherently
oscillating scalar fields behaves as like the pressureless
cold(T = 0) matter. If the interaction with the known par-
ticles is sufficiently weak and its mass is light enough, it
can be a cold dark matter(CDM) candidate. Since the in-
visible axion has the interactions and the mass originat-
ing from quantum effect, they can be small enough. So
it’s a natural candidate of this kind CDM.
The potential of the axion field can be parameterized
by V (θ ) = −C(T )cos(θ ), whereθ = a/Fa andC(T ) =
αinstGeV4(T/1GeV)−n. If the initial miss alignment an-
gle θ1 is small enough, it becomes the harmonic poten-
tial with the massma(T ) =
√
C(T )/F2a . At the temper-
atureT1 satisfyingma(T1) ≃ 3H(T1) = 1.66g1/2∗ T 2/MP,
the axion field starts rolling down. In our parametriza-
tion,T1 =
(
αinst
4.98g1/2∗
MP
Fa
)2/(4+n)
GeV. Under the adiabatic
assumption,H, ṁa/ma ≪ ma, the number of the axion is
conserved. So the axion energy density at present can be
easily calculated asρa ≃ ma(Tγ )ma(T1)
(
R3(T1)
R3(Tγ )
)
θ 21 .
TEMPERATURE DEPENDENT AXION
MASS
To give a prediction, we should knowαinst and n. Al-
though, we cannot solve it fully, all we have to know is
the mass in the two different regime in which we can esti-
mate it numerically or phenomenologically. Two regimes
are zero temperature hadronic phase and the temperature
enough higher thanΛQCD [3].
At the zero temperature axion mass is given by
√
Z/(1+Z)mπ fπ/Fa, whereZ = mu/md . At the high
enough temperature, axion mass can be obtained by the
perturbative instanton calculus under the dilute gas ap-
proximation [11]. The instanton density with the sizeρ
is
n(ρ ,T ) = mumdmsCN(ξ ρ)3
1
ρ5
(
4π2
g2
)2N
e−8π
2/g2(Λ)
×e{− 13λ 2(2N+N f )−12A(λ )[1+ 16 (N−N f )]},
whereξ = 1.33876 andCN = 0.097163 forN = 3. With
the 3-loop running coupling constant, the numerical re-
sults are well fitted at theT ∼ 1GeV by the form [2]
αinst = 3.96×10−12
(
mumd ms
3·6·103 MeV3
)(
ΛQCD
380 MeV
)7.97
,
n = 7.44259−0.564492
(
ΛQCD
380 MeV
)
.
POSSIBLE CORRECTIONS
Our rough estimation relies on two assumptions,
the harmonic limit(θ1 ≪ 1) and the adiabatic
approximation( ˙ma/ma,H ≪ ma). We considered possi-
ble effects of them on the axion energy density.
There is also a possible correction from the QCD
phase transition. However, it turns out to be negligible,
if one does not include fine tuning [2, 10].
anharmonic correction
In the previous section, we use the fact that the axion
number is conserved. However, the axion field is in the
coherent state which is not the number eigenstate. There-
fore the concept of the number conservation is not obvi-
ous except for the case of the non-interacting harmonic
limit. The conservation quantity, total number of axion
is actually an adiabatic invariant of the system under the
slowly varying potential. In the anharmonic limit,θ ∼ 1,
there is no interpretation like the number conservation.
The adiabatic invariant is given by the area in the
phase space swept by one period,
∮
p dq [15]. In our
case, its effect is appeared as a correction factor [2, 14]
f1(θ1) = 2
√
2
πθ21
∫ +θ1
−θ1
√
cosθ ′− cosθ1 dθ ′. Now the axion
energy density with the anharmonic correction isρa ≃
ma(Tγ)ma(T1)
(
R3(T1)
R3(Tγ )
)
θ 21 f1(θ1).
initial overshoot correction
There are two adiabatic conditions we assumed. One is
for the expansion rate to be much smaller than the axion
oscillation,H ≪ ma. Another is for the change rate of the
mass to be smaller than the oscillation. However, at the
time of starting to roll down, the previous one cannot be
satisfied because the expansion rate is comparable with
the mass, 3H ≃ ma at that time.
Therefore the adiabatic approximation does not work
in the first few oscillations, and the numerical analysis
must be done in this period. To include its effect in our
calculations, we defined the another temperatureT2 and
the new misalignment angleθ2 at the temperatureT2,
such that the adiabatic condition is well satisfied after
T2. We explicitly checked that the values after the first
half oscillation are good enough. We obtainedT2/T1 =
f2(θ1,n) andθ2θ2 = f3(θ1,n) numerically, and they are
presented in the fig.
As one can sees, the correction factors,f2 and f3,
becomes larger at the largeθ1. It is because of that the
anharmonic terms make the field stays longer time at the
around of the top of the potential. So this initial overshoot
correction includes other anharmonic correction. To split
them, it is useful to factor out overall constant as like
f2(θ1,n)≃ c2 f̃2(θ1,n) and f3(θ1,n)≃ c2 f̃3(θ1,n), where
the function with a tilde goes to 1 asθ1 is going to zero.
With the correctionsf1, f2 and f3, the relic density
becomes
Ωa ∝
ma(T2)θ 22
T 32
f1(θ2)
∝
θ 21 c23 f̃ 23 (θ1,n) f1(θ1c3 f̃3(θ1,n))
c3+n/22 f̃
3+n/2
2 (θ1,n)
= 1.846×θ 21 ×F(θ1,n).
where the function
F(θ1,n) = f̃ 23 (θ1,n) f1(θ1c3 f̃3(θ1,n))/ f̃
3+n/2
2 (θ1,n)
(1)
which is shown in Fig. 1. As one can see, there is a 1.846
enhancement in the axion relic density.
0.0 0.5 1.0 1.5 2.0 2.5 3.0
Θ10.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
FHΘ1,nL
FIGURE 1. The combined correction factorF(θ1,n).
RESULTS AND CONCLUSION
With all of corrections we considered, the relic density
of the axion is estimated as
ρa ≃ 1.449×10−11
(
Fa,GeV
1012
)
1
T1,GeV
θ 21
γ
F(θ1,n)(eV)4,
(2)
where we usedZ ≡ mu/md = 0.5, mπ0 =135.5 MeV,
fπ =93 MeV andg∗s(present)=3.91, f (θ1) is the anhar-
monic correction given in Fig. 1 andγ is the entropy in-
crease ratio.
The energy fraction of the axion is given by
Ωa ≃ 0.379×
( mumdms
3 ·6 ·103 MeV
)−0.092
×
(
θ 21 F(θ1)
γ
)(
0.701
h
)2
×
(
ΛQCD
380MeV
)−0.733( Fa
1012GeV
)1.184−0.010x
,(3)
wherex = (ΛQCD/380MeV)− 1. Of course, these ex-
pressions gives the bound given in Fig. 2.
ACKNOWLEDGMENTS
This work is supported in part by the Korea Research
Foundation, Grant No. KRF-2005-084-C00001.
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0 2.´1012 4.´1012 6.´1012 8.´1012 1.´1013
FaΓ

HGeVL
0.5
1.0
1.5
2.0
2.5
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Θ1
FIGURE 2. The bound from overclosure of the universe. The
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