IFT-UAM/CSIC-07-15
Axino warm dark matter and Ωb − ΩDM coincidence
Osamu Seto
Instituto de F́ısica Teórica, Universidad Autónoma de Madrid,
Cantoblanco, 28049 Madrid, Spain
Masahide Yamaguchi
Department of Physics and Mathematics,
Aoyama Gakuin University, Sagamihara 229-8558, Japan
Abstract
We show that axinos, which are dominantly generated by the decay of the next-to-lightest super-
symmetric particles produced from the leptonic Q-ball (L-ball), become warm dark matter suitable
for the solution of the missing satellite problem and the cusp problem. In addition, Ωb − ΩDM
coincidence is naturally explained in this scenario.
PACS numbers: 95.35.+d, 12.60.Jv, 98.80.Cq, 04.65.+e
http://arxiv.org/abs/0704.0510v2
I. INTRODUCTION
Recent observations of cosmic microwave background anisotropies such as Wilkinson Mi-
crowave Anisotropy Probe measured the abundance of components of the Universe very
precisely. However, their origins are still one of the major mysteries of cosmology and parti-
cle physics. The fact that the abundances of dark matter and baryon are of the same order
may give us a great hint for their origins.
In the minimal supersymmetric standard model (MSSM), flat directions consist of
squarks/sleptons and produce a non-zero baryon or lepton number through the Affleck-
Dine (AD) mechanism [1]. Then, Q-balls, which are non-topological solitons [2], can be
produced due to the instability and absorb almost all the produced baryon or lepton num-
bers [3]. In the gravity mediated supersymmetry breaking model, the mass per charge of
a Q-ball is larger than that of a nucleon so that they are unstable against the decay into
light fermions. Then, they can directly decay into baryons and the lightest supersymmetric
particle (LSP). In case that the charge of the produced Q-balls is large enough, Q-balls can
survive even after the freeze-out of weakly interacting massive particles (WIMPs). Thus,
the reason why energy densities of dark matter and baryon are almost the same magnitude
can be explained in this scenario [4, 5].
However, it was pointed out that LSPs are often overproduced by the decay of Q-balls if
the LSP is the lightest neutralino in the MSSM [6], which gives the stringent conditions on
the neutralino LSPs and AD fields. Only a few models free from this overproduction have
been proposed [7, 8]. Instead, the supergravity models in which the LSPs is a stable gravitino
are investigated [9]. In this scenario, Q-balls decay into the next-to-lightest supersymmetric
particle (NLSP) directly instead of the LSP gravitino. The LSP gravitinos are produced
by the decay of NLSP and becomes dominant over other gravitinos produced by thermal
processed [10, 11] and by the decay of thermal produced NLSP [12, 13] in case that Q-balls
can survive the evaporation. Then, it is found that such a gravitino dark matter scenario is
still viable if the late decay of NLSP does not spoil the success of Big Bang Nucleosynthesis
(BBN). Another interesting possibility is that the LSP is an axino, which is the fermionic
superpartner of an axion. Axinos are also produced by the decay of NLSPs produced from
the Q-ball [14] as well as by thermal processes and by the decay of thermally produced
NLSP [15, 16, 17, 18, 19].
Such gravitinos and axinos often become an ideal candidate for cold dark matter. The
models of cold dark matter (CDM) and dark energy combined with inflation-based scale-
invariant primordial density fluctuations have succeeded at explaining many properties of
the observed universe, especially the large scale structure of the universe. However, going
into the smaller scales, some observations on galactic and subgalactic (. Mpc) seem to con-
flict with predictions by high-resolution N-body simulations as well as analytic calculations
based on the standard CDM model. The first discrepancy is called the missing satellite
problem [20]. The CDM-based models predict an order of magnitude higher number of
halos than those actually observed within the Local Group. The other is called the cusp
problem [21]. The CDM-based models also predict overly cuspy mass profile for the CDM
halos compared to actual observations within the Local Group. In order to reconcile such
discrepancies, several authors proposed modifications to the standard CDM-based model
though the photoionization mechanism may overcome such difficulties [22]. One method is
to reduce the small-scale power of primordial density fluctuations, which can be realized
in a specific model of inflation [23]. Another is to change the properties of dark matter.
Spergel and Steinhardt introduced strong self-interaction among cold dark matter particles
(collisional CDM), which enhances satellite destruction and suppress cusp formation [24].
The warm dark matter [25], which can have relatively large velocity dispersion at the epoch
of the matter-radiation equality, can also reduce satellite production and cusp formation.
In this paper, we consider axinos dominantly generated by the decay of NLSPs produced
from the leptonic Q-ball (L-ball). Such axinos become warm dark matter suitable for the
solution of the missing satellite problem and the cusp problem. In addition, Ωb − ΩDM
coincidence is naturally explained through the Affleck-Dine mechanism and the subsequent
L-ball formation in this scenario. In the next section, we discuss Ωb − ΩDM coincidence
based on the Affleck-Dine mechanism and the subsequent L-ball formation. In section III,
we show that axinos in our scenario become warm dark matter suitable for the solution of
the missing satellite problem and the cusp problem. In the final section, we give concluding
remarks.
II. Ωb − ΩDM COINCIDENCE FROM AFFLECK-DINE LEPTOGENESIS
We now discuss baryogenesis via Affleck-Dine leptogenesis and dark matter production
from Q-ball decays within the framework of gravity mediated supersymmetry breaking.
A. Lepton asymmetry
The potential of the AD flat direction field is, in general, lifted by soft supersymmetric
(SUSY) breaking terms and non-renormalizable terms [26, 27]. The full potential of the AD
field is given by
V (φ) =
1 +K ln
− c1H2
c2H + Am3/2
nMn−3
+H.c.
|φ|2n−2
M2n−6
. (1)
Here, mφ is the soft SUSY breaking scalar mass for the AD field with radiative correction
K ln |φ|2. A flat direction dependent constant, K, takes values from −0.01 to −0.1 [28]. Λ
denotes a renormalization scale and −c1H2 represents the negative mass squared induced by
the SUSY breaking effect which comes from the energy density of the inflaton, with an order
unity coefficient c1 > 0 [27]. λ is the coupling of a nonrenormalizable term and M is some
large scale acting as its cut-off. Terms proportional to A and c2 are the A-terms coming
from the low energy SUSY breaking and the inflaton-induced SUSY breaking, respectively,
where m3/2 denotes the gravitino mass. Here, we omitted possible terms which may appear
by thermal effects [29, 30]. These terms are negligible as long as we consider a sufficient
low reheating temperature after inflation, as we will. Moreover, the model would face with
“gravitino problem” [31], if the reheating temperature after inflation is so high that these
thermal effect become effective, unless gravitino is LSP [9].
The charge number density for the AD field φ is given by nq = iq(φ̇
∗φ− φ∗φ̇) where q is
the baryonic (or leptonic) charge for the AD field. By use of the equation of motion of the
AD field, the charge density can be rewritten as
nq(t) ≃
a(t)3
dt′a(t′)3
2qλm3/2
Im(Aφn), (2)
with a(t) being the scale factor. When the AD field starts to oscillate around the origin, the
charge number density is induced by the relative phase between A-terms. By taking into
account s = 4π2g∗T
3/90, the charge to entropy ratio after reheating is estimated as
4M2PH
q|A|λm3/2
TR|φos|n
sin δ. (3)
Here, MP ≃ 2.4 × 1018 GeV is the reduced Planck mass, tos is the time of the start of
the oscillation and sin δ is the effective CP phase. In case that thermal corrections are
ineffective, Hos ≃ mφ, which yields
|φos| ≃
)1/(n−2)
. (4)
From now on, as a concrete example, we consider a LLē direction of n = 6 as the AD field
for our scenario. Since this is a pure leptonic direction, the lepton asymmetry generated by
the Affleck-Dine mechanism can be estimated as
≃ 1× 10−10 q|A| sin δ
( m3/2
100GeV
103GeV
)3/2(
100GeV
. (5)
B. Baryon asymmetry and LSP production from Q-balls
The produced lepton asymmetry is not directly released to thermal bath. Instead, L-balls
are formed due to the instability and almost all produced lepton numbers are absorbed into
L-balls [3].
First of all, we briefly summarize relevant properties of Q-balls in gravity mediated SUSY
breaking models. The radius of a Q-ball, R, is estimated as R2 ≃ 2/(|K|m2φ) [5]. Numerical
calculations provide a fitting formula for the Q-ball charge
Q ≃ β̄
|φos|
ǫ for ǫ & ǫc
ǫc for ǫ < ǫc
, (6)
≃ 2q|A|
sin δ (7)
where ǫc ≃ 10−2 and β̄ = 6× 10−3 [3]. The Q-ball charge can be evaluated as
Q ∼ 2× 1020
4× 10−1
)3/2(
, (8)
where we assumed ǫ > ǫc because it looks to be more natural than the other which can be
realized only for an accidental small sin δ . Furthermore, if ǫ < ǫc, additional “unnatural”
parameters are required for our scenario, as we will show.
A part of the charge of a Q-ball can evaporate by the interaction with particles in the
thermal bath. The evaporation of charge of Q-ball is done by the evaporation with the rate
Γevap ≡
= −4πRQDevneq ≃ −4πR2QDevµQT 2, (9)
with Dev . 1 and by the diffusion with the rate
Γdiff ≡
= −4πkRQDdiffneq ≃ −4πkRQDdiffµQT 2, (10)
where µQ is the chemical potential of Q-balls and the numerical constant k is very close to
unity so that we will drop it hereafter. Ddiff ≈ a/T is a diffusion constant [32] and a is a
particle dependent coefficient given by [33, 34]
4 for squark
6 for quark
100 for left− handed (s)lepton
380 for right− handed (s)lepton
. (11)
Here, we see that both the evaporation and the diffusion are efficient for low temperature,
from Eqs. (9) and (10) with the relation between the cosmic time and the temperature:
for T & TR
for T < TR
. (12)
Moreover, by comparing Γdiff and Γevap,
Γdiff
Γevap
, (13)
we can find that for low temperature
T . a
mφ ∼ 10mφ
, (14)
the diffusion is more crucial for estimation of the evaporated charge from Q-ball. Equation
(10) is rewritten as
≃ −4πRQDdiffµQT 2
. (15)
Integrating Eq. (15) from mφ, because the evaporation from Q-ball is suppressed by the
Boltzmann factor, we can estimate the total evaporated charge as
∆Q ≃ 32kRQ
T 2RMP
∼ 3.2× 2.4√
× 1019
for mφ & TR,
∆Q ∼ 3.6× 2.4√
× 1019
for mφ < TR. By taking Eq. (8) into account, we obtain
× 10−1
4× 10−1
)( mφ
for mφ < TR
1 and find that about 10% of Q-ball charge would be evaporated. Here, one
can see why the case of ǫ < ǫc is irrelevant for us. If ǫ < ǫc ≃ 10−2, ǫ is replaced with ǫc
in Eq. (18). Then these Q-balls cannot survive the evaporation unless the AD field mass is
extremely small as mφ = O(10) GeV or (λ1/3MP/M)3/2 ≪ 1 .
The evaporated charges are released into the thermal bath so that a part of them is
transformed into baryonic charges through the sphaleron effects [35]. Then, the resultant
baryon asymmetry is given as
∆Q× 10−30
103GeV
100GeV
≃ 10−10
formφ & TR
formφ < TR
. (19)
Interestingly, the baryon asymmetry does not depend on the effective CP phase sin δ unlike
usual Affleck-Dine baryogenesis, because the CP phase dependences in both lepton asym-
metry nL/s and the charge of Q-ball Q cancel each other. In addition, for mφ < TR, the
baryon asymmetry basically depends on only one free parameter, the reheating temperature
TR, because other parameters are not free but known in a sense.
When Q-balls decay, the supersymmetric particles are released from them. Since the
Q-ball consists of scalar leptons, the number of the produced supersymmetric particles is
given by
YNLSP = NQ
= 2× 10−9
4× 10−1
)1/2(
, (20)
where NQ is the number of produced NLSP particles per one leptonic charge.
Such produced NLSPs decay into axino LSP with a typical lifetime of O(0.1− 1) second.
Thus, NLSPs produced by the Q-ball decay become a source of axino production. Of course,
like gravitinos, axinos can also be produced by other processes such as thermal processes
(TP), namely the scatterings and decays in the thermal bath, and non-thermal processes
(NTP), say the late decay of NLSPs produced thermally. The relevant Boltzmann equations
can be written as
ṅNLSP + 3HnNLSP = −〈σv〉(n2NLSP − n
2) + γQ−ball − ΓNLSPnNLSP, (21)
ṅã + 3Hnã = 〈σv(i+ j → ã+ ...)〉ijninj + 〈σv(i → ã+ ...)〉ini + ΓNLSPnNLSP, (22)
1 For mφ & TR, the result is of the same magnitude but with different dependence on mφ and TR.
where γQ−ball denotes the contribution to NLSP production by Q-balls decay, 〈σv〉ij and
〈σv〉i are the scattering cross section and the decay rate for the thermal production of
axinos, and ΓNLSP is the decay rate of the NLSP. The total NLSP abundance, before its
decay, is given by
YNLSP = NQ
+ Y TPNLSP, (23)
where NQnL/s denotes the NLSP produced by L-ball decay and Y
NLSP is the abundance of
NLSP produced thermally and given by
Y TPNLSP ≃
T=mNLSP
mNLSP/Tf
〈σv〉ann
. (24)
Here 〈σv〉ann is the annihilation cross section and Tf ∼ mNLSP/20 is the freeze-out temper-
ature. The resultant total axino abundance is expressed as
Yã = Y
ã + Y
ã . (25)
Y NTPã = YNLSP = NQ
+ Y TPNLSP (26)
is the nonthermally produced axino through the NLSP decay and Y TPã denotes the axi-
nos produced by thermal processes. For nonthermally produced axinos, while the NLSP
abundance produced by L-ball decay is
= 2× 10−9
2× 10−9
, (27)
the typical value of Y TPNLSP is given by
Y TPNLSP ≃ 10−11
100GeV
mNLSP
10−10GeV−2
〈σv〉ann
. (28)
Thus, nonthermal production of axinos due to the thermal relic NLSPs decay, Y TPNLSP, can
be negligible compared to that from Q-ball produced NLSPs, NQnL/s. On the other hand,
axino production by thermal processes is dominated by scattering processes for the case
that the reheating temperature is larger than the masses of neutralinos and gluinos. In this
case, the abundance of such axinos is proportional to the reheating temperature TR and the
inverse square of Peccei-Quinn (PQ) scale fa and given by [18]
Y TPã ≃ 10−8
1011GeV
, (29)
where N is the number of vacua and N = 1(6) for the KSVZ (DFSZ) model [36, 37]. Thus,
for TR ≃ 1 TeV, if fa/N & several ×1011 GeV [38], Y TPã is subdominant compared with
Y NTPã ≃ NQnL/s.
If this is the case, the energy density of axino is given by ρã = mãnNLSP due to the
R-parity conservation. Recalling
≃ 3.9× 10−10
GeV, (30)
the density parameter of axinos is expressed as
( mã
0.2GeV
2× 10−9
. (31)
Thus, axinos with the sub-GeV mass can be dark matter in our scenario. Now, one can see
that the Ωb and ΩDM is related through the lepton asymmetry. In fact, from Eqs. (19) and
(31), we obtain a relation between the abundances of dark matter and baryon asymmetry,
1× 10−1
mã/mp
, (32)
where mp(≃ 1 GeV) is the mass of proton. One may find the similar relation in the case
of baryonic Q-ball (B-ball) [14]. The difference between the case of B-ball and L-ball is
that the required mass of LSP from L-ball can be an order of magnitude smaller than that
in B-ball where the mass of LSP dark mater must be ≃ 1 GeV, mainly because a part of
lepton asymmetry produced by the Affleck-Dine mechanism, that is, only evaporated charges
∆Q/Q are converted to baryon asymmetry so that the number density of NLSPs produced
by the L-ball decay become larger for a fixed baryon asymmetry. As shown in the next
section, such a difference of axino masses is crucial for solving the missing satellite problem
and the cusp problem.
Equation (31) with TR ≃ 1 TeV to explain the observed baryon asymmetry yields a quite
natural scale of the AD field mass
mφ ≃ 1TeV
4× 10−1
( mã
0.2GeV
. (33)
As mentioned above, in our scenario, NLSP decays into axino at late time. Such late decay
is potentially constrained by BBN. The lifetime of NLSP is given as
τχ ≡ τ(χ → ã+ γ) = 0.33sec
C2aY Y Z
1/128
1011GeV
102GeV
for the case that the lightest neutralino χ is NLSP in [18]. Here, CaY Y is the axion model
dependent coupling coefficient between axion multiplet and U(1)Y gauge field, Z11 denotes
the fraction of b-ino component in the lightest neutralino. According to Ref. [18], we can
summerize the constraints as follows. First of all, for τχ ≤ 0.1 sec., there is no constraint.
The corresponding mass of the NLSP neutralino is
mχ = 320GeV
0.1sec
)1/3(
C2aY YZ
)1/3(
fa/N√
10× 1011GeV
from Eq. (34). Thus, if mχ & 320 GeV, this model is free from problems by the late decay
of NLSP. This lower bound is a bit stringent than that in [18], because we need to take the
PQ scale somewhat larger as we mentioned. On the other hand, for 0.1 sec. < τχ < 1 sec.,
the lower bound on axino mass exists and can be roughly expressed as
0.1GeV
102GeV
(CaY Y Z11)
1011GeV
+ 6 ≃ −4
0.33sec.
by reading Fig. 4 in Ref. [18]. In this case, axino must be heavier than a few hundred MeV.
For τχ ≃ 1 sec., the corresponding mass of the NLSP neutralino and the lower bound of
axino mass is given by
mχ ≃ 150GeV,
mã & 320MeV. (37)
III. A SOLUTION TO THE MISSING SATELLITE PROBLEM AND THE CUSP
PROBLEM
An interesting consequence of such light axinos is their large velocity dispersion. There-
fore, they can potentially solve the missing satellite problem and the cusp problem as stated
in the introduction. For the scenario of dark matter particle produced by the late decay of
long-lived particle, it is shown that the missing satellite problem and the cusp problem can
be solved simultaneously if the lifetime of long-lived particle and the ratio of mass between
dark matter particle and the mother particle satisfy the following relation [39]:
6.3× 102mã
sec. . τχ .
1.0× 103mã
sec., (38)
where we identified dark matter with axino (LSP) and the long-lived particle with the lightest
neutralino (NLSP), respectively.
Combining Eq. (34) with Eq. (38), we have the following relation between the masses of
the axino (LSP) and the lightest neutralino (NLSP),
mχ ≃ (0.33− 0.83)
( mã
C2aY Y Z
1/128
1011GeV
GeV. (39)
For mã ≃ 1 GeV in the B-ball case [14], mχ becomes a few GeV with its lifetime τχ ≫ 103
second even if we take fa/N to be several times 10
11 GeV, which is excluded. On the other
hand, for mã = O(0.1) GeV in the L-ball case of this paper, mχ becomes O(100) GeV with
its lifetime τχ . 1 second if we take fa/N to be several times 10
11 GeV. Thus, we find
that axinos in this scenario can solve the missing satellite problem and the cusp problem
simultaneously for natural mass scales with mã = O(0.1) GeV and mχ = O(100) GeV.
IV. CONCLUDING REMARKS
In this paper, we show that Affleck-Dine leptogenesis can explain baryon asymmetry
and dark matter abundance simultaneously and that Ωb −ΩDM coincidence is explained for
sub-GeV mass of the LSP axino. Though the basic idea is the same as Ref. [14], where
B-balls are considered, the mass of LSP axino becomes an order of magnitude smaller in
this scenario. On the other hand, the PQ scale is determined as fa/N = a few ×1011 GeV,
which will be tested in the future if PQ scale can be measured by e.g., the manner proposed
in [40].
The other attractive point is that axinos considered in this paper can potentially solve the
missing satellite problem and the cusp problem simultaneously because they are relatively
light and have large velocity dispersion. We have shown that axions in our scenario can solve
both problems for natural mass scales with mã = O(0.1) GeV and mNLSP = O(100) GeV.
For simplicity, we concentrate on the case of the lightest neutralino NLSP with the mass
to be O(100) GeV. Such neutralinos are detectable in Large Hadron Collider. In addition,
the corresponding lepton asymmetry is almost the maximal value under the assumption
of (M3/λM3P ) ≃ 1. Hence, if this model is the simultaneous answer to both Ωb − ΩDM
coincidence, the missing satellite problem, and the cusp problem, we would discover the
NLSP neutralinos 2 with the mass of O(100) GeV and their decays into photons (and
negligible missing energy) with the lifetime of about (0.1− 1) second.
Acknowledgments
We thank M. Kawasaki and T. Takahashi for useful comments. We are grateful to John
McDonald for pointing out an error in the power of a coefficient in an earlier version. O.S.
would like to thank RESCEU at the University of Tokyo for the hospitality where this work
was initiated. The work of O.S. is supported by the MEC project FPA 2004-02015 and the
Comunidad de Madrid project HEPHACOS (No. P-ESP-00346). M.Y. is supported in part
by JSPS Grant-in-Aid for Scientific Research No. 18740157 and the project of the Research
Institute of Aoyama Gakuin University.
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	Introduction
	b - DM coincidence from Affleck-Dine leptogenesis 
	Lepton asymmetry
	Baryon asymmetry and LSP production from Q-balls
	A solution to the missing satellite problem and the cusp problem
	Concluding remarks
	Acknowledgments
	References

  We show that axinos, which are dominantly generated by the decay of the
next-to-lightest supersymmetric particles produced from the leptonic $Q$-ball
($L$-ball), become warm dark matter suitable for the solution of the missing
satellite problem and the cusp problem. In addition, $\Omega_b - \Omega_{DM}$
coincidence is naturally explained in this scenario.

Introduction
	b - DM coincidence from Affleck-Dine leptogenesis 
	Lepton asymmetry
	Baryon asymmetry and LSP production from Q-balls
	A solution to the missing satellite problem and the cusp problem
	Concluding remarks
	Acknowledgments
	References
