Adiabatic suppression of the axion abundance and isocurvature due to coupling to hidden monopoles
HJE
Adiabatic suppression of the axion abundance and isocurvature due to coupling to hidden monopoles
Masahiro Kawasaki 0 2 3 5
Fuminobu Takahashi 0 2 3 4
Masaki Yamada 0 1 2 4
0 574 Boston Avenue , Medford, MA 02155 , U.S.A
1 Institute of Cosmology, Department of Physics and Astronomy, Tufts University
2 Azaaoba 63, Aramaki, Aoba , Sendai, Miyagi 9808578 , Japan
3 Kavli IPMU (WPI), UTIAS, The University of Tokyo
4 Department of Physics, Tohoku University
5 Institute for Cosmic Ray Research, The University of Tokyo
The string theory predicts many light elds called moduli and axions, which cause a cosmological problem due to the overproduction of their coherent oscillation after in ation. One of the prominent solutions is an adiabatic suppression mechanism, which, however, is nontrivial to achieve in the case of axions because it necessitates a large e ective mass term which decreases as a function of time. The purpose of this paper is twofold. First, we provide an analytic method to calculate the cosmological abundance of coherent oscillation in a general situation under the adiabatic suppression mechanism. Secondly, we apply our method to some concrete examples, including the one where a string axion acquires a large e ective mass due to the Witten e ect in the presence of hidden monopoles.
Cosmology of Theories beyond the SM; Beyond Standard Model; Solitons

2
3
5
6
1 Introduction
Overproduction problem of light scalar elds
2.1
2.2
Moduli problem
Axion overproduction problem
Adiabatic suppression mechanism
3.1
Calculation of adiabatic invariant
3.2
Examples
3.1.1
3.1.2
3.2.1
3.2.2
3.2.3
Case with constant parameters
Case with timedependent parameters
Large Hubbleinduced mass
Application to an axion model
Application to a generic model
4
QCD axion and adiabatic suppression mechanism
4.1
4.2
4.3
4.4
Axion dynamics with monopole DM
Axion abundance
Isocurvature problem
Constraints on monopole DM
String axion with monopole annihilation
Discussion and conclusions A The Witten e ect on the QCD axion 1 3
many light elds called moduli and axions [1, 2]. They acquire a mass from supersymmetry
(SUSY) breaking e ects and other nonperturbative e ects. Since their interactions with
the SM particles are typically suppressed by the Planck scale, those light elds are
longlived and play a major role in cosmology. In fact, they are known to cause a catastrophic
problem for the evolution of the Universe.
During in ation, those light elds may be displaced from the lowenergy minima, since
This is the notorious cosmological moduli problem.1
A novel solution to the problem was suggested by Linde about two decades ago [22].
If the light
elds obtain a timedependent e ective mass much larger than the Hubble
parameter, they may follow the slowlymoving potential minimum adiabatically. As a
result, the oscillation energy is suppressed when the Hubble parameter becomes comparable
to their lowenergy mass.2 In fact, however, the suppression of the moduli abundance is
not so e cient as originally expected [29, 30], since some amount of moduli oscillations is
necessarily induced after in ation. This is because the in aton is lighter than the Hubble
parameter for slowroll in ation and later becomes much heavier than the Hubble parameter
and the moduli mass. The oscillation of the in aton becomes relevant for the moduli
dynamics especially when their mass scales become comparable. Also, we note that it is
nontrivial to realize the adiabatic suppression mechanism for axions, because the axion
potential is protected by a shift symmetry and is induced from nonperturbative e ects.
Recently, we found in ref. [31] that the adiabatic suppression mechanism works for
axions coupled to a hidden U(
1
)H gauge symmetry with a hidden monopole. In the presence
of hidden monopoles, the theta parameter in the hidden U(
1
)H has a physical e ect, known
as the Witten e ect, where the monopole acquires a hidden electric charge proportional to
the theta parameter [32]. Here, when the PecceiQuinn (PQ) symmetry is also anomalous
under the hidden U(
1
)H symmetry, the theta parameter of the hidden U(
1
)H is promoted to
the dynamical axion eld [33]. This implies that the hidden monopole acquires an electric
charge related to the axion
eld value. Since the nonzero value of electric charge is not
favored to minimize the total energy, the axion starts to cancel the theta parameter of the
hidden U(
1
)H to make the hidden electric charge of monopole absent. This means that the
axion acquires an e ective mass via the Witten e ect and its abundance can be suppressed
by the adiabatic suppression mechanism. Also, the axion isocurvature can be suppressed
by the Witten e ect [31, 34]. See e.g. refs. [35{49] for other scenarios to suppress the axion
isocurvature.
The purpose of the present paper is twofold. First, we provide an analytic method
to calculate the abundance of light elds under the adiabatic suppression mechanism. We
1Another important prediction of the string theory in cosmology is the string landscape. There are
an exponentially large number of vacua in the
eld space of moduli and string axions and the anthropic
selection of vacua may explain the
ne tuning of the cosmological constant. The slowroll in ation in the
2Thermal in ation is another possible solution to the cosmological moduli problem, where the moduli
density is diluted by a miniin ation around the TeV scale [23{26]. See refs. [27, 28] for recent works.
{ 2 {
calculate an approximately conserved adiabatic invariant, which represents the comoving
number density of light elds, and show that the adiabatic invariant is nonzero but
exponentially suppressed if we neglect the initial abundance which may be induced by the e ect
of the in aton oscillation. Our result is consistent with ref. [22], but our method can be
applied to a more generic situation. Secondly, we apply our calculation to some concrete
models, including the one with the Witten e ect on the string axion.
This paper is organized as follows. In the next section, we brie y review the
moduli problem and the axion overproduction problem. Then in section 3, we calculate the
abundance of coherent oscillation of scalar eld under the adiabatic suppression
mechanism. Then in the subsequent sections, we apply the calculation to scenarios where an
Universe, and the subsequent evolution of the Universe is inconsistent with that of our
Universe. In this section, we brie y explain the overproduction problem of moduli and axions.
2.1
Moduli problem
The string theory predicts many light singlet scalar elds in low energy e ective theory.
Its potential is generically written as
moduli start to oscillate around the lowenergy vacuum at H(t) ' m . The amplitude of
its oscillation is expected to be of order the Planck scale, so that the resulting abundance
moduli is stabilized by SUSY breaking e ects in the Kahler potential, their axionic partners
remain much lighter than the moduli. In this case, the moduli decay into a pair of axions
with a large branching ratio, leading to the overproduction of axions [51{53]. Similarly,
the moduli generically decay into a pair of gravitinos with a sizable branching fraction if
kinematically accessible [54{57]. Thus produced gravitinos may spoil the BBN or produce
too many lightest SUSY particles.
2.2
Axion overproduction problem
The potential of axion can be similarly written as follows around the lowenergy vacuum:
where a is the axion eld and ma is the axion mass at the potential minimum.
For the QCD axion, its mass is given as
V (a)
1
2
m2aa2;
majT =0 ' (1 + z)2 fa
;
z
m f
(2.2)
(2.3)
(2.4)
(2.5)
HJEP01(28)53
at the zero temperature, where fa is the axion decay constant, z (' 0:56) is the ratio of
uand dquark masses, m
(' 140 MeV) is the pion mass, and f (' 130 MeV) is the pion
decay constant. In a nite temperature plasma with T
QCD, the axion mass depends
on the temperature as
m2a(T ) ' cT
4
QCD
f 2
a
T
QCD
n
;
where cT ' 1:68
10 7, n = 6:68, and
QCD = 400 MeV [58, 59].
Let us consider a case in which the PQ symmetry is broken before in ation. In this case,
the axion stays at a certain phase during in ation because it is massless and is a ected
by the Hubble friction e ect. Since the coherent length is stretched by the exponential
expansion of the Universe during in ation, the axion VEV is spatially homogeneous. After
in ation ends and before the QCD phase transition, the axion mass is much smaller than
the Hubble parameter, so that it continues to stay at a certain VEV due to the Hubble
friction e ect. Then at a time around the QCD phase transition, the axion mass becomes
larger than the Hubble parameter and starts to oscillate around the low energy vacuum,
at which the axion VEV cancels the undesirable strong CP phase [3{5]. The temperature
{ 4 {
Tosc;0 '
QCD
' 1:2 GeV
90cT MP2l
2g (Tosc;0)fa2
fa
1018 GeV) is the reduced Planck mass. The parameter g (T ) is the
e ective number of relativistic particles in the plasma and we use g (Tosc;0)
energy density of the axion oscillation decreases with time just like matter after the axion
mass becomes constant, and hence it is a good candidate for cold DM. Neglecting the
anharmonic e ect [60], one obtains the axion abundance as [61]
of the Universe at the onset of oscillation is given as
(2.6)
(2.7)
(2.8)
(2.9)
ah2
' 0:2 i2ni
fa
where h is the Hubble parameter in units of 100 km=s=Mpc and ini (
< ini
) is the
initial misalignment angle. The observed DM abundance,
when the axion decay constant is given by
DMh2 ' 0:12, can be explained
1
2
{ 5 {
;
;
confronts the overproduction problem of the axion energy density.
3
Adiabatic suppression mechanism
The moduli and axion overproduction problems can be avoided when the VEV of these
elds changes adiabatically at the onset of oscillation. This can be achieved when they have
a large timedependent mass term in addition to the lowenergy mass term [22]. In this
section, we explain the adiabatic suppression mechanism and show that it results in the
exponential suppression of an adiabatic invariant, which describes the comoving number
density of particles.
Suppose that a light eld (moduli or axion) has a timedependent mass m~(t). The
potential of the light eld
is then given by
1
2
0 is of order the Planck scale for the case of moduli or of order the axion decay
constant for the case of axion.
We denote the mass of light eld as m (t), which may
depend on time in some cases. Note that the model considered in ref. [22] corresponds to
the case with m (t) = const: (
of motion is then given by
m ) and m~2(t) = C2H2(t) (C: constant). The equation
+ 3H _ =
m2 (t)
= 0. Then, at the time around m~(t) ' m (t), the minimum
0m~ 2(t)=(m (t)2 + m~2(t)). Here, the time scale of v(t)
is of order the Hubble parameter (v_ =v
H(t)) while that of the oscillation is of order
almost no oscillation is induced through this dynamics.
Note that we should take a particular care of the origin of the e ective mass term.
Suppose that, for example, the e ective mass term comes from a coupling between the
moduli eld and the in aton. During in ation, the in aton mass is lighter than the Hubble
parameter for successful slowroll in ation. After in ation, the in aton oscillates about
its potential minimum and its mass becomes larger than the Hubble parameter. This
implies that the e ective modulus mass becomes necessarily comparable to the in aton
mass sometime after in ation. Then, the modulus dynamics can be signi cantly a ected
by the in aton oscillations, leading to a production of the modulus eld after in ation [29].
The model used in ref. [22] confronts this issue. On the other hand, the examples we use
in section 4 and 5 are free from this issue, since the e ective axion mass appears after
in ation when the in aton mass is already much heavier than the e ective axion mass.
Below, we evaluate the abundance of the light eld by calculating the time evolution
of an adiabatic invariant. Our calculation not only reproduces the result of ref. [22] but
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
also is applicable to more generic cases.
3.1
Calculation of adiabatic invariant
We rewrite the equation of motion as
where
' + m2(t) ('
v(t)) = 0;
'
m2(t)
v(t)
3p=2
t
t0
m2 (t) +
m~ 2(t)
m2(t) 0
3
2
;
t
t0
1
p
3p=2
3
2
:
H2(t) + m~2(t)
The dynamics of such a eld is analogous to a motion of a particle with the Hamiltonian,
H =
2
1 2 +
1
2
m2(t) ('
v(t))2 :
{ 6 {
where
is a canonical momentum of the particle '. This is the Hamiltonian of a harmonic
oscillator with two timedependent parameters.
We can de ne an adiabatic invariant for a oneparticle system with a compact
trajectory. We assume that the typical time scale of its oscillation m(t) is much larger than H(t)
and those of the two parameters ( m_=m) 1 and (v_ =v) 1 are of order H(t). In this case, the
adiabatic invariant approximately conserves. It is explicitly written as
I =
where the integral is taken over the interval of one periodic motion assuming constant m
and v. In an oscillating homogeneous scalar
eld, the adiabatic invariant can be
interpreted as the comoving number density of particles, and its approximate conservation law
means that almost no particles are produced due to slowly changing parameters. However,
exponentially suppressed but nonzero amount of particles are produced throughout the
dynamics and we can calculate it in the following way. We follow and generalize a method
explained in ref. [62].
3.1.1
Case with constant parameters
Note that since we de ne ' as eq. (3.4), the adiabatic invariant is proportional to the
comoving number density of the eld .
Let us take the action as a function of ' and t such as
Noting that L =
H, where H is the Hamiltonian, we obtain
S('; t) =
Z ';t
Ldt0:
dS =
d'
Hdt:
First, let us consider a trivial case where the parameters m and v are constant in time.
The result is of course given by
where E is the energy,
= '_ is the canonical momentum, and (t) is given by mt. The
adiabatic invariant, which is exactly conserved in this case, is calculated as
r 2E
p
'(t) =
m2 sin (t) + v;
(t) =
2E cos (t);
I = E=m:
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
(3.14)
(3.15)
Note that should be rewritten in terms of ' by using eqs. (3.9) and (3.10). Since the energy is conserved for the case of constant m and v, it is given by
S('; t) = S0('; E)
Et;
S0('; E) =
('0; E)d'0;
Z
{ 7 {
where we explicitly write E dependence of S0 for later convenience. Here, we can take
S0('; E) as a function of I because E = mI. In the case of the harmonic oscillator, S0 is
explicitly calculated as
S0('; I) = p
2mI
Z r
1
('0
v)2d'0 = I
;
(3.16)
+
sin 2
2
=
=
I_ =
_ =
= m;
m
2I
H
';I
';I
{ 8 {
which imply that I = const: and
= mt as expected.
3.1.2
Case with timedependent parameters
Next, we consider a case where the parameters m and v depend on time. We perform a
canonical transformation of this system via the mother function of S0 given in eq. (3.16),
where the constant parameters m and v are replaced by the timedependent ones. Since the
mother function depends on time via the parameters m(t) and v(t), the new Hamiltonian
=
I
2m
p
sin 2 ;
2mI cos ;
where (@f (x; y)=@x)y is the partial derivative of f (x; y) with respect to x while y xed.
where
= ('; I) is given by the inverse of eq. (3.9).
Let us take S0 as a mother function of a canonical transformation:
where H0 is a new Hamiltonian. The last terms comes from the Legendre transformation
that change the dependence of the variable to I. Then the adiabatic invariant I becomes
a new canonical momentum and satis es
dt
dS0 ('; I) =
d'
dt
I
d
dt
H
is a new canonical variable. It coincides with the one used in eq. (3.9),
as expected. Since the mother function S0 is independent of time, the new Hamiltonian
is identi ed with the old one, i.e., H = E(I). The Hamilton equations of motion are thus
given by
(3.18)
(3.19)
(3.20)
(3.21)
(3.22)
(3.23)
(3.24)
The new canonical variables I and
are determined by eqs. (3.18) and (3.19). The
equations of motion are now given by
_ =
v_ ;
v_ ;
where (@E=@I)m;v (= m) is the frequency of the harmonic oscillator for the case of
timeindependent parameters. These are rewritten as
I_(t) =
_
' m;
I
m_
m
p
cos (
2
)
2mIv_ sin ;
where we neglect the second and third terms in eq. (3.26) because the inverse of the
oscillation time scale m is much larger than that of m_=m and v_ =v.
On some occasions, we are interested in the case where the adiabatic invariant I is
absent initially. Then the second term of the righthand side in eq. (3.27) mainly contributes
to the growth of I. Even if I is initially nonzero, we can neglect the rst term in the
righthand side for the case of I
(m= m_)2(v_ =v)2mv2
mv2. For example, suppose that
oscillates at t = t0 with an amplitude of 0
smaller than mv2
following integral:
' m 20(t=t0)3p for t
. Then we have I0 ' m 20
, which is much
t0. Therefore, what we need to calculate is the
pI(
1
)
pI(t0) =
Z 1
t0
dt
r m
2
v_ sin ;
where t0 is an arbitrary time much before m~(t) ' m (t). We need to specify a model to
3.2
3.2.1
calculate this integral.
Examples
Large Hubbleinduced mass
Here let us check that our method reproduces the result in ref. [22], by taking m
= const:
and m~(t) = CH(t) with C
1. This can be realized when we consider a modulus that
has interaction with an in aton with a cuto scale below the Planck scale. In this case, we
can take the limit of t0 ! 0 because the axion initially stays at a certain VEV due to the
Hubble friction e ect (i.e., I(t0) = 0 for t0 ! 0).
We decompose the oscillation factor such as sin
= (ei
e i )=2i, and calculate each
term separately. The integrand has poles and branch cuts in the complex tplane, so that we
replace the integration contour in the complex plane as shown in the left panel in gure 1:
i Z
p
v_ pmei dt =
i Z
p
2 2 C2+C3+C4+C5+C6
v_ pmei dt:
The integrand is given as
v_ pmei =
V
C~3=2
t3p=2 3=2
1 + zp2to2le
7=4
" 3p
2
+
3p
2
2
t
2 #
{ 9 {
tpole
0
C5
C4
C3
C2
C6
C1
Re(t)
Im(t)
0
tpole
C5
C4
C1
C3
↵
C2
C6
Re(t)
HJEP01(28)53
ma / tn=2 (right panel). The red lines represent branch cuts and the red points are poles.
where we de ne
The parameter is calculated from
On the imaginary axis of t, it is given by
(t) =
8
>
>
>
<
>
>
Z
>: C2+C3+C4
Z
C2
zpole
m2 + C~2=z02 for z < zpole
dz0
q
m2
C~2=z02 for z > zpole
where t = iz and tpole
izpole is the pole location in the complex tplane and z0
t0. The
integral for the contour C2 can be calculated as
zpole
~
C
V
C~=m
The second terms in eq. (3.36) are calculated as
i
Z z
z0
Z z
zpole
dz0
q
m2 +C~2=z02 =
C
~q
1 z2=zp2ole
dz0
q
m2
C~2=z02 = iC~qz2=zp2ole
~
C
2
C4 m(t0)dt0 = 0 because m(t) = 0 at t = tpole.
(3.33)
(3.34)
(3.35)
(3.36)
(3.37)
(3.38)
(3.40)
Now we can calculate eq. (3.30). First, it is easy to see that the integral of C2 in
eq. (3.30) is proportional to t03p=2 1=2, so that it can be neglected in the limit of t0 ! 0
for p > 1=3. Since we are interested in the radiation dominated era, where p = 1=2, this
condition is satis ed.
Second, the integral of the contour C3 has an oscillation term eq. (3.39) from ei . The
integrant v_
time scale of this oscillation term (/ C~) is mush larger than that of the prefactor in the
pm, so that the integral of C3 contour is suppressed by the oscillation term.
However, when t is su ciently close to the pole (jt
term becomes constant, and the integral gives a large contribution.
izpolej . zpole=C~2=3), the oscillation
Finally, the integral of the contour C5 and C6 has a damping term eq. (3.40) from ei ,
so that the integral of RC5+C6 ei is highly suppressed by the additional exponential factor.
However, when t is su ciently close to the pole (jt
term becomes constant, and the integral gives a large contribution.
izpolej . zpole=C~2=3), the damping
Therefore, the integral at the vicinity of the pole gives the main contribution of
eq. (3.30). We should calculate
Z
jt izpolej<zpole=C~2=3
v_ pmei dt
2V
' C~3=2
ei (t=izpole) (izpole)3p=2 3=2 Z
1 +
t
where the integral should be taken on the contour C3 + C4 + C5. The integral can be
calculated as
We can calculate R v_ pme i dt in the same way. Combining these results, we obtain
the adiabatic invariant such as
Z
jt izpolej<zpole=C~2=3
=
3
4 zpole 7=4
2
1 +
t
zpole
where we use C~ ' pC. All parameter dependences are consistent with the one derived by
Linde. Thus we can solve the moduli problem when it has an e ective mass term CH(t)
with C
C~3p+1
I ' m3p 1 0 0
2t 3pe
where we assume I(t0) = 0 and omit O(
1
) factors. Using I ' m '2 for CH
obtain
t0
t
3p=2
' ' p3=2C(3p+1)=2 0
3p=2
e
pC=2;
(3.44)
m , we
(3.45)
To sum up, the exponential factor comes from eq. (3.38) and the other factors should
be calculated around the vicinity of the pole t = izpole. These facts allow us to easily
estimate the resulting adiabatic invariant.
Finally, we rewrite the result in terms of the number density to entropy density ratio
for later convenience:
n
s
m
2=2
s
=
45
where we use p = 1=2, and g s and g are e ective relativistic degrees of freedom for entropy
and energy density, respectively.
3.2.2
Application to an axion model
Now we calculate the abundance in the case where m
depends on time such as the case
of QCD axion, where m (t)
Atpn=2 [see eq. (2.5)]. In this subsection, we assume m~(t) =
CH(t), which is considered in refs. [40, 48]. This can be realized when we introduce a term
L
c2RRMp2l cos
a
fa
R ;
where R is the Ricci scalar. In the radiation dominated era, this term gives m~(t) =
CH(t) with C = 1:6cR sMpl=fa, where s is the nestructure constant of SU(3)c gauge
In this case, i (t = ei zpole) is calculated as
The integral of eq. (3.30) is again dominated by the contribution in the vicinity of the
ei zpole < zpole=(C~p2 + pn)2=3. Then we estimate the adiabatic invariant such
like
interaction.
pole: t
as
As shown in the right panel in gure 1, the pole location is given by t = ei zpole, where
Therefore, an e ective mass of m~(t) = CH(t) suppresses the axion abundance e ciently
when C
(3.46)
(3.47)
Finally, we calculate the abundance in the case of m (t) = Atpn=2 and m~(t) = C0t d, where
d is a constant parameter. The following calculation should reproduce the previous result
if we take d = 1 and C0 = Cp.
In this case, m2(t) is given by
m2(t) = A2tpn +
H2(t) + C02t 2d:
We neglect the H2(t) term because we are interested in the change of adiabatic invariant
H2(t). Note that t0 cannot be taken to be 0 because
C02t 2d is larger than H2(t) only after a certain time. Thus we take, say, t0
zpole=10,
though our result is independent of this value for t0
zpole. Here, zpole is given by
3
2
3
2
1
p
A
=
2d + pn
:
zpole
C0 2=(2d+pn)
:
First, let us calculate the imaginary part of (tpole). The integral on the contours C2
dt0t0 d + ei(1 d) Z 1
dt0pt0 2d
t0pn
#
t0=zpole
The pole location is given by t = ei zpole, where
and C3 is calculated as
Im[ (tpole)] = Im
where in the second line we de ne
Z
C2+C3
' C0zp1olde Im
We can evaluate D numerically for given parameters and we nd that it is almost
independent of t0=zpole for t0=zpole
1 and is about unity for (2d + pn)
1.
The integral of eq. (3.30) is again dominated by the contribution in the vicinity of the
pole:
Then we estimate the adiabatic invariant such as
t
ei zpole < zpole
I ' 2d + pn zp3pole2d+1t0 3p 20e 2Im[ (tpole)]:
(3.55)
(3.56)
(3.57)
(3.58)
(3.59)
(3.60)
(3.61)
It is convenient to rewrite this result as the number density to entropy density ratio:
n
s
m
2=2
s
=
45
where we use p = 1=2, and g s and g are e ective relativistic degrees of freedom for entropy
and energy density, respectively. Note that Im[ (tpole)] / C0zp1olde ' m~ =H(t = zpole)
1.
HJEP01(28)53
4
QCD axion and adiabatic suppression mechanism
The QCD axion potential is protected by the PQ symmetry, and therefore, it is di cult
to give the axion a timedependent e ective mass term, which is needed to realize the
adiabatic suppression mechanism. In this section, we consider the cosmological history of
the QCD axion or string axion that acquires an e ective mass due to the Witten e ect,
using the result given in the previous section.
4.1
Axion dynamics with monopole DM
Here we consider a cosmological history of the axion in the presence of hidden monopoles
of U(
1
)H .
We assume that the PQ symmetry is anomalous under the hidden U(
1
)H ,
which then implies that the axion acquires an e ective mass due to the Witten e ect (see
appendix A). In this section, we consider the case where U(
1
)H is not broken, and monopole
is stable and can be DM [63].3
Suppose that monopoles are produced at a temperature of Tm (Tm
QCD) with
an initial number density of nM (Tm). Its number density decreases with time due to the
cosmic expansion, so that the ratio to the entropy density is constant:
nM = const:
In the presence of monopoles, the Witten e ect gives an e ective mass of the axion
[see eq. (A.19)] [32, 33]:
where
is a constant de ned by eq. (A.17) or eq. (A.18). Here, the axion decay constant
associated with U(
1
)H is denoted as fa;H , which may be di erent from fa by a rational
number because of the di erence of periodicity of the axion (or socalled the domain wall
number).4 Speci cally, they are related to each other by fa;H = (NDW=Na;H )fa, where
NDW and Na;H are the domain wall numbers of the axion in terms of the QCD instanton
3A scenario where 't HooftPolyakov monopole accounts for DM was discussed in refs. [64, 65], where
massive gauge bosons are another component of DM.
nism [66, 67].
4fa;H can be enhanced by many orders of magnitude with respect to fa in the clockwork
mechae ect and the Witten e ect, respectively. This implies that the ratio between the e ective
axion mass and the Hubble parameter increases with time such as
m2a;M(T )
H2(T )
= 2 YM H2fa;H
;
s
which is approximately proportional to T 1 in the radiation dominated era. The ratio
becomes unity at a temperature of T = Tosc;1 and time t = tosc;1, which is given by
Tosc;1 ' YM
' 2
8 MP2l
fa;H
109 GeV
mM
1 TeV
1
M h
2
0:12
fa;H
1016 GeV
1
;
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
(4.9)
where
M is the relic density of monopole:
2
M h ' 3:6 eV
mM YM
:
In the case of the 't HooftPolyakov monopole,
is given by eq. (A.18) (
=
H =(32 2
rcfa;H )), where
H is the
nestructure constant of U(
1
)H and rc is an
electric screening scale of an electrically charged particle.5 We may take rc = mW1 = 1=(ev) '
1= H mM , where mW is the mass of SU(
2
) gauge bosons, e is the hidden gauge coupling
constant, and v is the SU(
2
) breaking scale. Then we obtain
Tosc;1 ' 65 GeV H2
M h
0:12
2
fa;H
In this paper, we require that the ratio becomes unity before the QCD phase transition,
i.e., Tosc;1 & Tosc;0, which condition is rewritten as
fa . 9
for the case of the 't HooftPolyakov monopole. Note that fa may be di erent from fa;H
by a rational factor.
The Witten e ect should be su ciently small so that the axion VEV cancels the
undesired strong CP phase at present. We can check it as follows:
m2a;M
m2a tp
;
where the left hand side is evaluated at the present time tp. Note that
1. Since this
fraction is much smaller than 10 10 at present, the strong CP problem is solved even in
the presence of monopole.
5The monopole may be a fundamental particle as considered in refs. [68{70]. In this case, we should use
eq. (A.17) instead of eq. (A.18), which leads to a larger Witten e ect.
We consider a scenario where the PQ symmetry is broken before in ation. In this case, the
axion stays at a certain phase during and after in ation. After in ation ends and monopoles
are produced, the ratio of eq. (4.3) increases and reaches about unity at T = Tosc;1. Then
the axion starts to oscillate around the minimum (i.e., hai = ), which is determined by
the hidden sector.
The axion starts to oscillate by the Witten e ect before the QCD phase transition when
eq. (4.8) is satis ed. The resulting axion energy density is given by H2(Tosc;1) i2ni;Hfa2;H =2
at the onset of oscillation, where ( ini;H + ) is the initial misalignment angle of axion. This
gives the initial value of adiabatic invariant:
which represents the comoving number density of axion normalized at t = t0. Note that we
include the factor of (tosc;1=t0)3=2 because we normalize the adiabatic invariant at t = t0.
Although the e ective mass of the axion eq. (A.19) decreases with time, its comoving
number density, which is an adiabatic invariant, is approximately conserved after the onset
of oscillation. In particular, the e ect of the QCD instanton around the time of QCD phase
transition does not a ect the number density of axion so much. The resulting amount of
induced axion at the QCD phase transition is given by eq. (3.61) with d = 3p=2:
(4.10)
(4.11)
(4.12)
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
where D (' 1) is given by eq. (3.59) and
2C02
I ' n + 3 zpolet0 3=2 2e 2 C0zp1=o4leD:
0
0 = fa;H
C0 = H(tosc;1)tosc;1
3=4
Here and hereafter we use p = 1=2. The induced
I is smaller than I0 when
2C02zp1=o2le
n + 3
zpole
tosc;1
1=2
e 2 C0zp1=o4le . 1
H2(tosc;1)
$ m2a(tosc;1)
&
1
ln
H2(tosc;1)
m2a(tosc;1)
2(n+3)
;
where we use
= 2 =(n + 3) and assume ini;H
satis ed. However, we should note that we assume H2(tosc;1)
(= O(
1
)). This inequality is usually
m2a(tosc;1) in the above
calculation. Therefore, we conclude that the resulting axion abundance is determined by
I0 when H(tosc;1) & ma(tosc;1).
When the adiabatic suppression mechanism works, the present axion number density
is determined by I0 and is given as
The axion density parameter is thus given by
HJEP01(28)53
na
s
'
'
2
2
H
ah2 ' 2
Since the total abundance should be smaller than the observed DM abundance, we obtain
an upper bound on the axion decay constant:
fa;H . 4
We show contours of the axion and monopole abundance in
gure 2. We take fa =
1
1012 GeV (green curve), 2
1012 GeV (blue curve), and 4
1012 GeV (red curve). We
assume ini;H = 1,
H = 1, and NDW = Na;H . The observed DM abundance can be
explained at the intersection point between the contour curve and the diagonal magenta
dashed line for each value of fa. Note that the adiabatic suppression mechanism works for
the case of Tosc;1 & Tosc;0, so that we should restrict ourselves to the case of
ah2 . 0:22 i2ni;H H
1:3
Na;H
NDW
0:69
This is shown in gure 2 as the unshaded region. The adiabatic suppression mechanism
does not work in the lightgray shaded region, in which case the coherent oscillation of
axion is induced at the QCD phase transition and the resulting axion abundance is given
by the sum of eqs. (2.8) and (4.21).
Finally, we comment on the excited states of the monopole (i.e., JuliaZee dyons). At
the monopole production process, some excited states of a monopole with nonzero
hiddenelectric charges may be produced [71]. This is considered to be the case especially if the
axion
eld takes di erent values over the space, e.g., around the axionic cosmic string.
In the scenario considered in this subsection, however, the e ective theta parameter of
the hidden U(
1
)H (i.e., axion VEV) is constant in the whole Universe at the monopole
production process, so that we expect that only monopoles are produced at that time.
Thus the axion acquires the spaceindependent quadratic mass term.
(4.18)
(4.19)
(4.20)
(4.21)
(4.22)
(4.23)
H = 1, and NDW = Na;H . The diagonal (magenta) dashed line represents the observed DM
abundance. In the lightgray shaded region, the adiabatic suppression does not work and the
results should not be trusted.
4.3
Isocurvature problem
When the PQ symmetry is broken before in ation, the axion acquires quantum
uctuations during in ation [72]. This is quantitatively written in terms of uctuations of initial
misalignment angle
ini;H such as
where Hinf is the Hubble parameter during in ation. This results in isocurvature modes
in density perturbations whose amplitude is given by
ini;H
a
DM
2
Pad
Piso . 0:037;
Hinf ;
2 fa;H
Hinf
fa;H ini;H
2
:
fa;H & 3:4
104Hinf
a
1
DM ini;H
;
Since the CMB temperature anisotropies are predominantly adiabatic, the Planck
observation puts the constraint on the amplitude of isocurvature perturbations:
where the amplitude of adiabatic perturbations is measured as Pad ' 2:2
a result, the axion decay constant should satisfy
10 9 [73]. As
(4.24)
(4.25)
(4.26)
(4.27)
which can be rewritten by eq. (4.21) such as
Hinf . 1:2
1
1010 GeV H2 ini;H
Na;H
NDW
M h
2
0:12
fa;H
One can see that the isocurvature constraint on the in ation scale is greatly relaxed for
given fa and the initial misalignment [31]. For instance, in the ordinary scenario without
the Witten e ect, the upper bound on Hinf is of order 107 GeV for fa = 1012 GeV and
= O(
1
). This is because the axion abundance is suppressed by the Witten e ect and it can
be smaller than the observed DM abundance without netuning the initial misalignment
angle. The hidden monopoles are a prime candidate for the major DM component. Note
that, in this scenario, the perturbation of axion energy density,
a= a, is not suppressed,
and therefore, this e ect cannot be mimicked by making the initial displacement smaller
since it would enhance
a= a.
Constraints on monopole DM
When the hidden U(
1
)H is unbroken, the monopole can be DM [63, 64].6 Since monopoles
interact with hidden photons, they are interacting massive DM. The cross section is
calculated as [65]
T =
2
16
m2M vM4 log 1 +
8
' 0:4 cm2=g mM M
2
m4M v4
2
M M
v
10 km=s
4
mM
1 PeV
3
(4.29)
(4.30)
for monopole DM, where
M
g2=(4 ) (= 1= H ) is the nestructure constant for the
magnetic charge and we take the log factor as
40 in the second line.
There are several constraints on the selfscattering cross section depending on the DM
velocity v. The Bullet cluster gives the upper bound such as
T =mM < 1:25 cm2=g for
v
1000 km=s [74{76]. The most stringent constraint comes from the ellipticity of DM
halo, where observations indicate that DM halos are somewhat elliptical. Since the
selfinteraction of DM results in a spherical DM halo, the cross section is restricted above by
this consideration such as T =mM . 0:1
1 cm2=g for v
The selfscattering of DM is well motivated to solve astrophysical problems:
corecusp problem
and toobigtofail problem [79].
These can be solved when
T =mM
< 0:1
10 cm2=g for v
10
Also, the recent observation
of Abel 3827 indicates
T =mM = 1:5 cm2=g, which also motivates us to consider the
selfinteracting monopole DM [81] (see also ref. [82]). These can be addressed when the
mass of monopole is of order PeV scale for
H = 1.
So far we have not speci ed the nature of the monopole or the mechanism to
generate it. Here we brie y comment on them for the case of 't HooftPolyakov monopole
6When we consider 't HooftPolyakov monopole originated from the SSB of SU(
2
)H to U(
1
)H, there are
massive gauge bosons as well as monopoles. This theory has been investigated in refs. [64, 65], where they
found that the massive gauge bosons are a dominant component of DM.
originated from the spontaneous symmetry breaking (SSB) of SU(
2
)H to U(
1
)H , following
refs. [64, 65], where both monopoles and heavy gauge bosons are stable and are DM
candidates. Monopoles are produced by the KibbleZurek mechanism at the SSB of SU(
2
) gauge
symmetry and their abundance strongly depends on the critical exponents associated with
the phase transition. The produced monopoles experience the Brownian motion due to
collisions with heavygauge bosons while the attractive Coulomb force between the monopoles
and antimonopoles makes them drift towards each other. Eventually the monopoles and
antimonopoles may be captured by the attractive force and annihilate into heavygauge
bosons. The resulting monopole abundance becomes comparable to that of heavy gauge
bosons for the case of the hidden gauge coupling strength
H = O(0:1
1). The total
abundance of the monopoles and heavy gauge bosons can explain the observed DM
abundance for the case where the SSB of SU(
2
)H occurs at the energy scale of order 105 GeV.
This is actually a reasonable parameter that can also address the astrophysical problems
such as the corecusp problem and toobigtofail problem.
5
String axion with monopole annihilation
In this section, we consider the case that the monopoles disappear at a temperature of Tann
after the QCD phase transition but before the BBN epoch (i.e., 1 MeV . Tann <
QCD).
This can be realized by the SSB of U(
1
)H at the temperature of Tann because each monopole
and antimonopole pair is attached by a cosmic string associated with the SSB of U(
1
)H
and annihilate with each other [83]. In this case, the Witten e ect turns o and the axion
becomes massless at Tann. Since there is no monopole in the present Universe,7 we are
interested in the case that the axion is the dominant component of DM.
When we require that the monopoles do not dominate the Universe before they
disappear, we should satisfy
where
M
tot T =Tann
. 1
M
tot T =Tann
M h
2
107
Tann
100 MeV
Tosc;1
Tann
. 1:2
H
fa
1016 GeV
1
;
2
:
=
where we use g
gs and de ne
M h2 via the relation of eq. (4.6). Or, we should satisfy
where we assume 't HooftPolyakov monopole and use
1=( H mM ) [see eq. (A.18)].
On the other hand, we can also consider the case that the monopoles dominate the
Universe before they annihilate. In this case, the annihilation of monopole generates entropy
jwith cuto
with rc =
7We assume that the massive hidden photon decays into the SM particles through a kinetic mixing
between the hidden U(
1
)H and hypercharge soon after the monopole annihilation.
(5.1)
(5.2)
(5.3)
and dilute SM plasma. We de ne the dilution factor
by
sbefore
safter = Max
"
M
When the dilution factor is larger than unity, the axion abundance is diluted by a factor
of
1 at T = Tann.
In the case that monopoles disappear after the QCD phase transition, the adiabatic
suppression mechanism works around the QCD phase transition, so that the axion
abundance is determined by eq. (4.21) with a dilution factor:
Note that in this case, the monopole abundance can be larger than the DM abundance so
that the condition of eq. (4.22) can be avoided. In fact, the axion with a decay constant
of order the GUT scale can be consistent with the observed DM abundance without a
netuning of axion misalignment angle.
The isocurvature constraint is the same as eq. (4.27) with
a =
DM. It can be
rewritten as
Hinf . 3:0
1012 GeV ini;H
fa;H
(5.4)
(5.5)
(5.6)
(5.7)
(5.8)
Finally, let us check that the annihilation products of the monopoles give only a
negligible e ect on cosmology. The annihilation products are massive hidden photons, and its
energy density would dominate the Universe unless it decays into the SM particles. We
assume a nonzero kinetic mixing between the hidden U(
1
)H and hypercharge to make the
massive hidden photon decay into the SM particles. Denoting the kinetic mixing angle as
, one can estimate the lifetime of the hidden photon as
0
10 2 sec
Supernova cooling argument puts a signi cant constraint on the kinetic mixing such as
. 4
10 9 for m 0 . 5
10 GeV [84, 85]. The beam dump experiment E137 puts
a weaker constraint but is relevant for a slightly heavier hidden photon:
m 0 . 4
102 GeV [85, 86]. The lifetime of the hidden photon can be much shorter than
1 sec while satisfying these constraints, so that they can decay well before the BBN epoch
and do not give any signi cant e ect in the subsequent cosmology. Since there should be
an adjoint Higgs eld a in the hidden sector to break SU(
2
)H down to U(
1
)H , the kinetic
. 10 7 for
mixing originates from the following higherdimensional operator [87, 88]:
a
M
Ga F
v
M
G
3 F ;
where Ga and F
are SU(
2
)H and U(
1
)Y gauge elds, respectively. The parameter M is
a cuto scale. This operator leads to the kinetic mixing of v=M , which is of order 10 9 for
v = 105 GeV and M = 1014 GeV.
The adiabatic suppression mechanism is a novel mechanism to solve the moduli
problem [22]. For the mechanism to work, the moduli must have a timedependent mass much
larger than the Hubble parameter. In this case, the moduli follow the timedependent
minimum adiabatically, and thus the resultant oscillation amplitude is exponentially
suppressed if one neglects the initial abundance generated by the e ect of the in aton
oscillations [29, 30].
In this paper, we have provided an analytic method to calculate the resultant adiabatic
invariant which describes the comoving number density of homogeneous scalar
eld in
the adiabatic suppression mechanism. In particular, we have seen that the exponential
suppression comes from a pole of the mass of the scalar eld in the complex plane. The
parameter dependence of our results are consistent with the result of ref. [22], and moreover,
our method can be used to calculate the number density in more generic models.
Then we apply the result to axions. We have considered a model in which the axion
obtains an e ective mass due to the Witten e ect in the presence of monopoles, and as a
result, it starts to oscillate much before the epoch of QCD phase transition. This implies
that the energy density of axion coherent oscillation can be much smaller than the case
without the early oscillation. We have found that the axion energy density can be consistent
with the observed DM abundance even in the case that the axion decay constant is as large
as the GUT scale. Due to the early oscillation, both the axion energy density and its
uctuations are suppressed, so that the isocurvature problem can be ameliorated.
If U(
1
)H symmetry is not broken, the monopole is stable and also contribute to DM.
Since monopoles interact with themselves via U(
1
)H gauge interaction, they have a sizable
velocitydependent selfinteraction cross section. Such a selfinteracting DM may be
observationally preferred since it can relax the tension between the observed DM density pro le
and the prediction of CDM model. Note however that, if stable, the monopole abundance
has to be smaller than or equal to the observed DM density, so that the Witten e ect on
the axion is limited in this case. Thus we also considered the case that U(
1
)H symmetry
is spontaneously broken at an intermediate scale so that the monopoles annihilate due to
the tension of cosmic strings associated with U(
1
)H breaking. In this case, the monopole
abundance in the early Universe is not related to the DM abundance, and so, the Witten
e ect can be much more signi cant. In particular, the axion density can be su ciently
suppressed even if its decay constant is as large as GUT scale.
Acknowledgments
F.T. thanks Ken'ichi Saikawa for discussions.
This work is supported by MEXT
KAKENHI Grant Numbers 15H05889 (M.K. and F.T.), JP15K21733 (F.T.), and JSPS
KAKENHI Grant Numbers 17K05434(M.K.), 17H01131(M.K.), JP17H02875 (F.T.),
JP17H02878(F.T.), JP26247042(F.T.), and JP26287039 (F.T.), JSPS Research Fellowships
for Young Scientists (M.Y.), and World Premier International Research Center Initiative
(WPI Initiative), MEXT, Japan.
In this appendix we explain the Witten e ect on the axion and calculate its e ective mass
in the presence of monopole.
We introduce a hidden Abelian gauge symmetry U(
1
)H with a monopole with mass of
mg and a magnetic charge g. The Lagrangian of gauge elds is written as
where F~
1=2
F
and e is the gauge coupling constant of the hidden gauge theory.
In the presence of monopoles, the Maxwell's equations are given by
2
32 2
F F~ ;
F
+
e
2
F
= jM
= 0;
where jM is a monopole current. In the presence of monopoles, i.e., in the case of jM 6= 0,
there is no electromagnetic potential A
de ned in the whole region. We can de ne it only
in those regions where jM = 0. The topology of these regions is nontrivial and in fact A is
singular in the presence of monopoles. This implies that the term in the Abelian gauge
theory cannot be eliminated by integrating by parts in the presence of monopoles. As we
see below, the term is physical and leads to an e ect known as the Witten e ect [32].
The Gauss's laws are now modi ed as
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
where Ei
F0i and Bi
1=2 ijkF jk. Let us emphasize that E and B are of the hidden
electric and magnetic elds. We rewrite jM0 as
where nM+ (nM ) is the number density of (anti)monopoles. Then, eq. (A.5) implies that
the monopole also carries an electric charge q, which is proportional to :
e
2
r E +
8 2 r ( B) = 0;
r B = jM0 ;
jM0 = g(nM+
nM );
q =
e2g
8 2 :
q
e
+
eg
This implies that the usual charge quantization condition, q=e = n, is extended to
where n is an integer that is nonzero for dyons. Therefore, the monopole is a dyon in a
theory with a nonzero value of , which is known as the Witten e ect. Note that when we
replace
+ 2 , the value of n changes from n to n + 1 due to the Dirac's quantization
condition: g = 4 =e.8 This means that the periodicity of is modi ed in the presence of
monopole (dyon) such as
!
+ 2 and n ! n + 1.
Now we assume that PQ symmetry is anomalous in terms of U(
1
)H and the QCD
axion a couples with U(
1
)H . Thus we promote the theta angle of the hidden U(
1
)H theory
to an axion by the PQ mechanism such as
! a=fa;H
[33]:
L =
2
32 2
a
fa;H
F
F
~ :
where fa;H is the axion decay constant associated with U(
1
)H . Suppose that a and a +
2 nfa;H are physically identical where n is a positive integer. Then, the smallest n is called
the domain wall number, Na;H . Let us rst consider a monopole located at the origin of
a coordinate and calculate the e ect of the axion on the energy density of electromagnetic
eld. The Maxwell's equation of eq. (A.4) can be of course solved such as
HJEP01(28)53
where r^ is a normal vector along the radial direction. Together with eq. (A.5), this implies
that the electric eld is given by
B =
g r^
4 r2
;
E =
eg
fa;H
r^
r2
;
jw=o cuto
VM =
fa;H
=
e
8
;
a
fa;H
2
;
8In our convention, halfinteger electric charges are allowed.
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
(A.15)
(A.16)
(A.17)
where a(r = 0) = fa;H . This means that a nonzero axion eld value carries a large cost
in the electrostatic eld energy:
V =
d3r
Z
Z
=
4 r2dr
1
2 (ra)2 +
2
1 E2 ;
e
2
fa;H
where we use eg = 2 in the second line.
eq. (A.13):
We obtain the following axion con guration that minimizes the total energy of
a(r) = a0 exp ( r0=r) + fa;H ;
r0 =
e
16 2fa;H
;
where (a0 + fa;H ) denotes the asymptotic eld value of the axion. The total energy is
then given by
where we integrate eq. (A.13) from r = 0 to 1. However, the integral of eq. (A.13) may
have to be taken in the interval of [rc; 1) where rc is a cuto scale due to, e.g., an electric
screening e ect of an electrically charged particle. This is actually true for a U(
1
)H gauge
theory with a 't HooftPolyakov monopole, where there are charged massive gauge bosons
after the SSB of SU(
2
) gauge symmetry. In that case, rc is given by the inverse of the mass
of the charged gauge bosons. Then the parameter
Note that the resulting energy eq. (A.16) is positive whether the monopole charge is
positive or negative. As a result, the energy density of the axion ground state in a plasma
with monopoles and antimonopoles is given by U = nM V0, where nM = nM+ + nM .9 This
implies that the axion obtains an e ective mass of
jwith cuto
=
H
1
32 2 rcfa;H
;
fa;H
;
(A.18)
HJEP01(28)53
(A.19)
and has a VEV of
at the minimum of the potential in a plasma with monopoles due to
the Witten e ect. Here we explicitly write the temperature dependence of nM due to, say,
the expansion of the Universe.
Note that the axion VEV is determined such that the electric charge of monopole
(dyon) is absent. As we can see from eq. (A.8), the periodicity of
is absent if we x the
value of n. This may imply that the axion does not have periodic potential but has a mass
term of eq. (A.16) due to the Witten e ect. However, monopoles (dyons) can have excited
states with electric charges equal to or more than unity, which are known as JuliaZee
dyons [71]. Such excited states can be produced at the monopole production process if, e.g.,
the PQ symmetry is broken after in ation and cosmic strings form at the SSB. In this case,
the e ective
parameter (or axion VEV) changes 2 Na;H around the cosmic strings, where
Na;H is the domain wall number. Then, at the monopole production process the JuliaZee
dyons with charge n (
Na;H
1) may form in the domain of
2 (2 n
; 2 n + ). As
a result, the domain wall may form at the boundary of nearby domains due to the Witten
e ect. In this paper, we do not consider this case and focus on the case where the PQ
symmetry is broken before in ation.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
9We de ne nM as the sum of the number densities of monopoles and antimonopoles, so that the axion
mass squared is di erent from the one in ref. [33] by a factor two.
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