Can QCD Axion explain
the CMB Anisotropy ?
Kiyoharu Kawana (Seoul National University)
Based on Phys. Rev. D 102, 103513 (2020), 2007.06802 and 2105. 06803
Collaboration with Iso Satoshi (KEK) and Kengo Shimada (KEK)Cosmic Microwave Backgrounds• The Universe is filled with photons = Cosmic Microwave Backgrounds (CMB)• Its energy spectrum is well fitted by Planck distribution with temperature T0=2.73K
2

dn
1
p
=
dp
2⇡ 2 ep/T0

1• The expansion of the Universe tells us that our universe was hotter and SM particles werein thermal equilibrium at early Universe = Big Bang

• Besides, the CMB temperature does not much depend on direction in the sky→ One of the evidences of almost homogeneous and isotropic Universe• However, small fluctuation ΔT does exist
γ

Temperature fluctuationsγ
γCont’dPlanck 2018

Cont’d• The temperature fluctuation ΔT is usually measured by its angular (scale)dependences

Dl = l(l + 1)Cl /(2⇡)

Cl =

Z

d⌦

Z

!
!
0
!
!
0 h T ( n ) T ( n )i
0
d⌦
P
(
n
·
n
)
l
2
T0

Planck 2018

l

4π m
m*
Pl( n ⋅ n ′) =Yl ( n )Yl ( n ′)∑ 2l + 1m=−lRoughly, Pl~cos(lθ).
For fixed l, we are looking at θ~π/l
Large l Small θ

1. What can we learn from this result ? (next slide)2. How do we explain this fluctuations theoretically ?

Necessary Inputs for This talk① Amplitude of Power Spectrum• Cl is expressed as

∞

2
3
Cl =
(d ln k) k Pℛ(k)Δl(k)
π ∫0

where

l (k) : transfer function

which depends on cosmological models• For modes well outside horizon at recombination (l<<100), only gravitational red or blue shift
matters (Sachs-Wolfe eﬀects).
→ In this region, Δl is simply a spherical Bessel function
1Δl(k) = jl(krL), rL : comoving radius of the last scattering surface
3• Spherical Bessel function jl(x) has a peak around x~l, thus roughly
2 3
Cl ⇠ 2 k PR (k)
3 ⇡
k=l/r

L

Z 1
|

0

2

(d ln x)jl (x)
{z
}

/l

1 (l+1)2 3
k PR (k)) l(l + 1)Cl ⇠
9⇡
k=l/r

1

We can know PR(k) (primordial power spectrum) from Cl

LNecessary Inputs for This talkCont’d• Planck 2018 results3k
As (k) :=
P
(k)
=
A
(k
)
R
s
⇤
2⇡ 2

✓

k
k⇤

◆ns 1

,

9
As (k⇤ ) = 2.1+0.031
⇥
10
,
0.034

amplitude of fluctuations

ns = 0.965 ± 0.004 (68% CL)
Small scale dependence

k*=0.05 Mpc-1 (a reference scale)② Isocurvature perturbation ← diﬀerent type of fluctuationβiso(k*) :=Pℐ

Pℛ + Pℐ< 0.00107 (95 % CL for fully anti − correlated case)k=k*③ Non-Gaussianity constraints ← deviation from Gaussian fluctuations
local
fNL = 4 ± 20,

gNL = ( 5.8 ± 13) ⇥ 10

4(95%CL by Planck 2018)＊ I will explain the details later

Inflation is an Unique Solution ?
• The most popular solution for the CMB anisotropy is (single field) inflation because
it can solve a few cosmological problems simultaneously.

• Quantum fluctuation of inflaton <δφ2> is the origin of the CMB anisotropy.
• In particular, single field inflation automatically predicts adiabatic perturbations which isfavored by the CMB observations.

• This is simple and seems to be reasonable, but not uniquee.g.) Any (massless) scalar field can have fluctuation during the inflation.
After the reheating, it can convert its fluctuation into radiation = Curvaton scenario
D.H. Lyth and D. Wands, (’02)

Why QCD Axion ?

R.D. Peccei and H. R. Quinn (’77)
F. Wilczek (’78), S. Weinberg (’78),
M A. Shifman, A.I. Vainshtein, and V I. Zakharov (80)

• QCD axion is pseudo NG boson that appears as a result of SSB of some global
U(1) symmetry = Peccei-Quinn symmetry• Typically, it has interaction with fermions like e
iA/fA
redefine ψL or ψR to eliminate e

iA/fAψLψR + (h . c.). But, we can always• Due to the anomaly, the interaction between gauge fields appears instead
2

2

g
A
g
4
μν
4
′
μν
d x θ+
Tr[G̃ Gμν] :=d xA (x)Tr[G̃ Gμν]
32 ∫
fA )
32 ∫
(

• θ is now dynamical quantity and fixed at the minimum of the axion potential VA(A)If VA(A) has a minimum at A=0, this can be a dynamical solution of strong CP problem| θeff | ≲ 10

−10

nEDM collaboration (2020)

Axion Potential• Non-perturbative eﬀects of QCD produce axion potential.
Naive instanton calculation gives
2 2
VA(A) = mA0 fA[1 − cos(A/fA)],

2 2
m
f
π
A
2
mA0 =
(mu + md)2 f A2

mumd

Blue: instanton
Orange: based on chiral Lagrangian

C.f. Calculation based on chiral Lagrangian gives
VA(A) = − mπ2 fπ2

4mumd

A
1−
sin
(mu + md)2
( 2fA )
2

VA actually has minimum at A=0 !

• Finite temperature potential is also important to determine its abundance.
Lattice calculation gives
2 2
VA = mA (T ) fA [1cos(A/fA )],2
where mA (T ) = mA0 ⇥(Temperature dependent mass

b=1.02 for three light quarks(TQCD /T )4b
1

for T TQCD ,for T  TQCD ,[Sz. Borsanyi et al. Nature, 539(7627):69–71, 2016.]

Short Summary of this talk
• Consider a possibility whether QCD axion can explain the CMB anisotropy• The following situations are assumed:
1. Fluctuation from the inflaton is too small to explain the CMB anisotropy (trivial)
2. QCD axion is massless during the inflation, i.e. PQ symmetry is broken
3
H
t
inf
→ Axion also gets primordial fluctuations as well as the inflaton ⟨δA 2⟩ ∼2π

• The axion fluctuation is transferred to SM radiation by the energy conversion at around the
QCD phase transition, TQCD ~150 MeV• Constrain parameter space of QCD axion by the CMB observations
In particular, we see that some dilution of QCD axion is necessary→ As a simplest possibility, we study secondary (thermal) inflation

Talk Plan
1. Brief review of cosmological perturbations
2. Fluctuation of QCD Axion and its conversion to radiation
3. Isocurvature constraints and Non-Gaussianity
4. Secondary (thermal) Inflation scenario
5. Conclusion＊ In the following, background quantities are represented with bar, e.g. ā, ρ̄.

Talk Plan
1. Brief review of cosmological perturbations
2. Fluctuation of QCD Axion and its conversion to photons
3. Isocurvature constraints and Non-Gaussianity
4. Secondary (thermal) inflation model (if there is suﬃcient time)
5. Conclusion＊ In the following, background quantities are represented with bar like ā, ρ̄.

Curvature Perturbations ζ[J.M. Bardeen et al. (’83)]• We write the Friedmann-Robertson-Walker (FLRW) metric as2

2

2 2ψ(x)ds = − dt + a(t) eijδijdx dx ,ψ(x) : fluctuation of local e − folding number• We consider perturbations at linear order for simplicity• The uniform density curvature perturbations ζ is defined by• ζ is gauge invariant and conserved on super horizon scale if total pressure p is a function of
total energy density ρ, i.e. p=p(ρ) (barotropic)

ζ = − ψ (for uniform density slice i.e. δρ = 0)
＊But, the universe generally consists of various components.
So ζ is not a conserved quantity in general.

•Cont’dWe can also define a similar curvature perturbation for each component X• This is also gauge invariant and conserved on super horizon scale if X does not interact withother components.

• The original ζ can be also rewritten as・・・ ✳

• A perturbation is called adiabatic if all ζX are the same, i.e. ζX = ζY = ζadi
→ In particular, ζ also coincides with ζ from ✳
adi• Other perturbations are generally called non-adiabatic or entropic. Thus, it is characterized by
the diﬀerence between ζX’s
SAr := − 3(ζA − ζr)＊In particular, the fluctuation with δρ = 0 is called isocurvature perturbation.

Cont’dRelation between power spectrum and ζ• On superhorizon scale (l<<100) at recombination, the eﬀect of gravitational potential Φ
is dominant = Sachs-Wolfe eﬀect
δT 1
= Φ
T
3

(for super horizon)rec• In the matter dominated era, Φ is given by• Thus, we obtainδT1= − ζrT
5(for superhorizon scale)rec∴ Power spectrum P

Recall∞2
Cl =(d ln k) k 3Pℛ(k)Δl(k)π ∫0is determined by ⟨ζζ⟩(k)ℛ
We have to calculate ζ theoretically.

Talk Plan
1. Brief review of cosmological perturbations
2. Fluctuation of QCD Axion and its conversion to radiations
3. Constraints from Non-Gaussianity and Isocurvature
4. Secondary (thermal) inflation model (if time permits)
5. Conclusion

Before going into the details….
• Rough sketch of the cosmological history of QCD axion

VA

Get Primordial
Fluctuations δAThis energy is compensated byradiations. The fluctuations areconverted to radiation.
ζA → ζγA

Axion potential starts to increase
after the EW phase transition

We have to calculate
curvature perturbations
carefully for each epochs

VA

A

The increase of the potential
stops when T=TQCDCont’d(I) Primordial fluctuations by Inflation• Fourier modes

3
d
k
ik⋅x
† −ik⋅x
̂ =
̂
̂e
δA(x)
δ
Ã
(t)
a
e
+
δ
Ã
*
(t)
a
k
(
)
k
k
k
∫ (2π)dTheir Heisenberg equations are···
δÃk(t) + 3Hinf δÃk(t) +Meaning of these creation-annihilation
operators are explained later

2

k
2
+
m
δ
Ã
(t)
=
0
k
2
( a(t)
)
−1

−3/2

1 dχk
9
m2
1
+
+
1
−
−
2 ) (kη)2d(kη)2 kη d(kη)( 4 Hinf

χk = 0 ← Bessel diﬀerential equation• Or, by introducing conformal time η = − (a(t)Hinf ) and new function χk(t) ≡ (−η)
it can be rewritten as
d 2χk

≡ ν2

→ In general, there are two independent solutions Jν(kη) and Yν(kη)
Their arbitrary linear combination is also a solution

δÃk(t),Bunch-Davies Vacuum

1
Hinf[T. S. Bunch and P. C. W. Davies (’78)]

Cont’d

a(η)
→
0
When
,
(almost)allphysicalmodes
arewithinhorizon
i.e.
k
:=
k/a
>>
H
phys
inf
•
→ Spacetime is essentially the same as ﬂat Minkowski spacetime• Natural choice of mode function is the plane wave when t → 0 (η=- ).
t→0 ∫δAĤ (x) →3d kphys(2π)312ωk(t)

̂ae( k−i ∫ t dsωk(t)+ikphys⋅xphys• Hankel function satisﬁes this condition:•t∫idsωk(t)−ikphys⋅xphys†+ ak̂ e,)(1)χk(η) = Hν (−kη) := Jν(−kη) + iYν(−kη)†
Creation-annihilation operators (ak̂ , ak̂ ) deﬁne a vacuum state at initial timeak̂ | 0⟩BD = 0

for all k(Bunch-Davies (BD) Vacuum)

=δAkCont’dAxion Fluctuation at the end of Inflation
Time direction(1)(-kη) gradually begins to deviate from the plane wave.
However,
as
time
goes,
H
ν
•

• In particular, on the superhorizon limit(equilibrium distribution):k≪ 1 , δAk approaches constant valuea(t)Hinf
1

2
Hinf

k
| δAk | → 3
2k ( a(tf )H )2

Remember this

2m2
3H 2
inf

→

Hinf

2
Hinf

m=0 2k 3• As a result, the modes outside horizon accumulate, and produce a ﬁeld ﬂuctuation:2⟨δA(tf ) ⟩ ∼Hinf af

∫0

0

3

3
Hinf

d k
2
|
|
δA
(t
)
=
t
f
k
(2π)3
4π 2
outside horizon(for m = 0) (Bunch-Davies distribution)

Hinf

inside horizon

k/af〜 same as Minkowski spacetime

Before going into the details….
• Rough sketch of the cosmological history of QCD axion

VA

Get Primordial
Fluctuations δAThis energy is compensated byradiations. The fluctuations areconverted to radiation.
ζA → ζγA

Axion potential starts to increase
after the EW phase transition

VA

A

The increase of the potential
stops when T=TQCD

(II) (iii) Transfer of fluctuations from Axion to Radiation• After the EW phase transition (T~160GeV), the axion potential starts to increase
2 2
VA = mA (T ) fA [1cos(A/fA )],2
where mA (T ) = mA0 ⇥((TQCD /T )1

4b

• This increase of energy is compensated by radiation

for T TQCD ,for T  TQCD ,b~1.02 by Lattice calculation
Sz. Borsanyi et al. (2016)⇢A + ⇢r = 0 (energy conservation)• It is very hard to solve the energy conservation rigorously…
→ Let us assume that the energy transfer suddenly occurs at T=TQCD ~ 150MeVHigh T

TQCD

Low T

Within this approximation, energy conservation only at T=TQCD matters !

(II) Fluctuation before T=TQCDInflationCont’dRadiation dominatedζr=ζinfTR

TQCD=150MeVTime• After reheating, radiation obtains the fluctuation ζr = ζinf( ∼ 0) from inflaton, and it isconserved until T=TQCD• To calculate fluctuations after T=TQCD, we need to know ψ• Right before T=TQCD, only radiation energy exists. Therefore, it is natural to choose theuniform temperature slice i.e. δρr=0.

δρr
ζr := − ψ += − ψ = ζinf ∼ 0
4
ρ̄rIn this slicedef∴ ψ is given by ζini

at T=TQCD

Recallds2 =

dt2 + a(t)2 e (x) ij dxi dxjCont’d(III) Fluctuation after T=TQCD• Right after T=TQCD, we have

⇢ = ⇢A + ⇢r = 0

・・・①(energy conservation)ψ = ζinf• From the definition ζr = − ψ − δρr /(4ρr), δρr can be written asδρr = − 4ρr(ζr + ψ) = − 4Ωr ρ(ζr − ζinf) ・・・②

• Energy fluctuation of axion is⇢A =0
VA (A) = VA (A) A・・・③• By substituting ②③ into ①, we can solve ζr as
ζr = ζinf +Negligible

′
R VA

4 VA

A=A

R δA
δA ∼
,
2 A

where R :=determined by initial
fluctuation

ΩAΩrright after TQCD

Ratio of energy densities after the energy transfer

Result ζr = ζinf +

•

2
H
2
inf
Recall | Ak | =(2⇡)3 k 3

′
R VA

4 VA

A=A

R δA
δA ∼
,
2 A

where R :=ΩAΩrCont’dright after TQCD

from the primordial inflation• Thus, the CMB scalar amplitude is given byk3
2||
As(k*) :=ζr( 2π 2
)

1/2

2

1/2

R 2 | δAk |=( 22π 2 A2 )fA : axion decay constant,

=R×

Hinf

4πfAθ

−5

≃ 4.6 × 10

(Planck 2018)✓ : initial angle• As long as we consider radiation dominated era, R can not be larger than 1.
Hini
4⇡ ⇥ 4.6 ⇥ 10 5
=
R
fA ✓

4.8 ⇥ 104

• Besides, we have assumed that PQ symmetry is broken during inflation, i.e.
Thus, we obtain

4.8 × 10

−4

<

Hinf
fAθ

<θ

−1

Hinf
fA

<1

Talk Plan
1. Brief review of cosmological perturbations
2. Fluctuation of QCD Axion and its conversion to photons
3. Constraints from Non-Gaussianity and Isocurvature
4. Secondary (Thermal) inflation model (if time permits)
5. ConclusionNon-Gaussianity (NG)• In the following, we simply assume adiabatic fluctuation, i.e. ζ=ζX
• If the fluctuation ζ is Gaussian, all observables are determined by two point function ⟨ζζ⟩We represent such a Gaussian fluctuation as
Z
3
d
k
G
G
ik·x
ˆ
ˆ
ˆ
⇣(x) = ⇣ (x) =⇣e(Gaussian
distribution)k(2⇡)3• The deviation from this is characterized by the three point function:
3
(3)
̂
̂
̂
⟨ζ ζ ζ ⟩ = (2π) B(k , k , k )δ (k + k + k )k1 k2 k3

1

2

3

1

2

3

• Functional form of B(k1,k2,k3) depends on the statistical properties of ζ

→ Three famous types:① equilateral type ② orthogonal type ③ local type ←In axion scenario, this is produced＊ Planck 2018 gives constraints for each types of NG

k1

Local Type NG

k3

k2

B

(k1 , k2 , k3 ) =Recall3(3)ˆˆh⇣k1 ⇣k2 i := (2⇡) P⇣ (k1 )(k1 + k2 )• Local type is defined by
local

Cont’d⇤6 local ⇥fNL P⇣ (k1 )P⇣ (k2 ) + P⇣ (k2 )P⇣ (k3 ) + P⇣ (k3 )P⇣ (k1 )5• This type is produced when full ζ has the following functional form:ˆˆ⇣ = ⇣G• Current observational bounds are
local
fNL = 4 ± 20,

3 local ˆ2
93
ˆ
fNL ⇣G + gNL ⇣G + · · ·525gNL = ( 5.8 ± 13) ⇥ 10

4

∵ Just calculate <ζζζ> by using
Wick’s theorem(95%CL by Planck 2018)＊ As dimensionless parameters, those bounds themselves are not so strong.
But, they give (relatively) strong lower bounds on R=ΩA/Ωr in our scenario.

Cont’dNG in QCD Axion Scenario• Because we are now interested in higher order terms, we have to solve the energy
conservation low δρA + δρr = 0, nonlinearly.
→ As a result, ζr is given by a nonlinear function of δA (Gaussian)• The calculation is straightforward but tedious… So, I just show the results:
′′
V
V
10
A
A
local
fNL = −+
R,
3R [ VA′ 2
]
A=A

1 20
gNL =
6 ( 3R )

2

2
′′′ 2
VAVA
20 local 1 20
+
f
−
NL
3
(
)
′
3
6
3
VA A=A• In the case of sinusoidal potential, i.e. VA = V0(1 − cos(A/fA)), they becomecan become sizable
when R<0.1
small R is not allowed

10 cos ✓ 1 cos ✓
10
fNL =
,
2
3R
3
sin ✓
✓
◆2
✓ ◆
1 20
✓
20
2
gNL =
tan
+ fNL
6 3R
2
3

1
6

✓

20
3

◆2

.

Allowed Region

10 cos ✓ 1 cos ✓
10
fNL =
,
2
3R
3
sin ✓
✓
◆2
✓ ◆
1 20
✓
20
2
gNL =
tan
+ fNL
6 3R
2
3

Cont’d
1
6

✓

20
3

◆2

.

• Dark blue region represents the allowed region by fNLlocal• Orange region represents the allowed region by gNL

Typically, R has to be ≳ 0.1. But, smaller
values are allowed around θ=π/2, 3π/2 !
where V’’A vanishes

Isocurvature Constraint• Recall that isocurvature perturbations is measured by the diﬀerence of curvature
perturbations:

SAr = − 3(ζA − ζr)• After the QCD phase transition, SAr is conserved because there is no energy transfer
between axions and photons → ζA and ζr are both conserved• Besides, recall also that SAr is gauge invariant→ We can choose any gauge slice to calculate thisStep1) The easiest choice is the uniform temperature slice, i.e. δργ=0

δρr
ζr = − ψ += − ψ (uniform temperature slice)
4ρ̄r
def

Step2) From the definition of ζA, we haveζA = − ψ +
def

δρA

3ρ̄A

= ζr +δρA

3ρ̄A∴ SAr = −

δρA
ρ̄A

=−

′
VA

VA

δA
A=Ā

← well known result

Cont’d• The isocurvature perturbation is constrained by the combinationℐ := rASAγ,rA = ΩA /ΩDM |today• Now, isocurvature power spectrum can be calculated as
hIk Iq i =

✓

◆
2
0

VA
rA
VA

h Ak Aq i = (2⇡)

where PI =✓0
rA V A

2⇡ VA• The current observational bound

Hinf

◆2

3 (3)

'

✓

✓

◆
2
0

VA
(k + q) rA
VA
rA Hinf
⇡fA ✓

◆2

2
Hinf
2k 3

4 5

2 ⇡
:= 3 PIk

2 2
Use VA ∼ mA A /2

PI
PI
'
<
0.0011(95%
CL),
iso (k⇤ ) : =9
PR + PI k=k
2.1 ⇥ 10⇤)rA Hinf6< 1.5 ⇥ 10
⇡fA ✓If axion is DM i.e. rA=1 and θ=O(1),Hinf/fA has to be very small

All Results
1. Amplitude of the CMB power spectrum:

2. Isocurvature perturbations:

3. Non-Gaussianity:

p
RHinf
5
As =
= 4.6 ⇥ 10
4⇡fA ✓rA Hinf6< 1.5 ⇥ 10⇡fA ✓We can eliminate Hinf/fAθ

)rA < 8.2 ⇥ 10

3

R  8.2 ⇥ 10

Unfortunately, QCD axion can not
become DM in this scenario….

Around θ=π/2,3π/2, R ~0.01 is allowed.
In such regions, rA has to be more suppressed

3

inf

Cont’d• The parameters has to lie on the blue contour
to explain the CMB amplitude As
p
RHinf
As =
= 4.6 ⇥ 10 5
4⇡fA ✓• When we fix Hinf/fAθ, there is an upper boundof rA by isocurvature constraintrA Hinf< 1.5 ⇥ 10 6
⇡fA ✓rA needs to be very small << 0.01.
Thus, we need some dilution of axions→ Thermal (Secondary) Inflation (next)

Talk Plan
1. Brief review of cosmological perturbations
2. Fluctuation of QCD Axion and its conversion to photons
3. Constraints from Non-Gaussianity and Isocurvature
4. Secondary (thermal) inflation model
5. Conclusion

Axion Abundance• As is well known, QCD axion can become DM by misalignment mechanismΩA ∼ 0.3θ

2( 1012GeV )
fA

1.16

150MeV( TQCD )0.67∴ rA~1 in the standard cosmology of QCD axion−4• We need a dilution mechanism to obtain rA ≲ 10
• Secondary (thermal) inflation is the simplest possibility• Various new physics models predict secondary inflation

e.g. GUT model, [D. H. Lyth and D.Stewart (’95)]
Curvaton model, [Gong, Kitashima and Terada (’17)
Classically Conformal Models, [Iso, Shimada (’17)→ We are not interested in model constructions→ Let’s proceed our calculations based on a few (reasonable) assumptions(next slices)Assumptions

We call scalar field φ which causes the secondary inflation as flaton

1. QCD axion does not have any interactions with flaton→ Only exponential expansion aﬀects the axion abundance
2. Secondary inflation starts from Tinf ≳TQCD ~150MeV (this is not crucial)
3. Secondary inflation ends at very low temperature Tend << TQCD ~150MeV
4. The secondary reheat temperature TR is higher than Tosc~1GeV→ This guarantees that QCD axion is following the standard thermal history after
the secondary reheating.

Tinf

TQCD

Tend

TR

t
＊Based on these assumptions, only the time evolution of axion field A(t) aﬀects the abundance !

Axion Evolution during Secondary Inflation• Axion potential

VA = mA(T )2 fA2 [1 − cos(A)], mA(T ) = mA0 ×

4b(TQCD /T ){1

When T≥TQCD, VA is strongly suppressed VA ∝ (Tfor T ≥ TQCD
for T < TQCD
8

/T)
QCD

→ Axion field does not much evolve until T=TQCD• More precisely, the EOM of axion is
2
d
θ
dθ
··
·
2
8
8N
A + 3HTI A + mA0 fA(TQCD /T) sin(A/fA) = 0 →
+
3
+
3ηe
sin(θ)
=
0
dN 2
dN
where

N :=

t

∫t

dtHTI

QCD

e-folding from t=tQCD

•

2

2

1 mA0
100
10 GeV
1TeV−6
η :== 6.6 × 10
3 ( HTI )fA( gTI ) () ( TTI )12

4

Ratio between axion mass and Hubble of secondary inflation

Because of the temperature suppression e8N, slow roll approximation is gooddθ
8N≃ − ηe sin(θ)
dN

Cont’dThis is small even if η~O(1)• At t=tQCD (i.e.T=TQCD), we have

ηtan(θQCD /2) = tan(θini /2)exp − (1 − e −8|NEW|) ∼ tan(θini /2)e −η/8[ 8]η∴ θQCD ∼ θini − sin θini
8

∴ Axion field does not actually evolve before T=TQCD• After T=TQCD, however, the axion field does evolve because there is no temperature
suppression

2

d θ
dθ
dθ
EOM :
+
3
+
3η
sin(θ)=0→+ηsin(θ)
∼
0
(slow
roll)
dN 2
dN
dN

∴ θend = 2arctan [tan(θQCD /2)eΩA ∼ 0.3θini

2( 1012GeV )
fA

1.16

∼
2
tan(θ
/2)e
]
QCD

0.67

150MeV( TQCD )Conventional result−ηNendΩA ∼ 0.3θini

2−ηNend,( 1012GeV )Nend := log

fA

1.16

TQCD
Tend

150MeV( TQCD )0.67

× e −2ηNend

Cont’dResults
Nend=3

Nend=10• Solid lines correspond torA=const

Blue regions are allowed by Non-Gaussianity constraints→ Secondary Inflation can actually realize rA~10-4 while explaining CMB anisotropy !

Summary• We discussed a possibly whether QCD axion can explain the CMB fluctuation• In general, this scenario has to satisfy the following conditions (at least)(i) the CMB amplitude (ii) Isocurvature, and (iii) Non-Gaussianity• We found that QCD axion can explain the anisotropy if the current abundance is
suﬃciently small (~10-4).

This requires additional dilution in the early universe, such as secondary inflation• Future works: • Is it possible to construct another scenario that is compatible with axion DM ?
• Construct a concrete model of secondary inflation• Dark Sector, string axion, and moreThank you for your attention !

[K.K, S. Iso, K. Shimada, arXiv:2105.06803]

Backup Slides

Gauge slice (choice)′μ• We always have gauge degrees of freedom: x = f(x)• Correspondingly, operators change by their Lie derivative at the same coordinate point xμQ′(x) − Q(x) := δQ = ℒδxQ, e . g . δϕ = − δx ∂μϕ (for scalar quantity)• Typically, a scalar quantity is set to zero by using gauge dof
e.g.
1. Uniform density slice
2. Spatially flat gauge

total energy density has no fluctuation δρ=0
no fluctuation of scale factor, ψ=0

3. Comoving slicing (during inflation)

no fluctuation of inflaton δφ=0

Sachs-Wolfe Effects

φ(x)Observer• There are two contributions to temperature fluctuation• One is the gravitational blue(red)-shiftδTT= Φ( n rL)⋯(1)rec2

ds = − (1 + 2Φ)dt + ⋯• The other is the change of scale factor (time dilation)
δT
T

rec

δa
=−
a

rec

2
= − Hδt = − × tΦ( n rL)
3t

rec

2

∴ δt = Φt2= − Φ( n rL)⋯(2)
3

Matter dominated1∴ (1) + (2) = Φ( n rL)
3(Sachs-Wolfe Eﬀects)

Gravitational Potential and Curvature Perturbation• Define δD := δρDM / ρ̄DM
• Linear order solution during matter dominated era (in Synchronous gauge)
9q 2t 2
δDq =
ζ
q
2
10a

• On the other hand, from the Poisson equation222∇ Φ = 4πGδρDM → Φq = − (a/q) 4πGδρDq = − (a/q) 4πGρ̄DMδDq
2

2• Using Friedman equation H = (2/3t) = 8πGρ̄DM /33∴ Φq = − ζq (matter dominated era)5Conservation of ζX• Consider super horizon mode (i.e. neglect space dependence), and take the linear order
perturbation of energy conservation low··ρ + 3H(ρ + p ) = 0 → δρ
·
+
3H(δρ
+
δp
)
+
3
ψ(ρ
+
p
)
=
0
X
X
X
X
X
X
X
X

• Taking time derivative of ζX and using the above equation, we have
·
dζX
δρX
pX
d
H
=
−ψ −
=
δpX− · δρXdt
dt [3(ρ + p) ] ρX + pX [
ρX ](for superhorizon mode)• When X is a barotropic fluid i.e. p=p(ρ), we have·p·
δpX ρX δpX
dζXX·ρ = δρ ρ· = δρ → ∴ dt = 0
X
X X
X(for superhorizon modes)• This was also proven non-linearly by Lyth, Malik and Sasaki (‘05)

Isocurvature contributions• At recombination, the uniform density curvature perturbation isζ=∑X (1 + ωX)ΩXζX
1 + ωtot=(4Ωr + 3Ωm + 3Ωb)ζr + 3ΩAζA
4Ωr + 3Ωm + 3Ωb + 3ΩA= ζr +

3ΩA

4Ωr + 3Ωm + 3Ωb + 3ΩA(ζA − ζr) ∼ ζr +

Ωm + ΩA(ζA − ζr)∼−

Add and subtract 3ΩAζr in numerator• Thus, the temperature fluctuation on superhorizon scale is
2rA
δT
6
1
∼ − ζr − 2Φ = − ζr + ζ = ζr −
SAγ
T
5
5
5

ΩA

rA
3

SAγ

Matter dominated(for superhorizon scale)∴ Isocurvature is constrained by the combination ℐ := rASAγConsistency check
2
H
V
inf
inf
• Inflation gives
As =
∼
2
4
2
2
24π Mplϵ 8π ϵMpl
2 2
R
H
inf
QCD
• Our scenario gives
As
∼
2
2
2
16π f Aθ

∴

QCD
inf
As ≪ As→ϵ : slow roll parameter2

We can easily evade this

fAθ
1
≪ϵ
2 ( RMpl )For example, if we choose fA=1012GeV, θ=1, R=0.01, this becomes 8.5 × 10−10 ≪ ϵ
On the other hand, the current upper bound is0.05
ϵ≲∼ 0.00316

Other Non-Gaussianities• Equilateral type
k1

k2
k3

• Orthogonal type
k1

k2
k3

Calculations of Non-Gaussianity

Gaussian part ζr ∼

′
R VA

4 VA

δA
A=A