International workshop on“Axion physics and dark matter cosmology”@ Osaka University, December 20-21, 2017

Cosmological Problems of QCD Axion
Masahiro Kawasaki(ICRR & Kavli IPMU, The University of Tokyo)
Mainly Based on
MK, Saikawa, Sekiguchi, arXiv:1412.0789
MK, Yanagida, Yoshino, arXiv:1305.5338
Harigaya, Ibe, MK, Yanagida, arXiv:1507.00119
MK, Sonomoto, arXiv:1710.07269
MK, Sekiguchi, Yamaguchi, Yokoyama, in progress1. Axion

is the Nambu-Goldstone boson associate with U(1)• Axionbreaking and can be identified with the phase of PQ scalar

PQ= | |ei✓ = (⌘ + ')eia/⌘⌘ : breaking scale• Axion acquires mass through QCD non-perturbative eﬀect
ma0.6

105eV1Fa1012 GeV

Fa = /NDW
NDW : domain wall number• Axion is a good candidate for dark matter of the universe• Cosmological evolution of axion (PQ scalar)
PQ symmetry breaking after inflation
Formation of topological defects

Domain wall problem

PQ symmetry breaking before inflation
Isocurvature perturbations

Isocurvature problem

2

Today’s Talk

• Introduction• PQ symmetry breaking after inflation
Cosmological evolution of axion
Domain problem and comic axion density• PQ symmetry breaking before inflation
Isocurvature perturbation problem
Suppression of Isocurvature Perturbations• Conclusion
32. Cosmological Evolution of Axion (PQ after inflation)

T• U (1) symmetry is broken

TPQ

V( )

FaAxion is a phase direction of PQ scalarand massless= | |ei= | |eia/

ma = 0T =0

Formation of Cosmic Strings

TQCDV (a)• Axion acquires mass

NDW=2

through non-perturbative eﬀectUPQ(1) is broken to ZN DW
Coherent oscillation
Formation of Domain Walls

0π2πθ = a / (Fa NDW)
4

• Domain walls attach to strings
NDW ≥ 2wall

NDW=1

NDW=3string

Collapse
Stable and dominate the universe
Domain Wall Problem

Virenkin Everett (1982)
Barr Choi Kim (1986)

Axion overproduction
5

3. Cosmic Axion Density
3.1 Coherent axion oscillation

HV (a)

NDW = 1ma (T )0

• Axion field starts to oscillate at T = Toscillation of axion field gives a significant• Coherent
contribution to the cosmic density (h0.12 )⇤CDMa,osc h2

71042

Fa1010 GeV

1.192

spatial average
2= a /Fa : misalighnment angle at T
2

⌦a,osc h ' 0.12

if Fa ' 2 ⇥ 10

2πFa a6

including anharmonic eﬀect
11GeV
6

3.2 Axions from strings• Numerical Simulation

Hiramatsu, MK, Sekiguchi, Yamaguchi, Yokoyama (2010)
MK, Saikawa, Sekiguchi (2014)

network obeys• String
scaling solutionµ⇢string = ⇠ 2t(µ ⇠ ⌘ 2 : string tension)= 1.0 ± 0.5

scaling parameter ξstrings are produced when U(1)• Axionic
spontaneously broken

PQ symmetry is2

1.5

10.5

05

1015
proper time t/tcrit

2025

Figure 5: Time evolution of the scaling parameter ξ obtained by averaging over 2alizations.
[Note that data points are not homogeneously placed in t, bu√τ ∝ t.]• Scaling solution is established by emitting axions• Emitted axion energy ρ is estimated from ρ
we know average energy !
¯ a the axion density when it• Ifbecomes
non-relativistic is estimated as ⇢ = m (⇢/¯! )4.2

a, strNet energy spectrum of radiated axions

Figure 6 shows the energy spectra
stringof radiated axions at t1 = 12.25tcrit (left) and t2 = 2(right), that are estimated from the 20 realizations used in Section 4.1. The amplof energy spectrum at t1 is about (t1 /t2 )2 ≃ 0.24 times that at t2 . This is becaus
energy density of free axions scales as R(t)4 , without emission or absorption. We
clear exponential behavior at large k after the removal of the contamination from str
As we see in Section 4.1, the system of axionic strings are already in the scaling reg
Most of axions at this epoch are however emitted before the settlement into the regim
order to extract the energy spectrum of axions radiated during the scaling regime, weto diﬀerentiate the energy spectra at diﬀerent times. We define the diﬀerential spec
7of radiated axion between t1 and t2 ,aaa,stra• Energy Spectrum
peaked at k ~ horizon scale
exponentially suppressed at higher k•

Average energy✏ = 4.02 ± 0.702¯a =tMK, Saikawa, Sekiguchi (2014)• Cosmic density of produced axion

differential spectrum ΔPfree(k;12.25tcrit,25tcrit)

Density of Axions from Strings(horizon scale) 1

5103
2

10

11010010-1-2

10

0

50
100
150
comoving wavenumber k200

Figure 7: Diﬀerential energy spectrum of radiated axions between t1 =t2 = 25tcrit . Errors shown are estimated from the quadrature of those show
1.19errors alone (statistical and syste
Green (red) bars correspond to statisticala is arbitrary since the scale factor R(tcrit )/RNote 3that 2
the scale in y-axis
defined. DW102⌦a,string h = (7.3 ± 3.9) ⇥ 10a,osc h24N✓

F10 GeV◆just before tw . After tw , axions becomes non-relativistic due to the finite
1.19
axions exist as CDM in the Universe.
a a detailed derivation of the density parameter o3
In Appendix
B, we giveradiated from axionic
10strings, adopting the axion mass at finite temperatur
studies [6, 31]. By substituting Eqs. (24) and (27) obtained from our simul(50), we obtain

10F
10 GeV#$#8$

3.3 Axion from Domain Walls (NDW =1)• Axion energy density from collapsing

domain walls is estimated in the same
way as strings• Simulation of string-wall network
Lattice simulation with N(grid) = (512)3
Hiramatsu, MK, Saikawa, Sekiguchi (2012)

Scaling property
1.4

Average energy
Axions from collapsed domain walls
23⌦a,wall h =(5.4 ± 2.1) ⇥ 10
✓
◆1.19
Fa
⇥
1010 GeV

area parameter A• = ⇤QCD /Fa

1.2

κ = 0.40
κ = 0.35
κ = 0.30
κ = 0.25
κ = 0.20

10.8
0.6
0.4
0.2
0024681012

conformal time τ / τc9Cosmic Axion Density (NDW =1)• Total cosmic axion density⌦a,tot h2 = ⌦a,osc h2 + ⌦a,string h2 + ⌦a,wall h2✓◆1.19
Fa2= (1.7 ± 0.4) ⇥ 101010 GeVCDM h• Constraint on F

2

0.12a

Fa . (4.26.5) ⇥ 10

ma & (0.91.4) ⇥ 10 4 eV10GeV10

3.4 Axion from Walls (NDW ≥ 2)

• Wall-string networks are stable and soon dominate the universe
Domain Wall Problem
problem can be avoided by introducing a “bias” term
• Thewhich explicitly breaks PQ symmetry

Vbias =

•3ei

3+ h.c.

Sikivie (1982)⌅ : bias parameter: phase of bias term

Bias term lifts degenerated vacua
and leads to DW annihilation
large bias is favoredw biasw/o bias2V
10−π−π/20a = a/fa

π/2π

6 0 and spoils
term shifts the minimum of the potential h✓i =• Biasthe original idea of Peccei and Quinnsmall bias is favoredMore stringent constraint on Fa
Axion can be dark matter for smaller Fa

11

3.5 Summary: case of symmetry breaking after inflation
108109-2101010-3101011-4101012
10-5

parameter ranges
where axion canbe dark matter•Axion can be dark matter of the universe for Fa ~ 109 GeV or~5X1010 GeV and can be probed in the next generation experiments
123.6 Recent Progress MK, Sekiguchi, Yamaguchi, Yokoyama in progress• Our prediction heavily depends on scaling behavior• We need know the scaling behavior on longer time scale• We are updating our simulations ( from N =512 to 4096 )3grid2ngrid=20483
ngrid=40963
ξ=0.947
ξ=0.834+0.143log(1+ζ/500)

3yrainsimulation• New
suggests scaling
parameter ξ slightly
increases at large tstring parameter ξ

1.5milePr1

0.5

previous simulations
ζcr=9.6 (g*=103, η/M*=5x10-3)

0100

1000
ratio of horizon size to string width ζ=ζcr(t/tcr)• The previous result might have larger systematic errors

13

4. Axion in the Inflationary Universe (PQ before inflation)• If PQ symmetry is broken during or before inflation
Strings and domain walls are diluted away by inflation
No domain wall problem
Only coherent oscillation gives a significant contribution tothe cosmic density
a,osc0.19

2Fa1012 GeV

1.19Inflation makes
the same in the whole observable
universe (is a kind of free parameter )Isocuravture perturbation problem

144.1 Axion Isocurvature Fluctuations• Axion acquires fluctuations during inflation
a = Faa =aa (t) += m2a Fa2 [Hinf222(t,x)=m[a(t)+a(t,x)]/2aa+• For small fluctuation>a( h a2 i = Fa2 h ✓a2 i = (Hinf /2⇡)22(x)]/2a( a = Fa ✓ a⇢a✓a)'2⇢a✓⇤

15

4.1 Axion Isocurvature Fluctuations
fluctuations contribute to CDM• Axionisocurvature density perturbation
S=CDM

3=

CDM

•aaCDMaIsocurvature perturbations lead to
CMB angular power spectrum which
is diﬀerent from adiabatic one

constraint on amplitude of• Stringent
isocurvature perturbation

isoWMAP9

CMB angular Power spectrum

adiabatic

isocurvature

PS (k0 )P (k0 ) + PS (k0 )k0 = 0.002 Mpc 1

iso < 0.047 (95% CL)

PLANCK 2015
iso < 0.033 (95% CL)
16

Axion isocurvature fluctuations
10013

Fa=10 GeV

-1

1010-2-3abundance

10powerspectrum-4

10restorationof U(1)PQ-51010-6

Fa = 1013 GeV

10-7
10-8

68

10121416

1010101010
10
Hubble scale during inflation Hinf [GeV]

misalignment angle during inflation θmisalignment angle during inflation θ• Stringent constraints from CMB

Hikage, MK, Sekiguchi, T.Takahashi (2012)100

15
Fa=10 GeV

10-1
10-2
10-3-410abundancepowerspectrum-5

1010-6
10-710-8

Fa = 1015 GeV
1061081010101210141016
Hubble scale during inflation Hinf [GeV]

Constraint from power spectrum is updated including Planck data

low energy scale inflation models are allowed• OnlyHigh scale inflation ( H >10 GeV ) inconsistent with axion
inf•

If axion is dark matter

137Hinf < 2.2 ⇥ 10 GeV✓Fa1012 GeV

◆0.41

17

High scale inflationhigh scale inflation is favored because it is• Observationallytestable by observing B-mode polarization of CMBTensor mode (gravitational wave) produced during inflationr : tensor-to-scalar ratio

Hinf = 8.6 ⇥ 1013 GeV (r/0.1)1/2
r ~ 0.01 by experiments on the earthr ~ 0.001 by satellite experiments18

4.2 Suppressing Isocurvature Perturbationsof isocurvature perturbations is determined by• Amplitude
fluctuations of misalignment angle
NDW
✓a '⌘

✓

Hinf2⇡◆

1=Fa✓

Hinf2⇡

⇢a
✓a'2
⇢a✓⇤

◆

PQ field has a large value during inflation eﬀective PQ scale• IfbecomeslargeNDW| |aHinf2| |suppress isocurvature perturbations

FaaLinde (1991)

ImVHinfHinf⌘ ⌘=N F
DW aFaRe19

•Assuming PQ field has a large field value | | ' Mp during inflation
and axion is dark matter

Hinf = 8.6 ⇥ 1013 GeV (r/0.1)1/2

10-1

NDW=110-3

SN1987A

r10-2

10-4 810IsocurvatureΩa < ΩDM

1010

1012

1014

Fa [GeV]

1016

1018

axion DMmatter axion is consistent with high scale inflation whose• Darktensor mode is detectable in near future
20• However, PQ field oscillates after inflationLarge fluctuations of PQ field through parametric resonanceThis leads to non-thermal restoration of U(1)PQ symmetry

VPQ '2(| |2⌘ 2 )2 +|2| i&⌘
2h|

2| | |2•
avoid defect formation, PQ field must settle down to the• Tominimum before the fluctuations fully develop

symmetry is restore

Strings and domain walls are produced• Lattice simulation

100MK, Yanagida, Yoshino (2013)

initial value of PQ fieldv2 22| i⇠⌘
2 of h|initial value
PQfield| |10-52i| |i

breaking scalenormalized by | |i

⌘ & 10⌘ = 7 ⇥ 1013 GeV
2Lower bound on breaking scale η4| | i = Mp

⌦hRe i2 ↵⌦ Re 2 ↵Im

10-10

10-15

10-20

10-25

050

100150

⌧¯

200

⌧¯i

250

300

350

21

Constraint in high scale inflation models• Assume axion is dark matter

NDW 21014⌦a = ⌦CDM

misalignment

Hinf . 10GeV(r . 5 ⇥ 10 5 )Too small for tensor mode to be
detected

1013

Hinf GeV⇥

No domain wall and isocurvature
perturbation problems

12MK, Yanagida, Yoshino (2013)

SN 1987A

10121011no domain wall
|Mp
inflationi| <1010
108 109 10101011101210131014101510161017⌘v GeV⇥( If axion density < CDM density, high scale inflation (Hinf =1013 GeV)
consistent with axion for θa < O(10-(2-3) ) )22

4.3 Model without cosmological problems•

If PQ potential is controlled by VPQ ⇠ | |2n (noscillation is very slow

•

Model

parametric resonance is ineﬀective

m2 | |2 +V( )=3)

2|4 + 62 |Mp

24||6cH H 2 | |2during inflation| | ⇠ Mp due to | |6 and Hubble induced terms

after inflation, PQ field oscillates slowly until its amplitude✓◆✓◆becomes
52 ⇥ 1016

osc-4 = 10GeV47 ⇥ 10 86afterwards PQ field oscillates by quartic term
To avoid domain wall problem

osc-4. 10 ⌘423

10-1

Axion dark matter isallowed by high scale
inflation with|for NDW = 110-3

SN1987A

10-2rr < 0.1

NDW=1i | = Mp

IsocurvatureΩa < ΩDM

Hinf < 8 ⇥ 1013 GeV
10-4 810r < 0.02 for NDW = 610101014

10|
SN1987A

10r1018NDW=6-210-31016

Fa [GeV]-1Hinf < 4 ⇥ 1013 GeV

1012

10-4 810IsocurvatureΩa < ΩDM1010

i | = Mp1012

1014

Fa [GeV]1016

1018
24

4.4 SUSY Axion Model⌘• Superpotential W = h• PQ charges (+1), ( 1), (0)⌘ + | | (|• Potential V = h
2++2• Flat direction

0+ m2+ |+Vacuum |r± =±e±| =i✓±

a=2 2+2= Fa ,•• Axion

022|+m|+022|+ +||2 )|2( soft SUSY breaking masslog V0 =0m⌥⌘m±2+ ✓+2 ++2✓2Fa =q+2 ++2 /N

25

SUSY Axion Model• During inflation PQ fields settle down to the flat direction
⌘a'✓
✓'N
✓
If
•
++ +a+

Isocurvature perturbations✓a 'NDW+✓Hinf2⇡◆

Isocurvature perturbations
are suppressed

Parametric resonance takes place after inflation
Axion field can have large fluctuations
U(1)PQ is not restored because of the flat direction± 6= 0Domain walls without boundaries can be formed
if misalignment angle is completely random
26

SUSY Axion Model

have examined whether domain walls are formed• We
using lattice simulations

MK, Sonomoto (2017)

domain walls are formed although we do not search the• Nofull parameter space• Result suggests domain walls may not be formed
27

5. Conclusion

symmetry is broken after inflation, topological defects• IfarePQformed
and axions from them give a significant
contribution to the CDM density and axion can be dark
matter for Fa ~ 5X1010 GeV
domain wall number ≥ 2 there exist a serious domain wall
• For
problem which can be avoided by introducing a bias term
and axon can be dark matter for lower PQ scales ( Fa ~3X109 — 1010 GeV )PQ symmetry breaks before or during inflation, axion has• Ifisocurvaturedensity perturbations which are stringently
constrained by CMB observations.
PQ scalar has a large field value during inflation,• Ifisocurvature
perturbations can be suppressed and high scale
inflation is consistent with axion dark matter28