Dynamics of axion strings and Implications for axion dark matter - - PowerPoint PPT Presentation

Dynamics of axion strings and Implications for axion dark matter

Toyokazu Sekiguchi (RESCEU, University of Tokyo)

TAUP2019 Sep 10, 2019 - Toyama

Outline

• Introduction: production of axion dark matter in cosmology
• Field theoretic simulation of axion strings
• Results
• Long term dynamics of axion strings
• Axion radiation spectrum
• Impact on axion CDM abundance

Reference

• Masahiro Kawasaki, TS, Masahide Yamaguchi, & Jun’ichi Yokoyama, arXiv:1806.05566
• M. Kawasaki, Ken’ichi Saikawa, & TS [arXiv:1412.0789]
• Takashi Hiramatsu, M. Kawasaki, TS, M. Yamaguchi & J. Yokoyama [arXiv:1012.550]

Axion: motivation

Dynamical solution to Strong CP problem in QCD

c → CP violation

A global U(1)PQ is introduced in Peccei-Quinn mechanism. Due to quantum anomaly, θ acquires a physical degree of freedom (axion). The CP symmetry is restored at its potential minimum.

ℒ ⊃ θ 32πGa

μν ˜

Gaμν

The axion feebly interacts with SM particles only with couplings suppressed by the axion decay constant . The axion is virtually stable in the cosmological time scale. In relevant mechanisms, produced axions are non-relativistic (see below).

fa

Candidate of cold dark matter

Axion cosmology

Categorized into two scenarios.

Axion cosmology

Categorized into two scenarios.

U(1)PQ spontaneously breaks after inflation

The axion field is initially random and inhomogeneous. Axion strings form when U(1)PQ breaks down. Afterwards, domain walls (DW) also form at the QCD phase transition. If NDW=1, DWs are unstable and disappear. CDM axions are produced from these topological defects as well as misalignment.

T > fa T < fa

axion

Axion cosmology

Categorized into two scenarios.

U(1)PQ spontaneously breaks after inflation

The axion field is initially random and inhomogeneous. Axion strings form when U(1)PQ breaks down. Afterwards, domain walls (DW) also form at the QCD phase transition. If NDW=1, DWs are unstable and disappear. CDM axions are produced from these topological defects as well as misalignment.

T > fa T < fa

axion

U(1)PQ is never restored after inflation

The axion field is initially almost homogenous. The axion field starts to oscillate coherently from its initial misalignment from the potential minimum. CDM are produced predominantly from the misalignment mechanism.

−π

θi

πfa −πfa

axion

Axion cosmology

Categorized into two scenarios.

U(1)PQ spontaneously breaks after inflation

The axion field is initially random and inhomogeneous. Axion strings form when U(1)PQ breaks down. Afterwards, domain walls (DW) also form at the QCD phase transition. If NDW=1, DWs are unstable and disappear. CDM axions are produced from these topological defects as well as misalignment.

T > fa T < fa

axion

U(1)PQ is never restored after inflation

The axion field is initially almost homogenous. The axion field starts to oscillate coherently from its initial misalignment from the potential minimum. CDM are produced predominantly from the misalignment mechanism.

−π

θi

πfa −πfa

axion

Axion strings

real space field space

Topological defect associated to broken U(1)PQ

Axion strings

real space field space

Topological defect associated to broken U(1)PQ

• Tension

c Logarithmically divergent IR cutoff ~string separation L (~horizon scale t)

μ ≃ πf2

a ln (faL)

• Long range force
• Energy loss via axion emission

This is the process CDM axions are produced from strings. The efficiency is supposed to be decline as L increases.

Dabohlker and Quashnock ’90

Axion strings differ from local (abelian Higgs) ones:

Potentialc ∼ ∝ ln(faL)

Axion strings

real space field space

Topological defect associated to broken U(1)PQ

• Tension

c Logarithmically divergent IR cutoff ~string separation L (~horizon scale t)

μ ≃ πf2

a ln (faL)

• Long range force
• Energy loss via axion emission

This is the process CDM axions are produced from strings. The efficiency is supposed to be decline as L increases.

Dabohlker and Quashnock ’90

Axion strings differ from local (abelian Higgs) ones:

Potentialc ∼ ∝ ln(faL)

Presence of axion makes characteristic parameters of axion strings time-dependent.

Simulation of axion strings

Field theoretic simulation on a lattice: first principles calculation

c

with a wine bottle potential c

Grid number of our simulation c . Physical string simulations (not fat strings).

·· Φ + 3 2t · Φ − 1 R2Φ = ∂V[Φ] ∂Φ* V[Φ] = (|Φ|2 − f2

a

2 )

2

Ngrid = 40963

In radiation domination

c

.

R ∝ t

Oakforest-Pacs cluster

Simulation of axion strings

Field theoretic simulation on a lattice: first principles calculation

c

with a wine bottle potential c

Grid number of our simulation c . Physical string simulations (not fat strings).

·· Φ + 3 2t · Φ − 1 R2Φ = ∂V[Φ] ∂Φ* V[Φ] = (|Φ|2 − f2

a

2 )

2

Ngrid = 40963

In radiation domination

c

.

R ∝ t

horizon string

Why are string simulations hard?

Two different scales should be incorporated:

• String core widthc
• Horizon size c

➡ Dynamic range is limited: c

In reality, c at QCD PT.

∼ fa

−1

∼ t fa t ≲ N1/3

grid = O(103)

fa tQCD ∼ O(1030)

Simulation results need to be extrapolated by many orders of magnitude.

Oakforest-Pacs cluster

R=9.8

short loops long loops

R=10.0

short loops long loops

R=10.0

short loops long loops

R=10.2

short loops long loops

R=10.4

String velocity

Velocity estimator on a lattice

Lagrangian view point: c . Projection onto surfaces normal to string direction. c

v(x, t) ⋅ ∇Φ(x, t) + · Φ(x, t) = 0 v(x, t) = (∇Φ × ∇Φ*) × ( · Φ∇Φ* − · Φ*∇Φ) (∇Φ × ∇Φ*)2

0.2 0.4 0.6 0.8 1 10 100 1000 rms of velocity 〈v2〉1/2 physical time t/d ζ=9.5 (v/M*=5x10-3) ζ=23.9 (v/M*=2x10-3) ζ=47.7 (v/M*=1x10-3)

Yamaguchi & Yokoyama '02

Results: non-relativistic strings

Velocity rms c Significantly smaller than local strings (c ).

⟨v2⟩1/2 ≃ 0.25 ⟨v2⟩1/2 ≃ 0.5

• 2fa/MPl = 1 × 10−3

= 2 × 10−3 = 5 × 10−3

physical time fat rms of velocity ⟨v2⟩1/2

Energy loss of axion strings

Evolution of comoving length of “infinite” strings

Comoving length c of infinite strings evolves due to energy loss (in NR limit). c In simulation we define infinite strings as loops with circumference c .

L ( dL∞ dt )total = ( dL∞ dt )loop production + ( dL∞ dt )loop absorption + ( dL∞ dt )axion emission ℓ > t

2 1

• 1

−(

d ln L∞ d ln t )

scale factor R =

t/tPQ

preliminary

Contribution of each loss channel

c Axion emission predominates the energy loss of infinite strings.

−( d ln L∞ d ln t ) = 0.2 ± 0.05 (loop production) −0.05 ± 0.05 (loop absorption) 0.85 ± 0.17 (axion emission)

Energy loss of axion strings

Evolution of comoving length of “infinite” strings

Comoving length c of infinite strings evolves due to energy loss (in NR limit). c In simulation we define infinite strings as loops with circumference c .

L ( dL∞ dt )total = ( dL∞ dt )loop production + ( dL∞ dt )loop absorption + ( dL∞ dt )axion emission ℓ > t

How to compute axion abundance

c(Number of radiated axions) ≈

(Energy loss of strings) (Mean energy of radiated axions)

How to compute axion abundance

c(Number of radiated axions) ≈

(Energy loss of strings) (Mean energy of radiated axions)

string

String evolution: scaling behavior

horizon

Kibble (1985); Bennet (1986); Martin & Shellard (2002); …

Empirical law. The number of strings per horizon has been known to stay O(1) after relaxation.

How to compute axion abundance

c(Number of radiated axions) ≈

(Energy loss of strings) (Mean energy of radiated axions)

string

String evolution: scaling behavior

horizon

Kibble (1985); Bennet (1986); Martin & Shellard (2002); …

Empirical law. The number of strings per horizon has been known to stay O(1) after relaxation.

Energy spectrum of radiated axions

c in simulation

· ϕ

string cores

Computed from kinetic energy of axions: c with c Less string contamination than gradient part.

· ϕ = Im [ · Φ/Φ] Φ = | Φ |exp[iϕ/fa]

Not a trivial task in simulation —- string core should be removed. Pseudo-power spectrum estimator method

• statistical reconstruction used in CMB analysis
• masking & deconvolution

Hiramatsu et al. (2012)

Testing scaling law

c average number of strings per horizon

ξ ≡ ρstringt2 μ = ρstringt3 μt

String parameter

0.5 1 1.5 2 10 100 1000 string parameter ξ physical time t/d No loop removal Previous dynamic range ζ=9.5 (v/M*=5x10-3) ζ=23.9 (v/M*=2x10-3) ζ=47.7 (v/M*=1x10-3) previous result w/ ζ=9.5 (v/M*=5x10-3)

• (previous)

2fa/MPl = 1 × 10−3 = 2 × 10−3 = 5 × 10−3 = 1 × 10−3

with Ngrid=5123

physical time fat string parameter ξ

Testing scaling law

c average number of strings per horizon

ξ ≡ ρstringt2 μ = ρstringt3 μt

String parameter

0.5 1 1.5 2 10 100 1000 string parameter ξ physical time t/d No loop removal Previous dynamic range ζ=9.5 (v/M*=5x10-3) ζ=23.9 (v/M*=2x10-3) ζ=47.7 (v/M*=1x10-3) previous result w/ ζ=9.5 (v/M*=5x10-3)

• (previous)

2fa/MPl = 1 × 10−3 = 2 × 10−3 = 5 × 10−3 = 1 × 10−3

Deviation from the scaling law

We found c increases (logarithmically) in time. This confronts the scaling law, which predicts c .

c with c

Turn over from decrease to increase indicates the growth of c is not caused by initial condition. Time-dependence of characteristic parameters of axion strings in contrastive to local ones.

ξ ξ = (constant)

ξ(t) ∼ α log(fat) 0.15 ≲ α ≲ 0.2

ξ

with Ngrid=5123

physical time fat string parameter ξ

Comparison with other studies

• M. Gorghetto, E. Hardy & G. Villadoro,

arXiv:1708.07521

2 3 4 5 6 7 0.05 0.10 0.50 1 5 log(mr/H)

• physical

1 2 3 4 5 6

log(ηt)

0.0 0.5 1.0 1.5 2.0

ζ

• M. Hindmarsh, J. Lizarraga, A. Lopez-Eiguren &
• J. Urrestilla, arXiv:1708.07521

ξ

String-only simulations

Axion spectrum

Net energy spectrum of axion radiation emitted between t1 and t2.

ΔP(pphys; t1, t2) ≡ R(t1)4 dρaxion(t1) d ln pphys − R(t2)4 dρaxion(t2) d ln pphys

Differential energy spectrum

Axion spectrum

Net energy spectrum of axion radiation emitted between t1 and t2.

ΔP(pphys; t1, t2) ≡ R(t1)4 dρaxion(t1) d ln pphys − R(t2)4 dρaxion(t2) d ln pphys

Differential energy spectrum Results: soft spectrum

0.001 0.01 0.1 1 10 100 0.01 0.1 1 differential spectrum d(a4ρ)/dlnk(k,t2)-d(a4ρ)/dlnk(k,t1) physical wave number (k/a ma) (a1,a2)=(3,5) (a1,a2)=(5,7) (a1,a2)=(7,9)

physical wave number ( )

k/Rfa

• (R1, R2) = (3, 5)

= (5, 7) = (7, 9)

Differential spectrum

• (arbitrary units)

ΔP(p; t1, t2)

Axion spectrum

Net energy spectrum of axion radiation emitted between t1 and t2.

ΔP(pphys; t1, t2) ≡ R(t1)4 dρaxion(t1) d ln pphys − R(t2)4 dρaxion(t2) d ln pphys

Differential energy spectrum Results: soft spectrum

0.001 0.01 0.1 1 10 100 0.01 0.1 1 differential spectrum d(a4ρ)/dlnk(k,t2)-d(a4ρ)/dlnk(k,t1) physical wave number (k/a ma) (a1,a2)=(3,5) (a1,a2)=(5,7) (a1,a2)=(7,9)

physical wave number ( )

k/Rfa

• (R1, R2) = (3, 5)

= (5, 7) = (7, 9)

Differential spectrum

• (arbitrary units)

ΔP(p; t1, t2)

Radiated axions have typical momentum comparable to the Hubble rate.

c with c

Consistent with our previous results Hiramatsu et al. ’11. No apparent time-evolution of c .

2 ≲ ϵ ≲ 4 ϵ = 2πt/⟨p−1

phys⟩

ϵ

2 4 6 8 10 10 100 ratio of mean momentum to Hubble ε physical time t/d ζ=9.5 (v/M*=5x10-3) ζ=23.9 (v/M*=2x10-3) ζ=47.7 (v/M*=1x10-3) previous result w/ ζ=9.5 (v/M*=5x10-3)

• (previous)

2fa/MPl = 1 × 10−3 = 2 × 10−3 = 5 × 10−3 = 1 × 10−3

physical time fat mean momentum in units of Hubble ϵ

Comparison with other studies

Spectral shape is a long-standing controversy

ΔP(k) ∝ 1/kq−1

Peak at E ~ 1/H (IR dominated)

Davis '86; Davis & Shellard '88; Dabohlker and Quashnock ’90; Battye & Shellard ’95; …

q>1 q~1

Harari & Sikivie ’87; Hagmann & Sikivie ’91; Hagmann, Chang & Sikivie '01; …

Scale invariant q<1 UV dominated up to E ~ fa ? produced axions soft, many hard, few

Comparison with other studies

Spectral shape is a long-standing controversy

ΔP(k) ∝ 1/kq−1

Peak at E ~ 1/H (IR dominated)

Davis '86; Davis & Shellard '88; Dabohlker and Quashnock ’90; Battye & Shellard ’95; …

q>1 q~1

Harari & Sikivie ’87; Hagmann & Sikivie ’91; Hagmann, Chang & Sikivie '01; …

Scale invariant q<1 UV dominated up to E ~ fa ? produced axions soft, many hard, few

mr/2H log(mr/H)= 6

1 5 10 50 100 500 10-3 10-2 10-1 k/H F

physical

On the other hand, Gorghetto, Hardy & Villadoro ’18 indicates a hard spectrum (c ), which indicates misalignment dominates over strings.

q ≃ 0.7

Our result indicates soft axions are abundantly produced (c ). This suggests string contribution predominates the relic axion abundance.

q ≃ 2

Gorghetto, Hardy & Villadoro ’18

Extrapolation of simulation results to cfatQCD ≃ 1030

• Quasi-scaling behavior:
• Strings lose energy via axion emission:

c

• Number density of radiated axions

c

ρstring(t) ≃ ξ(t) t2 2πf2

a ln(fat)

1 R(t)4 dR(t)4ρaxion(t) dt ≃ ξ(t) t3 2πf2

a ln(fat)

naxion(t) ≃ 1 R(t)3 ∫

t

dt′ 1 R(t′)⟨p−1

phys⟩

d[R4(t′)ρaxion(t′)] dt′

Impact on axion abundance

Extrapolation of simulation results to cfatQCD ≃ 1030

• Quasi-scaling behavior:
• Strings lose energy via axion emission:

c

• Number density of radiated axions

c

ρstring(t) ≃ ξ(t) t2 2πf2

a ln(fat)

1 R(t)4 dR(t)4ρaxion(t) dt ≃ ξ(t) t3 2πf2

a ln(fat)

naxion(t) ≃ 1 R(t)3 ∫

t

dt′ 1 R(t′)⟨p−1

phys⟩

d[R4(t′)ρaxion(t′)] dt′

Impact on axion abundance

Axion abundance from strings

c with c , c .

[Ωaxionh2]string = 8.7 × ( ξ(tQCD) ϵ(tQCD) ) ( fa 1012 GeV)

1.19

ξ(tQCD) ≃ 10 ϵ(tQCD)[ ∝ ξ(tQCD)] ≃ 10

Emitted axion wavelength is proportional to string correlation length.

Axion counts for DM when cmaxion ≃ 200μeV .

Extrapolation of simulation results to cfatQCD ≃ 1030

• Quasi-scaling behavior:
• Strings lose energy via axion emission:

c

• Number density of radiated axions

c

ρstring(t) ≃ ξ(t) t2 2πf2

a ln(fat)

1 R(t)4 dR(t)4ρaxion(t) dt ≃ ξ(t) t3 2πf2

a ln(fat)

naxion(t) ≃ 1 R(t)3 ∫

t

dt′ 1 R(t′)⟨p−1

phys⟩

d[R4(t′)ρaxion(t′)] dt′

Impact on axion abundance

Axion abundance from strings

c with c , c .

[Ωaxionh2]string = 8.7 × ( ξ(tQCD) ϵ(tQCD) ) ( fa 1012 GeV)

1.19

ξ(tQCD) ≃ 10 ϵ(tQCD)[ ∝ ξ(tQCD)] ≃ 10

Emitted axion wavelength is proportional to string correlation length.

Axion counts for DM when cmaxion ≃ 200μeV .

ma(eV) 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 1 |Caγ|˜ %1/2

a

10−1 1 10 102 103

C A S T

BabyIAXO IAXO

Axion models

KSVZ BNL+UF

KLASH

ADMX

HAYSTAC ACTION / IAXO-DM ORGAN

ADMX CAPP

MADMAX A B R A C A D A B R A / D M

• R

a d i

• Irastorza and Redondo ’18

Conclusion

When the Peccei-Quinn U(1)PQ symmetry breaks spontaneously after inflation, axion strings form and can predominantly contribute to the abundance of the axion CDM. We have performed largest-ever (Ngrid=40983) string simulations. This allows us to examine long-term dynamics of axion cosmic strings. Our key findings:

• Deviation from scaling: string number per horizon grows in time logarithmically.
• Spectrum peaks at IR.

Our simulation implies the axion abundance is enhanced by a few times from the previous estimate. However, simulations from different groups are at variance.