Understanding the scaling behaviour of axion cosmic strings€¦ · Toyokazu Sekiguchi (IBS-CTPU) HU-CTPU Sapporo Summer Institute Aug 25, 2016 In collaboration with Masahiro Kawasaki

Understanding the scaling behaviour of axion cosmic strings

Toyokazu Sekiguchi (IBS-CTPU)

HU-CTPU Sapporo Summer Institute Aug 25, 2016

In collaboration with Masahiro Kawasaki (ICRR), Jun’ichi Yokoyama (U of Tokyo) and Masahide Yamaguchi (TITech)

Outline• Introduction: axion & axion cosmic strings

• Field-theoretic simulation of axion cosmic strings

References:• M. Kawasaki, TS, M. Yamaguchi, & J. Yokoyama, in prep. • M. Kawasaki, K. Saikawa, & TS [arXiv:1412.0789] • T. Hiramatsu, M. Kawasaki, K. Saikawa, & TS [arXiv:1202.5851,1207.3166] • T. Hiramatsu, M. Kawasaki, TS, M. Yamaguchi & J. Yokoyama [arXiv:1012.550]

• Understanding of scaling behaviour with one-scale string model

• Summary

Axion

CP violation

Strong CP problem in QCD

– Experimental bound: ; Why so small?

✓

32⇡2Ga

µ⌫Gaµ⌫ 2 L

✓ . 10�10

Solution: Peccei-Quinn mechanism Peccei & Quinn (1977)

– Anomalous U(1)PQ broken spontaneously at

– Pseudo-NG boson : axion → candidate of CDMa(x)

ma(x) ' 6µeV

✓fa

1012[GeV]

◆�1

✓(x) = ✓ � a(x)

NDWfa

⇠ fa

– Number of degenerate vacua: domain-wall (DW) number NDW

Axion cosmologyTwo possibilities:

• U(1)PQ is broken before inflation

• U(1)PQ is restored during or after inflation (max[Hinf, T]>fa)

- Random aini. Global strings and DWs form.

- CDM axions are also produced from these topological defects.

- (Almost) homogeneous

- CDM axions from coherent oscillation

- CDM isocurvature perturbations is generated → bound on Hinf

aini ⇠ ⇡fa

⌦axion

h2 ' 1.1⇥✓

fa1012GeV

◆1.2

fa < 1.5⇥ 1011GeV→

V (a)

⇡

a(x)/fa

�⇡

Formation of axion defects• : U(1)PQ is restoredT > fa V (�)

T > fa

T < fa

• : U(1)PQ breaks down

- Random distribution of phase: unif(-π,π)

- Formation of global strings

T = fa

time

Φ: PQ complex scalar

V (�)

(NDW=1)

• : QCD phase transition

- Axion acquires potential ~

- Formation of DWs

T = ⇤QCD

⇤

4QCD cos

✓a

fa

◆

For NDW=1 (e.g., KSVZ model), DWs has boundaries (edged by strings). DW-string system collapses in ~O(1) Hubble time.

Axionic stringsFormation of global string network

Network evolution: scaling solution

- aini is random in each causal patch

- Cosmological network of global strings forms

V (�)

- Number of strings in a horizon stays constant of O(1).

string

causal patch

Energy of string is released by radiating axions → axion CDMDavis (1986); Davis & Shellard (1989)

Controversies on estimation of axion CDM abundance

• # of strings per horizon volume

peak at E ~ 1/HDavis (1986); Davis & Shellard (1988); Battye & Shellard (1994); …

Soft Dominant contribution in Ωa from defects

peak at E ~ ma (~1/H)Nagasawa & Kawasaki (1994); …

dP/dlnE ~ constHarari & Sikivie (1987); Hagmann & Sikivie (1991); Hagmann, Chang & Sikivie (2001); …

E ~ ma ln(fa/ma) ~ O(10)ma

Chang, Hagmann Sikivie (1999); …

strings

DWs

Hard Coherent oscillation dominates

• spectrum of axion radiation from defects

• O(10): similar to like local strings

• ~1: significant backreaction from NG boson emission

Simulation of axion stringsPQ scalar on the lattice Φ(xi, yj, zk)

Field theoretic simulation: first principles calculation (↔ string based action)

(box L) > (horizon ∝a) >> (string width ∝1/a) > (lattice spacing=L/Ngrid)

Drawback: limited dynamical range

Measure of hierarchy between [string width]/[horizon size]:

in RD�+ 3H�� 1

a2r2� =

@V [�;T ]

@�⇤a(⌧) / ⌧

V [�;T ] =�

4(|�|2 � ⌘2)2 +

�

6T 2|�|2 fa =

p2⌘

In reality, ζ should be ~108 at PQ SSB (for fa=1010GeV). Simulation results needs to be extrapolated by orders of magnitude. Qualitative understanding of string network dynamics cannot be more important.

⇣(⌧) ⌘ ⌘H

3/ ⌧2 ≤ Ngrid

Numerical computation

• Due to the shortage of resolution, simulation time was limited. Parameters should have been highly tuned.

• We upgraded our simulation code for massively parallel computation on clusters with MPI+OpenMP hybrid method.

• Using ~1000 CPUs, ~2TB memory, resolution is enhanced to 40963

(500 times in memory, 4000 times in CPU time).

• Only small-scale (Ngrids≤5123 on PC) simulations were available so far.

• Approved as Collaborative Interdisciplinary Program at Tsukuba U Computer Center.

COMA clusters at Tsukuba

Setup & Initial conditionBackground: radiation domination

h�(k, ⌧)�(k0, ⌧)⇤i = !(k, ⌧)

e!(k,⌧)/T (⌧) � 1(2⇡)3�(3)(k� k0)

h�(k, ⌧)�(k0, ⌧)⇤i = 1

!(k, ⌧)

1

e!(k,⌧)/T (⌧) � 1(2⇡)3�(3)(k� k0)

Other correlations vanish.

Initial condition: thermal distribution at T~Tcr (=fa)

⌧ / R t =R⌧

2/ R2 2H =

1

t

V (�)T > fa

T < fa

Boundary condition: periodic

Box size (comoving) : 2/H(t) at the time of simulation ends

Identification of stringsIdentification of strings — non-trivial due to discrete nature of lattice

[field space][real space]

ReΦ

ImΦ

A

D

C

B

ReΦ= 0

ImΦ= 0

A

D

C

B stringquadrate convex Phase of Φ on all the vertices

of a convex penetrated by a string ranges > π.

Our method Hiramatsu + (2010):

Δθ>πOK even if string is moving.

Separation of loop and infinite strings Kawasaki+, in prep

• Grouping string points via Friends-of-Friends algorithm.

• A group of string points with L<2 π τ is regarded as a loop; otherwise as an infinite string.

• For the first time loops/infinite strings are separated automatically in field theoretic simulation of axion strings.

Velocity & relaxation

About a few Hubble time after PQ PT, strings become completely relaxed.

Mildly relativistic with v~0.5. Parameter independence is confirmed.

previous simulations


‣ Previous sim. marginally reached this.

String velocity Yamaguchi & Yokoyama 2002

‣ Projecting onto surface normal to the string direction, .ir�⇥r�⇤

‣ Lagrangian viewpointv(x, ⌧) ·r�(x, ⌧) + �(x, ⌧) = 0

v(x, ⌧) =(r�⇥r�⇤)⇥ (�r�⇤ � �⇤r�)

(r�⇥r�⇤)2

0

0.5

1

1.5

2

100 1000

rms

<v2 >1/

2 and

Lor

entz

fact

or <γ>

ζ(τ)=(τ/τcr)2ζ(τcr)

ζcr=30 (g*=102, η/M*=5x10-3)ζcr=25 (g*=103, η/M*=2x10-3)ζcr=15 (g*=102, η/M*=1x10-2)ζcr=9.5 (g*=103, η/M*=5x10-3)

Scaling behaviour


String parameter

‣ ξ~0.8(~constant)

‣ Effective # of strings per horizon volume

⇠ ⌘ ⇢lengtht2

- Consistent with previous analysis. (cf. ξ>10 for local string)

- Scaling behaviour is realized.

Loop fraction

‣ For the parameters probed, fraction of loops in total length is10-20%, nearly constant of time.

0

0.2

0.4

0.6

0.8

1

100 1000

leng

th fr

actio

n of

loop

s w

/ L<2

πτ



0

0.5

1

1.5

2

100 1000

strin

g pr

amet

ers

of in

finite

stri

ngs

w/ L

> 2π

τ





‣ There’s a tendency that for larger ζcr, loop production becomes more important.

Axion emission (1)Estimation of energy spectrum of axion radiation

‣ Not trivial — Need to remove string cores

- Masking & deconvolution- statistical reconstruction (used in CMB analysis)

Energy emission rate into axion radiation‣ Qualitatively consistent with one-scale

model:

1

10

100

100 1000

diffe

rent

ial r

adia

tion

ener

gy

d(R

4 ρra

d)/dτ i

n un

its o

f η2 /(R

τ cr3 )


ζ=30 (g*=102, η/M*=5x10-3)ζ=25 (g*=103, η/M*=2x10-3)ζ=15 (g*=102, η/M*=1x10-2)ζ=9.5 (g*=103, η/M*=5x10-3)

‣ Our method Hiramatsu+(2011): pseudo-power spectrum

da/dt in simulation slice

string cores

d⇢raddt

= �4H⇢rad + �NG⇢string

�NG = /L / t�1

➡

with

dR4⇢NG

d⌧/ ⌘2

R⌧3cr

Axion emission (2)Mean momentum of radiated axions

Relic abundance of axion CDM from strings

‣ The # of axions emitted before QCD PT should conserve through QCD PT.

0

2

4

6

8

10

100 1000

axio

n ra

diat

ion

para

met

er ε



‣〈k〉~a few Hubble

hki =Rdk k2|✓(k)|2

Rdk k|✓(k)|2

✏ ⌘ ⌧hki/4⇡

‣ Assuming scaling behaviour (i.e. ρstring), we can estimate the # of emitted axions:naxion

⇠ �⇢string

/hki

‣ CDM abundance from strings (> coherent oscillation & DW)

➡ consistent with Davis & Shellard (1988):

[⌦axion

h2]strings

= (1.7± 0.9)⇥✓

fa1012GeV

◆1.2

Energy dissipation of strings into radiation proceeds in an adiabatic way.

Understanding scaling behaviour (1)

One-scale model Kibble (1985); Bennet (1986); Martins & Shellard (1996), …

‣ Only one relevant scale: correlation length L ~ t/√ξ

‣ Great success in local strings.

‣ For global strings, however, backreaction from NG boson emission should be taken into account.


Thermodynamic equation based on velocity-dependent one-scale model‣ infinite strings

‣ loops

‣ axion radiation

d⇢1dt

=� 2H(1 + v21)⇢1 � cv1L1

⇢1 �

L1⇢1

d⇢ldt

=� 2H(1 + v2l )⇢l +cv1L1

⇢1 �

Ll⇢l

d⇢NG

dt=� 4H⇢NG �

⇢1L1

+⇢lLl

�

redshiftloop

production

axion emission

Mean momentum of emitted axions:⇢1L1

+⇢lLl

�' ⇢string

✏

t

0

2

4

6

8

10

100 1000

axio

n ra

diat

ion

para

met

er ε



with ε~3


From Eq for infinite strings:

From Eq for loop fraction:d

d ln t

⇢l⇢1

� B

A

�= �A

⇢l⇢1

� B

A

�

A ' ✏⇢1⇢l

p⇠ � cv1

p⇠ � (v21 � v2l )

B ' cv1p

⇠with

B/A should be ~0.1.

d ln ⇠

d ln t= � 1

ln ⇣+ 1� v21 �

p⇠(cv1 + ) ⇡ 0

0

0.2

0.4

0.6

0.8

1

100 1000

leng

th fr

actio

n of

loop

s w

/ L<2

πτ



0

0.5

1

1.5

2

100 1000

strin

g pr

amet

ers

of in

finite

stri

ngs

w/ L

> 2π

τ



These two conditions leads

cv1 ' (p

⇠�1

� ✏�1)(1� v21 � (ln ⇣)�1) ' 0.4

' ✏�1(1� v21 � (ln ⇣)�1) ' 0.2• Consistent with Yamaguchi &

Yokoyama (2002) • Loop production & NG emission

are comparably important.


Consistency check: axion emission rate

From one-scale model:

Putting 1

10

100

100 1000

diffe

rent

ial r

adia

tion

ener

gy

d(R

4 ρra

d)/dτ i

n un

its o

f η2 /(R

τ cr3 )



R4⇢NG

d⌧=

⌘2

R⌧3cr8⇡⇠✏ ln ⇣

⇠ ' 1

R4⇢NG

d⌧' 90

⌘2

R⌧3crwe obtain in agreement with direct estimation from sim.

✏ ' 3

We for the first time confirm the consistency of one-scale model description of global strings with significant backreaction.

' 0.2

• Our result consolidates the assumption of scaling behaviour essential for the estimation of axion abundance from cosmic strings. Since simulation result is consistent with our previous analysis, the derived constraints on fa has not changed, i.e., .

Summary• We considered a scenario where the Peccei-Quinn symmetry is restored in

the early Universe. A network of global strings forms. Due to significant backreaction from axion emission, evolution of the network is in nature different from local strings.

• Previous simulations (Ngrid<5123) are limited in simulation time and suffer from fine tuning of parameters. We performed a largest-ever simulation (Ngrid=40963) with massive parallelisation with varying parameters.

• By introducing a variety of new analysis methods, we can derive some relevant thermodynamic quantities characterising the string network. We found one-scale model with significant backreaction offers a consistent description of scaling behaviour of global strings.

fa . (4.6� 7.2)⇥ 1010GeV

Thank you for your attention!

Documents

Understanding the scaling behaviour of axion cosmic strings€¦ · Toyokazu Sekiguchi (IBS-CTPU) HU-CTPU Sapporo Summer Institute Aug 25, 2016 In collaboration with Masahiro Kawasaki