Understanding the scaling behaviour of axion cosmic strings

Transcription

1 Understanding the scaling behaviour of axion cosmic strings Toyokazu Sekiguchi (IBS-CTPU) In collaboration with Masahiro Kawasaki (ICRR), Jun ichi Yokoyama (U of Tokyo) and Masahide Yamaguchi (TITech) HU-CTPU Sapporo Summer Institute Aug 25, 2016

2 Outline Introduction: axion & axion cosmic strings Field-theoretic simulation of axion cosmic strings Understanding of scaling behaviour with one-scale string model Summary References: M. Kawasaki, TS, M. Yamaguchi, & J. Yokoyama, in prep. M. Kawasaki, K. Saikawa, & TS [arxiv: ] T. Hiramatsu, M. Kawasaki, K. Saikawa, & TS [arxiv: , ] T. Hiramatsu, M. Kawasaki, TS, M. Yamaguchi & J. Yokoyama [arxiv: ]

3 Axion Strong CP problem in QCD 32 2 Ga µ G aµ 2 L CP violation Experimental bound: ; Why so small? Solution: Peccei-Quinn mechanism Peccei & Quinn (1977) Anomalous U(1) PQ broken spontaneously at (x) = a(x) N DW f a a(x) f a Pseudo-NG boson : axion candidate of CDM 1 f a m a (x) ' 6 µev [GeV] Number of degenerate vacua: domain-wall (DW) number N DW

4 Axion cosmology Two possibilities: U(1) PQ is broken before inflation V (a) - (Almost) homogeneous a ini f a - CDM axions from coherent oscillation axion h 2 ' 1.1 f a GeV 1.2 f a < GeV a(x)/f a - CDM isocurvature perturbations is generated bound on H inf U(1) PQ is restored during or after inflation (max[h inf, T]>f a ) - Random a ini. Global strings and DWs form. - CDM axions are also produced from these topological defects.

5 Formation of axion defects T>f a T>f a V ( ) : U(1) PQ is restored T = f a : U(1) PQ breaks down - Random distribution of phase: unif(-π,π) T<f a - Formation of global strings T = QCD : QCD phase transition a - Axion acquires potential ~ 4 QCD cos f a Φ: PQ complex scalar V ( ) time - Formation of DWs For N DW =1 (e.g., KSVZ model), DWs has boundaries (edged by strings). DW-string system collapses in ~O(1) Hubble time. (NDW=1)

6 Axionic strings Formation of global string network V ( ) - a ini is random in each causal patch - Cosmological network of global strings forms Network evolution: scaling solution - Number of strings in a horizon stays constant of O(1). string causal patch Energy of string is released by radiating axions axion CDM Davis (1986); Davis & Shellard (1989)

7 Controversies on estimation of axion CDM abundance # of strings per horizon volume O(10): similar to like local strings ~1: significant backreaction from NG boson emission spectrum of axion radiation from defects Soft Dominant contribution in Ω a from defects Hard Coherent oscillation dominates strings peak at E ~ 1/H Davis (1986); Davis & Shellard (1988); Battye & Shellard (1994); dp/dlne ~ const Harari & Sikivie (1987); Hagmann & Sikivie (1991); Hagmann, Chang & Sikivie (2001); DWs peak at E ~ m a (~1/H) Nagasawa & Kawasaki (1994); E ~ m a ln(f a /m a ) ~ O(10)m a Chang, Hagmann Sikivie (1999);

8 Simulation of axion strings PQ scalar on the lattice Φ(xi, yj, zk) +3H 1 a 2 r2 [ ; T a( ) / in RD V [ ; T ]= ( 2 2 ) 2 + T 2 2 f a = p Field theoretic simulation: first principles calculation ( string based action) Drawback: limited dynamical range (box L) > (horizon a) >> (string width 1/a) > (lattice spacing=l/n grid ) Measure of hierarchy between [string width]/[horizon size]: ( ) H 3 / 2 N grid In reality, ζ should be ~10 8 at PQ SSB (for f a =10 10 GeV). Simulation results needs to be extrapolated by orders of magnitude. Qualitative understanding of string network dynamics cannot be more important.

9 Numerical computation Only small-scale (N grids on PC) simulations were available so far. 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. Approved as Collaborative Interdisciplinary Program at Tsukuba U Computer Center. COMA clusters at Tsukuba Using ~1000 CPUs, ~2TB memory, resolution is enhanced to (500 times in memory, 4000 times in CPU time).

10 Setup & Initial condition Background: radiation domination / R Boundary condition: periodic t = R 2 / R2 2H = 1 t Box size (comoving) : 2/H(t) at the time of simulation ends Initial condition: thermal distribution at T~T cr (=f a ) h (k, ) (k 0, ) i = 1!(k, ) 1 e!(k, )/T ( ) 1 (2 )3 (3) (k k 0 ) T<f a T>f a V ( ) h (k, ) (k 0, ) i =!(k, ) e!(k, )/T ( ) 1 (2 )3 (3) (k k 0 ) Other correlations vanish.

11 Identification of strings Identification of strings non-trivial due to discrete nature of lattice Our method Hiramatsu + (2010): Phase of Φ on all the vertices of a convex penetrated by a string ranges > π. convex [real space] [field space] OK even if string is moving. Δθ>π 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.

12 Velocity & relaxation 2 ζ cr =30 (g * =10 2, η/m * =5x10-3 ) ζ cr =25 (g * =10 3, η/m * =2x10-3 ) ζ cr =15 (g * =10 2, η/m * =1x10-2 ) ζ cr =9.5 (g * =10 3, η/m * =5x10-3 ) previous simulations String velocity Yamaguchi & Yokoyama 2002 v(x, ) = (r r ) ( r r ) (r r ) 2 rms <v 2 > 1/2 and Lorentz factor <γ> Lagrangian viewpoint v(x, ) r (x, )+ (x, ) =0 Projecting onto surface normal to the string direction, ir r. About a few Hubble time after PQ PT, strings become completely relaxed. Previous sim. marginally reached this. previous simulations ζ(τ)=(τ/τ cr ) 2 ζ(τ cr ) Mildly relativistic with v~0.5. Parameter independence is confirmed.

13 Scaling behaviour string prameters of infinite strings w/ L> 2πτ length fraction of loops w/ L<2πτ ζ cr =30 (g * =10 2, η/m * =5x10-3 ) ζ cr =25 (g * =10 3, η/m * =2x10-3 ) ζ cr =15 (g * =10 2, η/m * =1x10-2 ) ζ cr =9.5 (g * =10 3, η/m * =5x10-3 ) previous simulations previous simulations ζ(τ)=(τ/τ cr ) 2 ζ(τ cr ) ζ cr =30 (g * =10 2, η/m * =5x10-3 ) ζ cr =25 (g * =10 3, η/m * =2x10-3 ) ζ cr =15 (g * =10 2, η/m * =1x10-2 ) ζ cr =9.5 (g * =10 3, η/m * =5x10-3 ) previous simulations ζ(τ)=(τ/τ cr ) 2 ζ(τ cr ) String parameter Effective # of strings per horizon volume length t 2 ξ~0.8(~constant) - Scaling behaviour is realized. - Consistent with previous analysis. (cf. ξ>10 for local string) Loop fraction For the parameters probed, fraction of loops in total length is10-20%, nearly constant of time. There s a tendency that for larger ζ cr, loop production becomes more important.

14 Axion emission (1) Estimation of energy spectrum of axion radiation Not trivial Need to remove string cores Our method Hiramatsu+(2011): pseudo-power spectrum - statistical reconstruction (used in CMB analysis) - Masking & deconvolution string cores da/dt in simulation slice Energy emission rate into axion radiation Qualitatively consistent with one-scale model: d rad dt = 4H rad + NG string dr 4 NG d with / 2 R 3 cr NG = apple/l / t 1 differential radiation energy d(r 4 ρ rad )/dτ in units of η 2 3 /(Rτ cr ) ζ=30 (g * =10 2, η/m * =5x10-3 ) ζ=25 (g * =10 3, η/m * =2x10-3 ) ζ=15 (g * =10 2, η/m * =1x10-2 ) ζ=9.5 (g * =10 3, η/m * =5x10-3 ) ζ(τ)=(τ/τ cr ) 2 ζ(τ cr )

15 Axion emission (2) Mean momentum of radiated axions hki = R dk k 2 (k) 2 R dk k (k) 2 k ~a few Hubble consistent with Davis & Shellard (1988): Energy dissipation of strings into radiation proceeds in an adiabatic way. axion radiation parameter ε ζ=30 (g * =10 2, η/m * =5x10-3 ) ζ=25 (g * =10 3, η/m * =2x10-3 ) ζ=15 (g * =10 2, η/m * =1x10-2 ) ζ=9.5 (g * =10 3, η/m * =5x10-3 ) hki/ ζ(τ)=(τ/τ cr ) 2 ζ(τ cr ) Relic abundance of axion CDM from strings Assuming scaling behaviour (i.e. ρ string ), we can estimate the # of emitted axions: ṅ axion string /hki The # of axions emitted before QCD PT should conserve through QCD PT. CDM abundance from strings (> coherent oscillation & DW) 1.2 [ axion h 2 f a ] strings =(1.7 ± 0.9) GeV

16 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.

17 Understanding scaling behaviour (2) Thermodynamic equation based on velocity-dependent one-scale model infinite strings loops axion radiation d 1 dt = 2H(1 + v 2 1) 1 cv 1 L 1 1 d l dt = 2H(1 + v2 l ) l + cv 1 L 1 1 d NG dt = 4H loop NG apple redshift production apple 1 L 1 apple l L l axion emission apple 1 L 1 + l L l Mean momentum of emitted axions: apple 1 with ε~3 L 1 + l L l ' string t axion radiation parameter ε ζ=30 (g * =10 2, η/m * =5x10-3 ) ζ=25 (g * =10 3, η/m * =2x10-3 ) ζ=15 (g * =10 2, η/m * =1x10-2 ) ζ=9.5 (g * =10 3, η/m * =5x10-3 ) ζ(τ)=(τ/τ cr ) 2 ζ(τ cr )

18 Understanding scaling behaviour (3) From Eq for infinite strings: d ln d ln t = 1 ln +1 v2 1 From Eq for loop fraction: d d ln t apple l 1 with B A = A apple l 1 B A A ' apple 1 l p cv1 p (v 2 1 v 2 l ) B ' cv 1 p B/A should be ~0.1. p ( cv1 + apple) 0 string prameters of infinite strings w/ L> 2πτ length fraction of loops w/ L<2πτ ζ cr =30 (g * =10 2, η/m * =5x10-3 ) ζ cr =25 (g * =10 3, η/m * =2x10-3 ) ζ cr =15 (g * =10 2, η/m * =1x10-2 ) ζ cr =9.5 (g * =10 3, η/m * =5x10-3 ) ζ(τ)=(τ/τ cr ) 2 ζ(τ cr ) ζ cr =30 (g * =10 2, η/m * =5x10-3 ) ζ cr =25 (g * =10 3, η/m * =2x10-3 ) ζ cr =15 (g * =10 2, η/m * =1x10-2 ) ζ cr =9.5 (g * =10 3, η/m * =5x10-3 ) ζ(τ)=(τ/τ cr ) 2 ζ(τ cr ) These two conditions leads apple ' 1 (1 v 2 1 (ln ) 1 ) ' 0.2 cv 1 ' ( p 1 1 )(1 v 2 1 (ln ) 1 ) ' 0.4 Consistent with Yamaguchi & Yokoyama (2002) Loop production & NG emission are comparably important.

19 Understanding scaling behaviour (4) Consistency check: axion emission rate 100 From one-scale model: R 4 NG d = 2 R cr 3 8 apple ln differential radiation energy d(r 4 ρ rad )/dτ in units of η 2 3 /(Rτ cr ) Putting apple ' ' 1 ' 3 10 ζ=30 (g * =10 2, η/m * =5x10-3 ) ζ=25 (g * =10 3, η/m * =2x10-3 ) ζ=15 (g * =10 2, η/m * =1x10-2 ) ζ=9.5 (g * =10 3, η/m * =5x10-3 ) ζ(τ)=(τ/τ cr ) 2 ζ(τ cr ) we obtain R 4 NG d ' 90 2 R 3 cr in agreement with direct estimation from sim. We for the first time confirm the consistency of one-scale model description of global strings with significant backreaction.

20 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 (N grid <512 3 ) are limited in simulation time and suffer from fine tuning of parameters. We performed a largest-ever simulation (N grid = ) 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. 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 f a has not changed, i.e.,. f a. ( ) GeV

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