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]
c Axion: motivation Dynamical solution to Strong CP problem in QCD θ μν ˜ → CP violation 32 π G a G a μν ℒ ⊃ 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. Candidate of cold dark matter 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. f a In relevant mechanisms, produced axions are non-relativistic (see below).
Axion cosmology Categorized into two scenarios.
Axion cosmology Categorized into two scenarios. T > f a U(1) PQ spontaneously breaks after inflation T < f a 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 N DW =1, DWs are unstable and disappear. CDM axions are produced from these topological defects as well as misalignment. axion
Axion cosmology Categorized into two scenarios. T > f a U(1) PQ spontaneously breaks after inflation T < f a 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 N DW =1, DWs are unstable and disappear. CDM axions are produced from these topological defects as well as misalignment. 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. − π f a π f a − π axion θ i CDM are produced predominantly from the misalignment mechanism.
Axion cosmology Categorized into two scenarios. T > f a U(1) PQ spontaneously breaks after inflation T < f a 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 N DW =1, DWs are unstable and disappear. CDM axions are produced from these topological defects as well as misalignment. 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. − π f a π f a − π axion θ i CDM are produced predominantly from the misalignment mechanism.
Axion strings Topological defect associated to broken U(1) PQ real space field space
c Axion strings Topological defect associated to broken U(1) PQ Axion strings differ from local (abelian Higgs) ones: • Tension real space a ln ( f a L ) μ ≃ π f 2 Logarithmically divergent IR cutoff ~string separation L (~horizon scale t) • Long range force Potentialc ∼ ∝ ln( f a L ) field space • 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
c Axion strings Topological defect associated to broken U(1) PQ Axion strings differ from local (abelian Higgs) ones: • Tension real space a ln ( f a L ) μ ≃ π f 2 Logarithmically divergent IR cutoff ~string separation L (~horizon scale t) • Long range force Potentialc ∼ ∝ ln( f a L ) field space • 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 Presence of axion makes characteristic parameters of axion strings time-dependent.
c c Simulation of axion strings Field theoretic simulation on a lattice: first principles calculation Oakforest-Pacs cluster V [ Φ ] = ( | Φ | 2 − f 2 2 2 ) R 2 Φ = ∂ V [ Φ ] Φ + 3 Φ − 1 ·· · a In radiation domination with a wine bottle potential c ∂Φ * 2 t . R ∝ t Grid number of our simulation c N grid = 4096 3 . Physical string simulations (not fat strings).
c c Simulation of axion strings Field theoretic simulation on a lattice: first principles calculation Oakforest-Pacs cluster V [ Φ ] = ( | Φ | 2 − f 2 2 2 ) R 2 Φ = ∂ V [ Φ ] Φ + 3 Φ − 1 ·· · a In radiation domination with a wine bottle potential c ∂Φ * 2 t . R ∝ t Grid number of our simulation c N grid = 4096 3 . Physical string simulations (not fat strings). horizon Why are string simulations hard? Two different scales should be incorporated: − 1 • String core width c ∼ f a string • Horizon size c ∼ t f a t ≲ N 1/3 grid = O (10 3 ) ➡ Dynamic range is limited: c Simulation results need to be extrapolated by many orders of magnitude. In reality, c at QCD PT. f a t QCD ∼ O (10 30 )
long loops R=9.8 short loops
long loops R=10.0 R=10.0 short loops
long loops R=10.2 short loops
long loops R=10.4 short loops
� c � � String velocity Velocity estimator on a lattice Yamaguchi & Yokoyama '02 v ( x , t ) ⋅ ∇Φ ( x , t ) + · Lagrangian view point: c . Φ ( x , t ) = 0 Projection onto surfaces normal to string direction. v ( x , t ) = ( ∇Φ × ∇Φ *) × ( · Φ∇Φ * − · 1 Φ * ∇Φ ) ζ =9.5 (v/M * =5x10 -3 ) 2 f a / M Pl = 1 × 10 − 3 ζ =23.9 (v/M * =2x10 -3 ) = 2 × 10 − 3 ( ∇Φ × ∇Φ *) 2 ζ =47.7 (v/M * =1x10 -3 ) = 5 × 10 − 3 0.8 rms of velocity � ⟨ v 2 ⟩ 1/2 rms of velocity 〈 v 2 〉 1/2 0.6 Results: non-relativistic strings 0.4 ⟨ v 2 ⟩ 1/2 ≃ 0.25 Velocity rms c ⟨ v 2 ⟩ 1/2 ≃ 0.5 Significantly smaller than local strings (c ). 0.2 0 10 100 1000 physical time t/d physical time � f a t
c 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). L = ( + ( + ( ( dt ) total dt ) loop production dt ) loop absorption dt ) axion emission dL ∞ dL ∞ dL ∞ dL ∞ In simulation we define infinite strings as loops with circumference c . ℓ > t
c c 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). L = ( + ( + ( ( dt ) total dt ) loop production dt ) loop absorption dt ) axion emission dL ∞ dL ∞ dL ∞ dL ∞ In simulation we define infinite strings as loops with circumference c . ℓ > t 2 Contribution of each loss channel preliminary d ln t ) 1 d ln L ∞ 0.2 ± 0.05 (loop production) − ( d ln t ) = d ln L ∞ � − ( − 0.05 ± 0.05 (loop absorption) 0 0.85 ± 0.17 (axion emission) -1 Axion emission predominates the energy loss of infinite strings. scale factor � R = t / t PQ
How to compute axion abundance (Energy loss of strings) c (Number of radiated axions) ≈ (Mean energy of radiated axions)
How to compute axion abundance (Energy loss of strings) c (Number of radiated axions) ≈ (Mean energy of radiated axions) String evolution: scaling behavior Empirical law. The number of strings per horizon has been known to stay O(1) after relaxation. Kibble (1985); Bennet (1986); Martin & Shellard (2002); … horizon string
c How to compute axion abundance (Energy loss of strings) c (Number of radiated axions) ≈ (Mean energy of radiated axions) String evolution: scaling behavior Energy spectrum of radiated axions Empirical law. The number of strings per horizon has Computed from kinetic energy of axions: been known to stay O(1) after relaxation. · ϕ = Im [ · Φ / Φ ] with c Φ = | Φ | exp[ i ϕ / f a ] Kibble (1985); Bennet (1986); Martin & Shellard (2002); … Less string contamination than gradient part. horizon string cores · c in simulation ϕ Not a trivial task in simulation —- string core should be removed. Pseudo-power spectrum estimator method Hiramatsu et al. (2012) • statistical reconstruction used in CMB analysis • masking & deconvolution string
� � � � c Testing scaling law 2 ζ =9.5 (v/M * =5x10 -3 ) 2 f a / M Pl = 1 × 10 − 3 ζ =23.9 (v/M * =2x10 -3 ) = 2 × 10 − 3 ζ =47.7 (v/M * =1x10 -3 ) = 5 × 10 − 3 previous result w/ ζ =9.5 (v/M * =5x10 -3 ) = 1 × 10 − 3 (previous) 1.5 with N grid =512 3 string parameter � ξ String parameter string parameter ξ 1 ρ string t 2 ρ string t 3 ξ ≡ = μ μ t 0.5 average number of strings per horizon Previous dynamic range No loop removal 0 10 100 1000 physical time � f a t physical time t/d
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