Post-Inflationary Scenario

• Random initial axion phases
• PQ-symmetry breaking after inflation
• tilt of the Mexican hat potential
• axion roles down to potential minimum if possible, but topological defects
• roughly one string per Hubble patch
• strings annihilate or shrink  radiating axions
• What is their final phase space density?

Axion Strings 

strings = 1D-loops with phase changing 2pi around them

energy in phase gradient diverges - but decreasing modulus

phase

modulus

Scale Separation

inter-string separation

string core size

axion decay constant

Hubble scale

• Cosmic expansion increases separation (solution PRS strings)
• not feasible on 3D uniform grids
• possible solution: adaptive mesh refinement around 1D strings (1024^3 root grid and 10 additional levels -> 14 e-folds)

[Drew, Shellard, 2019,arXiv:1910.01718]

AMR can improve current

  extrapolations

[Gorghetto et.al., arXiv: 1806.04677, 2007.04990 ]

Initial Conditions

• Diffuse random initial phase
• longer diffusion time -> larger (better resolved) Hubble patches
• start with constant modulus

Adaptive Mesh Refinement (AMR)

• Tag cells for refinement using string plaquettes [ Leesa Fleury, Guy D. Moore,  2015, arXiv:1509.00026 ]
• create new grids with progressively more resolution on higher levels
• Use MPI communication to transfer interpolated data between grids and levels
• subcycling: use smaller time steps on higher resolved levels
• evolve with spectral method on root grid and finite differencing on higher levels

We extended AxioNyx for AMR string simulations

Axion Energy Spectrum

Energy spectra between high resolution uniform grid and AMR simulations match up to half the Nyquist frequency

Conclusion

• Axion string simulations suffer from large separation of scales between string core and network size
• extreme extrapolations over orders of magnitude are neccessary
• AMR simulations can cross important resolution thresholds to make extrapolations more predictive
• Important to constrain the final axion phase space density