Power spectrum of domain-wall network and its implications for isotropic and anisotropic cosmic birefringence

Recently, based on a novel analysis of the Planck satellite data, a hint of a uniform rotation of the polarization of cosmic microwave background photons, called isotropic cosmic birefringence, has been reported. The suggested rotation angle of polarization of about $0.2-0.4$ degrees strongly suggests that it is determined by the fine structure constant, which can be naturally explained over a very wide parameter range by the domain walls of axion-like particles. Interestingly, the axion-like particle domain walls predict not only isotropic cosmic birefringence but also anisotropic one that reflects the spatial distribution of the axion-like particle field on the last scattering surface. In this Letter, we perform lattice simulations of the formation and evolution of domain walls in the expanding universe and obtain for the first time the two-point correlation function and power spectrum of the scalar field that constitutes the domain walls. We find that while the power spectrum is generally consistent with analytical predictions based on random wall distributions, there is a predominant excess on the scale corresponding to the Hubble radius. Applying our results to the anisotropic cosmic birefringence, we predict the power spectrum of the rotation angles induced by the axion-like particle domain walls and show that it is within the reach of future observations of the cosmic microwave background.