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Recently, based on a novel analysis of the Planck satellite data, a hint of auniform rotation of the polarization of cosmic microwave background photons,called isotropic cosmic birefringence, has been reported. The suggestedrotation angle of polarization of about $0.2-0.4$ degrees strongly suggeststhat it is determined by the fine structure constant, which can be naturallyexplained over a very wide parameter range by the domain walls of axion-likeparticles. Interestingly, the axion-like particle domain walls predict not onlyisotropic cosmic birefringence but also anisotropic one that reflects thespatial distribution of the axion-like particle field on the last scatteringsurface. In this Letter, we perform lattice simulations of the formation andevolution of domain walls in the expanding universe and obtain for the firsttime the two-point correlation function and power spectrum of the scalar fieldthat constitutes the domain walls. We find that while the power spectrum isgenerally consistent with analytical predictions based on random walldistributions, there is a predominant excess on the scale corresponding to theHubble radius. Applying our results to the anisotropic cosmic birefringence, wepredict the power spectrum of the rotation angles induced by the axion-likeparticle domain walls and show that it is within the reach of futureobservations of the cosmic microwave background.

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