Exact CMB $B$-mode power spectrum from anisotropic cosmic birefringence

We calculate the cosmic microwave background $B$-mode power spectrum resulting from anisotropic cosmic birefringence, without relying on the thin approximation of the last scattering surface. Specifically, we consider the influence of anisotropic cosmic birefringence arising from massless axionlike particles. Comparing our results to those obtained using the thin approximation, we observe a suppression in the amplitude of the $B$-mode power spectrum by approximately an order of magnitude at large angular scales ($\ell \lesssim 10$) and by a factor of 2 at small angular scales ($\ell \gtrsim 100$) when not employing the thin approximation. We also constrain the amplitude of the angular power spectrum of the scale-invariant anisotropic cosmic birefringence using the SPTpol $B$-mode power spectrum. We find that the amplitude is constrained as $A_{\mathrm{CB}}\times {10}^4={1.03}_{-0.97}^{+0.91}\left(2\sigma \right)$.