Is cosmic birefringence due to dark energy or dark matter? Simulation-based inference

Simulation-based inference (SBI) is a powerful inference technique for cases where the exact functional form of the likelihood is not known. A prime example is the likelihood of cross-correlation power spectra of the cosmic microwave background (CMB) fields at low multipoles, $\ell\lesssim 10$. In this paper, we investigate a parity-violating cross-correlation between $E$- and $B$- mode polarization fields using SBI. The $EB$ correlation at low $\ell$ is essential to distinguish between possible axion dark energy and dark matter interpretations of `cosmic birefringence', a rotation of the plane of linear polarization of the CMB, recently reported from WMAP, Planck, and Atacama Cosmology Telescope data. We use neural likelihood estimation to infer the likelihood of the $EB$ correlation at low $\ell$ and show that it is highly non-Gaussian. We then employ neural posterior estimation to constrain the scalar field mass ($m_ϕ$), the cosmic birefringence amplitude ($gϕ_\mathrm{in}/2$), and the instrumental miscalibration angle ($α$), from simulated datasets. We find that the posterior on $m_ϕ$ shows two regimes, with a transition marked by $10^{-32}$ eV, highlighting a strong sensitivity to the scale dependence of cosmic birefringence. To quantify this behavior, we compute the probability $p(m_ϕ < 10^{-32}$ eV) for various fiducial values of $m_ϕ$. We find that $α$ and the contribution of lensed $B$ modes ultimately limit our ability to exclude the dark energy scenario fully.