Inflationary Axion Isocurvature in the CMB across All Ultralight Masses

Abstract

If the Peccei-Quinn symmetry of an ultralight axion is broken before the end of inflation, axion quantum fluctuations seed isocurvature perturbations, linking them to the tensor-to-scalar ratio $r$. We extend the effective time average (ETA) approach of the Boltzmann code ${\rm AxiECAMB}$ to accurately evolve these perturbations across the full axion mass range from dark energy ($m_a \lesssim H_0$) to dark matter ($m_a \gg 10^{-28}$ eV) types. We provide analytic fitting formulae for the axion abundance given the initial field value $\phi_{\rm ini}$, accurate at sub-percent level for $m_a \gg H_0$ and allowed dark matter fraction $f_{\rm dm}$. In the dark matter regime, the Planck bound on CDM isocurvature requires $r\ f_{\rm dm} < 0.08\,(m_a/10^{-27}\,{\rm eV})^{-1/2}$, which becomes stronger than the current BICEP tensor bound for $m_a f_{\rm dm}^2 \gtrsim 10^{-26.4}\,{\rm eV}$. For $10^{-32} \lesssim m_a/{\rm eV} \lesssim 10^{-28}$, Jeans suppression breaks the degeneracy with CDM isocurvature, leaving unique signatures, and in the dark energy regime ($m_a \lesssim H_0$), the isocurvature signal is even more highly suppressed, peaking only at the CMB quadrupole. We provide analytic scalings for both signatures. Given the tensor bound, any primary CMB detection in these two lightest regimes would indicate a non-inflationary origin of the isocurvature modes or a breakdown of the standard frozen-field misalignment scenario. A window of coexistence opens near $m_a \sim 10^{-25}$ eV and $f_{\rm dm} \gtrsim 0.1$ where both axion isocurvature and tensor modes could be discovered just below current bounds while simultaneously alleviating the $S_8$ tension.