Post-inflationary axion constraints from the Lyman-$\alpha$ forest

Among the most compelling cold dark matter candidates, the axion has recently been subject to a wide range of astrophysical studies aiming to constraints its properties. We present updated bounds on the isocurvature fraction, $f_{\rm{iso}}$, which parameterizes the contribution of isocurvature perturbations induced by post-inflationary produced axion-like particles (ALPs) to the ordinary power spectrum. We use new simulations based on the Sherwood-Relics suite to fit high-resolution Lyman-$\alpha$ forest flux power spectrum data. With the published noise model of the Lyman-$\alpha$ forest data, we find a tentative detection of $f_{\rm{iso}}$ = ${0.0064^{+0.0012}_{-0.0014}}$ (68% C.L), after accounting for the degenerate effect of IGM thermal evolution. With a more conservative modelling of the residual noise in the data, the upper bound is weakened to $f_{\rm{iso}}< 0.0084$ (95% C.L), which translates into an ALP temperature-independent mass $m_a > 1.73 \times 10^{-18}$eV. Our constraints are stronger than bounds derived from large-scale structure probes at higher and lower redshifts and are competitive with those derived from UV luminosity function data. Interestingly, the best current Lyman-$\alpha$ forest data prefers a non-zero contribution from isocurvature modes.