Axion dark matter from extended misalignment with a constant-$\omega_\phi$ pre-oscillatory phase and dark radiation

In this work, we extend the standard pre-inflationary misalignment mechanism for axion-like particles (ALPs) by introducing a pre-oscillatory phase with constant equation of state $\omega_\phi\in[-1,1]$, generated by a tracking potential. During the radiation-dominated era, the potential undergoes a rapid transition to the conventional cosine potential. The resulting change in the potential energy across the transition can drive the ALP into a kinetic misalignment phase ($\omega_\phi=1$) prior to the onset of oscillations. Motivated by persistent cosmological tensions, such as those in $H_0$ and $S_8$, we also investigate an ALP coupling to a dark radiation sector (DR), allowing for its decay. Using a Bayesian analysis, we constrain the ALP parameter space with current cosmological data. Our analysis shows that ALP-induced DR does not resolve the existing tensions. Instead, the data place robust constraints on the model, favoring negative values of $\omega_\phi$ and constraining the symmetry-breaking scale to $f_\phi\in[80,1.5\times10^{10}]~\mathrm{TeV}$, corresponding to ALP masses in the range $m_\phi\in[10^{-20},10^{-2}]~\mathrm{eV}$.