Audible Axions with a Booster: Stochastic Gravitational Waves from Rotating ALPs

1- Interesting framework to produce gravitational waves detectable by the current and future experiments, which could also address the dark matter and baryogenesis problem.

1- Oversimplification of the evolution of the complex field $P$ (lack of an UV-complete model).

The present paper discusses the production of gravitational waves generated by dark photons tachyonic instability, due to the evolution of axions or axion-like particles (ALP). Here the authors use the kinetic misalignment mechanism instead of the conventional misalignment mechanism. They present parameter space where tachyonic production of dark photons becomes efficient and the GW spectra for the different benchmark parameter points. Furthermore, the authors estimate relic abundance of the ALP and when the ALP is the QCD axion, with the possibility of produce the baryon asymmetry by the axiogenesis mechanism. Finally, the authors find it is possible that the dark photon can be the DM too.

The paper builds upon earlier work by the authors, it is well written and summary of the relevant mechanisms at work, and leads to interesting predictions that may serve as a benchmark in upcoming gravitational wave searches. It is therefore suitable for publication in SciPost. However, before it proceeds to publication, I would like to ask the authors to perform a minor revision of their manuscript in which they address the following questions and comments.

1-In Figure 4 left, the authors show the GW spectrum of the plus and minus helicities. They explain that one helicity is enhance over the other depending on the sign of the axion velocity. Naively, I would have expected the difference to be bigger, meaning almost zero for the minus helicity, not ~$\mathcal{O}(100)$. It would be nice to understand where this difference might be coming from analytically.

2-In figure 7 the authors show the excluded region for $f_{\phi} \gtrsim 3.4 \times 10^{7} \textrm{GeV} $ due to stellar constrains, but looking into references [47] and [48] these constraints are more stringer. I wonder why there are different.

3- In Table 1 the saxion mass should be $m_{S,0}$ instead of $m_{s}$

4-For very light saxion masses, the coupling $\lambda$ would be subject to quantum correction which will imposes some constrain in $y_{\psi}$. I think is worth of explaining this point.

1-Key ideas explained very clearly
2-Analytical estimates demonstrating the key parameter dependencies are backed up by numerical simulations
3-Testable model addressing open questions in cosmology

The present manuscript studies the generation of gravitational waves, dark matter and a matter-antimatter asymmetry from an axion(-like) particle coupled to dark photons. Extending earlier work on `audible axions' the authors allow for a large initial kinetic energy of the axion, sourced by PQ violating terms in the potential. Large parts of the presented model are testable via future gravitational wave observations, with complementary constraints arising from axion searches. The paper is well written and the key ideas are explained clearly. Analytical estimates demonstrating the key parameter dependencies are backed up by numerical simulations, resulting in a convincing picture. I only have a few questions (see below) before recommending this paper for publication.

1- Eq. (3.13) is UV divergent. How is this quantity regulated to ensure that the UV divergent vacuum contribution does not impact the axion dynamics?
2- If I understand correctly, the radial part of PQ field is taken to decrease due to Hubble friction before becoming constant once reaching the minimum of the Mexican hat potential. With the total PQ charge scaling as $S^2 \dot\theta \sim a^{-3}$, the decrease in $S$ becomes non-trivial once $\dot\theta$ is no longer constant due to energy transfer between the axion motion and the dark photons. Could this significantly impact any of the predictions?
3- With the arguments presented, I'm not entirely convinced that Eq. (5.3) is a lower bound. It rather seems to me like an estimate based on observations in the numerical simulations? It would be nice to have some analytical understanding as to why energy density in the axion field is well described by a kination-like scaling starting at $a_\text{GW}$.