This paper is available on arxiv under CC 4.0 license.

**Authors:**

(1) HARRISON WINCH, Department of Astronomy & Astrophysics, University of Toronto and Dunlap Institute for Astronomy and Astrophysics, University of Toronto;

(2) RENEE´ HLOZEK, Department of Astronomy & Astrophysics, University of Toronto and Dunlap Institute for Astronomy and Astrophysics, University of Toronto;

(3) DAVID J. E. MARSH, Theoretical Particle Physics and Cosmology, King’s College London;

(4) DANIEL GRIN, Haverford College;

(5) KEIR K. ROGERS, Dunlap Institute for Astronomy and Astrophysics, University of Toronto.

## Table of Links

- Abstract and Intro
- Methods
- Phenomenology
- Discussion and Future Work
- Conclusion
- Acknowledgments and References

## 4. DISCUSSION AND FUTURE WORK

Although comparison to LSS likelihoods from galaxy surveys, and CMB likelihoods for the lensing, temperature, and polarization power spectra are the most straightforward, the tightest current constraints on axions come from measurements of the Ly-α forest, as these are able to probe the MPS at much smaller scales than either galaxy surveys or the CMB (Rogers & Peiris 2021). However, comparing MPS predictions for extreme axions to data from the Ly-α forest is more difficult, as it requires hydrodynamical simulations of the small-scale nonlinear structure, which in principle could depend on the nonlinear behaviour of the extreme axion model. In this paper, we used the estimates of the linear MPS from the Ly-α forest data, which assumed CDM for the small-scale structure evolution, but this method is only valid in the lowaxion-density regime, where CDM makes up most of the dark matter. Some work has been done modeling the nonlinear Ly-α forest for extreme axions(Leong et al. 2019), but this simulation is computationally expensive. Ideally, the best approach would be to train an emulator to produce extreme axion predictions of the Ly-α data, similar to what was done in Rogers & Peiris (2021). When combined with our modified axionCAMB, this could allow for rapid computation and direct comparison to Ly-α forest data, which would give the most informative constraints on the small-scale behaviour of these extreme axion models. In addition, direct comparison to Ly-α observables would allow us to use higher resolution spectroscopic surveys, such as those done with Keck or VLT Lu et al. (1996); Irsiˇ c et al. ˇ (2017b).

Accurate simultaneous constraints on the axion mass, density fraction, and starting angle, would quantitatively address an important question that, so far, has only been approached qualitatively: namely, the required degree of fine-tuning for these extreme axion models to work. Figure 10 shows that a good agreement with data can be reached with axion starting angles that are close to the peak, seperated by less than 10%. Arvanitaki et al. (2020) have proposed a model that could drive the axion field to start near the peak at extremely early times, but the plausibility of these models would depend on exactly how much fine tuning is required. This required degree of fine-tuning depends on the axion mass and density fraction, as seen in Figures 9, 10, and 11, and could also depend on other cosmological parameters. With our modified axionCAMB, we could create estimates of the necessary degree of fine-tuning for a range of axion and cosmological parameters, helping to inform the plausibility of these models that produce starting angles close to π.

Another area worth exploring is comparing these constraints to forecast sensitivities by future CMB experiments, such as the Simons Observatory, and CMB-S4 (Hlozek et al. ˇ 2017; Lee et al. 2019; Dvorkin et al. 2022; Abazajian et al. 2022). Although Planck is already cosmic-variance limited for temperature at low-ℓ, there may be substantial improvements to be made with an experiment with better polarization and/or high-ℓ data (Aghanim et al. 2016). CMB lensing also offers the ability to probe the DM MPS at a range of scales(Rogers et al. 2023). We could also experiment with simultaneous constraints from CMB and MPS sources. Direct probes of the MPS can also be used to constrain the extreme axion model, including the Dark Energy Survey (which we used to constrain the vanilla axion model in Dentler et al. 2022), Euclid (Amendola et al. 2018), JWST (Parashari & Laha 2023), and the Vera Rubin Observatory (Mao et al. 2022).

Lastly, we could try constraining potentials beyond just the standard cosine shape. Models have been proposed with axions possessing quartic, hyperbolic cosine, or monodromic potentials (Cembranos et al. 2018; Urena L ˜ opez ´ 2019;Jaeckel et al. 2017). In addition, axion-like scalar fields with a variety of potentials have been proposed as an early dark energy component potentially capable of relieving the Hubble tension (Kamionkowski & Riess 2022; Poulin et al. 2023). Axion perturbations in all of these potentials could conceivably be modeled using our modified axionCAMB, since the potential function is implemented generically. The only requirement would be that the potential being tested must simplify to a quadratic at small ϕ values, in order for the particle DM approximation to be valid at late times.