The UV Sensitivity of Axion Monodromy Inflation

1 - This paper presents an interesting synergy between several research areas: axion monodromy motivated by UV physics leading to oscillatory couplings, enhanced cosmological collider signal in the presence of such oscillatory coupling, and observational prospect of upcoming cosmological surveys to discover the signal.

2 - The paper explains multiple theory results with reasonable clarity given its short length.

The authors present an interesting case study of cosmological collider signal enhanced by oscillatory feature in the inflaton potential, in the context of axion monodromy from string theory. The feature provides another energy scale in the system, and when it is sufficiently larger than Hubble, it allows heavy particles to imprint unsuppressed signals in cosmological observables, such as the curvature perturbation bispectrum. Depending on the parameter choice, the authors show that the bispectrum could be within reach of upcoming surveys.

The authors present comprehensive calculations from first principle, derive the background trajectory of both the inflaton and the heavy modulus, analyze the turning correction to the curvature and isocurvature degrees of freedom, and finally calculate the the shape and enhanced amplitude of the bispectrum signal. While the generalities of such feature-enhanced cosmological collider signal have been studied before, as the authors have cited in [19], this paper presents a concrete realization in a theoretically well-motivated model.

I do have some questions regarding the authors' claim that they find significant oscillatory couplings in not only the linear mixing between the curvature and isocurvature modes but also the cubic interaction, unlike in [19]. The authors calculate the mixing between the curvature and isocurvature modes using the EFT of inflation approach, and finally arrive at the Lagrangian in equation (9). First of all, I think the significance of the presence of the oscillatory cubic coupling for the cosmological collider signal is not very clear, since the authors say that the enhancement of the bispectrum is as expected from the analysis of [19]. Furthermore, I currently do not agree with some of the intermediate steps made to arrive at (9), and I request that my questions listed below are addressed before considering this paper for publication.

1- Above equation (9), it is claimed that the dominant oscillatory cubic interaction comes from the $\dot{\lambda}\pi \dot{\pi}\sigma$ term, using $\dot{\lambda} = \omega \lambda$. However, $\lambda(t)$, being the total field velocity $\dot{\Phi}_t$ (multiplied by the turning rate $\Omega$), is dominated by $\dot{\phi}_0$ and does not satisfy $\dot{\lambda} = \omega \lambda$; I would think that it should be suppressed by the ratio $\dot{\phi}_1/\dot{\phi}_0$.

2- Regardless of whether the $\dot{\lambda}\pi \dot{\pi}\sigma$ is enhanced with respect to $\lambda (\partial_{\mu}\pi)^2\sigma$ term, $\lambda$ consists of a leading, non-oscillatory $\dot{\phi}_0$ piece and subleading, oscillatory $\dot{\phi}_1$ and $\dot{\rho}_1$ pieces, so I think the cubic vertex in (9) should have similar $\bar{g}+g_3$ structure as the linear mixing $\bar{g}+g_2$. In [19] it is shown that an oscillatory linear mixing is essential to an enhanced bispectrum, hence it is important to include the oscillatory $g_2$ coupling even though it is subleading. It should be clarified whether it is the same case for $g_3$, where the subleading oscillatory piece is important compared to the leading, non-oscillatory piece for the cosmological collider signal.

3- Moreover, given that the authors have emphasized the presence of this oscillatory cubic interaction throughout the paper as an important distinction from the results in [19], I think the effect of the oscillatory cubic vertex $g_3$ in the cosmological collider signal should be discussed more explicitly, instead of being lumped together with $g_2$.

4- In the calculation of the background field trajectory around equation (6), the estimate of the size of $\rho$ oscillation $B\simeq b_* f^2/\Lambda$ relies on $\omega$ to be greater than but close to $m$, such that $\Xi^4 = 9\omega^2 H^2 + (\omega^2-m^2)^2 \approx 9 \omega^2 H^2$ instead of $\Xi^4 \approx \omega^4$. If $\omega \gg m$, $B$ would be further suppressed by a factor of $H/\omega$. Is the need for $\omega$ to be close to $m$ important for any of the subsequent analysis and cosmological collider signal?

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