Loops in inflation with strongly non-geodesic motion

TL;DR

The paper investigates one-loop corrections to scalar and tensor power spectra in an EFT of inflation with imaginary speed of sound, a setup that models strongly non-geodesic multi-field dynamics with heavy entropic perturbations. It develops a complete framework based on the EFT with a finite validity domain, employing the in-in formalism and a momentum cutoff to compute loop corrections from cubic and quartic vertices. The key findings are that scalar loops from two cubic vertices scale as $x^5$ (with additional enhancement at small $|c_s|$), while tensor loops from two cubic vertices are exponentially enhanced relative to the tree-level tensor spectrum, imposing stringent perturbativity bounds. Quartic-vertex loops are subleading for scalars but induce logarithmic IR divergences in tensors, which are regulated by an infrared cutoff. Overall, perturbativity in this class of models is achievable only within a limited region of parameter space, and the work provides a careful, generalizable prescription for handling loop integrals in EFTs with restricted validity domains.

Abstract

We study loop corrections in the effective field theory of inflation with imaginary speed of sound, which has been shown to provide an effective description of multi-field inflationary models characterized by strongly non-geodesic motion and heavy entropic perturbations. We focus on the one-loop corrections to the scalar and tensor power spectra, taking into account all relevant vertices at leading order in derivatives and in slow-roll. We find a power-law dependence of the scalar two-point function on the scale that defines the range of validity of the effective theory, analogous to the enhancement observed in tree-level correlation functions. Even more dramatic, the relative correction to the tensor spectrum is exponentially enhanced, albeit also suppressed in the slow-roll limit. In spite of these large effects, our results show that this class of models can satisfy the requirement of perturbative control and a consistent loop expansion within a range of parameters of phenomenological interest. On the other hand, models predicting large values of the power spectrum on small scales are found to be under strong tension. As a technical bonus, we carefully explain the prescription for the regularization and manipulation of loop integrals in this set-up, where one has a non-trivial domain of integration for time and momentum integrals owing to the regime of validity of the effective field theory. This procedure is general enough to be of potential applicability in other contexts.

Figures (10)

• Figure 1: The integration domain in the loop diagram with one quartic vertex.
• Figure 2: The integration domain for the momentum integrals in the loop diagram with two cubic vertices. The gray region corresponds to the domain after introducing the EFT cutoff, which may be divided into three parts (indicated by the green dashed lines) for the purpose of calculating the integrals. The two graphs show the two qualitatively distinct cases arising from this division.
• Figure 3: The integration domain for the momentum integrals in the loop diagram with two cubic vertices, in the $(t,s)$ plane (cf. \ref{['eq:ts variables']}). The gray region corresponds to the domain after introducing the EFT cutoff, which may be divided into four parts ($A$, $B$, $C$ and $D$) for the purpose of calculating the integrals (see the main text for the rationale behind this division).
• Figure 4: Leading scalar one-loop diagrams in the large-$x$ approximation. The sign $\pm$ associated with each mode indicates if it is growing or decaying, see Eq. \ref{['eq:growing-decaying modes']}.
• Figure 5: Consistency with perturbativity as dictated by the criterion $\mathcal{P}_{\zeta}^{\rm(1-loop)}/\mathcal{P}_{\zeta}^{\rm(tree)}<1$ (colored regions) as function of $x$ and $|c_s|$, for several values of $\mathcal{P}_\zeta$ (the tree-level dimensionless power spectrum). Solid lines:$\mathcal{P}_{\zeta}^{\rm(1-loop)}$ as given by Eq. \ref{['eq:1loop_cubic']} with $\mathcal{A}=1$, i.e. the case of the diagram with two insertions of $H_{\zeta\zeta\zeta}^{(1)}$. Dashed lines: The same in the case of the loop diagram with two insertions of $H_{\zeta\zeta\zeta}^{(2)}$, cf. Eq. \ref{['eq:1loop_cubic2']}. Note that in both cases the late-time limit $\eta\to0^-$ has been taken.