On the One Loop Corrections to Inflation II: The Consistency Relation

TL;DR

This paper analyzes one-loop quantum corrections to inflation in general monomial potentials, focusing on infrared-enhanced effects that accumulate with the total number of e-foldings. Using the Schwinger-Keldysh formalism and the super-horizon (SH) limits of the third- and fourth-order actions, it derives corrections to the background evolution and to the inflaton two-point function, finding that in chaotic $m^2\phi^2$ inflation the loop-corrected power spectrum can be boosted by up to $\sim 70\%$ for certain horizon-exit times, while for $\lambda\phi^4$ it remains at the few- to mid-percent level. The work discusses the physical implications for the tensor-to-scalar ratio and the tensor-scalar consistency relation, showing that loop effects can blur the predicted $(n_s,r)$ relation and thus influence model discrimination. It also connects these results to Weinberg's broader analysis of quantum contributions to cosmological correlators and emphasizes the need for tensor-loop calculations and possibly higher-order corrections for a complete perturbative picture.

Abstract

In this paper we extend our previous treatment of the one-loop corrections to inflation. Previously we calculated the one-loop corrections to the background and the two-point correlation function of inflaton fluctuations in a specific model of chaotic inflation. We showed that the loop corrections depend on the total number of e-foldings and estimated that the effect could be as large as a few percent in a lambda-phi-four model of chaotic inflation. In the present paper we generalize the calculations to general inflationary potentials. We find that effect can be as large as 70% in the simplest model of chaotic inflation with a quadratic inflationary potential. We discuss the physical interpretation of the effect in terms of the tensor-to-scalar consistency relation. Finally, we discuss the relation to the work of Weinberg on quantum contributions to cosmological correlators.

Figures (4)

• Figure 1: The seagull diagram, which dominates the one-loop contribution to the two-point correlation function of inflaton field fluctuations.
• Figure 2: The relative one-loop correction $\delta\epsilon/\epsilon$ to the slow-roll parameter $\epsilon$. In the left panel we have plottet the one-loop correction in the case of a low reheating temperature for $N_*=45$. In the right panel we have shown the same plot, but now with $N_*=60$.
• Figure 3: The relative one-loop correction $\delta \mathcal{P}/\mathcal{P}$ to the power spectrum of inflaton fluctuations. In the left panel we have plottet the one-loop correction for $N_*=45$. In the right panel we have shown the same plot, but with $N_*=60$. We see that in the present approximation the maximal one-loop correction is of the order $5\%$ to $15\%$ for $\lambda\phi^4$, while it is $50\%$ to $70\%$ for a typical model of $m^2\phi^2$ chaotic inflation.
• Figure 4: The likelihood contours of the tensor-to-scalar ratio vs. the scalar spectral index Hamann:2006pf. Two models are indicated. On the edge of the $95\%$ exclusion likelihood contour is the predictions from the $\lambda\phi^4$ model while in the middle of the $68\%$ exclusion likelihood contour the predictions of the $m^2\phi^2$ model is indicated. We have indicated the model predictions with $45<N_*<60$. If we did not take into account one-loop corrections, the predictions would be line-shaped between the squares. The full polygons indicates qualitatively the theoretical uncertainty when the one-loop correction to the two-point correlation function of inflaton fluctuations are included.