Enhancing inflationary model predictions via refined slow-roll dynamics

TL;DR

The paper addresses the sensitivity of inflationary predictions to the details of slow-roll dynamics and reheating by developing a minimal, model-agnostic framework that emphasizes accurate inflation-era perturbations over detailed reheating modelling. It implements three key improvements: (i) numerically solving the full background inflationary equations to replace leading-order slow-roll approximations, (ii) incorporating higher-order slow-roll corrections to $n_s$ and $r$, and (iii) exploring the onset of reheating at the bottom of the potential. When applied to Starobinsky inflation, these refinements shift the inferred values of the observables, notably increasing $N_k$ and altering the allowable ranges for $n_s$, $N_{\rm re}$, and $T_{\rm re}$ (e.g., $\Delta n_s$ up to $\sim\text{-few} \times10^{-3}$) and potentially tightening or ruling out models under future precision in $n_s$. The work demonstrates that modest corrections to background dynamics and perturbation theory can have substantial consequences for model viability, helping to resolve degeneracies in the inflationary paradigm and guiding robust predictions for upcoming CMB and large-scale structure probes.

Abstract

The inflationary paradigm not only addresses early universe puzzles but also aligns well with the observational constraints, with slow-roll inflationary models fitting the best. Evaluating these model predictions requires considering both slow-roll inflationary dynamics and the subsequent reheating epoch. This involves the quantitative analysis that takes into account the effective equation of state (EoS) and duration of reheating, connecting these with the perturbations generated deep during the inflationary era. Given the complexities involved, many approximations are often used for simplification. However, as future observations are expected to improve the accuracy of these observables significantly, this work takes a different approach. Instead of relying on approximations, and instead of looking into the complex effects of (pre-)reheating, we focus on the corrections arising purely from more accurate analytical evaluations of the perturbations generated during the inflationary era itself. This is because the reheating dynamics is model-dependent and lack a single concrete analytical description, and thus introduce large uncertainties, making robust predictions difficult. In this article, we mainly incorporate two improvements: the first is the accurate dynamics of the slow-roll evolution and, thus, the end of inflation, and the second is the higher-order slow-roll corrections to the perturbed observables. Our findings indicate that, by implementing these corrections, the theoretical predictions improve significantly. It also indicates that seemingly minor corrections can have significant effects on the perturbed observables, and these refined predictions can be compared with future observations to potentially rule out models and help resolve the degeneracy problem of the inflationary paradigm.

Figures (8)

• Figure 1: The Starobinsky potential given in Eq. \ref{['eq: staro pot']}.
• Figure 2: We plot the duration of reheating $N_{\rm re}$ and reheating temperature $T_{\rm re}$ given in Eqs. \ref{['eq: re_duration_final']} and \ref{['eq: re temp end']} as functions of the scalar spectral index $n_{s}$ given by Eq. \ref{['eq: starobinsky observable parameters']} using leading order slow-roll approximations for the case of Starobinsky inflation. Please note that different colors represent dynamics corresponding to different effective EoS parameter $w_{\rm re}$ as indicated in the figure. The blue-shaded region represents the $1\sigma$ constraint on the value of $n_s$ using ongoing observations Planck:2018jriPlanck:2018vygBICEP:2021xfzGalloni:2022mok with $n_{s} = 0.9672 \pm 0.0059$. The dark blue region shows the future projected bound on $n_s$ with a sensitivity of $10^{-3}$, assuming its central value remains unchanged. The temperature below the lighter red region is excluded due to the constraint from the electroweak scale, which is taken to be $100$ GeV.
• Figure 3: We plot the inflationary observables, tensor-to-scalar ratio ($r$) as a function of scalar spectral index ($n_s$) for the Starobinsky model using leading order slow-roll approximations given by Eq. \ref{['eq: starobinsky observable parameters']}. This evolution is presented considering the bound on duration $N_{ k}$ using the reheating regime. We consider different EoS parameters during reheating $w_{\rm re}$ and observe the constraint on the duration $N_{\rm k}$. Correspondingly, in the figure, blue line corresponds to the bound for $w_{\rm re}<1/3$, the gray line for $w_{\rm re}>1/3$ and the purple line for the future observational bound of $n_{\rm s}$. The blue-shaded region represents the $1\sigma$ constraint on the value of $n_{\rm s}$ using ongoing observations Planck:2018jriPlanck:2018vygBICEP:2021xfzGalloni:2022mok with $n_{\rm s} = 0.9672 \pm 0.0059$. The dark blue region shows the future projected bound on $n_{\rm s}$ with a sensitivity of $10^{-3}$, assuming its central value remains unchanged.
• Figure 4: We plot the duration of reheating $N_{\rm re}$ and reheating temperature $T_{\rm re}$ given by Eqs. \ref{['eq: re_duration_final']} and \ref{['eq: re temp end']} as function of the scalar spectral index $n_{s}$ parametrically for both analytical and numerical solution, given by Eqs. \ref{['eq: starobinsky observable parameters']} and \ref{['eq:cmb']}, respectively. The solid lines are for the numerical solution, and the dashed lines are for the analytically approximated solution. Please note that different colors represent dynamics corresponding to different effective equations of state parameter $w_{\rm re}$ as indicated in the figure. The blue-shaded region represents the $1\sigma$ constraint on the value of $n_s$ using ongoing observations Planck:2018jriPlanck:2018vygBICEP:2021xfzGalloni:2022mok with $n_{s} = 0.9672 \pm 0.0059$. The dark blue region shows the future projected bound on $n_s$ with a sensitivity of $10^{-3}$, assuming its central value remains unchanged. The temperature below the lighter red region is excluded due to the constraint from the electroweak scale, which is taken to be $100$ GeV.
• Figure 5: We plot the inflationary observables, tensor-to-scalar ratio ($r$) as a function of scalar spectral index ($n_s$) for the Starobinsky model using numerical solution given by Eq. \ref{['eq:cmb']}. This evolution is presented considering the bound on duration $N_{ k}$ using the reheating regime. We consider different EoS parameters during reheating $w_{\rm re}$ and observe the constraint on the duration $N_{\rm k}$. Correspondingly, in the figure, blue line corresponds to the bound for $w_{\rm re}<1/3$, the gray line for $w_{\rm re}>1/3$ and the purple line for the future observational bound of $n_{\rm s}$. The blue-shaded region represents the $1\sigma$ constraint on the value of $n_{\rm s}$ using ongoing observations Planck:2018jriPlanck:2018vygBICEP:2021xfzGalloni:2022mok with $n_{\rm s} = 0.9672 \pm 0.0059$. The dark blue region shows the future projected bound on $n_{\rm s}$ with a sensitivity of $10^{-3}$, assuming its central value remains unchanged.