Power law $α$-Starobinsky inflation

TL;DR

This work generalizes Starobinsky inflation by combining a power-law $R^{β}$ term with an $α$-attractor–like potential, deriving the Einstein-frame form and numerically computing scalar and tensor power spectra without slow-roll. Using ModeChord to evolve the background and perturbations and CosmoMC to fit Planck-2018, BK18, DES, and BAO data, the authors constrain $(α,β,M,N_{pivot})$ and find $β≈1.969$, $\log_{10} α ≈0.37$, $M≈3.3×10^{-5}$, and $N_{pivot}≈47$, with $r$–$n_s$ predictions consistent with data within $1σ$. Bayesian evidence via MCEvidence shows the power-law $α$-Starobinsky model is mildly favored over Starobinsky and other generalizations, suggesting a modest departure from the original model. The results imply that generalized Starobinsky-inspired inflation remains compatible with current observations and motivate further data to sharpen constraints and connect to no-scale supergravity frameworks.

Abstract

In this work we consider a generalization of Starobinsky inflation obtained by combining power law ($R^β$), and $α$-Starobinsky inflation ($E$-model). The Einstein frame potential for this model is that of power law Starobinsky inflation modified by a parameter $α$ in the exponential. After computing power spectra for scalar and tensor perturbations numerically, we perform MCMC analysis to put constraints on the potential parameters $α$, $β$ and $M$, and the number of e-foldings $N_{pivot}$ during inflation, using Planck-2018, BICEP/Keck (BK18), DES and BAO observations. We find $\log_{10}α= 0.37^{+0.82}_{-0.85}$, $β= 1.969^{+0.020}_{-0.023}$, $M=\left(3.54^{+2.62}_{-1.73}\right)\times 10^{-5}$ and $N_{pivot} = 47\pm{10}$. With these mean values of the potential parameters $α$ and $β$, and varying $N_{pivot}$ between $40$ to $55$, we also find that the $r-n_s$ predictions of our model lie well within the $1σ$ bounds of joint constraints from combined analysis of ACT, Planck-2018, BICEP and BAO observations. We compute the Bayesian evidences for our proposed model, power law Starobinsky inflation, $α$-Starobinsky inflation and Starobinsky inflation. Considering the Starobinsky model as the base model, we calculate the Bayes factor and find that our proposed model is mildly favored by the CMB and LSS observations.

Figures (3)

• Figure 1: Variation of potential (\ref{['alphabetapot']}) with the field for different values of $\alpha$ and $\beta$, as indicated. Both the potential and the field are expressed in $M_{Pl} = 1$ units.
• Figure 2: 1$\sigma$ and 2$\sigma$ confidence contours of (a) potential parameters and $N_{pivot}$, and (b) derived parameters $n_s$ and $r$ using Planck-2018 TT, TE, EE, lowE with lensing, BICEP (BK18) BICEP:2021xfz, Dark Energy Survey DES:2020mlx, and BAO data from BOSS/DR12 BOSS:2015zan, 6dFGS and SDSS. Marginalized probability distributions of the individual parameters are also displayed for power law $\alpha$-Starobinsky model.
• Figure 3: The $r-n_s$ predictions for power law $\alpha$-Starobinsky inflation (black line) along with the joint constraints from ACT, Planck, BICEP/Keck and BAO observations ACT:2025tim.