Constant-Roll Inflation as a Competitive Alternative to Slow-Roll Inflation

Abstract

Starting from the constant-roll condition, which involves the constant-roll parameter $\beta$, which is a fundamental constant in the constant-roll inflationary models, we derive the scalar power spectrum and obtain an analytic expression for the scalar spectral index $n_s$ in terms of $\beta$ and the number of remaining e-folds $\tilde{N}$. The resulting relation is non-monotonic in $\beta$, allowing two distinct constant-roll branches, one with $\beta <0$ and one with $\beta >0$, to reproduce nearly the same observed scalar tilt. We constrain $\beta$ using Planck 2018 CMB data, and further using the combined Planck 2018 + ACT DR6 data. For Planck 2018, the posterior distribution of $\beta$ exhibits a bimodal structure associated with the two analytical branches, while the combined Planck+ACT analysis tightens the allowed range of $\beta$ and shows continued consistency with the $\beta =0$ slow-roll-like spectral point. The data constrain $\beta$ to be small, but do not require it to vanish exactly. We also estimate the inflationary mass scale $M$ by using the observed scalar-amplitude normalization at the pivot scale. Both the negative and positive $\beta$ branches give a high-scale inflationary normalization of order $M\sim 10^{-5}{M}_{\mathrm{Pl}}$. We show that the $\beta \to 0$ limit is only spectrally slow-roll like, but does not reproduce standard slow-roll dynamics. Our results show that constant-roll inflation is observationally competitive with the usual slow-roll inflationary paradigm. The constant roll inflation model is supported by the latest CMB observations equally well or marginally better when compared with the slow roll inflation model.