Constraints on ultra-slow-roll inflation with the NANOGrav 15-Year Dataset

Ultra-slow-roll~(USR) inflation predicts an exponential amplification of scalar perturbations at small scales, which leads to a stochastic gravitational wave background~(SGWB) through the coupling of the scalar and tensor modes at the second-order expansion of the Einstein equation. In this work, we search for such a scalar-induced SGWB from the NANOGrav 15-year (NG15) dataset, and find that the SGWB from USR inflation could explain the observed data. The Bayes factors are $54\pm 5$ for the USR inflation model alone and $68\pm 6$ for the combined USR inflation plus supermassive black hole binaries (SMBHB) models. We place constraints on the amplitude of the scalar power spectrum to $P_{\mathrm{Rp}} > 10^{-1.95}$ at $95\%$ confidence level (C.L.) at the scale of $k\sim 20\, \mathrm{pc}^{-1}$. We find that $\log_{10} P_{\mathrm{Rp}}$ degenerates with the peak scale $\log_{10} k_{\mathrm{p}}$. We also obtain the parameter space allowed by the data in the USR inflationary scenario, where the $e$-folding numbers of the duration of the USR phase has a lower limit $\Delta N > 2.80$ ($95\%$ C.L.) when the USR phase ends at $N\approx 20$. With astrophysically motivated priors, the NG15 dataset fits both the USR inflation model and SMBHB model equally well.