Constraints on the inflaton potential from scalar-induced gravitational waves and primordial black holes

A plateau on the background inflaton potential $V_{\rm b}(\phi)$ can lead cosmic inflation into an ultraslow-roll phase, greatly enhancing the primordial power spectrum on small scales, and resulting in intensive scalar-induced gravitational waves (GWs) and abundant primordial black holes (PBHs). In this work, we construct an anti-symmetric perturbation $\delta V(\phi)$ on $V_{\rm b}(\phi)$ with three model parameters, the position, width, and slope of $\delta V(\phi)$, and constrain these parameters from the potential stochastic gravitational wave background (SGWB) in the NANOGrav 15-year data set. The GW spectrum from the merger of supermassive black hole binaries (SMBHBs) with two model parameters, the amplitude and spectral index, is also investigated for comparison. We perform Bayesian analysis in three steps with increasing number of model parameters, and obtain the allowed parameter ranges. When the constraints on PBH abundance are taken into account, these ranges become further narrower. We find that the increase of model parameters cannot significantly improve the Bayes factors, and the model with an almost perfect plateau on $V_{\rm b}(\phi)$ is favored. Moreover, the interpretation of the SGWB only via the GWs generated by SMBHBs is not preferred by the data. Two different forms of $V_{\rm b}(\phi)$ are considered, and the analogous results confirm the generality and robustness of our work.