Constraining ultra slow roll inflation using cosmological datasets

In recent years, the detection of gravitational waves by LIGO and PTA collaborations have raised the intriguing possibility of excess matter power at small scales. Such an increase can be achieved by ultra slow roll (USR) phase during inflationary epoch. We constrain excess power over small scales within the framework of such models using cosmological datasets, particularly of CMB anisotropies and Lyman-$\alpha$. We parameterize the USR phase in terms of the e-fold at the onset of USR (counted from the end of inflation) $\bar N_1$ and the duration of USR phase $\Delta N$. The former dictates the scale of enhancement in the primordial power spectrum, while the latter determines the amplitude of such an enhancement. From a joint dataset of CMB, SNIa and galaxy surveys, we obtain $\bar N_1 \lesssim 45$ with no bound on $\Delta N$. This in turn implies that the scales over which the power spectrum can deviate significantly from the nearly scale invariant behavior of a typical slow-roll model is $k \gtrsim 1 \, \rm Mpc^{-1}$. On the other hand, the Lyman-$\alpha$ data is sensitive to baryonic power spectrum along the line of sight. We consider a semi-analytic theoretical method and high spectral-resolution Lyman-$\alpha$ data to constrain the model. The Lyman-$\alpha$ data limits both the USR parameters: $\bar N_1 \lesssim 41$ and $\Delta N \lesssim 0.4$. This constrains the amplitude of the power spectrum enhancement to be less than a factor of hundred over scales $1 \lesssim k/{\rm Mpc^{-1}} \lesssim 100$, thereby considerably improving the constraint on power over these scales as compared to the bounds arrived at from CMB spectral distortion.