Primordial black holes and second order gravitational waves from inflationary $\alpha$-attractor models

I will talk about primordial black hole (PBH) formation using a type of inflection-point potential derived using the inflationary $\alpha$-attractor model. The inflection plateau region gives rise to a phase called ultra slow-roll (USR) where the curvature power spectrum $\mathcal{P}_{\zeta}$ is amplified upto $10^{-3}$ and $10^{-2}$ which produces the right conditions for PBH formation. The curvature perturbation modes that collapse to form PBHs are approximately $k\sim 10^{14}\text{Mpc}^{-1}$, giving rise to PBHs in the mass range $10^{16}$ to $10^{18}\text{g}$ which can be cosmologically relevant as a cold dark matter (CDM) candidate. The abundance of PBHs is derived using a modification of the Press-Schechter theory which takes into account the intrinsic non-Gaussianity of the overdensity perturbations $\delta\rho/\rho$. I will present the results with the nonvanishing skewness and kurtosis of the distribution of $\delta\rho/\rho$ and show that this $\alpha$-attractor inflationary model can produce PBHs that can comprise a cosmologically significant proportion of CDM. We will also see that second order gravitational waves can be produced from this model in the frequency range $10^{-2}$ to $10^{2}\text{Hz}$ at levels which can potentially be detected in the future.