Two-loop corrections in power spectrum in models of inflation with PBHs formation

Abstract: We calculate the two-loop corrections in primordial power spectrum in models of single field inflation incorporating an intermediate USR phase employed for PBHs formation. Among the total eleven one-particle irreducible Feynman diagrams, we calculate the corrections from the "double scoop" two-loop diagram involving two vertices of quartic Hamiltonians. We demonstrate that the fractional two-loop correction in power spectrum scales like the square of the fractional one-loop correction. We confirm our previous findings that the loop corrections become arbitrarily large in the setup where the transition from the intermediate USR to the final slow-roll phase is very sharp. This suggests that in order for the analysis to be under perturbative control against loop corrections, one requires a mild transition with a long enough relaxation period towards the final attractor phase.