Primary gravitational waves at high frequencies. Part II. Emergence of the exponential cut-off in the power spectrum

In standard slow roll inflation, the power spectrum (PS) of primary gravitational waves (PGWs) generated from the quantum vacuum rises as k 2 over wave numbers k which never leave the Hubble radius. In fact, over such small scales, the PS exhibits a similar behavior even when it is evaluated at any time after inflation. In a recent work, we had argued that the PS of PGWs has to be regularized in order to truncate the unphysical quadratic rise at large wave numbers or, equivalently, at high frequencies. Assuming instantaneous transitions from inflation to the epochs of radiation and matter domination, we had shown that the regularized PS oscillates with a constant amplitude about a vanishing mean over small scales during these epochs. We had also smoothed the transition (or, more precisely, the 'effective potential' governing the equation of motion of GWs) from inflation to radiation domination using a linear function and exactly evaluated the regularized PS of PGWs post inflation. In such a case, we had shown that, over small scales, while the regularized PS continues to oscillate about zero, its amplitude decreases as k -1 . In this work, using the Born approximation, we examine the behavior of the regularized PS of PGWs over small scales when they are evolved through smoother and smoother transitions from inflation to the epochs of radiation and matter domination. We explicitly illustrate that, at small scales or high frequencies, the suppression in the regularized PS of PGWs occurs more and more sharply as the transition is smoothed further and further. With the help of specific examples, we also show that, in the case of completely smooth transitions described by an infinitely differentiable 'effective potential', the regularized PS of PGWs exhibits an exponential suppression on small scales. We argue that the observation of the exponential drop in the PS of PGWs can, in principle, help us determine the energy scale as well as the time of the end of inflation. We clarify a few related issues and discuss the wider implications of the results we obtain.