Primary gravitational waves at high frequencies I: Origin of suppression in the power spectrum

[Abridged] The primary gravitational waves (PGWs) are generated in the early universe from the quantum vacuum during inflation. In slow roll inflation, the power spectrum (PS) of PGWs over large scales, which leave the Hubble radius during inflation, is nearly scale-invariant. However, over very small scales, which never leave the Hubble radius, the PS of PGWs behaves as k^2, where k denotes the wave number. We examine the PS of PGWs at such high wave numbers or frequencies when the PGWs are evolved post-inflation, through the epochs of radiation and matter domination. Firstly, we argue that the PS has to be regularized in order to truncate the unphysical k^2 rise at high frequencies. Assuming instantaneous transitions from inflation to the epochs of radiation and matter domination, we carry out the method of adiabatic regularization to arrive at the PS of PGWs over a wide range of frequencies. We show that the process of regularization truncates the k^2 rise and the PS of PGWs oscillates with a fixed amplitude about a vanishing mean value over small scales or, equivalently, at high frequencies. Secondly, we smooth the transition from inflation to radiation domination (to be precise, we smooth the 'effective potential' governing the equation of motion of PGWs) and examine the impact of the smoothing on the regularized PS of PGWs. With the help of a linear smoothing function, we explicitly show that the smoother transition leads to a power-law suppression in the amplitude of the oscillations (about the zero mean value) of the regularized PS of PGWs over small scales that never leave the Hubble radius during inflation. Our analysis indicates that, when transitions are involved, regularization as well as smooth transitions seem essential to ensure that the correlation functions of the PGWs in real space are well behaved. We discuss the directions in which our results need to be extended.