Gravitational Waves as a Probe to the Early Universe

We investigate primordial gravitational waves produced in the early universe within the Running Vacuum Model, which ensures a smooth transition from a primeval inflationary epoch to a radiation-dominant era, ultimately following the standard Hot Big Bang trajectory. In contrast to traditional methods, we approach the gravitational wave equation by reformulating it as an inhomogeneous equation and addressing it as a back-reaction problem. The effective potential, known as the Grishchuk potential, which drives cosmic expansion, is crucial in damping the amplitude of gravitational waves. Our findings indicate that this potential is contingent upon the maximum value of the reduced Hubble parameter, (Formula presented.), which is sensitive to the time at which there is a transition from vacuum energy dominance to radiation dominance. By varying (Formula presented.), we explore its influence on the scale factor and effective potential, revealing its connection to the spectrum of gravitational wave amplitudes that can be constrained by observational data.