Gravitational waves and the cosmological equation of state

Abstract

Primordial gravitational waves are amplified during eras when their wavelengths are pushed outside the cosmological horizon. This occurs in both inflationary and ``pre-big-bang'' or ``bounce'' cosmologies. The spectrum is expressed as a normalized energy density per unit logarithmic frequency, denoted Omega. The spectral index (logarithmic slope) of Omega is simply related to three properties of the early universe: (i) the gravitons' mean initial quantum occupation number N(n) (=1/2 for a vacuum state), where n is the (invariant) conformal frequency of the mode, and (ii) & (iii) the parameter gamma=p/rho of the cosmological equation of state during the epoch when the waves left the horizon (gamma=gamma_i) and when they reentered (gamma=gamma_f). In the case of an inflationary cosmology, the spectral index is equal to d(ln N)/d(ln n) + 2(gamma_i + 1)/(gamma_i + 1/3) + 2(gamma_f - 1/3)/(gamma_f + 1/3) and for bounce cosmologies it is equal to d(ln N)/d(ln n) + 4(gamma_i)/(gamma_i + 1/3) + 2(gamma_f - 1/3)/(gamma_f + 1/3) These expressions are compared against various more model-specific results given in the literature.