Resonant Gravitational Wave Amplification - Axion and Inflaton

The nature of the cosmological background of gravitational waves is a subject of great interest at present, with many possible sources ranging from quantum fluctuations in a string-dominated phase 1,2 or inflation , to oscillations of cosmic string loops . Here we focus instead on a powerful mechanism for distortion and amplification of any existing gravitational wave background namely damped parametric resonance due to oscillatory phases that the universe may have undergone. Examples are provided by reheating at the end of inflation , an oscillatory dilaton phase, or during coherent axion or moduli oscillations if they form a significant portion of the dark matter. We will discuss the amplification within the gauge-invariant Bardeen formalism. Using the gauge-invariant and covariant electric and magnetic parts of the Weyl tensor gives similar results . The evolution of the transversetraceless (TT) metric perturbations hab are naturally described by the Fourier mode functions hǫ,k, where ǫ = {+,×} are the polarisation states. The hk a satisfy: