Abstract
We consider the evolution of the gravitational wave spectrum for super-Hubble modes in interaction with a relativistic fluid, which is regarded as an effective description of fluctuations in a light scalar minimally coupled field, during the earliest epoch of the radiation dominated era after the end of inflation. We obtain the initial conditions for gravitons and fluid from quantum fluctuations at the end of inflation, and assume instantaneous reheating. We model the fluid by using relativistic causal hydrodynamics. There are two dimensionful parameters, the relaxation time $\tau$ and temperature. In particular we study the interaction between gravitational waves and the non trivial tensor (spin 2) part of the fluid energy-momentum tensor. Our main result is that the new dimensionful parameter $\tau$ introduces a new relevant scale which distinguishes two kinds of super-Hubble modes. For modes with $H^{-1}<\lambda<\tau$ the fluid-graviton interaction increases the amplitude of the primordial gravitational wave spectrum at the electroweak transition by a factor of about $1.3$ with respect to the usual scale invariant spectrum.
Highlights
In this paper we shall consider the evolution of the primordial gravitational wave background during the early radiation dominated era [1] [2] [3], from reheating after inflation up to the cosmological electroweak transition
We will use second order hydrodynamics [4] [5] as an effective theory for the matter fields, and obtain a linear theory for gravitons consistently coupled to the spin-2 component of the matter energy-momentum tensor
Our motivation in using hydrodynamics as an effective theory comes from the highly successful description of the early evolution of the fireball created in relativistic heavy ion collisions (RHICs) by these methods, even in early stages where it is unlikely that local thermal equilibrium has been established [6] [7]
Summary
In this paper we shall consider the evolution of the primordial gravitational wave background during the early radiation dominated era [1] [2] [3], from reheating after inflation up to the cosmological electroweak transition. For simplicity we shall not consider an explicit coupling of the fluid to other matter fields, the self and gauge interactions of the fluid will appear through the constitutive relations for the fluid, that is its relaxation time τ (to be discussed in Section VI), and its temperature. Under this approximation the equations of the model are the Einstein equations, energy-momentum conservation, and a Cattaneo-Maxwell equation for ζμν to be provided below. Appendix A discusses the conformal invariance of fluid equations in the limit of massless particles, and Appendix B clarifies some technical tools to calculate the Fourier transform of the noise kernel for scalar fields
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