Graviton Production in Elliptical and Hyperbolic Universes

TL;DR

This paper analyzes the creation of gravitons in elliptical and hyperbolic universes by solving the gravitational wave equation exactly for various equations of state and calculating the resulting particle production using Bogoliubov transformations.

Contribution

It provides exact solutions for the gravitational wave equation in curved universes and derives explicit expressions for the vacuum state and graviton production across different cosmological scenarios.

Findings

Exact hypergeometric solutions for gravitational wave equations.

Explicit Bogoliubov coefficients for graviton creation.

Applicability to various physical equations of state.

Abstract

The problem of cosmological graviton creation for homogeneous and isotropic universes with elliptical ($\vae =+1$) and hyperbolical ($\vae =-1$) geometries is addressed. The gravitational wave equation is established for a self-gravitating fluid satisfying the barotropic equation of state $p=(\gamma -1)\rho$, which is the source of the Einstein's equations plus a cosmological $\Lambda$-term. The time dependent part of this equation is exactly solved in terms of hypergeometric functions for any value of $\gamma$ and spatial curvature $\vae$. An expression representing an adiabatic vacuum state is then obtained in terms of associated Legendre functions whenever $\gamma\neq \frac{2}{3}\; \frac{(2n+1)}{(2n-1)}$, where n is an integer. This includes most cases of physical interest such as $\gamma =0,\;4/3\;,1$. The mechanism of graviton creation is reviewed and the Bogoliubov coefficients related to transitions between arbitrary cosmic eras are also explicitly evaluated.