Effective energy density of relic gravitons

TL;DR

This paper examines the challenges in defining gauge-invariant energy density and related quantities for relic gravitons, comparing different pseudo-tensor proposals across various cosmological regimes to identify the most consistent approach.

Contribution

It provides a systematic comparison of energy-momentum pseudo-tensors for relic gravitons, clarifying their physical implications in different regimes and proposing the most plausible parametrization.

Findings

Different pseudo-tensor proposals yield compatible results inside the horizon.

Significant discrepancies arise for super-horizon wavelengths.

A systematic comparison helps identify the most consistent energy-momentum description.

Abstract

The energy density, the pressure and the anisotropic stress of the relic gravitons do not have a unique gauge-invariant and frame-invariant expression since the equivalence principle ultimately forbids the localization of the energy-momentum of the gravitational field. All the strategies proposed so far suggest compatible descriptions inside the effective horizon but they lead to sharply different answers when the wavelengths of the gravitons exceed the Hubble radius by remaining shorter than the typical extension of causally connected patches. We present a coherent discussion of the energy-momentum pseudo-tensors of the relic gravitons in the different kinematical regimes with the aim of scrutinizing the mutual consistency of the competing suggestions. The various proposals are systematically compared by deducing and analyzing the explicit form of the observables in realistic physical situations. General lessons are drawn on the most plausible parametrization of the energy-momentum pseudo-tensor of the relic gravitons.