Quantum correlations for the metric

TL;DR

This paper explores the quantum correlations of the metric in cosmology, relating the correlation function to the effective action and providing a covariant framework for physical metric fluctuations.

Contribution

It introduces a covariant approach to metric correlations, distinguishing physical fluctuations from gauge artifacts, and relates the correlation function to the effective action in cosmological models.

Findings

Derived the correlation function as the inverse of the second derivative of the effective action.

Showed the relation between mode functions and the correlation function in Einstein-Hilbert approximation.

Separated physical metric fluctuations from gauge fluctuations, enabling gauge-independent analysis.

Abstract

We discuss the correlation function for the metric for homogeneous and isotropic cosmologies. The exact propagator equation determines the correlation function as the inverse of the second functional derivative of the quantum effective action. This formulation relates the metric correlation function employed in quantum gravity computations to cosmological observables as the graviton power spectrum. In the Einstein-Hilbert approximation for the effective action the on-shell graviton correlation function can be obtained equivalently from a product of mode functions which solve the linearized Einstein equations. In contrast, the product of mode functions, often employed in the context of cosmology, does not yield the correlation function for the vector and scalar components of the metric fluctuations. We divide the metric fluctuations into "physical fluctuations", which couple to a conserved energy momentum tensor, and gauge fluctuations. On the subspace of physical metric fluctuations the relation to physical sources becomes invertible, such that the effective action and its relation to correlation functions no longer needs to involve a gauge fixing term. The physical metric fluctuations have a similar status as the Bardeen potentials, while being formulated in a covariant way. We compute the effective action for the physical metric fluctuations for geometries corresponding to realistic cosmologies.