Zero Cosmological Constant from Normalized General Relativity

TL;DR

This paper introduces a normalized version of general relativity that naturally leads to a zero cosmological constant, providing insights into the nature of vacuum energy and the universe's large-scale structure.

Contribution

It proposes a normalized Einstein-Hilbert action invariant under constant shifts, linking the cosmological constant to a Machian average and analyzing its implications in various cosmological models.

Findings

Empty spacetime is Ricci flat.

The cosmological constant vanishes in the scalar field paradigm.

A non-zero cosmological constant can arise in modified normalized theories.

Abstract

Normalizing the Einstein-Hilbert action by the volume functional makes the theory invariant under constant shifts in the Lagrangian. The associated field equations then resemble unimodular gravity whose otherwise arbitrary cosmological constant is now determined as a Machian universal average. We prove that an empty space-time is necessarily Ricci tensor flat, and demonstrate the vanishing of the cosmological constant within the scalar field paradigm. The cosmological analysis, carried out at the mini-superspace level, reveals a vanishing cosmological constant for a Universe which cannot be closed as long as gravity is attractive. Finally, we give an example of a normalized theory of gravity which does give rise to a non-zero cosmological constant.