Cosmological constant as an integration constant

Abstract

The discrepancy between the observed value of the cosmological constant (CC) and its expected value from quantum field theoretical considerations motivates the search for a theory in which the CC is decoupled from the vacuum energy. In this article, we consider the viability of theories in which the Einstein equations are recovered (without additional constraints) and in which the CC is regarded as an integration constant. These theories include trace-free Einstein gravity, theories constructed from the Codazzi equation (which includes Cotton gravity and a gauge-gravity inspired theory), and conformal Killing gravity. We remark on a recent debate regarding Cotton gravity and find that while the Codazzi equation of that theory is indeed underdetermined, the solutions of the Codazzi equation trivialize to $$\lambda g_{ab}$$ λ g ab on generic backgrounds, and that in principle, one can close the system with the divergence-free condition and an appropriate choice of initial data. We also propose a full variational principle (full in the sense that variations in all variables are considered) for each of the aforementioned theories that can incorporate the matter sector; in this manner, we can obtain the trace-free Einstein equations without a unimodular constraint. The resulting actions require additional (auxiliary) fields and are therefore only expected to be effective, but they may provide a useful starting point in bottom up approaches to constructing more fundamental theories.