Cosmological constant as an integration constant

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 theories constructed from the derivatives of the Einstein equation, in which the CC is regarded as an integration constant. We focus on theories that can be written as a Codazzi equation (which includes Cotton gravity and a gauge-gravity inspired theory), and remark on a recent debate regarding the viability of Cotton gravity; we find that while the Codazzi equation is indeed underdetermined, the solutions of the Codazzi equation trivialize to λ gab on generic backgrounds. Moreover, we find that the divergence-free condition can in principle close the system for an appropriate choice of initial data. We also propose a full variational principle (full in the sense that variations with respect to all variables are considered) for the aforementioned theories that can incorporate the matter sector.