Gravitational Backreaction in de Sitter: A Canonical ADM Approach

Abstract

We develop a canonical ADM framework for gravitational backreaction during inflation. Our approach extends the formalism used to compute cosmological correlators by applying it to the evolution of the quantum background, which we identify with the metric one-point function. The aim is to assess the quantum stability of inflationary geometries under gravitational backreaction. Within this framework, the backreaction is determined by the expectation values of the Hamiltonian constraint and the equations of motion for the spatial metric. Working in the exact de Sitter limit, we apply the framework to a massless minimally coupled spectator scalar field and the physical graviton polarizations. At one loop, their backreaction renormalizes the relation between the cosmological constant and the Hubble parameter without inducing any secular departure from de Sitter evolution, in agreement with previous results. We perform the calculation using a hard momentum cutoff and show that a fixed physical cutoff allows the homogeneous background equations to be renormalized with time- and background-independent coefficients, in contrast to a fixed comoving cutoff. Finally, we demonstrate that the explicit representation of the backreaction depends on the gauge condition imposed on the metric fluctuations, as well as on the background-fluctuation split. Nevertheless, these different representations are related by a gauge transformation and are therefore physically equivalent.