Metric-affine cosmological models and the inverse problem of the calculus of variations. Part II: Variational bootstrapping of the $\Lambda$CDM model

TL;DR

This paper introduces a variational bootstrapping method to derive metric-affine gravity models that closely resemble the $\\Lambda$CDM cosmological model, enabling systematic construction of modified gravity theories from initial field equation guesses.

Contribution

It develops a novel variational bootstrapping approach to construct metric-affine gravity models aligned with the $\\Lambda$CDM paradigm, starting from educated guesses of field equations.

Findings

Successfully applied the method to derive corrected metric equations.

Obtained a broad class of quadratic metric-affine theories.

Identified models that include linear curvature terms.

Abstract

The method of variational bootstrapping, based on canonical variational completion, allows one to construct a Lagrangian for a physical theory depending on two sets of field variables, starting from a guess of the field equations for only one such set. This setup is particularly appealing for the construction of modified theories of gravity, since one can take lessons from GR for an "educated guess" of the metric field equations; the field equations for the other fields are then fixed by the obtained Lagrangian (up to terms that are completely independent from the metric tensor). In the present paper, we apply variational bootstrapping to determine metric-affine models which are, in a variational sense, closest to the $\Lambda$CDM model of cosmology. Starting from an "educated guess" that formally resembles the Einstein field equations with a cosmological "constant" (actually, a scalar function built from the metric and the connection) and a dark matter term, the method then allows to find "corrected" metric equations and to "bootstrap" the connection field equations. Lagrangians obtained via this method, though imposing some restricting criteria, still encompass a wide variety of metric-affine models. In particular, they allow for a subclass of quadratic metric-affine theories restricted to linear terms in the curvature tensor.