Analyzing modified unimodular gravity via Lagrange multipliers

Abstract: The so-called unimodular version of General Relativity is revisited. Unimodular gravity is constructed by fixing the determinant of the metric, what leads to the trace-free part of the equations instead of the usual Einstein field equations. Then, a cosmological constant naturally arises as an integration constant. While unimodular gravity turns out equivalent to General Relativity (GR) at the classical level, it provides important differences at the quantum level. Here we extend the unimodular constraint to some extensions of General Relativity that have drawn a lot of attention over the last years, as $f(R)$ gravity (or its scalar-tensor picture) and Gauss-Bonnet gravity. The corresponding unimodular version of such theories is constructed as well as the conformal transformation that relates the Einstein and Jordan frames for these non-minimally coupled theories. From the classical point of view, the unimodular versions of such extensions are completely equivalent to their originals, but an effective cosmological constant arises naturally, what may provide a richer description of the universe evolution. Here we analyze the case of Starobisnky inflation and compared with the original one.