On the Supersymmetric Completion of $R+R^2$ Gravity and Cosmology

TL;DR

This work clarifies how $R+R^2$ gravity can be consistently embedded in old-minimal $N=1$ supergravity. It shows that any faithful SUSY completion inevitably contains two dynamical chiral multiplets—the inflaton and the goldstino—and derives their couplings from a no-scale–type Kähler potential and a superpotential. The analysis distinguishes the true two-multiplet construction from the single-multiplet '$F(R)$' approach, demonstrating that the latter yields a bosonic action linear in $R$ with a propagating $X$ and does not realize $R^2$-type inflation. The paper also discusses how additional sgoldstino-dependent deformations can preserve inflationary observables, highlighting the fundamental role of the conformon in the superconformal formulation.

Abstract

We revisit and clarify the supersymmetric versions of $R+ R^2$ gravity, in view of the renewed interest to these models in cosmology. We emphasize that the content of the dual standard supergravity theory in the old minimal formulation necessarily includes two massive chiral multiplets, that we call the inflaton and the goldstino. We point out that the presence of these multiplets is model independent in the old minimal formulation and therefore any theory that contains a single chiral multiplet fails to be a supersymmetric generalization of the $R+R^2$ gravity. The supergravity interactions of the two chiral multiplets are encoded in a superpotential mass term and an arbitrary Kahler potential for the goldstino multiplet. The implication for cosmology of the supersymmetric $R+R^2$ gravity is also discussed.