Fourth order gravity: equations, history, and applications to cosmology

TL;DR

This work surveys fourth-order gravity arising from L(R) and related higher-derivative actions, tracing historic roots from Weyl to modern scalar-tensor mappings. It analyzes both cosmological implications (via scalar-field equivalence, ground states, and inflationary behavior) and weak-field phenomenology (Newtonian limits with Yukawa corrections) to assess viability and physical implications. A central result is the Bicknell theorem, establishing a conformal equivalence between L(R) gravity and minimally coupled scalar-tensor theories, enabling a unified view of inflationary dynamics and ground-state structure; this links Starobinsky-like inflation to higher-derivative terms. The Newtonian limit generically yields a sum of Newtonian and Yukawa potentials with masses determined by the higher-order coefficients, offering testable predictions at short distances and guiding constraints on the parameters of $R^2$, $R\Box R$, and higher-order terms; the analysis also highlights conditions under which de Sitter acts as an attractor, informing early-universe cosmology and the viability of higher-order gravity theories.

Abstract

The field equations following from a Lagrangian L(R) will be deduced and solved for special cases. If L is a non-linear function of the curvature scalar, then these equations are of fourth order in the metric. In the introduction we present the history of these equations beginning with the paper of H. Weyl from 1918, who first discussed them as alternative to Einstein's theory. In the third part, we give details about the cosmic no hair theorem, i.e., the details how within fourth order gravity with L= R + R^2 the inflationary phase of cosmic evolution turns out to be a transient attractor. Finally, the Bicknell theorem, i.e. the conformal relation from fourth order gravity to scalar-tensor theory, will be shortly presented.