The nearly Newtonian regime in Non-Linear Theories of Gravity

TL;DR

This paper reexamines the Newtonian limit of modified gravity models with higher order curvature terms, showing they can produce correct Newtonian behavior if coefficients are small, and discusses discrepancies with previous approaches.

Contribution

It introduces a new approach to analyze the Newtonian limit in non-linear gravity theories, clarifying conditions under which these models are viable and addressing previous conflicting results.

Findings

Models with negative powers of curvature can yield correct Newtonian limits.

Small coefficients of curvature powers are necessary for validity.

Discrepancies with previous metric approach analyses are highlighted.

Abstract

The present paper reconsiders the Newtonian limit of models of modified gravity including higher order terms in the scalar curvature in the gravitational action. This was studied using the Palatini variational principle in [Meng X. and Wang P.: Gen. Rel. Grav. {\bf 36}, 1947 (2004)] and [Dom\'inguez A. E. and Barraco D. E.: Phys. Rev. D {\bf 70}, 043505 (2004)] with contradicting results. Here a different approach is used, and problems in the previous attempts are pointed out. It is shown that models with negative powers of the scalar curvature, like the ones used to explain the present accelerated expansion, as well as their generalization which include positive powers, can give the correct Newtonian limit, as long as the coefficients of these powers are reasonably small. Some consequences of the performed analysis seem to raise doubts for the way the Newtonian limit was derived in the purely metric approach of fourth order gravity [Dick R.: Gen. Rel. Grav. {\bf 36}, 217 (2004)]. Finally, we comment on a recent paper [Olmo G. J.: Phys. Rev. D {\bf 72}, 083505 (2005)] in which the problem of the Newtonian limit of both the purely metric and the Palatini formalism is discussed, using the equivalent Brans--Dicke theory, and with which our results partly disagree.