$R + R^2$ Gravity as $R +$ Backreaction

TL;DR

This paper explores a perturbative approach to quadratic gravity, showing it preserves physical degrees of freedom and leads to repulsive barriers near the big bang, implying a bounce instead of a singularity.

Contribution

It introduces a perturbative interpretation of $R + eta R^2$ gravity that avoids common higher derivative issues and analyzes its cosmological implications.

Findings

Repulsive barriers near $a=0$ prevent singularities.

Universe can bounce from contraction to expansion.

Probability of big bang singularity is exponentially suppressed.

Abstract

Quadratic theory of gravity is a complicated constraint system. We investigate some consequences of treating quadratic terms perturbatively (higher derivative version of backreaction effects). This approach is shown to overcome some well known problems associated with higher derivative theories, i.e., the physical gravitational degree of freedom remains unchanged from those of Einstein gravity. Using such an interpretation of $R + \beta R^2$ gravity, we investigate a classical and Wheeler DeWitt evolution of $R + \beta R^2$ gravity for a particular sign of $\beta$, corresponding to non- tachyon case. Matter is described by a phenomenological $\rho \propto a(t)^{-n}$. It is concluded that both the Friedmann potential $U(a)$ ($ {\dot a}^2 + 2U(a) = 0 $) and the Wheeler DeWitt potential $W(a)$ ($\left[-{\partial^2\over \partial a^2} + 2W(a)\right]\psi (a) =0 $) develop repulsive barriers near $a\approx 0$ for $n>4$ (i.e., $ p > {1\over 3}\rho $). The interpretations is clear. Repulsive barrier in $U(a)$ implies that a contracting FRW universe ($k>0, k=0, k<0$) will bounce to an expansion phase without a total gravitational collapse. Repulsive barrier in $W(a)$ means that $a \approx 0$ is a classically forbidden region. Therefore, probability of finding a universe with the big bang singularity ($a=0$ ) is exponentially suppressed.