Aspects of cosmological expansion in F(R) gravity models

TL;DR

The paper investigates cosmological expansion in $F(R)$ gravity, focusing on two viable models, $F_{ m HSS}$ and $F_{ m AB}$, and questions the perturbative prediction of high-frequency Ricci-scalar oscillations with growing past amplitude. It combines nonperturbative numerical simulations with improved perturbative oscillator analyses to reveal nonlinear, asymmetric $R$ oscillations about the GR limit, and identifies a finite-time singularity when evolving backward in time. The authors show that, although these oscillations occur, their effects on the Hubble parameter $H$ and scale factor $a$ are strongly suppressed if $R F''(R)\ll 1$, and that appropriate initial conditions can avoid singular behavior, albeit with potential instability to perturbations away from a pure matter era. The work highlights the need for regularization or constraints in viable $F(R)$ theories and provides a framework for analyzing nonlinear Ricci-scalar dynamics in related models, with implications for late-time cosmology and local gravity tests.

Abstract

We study cosmological expansion in F(R) gravity using the trace of the field equations. High frequency oscillations in the Ricci scalar, whose amplitude increase as one evolves backward in time, have been predicted in recent works. We show that the approximations used to derive this result very quickly breakdown in any realistic model due to the non-linear nature of the underlying problem. Using a combination of numerical and semi-analytic techniques, we study a range of models which are otherwise devoid of known pathologies. We find that high frequency asymmetric oscillations and a singularity at finite time appear to be present for a wide range of initial conditions. We show that this singularity can be avoided with a certain range of initial conditions, which we find by evolving the models forwards in time. In addition we show that the oscillations in the Ricci scalar are highly suppressed in the Hubble parameter and scale factor.

Figures (11)

• Figure 1: (a) $R = R_{\rm GR} + \delta R$ as predicted in ref. st for the HSS model. The solid lines are the upper and lower envelopes of the solution, and the dashed line is $R_{\rm GR}$. We note the turning point in the lower envelope, at which point the amplitude of $\delta R$ can become larger than $R_{\rm GR}$; (b) $R = R_{\rm GR} + \delta R$ for the AB model, where once again we observe a turning point in the lower envelope.
• Figure 2: (a) the envelope functions for $R$ in the HSS model, with $\epsilon=0.1$, obtained by solving the full field equations numerically as described in the text, with perturbed initial conditions. We note that $R$ differs significantly from the linearized approximation $R = R_{\rm GR} + \delta R$, obtained in the previous section; (b) $\delta H = (H - H_{\rm GR})/H$. It is clear that $H$ oscillates, and the amplitude of these oscillations grows to the past; (c) the deviation of the scale factor from its General Relativistic limit, $\delta a = (a - a_{\rm GR})/a$. We see that $a$ will deviate from $a_{\rm GR}$ as we evolve backwards in time, however $\delta a$ is highly suppressed compared to the deviations in $R$; (d) $\delta a$ over a smaller range of $t$ close to the end point of the evolution. This shows the oscillatory behaviour of the scale factor which eventually develops.
• Figure 3: (a) the envelopes of $\delta R = (R - R_{\rm GR})/R$ for the HSS model, obtained by solving the full field equations numerically, with unperturbed initial conditions. Since $R = R_{\rm GR}$ is not a solution to the field equations, we find that $\delta R \neq 0$; (b) $\delta H = (H - H_{\rm GR})/H$. We see that $H$ oscillates, and the amplitude of these oscillations grows to the past. Further, $\delta H$ does not oscillate around zero, indicating that the Hubble parameter will deviate from $H_{\rm GR}$ as we evolve backwards in time; (c) $\delta H$ over a small time regime to explicitly show the oscillatory behaviour of $H$; (d) $\delta a = (a - a_{\rm GR})/a$. We see that $a$ will deviate from $a_{\rm GR}$ as we evolve backwards in time, however $\delta a$ is highly suppressed.
• Figure 4: (a) the envelope functions for $R$ in the AB model, with $\epsilon = 0.32$, obtained by solving the full field equations numerically as described in the text, with perturbed initial conditions. As in the HSS model, we see that $R$ differs significantly from the linearized approximation $R = R_{\rm GR} + \delta R$, obtained in the previous section; (b) $\delta H = (H - H_{\rm GR})/H$, which oscillates (not around $\delta H =0$); (c) $\delta a = (a - a_{\rm GR})/a$. We see that $a$ will deviate from $a_{\rm GR}$ as we evolve backwards in time, however $\delta a$ is highly suppressed; (d) $\delta a$ over a smaller range of $t$, which explicitly shows the oscillatory behaviour of the scale factor.
• Figure 5: (a) envelope functions for $\delta R = (R - R_{\rm GR})/R$ for the AB model, with unperturbed initial conditions. Since $R = R_{\rm GR}$ is not a solution to the field equations, we find that $\delta R$ oscillates; (b) $\delta H = (H - H_{\rm GR})/H$. It is clear that $H$ oscillates, and the amplitude of these oscillations grows to the past. Further, $\delta H$ does not oscillate around zero, indicating that the Hubble parameter will deviate from $H_{\rm GR}$ as we evolve backwards in time; (c) $\delta H$ over a small time regime, explicitly showing the oscillatory behaviour of $H$; (d) $\delta a = (a - a_{\rm GR})/a$. We see that $a$ will deviate from $a_{\rm GR}$ as we evolve backwards in time, however $\delta a$ is also highly suppressed.