Tracking $f(R)$ cosmology

TL;DR

This paper analyzes conditions for stable tracker solutions in metric $f(R)$ gravity theories, showing how these models can exhibit different curvature behaviors in Einstein and Jordan frames, with implications for dark energy modeling.

Contribution

It derives specific stability conditions for tracker solutions in metric $f(R)$ gravity and explores their behavior across Einstein and Jordan frames, revealing possible increasing or decreasing curvature scenarios.

Findings

Stable tracker solutions exist for specific parameter ranges.

Curvature can decrease in Einstein frame but increase in Jordan frame.

Certain $f(R)$ models exhibit tracking behavior compatible with observed cosmic acceleration.

Abstract

Metric $f(R)$ gravity theories are conformally equivalent to models of quintessence in which matter is coupled to dark energy. We derive a condition for stable tracker solution for metric $f(R)$ gravity in the Einstein frame. We find that tracker solutions with $-0.361<\omega_{\varphi}<1$ exist if $0<\Gamma<0.217$ and $\frac{d}{dt} \ln f'(\tilde{R})>0$, where $\Gamma=\frac{V_{\varphi\varphi}V}{V_{\varphi}^{2}}$ is dimensionless function, $\omega_{\varphi}$ is the equation of state parameter of the scalar field and $\tilde{R}$ refers to Jordan frame's curvature scalar. Also, we show that there exists $f(\tilde{R})$ gravity models which have tracking behavior in the Einstein frame and so the curvature of space time is decreasing with time while they lead to the solutions in the Jordan frame that the curvature of space time can be increasing with time.