Stability of cosmological solutions in $f(R)$ models of gravity

We reconcile seemingly conflicting statements in the literature about the behavior of cosmological solutions in modified theories of gravity where the Einstein-Hilbert Lagrangian for gravity is modified by the addition of a function of the Ricci scalar, $f(R)$. Using the example of $f(R)=\pm {\mu }^4/R$ we show that only such choices of $f(R)$ where ${\mathrm{d}}^{2}f/\mathrm{d}{R}^2>0$ have stable high-curvature limits and well-behaved cosmological solutions with a proper era of matter domination. The remaining models enter a phase dominated by both matter and scalar kinetic energy where the scalar curvature remains low.

Figure 1

Exact effective potentials with constant term subtracted, for a range of matter densities for scalar field, in CDTT (top panel) and mCDTT (bottom panel) models. Both models exhibit extrema at $R_{\mathrm{J}}={\kappa }^2{\rho }_{\mathrm{m}}^{\mathrm{J}}$: a maximum for CDTT and a minimum for mCDTT.

Figure 2

Evolution of the Jordan-frame Ricci scalar for cold dark matter with cosmological constant (LCDM, solid line; $\mu =H_0$), mCDTT (dashed line; $\alpha =2$), and CDTT (dotted line; $\alpha =2$). All models start with the same GR initial conditions. This solution is unstable in the CDTT model and the curvature rapidly relaxes to the $\varphi \mathrm{MDE}$ phase where it is much lower than for the corresponding GR solution. This phase replaces the standard matter domination for $f(R)$ models with $B<0$. mCDTT follows standard LCDM evolution until late times, where it asymptotes to a milder acceleration phase.