Cosmology of $f(R)$ gravity in the metric variational approach

We consider the cosmologies that arise in a subclass of $f(R)$ gravity with $f(R)=R+{\mu }^{2n+2}/(-R{)}^n$ and $n\in (-1,0)$ in the metric (as opposed to the Palatini) variational approach to deriving the gravitational field equations. The calculations of the isotropic and homogeneous cosmological models are undertaken in the Jordan frame and at both the background and the perturbation levels. For the former, we also discuss the connection to the Einstein frame in which the extra degree of freedom in the theory is associated with a scalar field sharing some of the properties of a “chameleon” field. For the latter, we derive the cosmological perturbation equations in general theories of $f(R)$ gravity in covariant form and implement them numerically to calculate the cosmic microwave background (CMB) temperature and matter power spectra of the cosmological model. The CMB power is shown to reduce at low $l$’s, and the matter power spectrum is almost scale independent at small scales, thus having a similar shape to that in standard general relativity. These are in stark contrast with what was found in the Palatini $f(R)$ gravity, where the CMB power is largely amplified at low $l$’s and the matter spectrum is strongly scale dependent at small scales. These features make the present model more adaptable than that arising from the Palatini $f(R)$ field equations, and none of the data on background evolution, CMB power spectrum, or matter power spectrum currently rule it out.

Figure 1

A phase portrait of the model. The trajectories in the 3-dimensional space spanned by $x_1$, $x_2$, $x_3$ are projected to the $x_1-x_2$ and $x_1-x_3$ planes, respectively. The critical points $P_2$, $P_3$ are shown. Only 2 different trajectories with different initial conditions are depicted for clearness and $n=-0.1$ is adopted. It is seen that $x_1$ oscillates around 0. Note that these are just for illustrations and do not represent realistic cosmologies.

Figure 2

The fractional energy densities ${\Omega }_{\mathrm{m}}$, ${\Omega }_{\mathrm{rad}}$, and ${\Omega }_{\mathrm{DE}}\equiv 1-{\Omega }_{\mathrm{m}}-{\Omega }_{\mathrm{rad}}$ as functions of redshift $z$. We plot the cases for $n=-0.01$, $-0.1$, $-0.2$, and $-0.5$, respectively. Note that in general ${\Omega }_{\mathrm{m}0}>0.3$ because we require $\kappa {\rho }_{\mathrm{m}0}/3H_0^2=0.3$ instead and because $F_0<1$.

Figure 3

The same as FIG. 2, but for the case of $n=0$ ($\Lambda \mathrm{CDM}$).

Figure 4

$F$ as a function of redshift for the cases $n=-0.01$, $-0.1$, $-0.2$, and $-0.5$, respectively, as indicated beside the curves.

Figure 5

The CMB power spectra of the $f(R)$ model. The lines from top to bottom represent the cases of $n=0$, $-0.01$, $-0.1$, $-0.2$, and $-0.5$, respectively. For all the curves we have adopted the parameters $H_0=70\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ and $\kappa {\rho }_{\mathrm{m}0}/3H_0^2=0.3$.

Figure 6

The potentials ${\varphi }_k$ (see definition in text) as functions of the scale factor $a$ for various $k$ values. Only the cases of $\Lambda \mathrm{CDM}$ ($n=0$), $n=-0.2$, and $n=-0.5$ are shown for clearness. The scales are chosen to be $k=0.0005,0.001,0.005,0.01{\mathrm{Mpc}}^{-1}$ as indicated beside the curves. For all the curves we have adopted the parameters $H_0=70\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ and $\kappa {\rho }_{\mathrm{m}0}=0.3$.

Figure 7

The matter power spectra of the $f(R)$ model. The lines from bottom to top represent the cases of $n=0$, $-0.01$, $-0.1$, $-0.2$, and $-0.5$, respectively. In all these plots we have adopted $H_0=70\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ and $\kappa {\rho }_{\mathrm{m}0}/3H_0^2=0.3$.

Figure 8

The dependence of the matter power spectra on $H_0$ (upper panel) and $\kappa {\rho }_{\mathrm{m}0}/3H_0^2$ (lower panel). Upper panel: the solid curves from bottom to top represent $H_0=65,68,70\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$, respectively ($\kappa {\rho }_{\mathrm{m}0}/3H_0^2=0.3$). Lower panel: the solid curves from bottom to top represent $\kappa {\rho }_{\mathrm{m}0}/3H_0^2=0.25$, 0.27, 0.30, respectively, ($H_0=70\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$). Dashed curves are the spectra for the $\Lambda \mathrm{CDM}$ model. $n=-0.5$ is adopted for illustration. We see that a smaller $\kappa {\rho }_{\mathrm{m}0}/3H_0^2$ can bring the shape of the spectrum closer to the $\Lambda \mathrm{CDM}$ one: in this case data on large-scale structure cannot place stringent constraints on the model.