Cosmology off(R)gravity in the metric variational approach

Abstract

We consider the cosmologies that arise in a subclass of $f(R)$ gravity with $f(R)=R+{\ensuremath{\mu}}^{2n+2}/(\ensuremath{-}R{)}^{n}$ and $n\ensuremath{\in}(\ensuremath{-}1,0)$ 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.