FRCAMB: An $f(R)$ Code for Anisotropies in the Microwave Background
  An $f(R)$ gravity model is proposed to realize a late time accelerated
expansion of our Universe. To test the viability of an $f(R)$ gravity model
through cosmic observations, the background evolution and the
Einstein-Boltzmann equation should be solved for studying the effects on the
cosmic microwave background power spectrum and on the matter power spectrum. In
the market, we already have the modified versions of {\bf CAMB} code, for
instance {\bf EFTCAMB} and {\bf MGCAMB}. However, in these publicly available
Einstein-Boltzmann codes, a specific background cosmology, for example the
$\Lambda$CDM or $w$CDM, is assumed. This assumption would be non-proper for a
specific $f(R)$ model where the background evolution may be different from a
$\Lambda$CDM cosmology. Therefore the main task for this paper is to present a
code to calculate the anisotropies in the microwave background for any $f(R)$
gravity model based on {\bf CAMB} code, i.e. {\bf FRCAMB}, where the background
and perturbation evolutions are included consistently. As results, one can
treat {\bf FRCAMB} as a blackbox to output the CMB power spectrum and matter
power spectrum, once an $f(R)$ function, its first two derivative with respect
to $R$, i.e. $f_R\equiv df/dR$, $f_{RR}\equiv d^2f/dR^2$ and the reasonable
values of the model parameters are inputted properly. As by-products, one can
also output the effective equation of state of $f(R)$ model, the evolution of
the dimensionless energy densities and other interesting cosmological
quantities.
