Cosmic microwave background with Brans-Dicke Gravity: I. Covariant Formulation

In the covariant cosmological perturbation theory, a 1+3 decomposition ensures that all variables in the frame-independent equations are covariant, gauge-invariant and have clear physical interpretations. We develop this formalism in the case of BransDicke gravity, and apply this method to the calculation of cosmic microwave background (CMB) anisotropy and large scale structures (LSS). We modify the publicly available Boltzmann code CAMB to calculate numerically the evolution of the background and adiabatic perturbations, and obtain the temperature and polarization spectra for the Brans-Dicke theory. In this paper, we describe our theoretical formalism in detail, and compare the CMB and LSS spectra in Brans-Dicke theory with those in the standard general relativity theory. Constraints on Brans-Dicke model with current observational data is presented in a companion paper [1](paper II).