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 and gauge invariant and that they have clear physical interpretations. We develop this formalism in the case of Brans-Dicke gravity, and apply this method to the calculation of CMB anisotropy and large scale structures. 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 of Brans-Dicke theory for both scalar and tensor modes; the tensor mode results for Brans-Dicke gravity are obtained numerically for the first time. We first present our theoretical formalism in detail, and then explicitly describe the techniques used in modifying the CAMB code. These techniques are also very useful for other gravity models. Next we compare the CMB and large scale structure spectra in Brans-Dicke theory with those in the standard general relativity theory. At last, we investigate the integrated Sachs-Wolfe effect and the CMB lensing effect in Brans-Dicke theory. Constraints on the Brans-Dicke model with current observational data are presented in a companion paper [F. Wu and X. Chen, following Article, Phys. Rev. D 82, 083003 (2010)] (paper II).