  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 Brans-Dicke 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 of the
Brans-Dicke theory for both scalar and tensor mode, the tensor mode result for
the Brans-Dicke gravity are obtained numerically for the first time. We first
present our theoretical formalism in detail, then explicitly describe the
techniques used in modifying the CAMB code. These techniques are also very
useful to other gravity models. Next we compare the CMB and LSS spectra in
Brans-Dicke theory with those in the standard general relativity theory. At
last, we investigate the ISW effect and the CMB lensing effect in the
Brans-Dicke theory. Constraints on Brans-Dicke model with current observational
data is presented in a companion paper (paper II).
