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In the covariant cosmological perturbation theory, a 1+3 decompositionensures that all variables in the frame-independent equations are covariant,gauge-invariant and have clear physical interpretations. We develop thisformalism in the case of Brans-Dicke gravity, and apply this method to thecalculation of cosmic microwave background (CMB) anisotropy and large scalestructures (LSS). We modify the publicly available Boltzmann code CAMB tocalculate numerically the evolution of the background and adiabaticperturbations, and obtain the temperature and polarization spectra of theBrans-Dicke theory for both scalar and tensor mode, the tensor mode result forthe Brans-Dicke gravity are obtained numerically for the first time. We firstpresent our theoretical formalism in detail, then explicitly describe thetechniques used in modifying the CAMB code. These techniques are also veryuseful to other gravity models. Next we compare the CMB and LSS spectra inBrans-Dicke theory with those in the standard general relativity theory. Atlast, we investigate the ISW effect and the CMB lensing effect in theBrans-Dicke theory. Constraints on Brans-Dicke model with current observationaldata is presented in a companion paper (paper II).

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