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).

Figure 1

The time evolution of the Brans-Dicke field $\phi$.

Figure 2

The evolution of the time derivative of the Brans-Dicke field ${\phi }^{\prime }$.

Figure 3

The time evolution of the effective Newtonian gravitational coupling $G_{\mathrm{eff}}$.

Figure 4

CMB temperature and polarization power spectra for Brans-Dicke theories with $\omega =\infty$, $\pm 75$ in the scalar mode.

Figure 5

$\Delta C_l=C_l(\omega =\infty )-C_l(\omega =75)$ for TT, TE, and EE correlations.

Figure 6

CMB temperature and polarization power spectra for the Brans-Dicke theories in the tensor mode. The solid, dotted, and dashed curves represent the Brans-Dicke model with $\omega =\infty$, 75, and $-75$, respectively. The tensor-to-scalar ratio $R$ is set to 0.1.

Figure 7

The matter power spectra at $z=0$. The solid, dotted, and dashed curves represent the Brans-Dicke model with $\omega =\infty$, 75, and $-75$, respectively.

Figure 8

The ISW effect of the TT power spectra in the scalar mode for Brans-Dicke gravity. $C_l^{\mathrm{ISW}}$ is the CMB TT power spectrum only considering the ISW effect. $\Delta C_l^{\mathrm{ISW}}=C_l^{\mathrm{ISW}}({\omega }^{\prime })-C_l^{\mathrm{ISW}}(\omega =\infty )$, ${\omega }^{\prime }=75$ for the red dotted curve, and ${\omega }^{\prime }=-75$ for the blue dashed curve.

Figure 9

The redshift distribution of the assumed sample with median redshift $z_m=0.33$.

Figure 10

The ISW effect from the CMB temperature and galaxy overdensity correlation. The upper panel is the angular power spectrum, while the bottom panel is the angular correlation function.

Figure 11

Comparison of unlensed and lensed CMB TT power spectra in the scalar mode for Brans-Dicke gravity with $\omega =75$. $\Delta C_l^{\mathrm{TT}}/C_l^{\mathrm{TT}}$ is the relative difference, where $\Delta C_l^{\mathrm{TT}}=C_l^{\mathrm{TT}}(\mathrm{\text{lensed}})-C_l^{\mathrm{TT}}(\mathrm{\text{unlensed}})$, and $C^{\mathrm{TT}}$ in the denominator is the unlensed TT power spectrum.