Observational cosmology. VI. The microwave background and the Sachs-Wolfe effect

The use of observational coordinates allows the formulation of redshift in a general cosmological space-time in a simple form, $\mathrm{}(1+z)=\frac{A_0\mathrm{dw}}{d\tau }$, where $A_0$ is a normalization constant, $w$ is the observational time coordinate, and $\tau$ is the proper time along the fundamental flow lines. This in turn allows easy calculation of the anisotropy of the cosmic microwave background radiation (CMBR) due to the Sachs-Wolfe (SW) effect. We reproduce the usual dominant first-order effect $\frac{\delta T_R}{T_R}=\frac{1}{3}({\delta }_{\mathrm{bB}}-{\delta }_{\mathrm{bA}})$, where ${\delta }_b$ is the density contrast of baryons on the last scattering surface; as implied by this equation, the observationally significant result is the difference of ${\delta }_b$ in two different directions $A$ and $B$ on the plane of the sky. In order to obtain the actual result, one also needs to study perturbations of the temperature on the background last-scattering surface and on its first-order counterpart. In addition to the usual dominant term in the SW effect, we obtain a second term when the pressure $p$ is significant at last scattering; this term depends on the difference in the pressure at the points of emission $A$ and $B$ on the last-scattering surface, and enters the CMBR anisotropy with a sign opposite to that of the usual term. For the adiabatic case, this pressure term can reduce the SW effect by up to 87% in a low-density universe. When $p=0\mathrm{}$ at the last scattering surface, the usual SW result is obtained. Our results are gauge invariant to first order. Other explicit contributions to the Sachs-Wolfe anisotropy in the observational-coordinate calculation are clearly higher order. We discuss the interpretation of these results and compare them to other calculations of the large-scale cosmic microwave background anisotropy.