1/3 factor in the CMB Sachs-Wolfe effect

Abstract

We point out that a pseudo Newtonian interpretation of the 1/3 factor in the Sachs-Wolfe effect, which relates the fluctuations in temperature and potential, $\ensuremath{\delta}T/T=(1/3)\ensuremath{\delta}\ensuremath{\Phi},$ is not supported by the general relativistic analysis. Dividing the full gravitational effect into separate parts depends on the choice of time slicing (gauge) and there exist infinitely many different choices. More importantly, interpreting the parts as being due to the gravitational redshift and the time dilation is not justified in the rigorous relativistic perturbation theory. We suggest regarding the 1/3 factor as a general relativistic result that applies in a restricted situation of adiabatic perturbation in the $K=0=\ensuremath{\Lambda}$ model with the last scattering occurring in the matter dominated era. For an isocurvature initial condition the corresponding result $\ensuremath{\delta}T/T=2\ensuremath{\delta}\ensuremath{\Phi}$ has a different numerical coefficient.