Revisiting Rotational Perturbations and the Microwave Background

Abstract

We consider general-relativistic rotational perturbations in homogeneous and isotropic Friedman - Robertson - Walker (FRW) cosmologies. Taking linear perturbations of FRW models, the general solution of the field equations contains tensorial, vectorial and scalar functions. The vectorial terms are in connection with rotations in the given model and due to the Sachs - Wolfe effect they produce contributions to the temperature fluctuations of the cosmic microwave background radiation (CMBR). In present paper we obtain the analytic time dependence of these contributions in a spatially flat, FRW model with pressureless ideal fluid, in the presence and the absence of a cosmological constant. We find that the solution can be separated into an integrable and a non-integrable part as is the situation in the case of scalar perturbations. Analyzing the solutions and using the results of present observations we estimate the order of magnitude of the angular velocity corresponding to the rotation tensor at the time of decoupling and today.