Impact of a causal primordial magnetic field on the Sachs Wolfe Effect

We present an analytical derivation of the Sachs Wolfe effect sourced by a primordial magnetic field, generated by a causal process, such as a first order phase transition in the early universe. As for the topological defects case, we apply the general relativistic junction conditions to match the perturbation variables before and after the phase transition, in such a way that the total energy momentum tensor is conserved across the transition. We find that the relevant contribution to the magnetic Sachs Wolfe effect comes from the metric perturbations at next-to-leading order in the large scale limit. The leading order term is strongly suppressed due to the presence of free-streaming neutrinos. We derive the neutrino compensation effect and confirm that the magnetic Sachs Wolfe spectrum from a causal magnetic field behaves as l(l+1)C_l^B ~ l^2 as found in the latest numerical analyses.