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We present an analytical derivation of the Sachs Wolfe effect sourced by aprimordial magnetic field, generated by a causal process, such as a first orderphase transition in the early universe. As for the topological defects case, weapply the general relativistic junction conditions to match the perturbationvariables before and after the phase transition, in such a way that the totalenergy momentum tensor is conserved across the transition. We find that therelevant contribution to the magnetic Sachs Wolfe effect comes from the metricperturbations at next-to-leading order in the large scale limit. The leadingorder term is strongly suppressed due to the presence of free-streamingneutrinos. We derive the neutrino compensation effect and confirm that themagnetic Sachs Wolfe spectrum from a causal magnetic field behaves asl(l+1)C_l^B ~ l^2 as found in the latest numerical analyses.

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