CMB temperature anisotropy at large scales induced by a causal
  primordial magnetic field
  We present an analytical derivation of the Sachs Wolfe effect sourced by a
primordial magnetic field. In order to consistently specify the initial
conditions, we assume that the magnetic field is generated by a causal process,
namely 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 which
generates the magnetic field, in such a way that the total energy momentum
tensor is conserved across the transition and Einstein's equations are
satisfied. We further solve the evolution equations for the metric and fluid
perturbations at large scales analytically including neutrinos, and derive the
magnetic Sachs Wolfe effect. 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 in
fact strongly suppressed due to the presence of free-streaming neutrinos. We
derive the neutrino compensation effect dynamically and confirm that the
magnetic Sachs Wolfe spectrum from a causal magnetic field behaves as
l(l+1)C_l^B \propto l^2 as found in the latest numerical analyses.
