CMB temperature anisotropy at large scales induced by a causal primordial magnetic field

TL;DR

This paper analytically derives the large-scale CMB temperature anisotropy caused by a causal primordial magnetic field, highlighting the role of neutrinos and the suppression of leading order effects.

Contribution

It provides an analytical derivation of the magnetic Sachs Wolfe effect for causal magnetic fields, including neutrino effects and the suppression of leading order contributions.

Findings

Magnetic Sachs Wolfe effect is dominated by next-to-leading order metric perturbations.

Neutrino free-streaming suppresses the leading order contribution.

Spectrum behaves as l(l+1)C_l^B ∝ l^2, consistent with numerical results.

Abstract

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.