Transfer matrices for magnetized CMB anisotropies

Large-scale magnetic fields can affect scalar cosmological perturbations whose evolution is described in the conformally Newtonian gauge and within the tight coupling approximation. The magnetized curvature perturbations present after matter radiation equality (and prior to decoupling) are computed in terms of an appropriate transfer matrix allowing a general estimate of the Sachs-Wolfe plateau. From the observation that CMB initial conditions should be (predominantly) adiabatic, the contribution of the magnetic field intensity can be constrained. Electronic address: [email protected] Large-scale magnetic fields are observed at a μ G level in galaxies, clusters and in some superclusters [1]. Compressional amplification (taking place during the gravitational collapse of the protogalaxy) allows to connect the observed magnetic field to a protogalactic field, present prior to gravitational collapse, of typical strength of 0.1 nG. A better understanding of the interplay between dynamo theory and the global conservation laws of magnetized plasmas has been recently achieved [2] also because of the improved comprehension of the solar dynamo action. It is then plausible, within the dynamo hypothesis, that the protogalactic field could be even much smaller than the nG and still explain some crucial properties of our magnetized Universe. Thanks to magnetic flux (and magnetic helicity) conservation in a conductive plasma, a magnetic field of nG strength at the epoch of galaxy formation can be as large as mG (i.e. roughly 6 orders of magnitude larger) at the epoch of photon decoupling, i.e. for zdec = 1100. If large-scale magnetic fields have primordial origin, they were present prior to matter-radiation equality affecting, potentially, CMB anisotropies [1]. Through the years, various studies have been devoted to the effect of large-scale magnetic fields on the vector and tensor CMB anisotropies [3] (see also [4] and references therein for some recent review articles). The implications of fully inhomogeneous magnetic fields on the scalar modes of the geometry remain comparatively less explored. By fully inhomogeneous we mean stochastically distributed fields that do not break the spatial isotropy of the background [4]. One of the aims of the present paper is to partially bridge this gap and to open the way for further developments. In short the idea is the following. The simplest set of initial conditions for CMB anisotropies, implies, in a ΛCDM framework, that a nearly scale-invariant spectrum of adiabatic fluctuations is present after matter-radiation equality but before decoupling for typical wavelengths larger than the Hubble radius at the corresponding epoch [5]. It became relevant, through the years, to relax the assumption of exact adiabaticity and to scrutinize the implications of a more general mixture of adiabatic and non-adiabatic initial conditions (see [6] and references therein). In this paper it will be argued, along a similar perspective, that large-scale magnetic fields slightly modify the adiabatic paradigm so that their typical strengths may be constrained. To achieve such a goal, the first step is to solve the evolution equations of magnetized cosmological perturbations well before equality. The second step is to follow the solution through equality (and up to decoupling). On a more technical ground, the second step amounts to the calculation of the so-called transfer matrix whose specific form is one of the the subjects of the present analysis. Consider then the system of cosmological perturbations of a flat Friedmann-RobertsonWalker (FRW) Universe, characterized by a conformal time scale factor a(τ), and consisting of a mixture of photons, baryons, CDM particles and massless neutrinos. In the conformally Newtonian gauge [6, 7] the scalar fluctuations of the metric tensor gμν are parametrized in terms of the two longitudinal fluctuations i.e. δg00 = 2a φ(τ, ~x) and δgij = 2a ψ(τ, ~x)δij . The Hamiltonian and momentum constraints, stemming from the (00) and (0i) components