Linear metric and temperature fluctuations of a charged plasma in a primordial magnetic field

We discuss tensor metric perturbations in a magnetic field around the homogeneous Juttner equilibrium of massless particles in an expanding universe. We solve the Liouville equation and derive the energy-momentum tensor up to linear terms in the metric and in the magnetic field.The term linear in the magnetic field is different from zero if the total charge of the primordial plasma is non-zero. We obtain an analytic formula for temperature fluctuations treating the tensor metric perturbations and the magnetic field as independent random variables. Assuming a cutoff on large momenta of the magnetic spectral function we show that the presence of the magnetic field can discriminate only low multipoles in the multipole expansion of temperature fluctuations. In such a case the term linear in the magnetic field can be more important than the quadratic one (corresponding to the fluctuations of the pure magnetic field).