Magnetic field back reaction on the matter power spectrum

At lowest order comoving magnetic fields which are frozen-into the expanding cosmic fluid do not evolve in time. At next-to-leading order the induction equation is sourced by the interaction term between the baryon velocity and the magnetic field amplitude which leads to a non-trivial evolution of the comoving magnetic field. Moreover, it induces non-trivial cross correlations between the adiabatic curvature mode and the magnetic mode. This cross correlation together with the evolution of the induced matter perturbation leads to interesting effects on the total matter power spectrum at small scales.