Tensor microwave anisotropies from a stochastic magnetic field

We derive an expression for the angular power spectrum of cosmic microwave background anisotropies due to gravity waves generated by a stochastic magnetic field and compare the result with current observations; we take into account the non-linear nature of the stress energy tensor of the magnetic field. For almost scale invariant spectra, the amplitude of the magnetic field at galactic scales is constrained to be of order ${10}^{-9}\hspace{0.5em}\mathrm{G}.$ If we assume that the magnetic field is damped below the Alfvén damping scale, we find that its amplitude at ${0.1h}^{-1}\hspace{0.5em}\mathrm{Mpc},$ $B_{\lambda },$ is constrained to be $B_{\lambda }<7.9\mathrm{}\times {10}^{-6}{e}^{3n}\hspace{0.5em}\mathrm{G},$ for $n<\mathrm{}-3/2,$ and $B_{\lambda }<9.5\mathrm{}\times {10}^{-8}{e}^{0.37n}\hspace{0.5em}\mathrm{G},$ for $n>\mathrm{}-3/2,$ where n is the spectral index of the magnetic field and $H_0=100h\hspace{0.5em}{\mathrm{km}\mathrm{}\mathrm{}\mathrm{s}}^{-1}\hspace{0.5em}{\mathrm{Mpc}}^{-1}$ is the Hubble constant today.