Lorentz-violating contributions of the Carroll-Field-Jackiw model to the CMB anisotropy

We study the finite temperature properties of the Maxwell-Carroll-Field-Jackiw (MCFJ) electrodynamics for a purely spacelike background. Starting from the associated finite temperature partition function, a modified black body spectral distribution is obtained. We thus show that, if the CMB radiation is described by this model, the spectrum presents an anisotropic angular energy density distribution. We show, at leading order, that the Lorentz-breaking contributions for the Planck's radiation law and for the Stefan-Boltzmann's law are nonlinear in frequency and quadratic in temperature, respectively. Using our results, we set up bounds for the Lorentz-breaking parameter, and show that Lorentz violation in the context of the MCFJ model is unable to yield the known CMB anisotropy (of 1 part in $10^{5})$.