Probing primordial magnetic fields with the cosmic microwave background

A primordial magnetic field (PMF) present before recombination can leave specific signatures in the cosmic microwave background (CMB) fluctuations. Of particular importance is its contribution to the B-mode polarization power spectrum. Indeed, vortical modes sourced by the PMF can dominate the B-mode power spectrum on small scales, as they survive damping up to a small fraction of the Silk length. Therefore, measurements of the B-mode polarization at high-l, such as the one recently performed by the South Pole Telescope (SPT), have the potential to provide stringent constraints on the PMF. We use the publicly released SPT B-mode polarization spectrum (2015 and 2019), along with the temperature and polarization data from the Planck satellite, to derive constraints on the magnitude B1Mpc, the spectral index nB and the energy scale at which the PMF was generated. We find that after marginalizing nB, Planck data constrains the magnetic amplitude to B1Mpc < 3.3 nG at 95% confidence level (CL), the SPT measurement improves the constraint to B1Mpc < 1.5 nG. The magnetic spectral index, nB, and the time of the generation of the PMF are unconstrained. For a nearly scale-invariant PMF, predicted by the simplest inflationary magnetogenesis models, the bound from Planck+SPT is B1Mpc < 1.2 nG at 95% CL for a non-helical PMF and B1Mpc < 1.1 nG for a maximally helical PMF. The bound from Planck data alone is B1Mpc < 1.7 nG at 95% CL when considering a maximally helical field. For a non-helical PMF with a spectral index of nB =2, expected for fields generated in post-inflationary phase transitions, the 95% CL bound is B1Mpc < 0.002 nG, corresponding to the magnetic fraction of the radiation density Ω_Bγ < 10-3 or the effective field Beff < 100 nG. We find that accounting for the helicity weakens the CMB constraints on PMF, allowing to have more magnetic power available on the 1Mpc comoving scale relevant to the formation of galactic magnetic fields. The patches for the Boltzmann code CAMB and the Markov Chain Monte Carlo engine CosmoMC, incorporating the PMF effects on CMB, are made publicly available.