Small-scale CMB anisotropies induced by the primordial magnetic fields

The primordial magnetic fields (PMFs) produced in the early universe are expected to be the origin of the large-scale cosmic magnetic fields. The PMFs are considered to leave a footprint on the cosmic microwave background (CMB) anisotropies due to both the electromagnetic force and gravitational interaction. In this paper, we investigate how the PMFs affect the CMB anisotropies on smaller scales than the mean-free-path of the CMB photons. We solve the baryon Euler equation with Lorentz force due to the PMFs, and we show that the vector-type perturbations from the PMFs induce the CMB anisotropies below the Silk scale as $\ell>3000$. Based on our calculations, we put a constraint on the PMFs from the combined CMB temperature anisotropies obtained by Planck and South Pole Telescope (SPT). We have found that the highly-resolved temperature anisotropies of the SPT 2017 bandpowers at $\ell \lesssim 8000$ favor the PMF model with a small scale-dependence. As a result, the Planck and SPT's joint-analysis puts a constraint on the PMF spectral index as $n_B
Comment: 18 pages, 5 figures, 1 table, published in JCAP