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 SPTs joint-analysis puts a constraint on the PMF spectral index as $n_B<-1.14$ at 95% confidence level (C.L.), and this is more stringent compared with the Planck-only constraint $n_B<-0.28$. We show that the PMF strength normalized on the co-moving 1 Mpc scale is also tightly constrained as $B_{1mathrm{Mpc}}<1.5$ nG with Planck and SPT at 95% C.L., while $B_{1mathrm{Mpc}}<3.2$ nG only with the Planck data at 95% C.L. We also discuss the effects on the cosmological parameter estimate when including the SPT data and CMB anisotropies induced by the PMFs.
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