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Constraints on the multi-lognormal magnetic fields from the observations of the cosmic microwave background and the matter power spectrum

Primordial magnetic fields (PMFs), which were generated in the early Universe before recombination, affect the motion of plasma and then the cosmic microwave background and the matter power spectrum. We consider constraints on PMFs with a characteristic correlation length from the observations of the anisotropies of the cosmic microwave background and the matter power spectrum. The spectrum of PMFs is modeled with multi-lognormal distributions, rather than power-law distribution, and we derive constraints on the strength $|\mathbf{B}_k|$ at each wave number $k$ along with the standard cosmological parameters in the flat Universe and the foreground sources. We obtain upper bounds on the field strengths at $k=10^{-1}, 10^{-2}, 10^{-4}$, and $10^{-5}$ Mpc$^{-1}$ as 4.7 nG, 2.1 nG, 5.3 nG, and 10.9 nG ($2σ$ C.L.) respectively, while the field strength at $k=10^{-3} $Mpc$^{-1}$ turns out to have a finite value as $|\mathbf{B}_{k = 10^{-3}}| = 6.2 \pm 1.3 $ nG ($1σ$ C.L.). This finite value is attributed to the finite values of BB mode data at $\ell > 300$ obtained from the QUAD experiment. If we do not include the BB mode data, we obtain only the upper bound on $B_{k=10^{-3}}$.