Primordial magnetic fields in the light of upcoming post-EoR Lyman-$α$ and 21-cm observations

Abstract: The Lorentz force exerted by a primordial magnetic field (PMF) on the coupled baryon-dark matter system may enhance total matter power at small scales after recombination. In the post-reionization (post-EoR) era, a weakly scale-dependent PMF of sub-nG strength is thus expected to influence the Lyman-$α$ (Ly$α$) power spectrum, the 21 cm power spectrum, and the Ly$α$-21 cm cross-spectrum at scales $k\gtrsim 1:h/\textrm{Mpc}$. We investigate the prospects of constraining the PMF sector via these three cosmological observables, by employing SNR estimation and Fisher forecast on the PMF amplitude $B_0$ and spectral index $n_{\rm B}$, for a next-generation DESI-like spectroscopic survey and two upcoming 21 cm facilities, namely SKA1-Mid and PUMA. Our results indicate the possibility of constraining both PMF parameters with $\lesssim10\%$ relative errors through the uncontaminated 21 cm auto-spectrum as well as the Ly$α$-21 cm cross-spectrum probed with the DESI-like+SKA1-Mid combination. Indicatively, the Ly$α$-21 cm cross-correlation via DESI-like+SKA1-Mid is predicted to constrain a fiducial scenario $B_0=0.8$ nG and $n_{\rm B}=-2.9$ with $1σ$ errors $ΔB_0\approx 0.07$ nG and $Δn_{\rm B}\approx0.02$. The DESI-like+PUMA setup is predicted to fare relatively worse due to its restriction to larger scales, resulting in comparatively one order of magnitude relaxed error bounds for similar fiducials. Since the Ly$α$-21 cm cross-signal is expected to be largely insensitive to foreground contamination (unlike the 21 cm auto-spectrum), it may serve as an optimal foreground-immune post-EoR probe to constrain a weakly scale-dependent sub-nG PMF via future DESI-like+SKA1-Mid observations.