Consistency of Planck, ACT and SPT constraints on magnetically assisted recombination and forecasts for future experiments

Primordial magnetic fields can change the recombination history of the universe by inducing clumping in the baryon density at small scales. They were recently proposed as a candidate model to relieve the Hubble tension. We investigate the consistency of the constraints on a clumping factor parameter $b$ in a simplistic model, using the latest CMB data from Planck, ACT DR4 and SPT-3G 2018. For the combined CMB data alone, we find no evidence for clumping being different from zero, though when adding a prior on $H_0$ based on the latest distance-ladder analysis of the SH0ES team, we report a weak detection of $b$. Our analysis of simulated datasets shows that ACT DR4 has more constraining power with respect to SPT-3G 2018 due to the degeneracy breaking power of the TT band powers (not included in SPT). Simulations also suggest that the TE,EE power spectra of the two datasets should have the same constraining power. However, the ACT DR4 TE,EE constraint is tighter than expectations, while the SPT-3G 2018 one is looser. While this is compatible with statistical fluctuations, we explore systematic effects which may account for such deviations. Overall, the ACT results are only marginally consistent with Planck or SPT-3G, at the $2-3\sigma$ level within $\Lambda$CDM+$b$ and $\Lambda$CDM, while Planck and SPT-3G are in good agreement. Combining the CMB data together with BAO and SNIa provides an upper limit of b<0.4 at 95\% c.l. (b<0.5 without ACT). Adding a SH0ES-based prior on the Hubble constant gives $b = 0.31^{+0.11}_{-0.15}$ and $H_0=69.28 \pm 0.56$ km/s/Mpc ($b = 0.41^{+0.14}_{-018}$ and $H_0=69.70 \pm 0.63$ km/s/Mpc without ACT). Finally, we forecast constraints on $b$ for the full SPT-3G survey, Simons Observatory, and CMB-S4, finding improvements by factors of 1.5 (2.7 with Planck), 5.9 and 7.8, respectively, over Planck alone.