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Primordial magnetic fields can change the recombination history of theuniverse by inducing clumping in the baryon density at small scales. They wererecently proposed as a candidate model to relieve the Hubble tension. Weinvestigate the consistency of the constraints on a clumping factor parameter$b$ in a simplistic model, using the latest CMB data from Planck, ACT DR4 andSPT-3G 2018. For the combined CMB data alone, we find no evidence for clumpingbeing different from zero, though when adding a prior on $H_0$ based on thelatest distance-ladder analysis of the SH0ES team, we report a weak detectionof $b$. Our analysis of simulated datasets shows that ACT DR4 has moreconstraining power with respect to SPT-3G 2018 due to the degeneracy breakingpower of the TT band powers (not included in SPT). Simulations also suggestthat the TE,EE power spectra of the two datasets should have the sameconstraining power. However, the ACT DR4 TE,EE constraint is tighter thanexpectations, while the SPT-3G 2018 one is looser. While this is compatiblewith statistical fluctuations, we explore systematic effects which may accountfor such deviations. Overall, the ACT results are only marginally consistentwith 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 CMBdata 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 and7.8, respectively, over Planck alone.

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