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.
