Context. The TOI-421 planetary system contains two sub-Neptune-type planets (Pb ~ 5.2 days, Teq,b ~ 900 K, and Pc ~ 16.1 days, Teq,c ~ 650 K) and is a prime target to study the formation and evolution of planets and their atmospheres. The inner planet is especially interesting as the existence of a hydrogen-dominated atmosphere at its orbital separation cannot be explained by current formation models without previous orbital migration.
Aims: We aim to improve the system parameters to further use them to model the interior structure and simulate the atmospheric evolution of both planets, to finally gain insights into their formation and evolution. We also investigate the possibility of detecting transit timing variations (TTVs).
Methods: We jointly analysed photometric data of three TESS sectors and six CHEOPS visits as well as 156 radial velocity data points to retrieve improved planetary parameters. We also searched for TTVs and modelled the interior structure of the planets. Finally, we simulated the evolution of the primordial H-He atmospheres of the planets using two different modelling frameworks.
Results: We determine the planetary radii and masses of TOI-421 b and c to be Rb = 2.64 ± 0.08 R⊕, Mb = 6.7 ± 0.6 M⊕, Rc = 5.09 ± 0.07 R⊕, and Mc = 14.1 ± 1.4 M⊕. Using these results we retrieved average planetary densities of ρb = 0.37 ± 0.05ρ⊕ and ρc = 0.107 ± 0.012 ρ⊕. We do not detect any statistically significant TTV signals. Assuming the presence of a hydrogen-dominated atmosphere, the interior structure modelling results in both planets having extensive envelopes. While the modelling of the atmospheric evolution predicts for TOI-421 b to have lost any primordial atmosphere that it could have accreted at its current orbital position, TOI-421 c could have started out with an initial atmospheric mass fraction somewhere between 10 and 35%.
Conclusions: We conclude that the low observed mean density of TOI-421 b can only be explained by either a bias in the measured planetary parameters (e.g. driven by high-altitude clouds) and/or in the context of orbital migration. We also find that the results of atmospheric evolution models are strongly dependent on the employed planetary structure model.

The raw and detrended photometric time series data are available at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (ftp://130.79.128.5) or via https://cdsarc.cds.unistra.fr/viz-bin/cat/J/A+A/686/A301

Based on data from CHEOPS Guaranteed Time Observations, collected under Programme ID CH_PR100024.