Field-level inference under pressure: the information content of the thermal Sunyaev-Zel'dovich field

The thermal Sunyaev--Zel'dovich (tSZ) field is highly non-Gaussian and contains cosmological information that cannot be fully captured by its angular power spectrum. In this work, we investigate the information content of this field using a convolutional neural network (CNN), and compare to simple and interpretable summary statistics, in a simulation-based inference (SBI) framework. We perform the first field-level analysis of the tSZ field, using a suite of $10{,}000$ simulated Compton-$y$ maps, where we jointly vary the cosmological parameters $Ω_c$ and $σ_8$ (the density fraction for cold dark matter and the matter clustering amplitude, respectively) and two parameters describing the amplitude and shape of the electron pressure profile, $P_0$ and $β$, that aim to capture astrophysical uncertainties. We consider both noiseless simulations and simulations including realistic Simons Observatory (SO) noise after component separation. We use SBI to compare the constraints obtained from the CNN with those from the three Minkowski functionals (MFs), individually and in combination, as well as the angular power spectrum. This is the first application of SBI to the MFs of the tSZ field. We find that the CNN provides tighter constraints than conventional summary statistics in the noiseless case. However, at SO-like noise levels, the combination of MFs and the power spectrum recovers constraints comparable to those obtained with the CNN. Our results highlight the potential of these interpretable summary statistics, as well as field-level inference, for extracting information from the tSZ field, motivating efforts toward their application to real data.