  We introduce GODMAX (Gas thermODynamics and Matter distribution using jAX), a
novel code designed to calculate correlations between the cosmological matter
distribution and various gas thermodynamic quantities. Utilizing the extensive
ANTILLES suite of 200 hydrodynamical simulations with a diverse range of
baryonic feedback strengths, we jointly fit the 3D profiles of total matter
distribution, electron density, and electron pressure across various halo
masses and redshifts. By accommodating significant variations in gas profiles
expected due to baryonic feedback, solving exact hydrostatic equilibrium
equation and offering flexible modeling of non-thermal pressure support, GODMAX
has the capability to jointly fit all these profiles within the measurement
uncertainties. This advancement enables, for the first time, robust joint
analyses of multiple cosmic probes, including the kinetic and thermal
Sunyaev-Zel'dovich effect, weak lensing, and X-ray observations. Furthermore,
the model accurately captures correlations between the total matter power
suppression due to baryonic feedback and local average thermodynamic
quantities, such as the baryon fraction and integrated tSZ effect, in high-mass
halos, aligning with observations from hydrodynamical simulations. Looking
ahead, we forecast the expected constraints on cosmological and baryonic
parameters from upcoming weak lensing catalogs from the LSST and tSZ maps from
the Simons Observatory. This analysis underscores the importance of
cross-correlations between weak lensing and tSZ in enhancing parameter
constraints by resolving major systematic uncertainties due to baryonic
physics. The GODMAX code leverages the JAX library, resulting in a fully
differentiable halo model with native GPU compilation support.
