Testing Gravity with Cosmic Microwave Background–Large-Scale Structure Cross-correlations

Precision measurements of the cosmic microwave background (CMB) and large–scale structure (LSS) have cemented the ΛCDM model as a successful baseline, yet some fundamental aspects remain unexplained, such as the physical origin of late–time cosmic acceleration. Forthcoming surveys enable model–level tests of scalar–tensor theories framed within the Effective Field Theory of Dark Energy (EFT). We develop a comprehensive Fisher–forecast framework that combines Planck and Simons Ob- servatory CMB temperature, polarization, and lensing (TT, TE, EE, ϕϕ) with Euclid tomographic photometric galaxy clustering (GCph) and weak lensing (WL), together with their cross–correlations. In addition, we combine these CMB×LSS forecasts with Type Ia supernova information, modeling a Vera Rubin Observatory–LSST–like survey. We account for survey specifications and a comprehensive nuisance sector (intrinsic alignments, multiplicative shear and magnification biases, galaxy bias) and consistently model relativistic light–cone effects, redshift–space distortions (RSD), and lensing magnification in number counts. We produce forecast constraints on EFT/model and standard cosmological parameters, map degeneracy directions, and quantify the incremental information from cross–probe combinations. Our results underscore the leverage of LSS×CMB cross–correlations for breaking growth–lensing degeneracies and highlight the need for self–consistent light–cone modeling when testing beyond–ΛCDM physics.