Cosmological gravity on all scales V: MCMC forecasts combining large scale structure and CMB lensing for binned phenomenological modified gravity

As cosmology rapidly approaches the data-dominated phase of stage IV large scale structure surveys, the modelling of nonlinear scales has become a serious challenge that faces the community, particularly when analysing models beyond $w$CDM. In this work, we emulate the matter power spectrum in a phenomenological parameterisation of modified gravity in which a time-varying effective gravitational constant $\mu$ and a gravitational slip $\eta$ are binned in redshift. We are able to achieve accuracy $<1\%$ in the modified gravity boost relative to COLA (COmoving Lagrangian Acceleration) simulations. We forecast the constraining power for each bin using a simulated $3\times 2$pt LSST Y10-like data vector and a $6\times 2$pt LSST Y10 x Simons Observatory cosmic microwave background (CMB) lensing data vector. We recover the characteristic degeneracy between $\mu$ and $\eta$ previously identified in Fisher forecasts and demonstrate that the best-constrained direction corresponds to the combination $\Sigma=\mu(1+\eta)/2$ which governs the lensing potential. We show that while large scale structure is sensitive to growth of structure at low redshift, CMB lensing extends the sensitivity to a higher redshift range. These results demonstrate that fast emulation of nonlinear modified-gravity effects enables full Bayesian analyses of model-agnostic gravity parameterisations with realistic survey data vectors and astrophysical systematics.