Here is a possible path to establishing how well CMB lensing will help us with DE, in the era of LSST and DESI measurements. Let's assume that we will have LSST measurements of galaxy clustering Cl(z), LSST measurements of shear Cl(z), and DESI measurements of BAO rs/DV(z).
We have a Fisher forecasting code available to use - David Alonso's GoFish code, that includes all of these measurements, including correlations with CMB lensing. The nuisance parameters it includes are galaxy biases for the clustering, and multiplicative biases for the shear. It also includes photo-z uncertainty, some intrinsic alignment uncertainty, and some accounting for baryonic effects.
Currently it spits our 'standard' DE parameters w0-wa, as well as curvature, neutrino mass etc. Here is a plan:
1. Jo's student Siddharth will use GoFish code to compare w0-wa errors for Planck+DESI-BAO+LSST-clustering versus Planck+S4+DESI-BAO + LSST-clustering (including all the cross-correlations and free galaxy bias parameters), and Planck+DESI-BAO + LSST-shear versus S4+DESI-BAO LSST-shear. These two tests will demonstrate the impact of adding S4 to the currently-funded experiments. The impact of S4 is expected to be a better handle on the bias parameters, and an extra higher-redshift slice.
2. Mat Madhavacheril will adapt that code to spit out sigma8(z) like for clusters, instead of w0-wa, as we expect to see a stronger effect of the contribution of CMB lensing at high redshift.
Note - aware that many others in the S4 lensing and DE groups have similar codes and alternative routes to doing this. Other contributions on this front welcome!