The gravitational potential reconstruction from peculiar velocity and weak lensing measurements

We present an analytic method for rapidly forecasting the accuracy of the gravitational potential reconstruction from measurements of radial peculiar velocities of every galaxy cluster with M>M-th in solid angle theta(2) and over a redshift range z(min) < z < z(max). These radial velocities can be determined from measurements of the kinetic and thermal Sunyaev-Zeldovich effects. For a shallow survey with 0.2 < z < 0.4, one mode of the potential (on length scales greater than or similar to60 Mpc) can be reconstructed for every approximately eight cluster velocity determinations. Deeper surveys require measurements of more clusters per a signal-to-noise ratio greater than 1. Accuracy is limited by the "undersampling noise" because of the nonobservation of a large fraction of mass that is not in galaxy clusters. Determining the potential will allow us to study in detail the relationship between galaxies and their surrounding large-scale density fields over a wide range of redshifts and to test the gravitational instability paradigm on very large scales. Observation of weak lensing by large-scale structure is complementary since lensing is sensitive to the tangential modes that do not affect the velocity.