Redshift Accuracy Requirements for Future Supernova and Number Count Surveys

We investigate the redshift accuracy of Type Ia supernova and cluster number count surveys required for the redshift uncertainties not to contribute appreciably to the dark energy parameter error budget. For the Supernova/Acceleration Probe experiment, we find that without the assistance of ground-based measurements individual supernova redshifts would need to be determined to about 0.002 or better, a challenging but feasible requirement for a low-resolution spectrograph. However, we find that accurate redshifts for z < 0.1 supernovae obtained with ground-based experiments are sufficient to protect the results against even relatively large redshift errors at high z. For the future cluster number count surveys such as with the South Pole Telescope, Planck, or DUET, we find that the purely statistical error in the photometric redshift is less important and that the irreducible systematic bias in redshift drives the requirements. The redshift bias must be kept below 0.001-0.005 per redshift bin (which is determined by the filter set), depending on the sky coverage and details of the definition of the minimal mass of the survey. Furthermore, we find that X-ray surveys have a more stringent required redshift accuracy than Sunyaev-Zeldovich (SZ) effect surveys since they use a shorter lever arm in redshift; conversely, SZ surveys benefit from their high-redshift reach only as long as some redshift information is available for distant (z 1) clusters.