Redshift Accuracy Requirements for Future Supernova and Number Count Surveys

We investigate the required redshift accuracy of type Ia supernova andcluster number-count surveys in order for the redshift uncertainties not tocontribute appreciably to the dark energy parameter error budget. For the SNAPsupernova experiment, we find that, without the assistance of ground-basedmeasurements, individual supernova redshifts would need to be determined toabout 0.002 or better, which is a challenging but feasible requirement for alow-resolution spectrograph. However, we find that accurate redshifts for z<0.1supernovae, obtained with ground-based experiments, are sufficient to immunizethe results against even relatively large redshift errors at high z. For thefuture cluster number-count surveys such as the South Pole Telescope, Planck orDUET, we find that the purely statistical error in photometric redshift is lessimportant, and that the irreducible, systematic bias in redshift drives therequirements. The redshift bias will have to be kept below 0.001-0.005 perredshift bin (which is determined by the filter set), depending on the skycoverage and details of the definition of the minimal mass of the survey.Furthermore, we find that X-ray surveys have a more stringent required redshiftaccuracy than Sunyaev-Zeldovich (SZ) effect surveys since they use a shorterlever arm in redshift; conversely, SZ surveys benefit from their high redshiftreach only so long as some redshift information is available for distant (z>1)clusters.Comment: Submitted to ApJ. Gaussianity assumption for observables fully justified. Photo-z's for SNe discusse