Analytical marginalization over photometric redshift uncertainties in cosmic shear analyses

As the statistical power of imaging surveys grows, it is crucial to account for all systematic uncertainties. This is normally done by constructing a model of these uncertainties and then marginalizing over the additional model parameters. The resulting high dimensionality of the total parameter spaces makes inferring the cosmological parameters significantly more costly using traditional Monte Carlo sampling methods. A particularly relevant example is the redshift distribution, p(z), of the source samples, which may require tens of parameters to describe fully. However, relatively tight priors can be usually placed on these parameters through calibration of the associated systematics. In this paper, we show, quantitatively, that a linearization of the theoretical prediction with respect to these calibrated systematic parameters allows us to analytically marginalize over these extra parameters, leading to a factor of similar to 30 reduction in the time needed for parameter inference, while accurately recovering the same posterior distributions for the cosmological parameters that would be obtained through a full numerical marginalization over 160 p(z) parameters. We demonstrate that this is feasible not only with current data and current achievable calibration priors but also for future Stage-IV data sets.