Analytic marginalization of $N(z)$ uncertainties in tomographic galaxy
  surveys
  We present a new method to marginalize over uncertainties in redshift
distributions, $N(z)$, within tomographic cosmological analyses applicable to
current and upcoming photometric galaxy surveys. We allow for arbitrary
deviations from the best-guess $N(z)$ governed by a general covariance matrix
describing the uncertainty in our knowledge of redshift distributions. In
principle, this is marginalization over hundreds or thousands of new parameters
describing potential deviations as a function of redshift and tomographic bin.
However, by linearly expanding the theory predictions around a fiducial model,
this marginalization can be performed analytically, resulting in a modified
data covariance matrix that effectively downweights the modes of the data
vector that are more sensitive to redshift distribution variations. We showcase
this method by applying it to the galaxy clustering measurements from the Hyper
Suprime-Cam first data release. We illustrate how to marginalize over
sample-variance of the calibration sample and a large general systematic
uncertainty in photometric estimation methods, and explore the impact of priors
imposing smoothness in the redshift distributions.
