Reconstructing the galaxy redshift distribution from angular cross power
  spectra
  The control of photometric redshift (photo-$z$) errors is a crucial and
challenging task for precision weak lensing cosmology. The spacial
cross-correlations (equivalently, the angular cross power spectra) of galaxies
between tomographic photo-$z$ bins are sensitive to the true redshift
distribution $n_i(z)$ of each bin and hence can help calibrate the photo-$z$
error distribution for weak lensing surveys. Using Fisher matrix analysis, we
investigate the contributions of various components of the angular power
spectra to the constraints of $n_i(z)$ parameters and demonstrate the
importance of the cross power spectra therein, especially when catastrophic
photo-$z$ errors are present. We further study the feasibility of
reconstructing $n_i(z)$ from galaxy angular power spectra using Markov Chain
Monte Carlo estimation. Considering an LSST-like survey with $10$ photo-$z$
bins, we find that the underlying redshift distribution can be determined with
a fractional precision ($\sigma(\theta)/\theta$ for parameter $\theta$) of
roughly $1\%$ and $10\%$ for the mean redshift and width of $n_i(z)$,
respectively.
