3-dimensional spherical analyses of cosmological spectroscopic surveys
  Spectroscopic redshift surveys offer great prospects for constraining the
dark sector in cosmology. Future surveys will however be both deep and wide and
will thus require an analysis in 3-dimensional spherical geometry. We review
and compare several methods which have been proposed in the literature for this
purpose, focusing in particular on implementations of the spherical harmonic
tomography (SHT) power spectrum $C^{i j}_{l}$ and the spherical Fourier Bessel
(SFB) power spectrum $C_{l} (k, k')$. Using a Fisher analysis, we compare the
forecasted constraints on cosmological parameters using these statistics. These
constraints typically rely on approximations such as the Limber approximation
and make specific choices in the numerical implementation of each statistic.
Using a series of toy models, we explore the applicability of these
approximations and study the sensitivity of the SHT and SFB statistics to the
details of their implementation. In particular, we show that overlapping
redshift bins may improve cosmological constraints using the SHT statistic when
the number of bins is small, and that the SFB constraints are quite robust to
changes in the assumed distance-redshift relation. We also find that the SHT
can be tailored to be more sensitive to modes at redshifts close to the survey
boundary, while the SFB appears better suited to capture information beyond the
smooth shape of the power spectrum. In this context, we discuss the pros and
cons of the different techniques and their impact on the design and analysis of
future wide field spectroscopic surveys.
