Three-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 three-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 $C_l^{ij}$ and the spherical Fourier Bessel (SFB) power spectrum $C_l(k,k^{\prime })$. 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.

Figure 1

Survey window function with mean redshift $\overline{z}=1.2$ and constant galaxy volume density $n_0=10^{-3}{(h^{-1}\mathrm{Mpc})}^{-3}$. This corresponds to a galaxy surface density of $n_{\mathrm{A}}=0.49{\mathrm{arcmin}}^{-2}$.

Figure 2

Spherical harmonic tomography power spectra $C_l^{ij}$ for the seven redshift bins and survey specified in Sec. 2, where $\overline{z}$ denotes the mean redshift of each bin. The left panel shows the auto-power spectra, while the absolute value of the cross-power spectra is shown on the right-hand side.

Figure 3

The SFB auto-power spectrum $C_l^{\prime }(k,k)$ as a function of wave vector $k$ and angular scale $l$.

Figure 4

Uncertainties ($1\sigma$) on cosmological parameters obtained with SHT power spectrum as a function of the number of redshift bins $N_{\mathrm{Bin}}$.

Figure 5

Uncertainties ($1\sigma$) on cosmological parameters obtained using the SFB power spectrum as a function of maximal wave number $k_{\max }$. Constraints on $w_0$ are not shown for clarity.

Figure 6

The SFB auto-power spectrum $C_l^{\prime }(\nu ,\nu )$ for two different distance redshift relations as a function of angular scale $l$. (a) Logarithmic distance-redshift relation. (b) Linear distance-redshift relation.

Figure 7

The marginalized uncertainty on $\beta$ as a function of bin variance i.e. overlap on the left as well as bin number on the right.

Figure 8

The marginalized uncertainty on the slope ${\widehat{\theta }}_1$ as a function of correlation coefficient $r$.