Figure 1

Survey window functions $\mathcal{W}$ and redshift distributions $P_z$. Upper panel: Gaussian window function (solid line) in Eq. (65) and top-hat window function (dotted line) in Eq. (66). Bottom panel: Corresponding redshift distributions obtained by using Eq. (21). The characteristic scales $r_0$ of the surveys are set equal to the comoving distance to $z=1$. The comoving number density is related to the survey window function as in Eq. (19).Reuse & Permissions

Figure 2

Spherical power spectra with Gaussian window function. In each panel, various curves with labels show the spherical power spectra of representative components of the full general relativistic formula ${\delta }_g^{\mathrm{obs}}$ in Eq. (34), which are further decomposed in Eqs. (37, 38, 39, 40, 41, 42, 43). The time evolution of the transfer and the weight functions is ignored, and their values are set equal to those at $z=0$. Each column plots the same spherical power spectra at angular multipoles, ranging from $l=2$ (top rows) to $l=10$ (bottom rows), illustrating their angular dependence. Vertical lines mark the characteristic wave number $k_c\equiv l/r_0$, below which spherical power spectra are suppressed due to the constraint $k\ge k_{\perp }\simeq k_c$. Gray curves show the three-dimensional power spectra at $z=0$, if there exist well-defined correspondences. For example, spherical power spectra for the density (solid line) and the redshift-space distortion (dashed line) in the first column reduce to the corresponding three-dimensional power spectra on small scales. The noise power spectrum (dotted line) is shown in the first column, assuming ${\tilde{n}}_g={10}^{-4}(h^{-1}\mathrm{Mpc}{)}^{-3}$. Compared to the contributions in the first two columns, the projected quantities in the third column are negligible, except the gravitational lensing convergence. The turnover of the spherical power spectra on large scales reflects the limit set by survey depth $\sim$ a few times $1/r_0$. These large-scale modes can be recovered by deconvolving the survey window function.Reuse & Permissions

Figure 3

Spherical power spectra with top-hat window function, in the same format as in Fig. 2. The general trend in the spherical power spectra are similar, but oscillations are imprinted in the spherical power spectra, due to the incomplete cancellation of the spherical Bessel function, imposed by the top-hat window function. The gray curves are identical to those in Fig. 2.Reuse & Permissions

Figure 4

Spherical power spectra with Gaussian window function. The time evolution of the transfer and the weight functions over the line-of-sight distance is explicitly computed. Compared to the spherical power spectra without evolution in Fig. 2, the spherical power spectra remain largely unaffected in shape, while the amplitude is naturally lower than the three-dimensional power spectra (gray) at $z=0$. Various curves are in the same format as in Fig. 2.Reuse & Permissions

Figure 5

Signal-to-noise ratio for a measurement of the spherical power spectrum of galaxies with constant bias $b=2$ (see text). Results are shown as a function of the maximum multipole $l_{\max }$. The curves show Eq. (85) for the top-hat (TW) and Gaussian (GW) survey window with several choices of $r_0$. The filled symbols show $S/N$ in the flat-sky approximation [Eq. (85)] for the Gaussian survey window with $r_0=2354$ (circle), 1177 (square), and $589h^{-1}\mathrm{Mpc}$ (triangle). Note that their abscise is arbitrary.Reuse & Permissions