The 2dF Galaxy Redshift Survey: Spherical Harmonics analysis of fluctuations in the final catalogue

TL;DR

This paper applies a spherical-harmonics and spherical-Bessel decomposition to the final 2dF Galaxy Redshift Survey to isolate radial and angular fluctuations and model redshift-space distortions without resorting to the far-field approximation. It employs a constant galaxy clustering bias model with luminosity corrections, uses mock Hubble Volume catalogues to validate parameter recovery and calibrate errors, and accounts for the survey window and small-scale velocities through a covariance-based likelihood. The analysis yields tight constraints on the linear distortion amplitude and the real-space power spectrum normalization, including $\\Omega_m^{0.6}\\sigma_8=0.46\pm0.06$, $\\beta(L_*,0)=0.58\pm0.08$, and $b(L_*,0)\\sigma_8=0.79\pm0.03$, with additional limits on power-spectrum shape parameters when marginalising over them. These results are consistent with WMAP and demonstrate the value of the spherical-harmonics approach as a robust, complementary method for extracting large-scale structure information from galaxy surveys.

Abstract

We present the result of a decomposition of the 2dFGRS galaxy overdensity field into an orthonormal basis of spherical harmonics and spherical Bessel functions. Galaxies are expected to directly follow the bulk motion of the density field on large scales, so the absolute amplitude of the observed large-scale redshift-space distortions caused by this motion is expected to be independent of galaxy properties. By splitting the overdensity field into radial and angular components, we linearly model the observed distortion and obtain the cosmological constraint Omega_m^{0.6} sigma_8=0.46+/-0.06. The amplitude of the linear redshift-space distortions relative to the galaxy overdensity field is dependent on galaxy properties and, for L_* galaxies at redshift z=0, we measure beta(L_*,0)=0.58+/-0.08, and the amplitude of the overdensity fluctuations b(L_*,0) sigma_8=0.79+/-0.03, marginalising over the power spectrum shape parameters. Assuming a fixed power spectrum shape consistent with the full Fourier analysis produces very similar parameter constraints.

Figures (11)

• Figure 1: Redshift distribution of the reduced galaxy catalogues for the two regions considered (solid circles), compared with the best fit redshift distribution for each of the form given by Eq. \ref{['eq:fz']}. The magnitude limit adopted for each sample is given in each panel.
• Figure 2: Likelihood contours for the recovered $b(L_*,0)\sigma_8$ and $\beta(L_*,0)$ assuming a fixed $\Lambda$CDM power spectrum shape. Solid lines in the top row show the recovered contours from the 2dFGRS, while the bottom row gives the average recovered contours from the $\Lambda$CDM mock catalogues. Contours correspond to changes in the likelihood from the maximum of $2\Delta\ln{\cal L}=2.3, 6.0, 9.2$. These values correspond to the usual two-parameter confidence of 68, 95 and 99 per cent. The open circle marks the ML position, while the solid circle marks the true parameters for the mock catalogues. The crosses give the ML positions for the 22 mock catalogues. Note that on average $57\,\%$ of the crosses lie within the $2\Delta\ln{\cal L}=2.3$ contour for the NGP and SGP mock catalogues. The chosen modes are not independent, although they are orthogonal, so we cannot assume that $\ln{\cal L}$ has a $\chi^2$ distribution. See Section \ref{['sec:confidence']} for a further discussion of the confidence intervals that we place on recovered parameters. The dashed lines plotted in the upper panels give the locus of models with constant redshift space power spectrum amplitude (see text for details).
• Figure 3: Likelihood contours for $\Omega_mh$ and $\Omega_b/\Omega_m$ assuming a $\Lambda$CDM power spectrum with $h=0.72$ and $n_s=1.0$. We have marginalised over the power spectrum amplitude and $\beta(L_*,0)$. Solid lines in the top row show the recovered contours from the 2dFGRS, while the bottom row gives the average recovered contours from the $\Lambda$CDM mock catalogues. Contours correspond to changes in the likelihood from the maximum of $2\Delta\ln{\cal L}=1.0, 2.3, 6.0, 9.2$. In addition to the contours plotted in Fig. \ref{['fig:sig8_beta']}, we also show the standard one-parameter 68 per cent confidence region to match with figure 5 in P01. The open circle marks the ML position. As in P01, we find a broad degeneracy in the ($\Omega_mh$, $\Omega_b/\Omega_m$) plane, which is weakly lifted with a low baryon fraction favoured for the 2dFGRS data. These parameter constraints are less accurate than those derived in C04 as we use less data, and we limit the number of modes used. ML positions for the 22 mock catalogues are shown by the crosses. It can be seen that a number of the mock catalogues have likelihood surfaces that are not closed, with the ML position being at one edge of the parameter space considered. However, these mocks all follow the general degeneracy between models with the same $P(k)$ shape.
• Figure 4: As Fig. \ref{['fig:sig8_beta']}, but now marginalising over the power spectrum shape as parameterized by $\Omega_mh$ and $\Omega_b/\Omega_m$. As can be seen, allowing for different power spectrum shapes only increases the errors on $b(L_*,0)\sigma_8$ and $\beta(L_*,0)$ slightly. The relative interdependence between the power spectrum shape and $b(L_*,0)\sigma_8$ and $\beta(L_*,0)$ is considered in more detail in Fig. \ref{['fig:param_degen']}.
• Figure 5: Contour plots showing changes in the likelihood from the maximum of $2\Delta\ln{\cal L}=1.0, 2.3, 6.0, 9.2$ for different parameter combinations for the combined likelihood from the 2dFGRS NGP and SGP catalogues, assuming a $\Lambda$CDM power spectrum with $h=0.72$ and $n_s=1.0$. There are four parameters in total, and in each plot we marginalise over the two other parameters. The primary degeneracy arises between $\Omega_mh$ and $\Omega_b/\Omega_m$, and corresponds to similar power spectrum shapes. $b(L_*,0)\sigma_8$ is also degenerate with $\Omega_mh$, although $\beta(L_*,0)$ is independent of the power spectrum shape.