Large-scale bias in the Universe II: redshift space bispectrum

TL;DR

This work develops a redshift-space bispectrum framework to break the $\Omega_0$–bias degeneracy by exploiting second-order perturbation theory (2OPT) plus an incoherent velocity-dispersion model for virialised motions. The authors derive the redshift-space bispectrum in terms of real-space power and bias parameters, quantify the large-scale Kaiser effect and small-scale Fingers-of-God damping, and implement a likelihood analysis using equilateral and degenerate triangle configurations across subvolumes. Validation with Hydra N-body simulations shows recovery of bias parameters $b_1$ and $b_2$ (via $c_1=1/b_1$, $c_2=b_2/b_1^2$) within errors in both real and redshift space, with clearly defined breakdown scales. They demonstrate the practicality of applying the method to future surveys (e.g., 2dF, SDSS) to attain few-percent accuracy on bias and, combined with $\beta$ measurements, tight constraints on $\Omega_0$; success hinges on accurate real-space power spectra and careful regional analysis to stay within the perturbative regime.

Abstract

The determination of the density parameter $Ω_0$ from the large-scale distribution of galaxies is one of the major goals of modern cosmology. However, if galaxies are biased tracers of the underlying mass distribution, linear perturbation theory leads to a degeneracy between $Ω_0$ and the linear bias parameter $b$, and the density parameter cannot be estimated. In Matarrese, Verde & Heavens (1997) we developed a method based on second-order perturbation theory to use the bispectrum to lift this degeneracy by measuring the bias parameter in an $Ω_0$-independent way. The formalism was developed assuming that one has perfect information on the positions of galaxies in three dimensions. In galaxy redshift surveys, the three-dimensional information is imperfect, because of the contaminating effects of peculiar velocities, and the resulting clustering pattern in redshift space is distorted. In this paper, we combine second-order perturbation theory with a model for collapsed, virialised structures, to extend the method to redshift space, and demonstrate that the method should be successful in determining with reasonable accuracy the bias parameter from state-of-the-art surveys such as the Anglo-Australian 2 degree field survey and the Sloan digital sky survey.

Figures (7)

• Figure 1: Scale dependence of the redshift distortions on the measurement of the bias parameter, which is overestimated by 1/ordinate if the distortions are ignored. The dashed line refers to the case where there is no small scale velocity dispersion ($\sigma=0$), the solid line is for $\sigma=200$ km s$^{-1}$, the dot-dashed line is for $\sigma=400$ km s$^{-1}$ and the dotted line is for $\sigma=700$ km s$^{-1}$. The wavenumber $k$ is in units of $h$ Mpc$^{-1}$.
• Figure 2: Redshift distortions effects on the bispectrum. The bispectrum is expressed in the dimensionless form $\chi^3$ that is the counterpart for the bispectrum of $\Delta^2$ for the power spectrum: $\chi^3 \equiv (2/\pi^2)^2 k^6 B$, where $k$ is the smallest wavenumber in the triangle. The dot-dashed line is the 2OPT-real space bispectrum, the dotted line is the redshift-space bispectrum obtained taking into account only the large scale boosting effect, the dashed line is the 2OPT bispectrum with the small scale smoothing effect included as in equation (8). The velocity dispersion used here ($\sigma = 650$ km s$^{-1}$) is the one that gives the best fit to the redshift-space power spectrum up to $k \simeq 0.8$. Finally the thick solid line with errorbars is the measured bispectrum from the redshift-space map. The agreement up to $k\simeq 0.7$ should be compared with the corresponding breakdown of 2OPT in real-space ($k=0.55$; MVH97). As before $k$ is in units of $h$ Mpc$^{-1}$.
• Figure 3: The real space joint likelihood for $c_1$ and $c_2$ as a function of the cutoff wavenumber employed, for a low-density CDM simulation with shape parameter $\Gamma=0.25$ and $\sigma_8=1.06$. The dotted and solid line contours contain 90 and 63 percent of the a posteriori probability $P(c_1,c_2 \mid DATA)$ assuming uniform prior. The wavenumber corresponding to $\Delta^2(k)=1$ is $k=0.17$. Second-order perturbation theory works well for the degenerate configuration up to $k_{\rm short} \simeq 0.66$; this corresponds to $\chi^3 \simeq 100$. This figure is relative to the final output ($\sigma_8=1.06$) of the low-$\Omega_0$ simulation.
• Figure 4: The real space joint likelihood for $c_1$ and $c_2$ for the biased catalogue.The dotted and solid line contours contain 90 and 63 percent of the a posteriori probability. In each panel $k$-vectors of the degenerate triangle configurations belong to a different shell in $k$ space. For the equilateral configuration the $k$-vectors limits are $0.3<k<0.5$ and for the degenerate the six shells are limited by: 0.62, 0.7, 0.8, 0.9, 1.0, 1.1. The first interval has been chosen in order to have about the same number of data as in the following interval and a signal of comparable strength. Second-order perturbation theory works well for the degenerate configuration up to $k_{\rm short} \simeq 0.8$ this corresponds to $\chi^3 \simeq 100$. The vertical dashed line shows the inverse of the effective bias of the power spectrum, $\sqrt{P_g/P}$ at $k=0.8$.
• Figure 5: Joint likelihood of $c_1=1/b_1$ and $c_2=b_2/b_1^2$, for a CDM N-body simulation ($\Omega_0=1$, $\sigma_8=0.64$, $\Gamma=0.25$, see text for further details), but in redshift space. Contours contain 68.3 and 90 per cent of the a posteriori probability $P(c_1,c_2\mid DATA)$ assuming uniform priors for $c_1$ and $c_2$.