Large-scale bias in the Universe: bispectrum method

TL;DR

The work tackles the degeneracy between $Ω_0$ and galaxy bias $b$ in linear theory by leveraging second-order perturbation theory to predict the galaxy bispectrum and its covariances. It develops unbiased and biased bispectrum expressions and a generating-functional framework to compute the six-point covariance (shot noise included), validating the approach with $N$-body simulations. By employing two triangle shapes (equilateral and degenerate) within a maximum-likelihood framework, it demonstrates the simultaneous estimation of $b_1$ and $b_2$, enabling, in combination with redshift-space distortion measures of $β=Ω_0^{0.6}/b$, an unambiguous constraint on $Ω_0$. The paper also addresses practical aspects such as selection functions, redshift distortions, and volume subdivision to optimize signal-to-noise, offering forecasts for future surveys that could yield percent-level bias constraints and thus robust cosmological inferences.

Abstract

Evidence that the Universe may be close to the critical density, required for its expansion eventually to be halted, comes principally from dynamical studies of large-scale structure. These studies either use the observed peculiar velocity field of galaxies directly, or indirectly by quantifying its anisotropic effect on galaxy clustering in redshift surveys. A potential difficulty with both such approaches is that the density parameter $Ω_0$ is obtained only in the combination $β= Ω_0^{0.6}/b$, if linear perturbation theory is used. The determination of the density parameter $Ω_0$ is therefore compromised by the lack of a good measurement of the bias parameter $b$, which relates the clustering of sample galaxies to the clustering of mass. In this paper, we develop an idea of Fry (1994), using second-order perturbation theory to investigate how to measure the bias parameter on large scales. The use of higher-order statistics allows the degeneracy between $b$ and $Ω_0$ to be lifted, and an unambiguous determination of $Ω_0$ then becomes possible. We apply a likelihood approach to the bispectrum, the three-point function in Fourier space. This paper is the first step in turning the idea into a practical proposition for redshift surveys, and is principally concerned with noise properties of the bispectrum, which are non-trivial. The calculation of the required bispectrum covariances involves the six-point function, including many noise terms, for which we have developed a generating functional approach which will be of value in calculating high-order statistics in general.

Figures (7)

• Figure 1: The weak dependence of $J$ on $\Omega_0$ for equilateral triangles.
• Figure 2: The power spectrum in the $N$-body simulation (solid), along with the linear power spectrum (dashed).
• Figure 3: The measured bispectrum from the simulation (solid), with the theoretical prediction from second-order perturbation theory shown dashed. Equilateral triangles are used, and no wavevector appears in more than one triangle. Error bars are errors in the mean for each bin.
• Figure 4: As in Fig. 3, but for 'degenerate' triangles $k$ in the abscissa is the smaller wavevector.
• Figure 5: Likelihood of $c_1 = 1/b_1$ where $b_1$ is the linear bias parameter, for the (unbiased) $N$-body simulation, for $0.55<k<1.1$, using degenerate triangles, and for equilateral triangles (dotted) for $k<0.55$.