The Bispectrum as a Signature of Gravitational Instability in Redshift-Space

TL;DR

This work investigates how gravitational instability and galaxy bias imprint on the galaxy bispectrum in redshift space. It develops a fully non-linear Eulerian redshift-space mapping and computes the leading tree-level bispectrum, then expands the result in multipoles to separate angular dependence and to probe the degeneracy between $\Omega$ and the linear bias $b$. Recognizing the limitations of perturbation theory in redshift space on mildly non-linear scales, the authors introduce a simple phenomenological damping model that multiplies the PT bispectrum by a velocity-dispersion factor, calibrated from redshift-space distortions of the power spectrum. Comparison with high-resolution N-body simulations shows that the model reproduces the scale- and configuration-dependent behavior of the redshift-space bispectrum, enabling robust constraints on $\Omega$, $b$, and $b_2$ from redshift surveys and clarifying the non-trivial interplay between bias and redshift distortions.

Abstract

The bispectrum provides a characteristic signature of gravitational instability that can be used to probe the Gaussianity of the initial conditions and the bias of the galaxy distribution. We study how this signature is affected by redshift distortions using perturbation theory and high-resolution numerical simulations. We obtain perturbative results for the multipole expansion of the redshift-space bispectrum which provide a natural way to break the degeneracy between bias and $Ω$ present in measurements of the redshift-space power spectrum. We propose a phenomenological model that incorporates the perturbative results and also describes the bispectrum in the transition to the non-linear regime. We stress the importance of non-linear effects and show that inaccurate treatment of these can lead to significant discrepancies in the determination of bias from galaxy redshift surveys. At small scales we find that the bispectrum monopole exhibits a strong configuration dependence that reflects the velocity dispersion of clusters. Therefore, the hierarchical model for the three-point function does not hold in redshift-space.

Figures (4)

• Figure 1: Top left panel shows the redshift-space tree-level bispectrum multipoles, $A_B^{(\ell)} \equiv B_{s}^{(\ell)}/B$, as a function of angle $\theta = \cos^{-1} \hat{{\hbox{k}}}_1 \cdot \hat{{\hbox{k}}}_2$, for configurations with $k_1/k_2=2$ and scale-free initial spectra, $P(k) \propto k^n$ ($n=-2$ solid, $n=0$ dotted). Top right panel shows the bispectrum quadrupole-to-monopole ratio $R_B = B_{s}^{(2)}/B_{s}^{(0)}$ for $r=k_1/k_2=10,2,1$ configurations as a function of $\theta$ ($n=-2$ solid, $n=0$ dotted) for $\Omega=1$ and $\Omega=0.3$. The bottom left plot shows the ratio of the redshift-space to real-space bispectrum for equilateral triangles, as a function of $\mu = \hat{{\hbox{k}}}_1 \cdot \hat{z}$ for different azimuthal angles $\phi$. The bottom right plot shows the hierarchical amplitude $Q_s$ for $k_1/k_2=2$ configurations with linear bias $b=2$, and quadratic bias $\gamma=b_2/b=1/2,0,-1/2$ (top to bottom), for $n=-2$. Solid curves represent the PT result, and dotted curves assume that bias and redshift-space mapping commute.
• Figure 2: The quadrupole-to-monopole ratio for the redshift-space power spectrum, $R_{\rm P}$, and equilateral bispectrum, $R_{\rm B}$. The symbols correspond to AP$^3$M $256^3$ particle N-body simulations (see text) at redshift $z=0,1$ averaged over four different observers, for SCDM (top) and $\Lambda$CDM (bottom). Squares denote measured values of $R_{\rm P}$, and triangles correspond to values of the ratio $R_{\rm B}$. The dashed and solid curves show the predictions of PT convolved with an exponential velocity dispersion model, Eqs. (\ref{['Ppheno']}-\ref{['Bpheno']}).
• Figure 3: The top left panel shows the power spectrum amplitude, $\Delta(k)\equiv 4 \pi k^3 P(k)$ in real (squares) and redshift (triangles) space. The top right panel shows the equilateral hierarchical amplitude $Q_{\rm eq}$ in real-space (squares) and the monopole of the redshift-space amplitude, $Q_{s\ \rm eq}$ (triangles). The bottom panels show the hierarchical amplitude $Q$ for $k_2/k_1= 2$ configurations, as a function of the angle $\theta$ between ${\hbox{k}}_1$ and ${\hbox{k}}_2$, in real (squares) and redshift space (triangles), for two different scales. In the bottom left plot, weakly non-linear distortions decrease the configuration dependence of $Q_s$, similar to the effects of bias in leading order PT. At smaller scales (lower right panel), the configuration dependence of $Q_s$ is greatly enhanced, reflecting the velocity dispersion of virialized clusters.
• Figure 4: Same as Fig \ref{['fig3']}, for the $\Lambda$CDM model. In the bottom right panel note the difference between the real- and redshift-space hierarchical amplitudes. Whereas $Q$ in real-space is approximately constant, its redshift-space counterpart depends strongly on configuration. Therefore, the hierarchical ansatz does not hold in redshift space.