The Bispectrum: From Theory to Observations

Abstract

The bispectrum is the lowest-order statistic sensitive to the shape of structures generated by gravitational instability and is a potentially powerful probe of galaxy biasing and the Gaussianity of primordial fluctuations. Although the evolution of the bispectrum is well understood theoretically from non-linear perturbation theory and numerical simulations, applications to galaxy surveys require a number of issues to be addressed. In this paper we consider the effect on the bispectrum of stochastic non-linear biasing, radial redshift distortions, non-Gaussian initial conditions, survey geometry and sampling. We find that: 1) bias stochasticity does not affect the use of the bispectrum to recover the mean biasing relation between galaxies and mass, at least for models in which the scatter is uncorrelated at large scales. 2) radial redshift distortions do not change significantly the monopole power spectrum and bispectrum compared to their plane-parallel values. 3) survey geometry leads to finite volume effects which must be taken into account in current surveys before comparison with theoretical predictions can be made. 4) sparse sampling and survey geometry correlate different triangles leading to a breakdown of the Gaussian likelihood approximation. We develop a likelihood analysis using bispectrum eigenmodes, calculated by Monte Carlo realizations of mock surveys generated with second-order Lagrangian perturbation theory and checked against N-body simulations. In a companion paper we apply these results to the analysis of the bispectrum of IRAS galaxies.

Figures (18)

• Figure 1: Recovering stochastic non-linear bias from bispectrum measurements. The smooth solid line is the bias relation obtained from the bispectrum with parameters $b_1$ and $b_2$ as shown in each panel. The true bias relation is represented by the central line (mean) and the $90\%$ scatter around it. The smoothing scale is $R=10$ Mpc/h .
• Figure 2: The top panel shows the power spectrum in redshift space as a function of scale for linear PT (dotted), 2LPT (solid), and N-body simulations (symbols). The bottom panel shows the reduced bispectrum for equilateral triangles in redshift space as a function of scale in tree-level PT (dotted), 2LPT (solid) and N-body simulations (symbols, with error bars from 4 different realizations).
• Figure 3: The bispectrum in redshift space for configurations with $k_2=2k_1=0.21$ h/Mpc as a function of the angle $\theta$ between ${ \hbox{k} }_1$ and ${ \hbox{k} }_2$. Dotted lines denote tree-level PT, solid lines correspond to 2LPT, and symbols with error bars show the result of 4 realizations of N-body simulations.
• Figure 4: The power spectrum in N-body simulations for $\Lambda$CDM (top set) and SCDM (bottom). Solid lines denote measurements in real space, whereas dotted lines and dashed lines (practically on top of each other) denote measurements in redshift space under radial mapping and in the plane-parallel approximation, respectively.
• Figure 5: The top left panel shows the ratio of bispectra for radial to plane-parallel redshift-space mapping as a function of angle as in Fig. \ref{['fig_pkradial']}. Triangles denote $k_2=2k_1=0.105$ h/Mpc, squares $k_2=2k_1=0.21$ h/Mpc. The remaining panels show sensitivity of $Q$ to change in parameters: $\Omega_m$ (top right), $\sigma_8$ (bottom left), $\Gamma$ (bottom right).