The bispectrum: From theory to observations

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 nonlinear 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 nonlinear 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 significantly change 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.