The Bispectrum: From Theory to Observations

The bispectrum is the lowest-order statistic sensitive to the shape ofstructures generated by gravitational instability and is a potentially powerfulprobe of galaxy biasing and the Gaussianity of primordial fluctuations.Although the evolution of the bispectrum is well understood theoretically fromnon-linear perturbation theory and numerical simulations, applications togalaxy surveys require a number of issues to be addressed. In this paper weconsider the effect on the bispectrum of stochastic non-linear biasing, radialredshift distortions, non-Gaussian initial conditions, survey geometry andsampling. We find that: 1) bias stochasticity does not affect the use of the bispectrumto recover the mean biasing relation between galaxies and mass, at least formodels in which the scatter is uncorrelated at large scales. 2) radial redshiftdistortions do not change significantly the monopole power spectrum andbispectrum compared to their plane-parallel values. 3) survey geometry leads tofinite volume effects which must be taken into account in current surveysbefore comparison with theoretical predictions can be made. 4) sparse samplingand survey geometry correlate different triangles leading to a breakdown of theGaussian likelihood approximation. We develop a likelihood analysis using bispectrum eigenmodes, calculated byMonte Carlo realizations of mock surveys generated with second-order Lagrangianperturbation theory and checked against N-body simulations. In a companionpaper we apply these results to the analysis of the bispectrum of IRASgalaxies.Comment: 46 pages, 18 figures. Accepted for publication in ApJ. Typos fixed and improved comparison with previous wor