Bispectrum
• Recall the 2-point correlation function. We can define 3-point and higher order correlation functions:

dP = (n0)n[1+ξ(n)]dV1...dVn

where n is the order of the correlation
• The bispectrum B123 is the FT of the 3-point correlation function (integrated over triangular regions). It is analogous to the power spectrum
• For Gaussian fluctuations in linear perturbation theory, they can be fully characterized by power spectrum (i.e. 2-point correlation function). In otherwords, if δ(k) is Gaussian distributed, then all statistical properties of the primordial fluctuations are characterized by P(k) in Fourier space (assuming linear perturbation theory holds true)
• Nonlinear gravitational instability induces 3-point and higher order correlations. Since we currently live in an epoch of extreme nonlinearity, we should expect such correlations to be non-zero
• Primordial non-Gaussian features will also produce 3-point and higher order correlations that can be observed. These correlations will be in addition to those predicted by nonlinear gravitational instability theory

Results from IRAS PSCz Galaxy Redshift Survey (Feldman et al. 2001)

• Observationally, the 3-point correlation function is non-zero (as expected)
• They statistically measure the bispectrum (B) and power spectrum (P) (from the appropriate correlation functions) over varying triangular regions to obtain reduced bispectrum amplitude:

Q ≡ B123/(P12+P23+P13)

• From theory, Q depends on bias b!! Thus once we measure Q and find a best-fit model, we know b
(we can combine with redshift-space distortion data and determine Ω)
• Punchline: the observed Q agrees with nonlinear gravitational instability theory, and severely limits the possibility of primordial non-Gaussianity
• Also, these guys find a bias: b = 0.84 +/- 0.28. The 2dF group finds a bias that is consistent with unity (i.e. b ∼ 1) within the range 0.1 < k < 0.5 h Mpc-1 (Lahav et al. 2002) which implies galaxies do indeed trace total mass on these small scales.
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