Primordial Non-Gaussianity, Scale-Dependent Bias, and the Bispectrum of Galaxies

Abstract

The three-point correlation function of cosmological fluctuations is a sensitive probe of the physics of inflation. We calculate the bispectrum, B g (k 1 , k 2 , k 3 ), Fourier transform of the three-point function of density peaks (e.g., galaxies), using two different methods: the Matarrese-Lucchin-Bonometto formula and the locality of galaxy bias. The bispectrum of peaks is not only sensitive to that of the underlying matter density fluctuations, but also to the four-point function. For a physically motivated, local form of primordial non-Gaussianity in the curvature perturbation,, where φ is a Gaussian field, we show that the galaxy bispectrum contains five physically distinct pieces: (1) non-linear gravitational evolution, (2) non-linear galaxy bias, (3) f NL , (4) f 2 NL , and (5) g NL . While (1), (2), and a part of (3) have been derived in the literature, (4) and (5) are derived in this paper for the first time. We also find that, in the high-density peak limit, (3) receives an enhancement of a factor of ∼15 relative to the previous calculation for the squeezed triangles (k 1 ≈ k 2 k 3 ). Our finding suggests that the galaxy bispectrum is more sensitive to f NL than previously recognized, and is also sensitive to a new term, g NL . For a more general form of local-type non-Gaussianity, the coefficient f 2 NL can be interpreted as τ NL , which allows us to test multi-field inflation models using the relation between the three-and four-point functions. The usual terms from Gaussian initial conditions, (1) and (2), have the smallest signals in the squeezed configurations, while the others have the largest signals; thus, we can distinguish them easily. We cannot interpret the effects of f NL on B g (k 1 , k 2 , k 3 ) as a scale-dependent bias, and thus replacing the linear bias in the galaxy bispectrum with the scaledependent bias known for the power spectrum results in an incorrect prediction. As the importance of primordial non-Gaussianity relative to the non-linear gravity evolution and galaxy bias increases toward higher redshifts, galaxy surveys probing a high-redshift universe are particularly useful for probing the primordial non-Gaussianity.

Structure formation from non-Gaussian initial conditions: Multivariate biasing, statistics, and comparison withN-body simulations

“…For unbiased tracers our result coincides with the matter power spectrum presented by [43]. However, in the most general (biased) case, it differs from what is obtained adopting the standard local bias expansion in terms of the density field [43][44][45][46]. The most remarkable feature is that the scale-dependent bias first discussed in Dal07 appears at leading order in our model for the power spectrum.…”

“…At tree level, our result corresponds to the usual bispectrum deriving from Gaussian initial conditions but where the linear bias b 10 is replaced by b 10 + b 01 /α(k). This is different from what has been found assuming univariate local biasing [45,46]. Therefore the analysis of three-point statistics represents a promising tool to test the different biasing schemes against observations.…”

“…Our paper differs from the recent work by [43][44][45][46] which is based on the assumption that fluctuations in the halo number counts only depend on the local mass density. By ignoring the expansion in the potential ϕ and only considering the dependence on the matter density perturbations δ, one would obtain different results which do not reduce to the model by Dal07 to leading order and do not match the simulations as well; in this case higher-order terms in the halo power spectrum grow bigger than the first-order contribution on large scales, thus casting doubts on the validity of the perturbative expansion.…”

General relativistic description of the observed galaxy power spectrum: Do we understand what we measure?

“…These very large-scale modes of the galaxy power spectrum are already measured with high signal-to-noise ratios from the SDSS, putting tight constraints on the amplitude of the primordial non-Gaussianity [11], comparable to the limits from the Wilkinson Microwave Anisotropy Probe (WMAP). Further theoretical work [11][12][13][14][15][16][17] shows that future galaxy surveys can measure the galaxy power spectrum on large scales with the accuracy enough to detect the small non-Gaussianity from the simplest single field inflationary models [18] and the * jyoo@cfa.harvard.edu nonlinear evolution of the Gaussian spectrum [19].…”

Halo/galaxy bispectrum with equilateral-type primordial trispectrum

“…The influence of scale-dependent bias sourced by the primordial non-Gaussianity appears not only in the halo/galaxy power spectrum but also in the halo/galaxy bispectrum and other polyspectra. Although it is well known that the latetime nonlinear gravitational evolution also gives the non-Gaussianity, if the amplitude of primordial non-Gaussianity is sufficiently large, the halo/galaxy bispectrum sourced by the primordial non-Gaussianity has a different scale dependence from the non-linear gravitational evolution and it can dominate on large scales [13][14][15][16][17][18][19][20][21]. Especially, when we consider the higher order local-type primordial non-Gaussianity, by combining the analysis of the halo/galaxy power spectrum with the bispectrum it is expected that we could get much tighter constraint on the primordial non-Gaussianity.…”