J. N. Fry

Biasing and hierarchical statistics in large-scale structure

In this paper we consider the consequences for galaxy formation of an arbitrary, effectively local biasing transformation of a hierarchical underlying matter distribution. We show that a general form of such a transformation preserves the hierarchical properties and the shape of the dispersion in the limit of small fluctuations, i.e., on large scales, although the values of the hierarchical amplitudes may change arbitrarily. We present expressions for the induced hierarchical amplitudes S(g,j) of the galaxy distribution in terms of the matter amplitudes S(j) and biasing parameters for j = 3-7. For higher order correlations, j greater than 2, restricting to a linear bias is not a consistent approximation even at very large scales. To draw any conclusions from the galaxy distribution about matter correlations of order j, properties of biasing must be specified completely to order j - 1.

The Galaxy correlation hierarchy in perturbation theory

I calculate the evolution of cosmological density correlation functions to lowest nonvanishing order in perturbation theory for an initially random Gaussian distribution. The three-point function so obtained scales with size as in the continuous hierarchy model, but there is a residual nontrival dependence on shape of the reduced three-point amplitude Q. The average value Q-bar = 34/21-(1/6)..gamma.. is consistent with observations in the nonperturbative regime. The four-point function is also hierarchical in form, with amplitudes R/sub a/ = (34/21)/sup 2/ and R-bar/sub b/ = 682/189. The perturbation expansion in fact gives kappa/sub N/proportionalxi/sup N/-1 for the reduced correlation function kappa/sub N/ to all orders N. A graphical technique enumerates the terms which appear in kappa/sub N/.

Deriving the Nonlinear Cosmological Power Spectrum and Bispectrum from Analytic Dark Matter Halo Profiles and Mass Functions

We present an analytic model for the fully nonlinear two- and three-point correlation functions of the cosmological mass density field, and their Fourier transforms, the mass power spectrum and bispectrum. The model is based on physical properties of dark matter halos, with the three main model inputs being analytic halo density profiles, halo mass functions, and halo-halo spatial correlations, each of which has been well studied in the literature. We demonstrate that this new model can reproduce the power spectrum and bispectrum computed from cosmological simulations of both an n = -2 scale-free model and a low-density cold dark matter model. To enhance the dynamic range of these large simulations, we use the synthetic-halo replacement technique of Ma & Fry, in which the original halos with numerically softened cores are replaced by synthetic halos of realistic density profiles. At high wavenumbers, our model predicts a slope for the nonlinear power spectrum different from the often-used fitting formulas in the literature based on the stable-clustering assumption. Our model also predicts a three-point amplitude, Q, that is scale dependent, in contrast to the popular hierarchical clustering assumption. This model provides a rapid way to compute the mass power spectrum and bispectrum over all length scales where the input halo properties are valid. It also provides a physical interpretation of the clustering properties of matter in the universe.

The Evolution of Bias

Bias in the galaxy distribution is often conceived as something applied at the present or as independent of time. In fact, a bias arising physically in the process of galaxy formation will evolve afterward, as galaxies move under the influence of gravity. I calculate the evolution of bias in a model in which galaxies are formed at a fixed time by a process that may depend nonlinearly on density but follow motions determined by the gravitational potential thereafter. The equation of continuity then determines the evolution. An initial bias decays with time, and in the long term, the galaxy distribution relaxes to that of the mass, but with galaxy formation occurring at a modest redshift, an appreciable bias may remain to the present. The evolution of bias changes somewhat the dependence of the bispectrum amplitude on configuration shape, but a weak dependence on configuration shape still corresponds to a large bias.

Nonlinear Evolution of the Bispectrum of Cosmological Perturbations

The bispectrum B(k1, k2, k3), the three-point function of density fluctuations in Fourier space, is the lowest order statistic that carries information about the spatial coherence of large-scale structures. For Gaussian initial conditions, when the density fluctuation amplitude is small (δ 1), tree-level (leading order) perturbation theory predicts a characteristic dependence of the bispectrum on the shape of the triangle formed by the three wave vectors. This configuration dependence provides a signature of gravitational instability, and departures from it in galaxy catalogs can be interpreted as due to bias, that is, nongravitational effects. On the other hand, N-body simulations indicate that the reduced three-point function becomes relatively shape-independent in the strongly nonlinear regime (δ 1). In order to understand this nonlinear transition and assess the domain of reliability of shape dependence as a probe of bias, we calculate the one-loop (next-to-leading order) corrections to the bispectrum in perturbation theory. We compare these results with measurements in numerical simulations with scale-free and cold dark matter initial power spectra. We find that the one-loop corrections account very well for the departures from the tree-level results measured in numerical simulations on weakly nonlinear scales (δ 1). In this regime, the reduced bispectrum qualitatively retains its tree-level shape, but the amplitude can change significantly. At smaller scales (δ 1), the reduced bispectrum in the simulations starts to flatten, an effect that can be partially understood from the one-loop results. In the strong clustering regime, where perturbation theory breaks down entirely, the simulation results confirm that the reduced bispectrum has almost no dependence on triangle shape, in rough agreement with the hierarchical Ansatz.

The Bispectrum of IRAS redshift catalogs

We compute the bispectrum for the galaxy distribution in the IRAS QDOT, 2Jy, and 1.2Jy redshift catalogs for wavenumbers 0.05 1 at large scales, \chi^2 non-Gaussian initial conditions are ruled out at the 95% confidence level. The IRAS data do not distinguish between Lagrangian or Eulerian local bias.We compute the bispectrum for the galaxy distribution in the IRAS QDOT, 2 Jy, and 1.2 Jy redshift catalogs for wavenumbers 0.05 ≤ k ≤ 0.2 h Mpc-1 and compare the results with predictions from gravitational instability in perturbation theory. Taking into account redshift-space distortions, nonlinear evolution, the survey selection function, and discreteness and finite-volume effects, all three catalogs show evidence for the dependence of the bispectrum on the configuration shape predicted by gravitational instability. Assuming Gaussian initial conditions and local biasing parameterized by linear and nonlinear bias parameters b1 and b2, a likelihood analysis yields 1/b1 = 1.32, 1.15 and b2/b = -0.57, -0.50 for the 2 and 1.2 Jy samples, respectively. This implies that IRAS galaxies trace dark matter increasingly weakly as the density contrast increases, consistent with their being underrepresented in clusters. In a model with χ2 non-Gaussian initial conditions, the bispectrum displays an amplitude and scale dependence different from that found in the Gaussian case; if IRAS galaxies do not have bias b1 > 1 at large scales, χ2 non-Gaussian initial conditions are ruled out at the 95% confidence level. The IRAS data do not distinguish between Lagrangian and Eulerian local bias.

Constraints on Galaxy Bias, Matter Density, and Primordial Non-Gaussianity from the PSCz Galaxy Redshift Survey

We compute the bispectrum for the IRAS PSCz catalog and find that the galaxy distribution displays the characteristic signature of gravity. Assuming Gaussian initial conditions, we obtain galaxy biasing parameters 1/b(1) = 1.20(+0.18)(-0.19) and b(2)/b(2)(1) = -0.42+/-0.19, with no sign of scale-dependent bias for k < or = 0.3h Mpc(-1). These results impose stringent constraints on non-Gaussian initial conditions. For dimensional scaling models with chi(2)(N) statistics, we find N > 49, which implies a constraint on primordial skewness B3 < 0.35.

The Bispectrum of IRAS Galaxies

We compute the bispectrum for the galaxy distribution in the IRAS QDOT, 2Jy, and 1.2Jy redshift catalogs for wavenumbers 0.05 1 at large scales, \chi^2 non-Gaussian initial conditions are ruled out at the 95% confidence level. The IRAS data do not distinguish between Lagrangian or Eulerian local bias.We compute the bispectrum for the galaxy distribution in the IRAS QDOT, 2 Jy, and 1.2 Jy redshift catalogs for wavenumbers 0.05 ≤ k ≤ 0.2 h Mpc-1 and compare the results with predictions from gravitational instability in perturbation theory. Taking into account redshift-space distortions, nonlinear evolution, the survey selection function, and discreteness and finite-volume effects, all three catalogs show evidence for the dependence of the bispectrum on the configuration shape predicted by gravitational instability. Assuming Gaussian initial conditions and local biasing parameterized by linear and nonlinear bias parameters b1 and b2, a likelihood analysis yields 1/b1 = 1.32, 1.15 and b2/b = -0.57, -0.50 for the 2 and 1.2 Jy samples, respectively. This implies that IRAS galaxies trace dark matter increasingly weakly as the density contrast increases, consistent with their being underrepresented in clusters. In a model with χ2 non-Gaussian initial conditions, the bispectrum displays an amplitude and scale dependence different from that found in the Gaussian case; if IRAS galaxies do not have bias b1 > 1 at large scales, χ2 non-Gaussian initial conditions are ruled out at the 95% confidence level. The IRAS data do not distinguish between Lagrangian and Eulerian local bias.

Halo Profiles and the Nonlinear Two- and Three-Point Correlation Functions of Cosmological Mass Density.

We investigate the nonlinear two- and three-point correlation functions of the cosmological density field in Fourier space and test the popular hierarchical clustering model that assumes a scale-independent three-point amplitude Q. In high-resolution N-body simulations of both the n=-2 scale-free model and the cold dark matter model, we find that Q at late times is not constant but increases with wavenumber far into the nonlinear regime. Self-similar scaling also does not hold as rigorously for the three-point function as for the two-point function in the n=-2 simulation; this is perhaps a manifestation of the finite simulation volume. We suggest that a better understanding of the behavior of the two- and three-point correlation functions in the nonlinear regime lies in the link to the density profiles of dark matter halos. We demonstrate and quantify how the slopes of the correlation functions are affected by the slope of the halo profiles using simple halo shapes and analytic clustering models.

Skewness in large-scale structure and non-Gaussian initial conditions

We compute the skewness of the galaxy distribution arising from the nonlinear evolution of arbitrary non-Gaussian intial conditions to second order in perturbation theory including the effects of nonlinear biasing. The result contains a term identical to that for a Gaussian initial distribution plus terms which depend on the skewness and kurtosis of the initial conditions. The results are model dependent; we present calculations for several toy models. At late times, the leading contribution from the initial skewness decays away relative to the other terms and becomes increasingly unimportant, but the contribution from initial kurtosis, previously overlooked, has the same time dependence as the Gaussian terms. Observations of a linear dependence of the normalized skewness on the rms density fluctuation therefore do not necessarily rule out initially non-Gaussian models. We also show that with non-Gaussian initial conditions the first correction to linear theory for the mean square density fluctuation is larger than for Gaussian models.