Higher-Order Angular Galaxy Correlations in the SDSS: Redshift and Color Dependence of non-Linear Bias

We present estimates of the N-point galaxy, area-averaged, angular correlation functions $\bar{\omega}_{N}$($\theta$) for $N$ = 2,...,7 for galaxies from the fifth data release of the Sloan Digital Sky Survey. Our parent sample is selected from galaxies with $18 \leq r < 21$, and is the largest ever used to study higher-order correlations. We subdivide this parent sample into two volume limited samples using photometric redshifts, and these two samples are further subdivided by magnitude, redshift, and color (producing early- and late-type galaxy samples) to determine the dependence of $\bar{\omega}_{N}$($\theta$) on luminosity, redshift, and galaxy-type. We measure $\bar{\omega}_{N}$($\theta$) using oversampling techniques and use them to calculate the projected, $s_{N}$. Using models derived from theoretical power-spectra and perturbation theory, we measure the bias parameters $b_1$ and $c_2$, finding that the large differences in both bias parameters ($b_1$ and $c_2$) between early- and late-type galaxies are robust against changes in redshift, luminosity, and $\sigma_8$, and that both terms are consistently smaller for late-type galaxies. By directly comparing their higher-order correlation measurements, we find large differences in the clustering of late-type galaxies at redshifts lower than 0.3 and those at redshifts higher than 0.3, both at large scales ($c_2$ is larger by $\sim0.5$ at $z > 0.3$) and small scales (large amplitudes are measured at small scales only for $z > 0.3$, suggesting much more merger driven star formation at $z > 0.3$). Finally, our measurements of $c_2$ suggest both that $\sigma_8 < 0.8$ and $c_2$ is negative.Comment: 46 pages, 19 figures, Accepted to Ap

Note on Redshift Distortion in Fourier Space

We explore features of redshift distortion in Fourier analysis of N-body simulations. The phases of the Fourier modes of the dark matter density fluctuation are generally shifted by the peculiar motion along the line of sight, the induced phase shift is stochastic and has probability distribution function (PDF) symmetric to the peak at zero shift while the exact shape depends on the wave vector, except on very large scales where phases are invariant by linear perturbation theory. Analysis of the phase shifts motivates our phenomenological models for the bispectrum in redshift space. Comparison with simulations shows that our toy models are very successful in modeling bispectrum of equilateral and isosceles triangles at large scales. In the second part we compare the monopole of the power spectrum and bispectrum in the radial and plane-parallel distortion to test the plane-parallel approximation. We confirm the results of Scoccimarro (2000) that difference of power spectrum is at the level of 10%, in the reduced bispectrum such difference is as small as a few percents. However, on the plane perpendicular to the line of sight of k_z=0, the difference in power spectrum between the radial and plane-parallel approximation can be more than 10%, and even worse on very small scales. Such difference is prominent for bispectrum, especially for those configurations of tilted triangles. The non-Gaussian signals under radial distortion on small scales are systematically biased downside than that in plane-parallel approximation, while amplitudes of differences depend on the opening angle of the sample to the observer. The observation gives warning to the practice of using the power spectrum and bispectrum measured on the k_z=0 plane as estimation of the real space statistics.Comment: 15 pages, 8 figures. Accepted for publication in ChJA

Topology of non-linear structure in the 2dF Galaxy Redshift Survey

We study the evolution of non-linear structure as a function of scale in samples from the 2dF Galaxy Redshift Survey, constituting over 221 000 galaxies at a median redshift of z=0.11. The two flux-limited galaxy samples, located near the southern galactic pole and the galactic equator, are smoothed with Gaussian filters of width ranging from 5 to 8 Mpc/h to produce a continuous galaxy density field. The topological genus statistic is used to measure the relative abundance of overdense clusters to void regions at each scale; these results are compared to the predictions of analytic theory, in the form of the genus statistic for i) the linear regime case of a Gaussian random field; and ii) a first-order perturbative expansion of the weakly non-linear evolved field. The measurements demonstrate a statistically significant detection of an asymmetry in the genus statistic between regions corresponding to low- and high-density volumes of the universe. We attribute the asymmetry to the non-linear effects of gravitational evolution and biased galaxy formation, and demonstrate that these effects evolve as a function of scale. We find that neither analytic prescription satisfactorily reproduces the measurements, though the weakly non-linear theory yields substantially better results in some cases, and we discuss the potential explanations for this result.Comment: 13 pages, matching proof to be published in MNRAS; new version adds reference and corrects figure

Topology of structure in the Sloan Digital Sky Survey: model testing

We measure the three-dimensional topology of large-scale structure in the Sloan Digital Sky Survey (SDSS). This allows the genus statistic to be measured with unprecedented statistical accuracy. The sample size is now sufficiently large to allow the topology to be an important tool for testing galaxy formation models. For comparison, we make mock SDSS samples using several state-of-the-art N-body simulations: the Millennium run of Springel et al. (2005)(10 billion particles), Kim & Park (2006) CDM models (1.1 billion particles), and Cen & Ostriker (2006) hydrodynamic code models (8.6 billion cell hydro mesh). Each of these simulations uses a different method for modeling galaxy formation. The SDSS data show a genus curve that is broadly characteristic of that produced by Gaussian random phase initial conditions. Thus the data strongly support the standard model of inflation where Gaussian random phase initial conditions are produced by random quantum fluctuations in the early universe. But on top of this general shape there are measurable differences produced by non-linear gravitational effects (cf. Matsubara 1994), and biasing connected with galaxy formation. The N-body simulations have been tuned to reproduce the power spectrum and multiplicity function but not topology, so topology is an acid test for these models. The data show a ``meatball'' shift (only partly due to the Sloan Great Wall of Galaxies; this shift also appears in a sub-sample not containing the Wall) which differs at the 2.5\sigma level from the results of the Millennium run and the Kim & Park dark halo models, even including the effects of cosmic variance.Comment: 13 Apj pages, 7 figures High-resolution stereo graphic available at http://www.astro.princeton.edu/~dclayh/stereo50.ep

The defect variance of random spherical harmonics

The defect of a function $f:M\rightarrow \mathbb{R}$ is defined as the difference between the measure of the positive and negative regions. In this paper, we begin the analysis of the distribution of defect of random Gaussian spherical harmonics. By an easy argument, the defect is non-trivial only for even degree and the expected value always vanishes. Our principal result is obtaining the asymptotic shape of the defect variance, in the high frequency limit. As other geometric functionals of random eigenfunctions, the defect may be used as a tool to probe the statistical properties of spherical random fields, a topic of great interest for modern Cosmological data analysis.Comment: 19 page

Limits on Primordial Non-Gaussianity from Minkowski Functionals of the WMAP Temperature Anisotropies

We present an analysis of the Minkowski Functionals (MFs) describing the WMAP three-year temperature maps to place limits on possible levels of primordial non-Gaussianity. In particular, we apply perturbative formulae for the MFs to give constraints on the usual non-linear coupling constant fNL. The theoretical predictions are found to agree with the MFs of simulated CMB maps including the full effects of radiative transfer. The agreement is also very good even when the simulation maps include various observational artifacts, including the pixel window function, beam smearing, inhomogeneous noise and the survey mask. We find accordingly that these analytical formulae can be applied directly to observational measurements of fNL without relying on non-Gaussian simulations. Considering the bin-to-bin covariance of the MFs in WMAP in a chi-square analysis, we find that the primordial non-Gaussianity parameter is constrained to lie in the range -70<fNL<91 at 95% C.L. using the Q+V+W co-added maps.Comment: 9 pages, 4 figures, accpeted for publication in MNRA

Primordial non-Gaussianity: large-scale structure signature in the perturbative bias model

I compute the effect on the power spectrum of tracers of the large-scale mass-density field (e.g., galaxies) of primordial non-Gaussianity of the form Phi=phi+fNL (phi-)+gNL phi^3+..., where Phi is proportional to the initial potential fluctuations and phi is a Gaussian field, using beyond-linear-order perturbation theory. I find that the need to eliminate large higher-order corrections necessitates the addition of a new term to the bias model, proportional to phi, i.e., delta_g=b_delta delta+b_phi fNL phi+..., with all the consequences this implies for clustering statistics, e.g., P_gg(k)=b_delta^2 P_deltadelta(k)+2 b_delta b_phi fNL P_phidelta(k)+b_phi^2 fNL^2 P_phiphi(k)+... . This result is consistent with calculations based on a model for dark matter halo clustering, showing that the form is quite general, not requiring assumptions about peaks, or the formation or existence of halos. The halo model plays the same role it does in the usual bias picture, giving a prediction for b_phi for galaxies known to sit in a certain type of halo. Previous projections for future constraints based on this effect have been very conservative -- there is enough volume at z<~2 to measure fNL to ~+-1, with much more volume at higher z. As a prelude to the bias calculation, I point out that the beyond-linear (in phi) corrections to the power spectrum of mass-density perturbations are naively infinite, so it is dangerous to assume they are negligible; however, the infinite part can be removed by a renormalization of the fluctuation amplitude, with the residual k-dependent corrections negligible for models allowed by current constraints.Comment: 11 pg, 2 fig, v2: added illustrative figure, minor improvements, v3: added references, version accepted by PR

A geometric description of the non-Gaussianity generated at the end of multi-field inflation

In this paper we mainly focus on the curvature perturbation generated at the end of multi-field inflation, such as the multi-brid inflation. Since the curvature perturbation is produced on the super-horizon scale, the bispectrum and trispectrum have a local shape. The size of bispectrum is measured by $f_{NL}$ and the trispectrum is characterized by two parameters $\tau_{NL}$ and $g_{NL}$. For simplicity, the trajectory of inflaton is assumed to be a straight line in the field space and then the entropic perturbations do not contribute to the curvature perturbation during inflation. As long as the background inflaton path is not orthogonal to the hyper-surface for inflation to end, the entropic perturbation can make a contribution to the curvature perturbation at the end of inflation and a large local-type non-Gaussiantiy is expected. An interesting thing is that the non-Gaussianity parameters are completely determined by the geometric properties of the hyper-surface of the end of inflation. For example, $f_{NL}$ is proportional to the curvature of the curve on this hyper-surface along the adiabatic direction and $g_{NL}$ is related to the change of the curvature radius per unit arc-length of this curve. Both $f_{NL}$ and $g_{NL}$ can be positive or negative respectively, but $\tau_{NL}$ must be positive and not less than $({6\over 5}f_{NL})^2$.Comment: 19 pages, 4 figures; refs added; a correction to \tau_{NL} for n-field inflation added, version accepted for publication in JCA

The Trispectrum in the Multi-brid Inflation

The trispectrum is at least as important as the bispectrum and its size can be characterized by two parameters $\tau_{NL}$ and $g_{NL}$. In this short paper, we focus on the Multi-brid inflation, in particular the two-brid inflation model in arXiv.0805.0974, and find that $\tau_{NL}$ is always positive and roughly equals to $({6\over 5}f_{NL})^2$ for the low scale inflation, but $g_{NL}$ can be negative or positive and its order of magnitude can be the same as that of $\tau_{NL}$ or even largerComment: 12 pages; minor correction, refs added; further refs added, version for publication in JCA

Curvaton Dynamics and the Non-Linearity Parameters in Curvaton Model

We investigate the curvaton dynamics and the non-linearity parameters in curvaton model with potential slightly deviating from the quadratic form in detail. The non-linearity parameter $g_{NL}$ will show up due to the curvaton self-interaction. We also point out that the leading order of non-quadratic term in the curvaton potential can be negative, for example in the axion-type curvaton model. If a large positive $g_{NL}$ is detected, the axion-type curvaton model will be preferred.Comment: 14 pages, 4 figures; refs adde