non-Gaussian Initial Conditions Research Articles

The Effects of Non‐Gaussian Initial Conditions on the Structure and Substructure of Cold Dark Matter Halos

We study the structure and substructure of halos obtained in N-body simulations for a ΛCDM cosmology with non-Gaussian initial conditions. The initial statistics are lognormal in the gravitational potential field with positive (LNp) and negative (LNn) skewness; the sign of the skewness is conserved by the density field, and the power spectrum is the same for all the simulations. Our aim is not to test a given non-Gaussian statistics but to explore the generic effect of positive- and negative-skew statistics on halo properties. From our low-resolution simulations, we find that LNp (LNn) halos are systematically more (less) concentrated than their Gaussian counterparts. This result is confirmed by our Milky Way- and cluster-sized halos resimulated with high resolution. In addition, they show inner density profiles that depend on the statistics: the innermost slopes of LNp (LNn) halos are steeper (shallower) than those obtained from the corresponding Gaussian halos. A subhalo population embedded in LNp halos is more susceptible to destruction than its counterpart inside Gaussian halos. On the other hand, subhalos in LNn halos tend to survive longer than subhalos in Gaussian halos. The spin parameter probability distribution of LNp (LNn) halos is skewed to smaller (larger) values with respect to the Gaussian case. Our results show how the statistics of the primordial density field can influence some halo properties, opening the possibility of constraining, albeit indirectly, the primordial statistics at small scales.

Discrete‐Time Risk‐Sensitive Filters with Non‐Gaussian Initial Conditions and Their Ergodic Properties

ABSTRACTIn this paper, we study asymptotic stability properties of risk‐sensitive filters with respect to their initial conditions. In particular, we consider a linear time‐invariant systems with initial conditions that are not necessarily Gaussian. We show that in the case of Gaussian initial conditions, the optimal risk‐sensitive filter asymptotically converges to a suboptimal filter initialized with an incorrect covariance matrix for the initial state vector in the mean square sense provided the incorrect initializing value for the covariance matrix results in a risk‐sensitive filter that is asymptotically stable, that is, results in a solution for a Riccati equation that is asymptotically stabilizing. For non‐Gaussian initial conditions, we derive the expression for the risk‐sensitive filter in terms of a finite number of parameters. Under a boundedness assumption satisfied by the fourth order absolute moment of the initial state variable and a slow growth condition satisfied by a certain Radon‐Nikodym derivative, we show that a suboptimal risk‐sensitive filter initialized with Gaussian initial conditions asymptotically approaches the optimal risk‐sensitive filter for non‐Gaussian initial conditions in the mean square sense. Some examples are also given to substantiate our claims.

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 ofIRASRedshift Catalogs

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.

The Bispectrum: From Theory to Observations

The bispectrum is the lowest order statistic sensitive to the shape of structures generated by gravitational instability and is a potentially powerful probe of galaxy biasing and the Gaussianity of primordial fluctuations. Although the evolution of the bispectrum is well understood theoretically from nonlinear perturbation theory and numerical simulations, applications to galaxy surveys require a number of issues to be addressed. In this paper we consider the effect on the bispectrum of stochastic nonlinear biasing, radial redshift distortions, non-Gaussian initial conditions, survey geometry, and sampling. We find that: (1) Bias stochasticity does not affect the use of the bispectrum to recover the mean biasing relation between galaxies and mass, at least for models in which the scatter is uncorrelated at large scales. (2) Radial redshift distortions do not significantly change the monopole power spectrum and bispectrum compared to their plane-parallel values. (3) Survey geometry leads to finite-volume effects, which must be taken into account in current surveys before comparison with theoretical predictions can be made. (4) Sparse sampling and survey geometry correlate different triangles, leading to a breakdown of the Gaussian likelihood approximation. We develop a likelihood analysis using bispectrum eigenmodes, calculated by Monte Carlo realizations of mock surveys generated with second-order Lagrangian perturbation theory and checked against N-body simulations. In a companion paper, we apply these results to the analysis of the bispectrum of IRAS galaxies.

Gravitational Clustering from \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage[OT2,OT1]{fontenc} \newcommand\cyr{ \renewcommand\rmdefault{wncyr} \renewcommand\sfdefault{wncyss} \renewcommand\encodingdefault{OT2} \normalfont \selectfont} \DeclareTextFontCommand{\textcyr}{\cyr} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \landscape $\chi ^{2}$ \end{document} Initial Conditions

We consider gravitational clustering from primordial non-Gaussian fluctuations provided by a χ2 model, as motivated by some models of inflation. The emphasis is on signatures that can be used to constrain this type of model from large-scale structure galaxy surveys. Non-Gaussian initial conditions provide additional nonlinear couplings otherwise forbidden by symmetry that cause nonlinear gravitational corrections to become important at larger scales than in the Gaussian case. In fact, the lack of hierarchical scaling in the initial conditions is partially restored by gravitational evolution at scales k > 0.1 h Mpc-1. However, the bispectrum shows much larger amplitude and residual scale dependence not present in evolution from Gaussian initial conditions, which can be used to test this model against observations. We include the effects of biasing and redshift distortions essential to compare this model with galaxy redshift surveys. We also discuss the effects of primordial non-Gaussianity on the redshift-space power spectrum and show that it changes the shape of the quadrupole to monopole ratio through nonlinear corrections to infall velocities.

On asymptotic stability of continuous-time risk-sensitive filters with respect to initial conditions

In this paper, we consider the problem of risk-sensitive filtering for continuous-time stochastic linear Gaussian time-invariant systems. In particular, we address the problem of forgetting of initial conditions. Our results show that suboptimal risk-sensitive filters initialized with arbitrary Gaussian initial conditions asymptotically approach the optimal risk-sensitive filter for a linear Gaussian system with Gaussian but unknown initial conditions in the mean square sense at an exponential rate, provided the arbitrary initial covariance matrix results in a stabilizing solution of the ( H ∞-like) Riccati equation associated with the risk-sensitive problem. More importantly, in the case of non-Gaussian initial conditions, a suboptimal risk-sensitive filter asymptotically approaches the optimal risk-sensitive filter in the mean square sense under a boundedness condition satisfied by the fourth order absolute moment of the initial non-Gaussian density and a slow growth condition satisfied by a certain Radon–Nikodym derivative.

Skewness as a probe of non-Gaussian initial conditions

We compute the skewness of the matter distribution arising from nonlinear evolution and from non-Gaussian initial perturbations. We apply our result to a very generic class of models with non-Gaussian initial conditions and we estimate analytically the ratio between the skewness due to nonlinear clustering and the part due to the intrinsic non-Gaussianity of the models. We finally extend our estimates to higher moments.

Predictions for the clustering properties of the Lyman-alpha forest -- I. One-point statistics

We present predictions for the one-point probability distribution and cumulants of the transmitted QSO flux in the high redshift Lyman-$\alpha$ Forest. We make use of the correlation between the Lyman-$\alpha$ optical depth and the underlying matter density predicted by gravitational instability theory and seen in numerical hydrodynamic simulations. We have modelled the growth of matter fluctuations using the non-linear shear-free dynamics, an approximation which reproduces well the results of perturbation theory for the cumulants in the linear and weakly non-linear clustering regime. As high matter overdensities tend to saturate in spectra, the statistics of the flux distribution are dominated by weakly non-linear overdensities. As a result, our analytic approach can produce accurate predictions, when tested against N-body simulation results, even when the underlying matter field has rms fluctuations larger than unity. Our treatment can be applied to either Gaussian or non-Gaussian initial conditions. Here we concentrate on the former case, but also include a study of a specific non-Gaussian model. We discuss how the methods and predictions we present can be used as a tool to study the generic clustering properties of the \lya forest at high-redshift. With such an approach, rather than concentrating on simulating specific cosmological models, we may be in the position to directly test our assumptions for the Gaussian nature of the initial conditions, and the gravitational instability origin of structure itself. In a separate paper we present results for two-point statistics.

The Projected Three-Point Correlation Function: Theory and Observations

We report results for the angular three-point galaxy correlation function in the APM survey and compare them with theoretical expectations. For the first time, these measurements extend to sufficiently large scales to probe the weakly non-linear regime. On large scales, the results are in good agreement with the predictions of non-linear cosmological perturbation theory, for a model with initially Gaussian fluctuations and linear power spectrum $P(k)$ consistent with that inferred from the APM survey. These results reinforce the conclusion that large-scale structure is driven by non-linear gravitational instability and that APM galaxies are relatively unbiased tracers of the mass on large scales; they also provide stringent constraints upon models with non-Gaussian initial conditions and strongly exclude the standard cold dark matter model.

Adaptive distributed partitioning filters: non-gaussian initial conditions

The problem of the state estimation of partially unknown, linear systems with non-Gaussian initial conditions in a multisensor environment is addressed in this paper. Two distributed algorithms are presented which can locally process the data collected by different local sensor subsystems. The local estimates are forwarded to a central processing center where the overall optimal estimate is obtained. The proposed algorithms are computationally attractive as well as theoretically interesting

Cosmological perturbation theory and the spherical collapse model — II. Non-Gaussian initial conditions

In Paper I of this series, we introduced the spherical collapse (SC) approximation in Lagrangian space as a way of estimating the cumulants ξJ of density fluctuations in cosmological perturbation theory (PT). Within this approximation, the dynamics is decoupled from the statistics of the initial conditions, so we are able to present here the cumulants for generic non-Gaussian initial conditions, which can be estimated to arbitrary order including the smoothing effects. The SC model turns out to recover the exact leading-order non-linear contributions up to terms involving non-local integrals of the J-point functions. We argue that for the hierarchical ratios SJ, these non-local terms are subdominant and tend to compensate each other. The resulting predictions show a non-trivial time evolution that can be used to discriminate between models of structure formation. We compare these analytic results with non-Gaussian N-body simulations, which turn out to be in very good agreement up to scales where σ ≲ 1.

Power fluctuations in the wavelet space: large-scale CMB non-gaussian statistics

We analyse the large-scale coherence of the CMB anisotropy field with non-gaussian initial conditions using 2-point function of the power fluctuations in the wavelet space. Employing the multivariate Edgeworth expansion (MEE) we constrain the normalization for the cosmic string mass-per-unit-length μ, obtaining Gμ c 2 = 1.075 −0.375 +0.455 × 10 −6 at 68% CL in the standard gaussian statistics. This value is consistent with the results obtained from the simulations of the evolution of the string network and by other large scale studies.

Adaptive control of linear systems with Markov perturbations

The stochastic model considered is a linear jump diffusion process X for which the coefficients and the jump processes depend on a Markov chain Z with finite state space. First, we study the optimal filtering and control problem for these systems with non-Gaussian initial conditions, given noisy observations of the state X and perfect measurements of Z. We derive a new sufficient condition which ensures the existence and the uniqueness of the solution of the nonlinear stochastic differential equations satisfied by the output of the filter. We study a quadratic control problem and show that the separation principle holds. Next, we investigate an adaptive control problem for a state process X defined by a linear diffusion for which the coefficients depend on a Markov chain, the processes X and Z being observed in independent white noises. Suboptimal estimates for the process X, Z and approximate control law are investigated for a large class of probability distributions of the initial state. Asymptotic properties of these filters and this control law are obtained. Upper bounds for the corresponding error are given.

Non-linear evolution of the tidal angular momentum of protostructures — II. Non-Gaussian initial conditions

The formalism that describes the non-linear growth of the angular momentum L of protostructures from tidal torques in a Friedmann Universe, as developed in a previous paper, is extended to include non-Gaussian initial conditions. We restrict our analysis here to a particular class of non-Gaussian primordial distributions, namely multiplicative models. In such models, strongly correlated phases are produced by obtaining the gravitational potential via a nonlinear local transformation of an underlying Gaussian random field. The dynamical evolution of the system is followed by describing the trajectories of fluid particles using second-order Lagrangian perturbation theory. In the Einstein-de Sitter universe, the lowest-order perturbative correction to the variance of the linear angular momentum of collapsing structures grows as t^8/3 for generic non-Gaussian statistics, which contrasts with the t^10/3 growth rate characteristic of Gaussian statistics. This is a consequence of the fact that the lowest-order perturbative spin contribution in the non-Gaussian case arises from the third moment of the gravitational potential, which is identically zero for a Gaussian field. Evaluating these corrections at the maximum expansion time of the collapsing structure, we find that these non-Gaussian and non-linear terms can be as high as the linear estimate, without the degree of non-Gaussianity as quantified by skewness and kurtosis of the density field being unacceptably large. The results suggest that higher-order terms in the perturbative expansion may contribute significantly to galactic spin which contrasts with the straightforward Gaussian case.

Partitioning nonlinear filters for non-gaussian initial conditions and correlated noises

Partitioning nonlinear filters for non-gaussian initial conditions and correlated noises

Finite-dimensional filters with nonlinear drift. V: solution to Kolmogorov equation arising from linear filtering with non-Gaussian initial condition

Despite its usefulness, the Kalman-Bucy filter is not perfect. One of its weaknesses is that it needs a Gaussian assumption on the initial data. Recently Yau and Yau introduced a new direct method to solve the estimation problem for linear filtering with non-Gaussian initial data. They factored the problem into two parts: (1) the on-line solution of a finite system of ordinary differential equations (ODEs), and (2) the off-line calculation of the Kolmogorov equation. Here we derive an explicit closed-form solution of the Kolmogorov equation. We also give some properties and conduct a numerical study of the solution.

Limit Gaussian Behavior of the Solutions of the Multidimensional Burger'; Equation with Weak-Dependent Initial Conditions

Asymptotic normality of the solutions of the initial-value problems for multidimensional Burgers' equation with weakly dependent possible non-Gaussian initial condition is proved.

Local Lagrangian approximations for the evolution of the density distribution function in large-scale structure

We examine local Lagrangian approximations for the gravitational evolution of the density distribution function. In these approximations, the final density at a Lagrangian point q at a time t is taken to be a function only of t and of the initial density at the same Lagrangian point. A general expression is given for the evolved density distribution function for such approximations, and we show that the vertex generating function for a local Lagrangian mapping applied to an initially Gaussian density field bears a simple relation to the mapping itself. Using this result, we design a local Lagrangian mapping which reproduces nearly exactly the hierarchical amplitudes given by perturbation theory for gravitational evolution. When extended to smoothed density fields and applied to Gaussian initial conditions, this mapping produces a final density distribution function in excellent agreement with full numerical simulations of gravitational clustering. We also examine the application of these local Lagrangian approximations to non-Gaussian initial conditions.

Skewness of the large-scale velocity divergence from non-Gaussian initial conditions

We compute the skewness $t_3$ and the corresponding hierarchical amplitude $T_3$ of the divergence of the velocity field for arbitrary non-Gaussian initial conditions. We find that $T_3$ qualitatively resembles the corresponding hierarchical amplitude for the density field, $S_3$, in that it contains a term proportional to the initial skewness, which decays inversely as the linear growth factor, plus a constant term which differs from the corresponding Gaussian term by a complex function of the initial three- and four- point functions. We extend the results for $S_3$ and $T_3$ with non-Gaussian initial conditions to evolved fields smoothed with a spherical tophat window function. We show that certain linear combinations, namely $S_3 + {1 \over 2} T_3$, $S_3 + T_3$, and $s_3 + t_3$, lead to expressions which are much simpler, for non-Gaussian initial conditions, than $S_3$ and $T_3$ (or $s_3$ and $t_3$) considered separately.