Scale-free Power Spectrum Research Articles

Universal profile of dark matter haloes and the spherical infall model

I propose a modification of the spherical infall model for the evolution of density fluctuations with initially Gaussian probability distribution and scale-free power spectra in the Einstein—de Sitter universe as developed by Hoffman & Shaham. I introduce a generalized form of the initial density distribution around an overdense region and cut it off at half the interpeak separation, accounting in this way for the presence of the neighbouring fluctuations. Contrary to the original predictions of Hoffman & Shaham, the resulting density profiles within virial radii no longer have a power-law shape, but their steepness increases with distance. The profiles of haloes of galactic mass are well fitted by the universal profile formula of changing slope obtained as a result of N-body simulations by Navarro, Frenk & White. The trend of steeper profiles for smaller masses and higher spectral indices is also reproduced. The agreement between the model and simulations is better for smaller masses and lower spectral indices, which suggests that galaxies form mainly by accretion, while formation of clusters involves merging.

The Cluster [ITAL]L[/ITAL][TINF]X[/TINF]-σ Relation Has Implications for Scale-Free Cosmologies

I show that the cluster LX-sigma relation should be sensitive to cosmologies with a scale-free power spectrum of initial density fluctuations, P(k) ~ k^n. I derive the dependence, and argue that a conservative interpretation of current observations implies n < -2.0 and n < -1.1 at the one-sided 90% and 99% confidence levels, respectively. This result, which agrees with constraints on n from the x-ray cluster temperature function, should be roughly independent of the value of Omega or Lambda.

Mass growth and density profiles of dark matter haloes in hierarchical clustering

We develop a model for the growth of dark matter haloes and use it to study their evolved density profiles. In this model, haloes are spherical and form by quiescent accretion of matter in clumps, which are called satellites. The halo mass as a function of redshift is given by the mass of the most massive progenitor, and is determined from Monte Carlo realizations of the merger-history tree. Inside the halo, satellites move under the action of the gravitational force of the halo and a dynamical friction drag force. The associated equation of motion is solved numerically. The energy lost to dynamical friction is transferred to the halo in the form of kinetic energy. As satellites sink into the halo, they continually lose matter as a result of tidal stripping. The stripped matter moves inside the halo free of dynamical friction. The evolved density profiles are steeper than those obtained by assuming that, once they have been accreted on to the parent halo, satellites remain at a fixed distance from the halo centre. We find that the final density profile depends mainly on the rate of infall of matter on to the halo. This, in turn, depends on the initial fluctuation field as well as on cosmology. For mass scales where the effective spectral index of the initial density field is less than −1, the model predicts a profile that can only be approximately matched by the one-parameter family of curves suggested by Navarro, Frenk and White. For scale-free power spectra with initial slope n, the density profile, within about 1 per cent of the virial radius, is ρ∝ r−β, with 3(3+n)/(5+n)≤β≤ 3(3+n)/(4+n).

Steps toward the Power Spectrum of Matter. III. The Primordial Spectrum

We compare the observed power spectrum of matter found in our earlier papers with analytical power spectra. We extrapolate the observed power spectra on small scales to find the linear power spectrum of matter. We consider spatially flat cold and mixed dark matter models with the cosmological constant as well as open models. We fix the Hubble constant and the baryon density in the middle of the allowed range and vary the density parameter and the cosmological constant. We determine the primordial power spectrum of matter using the power spectrum of matter and the transfer functions of analytical models. We take two different spectra suggested by observations: one with a sharp maximum at 120 h-1 Mpc and a second one with a broader maximum, as found for regions with rich and medium-rich superclusters of galaxies, respectively. For both models, the primordial power spectra have a break in amplitude; in the case of the spectrum with a sharp maximum the break is sharp. We conclude that a scale-free primordial power spectrum is excluded if presently available data on the distribution of clusters and galaxies represent the true mass distribution of the universe.

Evolution of peaks in weakly non-linear density field and dark halo profiles

Using the two-point Edgeworth series up to second order in the linear rms density fluctuation we construct the weakly non-linear conditional probability distribution function for the density field around an overdense region. This requires calculating the two-point analogues of the skewness parameter S3. We test the dependence of the two-point skewness on distance from the peak for scale-free power spectra and Gaussian smoothing. The statistical features of such a conditional distribution are given as the values obtained within linear theory corrected by the terms that arise as a result of weakly non-linear evolution. The expected density around the peak is found to be always below the linear prediction while its dispersion is always larger than in the linear case. For large enough overdensities the weakly non-linear corrections can be more significant than the peak constraint introduced by Bardeen et al. We apply these results to the spherical model of collapse as developed by Hoffman & Shaham and find that in general the effect of weakly non-linear interactions is to decrease the scale from which a peak gathers mass and therefore also the mass itself. In the case of an open universe this results in steepening of the final profile of the virialized proto-object.

Cosmological Perturbations: Entering the Nonlinear Regime

We consider next-to-leading-order (one-loop) nonlinear corrections to the bispectrum and skewness of cosmological density fluctuations induced by gravitational evolution, focusing on the case of Gaussian initial conditions and scale-free initial power spectra, P(k) kn. As has been established by comparison with numerical simulations, leading order (tree-level) perturbation theory describes these quantities at the largest scales. The one-loop perturbation theory provides a tool to probe the transition to the nonlinear regime on smaller scales. In this work, we find that, as a function of spectral index n, the one-loop bispectrum follows a pattern analogous to that of the one-loop power spectrum, which shows a change in behavior at a critical index nc ? -1.4, where nonlinear corrections vanish. The tree-level perturbation theory predicts a characteristic dependence of the bispectrum on the shape of the triangle defined by its arguments. For n nc, one-loop corrections increase this configuration dependence of the leading order contribution; for n nc, one-loop corrections tend to cancel the configuration dependence of the tree-level bispectrum, in agreement with known results from n = -1 numerical simulations. A similar situation is shown to hold for the Zeldovich approximation, where nc ? -1.75. We obtain explicit analytic expressions for the one-loop bispectrum for n = -2 initial power spectra, for both the exact dynamics of gravitational instability and the Zeldovich approximation. We also compute the skewness factor, including local averaging of the density field, for n = -2: S -->3(R)=4.02+3.83?$2{r G}$ -->(R) for Gaussian smoothing and S -->3(R)=3.86+3.18?$2{r TH}$ -->(R) for top-hat smoothing, where ?2(R) is the variance of the density field fluctuations smoothed over a window of radius R. A comparison with fully nonlinear numerical simulations implies that, for n < -1, the one-loop perturbation theory can extend our understanding of nonlinear clustering down to scales where the transition to the stable clustering regime begins.

A Lagrangian dynamical theory for the mass function of cosmic structures -- I. Dynamics

A new theory for determining the mass function of cosmic structures is presented. It relies on a realistic treatment of collapse dynamics. Gravitational collapse is analysed in the Lagrangian perturbative framework. Lagrangian perturbations provide an approximation of truncated type, i.e. small-scale structure is filtered out. The collapse time is suitably defined as the instant at which orbit crossing takes place. The convergence of the Lagrangian series in predicting the collapse time of a homogeneous ellipsoid is demonstrated; it is also shown that third-order calculations are necessary in predicting collapse. Then, the Lagrangian prediction, with a correction for quasi-spherical perturbations, can be used to determine the collapse time of a homogeneous ellipsoid in a fast and precise way. Furthermore, ellipsoidal collapse can be considered as a particular truncation of the Lagrangian series. Gaussian fields with scale-free power spectra are then considered. The Lagrangian series for the collapse time is found to converge when the collapse time is not large. In this case, ellipsoidal collapse gives a fast and accurate approximation of the collapse time; spherical collapse is found to poorly reproduce the collapse time, even in a statistical sense. Analytical fits of the distribution functions of the inverse collapse times, as predicted by the ellipsoid model and by third-order Lagrangian theory, are given. These will be necessary for a determination of the mass function, which will be given in Paper II.

The weakly non-linear density-velocity relation

We rigorously derive weakly nonlinear relation between cosmic density and velocity fields up to third order in perturbation theory. The density field is described by the mass density contrast, $\de$. The velocity field is described by the variable $\te$ proportional to the velocity divergence, $\te = - f(\Omega)^{-1} H_0^{-1} \nabla\cdot\bfv$, where $f(\Omega) \simeq \Omega^{0.6}$, $\Omega$ is the cosmological density parameter and $H_0$ is the Hubble constant. Our calculations show that mean $\de$ given $\te$ is a third order polynomial in $\te$, $\lan \de \ran|_{\te} = a_1 \te + a_2 (\te^2 - \s_\te^2) + a_3 \te^3$. This result constitutes an extension of the formula $\lan \de \ran|_{\te} = \te + a_2 (\te^2 - \s_\te^2)$, found by Bernardeau~(1992) which involved second order perturbative solutions. Third order perturbative corrections introduce the cubic term. They also, however, cause the coefficient $a_1$ to depart from unity, in contrast with the linear theory prediction. We compute the values of the coefficients $a_p$ for scale-free power spectra, as well as for standard CDM, for Gaussian smoothing. The coefficients obey a hierarchy $a_3 \ll a_2 \ll a_1$, meaning that the perturbative series converges very fast. Their dependence on $\Omega$ is expected to be very weak. The values of the coefficients for CDM spectrum are in qualitative agreement with the results of N-body simulations by Ganon et al. (1996). The results provide a method for breaking the $\Omega$-bias degeneracy in comparisons of cosmic density and velocity fields such as IRAS-POTENT.

The structure and dynamical evolution of dark matter haloes

(Shortened) We use N-body simulations to investigate the structure and dynamical evolution of dark matter halos in galaxy clusters. Our sample consists of nine massive halos from an EdS universe with scale free power spectrum and n = -1. Halos are resolved by ~20000 particles each, with a dynamical resolution of 20-25 kpc. Large scale tidal fields are included up to L=150 Mpc using background particles. The halo formation process can be characterized by the alternation of two dynamical configurations: a merging phase and a relaxation phase, defined by their signature on the evolution of the total mass and rms velocity. Halos spend on average one 1/3 of their evolution in the merging phase and 2/3 in the relaxation phase. Using this definition, we study the density profiles and their change during the halo history. The average density profiles are fitted by the NFW analytical model with an rms residual of 17% between the virial radius Rv and 0.01 Rv. The Hernquist (1990) profiles fits the same halos with an rms residual of 26%. The trend with mass of the scale radius of these fits is marginally consistent with that found by Cole & Lacey (1996): in comparison our halos are more centrally concentrated, and the relation between scale radius and halo mass is slightly steeper. We find a moderately large scatter in this relation, due both to dynamical evolution within halos and to fluctuations in the halo population. We analyze the dynamical equilibrium of our halos using the Jeans' equation, and find that on average they are approximately in equilibrium within their virial radius. Finally, we find that the projected mass profiles of our simulated halos are in very good agreement with the profiles of three rich galaxy clusters derived from strong and weak gravitational lensing observations.

Sensitivity of galaxy cluster morphologies to Ω0 and P(k)

We examine the sensitivity of the spatial morphologies of galaxy clusters to Omega_0 and P(k) using high-resolution N-body simulations with large dynamic range. Variants of the standard CDM model are considered having different spatial curvatures, SCDM (Omega_0=1), OCDM (Omega_0=0.35), LCDM (Omega_0=0.35, lambda_0=0.65), and different normalizations, sigma_8. We also explore critical density models with different spectral indices, n, of the scale-free power spectrum, P(k) propto k^n. Cluster X-ray morphologies are quantified with power ratios (PRs), where we take for the X-ray emissivity j_gas propto rho^2_DM, which we argue is a suitable approximation for analysis of PRs. We find that Omega_0 primarily influences the means of the PR distributions whereas the power spectrum (n and sigma_8) primarily affects their variances: log_10(P_3/P_0) is the cleanest probe of Omega_0 since its mean is very sensitive to Omega_0 but very insensitive to P(k). The PR means easily distinguish the SCDM and OCDM models, while the SCDM and LCDM means show a more modest, but significant, difference (3 sigma). The OCDM and LCDM models are largely indistinguishable in terms of the PRs. Finally, we compared these models to a sample of ROSAT clusters and find that the cluster morphologies strongly disfavor Omega_0=1, CDM while favoring low density, CDM models with or without a cosmological constant.

Nonlinear Growth of Two-dimensional Cosmological Density Fluctuations and Type of Singularity in Accord with the Catastrophe Theory

view Abstract Citations (3) References (21) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Nonlinear Growth of Two-dimensional Cosmological Density Fluctuations and Type of Singularity in Accord with the Catastrophe Theory Gouda, Naoteru Abstract Nonlinear growth of cosmological density fluctuations from the initial small fluctuations with a non- scale-free or scale-free power spectrum is analyzed in the two-dimensional system by the N-body simulations. It is found that after the first appearance of singularities of density (caustics), the power spectrum of the density fields obeys the power law even on small scales below the cutoff scale of the initial power spectrum. The value of the power index on these scales is almost -2 irrespective of the detailed initial conditions in the case that the initial power index n of the power spectrum on large scales is small and/or the cutoff scale is large. In these cases, the filament structures of the particle distribution (the "filament" shape of caustics of particle trajectories) clearly appear everywhere. It is shown that the power index on the small scales below the typical size of the thickness of filaments is determined only by the type of the singularity (caustics) irrespective of the detailed initial conditions. The type of the singularity is classified in accord with the catastrophe theory. In the above case, singularities of A2 type stably appear isotropically everywhere after the first appearance of caustics, and this is the reason why the value of the power index is almost -2. Publication: The Astrophysical Journal Pub Date: October 1996 DOI: 10.1086/177795 Bibcode: 1996ApJ...469..455G Keywords: COSMOLOGY: THEORY; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE full text sources ADS |

Loop Corrections in Nonlinear Cosmological Perturbation Theory

Using a diagrammatic approach to Eulerian perturbation theory, we analytically calculate the variance and skewness of the density and velocity divergence induced by gravitational evolution from Gaussian initial conditions, including corrections *beyond* leading order. Except for the power spectrum, previous calculations in cosmological perturbation theory have been confined to leading order (tree level)-we extend these to include loop corrections. For scale-free initial power spectra, the one-loop variance \sigma^2 = \sigma^2_l + 1.82 \sigma^4_l and the skewness S_3 = 34/7 + 9.8 \sigma^2_l, where \sigma_l is the rms fluctuation of the linear density field. We also compute loop corrections to the variance, skewness, and kurtosis for several non-linear approximation schemes, where the calculation can be easily generalized to 1-point cumulants of higher order and arbitrary number of loops. We find that the Zel'dovich approximation gives the best approximation to the loop corrections of exact perturbation theory, followed by the Linear Potential approximation (LPA) and the Frozen Flow approximation (FFA), in qualitative agreement with the relative behavior of tree-level results. In LPA and FFA, loop corrections are infrared divergent for spectral indices n < 0; this is related to the breaking of Galilean invariance in these schemes.

Loop Corrections in Nonlinear Cosmological Perturbation Theory. II. Two‐Point Statistics and Self‐Similarity

We calculate the lowest order nonlinear contributions to the power spectrum, two-point correlation function, and smoothed variance of the density field, for Gaussian initial conditions and scale-free initial power spectra, P(k) ~ kn. These results extend and, in some cases, correct previous work in the literature on cosmological perturbation theory. Comparing with the scaling behavior observed in N-body simulations, we find that the validity of nonlinear perturbation theory depends strongly on the spectral index n. For n < −1, we find excellent agreement over scales where the variance σ2(R) 10; however, for n ≥ −1, perturbation theory predicts deviations from self-similar scaling (which increase with n) not seen in numerical simulations. This anomalous scaling suggests that the principal assumption underlying cosmological perturbation theory, namely, that large-scale fields can be described perturbatively even when fluctuations are highly nonlinear on small scales, breaks down beyond leading order for spectral indices n ≥ −1. For n < −1, the power spectrum, variance, and correlation function in the scaling regime can be calculated using dimensional regularization.

Secondary Infall: Theory versus Simulations

The applicability of the highly idealized secondary infall model to `realistic' initial conditions is investigated. The collapse of proto-halos seeded by $3\sigma$ density perturbations to an Einstein--de Sitter universe is studied here for a variety of scale-free power spectra with spectral indices ranging from $n=1$ to $-2$. Initial conditions are set by the constrained realization algorithm and the dynamical evolution is calculated both analytically and numerically. The analytic calculation is based on the simple secondary infall model where spherical symmetry is assumed. A full numerical simulation is performed by a Tree N-body code where no symmetry is assumed. A hybrid calculation has been performed by using a monopole term code, where no symmetry is imposed on the particles but the force is approximated by the monopole term only. The main purpose of using such code is to suppress off-center mergers. In all cases studied here the rotation curves calculated by the two numerical codes are in agreement over most of the mass of the halos, excluding the very inner region, and these are compared with the analytically calculated ones. The main result obtained here, reinforces the foundings of many N-body experements, is that the collapse proceeds 'gently' and not {\it via} violent relaxation. There is a strong correlation of the final energy of individual particles with the initial one. In particular we find a preservation of the ranking of particles according to their binding energy. In cases where the analytic model predicts non-increasing rotation curves its predictions are confirmed by the simulations. Otherwise, sensitive dependence on initial conditions is found and the analytic model fails completely.

Kurtosis as a non-Gaussian signature of the large-scale velocity field

view Abstract Citations (13) References (18) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Kurtosis as a Non-Gaussian Signature of the Large-Scale Velocity Field Catelan, Paolo ; Moscardini, Lauro Abstract We discuss the nonlinear growth of the kurtosis of the smoothed peculiar velocity field (along an arbitrary direction), in an Einstein-de Sitter universe, induced by Gaussian primordial density fluctuations. Applying the perturbative theory, we show that, for different cosmological models, a departure from the original Gaussian distribution is gravitationally induced only on scales smaller than 20-30 Mpc. Models with scale-free power spectrum P(k) is proportional to k^n^, with - 1 < n <= 1, are also considered. A new statistics, independent of the power spectrum normalization and of time, is introduced to measure the kurtosis of the velocity field. Publication: The Astrophysical Journal Pub Date: November 1994 DOI: 10.1086/174875 arXiv: arXiv:astro-ph/9403035 Bibcode: 1994ApJ...436....5C Keywords: Cosmology; Equations Of Motion; Galactic Clusters; Kurtosis; Mathematical Models; Monte Carlo Method; Red Shift; Velocity Distribution; Galaxies; Normal Density Functions; Astrophysics; COSMOLOGY: THEORY; GALAXIES: CLUSTERING; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE; Astrophysics E-Print: 10 pages in Latex (plus 2 figures in a separate file), PD-cosmo/9401 full text sources arXiv | ADS |

Kurtosis and large-scale structure

view Abstract Citations (27) References (23) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Kurtosis and Large-Scale Structure Catelan, Paolo ; Moscardini, Lauro Abstract We discuss the nonlinear growth of the excess kurtosis parameter of the smoothed density fluctuation field δ, S_4_ = [<δ^4^> - 3<δ^2^>^2^]/<δ^2^>^3^ in an Einstein de Sitter universe. We assume Gaussian primordial density fluctuations with scale-free power spectrum P(k) is proportional to k^n^ and analyze the dependence of S_4_ on primordial spectral index n, after smoothing with a Gaussian filter. As already known for the skewness ratio S_3_, the kurtosis parameter is a decreasing function of n, both in exact perturbative theory and in the Zel'dovich approximation. The parameter S_4_ provides a powerful statistics to test different cosmological scenarios. Publication: The Astrophysical Journal Pub Date: May 1994 DOI: 10.1086/174034 arXiv: arXiv:astro-ph/9308002 Bibcode: 1994ApJ...426...14C Keywords: Astronomical Models; Cosmology; Density Distribution; Inflating; Kurtosis; Mass Distribution; Power Spectra; Skewness; Universe; Approximation; Differential Equations; Equations Of Motion; Fourier Transformation; Nonlinear Evolution Equations; Normal Density Functions; Statistical Analysis; Astrophysics; GALAXIES: CLUSTERING; COSMOLOGY: THEORY; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE; Astrophysics E-Print: 11 pages in Latex (plus 1 figure), SISSA 127/93/A full text sources arXiv | ADS |

The cosmological dependence of cluster density profiles

We use N-body simulations to study the shape of mean cluster density and velocity profiles in the nonlinear regime formed via gravitational instability. The dependence of the final structure on both cosmology and initial density field is examined, using a grid of cosmologies and scale-free initial power spectra P(k) varies as k(exp n). Einstein-de Sitter, open (Omega(sub 0) = 0.2 and 0.1) and flat, low density (Omega(sub 0) = 0.2 lambda(sub 0) = 0.8) models are examined, with initial spectral indices n = -2, -1 and 0. For each model, we stack clusters in an appropriately scaled manner to define an average density profile in the nonlinear regime. The profiles are well fit by a power law rho(r) varies as r(exp -alpha) for radii whereat the local density contrast is between 100 and 3000. This covers 99% of the cluster volume. We find a clear trend toward steeper slopes (larger alphas) with both increasing n and decreasing Omega(sub 0). The Omega(sub 0) dependence is partially masked by the n dependence; there is degeneracy in the values of alpha between the Einstein-de Sitter and flat, low-density cosmologies. However, the profile slopes in the open models are consistently higher than the Omega = 1 values for the range of n examined. Cluster density profiles are thus potentially useful cosmological diagnostics. We find no evidence for a constant density core in any of the models, although the density profiles do tend to flatten at small radii. Much of the flattening is due to the force softening required by the simulations. An attempt is made to recover the unsoftened profiles assuming angular momentum invariance. The recovered profiles in Einstein-de Sitter cosmologies are consistent with a pure power law up to the highest density contrasts (10(exp 6)) accessible with our resolution. The low-density models show significant deviation from a power law above density contrasts approximately 10(exp 5). We interpret this curvature as reflecting the non-scale-invariant nature of the background cosmology in these models. These results are at the limit of our resolution and so should be tested in the future using simulations with larger numbers of particles. Such simulations will also provide insight on the broader problem of understanding, in a statistical sense, the full phase space structure of collapsed, cosmological halos.

Cosmological N-body simulations with a tree code - Fluctuations in the linear and nonlinear regimes

The evolution of gravitational systems is studied numerically in a cosmological context using a hierarchical tree algorithm with fully periodic boundary conditions. The simulations employ 262,144 particles, which are initially distributed according to scale-free power spectra. The subsequent evolution is followed in both flat and open universes. With this large number of particles, the discretized system can accurately model the linear phase. It is shown that the dynamics in the nonlinear regime depends on both the spectral index n and the density parameter Omega. In Omega = 1 universes, the evolution of the two-point correlation function Xi agrees well with similarity solutions for Xi greater than about 100 but its slope is steeper in open models with the same n. 28 refs.

Isocurvature baryon-dominated open universe - Structure of galactic halos

Structure of galactic halos in the isocurvature baryon-dominated open universe models is calculated. The halos' rotation curves are calculated analytically, based on the secondary infall paradigm. The nonvanishing curvature of the expanding universe introduces scale dependence of final virial structure of the halos, assuming they are formed dissipationlessly by hierarchical clustering. For primordial scale-free power spectra, with a spectral index in the range of n = 0 to -1, and Omega(0) of about 0.1, the predicted galactic halos rotation curves are found to be steeply decreasing with radius. Given the observed amplitude and extent of the rotation curves of giant spiral galaxies and the nucleosynthesis bounds on the baryon mean density, the allowed range of parameters yields only halos with falling rotation curves, in clear disagreement with the observed flat galactic rotation curves. 33 refs.