Press-Schechter Approximation Research Articles

The Statistics of Primordial Black Holes in a Radiation-Dominated Universe: Recent and New Results

We review the nonlinear statistics of Primordial Black Holes that form from the collapse of over-densities in a radiation-dominated Universe. We focus on the scenario in which large over-densities are generated by rare and Gaussian curvature perturbations during inflation. As new results, we show that the mass spectrum follows a power law determined by the critical exponent of the self-similar collapse up to a power spectrum dependent cutoff, and that the abundance related to very narrow power spectra is exponentially suppressed. Related to this, we discuss and explicitly show that both the Press–Schechter approximation and the statistics of mean profiles lead to wrong conclusions for the abundance and mass spectrum. Finally, we clarify that the transfer function in the statistics of initial conditions for Primordial Black Holes formation (the abundance) does not play a significant role.

Path integral approach for merger rates of dark matter haloes

We use path integrals in order to estimate merger rates of dark matter haloes using the Extended Press-Schechter approximation (EPS) for the Spherical Collapse (SC) and the Ellipsoidal Collapse (EC) models. Merger rates have been calculated for masses in the range $10^{10}M_{\odot}\mathrm{h}^{-1}$ to $10^{14}M_{\odot}\mathrm{h}^{-1}$ and for redshifts $z$ in the range 0 to 3. A detailed comparison between these models is presented. Path approach gives a better agreement with the exact solutions for constrained distributions than the approach of \cite{shto02}. Although this improvement seems not to be very large, our results show that the path approach is a step to the right direction. Differences between the two widely used barriers, spherical and ellipsoidal, depend crucially on the mass of the descendant halo. These differences become larger for decreasing mass of the descendant halo. The use of additional terms in the expansion used in the path approach, other improvements as well as detailed comparisons with the predictions of N-body simulations, that could improve our understanding about the important issue of structure formation, are under study.

Merger rates of dark matter haloes: a comparison between EPS and N-body results

We calculate merger rates of dark matter haloes using the Extended Press-Schechter approximation (EPS) for the Spherical Collapse (SC) and the Ellipsoidal Collapse (EC) models. Merger rates have been calculated for masses in the range $10^{10}M_{\odot}\mathrm{h}^{-1}$ to $10^{14}M_{\odot}\mathrm{h}^{-1}$ and for redshifts $z$ in the range 0 to 3 and they have been compared with merger rates that have been proposed by other authors as fits to the results of N-body simulations. The detailed comparison presented here shows that the agreement between the analytical models and N-body simulations depends crucially on the mass of the descendant halo. For some range of masses and redshifts either SC or EC models approximate satisfactory the results of N-body simulations but for other cases both models are less satisfactory or even bad approximations. We showed, by studying the parameters of the problem that a disagreement --if it appears-- does not depend on the values of the parameters but on the kind of the particular solution used for the distribution of progenitors or on the nature of EPS methods. Further studies could help to improve our understanding about the physical processes during the formation of dark matter haloes.

The cosmological history of accretion onto dark halos and supermassive black holes

Aims. We investigate the cosmological growth of dark halos and follow the consequences of coeval growth for the accretion history of associated supermassive black holes. Methods. The Press-Schechter approximation is used to obtain an analytic expression for the mean rate of growth of dark matter halos. Dark halo accretion rates are compared with numerical work and the consequences for understanding AGN evolution are described. Results. The mean accretion rate onto dark matter halos is shown to have a simple analytic form that agrees with previous numerical work and that may easily be calculated for a wide range of halo mass, redshift and cosmological parameters. The result offers a significant improvement over published fitting formulae deduced from merger trees. We then consider the growth of associated supermassive black holes, and make a basic test of the simple hypothesis of Pure Coeval Evolution (PCE) in which, on average, black hole growth tracks dark halo growth. We demonstrate that both the absolute value of the integrated AGN bolometric luminosity density and its cosmological evolution derived from hard X-ray surveys are well-reproduced by PCE. Excellent agreement is found at , although the observed luminosity density drops by a factor 2 compared with PCE by z=0: black hole growth appears to decouple from halo growth at low redshifts, and this may be related to the phenomenon of downsizing. Overall, AGN evolution appears either to be caused by or to be closely linked to the slow-down in the growth of cosmic structure. We also discuss the mean Eddington ratio averaged over all galaxies, which is predicted to show strong evolution to higher values with redshift

Cosmological perturbations and the reionization epoch

We investigate the dependence of the epoch of reionization on the properties of cosmological perturbations, in the context of cosmologies permitted by WMAP. We compute the redshift of reionization using a simple model based on the Press–Schechter approximation. For a power-law initial spectrum we estimate that reionization is likely to occur at a redshift zreion= 17+10−7, consistent with the WMAP determination based on the temperature-polarization cross power spectrum. We estimate the delay in reionization if there is a negative running of the spectral index, as weakly indicated by WMAP. We then investigate the dependence of the reionization redshift on the nature of the initial perturbations. We consider chi-squared probability distribution functions with various degrees of freedom, motivated both by non-standard inflationary scenarios and by defect models. We find that in these models reionization is likely to occur much earlier, and to be a slower process, than in the case of initial Gaussian fluctuations. We also consider a hybrid model in which cosmic strings make an important contribution to the seed fluctuations on scales relevant for reionization. We find that in order for that model to agree with the latest WMAP results, the string contribution to the matter power spectrum on the standard 8 h−1 Mpc scale is likely to be at most at the level of 1 per cent, which imposes tight constraints on the value of the string mass per unit length.

The Origin of Angular Momentum in Dark Matter Halos

We propose a new explanation for the origin of angular momentum in galaxies and their dark halos, in which the halos obtain their spin through the cumulative acquisition of angular momentum from satellite accretion. In our model, the buildup of angular momentum is a random walk process associated with the mass assembly history of the halo's major progenitor. We assume no correlation between the angular momenta of accreted objects. The main role of tidal torques in this approach is to produce the random tangential velocities of merging satellites. Using the extended Press-Schechter approximation, we calculate the growth of mass, angular momentum, and spin parameter λ for many halos. Our random walk model reproduces the key features of the angular momentum of halos found in ΛCDM N-body simulations: a lognormal distribution in λ with an average of ≈ 0.045 and dispersion σλ = 0.56, independent of mass and redshift. The evolution of the spin parameter in individual halos in this model is quite different from the steady increase with time of angular momentum in the tidal torque picture. We find both in N-body simulations and in our random walk model that the value of λ changes significantly with time for a halo's major progenitor. It typically has a sharp increase due to major mergers and a steady decline during periods of gradual accretion of small satellites. The model predicts that, on average, the λ of ~1012 M☉ halos that had major mergers after redshift z = 3 should be substantially larger than the λ of those that did not. Perhaps surprisingly, this suggests that halos that host later forming elliptical galaxies should rotate faster than halos of spiral galaxies.

Evolution of the Cluster Mass Function: Gpc3Dark Matter Simulations

High-resolution N-body simulations of four popular cold dark matter cosmologies (LCDM, OCDM, QCDM, and tilted SCDM), each containing ~105 clusters of galaxies of mass M1.5 > 5 × 1013 h-1 M☉ in a Gpc3 volume, are used to determine the evolution of the cluster mass function from z = 3 to 0. The large volume and high resolution of these simulations allow an accurate measure of the evolution of cosmologically important (but rare) massive clusters at high redshift. The simulated mass function is presented for cluster masses within several radii typically used observationally (R = 0.5, 1.0, and 1.5 h-1 Mpc, both comoving and physical) in order to enable direct comparison with current and future observations. The simulated evolution is compared with current observations of massive clusters at redshifts 0.3 z 0.8. The Ωm = 1 tilted SCDM model, which exhibits very rapid evolution of the cluster abundance, produces too few clusters at z 0.3 and no massive clusters at z 0.5, in stark contradiction to observations. The Ωm = 0.3 models—LCDM, OCDM, and QCDM—all exhibit considerably weaker evolution and are consistent with current data. Among these low-density models, OCDM evolves the least. These trends are enhanced at high redshift and can be used to discriminate between flat and open low-density models. The simulated mass functions are compared with the Press-Schechter approximation. Standard Press-Schechter predicts too many low-mass clusters at z = 0, and too few clusters at higher redshift. We modify the approximation by a simple parameterization of the density contrast threshold for collapse, which has a redshift dependence. This modified Press-Schechter approximation provides a good fit to the simulated mass functions.

Quasar Clustering and the Lifetime of Quasars

Although the population of luminous quasars rises and falls over a period of 10^9 years, the typical lifetime of individual quasars is uncertain by several orders of magnitude. We show that quasar clustering measurements can substantially narrow the range of possible lifetimes with the assumption that luminous quasars reside in the most massive host halos. If quasars are long-lived, then they are rare phenomena that are highly biased with respect to the underlying dark matter, while if they are short-lived they reside in more typical halos that are less strongly clustered. For a given quasar lifetime, we calculate the minimum host halo mass by matching the observed space density of quasars, using the Press-Schechter approximation. We use the results of Mo & White to calculate the clustering of these halos, and hence of the quasars they contain, as a function of quasar lifetime. A lifetime of t_Q = 4 x 10^7 years, the e-folding timescale of an Eddington luminosity black hole with accretion efficiency eps=0.1, corresponds to a quasar correlation length r_0 ~ 10 Mpc/h in low-density cosmological models at z=2-3; this value is consistent with current clustering measurements, but these have large uncertainties. High-precision clustering measurements from the 2dF and Sloan quasar surveys will test our key assumption of a tight correlation between quasar luminosity and host halo mass, and if this assumption holds then they should determine t_Q to a factor of three or better. An accurate determination of the quasar lifetime will show whether supermassive black holes acquire most of their mass during high-luminosity accretion, and it will show whether the black holes in the nuclei of typical nearby galaxies were once the central engines of high-luminosity quasars.

Scaling laws in gravitational clustering for counts-in-cells and mass functions

We present in this article an analysis of some of the properties of the density field realized in numerical simulations for power-law initial power-spectra in the case of a critical density universe. We compare our numerical results in the non-linear regime with the predictions of a specific scaling model, focusing on its much wider range of applicability, which is one of its main advantages over the standard Press-Schechter approximation. We first check that the two-point correlation functions agree with the stable-clustering ansatz. Next we show that the statistics of the counts-in-cells obey the scaling law predicted by our scaling model. Then, we turn to mass functions of overdense and underdense regions. We first consider the mass function of just collapsed objects defined by a density threshold $\Delta~177$. We note that the usual Press-Schechter prescription agrees reasonably well with the simulations (although there are some discrepancies) while the numerical results are also consistent with the scaling model. Then, we consider more general mass functions defined by different density thresholds which can even be negative. This is out of reach of the Press-Schechter approach while our scaling model can handle these mass functions and it shows a reasonably good agreement with numerical results. Finally, we consider objects defined by a constant radius condition. Thus, we find that the scaling model allows one to study many different classes of objects and it clarifies the links between various statistical tools.

Cosmological Simulations with Scale‐Free Initial Conditions. I. Adiabatic Hydrodynamics

We analyze hierarchical structure formation based on scale-free initial conditions in an Einstein-de Sitter universe, including a baryonic component with Ωbary = 0.05. We present three independent, smoothed particle hydrodynamics (SPH) simulations, performed at two resolutions (323 and 643 dark matter and baryonic particles) and with two different SPH codes (TreeSPH and P3MSPH). Each simulation is based on identical initial conditions, which consist of Gaussian-distributed initial density fluctuations that have a power spectrum P(k) ∝ k-1. The baryonic material is modeled as an ideal gas subject only to shock heating and adiabatic heating and cooling; radiative cooling and photoionization heating are not included. The evolution is expected to be self-similar in time, and under certain restrictions we identify the expected scalings for many properties of the distribution of collapsed objects in all three realizations. The distributions of dark matter masses, baryon masses, and mass- and emission-weighted temperatures scale quite reliably. However, the density estimates in the central regions of these structures are determined by the degree of numerical resolution. As a result, mean gas densities and Bremsstrahlung luminosities obey the expected scalings only when calculated within a limited dynamic range in density contrast. The temperatures and luminosities of the groups show tight correlations with the baryon masses, which we find can be well represented by power laws. The Press-Schechter (PS) approximation predicts the distribution of group dark matter and baryon masses fairly well, though it tends to overestimate the baryon masses. Combining the PS mass distribution with the measured relations for T(M) and L(M) predicts the temperature and luminosity distributions fairly accurately, though there are some discrepancies at high temperatures/luminosities. In general the three simulations agree well for the properties of resolved groups, where a group is considered resolved if it contains more than 32 particles.

Determining the Amplitude of Mass Fluctuations in the Universe

We present a method for determining the amplitude of mass fluctuations on 8 h-1 Mpc scale, σ8. The method utilizes the rate of evolution of the abundance of rich clusters of galaxies. Using the Press-Schechter approximation, we show that the cluster abundance evolution is a strong function of σ8: dlogn/dz∝-1/σ28; low-σ8 models evolve exponentially faster than high-σ8 models, for a given mass cluster. For example, the number density of Coma-like clusters decreases by a factor of ~103 from z=0 to z0.5 for σ8=0.5 models, while the decrease is only a factor of ~5 for σ81. The strong exponential dependence on σ8 arises because clusters represent rarer density peaks in low-σ8 models. We show that the evolution rate at z1 is insensitive to the density parameter Ω or to the exact shape of the power spectrum. Cluster evolution therefore provides a powerful constraint on σ8. Using available cluster data to z~0.8, we find σ8=0.83±0.15. This amplitude implies a bias parameter bσ−18=1.2±0.2, i.e., a nearly unbiased universe with mass approximately tracing light on large scales. When combined with the present-day cluster abundance normalization, σ8Ω0.50.5, the cosmological density parameter can be determined: Ω0.3±0.1.

Damped lyman-alpha systems versus cold + hot dark matter

view Abstract Citations (115) References (43) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Damped Lyman-Alpha Systems versus Cold + Hot Dark Matter Klypin, Anatoly ; Borgani, Stefano ; Holtzman, Jon ; Primack, Joel Abstract Although the cold + hot dark matter (CHDM) cosmology provides perhaps the best fit of any model to all the available data at the current epoch (z = 0), CHDM produces structure at relatively low redshifts and thus is very sensitive to the observed numbers of massive objects at high redshifts. Damped Lyα systems are abundant in quasar absorption spectra and provide possibly the most significant evidence for early structure formation, and thus a stringent constraint on CHDM. Using the numbers of halos in N-body simulations to normalize Press-Schechter estimates of the number densities of protogalaxies as a function of redshift, we find that CHDM with {OMEGA}_c_/{OMEGA}nu_/{OMEGA}_b_ = 0.6/0.3/0.1 is compatible with the damped Lyα data only at < 2.5, but that it is probably incompatible with the z > 3 damped Lyα data. The situation is uncertain because there is very little data for z > 3. The predictions of CHDM are quite sensitive to the hot (neutrino) fraction, and we find that {OMEGA}_c_/{OMEGA}nu_/{OMEGA}_b_ = 0.725/0.20/0.075 (and possibly even {OMEGA}_c_/{OMEGA}nu_/{OMEGA}_b_ = 0.675/0.25/0.075) is compatible with the z > 3 data. With one massive neutrino species, using {OMEGA}nu_ = 0.20 instead of 0.30 corresponds to lowering the neutrino mass from 7.0 to 4.7 eV, for H_0_ = 50 km s^-1^ Mpc^-1^ and T - 2.726 K. In CHDM, the higher redshift damped Lyα systems are predicted to have lower masses (~3 x 10^10^ M_sun_ at z = 3), a prediction which can be checked by measuring the velocity widths of the associated metal-line systems. Predictions for high-z objects crucially depend on the effects of limited resolution and the finite box size in N-body simulations or on the parameters of the Press-Schechter approximation, if it is used. By analyzing our numerical simulations with vastly different resolutions and box sizes as well as those of Ma & Bertschinger (1994), we show that for the CHDM models with {OMEGA}nu_ = 0.2-0.3 the Press-Schechter approximation should be used with Gaussian filter with δ_c_ = 1.5 if halos are defined with the mean overdensity larger than 200. If one tries to recover the total mass of a collapsed halo, a better value for the collapse parameter is δ_c_ = 1.40. We argue that nonlinear effects due to waves both longer and shorter than those considered in numerical simulations could probably result in δ_c_ as low as δ_c_ = 1.3. Publication: The Astrophysical Journal Pub Date: May 1995 DOI: 10.1086/175578 arXiv: arXiv:astro-ph/9405003 Bibcode: 1995ApJ...444....1K Keywords: Computational Astrophysics; Cosmology; Damping; Dark Matter; Lyman Alpha Radiation; Universe; Absorption Spectra; Approximation; Galactic Evolution; Galactic Halos; Many Body Problem; Neutrinos; Quasars; Simulation; Space Density; Astrophysics; COSMOLOGY: DARK MATTER; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE; COSMOLOGY: THEORY; GALAXIES: FORMATION; GALAXIES: QUASARS: ABSORPTION LINES; Astrophysics; High Energy Physics - Phenomenology E-Print: submitted to ApJ 16 pages including 3 figures, uuencoded compressed postscript, SCIPP 94/09 full text sources arXiv | ADS |