Press-Schechter Mass Function Research Articles

Dark matter halo mass functions and density profiles from mass and energy cascade

Halo abundance and structure play a central role for modeling structure formation and evolution. Without relying on a spherical or ellipsoidal collapse model, we analytically derive the halo mass function and cuspy halo density (inner slope of −4/3) based on the mass and energy cascade theory in dark matter flow. The hierarchical halo structure formation leads to halo or particle random walk with a position-dependent waiting time tau _g. First, the inverse mass cascade from small to large scales leads to the halo random walk in mass space with tau _gpropto m_h^{-lambda }, where m_h is the halo mass and lambda is a halo geometry parameter with predicted value of 2/3. The corresponding Fokker-Planck solution for halo random walk in mass space gives rise to the halo mass function with a power-law behavior on small scale and exponential decay on large scale. This can be further improved by considering two different lambda for haloes below and above a critical mass scale m_h^*, i.e. a double-lambda halo mass function. Second, a double-gamma density profile can be derived based on the particle random walk in 3D space with a position-dependent waiting time tau _g propto Phi (r)^{-1} propto r^{-gamma }, where Phi is the gravitational potential and r is the particle distance to halo center. Theory predicts gamma =2/3 that leads to a cuspy density profile with an inner slope of −4/3, consistent with the predicted scaling laws from energy cascade. The Press-Schechter mass function and Einasto density profile are just special cases of proposed models. The small scale permanence can be identified due to the scale-independent rate of mass and energy cascade, where density profiles of different halo masses and redshifts converge to the -4/3 scaling law (rho _h propto r^{-4/3}) on small scales. Theory predicts the halo number density scales with halo mass as propto m_h^{-1.9}, while the halo mass density scales as propto m_h^{4/9}. Results were compared against the Illustris simulations. This new perspective provides a theory for nearly universal halo mass functions and density profiles.

Halo mass function in scale invariant models

ABSTRACT Sheth–Tormen mass function has been widely used to quantify the abundance of dark matter haloes. It is a significant improvement over the Press–Schechter mass function as it uses ellipsoidal collapse in place of spherical collapse. Both of these mass functions can be written in a form that is universal, i.e. independent of cosmology and power spectrum when scaled in suitable variables. However, cosmological simulations have shown that this universality is approximate. In this paper, we investigate the power spectrum dependence of halo mass function through a suite of dark-matter-only N-body simulations of seven power-law models in an Einstein–de Sitter cosmology. This choice of cosmology and a power-law power spectrum ensures the self-similar evolution of dark matter distribution, allowing us to isolate the power spectrum dependence of mass function. We find that the mass function shows a clear non-universality. We present fits for the parameters of the Sheth–Tormen mass function for a range of power-law power spectrum indices. We find a mild evolution in the overall shape of the mass function with the epoch. Finally, we extend our result to LCDM cosmology. We show that the Sheth–Tormen mass function with parameter values derived from a matched power-law Einstein–de Sitter cosmology provides a better fit to the LCDM mass function than the standard Sheth–Tormen mass function. Our results indicate that an improved analytical theory is required to provide better fits to the mass function.

A redefinition of the halo boundary leads to a simple yet accurate halo model of large-scale structure

ABSTRACT We present a model for the halo–mass correlation function that explicitly incorporates halo exclusion and allows for a redefinition of the halo boundary in a flexible way. We assume that haloes trace mass in a way that can be described using a single scale-independent bias parameter. However, our model exhibits scale-dependent biasing due to the impact of halo-exclusion, the use of a ‘soft’ (i.e. not infinitely sharp) halo boundary, and differences in the one halo term contributions to ξhm and ξmm. These features naturally lead us to a redefinition of the halo boundary that lies at the ‘by eye’ transition radius from the one-halo to the two-halo term in the halo–mass correlation function. When adopting our proposed definition, our model succeeds in describing the halo–mass correlation function with $\approx 2{{\ \rm per\ cent}}$ residuals over the radial range 0.1 h−1 Mpc < r < 80 h−1 Mpc, and for halo masses in the range 1013 h−1 M⊙ < M < 1015 h−1 M⊙. Our proposed halo boundary is related to the splashback radius by a roughly constant multiplicative factor. Taking the 87 percentile as reference we find rt/Rsp ≈ 1.3. Surprisingly, our proposed definition results in halo abundances that are well described by the Press–Schechter mass function with δsc = 1.449 ± 0.004. The clustering bias parameter is offset from the standard background-split prediction by $\approx 10{{\ \rm per\ cent}}\!-\!15{{\ \rm per\ cent}}$. This level of agreement is comparable to that achieved with more standard halo definitions.

FORMATION OF DARK MATTER HALOES IN A HOMOGENEOUS DARK ENERGY UNIVERSE

Several independent cosmological tests have shown evidences that the energy density of the universe is dominated by a dark energy component, which causes the present accelerated expansion. The large scale structure formation can be used to probe dark energy models, and the mass function of dark matter haloes is one of the best statistical tools to perform this study. We present here a statistical analysis of mass functions of galaxies under a homogeneous dark energy model, proposed in the work of Percival (2005), using an observational flux-limited X-ray cluster survey, and CMB data from WMAP. We compare, in our analysis, the standard Press–Schechter (PS) approach (where a Gaussian distribution is used to describe the primordial density fluctuation field of the mass function), and the PL (power–law) mass function (where we apply a non-extensive q-statistical distribution to the primordial density field). We conclude that the PS mass function cannot explain at the same time the X-ray and the CMB data (even at 99% confidence level), and the PS best fit dark energy equation of state parameter is ω = -0.58, which is distant from the cosmological constant case. The PL mass function provides better fits to the HIFLUGCS X-ray galaxy data and the CMB data; we also note that the ω parameter is very sensible to modifications in the PL free parameter, q, suggesting that the PL mass function could be a powerful tool to constrain dark energy models.

THE HALO MASS FUNCTION FROM EXCURSION SET THEORY. II. THE DIFFUSING BARRIER

In excursion set theory the computation of the halo mass function is mapped into a first-passage time process in the presence of a barrier, which in the spherical collapse model is a constant and in the ellipsoidal collapse model is a fixed function of the variance of the smoothed density field. However, N-body simulations show that dark matter halos grow through a mixture of smooth accretion, violent encounters and fragmentations, and modeling halo collapse as spherical, or even as ellipsoidal, is a significant oversimplification. We propose that some of the physical complications inherent to a realistic description of halo formation can be included in the excursion set theory framework, at least at an effective level, by taking into account that the critical value for collapse is not a fixed constant $\delta_c$, as in the spherical collapse model, nor a fixed function of the variance $\sigma$ of the smoothed density field, as in the ellipsoidal collapse model, but rather is itself a stochastic variable, whose scatter reflects a number of complicated aspects of the underlying dynamics. Solving the first-passage time problem in the presence of a diffusing barrier we find that the exponential factor in the Press-Schechter mass function changes from $\exp\{-\delta_c^2/2\sigma^2\}$ to $\exp\{-a\delta_c^2/2\sigma^2\}$, where $a=1/(1+D_B)$ and $D_B$ is the diffusion coefficient of the barrier. The numerical value of $D_B$, and therefore the corresponding value of $a$, depends among other things on the algorithm used for identifying halos. We discuss the physical origin of the stochasticity of the barrier and we compare with the mass function found in N-body simulations, for the same halo definition.[Abridged]

Models of high redshift luminosity functions and galactic outflows: The dependence on halo mass function

The form of the halo mass function is a basic ingredient in any semi-analytical galaxy formation model. We study the existing forms of the mass functions in the literature and compare their predictions for semi-analytical galaxy formation models. Two methods are used in the literature to compute the net formation rate of halos, one by simply taking the derivative of the halo mass function and the other using the prescription due to Sasaki [Sasaki, S., 1994. PASJ 46, 427]. For the historically used Press–Schechter (PS) mass function, we compare various model predictions, using these two methods. However, as the Sasaki formalism cannot be easily generalized for other mass functions, we use the derivative while comparing model predictions of different mass functions. We show that the reionization history and UV luminosity function of Lyman break galaxies (LBGs) predicted by the PS mass function differs from those using any other existing mass function, like Sheth–Tormen (ST) mass function. In particular the reionization efficiency of molecular cooled halos has to be substantially reduced when one uses the ST and other mass functions obtained from the simulation instead of the PS mass function. Using χ 2 -minimization, we find that the observed UV luminosity functions of LBGs at 3.0 ⩽ z ⩽ 7.4 are better reproduced by models using the ST mass function compared to models that use the PS mass function. On the other hand, the volume filling factor of the metals expelled from the galaxies through supernovae driven outflows differs very little between models with different mass functions. It depends on the way we treat merging outflows. We also show that the porosity weighted average quantities related to the outflow are not very sensitive to the differences in the halo mass function.

Statistical Properties of the Burgers Equation with Brownian Initial Velocity

We study the one-dimensional Burgers equation in the inviscid limit for Brownian initial velocity (i.e. the initial velocity is a two-sided Brownian motion that starts from the origin x=0). We obtain the one-point distribution of the velocity field in closed analytical form. In the limit where we are far from the origin, we also obtain the two-point and higher-order distributions. We show how they factorize and recover the statistical invariance through translations for the distributions of velocity increments and Lagrangian increments. We also derive the velocity structure functions and we recover the bifractality of the inverse Lagrangian map. Then, for the case where the initial density is uniform, we obtain the distribution of the density field and its $n$-point correlations. In the same limit, we derive the $n-$point distributions of the Lagrangian displacement field and the properties of shocks. We note that both the stable-clustering ansatz and the Press-Schechter mass function, that are widely used in the cosmological context, happen to be exact for this one-dimensional version of the adhesion model.

Halos and Voids in a Multifractal Model of Cosmic Structure

On the one hand, the large-scale structure of matter is arguably scale invariant, and on the other hand, halos and voids are recognized as prominent features of that structure. To unify both approaches, we propose to model the dark matter distribution as a set of fractal distributions of halos of different kinds. This model relies on the concept of a multifractal as the most general scaling distribution and on a plausible notion of a halo as a singular mass concentration in a multifractal. Voids arise as complementary to halos, namely, as formed by regular mass depletions. To provide halos with definite size and masses, we coarse-grain the dark matter distribution using a natural length derived from the lower scaling limit. This allows us to relate the halo mass function to the multifractal spectrum. Hence, we find that a lognormal model of the mass distribution nicely fits in this picture and, moreover, the Press-Schechter mass function can be recovered as a bifractal limit. To support our model of fractal distributions of halos, we perform a numerical study of the distribution produced in cosmological N-body simulations. In the Virgo ΛCDM GIF2 simulation, we indeed find fractal distributions of halos with various dimensions and a halo mass function of bifractal type. However, this mass function is just beyond the Press-Schechter's range, and we interpret it instead as caused by the undersampling of the distribution at the scale of halos, due to discretization.

Capturing Halos at High Redshifts

We study the evolution of the mass function of dark matter halos in the concordance ΛCDM model at high redshift. We employ overlapping (multiple realization) numerical simulations to cover a wide range of halo masses, 107-1015 h-1 M☉, with redshift coverage beginning at z = 20. The Press-Schechter mass function is significantly discrepant from the simulation results at high redshifts. Of the more recently proposed mass functions, our results are in best agreement with those of Warren and coworkers. The statistics of the simulations—along with good control over systematics—allow for fits accurate to the level of 20% at all redshifts. We provide a concise discussion of various issues in defining and computing the halo mass function and how these are addressed in our simulations.

Double distribution of dark matter halos with respect to mass and local overdensity

We present a double distribution function of dark matter halos, with respect to both object mass and local over- (or under-) density. This analytical tool provides a statistical treatment of the properties of matter surrounding collapsed objects, and can be used to study environmental effects on hierarchical structure formation. The size of the ``local environment'' of a collapsed object is defined to depend on the mass of the object. The Press-Schechter mass function is recovered by integration of our double distribution over the density contrast. We also present a detailed treatment of the evolution of overdensities and underdensities in ${\ensuremath{\Omega}}_{\mathrm{m}}+{\ensuremath{\Omega}}_{\ensuremath{\Lambda}}=1$ and ${\ensuremath{\Omega}}_{\mathrm{m}}=1$ universes according to the spherical evolution model. We explicitly distinguish between true and linearly extrapolated overdensities and provide conversion relations between the two quantities.

The 2dF QSO Redshift Survey - XIV. Structure and evolution from the two-point correlation function

In this paper we present a clustering analysis of quasi-stellar objects (QSOs) using over 20000 objects from the final catalogue of the 2dF QSO Redshift Survey (2QZ), measuring the redshift-space two-point correlation function, ξ(s). When averaged over the redshift range 0.3 < z < 2.2 we find that ξ(s) is flat on small scales, steepening on scales above ~25h-1 Mpc. In a WMAP/2dF cosmology (Ωm= 0.27, ΩΛ= 0.73) we find a best-fitting power law with s0= 5.48+0.42-0.48h-1 Mpc and γ= 1.20 +/- 0.10 on scales s= 1 to 25h-1 Mpc. We demonstrate that non-linear redshift-space distortions have a significant effect on the QSO ξ(s) at scales less than ~10h-1 Mpc. A cold dark matter model assuming WMAP/2dF cosmological parameters is a good description of the QSO ξ(s) after accounting for non-linear clustering and redshift-space distortions, and allowing for a linear bias at the mean redshift of bQ(z= 1.35) = 2.02 +/- 0.07. We subdivide the 2QZ into 10 redshift intervals with effective redshifts from z= 0.53 to 2.48. We find a significant increase in clustering amplitude at high redshift in the WMAP/2dF cosmology. The QSO clustering amplitude increases with redshift such that the integrated correlation function, , within 20h-1 Mpc is and . We derive the QSO bias and find it to be a strong function of redshift with bQ(z= 0.53) = 1.13 +/- 0.18 and bQ(z= 2.48) = 4.24 +/- 0.53. We use these bias values to derive the mean dark matter halo (DMH) mass occupied by the QSOs. At all redshifts 2QZ QSOs inhabit approximately the same mass DMHs with MDH= (3.0 +/- 1.6) × 1012h-1 Msolar, which is close to the characteristic mass in the Press-Schechter mass function, M*, at z= 0. These results imply that L*Q QSOs at z~ 0 should be largely unbiased. If the relation between black hole (BH) mass and MDH or host velocity dispersion does not evolve, then we find that the accretion efficiency (L/LEdd) for L*Q QSOs is approximately constant with redshift. Thus the fading of the QSO population from z~ 2 to ~0 appears to be due to less massive BHs being active at low redshift. We apply different methods to estimate, tQ, the active lifetime of QSOs and constrain tQ to be in the range 4 × 106-6 × 108 yr at z~ 2. We test for any luminosity dependence of QSO clustering by measuring ξ(s) as a function of apparent magnitude (equivalent to luminosity relative to L*Q). However, we find no significant evidence of luminosity-dependent clustering from this data set.

Constraints on the Cluster Mass‐Temperature Relation Imposed by the Diffuse X‐Ray Background

We show the constraints on the dynamical properties of groups and clusters of galaxies imposed by the diffuse X-ray background (XRB) emission. We first estimate the total integrated diffuse X-ray emission associated with groups and clusters of galaxies using the X-ray luminosity function derived from the Press-Schechter mass function and the mass-temperature (M-T) and X-ray luminosity-temperature relations of the galaxy groups and clusters. We have applied different cosmological models and M-T relations in estimating the integrated emission. For most of the models associated with the M-T relations derived from X-ray mass estimates, we find that the integrated emission significantly exceeds the unresolved diffuse XRB emission. We suggest that using the X-ray surface brightness profiles as a cluster mass estimator may significantly underestimate the real mass of galaxy clusters. Our methods provide strong constraints on the distribution and the dynamical evolution models of the galaxy clusters.

Resolved SZE Cluster Count

We study the counts of resolved SZE (Sunyaev-Zel'dovich eect) clus- ters expected from an interferometric survey in dierent cosmological models under dierent conditions. The self-similar universal gas model and Press-Schechter mass function are used. We take the observing frequency to be 90GHz, and consider two dish diameters, 1.2m and 2.5m. We calculate the number density of the galaxy clusters dN/(dd z) at a high flux limit S lim = 100mJy and at a relative low S lim = 10mJy. The total numbers of SZE clusters N in two low- 0 models are compared. The results show that the influence of the resolved eect depends not only on D, but also on S lim : at a given D, the eect

Autocorrelation Function of the Soft X‐Ray Background Produced by Warm‐Hot Gas in Dark Halos

We calculate the angular two-point autocorrelation function (ACF) of the soft X-ray background (SXRB) produced by the warm-hot intergalactic medium (WHIM) associated with dark halos, motivated primarily by searching for missing baryons and distinguishing different physical processes of the WHIM in dark halos. We employ a purely analytic model for the halo population that is completely determined by the universal density profile and the Press-Schechter mass function. We then adopt a phenomenological approach to nongravitational processes of the WHIM such as preheating and radiative cooling. It shows that the power spectra of the SXRB predicted by three WHIM models, namely, the self-similar model, preheating model, and cooling model, demonstrate remarkably different signatures in both amplitude and shape, with the peak locations moving from l ? 4 ? 104 for the self-similar model to a smaller value of l ? (3-5) ? 103 when nongravitational processes are taken into account. The corresponding ACFs for preheating and cooling models become shallower too as compared to the prediction of the self-similar model. This may permit an effective probe of the physical processes of the WHIM in massive halos in conjunction with the observationally determined power spectrum or ACF of the SXRB from diffuse WHIM. However, a direct comparison of our theoretical predictions with existing data (e.g., the ACF determined from ROSAT observations) is still difficult because of the dominant contribution of active galactic nuclei in the soft X-ray sky. We discuss briefly the implications of our results for resolving the missing baryon problem in the local universe.

Schechter versus Schechter: Subarcsecond Gravitational Lensing and Inner Halo Profiles

Subarcsecond lensing statistics depend sensitively on the inner mass profiles of low-mass objects and the faint-end slopes of the Schechter luminosity function and the Press-Schechter mass function. By requiring the luminosity and mass functions to give consistent predictions for the distribution of image separation below 1'', we show that dark matter halos with masses below 1012 M☉ cannot have a single type of profile, be it the singular isothermal sphere (SIS) or the shallower universal dark matter profile. Instead, consistent results are achieved if we allow a fraction of the halos at a given mass to be luminous with the SIS profile and the rest to be dark with an inner logarithmic slope shallower than -1.5 to compensate for the steeper faint-end slope of the mass function compared with the luminosity function. We quantify how rapidly the SIS fraction must decrease with decreasing halo mass, thereby providing a statistical measure for the effectiveness of feedback processes on the baryon content in low-mass halos.

Searching for Dark Matter Halos in the Suprime-Cam 2 Square Degree Field

We report the first result of a weak gravitational lensing survey on a 2.1 deg2 Rc-band image taken with a wide-field camera (Suprime-Cam) on the prime focus of the 8.2 m Subaru Telescope. The weak lensing mass reconstruction is applied to the data to search for dark matter halos of cluster scale M ≥ 1014 M☉. The reconstructed convergence field is divided by 1 σ noise to obtain the signal-to-noise ratio (S/N) map of the detection. Local maxima and minima are searched on the S/N map, and the probability distribution function (PDF) of the peaks is created to compare with model predictions. We found excess over noise PDF created from the randomized realization on both the positive and negative sides. Negative peaks imply the presence of voids in the dark matter distribution, and this is the first report of the detection. Positive peaks, on the other hand, represent the dark matter halos, and the number count of the halos on the 2.1 deg2 image is 4.9 ± 2.3 for S/N > 5, where the Gaussian smoothing radius of the convergence map is 1'. The result is consistent with the prediction assuming the Press-Schechter mass function and the Navarro-Frenk-White halo profile under the cluster-normalized cold dark matter cosmology. Although the present statistics are not enough because of the limited field of view, this work demonstrates that dark matter halo count via weak lensing is a promising way to test the paradigm of structure formation and the cosmological model.

Weak-lensing halo numbers and dark-matter profiles

Integral measures of weak gravitational lensing by dark-matter haloes, like the aperture mass, are sensitive to dierent physical halo properties dependent on the halo mass density prole. For isothermal proles, the relation between aperture mass and virial mass is steeper than for haloes with the universal NFW prole. Consequently, the halo mass range probed by the aperture mass is much wider for NFW than for isothermal haloes. We use recent modications to the Press-Schechter mass function in CDM models normalised to the local abundance of rich clusters, to predict the properties of the halo sample expected to be accessible with the aperture mass technique. While10 haloes should be detected per square degree if the haloes have NFW proles, their number density is lower by approximately an order of magnitude if they have isothermal proles. These results depend only very mildly on the cosmological background model. We conclude that counts of haloes with a signicant weak-lensing signal are a powerful discriminator between dierent dark-matter proles.

The X‐Ray Background from Groups and Clusters of Galaxies

We present an estimate of the X-ray background (XRB) spectrum from the warm-hot intergalactic medium (IGM) associated with groups and clusters of galaxies, using purely the observationally determined X-ray luminosity function (XLF) and X-ray luminosity (Lx) - temperature (T) relations for groups and clusters. As compared with previous semi-analytic models based on the Press-Schechter mass function, our approach provides a much simpler and more realistic way to evaluate the XRB from groups and clusters in the sense that we make no assumption about the dynamical and heating properties of the IGM, and the intrinsic dispersion in the Lx-T relations due to different physical mechanisms among different groups and clusters can be also included. It shows that the resulting 0.1-10 keV XRB spectrum by summing up the X-ray emission from all groups and clusters is consistent with current upper limits placed on the contribution from diffuse gas to the XRB. This may have profound implications for our understanding of the missing baryons in the universe.

Constraints on the Warm Dark Matter Model from Gravitational Lensing

Formation of sub-galactic halos is suppressed in warm dark matter (WDM) model due to thermal motion of WDM particles. This may provide a natural resolution to some puzzles in standard cold dark matter (CDM) theory such as the cusped density profiles of virialized dark halos and the overabundance of low mass satellites. One of the observational tests of the WDM model is to measure the gravitationally lensed images of distant quasars below sub-arcsecond scales. In this Letter, we report a comparison of the lensing probabilities of multiple images between CDM and WDM models using a singular isothermal sphere model for the mass density profiles of dark halos and the Press-Schechter mass function for their distribution and cosmic evolution. It is shown that the differential probability of multiple images with small angular separations down to 10 milliarcseconds should allow one to set useful constraints on the WDM particle mass. We discuss briefly the feasibility and uncertainties of this method in future radio surveys (e.g. VLBI) for gravitational lensing.

Expectations for Sunyaev‐Zeldovich Cluster Counts: Mass Function versus X‐Ray Luminosity Function

We present a comparison of the Sunyaev-Zeldovich (SZ) cluster counts predicted by the Press-Schechter (PS) mass function (MF) and the X-ray luminosity function (XLF) of clusters. The employment of the cluster XLF, together with the observationally determined X-ray luminosity-temperature (LX-T) relation, may allow us to estimate the SZ cluster counts in a more realistic manner, although such an empirical approach depends sensitively on our current knowledge of the dynamical properties of intracluster gas and its cosmic evolution. Using both the nonevolving and evolving XLFs of clusters suggested by X-ray observations, we calculate the expectations for SZ surveys of clusters with X-ray luminosity LX ≥ 3 × 1044 ergs s-1 and LX ≥ 1 × 1043 ergs s-1 in the 0.5-2.0 band, respectively. The nonevolving XLF results in a significant excess of SZ cluster counts at high redshifts as compared with the evolving XLF, while a slightly steeper LX-T relation than the observed one is needed to reproduce the distributions of SZ clusters predicted by the standard PS formalism. It is pointed out that uncertainties in the cosmological application of future SZ cluster surveys via the standard PS formalism should be carefully studied.