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]Comment: 7 pages, 5 figures. v3: significant conceptual improvements. More detailed comparison with N-body simulations. References adde

Characterizing the Shapes of Galaxy Clusters Using Moments of the Gravitational Lensing Shear

We explore the use of the tangential component of weak lensing shear to characterize the ellipticity of clusters of galaxies. We introduce an ellipticity estimator, and quantify its properties for isolated clusters from LCDM N-body simulations. We compare the N-body results to results from smooth analytic models. The expected distribution of the estimator for mock observations is presented, and we show how this distribution is impacted by contaminants such as noise, line of sight projections, and misalignment of the central galaxy used to determine the orientation of the triaxial halo. We examine the radial profile of the estimator and discuss tradeoffs in the observational strategy to determine cluster shape.Comment: 17 pages, 6 figures; added references, corrected typos, matches published versio

The overdensity and masses of the friends-of-friends halos and universality of the halo mass function

The friends-of-friends algorithm (hereafter, FOF) is a percolation algorithm which is routinely used to identify dark matter halos from N-body simulations. We use results from percolation theory to show that the boundary of FOF halos does not correspond to a single density threshold but to a range of densities close to a critical value that depends upon the linking length parameter, b. We show that for the commonly used choice of b = 0.2, this critical density is equal to 81.62 times the mean matter density. Consequently, halos identified by the FOF algorithm enclose an average overdensity which depends on their density profile (concentration) and therefore changes with halo mass contrary to the popular belief that the average overdensity is ~180. We derive an analytical expression for the overdensity as a function of the linking length parameter b and the concentration of the halo. Results of tests carried out using simulated and actual FOF halos identified in cosmological simulations show excellent agreement with our analytical prediction. We also find that the mass of the halo that the FOF algorithm selects crucially depends upon mass resolution. We find a percolation theory motivated formula that is able to accurately correct for the dependence on number of particles for the mock realizations of spherical and triaxial Navarro-Frenk-White halos. However, we show that this correction breaks down when applied to the real cosmological FOF halos due to presence of substructures. Given that abundance of substructure depends on redshift and cosmology, we expect that the resolution effects due to substructure on the FOF mass and halo mass function will also depend on redshift and cosmology and will be difficult to correct for in general. Finally, we discuss the implications of our results for the universality of the mass function.Comment: 19 pages, 17 figures, submitted to ApJ supplemen

The Large Scale Bias of Dark Matter Halos: Numerical Calibration and Model Tests

We measure the clustering of dark matter halos in a large set of collisionless cosmological simulations of the flat LCDM cosmology. Halos are identified using the spherical overdensity algorithm, which finds the mass around isolated peaks in the density field such that the mean density is Delta times the background. We calibrate fitting functions for the large scale bias that are adaptable to any value of Delta we examine. We find a ~6% scatter about our best fit bias relation. Our fitting functions couple to the halo mass functions of Tinker et. al. (2008) such that bias of all dark matter is normalized to unity. We demonstrate that the bias of massive, rare halos is higher than that predicted in the modified ellipsoidal collapse model of Sheth, Mo, & Tormen (2001), and approaches the predictions of the spherical collapse model for the rarest halos. Halo bias results based on friends-of-friends halos identified with linking length 0.2 are systematically lower than for halos with the canonical Delta=200 overdensity by ~10%. In contrast to our previous results on the mass function, we find that the universal bias function evolves very weakly with redshift, if at all. We use our numerical results, both for the mass function and the bias relation, to test the peak-background split model for halo bias. We find that the peak-background split achieves a reasonable agreement with the numerical results, but ~20% residuals remain, both at high and low masses.Comment: 11 pages, submitted to ApJ, revised to include referee's coment

Detection of lensing substructure using ALMA observations of the dusty galaxy SDP.81

We study the abundance of substructure in the matter density near galaxies using ALMA Science Verification observations of the strong lensing system SDP.81. We present a method to measure the abundance of subhalos around galaxies using interferometric observations of gravitational lenses. Using simulated ALMA observations, we explore the effects of various systematics, including antenna phase errors and source priors, and show how such errors may be measured or marginalized. We apply our formalism to ALMA observations of SDP.81. We find evidence for the presence of a $M=10^{8.96\pm 0.12} M_{\odot}$ subhalo near one of the images, with a significance of $6.9\sigma$ in a joint fit to data from bands 6 and 7; the effect of the subhalo is also detected in both bands individually. We also derive constraints on the abundance of dark matter subhalos down to $M\sim 2\times 10^7 M_{\odot}$, pushing down to the mass regime of the smallest detected satellites in the Local Group, where there are significant discrepancies between the observed population of luminous galaxies and predicted dark matter subhalos. We find hints of additional substructure, warranting further study using the full SDP.81 dataset (including, for example, the spectroscopic imaging of the lensed carbon monoxide emission). We compare the results of this search to the predictions of $\Lambda$CDM halos, and find that given current uncertainties in the host halo properties of SDP.81, our measurements of substructure are consistent with theoretical expectations. Observations of larger samples of gravitational lenses with ALMA should be able to improve the constraints on the abundance of galactic substructure.Comment: 18 pages, 13 figures, Comments are welcom

Mass Function Predictions Beyond LCDM

The mass distribution of halos, as specified by the halo mass function, is a key input for several cosmological probes. The sizes of $N$-body simulations are now such that, for the most part, results need no longer be statistics-limited, but are still subject to various systematic uncertainties. We investigate and discuss some of the reasons for these differences. Quantifying error sources and compensating for them as appropriate, we carry out a high-statistics study of dark matter halos from 67 $N$-body simulations to investigate the mass function and its evolution for a reference $\Lambda$CDM cosmology and for a set of $w$CDM cosmologies. For the reference $\Lambda$CDM cosmology (close to WMAP5), we quantify the breaking of universality in the form of the mass function as a function of redshift, finding an evolution of as much as 10% away from the universal form between redshifts $z=0$ and $z=2$. For cosmologies very close to this reference we provide a fitting formula to our results for the (evolving) $\Lambda$CDM mass function over a mass range of $6\cdot 10^{11}-3\cdot 10^{15}$ M$_{\odot}$ to an estimated accuracy of about 2%. The set of $w$CDM cosmologies is taken from the Coyote Universe simulation suite. The mass functions from this suite (which includes a $\Lambda$CDM cosmology and others with $w\simeq-1$) are described by the fitting formula for the reference $\Lambda$CDM case at an accuracy level of 10%, but with clear systematic deviations. We argue that, as a consequence, fitting formulae based on a universal form for the mass function may have limited utility in high precision cosmological applications.Comment: 19 pages; 18 figures; accepted for publication in the Ap

Mass functions and bias of dark matter halos

We revisit the study of the mass functions and the bias of dark matter halos. Focusing on the limit of rare massive halos, we point out that exact analytical results can be obtained for the large-mass tail of the halo mass function. This is most easily seen from a steepest-descent approach, that becomes asymptotically exact for rare events. We also revisit the traditional derivation of the bias of massive halos, associated with overdense regions in the primordial density field. We check that the theoretical large-mass cutoff agrees with the mass functions measured in numerical simulations. For halos defined by a nonlinear threshold $\delta=200$ this corresponds to using a linear threshold $\delta_L\simeq 1.59$ instead of the traditional value $\simeq 1.686$. We also provide a fitting formula that matches simulations over all mass scales and obeys the exact large-mass tail. Next, paying attention to the Lagrangian-Eulerian mapping (i.e. corrections associated with the motions of halos), we improve the standard analytical formula for the bias of massive halos. We check that our prediction, which contains no free parameter, agrees reasonably well with numerical simulations. In particular, it recovers the steepening of the dependence on scale of the bias that is observed at higher redshifts, which published fitting formulae did not capture. This behavior mostly arises from nonlinear biasing.Comment: 15 page

Collapse Barriers and Halo Abundance: Testing the Excursion Set Ansatz

Our heuristic understanding of the abundance of dark matter halos centers around the concept of a density threshold, or "barrier", for gravitational collapse. If one adopts the ansatz that regions of the linearly evolved density field smoothed on mass scale M with an overdensity that exceeds the barrier will undergo gravitational collapse into halos of mass M, the corresponding abundance of such halos can be estimated simply as a fraction of the mass density satisfying the collapse criterion divided by the mass M. The key ingredient of this ansatz is therefore the functional form of the collapse barrier as a function of mass M or, equivalently, of the variance sigma^2(M). Several such barriers based on the spherical, Zel'dovich, and ellipsoidal collapse models have been extensively discussed. Using large scale cosmological simulations, we show that the relation between the linear overdensity and the mass variance for regions that collapse to form halos by the present epoch resembles expectations from dynamical models of ellipsoidal collapse. However, we also show that using such a collapse barrier with the excursion set ansatz predicts a halo mass function inconsistent with that measured directly in cosmological simulations. This inconsistency demonstrates a failure of the excursion set ansatz as a physical model for halo collapse. We discuss implications of our results for understanding the collapse epoch for halos as a function of mass, and avenues for improving consistency between analytical models for the collapse epoch and the results of cosmological simulations.Comment: Version accepted by ApJ, scheduled for May 2009, v696. High-res version available at http://kicp.uchicago.edu/~brant/astro-ph/excursion_set_ansatz/robertson_excursion_set_ansatz.pd

The evolution of substructure II: linking dynamics to environment

We present results from a series of high-resolution N-body simulations that focus on the formation and evolution of eight dark matter halos, each of order a million particles within the virial radius. We follow the time evolution of hundreds of satellite galaxies with unprecedented time resolution, relating their physical properties to the differing halo environmental conditions. The self-consistent cosmological framework in which our analysis was undertaken allows us to explore satellite disruption within live host potentials, a natural complement to earlier work conducted within static potentials. Our host halos were chosen to sample a variety of formation histories, ages, and triaxialities; despite their obvious differences, we find striking similarities within the associated substructure populations. Namely, all satellite orbits follow nearly the same eccentricity distribution with a correlation between eccentricity and pericentre. We also find that the destruction rate of the substructure population is nearly independent of the mass, age, and triaxiality of the host halo. There are, however, subtle differences in the velocity anisotropy of the satellite distribution. We find that the local velocity bias at all radii is greater than unity for all halos and this increases as we move closer to the halo centre, where it varies from 1.1 to 1.4. For the global velocity bias we find a small but slightly positive bias, although when we restrict the global velocity bias calculation to satellites that have had at least one orbit, the bias is essentially removed.Comment: 14 pages, 14 figures, MNRAS in pres

The Effect of Environment on Shear in Strong Gravitational Lenses

Using new photometric and spectroscopic data in the fields of nine strong gravitational lenses that lie in galaxy groups, we analyze the effects of both the local group environment and line-of-sight galaxies on the lens potential. We use Monte Carlo simulations to derive the shear directly from measurements of the complex lens environment, providing the first detailed independent check of the shear obtained from lens modeling. We account for possible tidal stripping of the group galaxies by varying the fraction of total mass apportioned between the group dark matter halo and individual group galaxies. The environment produces an average shear of gamma = 0.08 (ranging from 0.02 to 0.17), significant enough to affect quantities derived from lens observables. However, the direction and magnitude of the shears do not match those obtained from lens modeling in three of the six 4-image systems in our sample (B1422, RXJ1131, and WFI2033). The source of this disagreement is not clear, implying that the assumptions inherent in both the environment and lens model approaches must be reconsidered. If only the local group environment of the lens is included, the average shear is gamma = 0.05 (ranging from 0.01 to 0.14), indicating that line-of-sight contributions to the lens potential are not negligible. We isolate the effects of various theoretical and observational uncertainties on our results. Of those uncertainties, the scatter in the Faber-Jackson relation and error in the group centroid position dominate. Future surveys of lens environments should prioritize spectroscopic sampling of both the local lens environment and objects along the line of sight, particularly those bright (I < 21.5) galaxies projected within 5' of the lens.Comment: Accepted for publication in The Astrophysical Journal; 28 pages, 9 figures, 5 table