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Register a new userThe dynamical structure of broken power-law and double power-law models for dark matter haloes
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Galaxy kinematics and gravitational lensing are two complementary ways to constrain the distribution of dark matter on galaxy scales. The typical dark matter density profiles adopted in dynamical studies cannot easily be adopted in lensing studies. Ideally, a mass model should be used that has the global characteristics of realistic dark matter distributions, and that allows for an analytical calculation of the magnifications and deflection angles. A simple model with these properties, the broken-power-law (BPL) model, has very recently been introduced. We examine the dynamical structure of the family of BPL models. We derive simple closed expressions for basic dynamical properties, and study the distribution function under the assumption of velocity isotropy. We find that none of the BPL models with realistic parameters has an isotropic distribution function that is positive over the entire phase space, implying that the BPL models cannot be supported by an isotropic velocity distribution, or models with a more radially anisotropic orbital structure. This result limits the attractiveness of the BPL family as a tool for lensing studies to some degree. More generally, we find that not all members of the general family of double power-law or Zhao models, often used to model dark matter haloes, can be supported by an isotropic or radially anisotropic distribution function. In other words, the distribution function may become negative even for spherically symmetric models with a well-behaved density profile.
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Simple but flexible dynamical models are useful for many purposes, including serving as the starting point for more complex models or numerical simulations of galaxies, clusters, or dark matter haloes. We present SpheCow, a new light-weight and flexible code that allows one to easily explore the structure and dynamics of any spherical model. Assuming an isotropic or Osipkov-Merritt anisotropic orbital structure, the code can automatically calculate the dynamical properties of any model with either an analytical density profile or an analytical surface density profile as starting point. We have extensively validated SpheCow using a combination of comparisons to analytical and high-precision numerical calculations, as well as the calculation of inverse formulae. SpheCow contains readily usable implementations for many standard models, including the Plummer, Hernquist, NFW, Einasto, Sersic, and Nuker models. The code is publicly available as a set of C++ routines and as a Python module, and it is designed to be easily extendable, in the sense that new models can be added in a straightforward way. We demonstrate this by adding two new families of models in which either the density slope or the surface density slope is described by an algebraic sigmoid function. We advocate the use of the SpheCow code to investigate the full dynamical structure for models for which the distribution function cannot be expressed analytically and to explore a much wider range of models than is possible using analytical models alone.
A large body of work based on collisionless cosmological N-body simulations going back over two decades has advanced the idea that collapsed dark matter haloes have simple and approximately universal forms for their mass density and pseudo-phase space density (PPSD) distributions. However, a general consensus on the physical origin of these results has not yet been reached. In the present study, we explore to what extent the apparent universality of these forms holds when we vary the initial conditions (i.e., the primordial power spectrum of density fluctuations) away from the standard CMB-normalised case, but still within the context of LCDM with a fixed expansion history. Using simulations that vary the initial amplitude and shape, we show that the structure of dark matter haloes retains a clear memory of the initial conditions. Specifically, increasing (lowering) the amplitude of fluctuations increases (decreases) the concentration of haloes and, if pushed far enough, the density profiles deviate strongly from the NFW form that is a good approximation for the CMB-normalised case. Although, an Einasto form works well. Rather than being universal, the slope of the PPSD (or pseudo-entropy) profile steepens (flattens) with increasing (decreasing) power spectrum amplitude and can exhibit a strong halo mass dependence. Our results therefore indicate that the previously identified universality of the structure of dark matter haloes is mostly a consequence of adopting a narrow range of (CMB-normalised) initial conditions for the simulations. Our new suite provides a useful test-bench against which physical models for the origin of halo structure can be validated.
Biorthonormal basis function expansions are widely used in galactic dynamics, both to study problems in galactic stability and to provide numerical algorithms to evolve collisionless stellar systems. They also provide a compact and efficient description of the structure of numerical dark matter haloes in cosmological simulations. We present a two-parameter family of biorthonormal double-power-law potential-density expansions. Both the potential and density are given in closed analytic form and may be rapidly computed via recurrence relations. We show that this family encompasses all the known analytic biorthonormal expansions: the Zhao expansions (themselves generalizations of ones found earlier by Hernquist & Ostriker and by Clutton-Brock) and the recently discovered Lilley, Sanders, Evans & Erkal expansion. Our new two-parameter family includes expansions based around many familiar spherical density profiles as zeroth-order models, including the $gamma$ models and the Jaffe model. It also contains a basis expansion that reproduces the famous Navarro-Frenk-White (NFW) profile at zeroth order. The new basis expansions have been found via a systematic methodology which has wide applications in finding further examples. In the process, we also uncovered a novel integral transform solution to Poissons equation.
The detection of microlensing has opened the way for the development of new methods in galactic astronomy. This series of papers investigates what microlensing can teach us about the structure and shape of the dark halo. In this paper we present formulas for the microlensing rate, optical depth and event duration distributions for a simple set of axisymmetric disk-halo models. The halos are based on the power--law models which have simple velocity distributions. Using these models, we show that there is a large uncertainty in the predicted microlensing rate because of uncertainty in the halo parameters. For example, models which reproduce the measured galactic observables to within their errors still differ in microlensing rate towards the Magellanic Clouds by more than a factor of ten. We find that while the more easily computed optical depth correlates well with microlensing rate, the ratio of optical depth to rate can vary by a factor of two (or greater if the disk is maximal). Comparison of microlensing rates towards the Large and Small Magellanic Clouds (LMC and SMC) and M31 can be used to aid determinations of the halo flattening and rotation curve slope. For example, the ratio of microlensing rates towards the LMC and SMC is $sim 0.7-0.8$ for E0 halos and $sim 1.0 - 1.2$ for E7 halos (c.f. Sackett & Gould 1993). Once the flattening has been established, the ratio of microlensing rates towards M31 and the LMC may help to distinguish between models with rising, flat or falling rotation curves. Comparison of rates along LMC and galactic bulge lines-of-sight gives useful information on the halo core radius, although this may not be so easy to extract in practice. Maximal disk models provide substantially smaller halo optical depths, shorter event durations and even larger model uncertainties.
We study a one-dimensional particles system, in the overdamped limit, where nearest particles attract with a force inversely proportional to a power of their distance and coalesce upon encounter. The detailed shape of the distribution function for the gap between neighbouring particles serves to discriminate between different laws of attraction. We develop an exact Fokker-Planck approach for the infinite hierarchy of distribution functions for multiple adjacent gaps and solve it exactly, at the mean-field level, where correlations are ignored. The crucial role of correlations and their effect on the gap distribution function is explored both numerically and analytically. Finally, we analyse a random input of particles, which results in a stationary state where the effect of correlations is largely diminished.
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