Individual Halos Research Articles

Where do galaxies end? Comparing measurement techniques of hydrodynamic-simulation galaxies’ integrated properties

Using the suite of high-resolution zoom re-simulations of individual haloes by Martig et al., and the large-scale simulation \emph{MassiveBlack-II}, we examine the differences in measured galaxy properties from techniques with various aperture definitions of where galaxies end. We perform techniques popular in the literature and present a new technique of our own, where the aperture radius is based on the baryonic mass profiles of simulated (sub)haloes. For the average Milky-Way-mass system, we find the two most popular techniques in the literature return differences of order 30 per cent for stellar mass, a factor of 3 for gas mass, 40 per cent for star formation rate, and factors of several for gas accretion and ejection rates. Individual cases can show variations greater than this, with the severity dependent on the concentration of a given system. The average difference in integrated properties for a more general galaxy population are not as striking, but are still significant for stellar and gas mass, especially for optical-limit apertures. The large differences that can occur are problematic for comparing results from various publications. We stress the importance of both defining and justifying a technique choice and discourage using popular apertures that use an exact fraction of the virial radius, due to the unignorable variation in galaxy-to-(sub)halo size. Finally, we note that technique choice does not greatly affect simulated galaxies from lying within the scatter of observed scaling relations, but it can alter the derived best-fit slope for the Kennicutt-Schmidt relation.

DETECTION OF ULTRAVIOLET HALOS AROUND HIGHLY INCLINED GALAXIES

We report the discovery of diffuse ultraviolet light around late-type galaxies out to 5-20 kpc from the midplane using Swift and GALEX images. The emission is consistent with the stellar outskirts in the early-type galaxies but not in the late-type galaxies, where the emission is quite blue and consistent with a reflection nebula powered by light escaping from the galaxy and scattering off dust in the halo. Our results agree with expectations from halo dust discovered in extinction by Menard et al. (2010) to within a few kpc of the disk and imply a comparable amount of hot and cold gas in galaxy halos (a few x 10^8 Msun within 20 kpc) if the dust resides primarily in Mg II absorbers. The results also highlight the potential of UV photometry to study individual galaxy halos.

REIONIZATION HISTORIES OF MILKY WAY MASS HALOS

We investigate the connection between the epoch of reionization and the present day universe, by examining the extended mass reionization histories of dark matter halos identified at z=0. We combine an N-body dark matter simulation of a 600 Mpc volume with a three-dimensional, seminumerical reionization model. This provides reionization redshifts for each particle, which can then be connected with the properties of their halos at the present time. We find that the vast majority of present-day halos with masses larger than ~ few x 10^11 Msun reionize earlier than the rest of the universe. We also find significant halo-to-halo diversity in mass reionization histories, and find that in realistic inhomogenous models, the material within a given halo is not expected to reionize at the same time. In particular, the scatter in reionization times within individual halos is typically larger than the scatter among halos. From our fiducial reionization model, we find that the typical 68% scatter in reionization times within halos is ~ 115 Myr for 10^(12 \pm 0.25) Msun halos, decreasing slightly to ~ 95 Myr for 10^(15 \pm 0.25) Msun halos. We find a mild correlation between reionization history and environment: halos with shorter reionization histories are typically in more clustered environments, with the strongest trend on a scale of ~ 20 Mpc. Material in Milky Way mass halos with short reionization histories is preferentially reionized in relatively large HII regions, implying reionization mostly by sources external to the progenitors of the present-day halo. We investigate the impact on our results of varying the reionization model parameters, which span a range of reionization scenarios with varying timing and morphology.

The imprint of dark matter haloes on the size and velocity dispersion evolution of early-type galaxies

Early-type galaxies (ETGs) are observed to be more compact, on average, at $z \gtrsim 2$ than at $z\simeq 0$, at fixed stellar mass. Recent observational works suggest that such size evolution could reflect the similar evolution of the host dark matter halo density as a function of the time of galaxy quenching. We explore this hypothesis by studying the distribution of halo central velocity dispersion ($\sigma_0$) and half-mass radius ($r_{\rm h}$) as functions of halo mass $M$ and redshift $z$, in a cosmological $\Lambda$-CDM $N$-body simulation. In the range $0\lesssim z\lesssim 2.5$, we find $\sigma_0\propto M^{0.31-0.37}$ and $r_{\rm h}\propto M^{0.28-0.32}$, close to the values expected for homologous virialized systems. At fixed $M$ in the range $10^{11} M_\odot \lesssim M\lesssim 5.5 \times 10^{14} M_\odot$ we find $\sigma_0\propto(1+z)^{0.35}$ and $r_{\rm h}\propto(1+z)^{-0.7}$. We show that such evolution of the halo scaling laws is driven by individual haloes growing in mass following the evolutionary tracks $\sigma_0\propto M^{0.2}$ and $r_{\rm h}\propto M^{0.6}$, consistent with simple dissipationless merging models in which the encounter orbital energy is accounted for. We compare the $N$-body data with ETGs observed at $0\lesssim z\lesssim3$ by populating the haloes with a stellar component under simple but justified assumptions: the resulting galaxies evolve consistently with the observed ETGs up to $z \simeq 2$, but the model has difficulty reproducing the fast evolution observed at $z\gtrsim 2$. We conclude that a substantial fraction of the size evolution of ETGs can be ascribed to a systematic dependence on redshift of the dark matter haloes structural properties.

Stellar haloes in Milky Way mass galaxies: from the inner to the outer haloes

We present a comprehensive study of the chemical properties of the stellar haloes of Milky-Way mass galaxies, analysing the transition between the inner to the outer haloes. We find the transition radius between the relative dominance of the inner-halo and outer-halo stellar populations to be ~15-20 kpc for most of our haloes, similar to that inferred for the Milky Way from recent observations. While the number density of stars in the simulated inner-halo populations decreases rapidly with distance, the outer-halo populations contribute about 20-40 per cent in the fiducial solar neighborhood, in particular at the lowest metallicities. We have determined [Fe/H] profiles for our simulated haloes; they exhibit flat or mild gradients, in the range [-0.002, -0.01 ] dex/kpc. The metallicity distribution functions exhibit different features, reflecting the different assembly history of the individual stellar haloes. We find that stellar haloes formed with larger contributions from massive subgalactic systems have steeper metallicity gradients. Very metal-poor stars are mainly contributed to the halo systems by lower-mass satellites. There is a clear trend among the predicted metallicity distribution functions that a higher fraction of low-metallicity stars are found with increasing radius. These properties are consistent with the range of behaviours observed for stellar haloes of nearby galaxies.

How to zoom: bias, contamination and Lagrange volumes in multimass cosmological simulations

We perform a suite of multimass cosmological zoom simulations of individual dark matter halos and explore how to best select Lagrangian regions for resimulation without contaminating the halo of interest with low-resolution particles. Such contamination can lead to significant errors in the gas distribution of hydrodynamical simulations, as we show. For a fixed Lagrange volume, we find that the chance of contamination increases systematically with the level of zoom. In order to avoid contamination, the Lagrangian volume selected for resimulation must increase monotonically with the resolution difference between parent box and the zoom region. We provide a simple formula for selecting Lagrangian regions (in units of the halo virial volume) as a function of the level of zoom required. We also explore the degree to which a halo's Lagrangian volume correlates with other halo properties (concentration, spin, formation time, shape, etc.) and find no significant correlation. There is a mild correlation between Lagrange volume and environment, such that halos living in the most clustered regions have larger Lagrangian volumes. Nevertheless, selecting halos to be isolated is not the best way to ensure inexpensive zoom simulations. We explain how one can safely choose halos with the smallest Lagrangian volumes, which are the least expensive to resimulate, without biasing one's sample.

Exploring the non-linear density field in the Millennium Simulations with tessellations – I. The probability distribution function

We use the Delaunay Tessellation Field Estimator (DTFE) to study the one-point density distribution functions of the Millennium (MS) and Millennium-II (MS-II) simulations. The DTFE technique is based directly on the particle positions, without requiring any type of smoothing or analysis grid, thereby providing high sensitivity to all non-linear structures resolved by the simulations. In order to identify the detailed origin of the shape of the one-point density probability distribution function (PDF), we decompose the simulation particles according to the mass of their host FoF halos, and examine the contributions of different halo mass ranges to the global density PDF. We model the one-point distribution of the FoF halos in each halo mass bin with a set of Monte Carlo realizations of idealized NFW dark matter halos, finding that this reproduces the measurements from the N-body simulations reasonably well, except for a small excess present in simulation results. This excess increases with increasing halo mass. We show that its origin lies in substructure, which becomes progressively more abundant and better resolved in more massive dark matter halos. We demonstrate that the high density tail of the one-point distribution function in less massive halos is severely affected by the gravitational softening length and the mass resolution. In particular, we find these two parameters to be more important for an accurate measurement of the density PDF than the simulated volume. Combining our results from individual halo mass bins we find that the part of the one-point density PDF originating from collapsed halos can nevertheless be quite well described by a simple superposition of a set of NFW halos with the expected cosmological abundance over the resolved mass range. The transition region to the low-density unbound material is however not well captured by such an analytic halo model.

Dark matter in disc galaxies – I. A Markov Chain Monte Carlo method and application to DDO 154

We present a new method to constrain the dark matter halo density profiles of disk galaxies. Our algorithm employs a Markov Chain Monte Carlo (MCMC) approach to explore the parameter space of a general family of dark matter profiles. We improve upon previous analyses by considering a wider range of halo profiles and by explicitly identifying cases in which the data are insufficient to break the degeneracies between the model parameters. We demonstrate the robustness of our algorithm using artificial data sets and show that reliable estimates of the halo density profile can be obtained from data of comparable quality to those currently available for low surface brightness (LSB) galaxies. We present our results in terms of physical quantities which are constrained by the data, and find that the logarithmic slope of the halo density profile at the radius of the innermost data point of a measured rotation curve can be strongly constrained in LSB ([Vstar/Vobs]max ~ 0.16) galaxies. High surface brightness galaxies ([Vstar/Vobs]max ~ 0.79) require additional information on the mass-to-light ratio of the stellar population - our approach naturally identifies those galaxies for which this is necessary. We apply our method to observed data for the dwarf irregular galaxy DDO 154 and recover a logarithmic halo slope of -0.39 +- 0.11 at a radius of 0.14 kpc. Our analysis validates earlier estimates which were based on the fitting of a limited set of individual halo models, but constitutes a more robust constraint than was possible using other techniques since it marginalises over a wide range of halo profiles.

Digital Filtering and Shape Identification in the Analysis of WAXD Patterns of Semicrystalline Polymers

Most frequently, the degree of crystallinity of polymers is determined using Wide Angle X-ray Diffraction (WAXD) technique. The method consists in the resolution of WAXD diffraction curve of a polymer into individual crystalline peaks and amorphous halo. This work presents a procedure, which was elaborated to help in a quick determination of the angular positions of crystalline peaks present in the diffraction curve of investigated polymer. The positions of peaks are determined using numerical differentiation. Using these data the computer program WAXSFIT identifies investigated polymer and prepares a set of starting parameters which are used in the calculations of the degree of crystallinity.

The spatial distribution of galactic satellites in the Λ cold dark matter cosmology

We investigate the spatial distribution of galactic satellites in high resolution simulations of structure formation in the LCDM model: the Aquarius dark matter simulations of individual halos and the Millennium II simulation of a large cosmological volume. To relate the simulations to observations of the Milky Way we use two alternative models to populate dark halos with "visible" galaxies: a semi-analytic model of galaxy formation and an abundance matching technique. We find that the radial density profile of massive satellites roughly follows that of the dark matter halo (unlike the distribution of dark matter subhalos). Furthermore, our two galaxy formation models give results consistent with the observed profile of the 11 classical satellites of the Milky Way. Our simulations predict that larger, fainter samples of satellites should still retain this profile at least up to samples of 100 satellites. The angular distribution of the classical satellites of the Milky Way is known to be highly anisotropic. Depending on the exact measure of flattening, 5--10 per cent of satellite systems in our simulations are as flat as the Milky Way's and this fraction does not change when we correct for possible obscuration of satellites by the Galactic disk. A moderate flattening of satellite systems is a general property of LCDM, best understood as the consequence of preferential accretion along filaments of the cosmic web. Accretion of a single rich group of satellites can enhance the flattening due to such anisotropic accretion. We verify that a typical Milky Way-mass CDM halo does not acquire its 11 most massive satellites from a small number of rich groups. Single--group accretion becomes more likely for less massive satellites. Our model predictions should be testable with forthcoming studies of satellite systems in other galaxies and surveys of fainter satellites in the Milky Way.

WARM-HOT GAS IN GROUPS AND GALAXIES TOWARD H2356-309

We present a detailed analysis of the galaxy and group distributions around three reported X-ray absorption line systems in the spectrum of the quasar H2356-309. Previous studies associated these absorbers with known large-scale galaxy structures (i.e., walls and filaments) along the line of sight. Such absorption lines typically trace 10^{5-7} K gas, and may be evidence of the elusive warm-hot intergalactic medium (WHIM) thought to harbor the bulk of the low-redshift "missing baryons;" alternatively, they may be linked to individual galaxies or groups in the filaments. Here we combine existing galaxy survey data with new, multi-object Magellan spectroscopy to investigate the detailed galaxy distribution near each absorber. All of these three absorption systems are within the projected virial radii of nearby galaxies and/or groups, and could therefore arise in these virialized structures rather than (or in addition to) the WHIM. However, we find no additional galaxies near a fourth "void" absorber recently found in the spectrum, suggesting that this system may indeed trace gas unassociated with any individual halo. Though the number of known systems is still small, spatial coincidences suggest that some X-ray absorbers lie in galaxy and/or group environments, though others could still trace the large-scale filamentary WHIM gas predicted by simulations.

Numerical simulations of the dark universe: State of the art and the next decade

ABSTRACT We present a review of the current state of the art of cosmological dark matter simulations, with particular emphasis on the implications for dark matter detection efforts and studies of dark energy. This review is intended both for particle physicists, who may find the cosmological simulation literature opaque or confusing, and for astro-physicists, who may not be familiar with the role of simulations for observational and experimental probes of dark matter and dark energy. Our work is complementary to the contribution by Baldi in this issue, which focuses on the treatment of dark energy and cosmic acceleration in dedicated N-body simulations. Truly massive dark matter-only simulations are being conducted on national supercomputing centers, employing from several billion to over half a trillion particles to simulate the formation and evolution of cosmologically representative volumes (cosmic scale) or to zoom in on individual halos (cluster and galactic scale). These simulations cost millions of core-hours, require tens to hundreds of terabytes of memory, and use up to petabytes of disk storage. Predictions from such simulations touch on almost every aspect of dark matter and dark energy studies, and we give a comprehensive overview of this connection. We also discuss the limitations of the cold and collisionless DM-only approach, and describe in some detail efforts to include different particle physics as well as baryonic physics in cosmological galaxy formation simulations, including a discussion of recent results highlighting how the distribution of dark matter in halos may be altered. We end with an outlook for the next decade, presenting our view of how the field can be expected to progress.

GALAXY-MASS CORRELATIONS ON 10 Mpc SCALES IN THE DEEP LENS SURVEY

We examine the projected correlation of galaxies with mass from small scales (<few hundred kpc) where individual dark matter halos dominate, out to 15 Mpc where correlated large-scale structure dominates. We investigate these profiles as a function of galaxy luminosity and redshift. Selecting 0.8 million galaxies in the Deep Lens Survey, we use photometric redshifts and stacked weak gravitational lensing shear tomography out to radial scales of 1 degree from the centers of foreground galaxies. We detect correlated mass density from multiple halos and large-scale structure at radii larger than the virial radius, and find the first observational evidence for growth in the galaxy-mass correlation on 10 Mpc scales with decreasing redshift and fixed range of luminosity. For a fixed range of redshift, we find a scaling of projected halo mass with rest-frame luminosity similar to previous studies at lower redshift. We control systematic errors in shape measurement and photometric redshift, enforce volume completeness through absolute magnitude cuts, and explore residual sample selection effects via simulations.

WHAT DO DARK MATTER HALO PROPERTIES TELL US ABOUT THEIR MASS ASSEMBLY HISTORIES?

Individual dark matter halos in cosmological simulations vary widely in their detailed structural properties such as shape, rotation, substructure and degree of internal relaxation. Recent non-parametric (principal component) analyses suggest that a few principal components explain a large fraction of the scatter in halo properties. The main principal component is closely linked with concentration, which in turn is known to be related to the mass accretion history of the halo. Here we examine more generally the connection between mass accretion history and structural parameters. The space of mass accretion histories has principal components of its own. We find that the strongest two can be interpreted as the overall age of the halo and the acceleration or deceleration of growth at late times. These two components only account for $\sim70$%\ of the scatter in mass accretions histories however, due to the stochastic effect of major mergers. Relating structural parameters to formation history, we find that concentration correlates strongly with the early history of the halo, while relaxation correlates with the late history. We examine the inferences about formation history that can be drawn by splitting haloes into subsamples, based on observable properties such as concentration and shape at some final time. This approach suggests interesting possibilities, such as the possibility of defining young and old samples of galaxy clusters in a rigorous, quantitative way, or testing the dynamical assumptions of galaxy formation models empirically.

Moving mesh cosmology: numerical techniques and global statistics

We present the first hydrodynamical simulations of structure formation using the new moving mesh code AREPO and compare the results with GADGET simulations based on a traditional smoothed particle hydrodynamics (SPH) technique. The two codes share the same Tree-PM gravity solver and include identical sub-resolution physics, but employ different methods to solve the equations of hydrodynamics. This allows us to assess the impact of hydro-solver uncertainties on the results of cosmological studies of galaxy formation. We focus on predictions for global baryon statistics, such as the cosmic star formation rate density, after we introduce our simulation suite and numerical methods. Properties of individual galaxies and haloes are examined by Keres et al. (2011), while a third paper by Sijacki et al. (2011) uses idealised simulations to analyse the differences between the hydrodynamical schemes. We find that the global baryon statistics differ significantly between the two simulation approaches. AREPO shows higher star formation rates at late times, lower mean temperatures, and different gas mass fractions in characteristic phases of the intergalactic medium, in particular a reduced amount of hot gas. Although both codes use the same implementation of cooling, more gas cools out of haloes in AREPO compared with GADGET towards low redshifts. We show that this is caused by a higher heating rate with SPH in the outer parts of haloes, owing to viscous dissipation of SPH's inherent sonic velocity noise and SPH's efficient damping of subsonic turbulence injected in the halo infall region, and because of a higher efficiency of gas stripping in AREPO. As a result of such differences, AREPO leads also to more disk-like morphologies compared to GADGET. Our results indicate that inaccuracies in hydrodynamic solvers can lead to comparatively large systematic differences.

A FIRST LOOK AT GALAXY FLYBY INTERACTIONS. I. CHARACTERIZING THE FREQUENCY OF FLYBYS IN A COSMOLOGICAL CONTEXT

Hierarchical structure formation theory is based on the notion that mergers drive galaxy evolution, so a considerable framework of semi-analytic models and N-body simulations has been constructed to calculate how mergers transform a growing galaxy. However, galaxy mergers are only one type of major dynamical interaction between halos -- another class of encounter, a close flyby, has been largely ignored. We use cosmological N-body simulations to reconstruct the entire dynamical interaction history of dark matter halos. We present a careful method of identifying and tracking a dark matter halo which resolves the typical classes of anomalies that occur in N-body data. This technique allows us to robustly follow halos and several hierarchical levels of subhalos as they grow, dissolve, merge, and flyby one another -- thereby constructing both a census of the dynamical interactions in a volume and an archive of the dynamical evolution of an individual halo. In addition to a census of mergers, our tool characterizes the frequency of close flyby interactions in the Universe. We find that the number of close flyby interactions is comparable to, or even surpasses, the number of mergers for halo masses $\gtrsim 10^{11}\,\Msun$ at $z \lesssim 2$. Halo flybys occur so frequently to high mass halos that they are continually perturbed, unable to reach a dynamical equilibrium. In particular, we find that Milky Way type halos undergo a similar number of flybys as mergers irrespective of mass-ratio for $z\lesssim 2$. We also find tentative evidence that at high redshift, $z \gtrsim 14$, flybys are as frequent as mergers. Our results suggest that close halo flybys can play an important role in the evolution of the earliest dark matter halos and their galaxies, and can still influence galaxy evolution at the present epoch. [Abridged]

The spatial and velocity bias of linear density peaks and protohaloes in the Λ cold dark matter cosmology

We use high resolution N-body simulations to investigate the Lagrangian bias of cold dark matter haloes within the LCDM cosmology. Our analysis focuses on "proto-haloes", which we identify in the simulation initial conditions with the subsets of particles belonging to individual redshift-zero haloes. We then calculate the number-density and velocity-divergence fields of proto-haloes and estimate their auto spectral densities. We also measure the corresponding cross spectral densities with the linear matter distribution. We use our results to test a Lagrangian-bias model presented by Desjacques and Sheth which is based on the assumption that haloes form out of local density maxima of a specific height. Our comparison validates the predicted functional form for the scale-dependence of the bias for both the density and velocity fields. We also show that the bias coefficients are accurately predicted for the velocity divergence. On the contrary, the theoretical values for the density bias parameters do not accurately match the numerical results as a function of halo mass. This is likely due to the simplistic assumptions that relate virialized haloes to density peaks of a given height in the model. We also detect appreciable stochasticity for the Lagrangian density bias, even on very large scales. These are not included in the model at leading order but correspond to higher order corrections.

The shape of dark matter haloes in the Aquarius simulations: Evolution and memory

We use the high resolution cosmological N-body simulations from the Aquarius project to investigate in detail the mechanisms that determine the shape of Milky Way-type dark matter haloes. We find that, when measured at the instantaneous virial radius, the shape of individual haloes changes with time, evolving from a typically prolate configuration at early stages to a more triaxial/oblate geometry at the present day. This evolution in halo shape correlates well with the distribution of the infalling material: prolate configurations arise when haloes are fed through narrow filaments, which characterizes the early epochs of halo assembly, whereas triaxial/oblate configurations result as the accretion turns more isotropic at later times. Interestingly, at redshift z = 0, clear imprints of the past history of each halo are recorded in their shapes at different radii, which also exhibit a variation from prolate in the inner regions to triaxial/oblate in the outskirts. Provided that the Aquarius haloes are fair representatives of Milky Way-like 1012M☉ objects, we conclude that the shape of such dark matter haloes is a complex, time-dependent property, with each radial shell retaining memory of the conditions at the time of collapse.

Galaxy formation in semi-analytic models and cosmological hydrodynamic zoom simulations

We present a detailed comparison between numerical cosmological hydrodynamic zoom simulations and semi-analytic models (SAMs) run within merger trees extracted from the simulations. The high-resolution simulations represent 48 individual halos with virial masses in the range 2.4*10^11M_sun < M_Halo < 3.3*10^13M_sun. They include radiative H & He cooling, photo-ionization, star formation and thermal SN feedback. We compare with different SAM versions including only this complement of physical processes, and also ones including supernova driven winds, metal cooling, and feedback from AGN. Our analysis is focused on the cosmic evolution of the baryon content in galaxies and its division into various components (stars, cold gas, and hot gas). Both the SAMs and simulations are compared with observational relations between halo mass and stellar mass, and between stellar mass and star formation rate, at low and high redshift. The simulations turn out to have much higher star formation efficiencies (by about a factor of ten) than the SAMs. Therefore the cold gas is consumed much more rapidly in the simulations and stars form much earlier. Also, simulations show a transition between stellar mass growth that is dominated by in situ formation of stars to growth that is predominantly through accretion of stars formed in external galaxies. In SAMs, stellar growth is always dominated by in situ star formation. In addition, SAMs overpredict the overall gas accretion rates relative to the simulations, and overestimate the fraction of "hot" relative to "cold" accretion. We discuss the reasons for these discrepancies, and identify several physical processes that are missing in our SAM. We also highlight physical processes that are neglected in the simulations studied here, but which appear to be crucial in order to understand the properties of real galaxies.

Probing the shape and history of the Milky Way halo with orbital spectral analysis

The phase space coordinates of individual halo stars obtained by Galactic surveys enable the computation of their full 3-dimensional orbits. Spectral analysis of halo orbits can be used to construct "frequency maps" which provide a compact representation of the 6-dimensional phase space distribution function. Frequency maps identify important major orbit families, and the orbital abundances reflect the shape and orientation of the dark matter halo relative to the disk. We apply spectral analysis to halo orbits in a series of controlled simulations of disk galaxies. Although the shape of the simulated halo varies with radius, frequency maps of local samples of halo orbits confined to the inner halo contain most of the information about the global shape of the halo and its major orbit families. Quiescent or adiabatic disk formation results in significant trapping of halo orbits in resonant orbit families (i.e. orbits with commensurable frequencies). If a good estimate of the Galactic potential in the inner halo (within ~50kpc) is available, the appearance of strong, stable resonances in frequency maps of halo orbits will allow us to determine the degree of resonant trapping induced by the disk potential. The locations and strengths of these resonant families are determined both by the global shape of the halo and its distribution function. Identification of such resonances in the Milky Way's stellar halo would therefore provide evidence of an extended period of adiabatic disk growth. If the Galactic potential is not known exactly, a measure of the diffusion rate of large sample of 10^4 halo orbits can help distinguish between the true potential and an incorrect potential. The orbital spectral analysis methods described in this paper provide a strong complementarity to existing methods for constraining the potential of the Milky Way halo and its stellar distribution function (ABRIDGED).