# Is the halo virial mass based on only dark matter mass or also include baryonic mass?

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Dark matter-only N-body simulations are ubiquitous and there exist many algorithms/codes to identify and measure the properties of dark matter halos in these simulations. One basic quantity describing dark matter halos is the virial mass (which itself can be defined in many ways, usually by finding the radius within which the mass density equals some number times the current mean cosmological density).

My question is: how can we trust the virial mass (or radius) measured in dark matter-only simulations, when they do not account for the presence of baryons? In principle, the virial mass should be the total gravitating halo mass: DM+baryons. Wouldn't we expect the radius within which the density equals some pre-defined number to change if we also had baryons distributed around dark matter overdensities? How do "halo finding" codes get around this problem when run on dark matter-only simulations?

More practically: is the "virial mass" of halos reported by halo finding codes only the dark matter mass, or the "total" (DM+baryons) mass? If the latter, I guess these codes have to assume something under the hood -- for example that every dark matter particle in the N-body simulation has associated with it the universal baryon fraction $$f_bsim15\%$$.

## Is the halo virial mass based on only dark matter mass or also include baryonic mass? - Astronomy

Astrophysical observations ranging from the scale of the horizon (∼ 15,000 Mpc) to the typical spacing between galaxies (∼ 1 Mpc) are all consistent with a Universe that was seeded by a nearly scale-invariant fluctuation spectrum and that is dominated today by dark energy (∼ 70 %) and Cold Dark Matter (∼ 25%), with baryons contributing only ∼ 5% to the energy density (Planck Collaboration et al. 2016, Guo et al. 2016). This cosmological model has provided a compelling backbone to galaxy formation theory, a field that is becoming increasingly successful at reproducing the detailed properties of galaxies, including their counts, clustering, colors, morphologies, and evolution over time (Vogelsberger et al. 2014, Schaye et al. 2015). As described in this review, there are observations below the scale of ∼ 1 Mpc that have proven more problematic to understand in the 𹯍M framework. It is not yet clear whether the small-scale issues with 𹯍M will be accommodated by a better understanding of astrophysics or dark matter physics, or if they will require a radical revision of cosmology, but any correct description of our Universe must look very much like 𹯍M on large scales. It is with this in mind that we discuss the small-scale challenges to the current paradigm. For concreteness, we assume that the default 𹯍M cosmology has parameters h = H0 / (100 km s 𢄡 Mpc 𢄡 ) = 0.6727, Ωm = 0.3156, ΩΛ = 0.6844, Ωb = 0.04927, σ8 = 0.831, and ns = 0.9645 (Planck Collaboration et al. 2016).

Given the scope of this review, we must sacrifice detailed discussions for a more broad, high-level approach. There are many recent reviews or overview papers that cover, in more depth, certain aspects of this review. These include Frenk & White (2012), Peebles (2012), and Primack (2012) on the historical context of 𹯍M and some of its basic predictions Willman (2010) and McConnachie (2012) on searches for and observed properties of dwarf galaxies in the Local Group Feng (2010), Porter, Johnson & Graham (2011), and Strigari (2013) on the nature of and searches for dark matter Kuhlen, Vogelsberger & Angulo (2012) on numerical simulations of cosmological structure formation and Brooks (2014), Weinberg et al. (2015) and Del Popolo & Le Delliou (2017) on small-scale issues in 𹯍M. Additionally, we will not discuss cosmic acceleration (the Λ in 𹯍M) here that topic is reviewed in Weinberg et al. (2013). Finally, space does not allow us to address the possibility that the challenges facing 𹯍M on small scales reflects a deeper problem in our understanding of gravity. We point the reader to reviews by Milgrom (2002), Famaey & McGaugh (2012), and McGaugh (2015), which compare Modified Newtonian Dynamics (MOND) to 𹯍M and provide further references on this topic.

This is a review on small-scale challenges to the 𹯍M model. The past ∼ 12 years have seen transformative discoveries that have fundamentally altered our understanding of “small scales” – at least in terms of the low-luminosity limit of galaxy formation.

Prior to 2004, the smallest galaxy known was Draco, with a stellar mass of M ≃ 5 × 10 5 M. Today, we know of galaxies 1000 times less luminous. While essentially all Milky Way satellites discovered before 2004 were found via visual inspection of photographic plates (with the exceptions of the Carina and Sagittarius dwarf spheroidal galaxies), the advent of large-area digital sky surveys with deep exposures and accurate star-galaxy separation algorithms has revolutionized the search for and discovery of faint stellar systems in the Milky Way (see Willman 2010 for a review of the search for faint satellites). The Sloan Digital Sky Survey (SDSS) ushered in this revolution, doubling the number of known Milky Way satellites in the first five years of active searches. The PAndAS survey discovered a similar population of faint dwarfs around M31 (Richardson et al. 2011). More recently the DES survey has continued this trend (Koposov et al. 2015, Drlica-Wagner et al. 2015). All told, we know of ∼ 50 satellite galaxies of the Milky Way and ∼ 30 satellites of M31 today (McConnachie 2012, updated on-line catalog), most of which are fainter than any galaxy known at the turn of the century. They are also extremely dark-matter-dominated, with mass-to-light ratios within their stellar radii exceeding ∼ 1000 in some cases (Walker et al. 2009, Wolf et al. 2010).

Given this upheaval in our understanding of the faint galaxy frontier over the last decade or so, it is worth pausing to clarify some naming conventions. In what follows, the term 𠇍warf” will refer to galaxies with M ≲ 10 9 M. We will further subdivide dwarfs into three mass classes: Bright Dwarfs (M ≈ 10 7𢄩 M), Classical Dwarfs (M ≈ 10 5𢄧 M), and Ultra-faint Dwarfs (M ≈ 10 2𢄥 M). Note that another common classification for dwarf galaxies is between dwarf spheroidals (dSphs) and dwarf drregulars (dIrrs). Dwarfs with gas and ongoing star formation are usually labeled dIrr. The term dSph is reserved for dwarfs that lack gas and have no ongoing star formation. Note that the vast majority of field dwarfs (meaning that they are not satellites) are dIrrs. Most dSph galaxies are satellites of larger systems.

Figure 1 illustrates the morphological differences among galaxies that span these stellar mass ranges. From top to bottom we see three dwarfs each that roughly correspond to Bright, Classical, and Ultra-faint Dwarfs, respectively.

Classical Dwarfs: M ≈ 10 5𢄧 M
– the faintest galaxies known prior to SDSS

Ultra-faint Dwarfs: M ≈ 10 2𢄥 M
– detected within limited volumes around M31 and the Milky Way

With these definitions in hand, we move to the cosmological model within which we aim to explain the counts, stellar masses, and dark matter content of these dwarfs.

The 𹯍M model of cosmology is the culmination of century of work on the physics of structure formation within the framework of general relativity. It also indicates the confluence of particle physics and astrophysics over the past four decades: the particle nature of dark matter directly determines essential properties of non-linear cosmological structure. While the 𹯍M model is phenomenological at present – the actual physics of dark matter and dark energy remain as major theoretical issues – it is highly successful at explaining the large-scale structure of the Universe and basic properties of galaxies that form within dark matter halos.

In the 𹯍M model, cosmic structure is seeded by primordial adiabatic fluctuations and grows by gravitational instability in an expanding background. The primordial power spectrum as a function of wavenumber k is nearly scale-invariant 1 , P(k) ∝ k n with n ≃ 1. Scales that re-enter the horizon when the Universe is radiation-dominated grow extremely slowly until the epoch of matter domination, leaving a scale-dependent suppression of the primordial power spectrum that goes as k 𢄤 at large k. This suppression of power is encapsulated by the “transfer function” T(k), which is defined as the ratio of amplitude of a density perturbation in the post-recombination era to its primordial value as a function of perturbation wavenumber k. This processed power spectrum is the input for structure formation calculations the dimensionless processed power spectrum, defined by

therefore rises as k 4 for scales larger than the comoving horizon at matter-radiation equality (corresponding to k = 0.008 Mpc 𢄡 ) and is approximately independent of k for scales that re-enter the horizon well before matter-radiation equality. Here, d(a) is the linear growth function, normalized to unity at a = 1. The processed z = 0 (a = 1) linear power spectrum for 𹯍M is shown by the solid line in Figure 2. The asymptotic shape behavior is most easily seen in the bottom panel, which spans the wave number range of cosmological interest. For a more complete discussion of primordial fluctuations and the processed power spectrum we recommend that readers consult Mo, van den Bosch & White (2010).

It is useful to associate each wavenumber with a mass scale set by its characteristic length rl = λ / 2 = π / k. In the early Universe, when δ ≪ 1, the total amount of matter contained within a sphere of comoving Lagrangian radius rl at z = 0 is

The mapping between wave number and mass scale is illustrated by the top and bottom axis in Figure 2. The processed linear power spectrum for 𹯍M shown in the bottom panel (solid line) spans the horizon scale to a typical mass cutoff scale for the most common cold dark matter candidate (∼ 10 𢄦 M see discussion in Section 1.6). A line at Δ = 1 is plotted for reference, showing that fluctuations born on comoving length scales smaller than rl ≈ 10 h 𢄡 Mpc ≈ 14 Mpc have gone non-linear today. The top panel is zoomed in on the small scales of relevance for this review (which we define more precisely below). Typical regions on these scales have collapsed into virialized objects today. These collapsed objects – dark matter halos – are the sites of galaxy formation.

1.3.1. Global properties. Soon after overdense regions of the Universe become non-linear, they stop expanding, turn around, and collapse, converting potential energy into kinetic energy in the process. The result is virialized dark matter halos with masses given by

where Δ ∼ 300 is the virial over-density parameter, defined here relative to the background matter density. As discussed below, the value of Mvir is ultimately a definition that requires some way of defining a halo's outer edge (Rvir). This is done via a choice for Δ. The numerical value for Δ is often chosen to match the over-density one predicts for a virialized dark matter region that has undergone an idealized spherical collapse (Bryan & Norman 1998), and we will follow that convention here. Note that given a virial mass Mvir, the virial radius, Rvir, is uniquely defined by Equation 4. Similarly, the virial velocity

is also uniquely defined. The parameters Mvir, Rvir, and Vvir are equivalent mass labels – any one determines the other two, given a specified over-density parameter Δ.

 Galaxy Clusters: Mvir ≈ 10 15 M⊙ Vvir ≈ 1000 km s 𢄡 Milky Way Mvir ≈ 10 12 M⊙ Vvir ≈ 100 km s 𢄡 Smallest Dwarfs Mvir ≈ 10 9 M⊙ Vvir ≈ 10 km s 𢄡

One nice implication of Equation 4 is that a present-day object with virial mass Mvir can be associated directly with a linear perturbation with mass Ml. Equating the two gives

We see that a collapsed halo of size Rvir is approximately 7 times smaller in physical dimension than the comoving linear scale associated with that mass today.

With this in mind, Equations 3-6 allow us to self-consistently define “small scales” for both the linear power spectrum and collapsed objects: M ≲ 10 11 M. As we will discuss, potential problems associated with galaxies inhabiting halos with Vvir ≃ 50 km s 𢄡 may point to a power spectrum that is non-CDM-like at scales rl ≲ 1 Mpc.

 WE DEFINE “SMALL SCALES” AS THOSE SMALLER THAN:

As alluded to above, a common point of confusion is that the halo mass definition is subject to the assumed value of Δ, which can vary by a factor of ∼ 3 depending on the author. For the spherical collapse definition, Δ ≃ 333 at z = 0 (for our fiducial cosmology) and asymptotes to Δ = 178 at high redshift (Bryan & Norman 1998). Another common choice is a fixed Δ = 200 at all z (often labeled M200m in the literature). Finally, some authors prefer to define the virial overdensity as 200 times the critical density, which, according to Equation 4 would mean Δ(z) = 200 ρc(z)/ρm(z). Such a mass is commonly labeled “M200” in the literature. For most purposes (e.g., counting halos), the precise choice does not matter, as long as one is consistent with the definition of halo mass throughout an analysis: every halo has the same center, but its outer radius (and mass contained within that radius) shifts depending on the definition. In what follows, we use the spherical collapse definition (Δ = 333 at z = 0) and adhere to the convention of labeling that mass “Mvir”.

Before moving on, we note that it is also possible (and perhaps even preferable) to give a halo a “mass” label that is directly tied to a physical feature associated with a collapsed dark matter object rather than simply adopting a Δ. More, Diemer & Kravtsov (2015) have advocated the use of a “splash-back” radius , where the density profile shows a sharp break (this typically occurs at ∼ 2 Rvir). Another common choice is to tag halos based not on a mass but on Vmax, which is the peak value of the circular velocity Vc(r) = √ G M(< r) / r as one steps out from the halo center. For any individual halo, the value of Vmax (≳ Vvir) is linked to the internal mass profile or density profile of the system, which is the subject of the next subsection. As discussed below, the ratio Vmax / Vvir increases as the halo mass decreases.

ROBUST PREDICTIONS FROM CDM-ONLY SIMULATIONS
A defining characteristic of CDM-based hierarchical structure formation is that the smallest scales collapse first – a fact that arises directly from the shape of the power spectrum (Figure 1) and that lies at the heart of many robust predictions for the counts and structure of dark matter halos today. As discussed below, baryonic processes can alter these predictions to various degrees, but pure dark matter simulations have provided a well-defined set of basic predictions used to benchmark the theory.

The dark matter profiles of individual halos are cuspy and dense [Figure 3]
The density profiles of individual 𹯍M halos increase steadily towards small radii, with an overall normalization and detailed shape that reflects the halo's mass assembly. At fixed mass, early-forming halos tend to be denser than later-forming halos. As with the mass function, both the shape and normalization of dark matter halo density structure is predicted by 𹯍M, with a well-quantified prediction for the scatter in halo concentration at fixed mass.

There are many more small halos than large ones [Figure 4]
The comoving number density of dark matter halos rises steeply towards small masses, dn / dMM α with α ≃ 𢄡.9. At large halo masses, counts fall off exponentially above the mass scale that is just going non-linear today. Importantly, both the shape and normalization of the mass function is robustly predicted by the theory.

Substructure is abundant and almost self-similar [Figure 5]
Dark matter halos are filled with substructure, with a mass function that rises as dN / dmm αs with αs ≃ 𢄡.8 down to the low-mass free-streaming scale (m ≪ 1 M for canonical models). Substructure reflects the high-density cores of smaller merging halos that survive the hierarchical assembly process. Substructure counts are nearly self-similar with host mass, with the most massive subhalos seen at mmax ∼ 0.2 Mhost.

1.3.2. Abundance. In principle, the mapping between the initial spectrum of density fluctuations at z → ∞ and the mass spectrum of collapsed (virialized) dark matter halos at later times could be extremely complicated: as a given scale becomes non-linear, it could affect the collapse of nearby regions or larger scales. In practice, however, the mass spectrum of dark matter halos can be modeled remarkably well with a few simple assumptions. The first of these was taken by Press & Schechter (1974), who assumed that the mass spectrum of collapsed objects could be calculated by extrapolating the overdensity field using linear theory even into the highly non-linear regime and using a spherical collapse model (Gunn & Gott (1972)). In the Press-Schechter model, the dark matter halo mass function – the abundance of dark matter halos per unit mass per unit volume at redshift z, often written as n(M, z) – depends only on the rms amplitude of the linear dark matter power spectrum, smoothed using a spherical tophat filter in real space and extrapolated to redshift z using linear theory. Subsequent work has put this formalism on more rigorous mathematical footing (Bond et al. 1991, Cole 1991, Sheth, Mo & Tormen 2001), and this extended Press-Schechter (EPS) theory yields abundances of dark matter halos that are perhaps surprisingly accurate (see Zentner 2007 for a comprehensive review of EPS theory). This accuracy is tested through comparisons with large-scale numerical simulations.

Simulations and EPS theory both find a universal form for n(M, z): the comoving number density of dark matter halos is a power law with log slope of α ≃ 𢄡.9 for MM * and is exponentially suppressed for MM * , where M * = M * (z) is the characteristic mass of fluctuations going non-linear at the redshift z of interest 2 . Importantly, given an initial power spectrum of density fluctuations, it is possible to make highly accurate predictions within 𹯍M for the abundance, clustering, and merger rates of dark matter halos at any cosmic epoch.

1.3.3. Internal structure. Dubinski & Carlberg (1991) were the first to use N-body simulations to show that the internal structure of a CDM dark matter halo does not follow a simple power-law, but rather bends from a steep outer profile to a mild inner cusp obeying ρ(r) ∼ 1 / r at small radii. More than twenty years later, simulations have progressed to the point that we now have a fairly robust understanding of the structure of 𹯍M halos and the important factors that govern halo-to-halo variance (e.g., Navarro et al. 2010, Diemer & Kravtsov 2015, Klypin et al. 2016), at least for dark-matter-only simulations.

To first approximation, dark matter halo profiles can be described by a nearly universal form over all masses, with a steep fall-off at large radii transitioning to mildly divergent cusp towards the center. A common way to characterize this is via the NFW functional form (Navarro, Frenk & White 1997), which provides a good (but not perfect) description dark matter profiles:

Here, r𢄢 is a characteristic radius where the log-slope of the density profile is 𢄢, marking a transition point from the inner 1 / r cusp to an outer 1 / r 3 profile. The second parameter, ρ𢄢, sets the value of ρ(r) at r = r𢄢. In practice, dark matter halos are better described the three-parameter Einasto (1965) profile (Navarro et al. 2004, Gao et al. 2008). However, for the small halos of most concern for this review, NFW fits do almost as well as Einasto in describing the density profiles of halos in simulations (Dutton & Macciò 2014). Given that the NFW form is slightly simpler, we have opted to adopt this approximation for illustrative purposes in this review.

As Equation 7 makes clear, two parameters (e.g., ρ𢄢 and r𢄢) are required to determine a halo's NFW density profile. For a fixed halo mass Mvir (which fixes Rvir), the second parameter is often expressed as the halo concentration: c = Rvir / r𢄢. Together, a Mvirc combination completely specifies the profile. In the median, and over the mass and redshift regime of interest to this review, halo concentrations increase with decreasing mass and redshift: cMvir 𢄪 (1 + z) 𢄡 , with a ≃ 0.1 (Bullock et al. 2001). Though halo concentration correlates with halo mass, there is significant scatter (∼ 0.1 dex) about the median at fixed Mvir (Jing 2000, Bullock et al. 2001). Some fraction of this scatter is driven by the variation in halo mass accretion history (Wechsler et al. 2002, Ludlow et al. 2016), with early-forming halos having higher concentrations at fixed final virial mass.

The dependence of halo profile on a mass-dependent concentration parameter and the correlation between formation time and concentration at fixed virial mass are caused by the hierarchical build-up of halos in 𹯍M: low-mass halos assemble earlier, when the mean density of the Universe is higher, and therefore have higher concentrations than high-mass halos (e.g., Navarro, Frenk & White 1997, Wechsler et al. 2002). At the very smallest masses, the concentration-mass relation likely flattens, reflecting the shape of the dimensionless power spectrum (see our Figure 1 and the discussion in Ludlow et al. 2016) at the highest masses and redshifts, characteristic of very rare peaks, the trend seems to reverse (a < 0 Klypin et al. 2016).

The right panel of Figure 3 summarizes the median NFW density profiles for z = 0 halos with masses that span those of large galaxy clusters (Mvir = 10 15 M) to those of the smallest dwarf galaxies (Mvir = 10 8 M). We assume the cMvir relation from Klypin et al. 2016. These profiles are plotted in physical units (unscaled to the virial radius) in order to emphasize that higher mass halos are denser at every radius than lower mass halos (at least in the median). However, at a fixed small fraction of the virial radius, small halos are slightly denser than larger ones. This is a result of the concentration-mass relation. Under the ansatz of abundance matching (Section 1.5, Figure 6), galaxy sizes (half-mass radii) track a fixed fraction of their host halo virial radius: rgal ≃ 0.015 Rvir (Kravtsov 2013). This relation is plotted as a dotted line such that the dotted line intersects each solid line at that r = 0.015 Rvir, where Rvir is that particular halo’s virial radius. We see that small halos are slightly denser at the typical radii of the galaxies they host than are larger halos. Interestingly, however, the density range is remarkably small, with a local density of dark matter increasing by only a factor of ∼ 6 over the full mass range of halos that are expected to host galaxies, from the smallest dwarfs to the largest cD galaxies in the universe.

On the left we show the same halos, now presented in terms of the implied circular velocity curves: Vc ≡ √ GM(< r) / r . The dotted line in left panel intersects Vvir at Rvir for each value of Mvir. The dashed line does the same for Vmax and its corresponding radius Rmax. Higher mass systems, with lower concentrations, typically have VmaxVvir, but for smaller halos the ratio is noticeably different than one and can be as large as ∼ 1.5 for high-concentration outliers. Note also that the lowest mass halos have RmaxRvir and thus it is the value of Vmax (rather than Vvir) that is more closely linked to the observable 𠇏lat” region of a galaxy rotation curve. For our “small-scale” mass of Mvir = 10 11 M, typically Vmax ≃ 1.2 Vvir ≃ 60 km s 𢄡 .

It was only just before the turn of the century that N-body simulations set within a cosmological CDM framework were able to robustly resolve the substructure within individual dark matter halos (Ghigna et al. 1998, Klypin et al. 1999a). It soon became clear that the dense centers of small halos are able to survive the hierarchical merging process: dark matter halos should be filled with substructure. Indeed, subhalo counts are nearly self-similar with host halo mass. This was seen as welcome news for cluster-mass halos, as the substructure could be easily identified with cluster galaxies. However, as we will discuss in the next section, the fact that Milky-Way-size halos are filled with substructure is less clearly consistent with what we see around the Galaxy.

Quantifying subhalo counts, however, is not so straightforward. Counting by mass is tricky because the definition of “mass” for an extended distribution orbiting within a collapsed halo is even more fraught with subjective decisions than virial mass. When a small halo is accreted into a large one, mass is preferentially stripped from the outside. Typically, the standard virial overdensity �ge” is subsumed by the ambient host halo. One option is to compute the mass that is bound to the subhalo, but even these masses vary from halo finder to halo finder. The value of a subhalo's Vmax is better defined, and often serves as a good tag for quantifying halos.

Another option is to tag bound subhalos using the maximum virial mass that the halos had at the time they were first accreted 3 onto a host, Mpeak. This is a useful option because stars in a central galaxy belonging to a halo at accretion will be more tightly bound than the dark matter. The resultant satellite's stellar mass is most certainly more closely related to Mpeak than the bound dark matter mass that remains at z = 0. Moreover, the subsequent mass loss (and even Vmax evolution) could change depending on the baryonic content of the host because of tidal heating and other dynamical effects (D'Onghia et al. 2010). For these reasons, we adopt Msub = Mpeak for illustrative purposes here.

The magenta lines in Figure 4 show the median subhalo mass functions (Msub = Mpeak) for four characteristic host halo masses (Mvir = 10 12� M) according to the results of Rodríguez-Puebla et al. (2016). These lines are normalized to the right-hand vertical axis. Subhalos are counted only if they exist within the virial radius of the host, which means the counting volume increases as ∝ MvirRvir 3 for these four lines. For comparison, the black line (normalized to the left vertical axis) shows the global halo mass function (as estimated via the fitting function from Sheth, Mo & Tormen 2001). The subhalo mass function rises with a similar (though slightly shallower) slope as the field halo mass function and is also roughly self-similar in host halo mass.

How do we associate dark matter halos with galaxies? One simple approximation is to assume that each halo is allotted its cosmic share of baryons fb = Ωb / Ωm ≈ 0.15 and that those baryons are converted to stars with some constant efficiency є: M = є fb Mvir. Unfortunately, as shown in Figure 5, this simple approximation fails miserably. Galaxy stellar masses do not scale linearly with halo mass the relationship is much more complicated. Indeed, the goal of forward modeling galaxy formation from known physics within the 𹯍M framework is an entire field of its own (galaxy evolution Somerville & Davé 2015). Though galaxy formation theory has progressed significantly in the last several decades, many problems remain unsolved.

Other than forward modeling galaxy formation, there are two common approaches that give an independent assessment of how galaxies relate to dark matter halos. The first involves matching the observed volume density of galaxies of a given stellar mass (or other observable such as luminosity, velocity width, or baryon mass) to the predicted abundance of halos of a given virial mass. The second way is to measure the mass of the galaxy directly and to infer the dark matter halo properties based on this dynamical estimator.

1.5.1. Abundance matching. As illustrated in Figure 5, the predicted mass function of collapsed dark matter halos has a considerably different normalization and shape than the observed stellar mass function of galaxies. The difference grows dramatically at both large and small masses, with a maximum efficiency of є≃ 0.2 at the stellar mass scale of the Milky Way (M ≈ 10 10.75 M). This basic mismatch in shape has been understood since the earliest galaxy formation models set within the dark matter paradigm (White & Rees 1978) and is generally recognized as one of the primary constraints on feedback-regulated galaxy formation (White & Frenk 1991, Benson et al. 2003, Somerville & Davé 2015).

At the small masses that most concern this review, dark matter halo counts follow dn / dMM α with a steep slope αDM ≃ 𢄡.9 compared to the observed stellar mass function slope of αg = 𢄡.47 (Baldry et al. 2012, which is consistent with the updated GAMA results shown in Figure 5). Current surveys that cover enough sky to provide a global field stellar mass function reach a completeness limit of M ≈ 10 7.5 M. At this mass, galaxy counts are more than two orders of magnitude below the naive baryonic mass function fb Mvir. The shaded band illustrates how the stellar mass function would extrapolate to the faint regime spanning a range of faint-end slopes α that are marginally consistent with observations at the completeness limit.

Figure 6 allows us to read off the virial mass expectations for galaxies of various sizes. We see that Bright Dwarfs at the limit of detection in large sky surveys (M ≈ 10 8 M) are naively associated with Mvir ≈ 10 11 M halos. Galaxies with stellar masses similar to the Classical Dwarfs at M ≈ 10 6 M are associated with Mvir ≈ 10 10 M halos. As we will discuss in Section 3, galaxies at this scale with M/Mvir ≈ 10 𢄤 are at the critical scale where feedback from star formation may not be energetic enough to alter halo density profiles significantly. Finally, Ultra-faint Dwarfs with M ≈ 10 4 M, Mvir ≈ 10 9 M, and M / Mvir ≈ 10 𢄥 likely sit at the low-mass extreme of galaxy formation.

1.5.2. Kinematic Measures. An alternative way to connect to the dark matter halo hosting a galaxy is to determine the galaxy’s dark matter mass kinematically. This, of course, can only be done within a central radius probed by the baryons. For the small galaxies of concern for this review, extended mass measurements via weak lensing or hot gas emission is infeasible. Instead, masses (or mass profiles) must be inferred within some inner radius, defined either by the stellar extent of the system for dSphs and/or the outer rotation curves for rotationally-supported gas disks.

Bright dwarfs, especially those in the field, often have gas disks with ordered kinematics. If the gas extends far enough out, rotation curves can be extracted that extend as far as the flat part of the galaxy rotation curve Vflat. If care is taken to account for non-trivial velocity dispersions in the mass extraction (e.g., Kuzio de Naray, McGaugh & de Blok 2008), then we can associate VflatVmax.

Owing to the difficulty in detecting them, the faintest galaxies known are all satellites of the Milky Way or M31 and are dSphs. These lack rotating gas components, so rotation curve measurements are impossible. Instead, dSphs are primarily stellar dispersion-supported systems, with masses that are best probed by velocity dispersion measurements obtained star-by-star for the closest dwarfs (e.g., Walker et al. 2009, Simon et al. 2011, Kirby et al. 2014). For systems of this kind, the mass can be measured accurately within the stellar half-light radius (Walker et al. 2009). The mass within the de-projected (3D) half-light radius (r1/2) is relatively robust to uncertainties in the stellar velocity anisotropy and is given by M(< r1/2) = 3 σ 2 r1/2 / G, where σ is the measured, luminosity-weighted line-of-sight velocity dispersion (Wolf et al. 2010). This formula is equivalent to saying that the circular velocity at the half-light radius is V1/2 = Vc(r1/2 ) = √ 3 σ. The value of V1/2 (≤ Vmax) provides a one-point measurement of the host halo's rotation curve at r = r1/2.

Although the idea of �rk” matter had been around since at least Zwicky (1933), it was not until rotation curve measurements of galaxies in the 1970s revealed the need for significant amounts of non-luminous matter (Freeman 1970, Rubin, Thonnard & Ford 1978, Bosma 1978, Rubin, Ford & Thonnard 1980) that dark matter was taken seriously by the broader astronomical community (and shortly thereafter, it was recognized that dwarf galaxies might serve as sensitive probes of dark matter Aaronson 1983, Faber & Lin 1983, Lin & Faber 1983). Very quickly, particle physicists realized the potential implications for their discipline as well. Dark matter candidates were grouped into categories based on their effects on structure formation. “Hot” dark matter (HDM) particles remain relativistic until relatively late in the Universe’s evolution and smooth out perturbations even on super-galactic scales “warm” dark matter (WDM) particles have smaller initial velocities, become non-relativistic earlier, and suppress perturbations on galactic scales (and smaller) and CDM has negligible thermal velocity and does not suppress structure formation on any scale relevant for galaxy formation. Standard Model neutrinos were initially an attractive (hot) dark matter candidate by the mid-1980s, however, this possibility had been excluded on the basis of general phase-space arguments (Tremaine & Gunn 1979), the large-scale distribution of galaxies (White, Frenk & Davis 1983), and properties of dwarf galaxies (Lin & Faber 1983). The lack of a suitable Standard Model candidate for particle dark matter has led to significant work on particle physics extensions of the Standard Model. From a cosmology and galaxy formation perspective, the unknown particle nature of dark matter means that cosmologists must make assumptions about dark matters origins and particle physics properties and then investigate the resulting cosmological implications.

 Cold Dark Matter (CDM) m ∼ 100 GeV, vth z=0 ≈ 0 km s 𢄡 Warm Dark Matter (WDM) m ∼ 1 keV, vth z=0 ∼ 0.03 km s 𢄡 Hot Dark Matter (HDM) m ∼ 1 eV, vth z=0 ∼ 30 km s 𢄡

A general class of models that are appealing in their simplicity is that of thermal relics. Production and destruction of dark matter particles are in equilibrium so long as the temperature of the Universe kT is larger than the mass of the dark matter particle mDM c 2 . At lower temperatures, the abundance is exponentially suppressed, as destruction (via annihilation) dominates over production. At some point, the interaction rate of dark matter particles drops below the Hubble rate, however, and the dark matter particles 𠇏reeze out” at a fixed number density (see, e.g., Kolb & Turner 1994 this is also known as chemical decoupling). Amazingly, if the annihilation cross section is typical of weak-scale physics, the resulting freeze-out density of thermal relics with m ∼ 100 GeV is approximately equal to the observed density of dark matter today (e.g., Jungman, Kamionkowski & Griest 1996). This subset of thermal relics is referred to as weakly-interacting massive particles (WIMPs). The observation that new physics at the weak scale naturally leads to the correct abundance of dark matter in the form of WIMPs is known as the “WIMP miracle” (Feng & Kumar 2008) and has been the basic framework for dark matter over the past 30 years.

WIMPs are not the only viable dark matter candidate, however, and it is important to note that the WIMP miracle could be a red herring. Axions, which are particles invoked to explain the strong CP problem of quantum chromodynamics (QCD), and right-handed neutrinos (often called sterile neutrinos), which are a minimal extension to the Standard Model of particle physics that can explain the observed baryon asymmetry and why neutrino masses are so small compared to other fermions, are two other hypothetical particles that may be dark matter (among a veritable zoo of additional possibilities see Feng 2010 for a recent review). While WIMPs, axions, and sterile neutrinos are capable of producing the observed abundance of dark matter in the present-day Universe, they can have very different effects on the mass spectrum of cosmological perturbations.

While the cosmological perturbation spectrum is initially set by physics in the very early universe (inflation in the standard scenario), the microphysics of dark matter affects the evolution of those fluctuations at later times. In the standard WIMP paradigm, the low-mass end of the CDM hierarchy is set by first collisional damping (subsequent to chemical decoupling but prior to kinetic decoupling of the WIMPs), followed by free-streaming (e.g., Hofmann, Schwarz & Stöcker 2001, Bertschinger 2006). For typical 100 GeV WIMP candidates, these processes erase cosmological perturbations with M ≲ 10 𢄦 M (i.e., Earth mass Green, Hofmann & Schwarz 2004). Free-streaming also sets the low-mass end of the mass spectrum in models where sterile neutrinos decouple from the plasma while relativistic. In this case, the free-streaming scale can be approximated by the (comoving) size of the horizon when the sterile neutrinos become non-relativistic. The comoving horizon size at z = 10 7 , corresponding to m ≈ 2.5 keV, is approximately 50 kpc, which is significantly smaller than the scale derived above for L * galaxies. keV-scale sterile neutrinos are therefore observationally-viable dark matter candidates (see Adhikari et al. 2016 for a recent, comprehensive review). QCD axions are typically ∼ µeV-scale particles but are produced out of thermal equilibrium (Kawasaki & Nakayama 2013). Their free-streaming scale is significantly smaller than that of a typical WIMP (see Section 3.2.1).

The previous paragraphs have focused on the effects of collisionless damping and free-streaming – direct consequences of the particle nature of dark matter – in the linear regime of structure formation. Dark matter microphysics can also affect the non-linear regime of structure formation. In particular, dark matter self-interactions – scattering between two dark matter particles – will affect the phase space distribution of dark matter. Within observational constraints, dark matter self-interactions could be relevant in the dense centers of dark matter halos. By transferring kinetic energy from high-velocity particles to low-velocity particles, scattering transfers “heat” to the centers of dark matter halos, reducing their central densities and making their velocity distributions nearly isothermal. This would have a direct effect on galaxy formation, as galaxies form within the centers of dark matter halos and the motions of their stars and gas trace the central gravitational potential. These effects are discussed further in Section 3.2.2.

The particle nature of dark matter is therefore reflected in the cosmological perturbation spectrum, in the abundance of collapsed dark matter structures as a function of mass, and in the density and velocity distribution of dark matter in virialized dark matter halos.

THREE CHALLENGES TO BASIC 𹯍M PREDICTIONS
There are three classic problems associated with the small-scale predictions for dark matter in the 𹯍M framework. Other anomalies exist, including some that we discuss in this review, but these three are important because 1) they concern basic predictions about dark matter that are fundamental to the hierarchical nature of the theory and 2) they have received significant attention in the literature.

Missing Satellites and Dwarfs [Figures 4-8]
The observed stellar mass functions of field galaxies and satellite galaxies in the Local Group is much flatter at low masses than predicted dark matter halo mass functions: dn / dMM αg with αg ≃ 𢄡.5 (vs. α ≃ 𢄡.9 for dark matter). The issue is most acute for Galactic satellites, where completeness issues are less of a concern. There are only ∼ 50 known galaxies with M > 300 M within 300 kpc of the Milky Way compared to as many as ∼ 1000 dark subhalos (with Msub > 10 7 M) that could conceivably host galaxies. One solution to this problem is to posit that galaxy formation becomes increasingly inefficient as the halo mass drops. The smallest dark matter halos have simply failed to form stars altogether.

Low-density Cores vs. High-density Cusps [Figure 9]
The central regions of dark-matter dominated galaxies as inferred from rotation curves tend to be both less dense (in normalization) and less cuspy (in inferred density profile slope) than predicted for standard 𹯍M halos (such as those plotted in Figure 3). An important question is whether baryonic feedback alters the structure of dark matter halos.

Too-Big-to-Fail [Figure 10]
The local universe contains too few galaxies with central densities indicative of Mvir ≃ 10 10 M halos. Halos of this mass are generally believed to be too massive to have failed to form stars, so the fact that they are missing is hard to understand. The stellar mass associated with this halo mass scale (M≃ 10 6 M, Figure 6) may be too small for baryonic processes to alter their halo structure (see Figure 13).

1 Recent measurements find n = 0.968 ± 0.006 (Planck Collaboration et al. 2016), i.e., small but statistically different from true scale invariance. Back.

2 The black line in Figure 4 illustrates the mass function of 𹯍M dark matter halos. Back.

3 This maximum mass is similar to the virial mass at the time of accretion, though infalling halos can begin losing mass prior to first crossing the virial radius. Back.

## Title: REPRODUCING THE STELLAR MASS/HALO MASS RELATION IN SIMULATED CDM GALAXIES: THEORY VERSUS OBSERVATIONAL ESTIMATES

We examine the present-day total stellar-to-halo mass (SHM) ratio as a function of halo mass for a new sample of simulated field galaxies using fully cosmological, CDM, high-resolution SPH + N-body simulations. These simulations include an explicit treatment of metal line cooling, dust and self-shielding, H-based star formation (SF), and supernova-driven gas outflows. The 18 simulated halos have masses ranging from a few times 10 to nearly 10 M. At z = 0, our simulated galaxies have a baryon content and morphology typical of field galaxies. Over a stellar mass range of 2.2 Multiplication-Sign 10-4.5 Multiplication-Sign 10 M we find extremely good agreement between the SHM ratio in simulations and the present-day predictions from the statistical abundance matching technique presented in Moster et al. This improvement over past simulations is due to a number systematic factors, each decreasing the SHM ratios: (1) gas outflows that reduce the overall SF efficiency but allow for the formation of a cold gas component (2) estimating the stellar masses of simulated galaxies using artificial observations and photometric techniques similar to those used in observations and (3) accounting for a systematic, up to 30% overestimate in totalmore » halo masses in DM-only simulations, due to the neglect of baryon loss over cosmic times. Our analysis suggests that stellar mass estimates based on photometric magnitudes can underestimate the contribution of old stellar populations to the total stellar mass, leading to stellar mass errors of up to 50% for individual galaxies. These results highlight that implementing a realistic high density threshold for SF considerably reduces the overall SF efficiency due to more effective feedback. However, we show that in order to reduce the perceived tension between the SF efficiency in galaxy formation models and in real galaxies, it is very important to use proper techniques to compare simulations with observations. « less

## Title: Illuminating dark matter halo density profiles without subhaloes

Cold dark matter haloes consist of a relatively smooth dark matter component as well as a system of bound subhaloes. It is the prevailing practice to include all mass, including mass in subhaloes, in studies of halo density profiles in simulations. However, often in observational studies satellites are treated as having their own distinct dark matter density profiles in addition to the profile of the host. This difference can make comparisons between theoretical and observed results difficult. In this work, we investigate density profiles of the smooth components of host haloes by excluding mass contained within subhaloes. We find that the density profiles of the smooth halo component (without subhaloes) differ substantially from the conventional halo density profile, declining more rapidly at large radii. We also find that concentrations derived from smooth density profiles exhibit less scatter at fixed mass and a weaker mass dependence than standard concentrations. Both smooth and standard halo profiles can be described by a generalized Einasto profile, an Einasto profile with a modified central slope, with smaller residuals than either a Navarro–Frenk–White or Einasto profile. Furthermore, these results hold for both Milky Way-mass and cluster-mass haloes. This new characterization of smooth halo profiles can bemore » useful for many analyses, such as lensing and dark matter annihilation, in which the smooth and clumpy components of a halo should be accounted for separately. « less

## Search this blog

I haven’t written much here of late. This is mostly because I have been busy, but also because I have been actively refraining from venting about some of the sillier things being said in the scientific literature. I went into science to get away from the human proclivity for what is nowadays called “fake news,” but we scientists are human too, and are not immune from the same self-deception one sees so frequently exercised in other venues.

So let’s talk about something positive. Current grad student Pengfei Li recently published a paper on the halo mass function. What is that and why should we care?

One of the fundamental predictions of the current cosmological paradigm, ΛCDM, is that dark matter clumps into halos. Cosmological parameters are known with sufficient precision that we have a very good idea of how many of these halos there ought to be. Their number per unit volume as a function of mass (so many big halos, so many more small halos) is called the halo mass function.

An important test of the paradigm is thus to measure the halo mass function. Does the predicted number match the observed number? This is hard to do, since dark matter halos are invisible! So how do we go about it?

Galaxies are thought to form within dark matter halos. Indeed, that’s kinda the whole point of the ΛCDM galaxy formation paradigm. So by counting galaxies, we should be able to count dark matter halos. Counting galaxies was an obvious task long before we thought there was dark matter, so this should be straightforward: all one needs is the measured galaxy luminosity function – the number density of galaxies as a function of how bright they are, or equivalently, how many stars they are made of (their stellar mass). Unfortunately, this goes tragically wrong.

Fig. 5 from the review by Bullock & Boylan-Kolchin. The number density of objects is shown as a function of their mass. Colored points are galaxies. The solid line is the predicted number of dark matter halos. The dotted line is what one would expect for galaxies if all the normal matter associated with each dark matter halo turned into stars.

This figure shows a comparison of the observed stellar mass function of galaxies and the predicted halo mass function. It is from a recent review, but it illustrates a problem that goes back as long as I can remember. We extragalactic astronomers spent all of the 󈨞s obsessing over this problem. [I briefly thought that I had solved this problem, but I was wrong.] The observed luminosity function is nearly flat while the predicted halo mass function is steep. Consequently, there should be lots and lots of faint galaxies for every bright one, but instead there are relatively few. This discrepancy becomes progressively more severe to lower masses, with the predicted number of halos being off by a factor of many thousands for the faintest galaxies. The problem is most severe in the Local Group, where the faintest dwarf galaxies are known. Locally it is called the missing satellite problem, but this is just a special case of a more general problem that pervades the entire universe.

Indeed, the small number of low mass objects is just one part of the problem. There are also too few galaxies at large masses. Even where the observed and predicted numbers come closest, around the scale of the Milky Way, they still miss by a large factor (this being a log-log plot, even small offsets are substantial). If we had assigned “explain the observed galaxy luminosity function” as a homework problem and the students had returned as an answer a line that had the wrong shape at both ends and at no point intersected the data, we would flunk them. This is, in effect, what theorists have been doing for the past thirty years. Rather than entertain the obvious interpretation that the theory is wrong, they offer more elaborate interpretations.

Faced with the choice between changing one’s mind and proving that there is no need to do so, almost everybody gets busy on the proof.

J. K. Galbraith

Theorists persist because this is what CDM predicts, with or without Λ, and we need cold dark matter for independent reasons. If we are unwilling to contemplate that ΛCDM might be wrong, then we are obliged to pound the square peg into the round hole, and bend the halo mass function into the observed luminosity function. This transformation is believed to take place as a result of a variety of complex feedback effects, all of which are real and few of which are likely to have the physical effects that are required to solve this problem. That’s way beyond the scope of this post all we need to know here is that this is the “physics” behind the transformation that leads to what is currently called Abundance Matching.

Abundance matching boils down to drawing horizontal lines in the above figure, thus matching galaxies with dark matter halos with equal number density (abundance). So, just reading off the graph, a galaxy of stellar mass M* = 10 8 M resides in a dark matter halo of 10 11 M, one like the Milky Way with M* = 5 x 10 10 M resides in a 10 12 M halo, and a giant galaxy with M* = 10 12 M is the “central” galaxy of a cluster of galaxies with a halo mass of several 10 14 M. And so on. In effect, we abandon the obvious and long-held assumption that the mass in stars should be simply proportional to that in dark matter, and replace it with a rolling fudge factor that maps what we see to what we predict. The rolling fudge factor that follows from abundance matching is called the stellar mass–halo mass relation. Many of the discussions of feedback effects in the literature amount to a post hoc justification for this multiplication of forms of feedback.

This is a lengthy but insufficient introduction to a complicated subject. We wanted to get away from this, and test the halo mass function more directly. We do so by use of the velocity function rather than the stellar mass function.

The velocity function is the number density of galaxies as a function of how fast they rotate. It is less widely used than the luminosity function, because there is less data: one needs to measure the rotation speed, which is harder to obtain than the luminosity. Nevertheless, it has been done, as with this measurement from the HIPASS survey:

The number density of galaxies as a function of their rotation speed (Zwaan et al. 2010). The bottom panel shows the raw number of galaxies observed the top panel shows the velocity function after correcting for the volume over which galaxies can be detected. Faint, slow rotators cannot be seen as far away as bright, fast rotators, so the latter are always over-represented in galaxy catalogs.

The idea here is that the flat rotation speed is the hallmark of a dark matter halo, providing a dynamical constraint on its mass. This should make for a cleaner measurement of the halo mass function. This turns out to be true, but it isn’t as clean as we’d like.

Those of you who are paying attention will note that the velocity function Martin Zwaan measured has the same basic morphology as the stellar mass function: approximately flat at low masses, with a steep cut off at high masses. This looks no more like the halo mass function than the galaxy luminosity function did. So how does this help?

To measure the velocity function, one has to use some readily obtained measure of the rotation speed like the line-width of the 21cm line. This, in itself, is not a very good measurement of the halo mass. So what Pengfei did was to fit dark matter halo models to galaxies of the SPARC sample for which we have good rotation curves. Thanks to the work of Federico Lelli, we also have an empirical relation between line-width and the flat rotation velocity. Together, these provide a connection between the line-width and halo mass:

The relation Pengfei found between halo mass (M200) and line-width for the NFW (ΛCDM standard) halo model fit to rotation curves from the SPARC galaxy sample.

Once we have the mass-line width relation, we can assign a halo mass to every galaxy in the HIPASS survey and recompute the distribution function. But now we have not the velocity function, but the halo mass function. We’ve skipped the conversion of light to stellar mass to total mass and used the dynamics to skip straight to the halo mass function:

The halo mass function. The points are the data these are well fit by a Schechter function (black line this is commonly used for the galaxy luminosity function). The red line is the prediction of ΛCDM for dark matter halos.

The observed mass function agrees with the predicted one! Test successful! Well, mostly. Let’s think through the various aspects here.

First, the normalization is about right. It does not have the offset seen in the first figure. As it should not – we’ve gone straight to the halo mass in this exercise, and not used the luminosity as an intermediary proxy. So that is a genuine success. It didn’t have to work out this well, and would not do so in a very different cosmology (like SCDM).

Second, it breaks down at high mass. The data shows the usual Schechter cut-off at high mass, while the predicted number of dark matter halos continues as an unabated power law. This might be OK if high mass dark matter halos contain little neutral hydrogen. If this is the case, they will be invisible to HIPASS, the 21cm survey on which this is based. One expects this, to a certain extent: the most massive galaxies tend to be gas-poor ellipticals. That helps, but only by shifting the turn-down to slightly higher mass. It is still there, so the discrepancy is not entirely cured. At some point, we’re talking about large dark matter halos that are groups or even rich clusters of galaxies, not individual galaxies. Still, those have HI in them, so it is not like they’re invisible. Worse, examining detailed simulations that include feedback effects, there do seem to be more predicted high-mass halos that should have been detected than actually are. This is a potential missing gas-rich galaxy problem at the high mass end where galaxies are easy to detect. However, the simulations currently available to us do not provide the information we need to clearly make this determination. They don’t look right, so far as we can tell, but it isn’t clear enough to make a definitive statement.

Finally, the faint-end slope is about right. That’s amazing. The problem we’ve struggled with for decades is that the observed slope is too flat. Here a steep slope just falls out. It agrees with the ΛCDM down to the lowest mass bin. If there is a missing satellite-type problem here, it is at lower masses than we probe.

That sounds great, and it is. But before we get too excited, I hope you noticed that the velocity function from the same survey is flat like the luminosity function. So why is the halo mass function steep?

When we fit rotation curves, we impose various priors. That’s statistics talk for a way of keeping parameters within reasonable bounds. For example, we have a pretty good idea of what the mass-to-light ratio of a stellar population should be. We can therefore impose as a prior that the fit return something within the bounds of reason.

One of the priors we imposed on the rotation curve fits was that they be consistent with the stellar mass-halo mass relation. Abundance matching is now part and parcel of ΛCDM, so it made sense to apply it as a prior. The total mass of a dark matter halo is an entirely notional quantity rotation curves (and other tracers) pretty much never extend far enough to measure this. So abundance matching is great for imposing sense on a parameter that is otherwise ill-constrained. In this case, it means that what is driving the slope of the halo mass function is a prior that builds-in the right slope. That’s not wrong, but neither is it an independent test. So while the observationally constrained halo mass function is consistent with the predictions of ΛCDM we have not corroborated the prediction with independent data. What we really need at low mass is some way to constrain the total mass of small galaxies out to much larger radii that currently available. That will keep us busy for some time to come.

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## The danger of hyper-specialization

The explosive expansion of knowledge that started in the mid 1800s led to hyper-specialization inside and outside academia. Even within a single discipline, say philosophy or physics, professionals often don't understand one another. As I wrote here before, "This fragmentation of knowledge inside and outside of academia is the hallmark of our times, an amplification of the clash of the Two Cultures that physicist and novelist C.P. Snow admonished his Cambridge colleagues in 1959." The loss is palpable, intellectually and socially. Knowledge is not adept to reductionism. Sure, a specialist will make progress in her chosen field, but the tunnel vision of hyper-specialization creates a loss of context: you do the work not knowing how it fits into the bigger picture or, more alarmingly, how it may impact society.

Many of the existential risks we face today — AI and its impact on the workforce, the dangerous loss of privacy due to data mining and sharing, the threat of cyberwarfare, the threat of biowarfare, the threat of global warming, the threat of nuclear terrorism, the threat to our humanity by the development of genetic engineering — are consequences of the growing ease of access to cutting-edge technologies and the irreversible dependence we all have on our gadgets. Technological innovation is seductive: we want to have the latest "smart" phone, 5k TV, and VR goggles because they are objects of desire and social placement.

CRSQ Volume 36 No. 4 March 2000

Dark matter has never been directly observed. Its presence is indicated by unexplained gravitational effects on stars and galaxies. It is sought within galaxies, in galaxy clusters and throughout space. Surprisingly, dark matter appears to comprise the bulk of the entire universe. This article surveys the evidence along with possible micro and macroscopic dark matter candidates. The entire idea is then evaluated from the creation perspective. There are also theological implications.

Astronomy is sometimes presented as successfully having answered the most basic questions about the universe. One popular book attempts to explain the complete evolution of the universe to within 10 -43 seconds of its origin (Hawking, 1996). This number is called the Planck time, when some cosmologists think the present laws of physics originated. In truth, of course, many fundamental questions remain. We actually know very little about our neigh- boring planets, and much less about the deep space beyond. Consider just a few of the mysteries in modern astronomy:

• Origin of the moon
• Star and galaxy formation .Source of cosmic rays
• Nature of quasars and their distances .Absence of evolved life elsewhere
• Dark matter

The last entry is explored in this article. Dark matter is so called because it emits no detectable radiation. In our current understanding of astronomy and physics, dark matter must comprise the majority of mass in the universe, between 90-99 percent. Yet it has never been detected with certainty, if it indeed exists. Carl Sagan described it as dark, quintessential, deeply mysterious stuff wholly unknown on earth (Sagan, 1994, p. 399). Astronomers therefore have no idea of the composition of the bulk of the entire universe. So much for a fundamental understanding of the physical universe! Dark matter is an apt topic for review and for a creationist evaluation. There is a large number of dark matter discussions and references currently available on the internet.

What Dark Matter is Not

The term dark matter does not refer to dark nebulae. These are abundant interstellar clouds of dust, which block the light from background stars, and therefore appear as dark silhouettes. Nearby examples include Orion's Horsehead nebula, the Great Rift in Cygnus and the Coal Sack near the Southern Cross.

Some students of the Bible have described a particular dark region in the northern sky. This is further proposed as the literal direction toward heaven. The idea is based on Job 26:7 which describes the north stretched out "over the empty place." However, there is no such northern region of space which is devoid of stars or galaxies. The Job reference simply describes the vastness of space in which stars exist, including the northern sky (DeYoung, 1986).

Neither does dark matter refer to dark line spectra. Stellar light spectra typically show dark lines where certain wavelengths have been absorbed by the star's own atmospheric gases. Also, dark matter does not refer to black holes. Finally, dark matter does not involve Olber's paradox, the profound question of why the sky is dark at night, in spite of seemingly endless stars.

The Location of Dark Matter

There are several alternate names for dark matter including missing mass, hidden matter, shadow matter and hot or cold dark matter. Why is it thought to exist and where must it be located? Dark matter will be discussed in three hierarchical categories. It is sought first within single galaxies. Next, the invisible material is thought to "glue" galaxy groups or clusters together. Finally, popular versions of the big bang model require immense amounts of dark matter existing throughout space. These three reservoirs for dark matter will be considered in the following sections.

In our solar system, inner planets travel faster than the outer planets. Mercury has an orbital velocity of about 107 ,000 miles/hour (172,000 km/h), while Pluto's velocity averages nearly ten times less, only 10,000 miles/hour (16,000 km/h). This variation in speed follows directly from Kepler's third law of planetary motion,

Here T, r, and v are the planet's orbital period, average radial distance from the sun and velocity. G is the universal gravity constant and M is the solar mass.

The stars within galaxies also experience orbital motion. In this case, however, the main gravitational mass is not concentrated at the center as in the solar system. Instead, stellar masses are spread throughout the galaxy, with vast orbits about the center. Isaac Newton proved that for such orbits, only mass lying within a star's orbit affects the star's motion. The gravity force from external mass cancels completely. This is exactly true only for a uniform spherical or circular distribution of mass. However a galaxy provides a fair approximation, including the disk-shaped spiral galaxies.

Equations 1 and 2 can be applied to an entire galaxy of revolving stars. Each star responds to the total gravitation of the partial mass of the galaxy that lies within its orbit. And it is exactly as if this mass were all positioned at the center of the galaxy. The orbital velocity of a star can be measured from the Doppler shift of its light. If the star is near the outer edge of a galaxy, taking r as the galaxy radius and knowing v, equation 2 then gives the total mass M of the galaxy. And here a major problem arises. In every case, the calculated galaxy mass is at least 5-10 times the mass of all the visible stars and other matter in the galaxy .This missing mass component is considered to be invisible dark matter.

There is a second way to look at the mass problem with spiral galaxies. One can plot a rotation curve, a graph of velocity versus distance from the galaxy center for component stars. Equation 2 shows that, for the solar system, planet velocity decreases as the inverse square root of distance from the sun. Because of the mass distribution of a spiral galaxy, equation 2 is not followed precisely. However, star velocities should still decrease as their outward distance increases. Instead, however, measurements show flat rotation curves for galaxies (Figure 1). That is, the orbital velocities of remote galaxy stars are largely constant, or even increase slightly with distance. The unexpected nature of the rotation curve was first noted by astronomers Vera Rubin and Kent Floyd in 1970, for the Andromeda galaxy. Said in another way, the outer galaxy stars revolve unexpectedly fast. If galaxies are stable, this implies a large amount of dark matter affecting stellar motion. Otherwise, spiral galaxies should be flying apart. One possible mass distribution is a giant halo of invisible matter surrounding and permeating entire galaxies. X-ray telescopes, including Rosat, have shown possible evidence for this halo around the Milky Way. However the question remains, how can 90 percent of galaxy mass remain invisible to optical telescopes?

 Figure 1. A schematic rotation curve for a typical spiral galaxy. The dashed line shows the trend of star velocities expected from equation 2, if most of the mass of a galaxy were within 40 million light years of the center. The solid line shows actual measurements (Hawley and Holcomb., 1998, p. 390).

The terms missing mass and dark matter were first suggested in 1933 by Cal Tech astrophysicist Fritz Zwicky. He observed the Coma cluster, a group of at least 1,000 galaxies located 400 million light years distant. These galaxies are assumed to be gravitationally bound together. Zwicky noticed that the galaxies had random velocities, and moved much faster than expected. In fact the galaxy cluster should have disintegrated by now. This anomalous motion is likewise true of our own local group of galaxies. This local group consists of the Milky Way, Andromeda, Magellanic Clouds and about 30 other galaxies, all lying within about a three million light year region.
Why have not these galaxies within clusters escaped from each other? As before, an invisible binding mass of galaxy groups is considered as dark matter. The galaxy motions suggest that the dark matter mass totals at least ten times that of all the visible galaxies. This shortfall in mass is much greater for the galaxy clusters than that within individual galaxies. The Coma cluster is found to be 90 percent missing. Another example, the Virgo cluster is 98 percent missing. That is, there is assumed to be 50 times more mass than is actually observed. Figure 2 is a photograph negative of the center of the galaxy cluster Abell 1060. The smaller pinpoints of light are nearby Milky Way galaxy stars. The spirals and elliptical galaxies shown in the figure are a few of Abell 1060's two hundred galaxy members. The cluster is about 220 million light years away.

 Figure 2. The central part of the galaxy cluster Abell 1060. The photo is from the Royal Observatory, Edinburgh, England (Bruck, 1990, p. 96).

Visible gas clouds within galaxy clusters also have added to the dark matter requirement. X-ray observations reveal vast clouds of hot, low-density gas within the clusters, Elsewhere, similar clouds appear to suffuse regions of space far from galaxy clusters. These energetic gas molecules are moving so fast that the observed clouds would quickly leak away and dissipate on an evolutionary time scale. It is therefore concluded that the clouds must contain much more matter than we see, binding them together.

In several cases, gravitational lenses appear to give multiple images of the same quasar. Quasars are thought to be far distant, very bright sources of light. Apparently, separate light signals from a distant quasar can be bent by an intervening galaxy and then directed toward the earth. In some cases the light-deflecting galaxies also can be seen. On a more local scale, similar gravity lensing sometimes appears to occur for stars within or near the Milky Way galaxy. However in these cases there is no observed intervening object. Dark matter is suggested as the cause of this light deflection. Several gravity lens surveys are currently underway. It is hoped that positive results may help determine the size of dark matter objects (Holz, 1999).

The virial theorem can be used to calculate the mass of a single galaxy or a galaxy cluster. It applies if a system is gravitationally stable, without collapse or disintegration taking place. The theorem states that the total gravitational potential energy of the star system equals exactly twice the total kinetic energy. If this condition is not met, the component objects either will cascade inward or escape, depending on the direction of imbalance.

From the virial theorem comes the mass formula for a galaxy cluster (Bruck, 1990, p. 99),

Here V is the average of the squared radial velocities observed for member galaxies within the cluster. R is an estimate of the geometric radius of the entire cluster and G is the gravity constant.

As an example, consider Abell 1060, a cluster of about 200 galaxies located 220 million light years away (Figure 2). Its V is about 7.14 x 10 5 m/s (0.24% light speed) and R is 6.1 x 1022 m ( 6.5 million light years ). The virial mass result is 14 x 10 44 kg, or about 7 x 10 14 solar masses. If there are 200 galaxies, each then averages 3.5 x 10 12 solar masses. This is about 10 times higher than the known mass of Andromeda and the Milky Way galaxies. The Abell 1060 galaxies probably do not contain this much extra mass. Instead the mass may exist as dark matter spread between the galaxies. A similar numerical discrepancy exists for every galaxy cluster, assuming they obey the virial condition.

Some astrophysicists have proposed that galaxy clusters are not gravitationally bound after all, so the virial theorem does not apply. The use of the theorem to calculate unseen mass has been called "totally unreasonable" (Burbidge, et al., 1999, p. 42). However if the galaxies are disrupting, then the clusters must be far younger than the multi-billion year age usually assigned to them (Bowers and Deemings, 1984, p. 504).

Dark matter is also required on the largest scale of all, that of the entire universe. In this case it is tied to versions of the big bang theory in at least two ways. First, dark matter is enlisted to explain the large-scale structure of the universe. In this view, the initial universe expansion from a singularity must have experienced positional variations in temperature or energy density. This resulted in a "clumpiness" of matter, with subsequent formation of gas clouds, stars and galaxies. The initial clumps grew larger in this way because of the gravity attraction of invisible cold dark matter concentrations.

Dark matter is also involved in the popular inflationary big bang model which predicts that the curvature of the universe must be flat (Figure 3). This means that the density of matter is exactly balanced between a universe which eventually collapses (a closed, finite universe), and one which expands forever (an open, infinite universe). The required critical density for a flat universe is about 10 -26 g/cm 3 . This corresponds to approximately 10 hydrogen atoms per cubic meter of space. Observed density estimates, although crude, lead to a value 10-100 times smaller than the critical density. Therefore a great amount of dark matter is needed to result in a flat, closed universe with zero curvature.

 Figure 3. Three views of the expansion of the universe over evolutionary time. The universe is either a: closed, eventually collapsing on itself, b: flat, ceasing expansion after infinite time or c: open, expanding outward forever.

This is an unanswered question since dark matter has never been directly observed, and may not even exist. Nevertheless, many possible candidates have been suggested (Trimble, 1987). Several will be listed and briefly evaluated here.

Non-luminous stars include primordial black holes black, brown or red dwarfs and energy-depleted white dwarfs. An immense number of these unlit stars would be needed to supply the necessary dark matter. If they average 0.1 solar mass and comprise 90 percent of the total known universe mass, then there must be at least 10 25 such stars. The basic problem is that none have been detected and identified with certainty. Surely, such an astonishing number of non-luminous stars easily should be detected with modern instruments. Even black holes themselves remain as theoretical constructs which have not been verified with certainty.

The brown dwarfs are a special case of failed, low mass stars which never ignited their internal nuclear fusion reactions. They are sometimes pictured as large gaseous planets, somewhat like Jupiter. They have also been called MACHOS, or massive compact halo objects. Efforts have been made to detect brown dwarfs indirectly by their eclipsing of normal stars. That is, one watches for a distant star to temporarily disappear when covered. The background starlight might also be distorted in a microlensing effect. No clear brown dwarf evidence has been found in this way, in spite of detailed searches (Hawley and Holcomb, 1998, p. 391). There maybe many brown dwarf stars, or they may be vary rare.

Diffuse matter would consist of unseen dust or gas particles that are widely dispersed. It has been described as molecular clouds, intergalactic matter, and as halo or coronal material which surrounds and permeates galaxies. The Milky Way contains l0 11 solar masses. The required invisible dark matter is 100 times greater, 10 13 solar masses worth. This would be an incredible amount of unseen diffuse matter.

Neutrinos are an abundant product of nuclear fusion, the process thought to energize stars including our sun. Creationists have suggested that gravity contraction may also be occurring within stars (Steidl, 1983). Whatever the combination, some nuclear fusion does occur with resulting neutrino production. From the sun, this sends a continuous flood of neutrinos toward earth with a flux as great as 10 12 neutrinos/cm 2. s.

Solar neutrinos have been detected, although only at about one-third of their expected number. Neutrinos require large, highly specialized detectors since the particles are very elusive and unreactive. Most travel directly through the earth's 8,000 mile diameter without any atomic collisions occurring.

Thus far, laboratory studies of neutrinos show zero mass. However, there is a suggestion that neutrinos might oscillate between different forms as they travel along at light speed. This behavior could mask a vanishingly small but finite mass. With their large abundance throughout space, neutrinos could thus comprise much of the sought-after dark matter. The proposed dynamical behavior for neutrinos might also explain their low abundance as measured from the sun. The problem remains, however, that no one has observed any mass whatsoever for neutrinos. The idea of neutrino mass appears to be a desperate hope for solving the embarrassing mass deficit.

Exotic particles are wisps of localized energy in space that have been theorized but never observed. In physics jargon, exotic particles are nonbarionic. They have much less mass than normal baryons such as neutrons and protons. Exotic particles are often given fanciful acronyms. WIMPS, or weakly interacting massive particles, are predicted by certain theoretical physics models. No one knows whether such dark matter particles exist. Nevertheless, they have been enlisted to help solve the solar neutrino problem. It is proposed that WIMPS inside the sun might help spread heat throughout the solar core. Solar energy could then be generated at a slightly lower temperature, with fewer neutrinos produced than now expected.

A WIMP detector has been built in England, deep underground. It consists of 200,000 liters of pure water. Scientists hope that an occasional WIMP particle speeding in from space might interact with a hydrogen atom in a detectable way. Results thus far have not been encouraging (Seife, 1999).

Axions, whimsically named for a laundry detergent, are another type of hypothesized subatomic particle which contribute mass to space. Other proposed but unobserved particles include photinos, neutralinos, gravitons, mini-black holes and antimatter. Astronomers also speak of bowling balls, a shorthand title for ordinary space matter in some hard-to-detect form.

Other exotic particle candidates include cosmic strings or membranes, preons and monopoles. There is certainly no shortage of suggestions to identify dark matter. In reality, however, the dark matter mystery remains completely unsolved after seven decades of intense study.

We have seen that dark matter is required if the laws of motion and gravity hold for galaxies, and if galaxy systems are stable. Since creationists are not locked into the big bang theory or evolutionary time, there are several options to consider. They will be discussed here as questions.

Are the laws of nature universal? This question allows for entirely different, unknown laws operating elsewhere in space. Dark matter then might be only an illusion, based on our local understanding of physics. However, there is no reason to expect such an unknowable multiverse instead of a universe. Instead, light signals coming from deep space, in all the intricate details of their spectra, appear much like light sources within our laboratories. Therefore the dark matter problem cannot easily be solved by rejecting known physics. Newton's and Kepler's laws of motion and gravity appear to be universal in their extent and application.

Are galaxies stable? If dark matter is lacking in galaxies, then over time they will simply disintegrate. This would be a major problem for evolutionary time, since galaxies then should no longer exist. In the recent creation view, however, little galaxy change would be noticeable since the creation event. After all, galaxies average 100,000 light years in diameter. In just 10,000 years, galaxy enlargement would be minimal.

Still, there is little reason to expect that galaxies are unstable in this way. With few exceptions, mainly within the solar system, transients and instabilities are not found in space studies. Instead, the created universe is marked by great durability .Consider our sun, which has sufficient hydrogen fuel to last for billions of years into the future, although the Creator, of course, may have other plans. Galaxies can be assumed to be stable, and thus must contain some form of dark matter.

Are galaxy clusters stable? Clusters are an entirely separate category from individual galaxies. There is little reason from a creation perspective why these clusters need to be bound together by unseen matter. The Creator may simply have placed these clusters throughout space much as we see them, with random galaxy velocities. Even if unbound, these clusters would only dissipate on a billion year time scale because of their vast size. Galaxy clusters may well be unstable in the long term.

Must the universe be flat? The creation view has no such requirement. The flatness requirement arises only with the big bang theory. The Creator, with equal ease, could have made a closed, flat or open universe. However, I suggest that it may well be open, with a lack of large scale dark matter. The simple reason may be to frustrate all natural origin theories, most of which call for a closed or flat universe. Something similar occurs for the planets. We find sufficient created variety and uniqueness in the solar system to cancel all natural attempts at an explanation, including the popular nebular hypothesis.

What then is dark matter? I have suggested that dark matter exists within galaxies, if not elsewhere. We have considered various physical micro and macro-size possibilities. But there is another option. Perhaps the dark matter we seek is in reality the unseen hand of the Creator. We know from Colossians 1: 17 that God in some way holds all things together. Therefore at some point, physical reality must mesh with the spiritual. And that point may lie in the unexplicable problems of modern science.

The law of gravity has been known since it was first explained by Isaac Newton in 1687. On a deep level, however, gravity remains a mystery .That is, we have no idea how objects physically communicate their positions and interact with each other. This ignorance about gravity or dark matter, for both creationists and non-creationists alike, should be a humbling experience. We know very little about physical reality, since we presently "see through a glass darkly" (I Corinthians 13: 12). Creationists look forward to the future, when our understanding will be made complete.

Arny, T. 1998 Explorations -An introduction to astronomy. McGraw Hill, New York.

Burbidge, G., F. Hoyle and J.V. Narlikar. 1999. A different approach to cosmology. Physics Today 52(4):38-44.

Bowers, R.L. and T. Deemings. 1984. Astrophysics II. Jones and Bartlett Publishers, Boston.

Bruck, M. T .1990. Exercises in practical astronomy using photographs. Adam Hilger, New York.

DeYoung, D. 1986. Is there an 'Empty Place' in the North? Creation Research Society Quarterly 23(3): 129-131.

Hawking, Stephen. 1996. The illustrated brief history of time. Bantam Books, New York.

Hawley, J.F. and K.A. Holcomb. 1998. Foundations of modem cosmology. Oxford University Press, New York.

Holz, D.E. 1999. Shedding light on dark matter. Nature 400( 6747):819-820.

Sagan, C. 1994. Pale blue dot. Random House, New York.

Seife, C. 1999. Deep in the coal mine something stirred. New Scientist 163(2201):16.

Steidl, P. 1983. Solar neutrinos and a young sun. In Mulfinger, G., Editor. Design and origins in astronomy. Creation Research Society Books, St. Joseph, MO. pp. 113-125.

Trimble, V. 1987. Existence and nature of dark matter in the universe. Annual Review of Astronomy physics 25:425-72.

## The Connection Between Galaxies and Their Dark Matter Halos

In our modern understanding of galaxy formation, every galaxy forms within a dark matter halo. The formation and growth of galaxies over time is connected to the growth of the halos in which they form. The advent of large galaxy surveys as well as high-resolution cosmological simulations has provided a new window into the statistical relationship between galaxies and halos and its evolution. Here, we define this galaxy–halo connection as the multivariate distribution of galaxy and halo properties that can be derived from observations and simulations. This galaxy–halo connection provides a key test of physical galaxy-formation models it also plays an essential role in constraints of cosmological models using galaxy surveys and in elucidating the properties of dark matter using galaxies. We review techniques for inferring the galaxy–halo connection and the insights that have arisen from these approaches. Some things we have learned are that galaxy-formation efficiency is a strong function of halo mass at its peak in halos around a pivot halo mass of 10 12 M, less than 20% of the available baryons have turned into stars by the present day the intrinsic scatter in galaxy stellar mass is small, less than 0.2 dex at a given halo mass above this pivot mass below this pivot mass galaxy stellar mass is a strong function of halo mass the majority of stars over cosmic time were formed in a narrow region around this pivot mass. We also highlight key open questions about how galaxies and halos are connected, including understanding the correlations with secondary properties and the connection of these properties to galaxy clustering.

## Paper Explainer: Gravitational probes of dark matter physics

This is description of my recent paper with Annika Peter of OSU, where we consider how to use gravitational measurements of dark matter (that is: astronomy and cosmology) to understand the particle physics of dark matter. I have a separate post on a very fun section of the paper, where we think about what you could learn about the Standard Model if you were made of dark matter and could see dark matter (but couldn't see baryons). Beyond that motivating thought experiment, this monster of a paper (126 pages and over 600 citations) is trying to cover a lot of ground.

The central thesis of the paper is that there is a huge potential to learn about the properties of dark matter: things like mass, interactions, production, etc using measurements from astronomy. This is not a completely novel idea: we know a great deal about dark matter from astronomy and cosmology (for example: dark matter is "cold" and not "hot"). However, there is an immense opportunity in the near future to do far more, thanks to improvements in simulation and some powerful new astronomical surveys which will be occurring.

To take advantage of all this new astronomical data, we particle physicists need to start doing our homework. One of the issues with melding the results of astrophysics with the questions that particle physicists want to answer about dark matter is that the two groups speak different languages. For particle physicists, we're interested in the nitty gritty of how dark matter (and any other particle that interacts with dark matter) fits together at a really granular level. We don't really know as much about what big astronomical surveys do, and how they might impact our theories. Astronomers are much more big-picture, as you'd expect from people who look at objects that are, at minimum, some $10^<50>$ times larger than individual particles. They're not accustomed to throwing around models of particles where everything is a free parameter. Neither approach is worse or better than the other, it's just two different sets of training and problem-solving.

Adding to this is that particle physicists did entirely too good a job of selling a particular type of dark matter (the Weakly Interacting Massive Particle, as found in supersymmetry) to our astronomer colleagues. WIMPs won't change astronomical results that much, they're nearly the most vanilla of dark matter candidates, but they are just a candidate for what dark matter might be. The problem is that we seem to have convinced a non-negligible number of astronomers that all dark matter acts like a WIMP, so they're not looking at what interesting models of dark matter their results might constrain. So you have one group (particle physicists) who don't know what astronomers are capable of doing, and another group (astronomers) who don't know that what they're doing can be repurposed to measure dark matter properties in novel ways.

One of the major impetuses for the paper was trying to come up with a common language that both groups can use to communicate what they are talking about when they talk about dark matter. The issue boils down to the vast possibilities that could be lying in the physics of the dark sector. Dark matter might be something as "boring" as a WIMP or it might be an entire mirrored copy of the Standard Model, with all the complexity and baroqueness that would imply. But if you're trying to build bridges between fields, you need to be able to boil all of that down to one or two parameters, at least to start. Such a simplification will miss much of what is interesting about a particular model of dark matter, but it will hopefully allow both groups to find some common ground.

There are many ways to try to capture the physics of dark matter in a set of parameters. But when Prof. Peter and I surveyed the options, we found nothing really to our liking. So we made our own. We wanted two parameters: one particle physics and one astrophysics. Then we could imagine classifying dark matter theories according to these two parameters.

The two parameters we picked we call $Lambda^<-1>$ and $M_< m halo>$.

The first, $Lambda^<-1>$, is the particle physics parameter: it has units of inverse energy. Basically, it is a measure of the interaction between dark matter (whatever it is), and the Standard Model. If you imagine that dark matter exchanges some particle with the stuff we know about, that particle has some couplings (measuring "how strongly" it interacts with both dark matter and the Standard Model), and some mass. Very roughly, one expects any rate of interaction to be proportional to the combination

where $g^2$ is (very heuristically) the combination of the Standard Model and dark matter couplings, and $M$ is the mass of whatever particle is allowing the dark matter to "talk" to the Standard Model. The smaller this number is, the harder it is for the Standard Model to interact with dark matter. The couplings $g^2$ are just numbers, and mass $M$ can be written (by particle physicists) as an energy (via $E=mc^2$), so this combination has units of inverse energy. So we just call it $Lambda^<-1>$, and viola, we have a parameter that captures some measure of one of the central questions particle physicists have about dark matter, namely "how hard is it to see dark matter in an experiment?"

Of course, the real answer to that question has to do with which Standard Model particles the dark matter interacts with, and lots of other issues besides. But hey, we're compressing everything down into two parameters, we have to simplify somewhere.

The second parameter, $M_< m halo>$ is the astrophysics parameter. It is defined as

Well, first we have to define what we mean by "halo" and then what we mean by "pure cold dark matter." We actually spend a significant amount of time in the paper carefully describing what a dark matter halo is and how astronomers parametrize it. Very generally, we know that in the early Universe, the matter and energy was distributed very uniformly. But not perfectly so, and those regions where there was a slight overdensity grew (because gravity is attractive), and become more and more dense. Critically though, there was a distribution of these overdensities in physical extent: there were overdensities that were only a few parsecs across, overdensities of a few 10's of parsecs, hundreds, kiloparsecs, and so on. Larger halos contain smaller halos, and so on down the scale.

Primer on dark matter halos from our paper. The "virial mass" is the mass of the dark matter contained in the halo (and maps to our astrophysical parameter). However, astronomers cannot measure mass directly, and so use proxies like orbital velocities of visible objects (galaxies, stars, and gas) to estimate mass. Presumably the hierarchical structure of halos continues down to ever-smaller scales, but below dwarf galaxies, no visible bodies are known to be embedded in the dark matter halos.

Naively, you'd expect the larger dark matter halos to contain more baryonic material, and this is more or less what we see. Groups of galaxies are found in dark matter halos massing 100 trillion to a quadrillion Suns ($10^<14-15>,M_odot$). These contain Milky Way-type galaxies, which mass a trillion Suns ($10^<12>,M_odot$). Galaxies like our own have satellite galaxies called "dwarfs" which also appear "in the field" (far from parent galaxies). The smallest known dwarfs might have a dark matter mass of a hundred million Suns ($10^8,M_odot$). Presumably, there are even smaller halos of dark matter, which contain baryonic gas. However, such small objects are not yet seen directly, because they do not contain many (or perhaps any) stars, and thus are not apparent in the surveys astronomers have performed. One of the characteristics of cold dark matter is that this hierarchy of clustered halos should continue down, to arbitrary small scales. There should be dark matter halos of a $100,M_odot$ or even the mass of a single Sun: $M_odot$. We've never seen them though, so we don't know if they really exist. But that's the prediction from what we can infer about the distribution of these primordial density perturbations.

But if you start modifying the dark matter particle physics, you're going to erase or modify this hierarchy of halos. I called dark matter "cold" a couple of times here, and that's a good example. Cold dark matter just means dark matter that was not moving at relativistic speeds early in the history of the Universe (in particular, when the Universe was composed of approximately equal densities of matter and radiation). Material moving at or near the speed of light can't be trapped inside gravitational perturbations short of black holes, so if dark matter wasn't "cold" (i.e., if it was "hot"), then it would move out of these initial density perturbations, erasing them. However, they can only move so far before the Universe cools enough for them to become non-relativistic, so they only erase the density perturbations that are sufficiently small - that is, dark matter halos today that are very massive should be fine, but the smaller ones shouldn't exist. Thus, the presence of dwarf galaxies limits dark matter to being "cold" (or at most "lukewarm"). We can characterize a model of hot or warm dark matter by the minimum dark matter halo mass that survives: this would be the parameter $M_< m halo>$ for those models.

But other models of dark matter will also deviate from the predictions of cold dark matter at some scale. Even particles as "vanilla" as a WIMP would terminate the hierarchy of halos at some scale, it's just that this scale is expected to be tiny (often $10^<-6>,M_odot$, which is about the mass of the Earth). We can't detect such tiny halos yet (or maybe ever), because they contain next to no baryons, but if we could, that would provide powerful limitations on the type of dark matter particle physics that could be at play.

Hot dark matter (or WIMPs) alter the prediction of cold dark matter at very early times that is, they set a $M_< m halo>$ primordially. But you could also imagine that dark matter particle physics could induce subtle changes over time in the structure of dark matter halos at some scale $M_< m halo>$ over time, that is, an evolutionary deviation. One example would be a model of self-interacting dark matter (SIDM), where dark matter particles can ricochet off each other at non-negligible rates. This would allow energy to move efficiently through a dark matter halo, heating some parts and cooling others. That wouldn't necessarily erase structure, but that would introduce some characteristic $M_< m halo>$ where you should see some difference from a model without self-interactions.

So these two parameters, $Lambda^<-1>$ and $M_< m halo>$, allow you to capture a lot of interesting dark matter physics in a plot-able space. The larger $Lambda^<-1>$ is, the easier it will be for a particle physicist to find dark matter in their experiments (all else being equal). The larger $M_< m halo>$ is, the easier it would be for astronomers (though we think a lot more work needs to be done to make this possible for general theories of dark matter, thus the paper). We made a plot estimating where many popular models of dark matter fall in our parameter space.

Estimates of particle physics and astrophysics parameters for a number of dark matter models. See paper for details.

Our take-away from all this is that there is a huge range of dark matter theories that have far too low of a $Lambda^<-1>$ to be reasonable searched for in particle physics experiments. The only way such models would be apparent is in astrophysics. Further, even if a model has a measurable $Lambda^<-1>$, that is just the dark matter-baryonic interaction there may be internal interactions and physics at play that would be completely invisible to, say, the LHC. Again, the only hope to find such physics is by turning to astrophysics.

Having introduced what we hope is a useful way of thinking about dark matter for both particle physicists and astronomers, we turn in the paper to what we might hope to learn from astronomical surveys, and what work needs to be done for the power of these searches to be full realized. We start by discussing a set of "hints" of non-trivial dark matter physics that have been of great interest for the last 10 years. These hints are generally called the "Crisis in Small Scale Structure," and basically they boil down to a number of discrepancies between our expectations of cold dark matter and our observations in halos of dwarf galaxy mass and above. That is, the Crisis might be pointing towards an existing $M_< m halo>$ of $10^<8-11>,M_odot$.

A summary of the hints for deviations from predictions of cold dark matter at particular halo mass scales (BTF is "baryonic Tully-Fisher relation" and "TBTF" is "too Big to Fail."), compared to the halo masses where baryonic effects are expected to exist and must be correctly accounted for.

The problem however, is that we can't be sure yet that we really understand both the prediction of cold dark matter (that is, the distribution of halos without any non-trivial physics), and we can't be sure that we're correctly mapping the observed properties of collections of stars and gas in to a mass for the halo. The former problem is that these class of galaxies are exactly where we expect baryons to start becoming important. Baryons can form stars. They can cool and form disks in galaxies. That allows energy to be injected into the halo (via supernova, for example), and increases the effects of gravitational tides. Without correctly modeling all those effects, which is computationally expensive, we can't be sure that what we're seeing isn't just normal gravitational physics in the dark sector combined with baryons.

Secondly, we need to remember that astronomers can't see dark matter directly. They only see stars and gas. They use the motion of those stars to infer the local gravitational field, and from that get a map of dark matter (this ignores gravitational lensing, which gets a more direct picture of the gravitational effects of dark matter). From the velocities of stars then, we must map to a particular halo mass. If that mapping is off for one reason or another, then the data will show a deviation from cold dark matter predictions where none exist.

After surveying the literature, this is basically where I come down, and hopefully the paper will make that clear to readers: there is room for new physics at small scales in dark matter. But the "Crisis" can probably be resolved without appealing to new physics: it most likely is a combination of baryonic effects and systematics in the measurements that can be corrected for. So we don't need new physics yet, but we need to keep looking. The Crisis in Small Scale Structure is both a perfect example of the sort of astrophysics-informs-particle-physics that we are interested in, and a cautionary tale.

So if the measurements resulting in the Crisis in Small Scale Structure aren't evidence of new physics, what are we going to do?

The rest of our paper is a long discussion of the many opportunities astrophysics will be affording to measure $M_< m halo>$ down to ever-smaller scales.

Our first point is that we really need to nail down the effects of baryons, which involves improving our simulations of dark matter structure formation. This is already in progress. There is a difficulty, in that these simulations are very computer-intensive: millions of hours of CPU time on large clusters. If you want to test the effect of new dark matter physics, you really should do that in conjunction with these baryonic simulations. However, since they're so expensive to run, simulators can't just throw in some crazy new model from a theorist to see what happens. So we need to figure out ways to get results about how new theories of dark matter would affect structure without using simulation that are at least approximately correct. If we have an interesting idea that these estimates suggest would have some very interesting result, then we can lobby our simulation-oriented colleagues to do a more expensive and accurate computation with.

Then we particle physicists need to start becoming much more aware of the opportunities coming our way from observations. We included a sketch of astronomical probes that are or will be available, and the scale of $M_< m halo>$ that we could hope they will be sensitive to.

Picking out some (since the full discussion is over 20 pages in the paper), at very large scales, we can look for novel effects of dark matter in the expansion history of the Universe. There are already some slight hints that the standard cosmology might not fit all the data, but these are not at a high enough statistical significance to take too seriously yet. But in the next few years, we'll get separate measurements of that cosmological history, that hopefully will resolve this question one way or the other. If discrepancies remain, then dark matter physics should be one of the first places to turn to for a resolution.

At smaller scale, recently astronomers have started identifying "ultradiffuse" galaxies that are difficult to fit into the current story of galaxy evolution. These are being discovered now as we map more of the sky in greater detail, and these objects might help us resolve the existing Crisis in Small Scale Structure more satisfactorily. And who knows, maybe reveal a new Crisis that requires new physics.

At smaller scales ($sim 10^9,M_odot$, i.e., dwarf galaxies), the more accurate surveys are both discovering new galaxies and resolving their velocity structure (and thus their dark matter density) more accurately. Again, this will be important for the Crisis at Small Scales, as well as testing many models of non-trivial dark matter. There are enormous data sets relevant to this scale of $M_< m halo>$, and more will be on the way.

Below this mass scale are the halos we expect to exist from cold dark matter, but have never seen. We have some ideas to push down our measurements, but really, this is the final frontier and new innovations are required. My personal obsession in terms of dark matter astrophysics right now is the Gaia space satellite, which is mapping the position and location of billions of nearby stars. This will allow us to do many things, but my day-dream is that it would allow us to pick up evidence of small dark matter halos drifting through the nearest kiloparsec of the galaxy or so. Whether that happens, or if Gaia can detect it if it does, is so far an open question.

As is probably apparent from the length of the paper, we thought for a long time about these sorts of problems, and wrote for a long time too. Dark matter is one of the biggest open questions in physics today, and we think that astrophysics might contain the key. It might not, but if we don't look, we'll never know. There's lots to do, and lots of fun to be had. And that's why we do science, right? Because it's fun.