11.3 Dark Matter Distribution and Halo Models

Halo Density Profiles

Dark matter halos are the invisible scaffolding around galaxies. They extend far beyond the visible disk, and their density profiles tell us how dark matter is distributed from the center outward. Getting these profiles right is essential for modeling galaxy dynamics, rotation curves, and structure formation.

NFW Profile and Isothermal Sphere Model

The NFW (Navarro-Frenk-White) profile is the standard model for halo density, derived from N-body cosmological simulations. It captures two key behaviors: a steep density rise toward the center and a more gradual falloff at large radii.

$\rho(r) = \frac{\rho_0}{(r/r_s)(1 + r/r_s)^2}$

• $\rho_0$ is a characteristic density (not the actual central density, which formally diverges in this profile)
• $r_s$ is the scale radius, marking the transition between the inner slope ($\rho \propto r^{-1}$) and the outer slope ($\rho \propto r^{-3}$)

The singular isothermal sphere (SIS) is a simpler, older model that assumes constant velocity dispersion throughout the halo:

$\rho(r) \propto r^{-2}$

This naturally produces flat rotation curves, which is why it was historically popular. However, it's a single power law with no transition between inner and outer behavior, so it's less physically realistic than NFW. You'll still see it used where analytic simplicity matters (e.g., lensing estimates).

Alternative Profiles and the Core-Cusp Problem

The Einasto profile adds flexibility by introducing a shape parameter $\alpha$:

$\rho(r) = \rho_0 \exp\left(-\frac{2}{\alpha}\left[\left(\frac{r}{r_s}\right)^\alpha - 1\right]\right)$

Unlike NFW, the Einasto profile doesn't have a fixed inner slope. By varying $\alpha$, it can fit a wider range of simulated halos, and some studies find it provides a statistically better fit than NFW, especially for the most massive halos.

The core-cusp problem is one of the most persistent tensions in dark matter astrophysics:

• CDM-only simulations consistently predict cusps: steeply rising density toward the halo center ($\rho \propto r^{-1}$ or steeper)
• Observations of dwarf and low-surface-brightness galaxies often show cores: roughly constant-density central regions

This discrepancy has a few proposed resolutions:

1. Baryonic feedback: Repeated episodes of gas inflow and supernova-driven outflow can gravitationally redistribute dark matter, flattening the cusp into a core
2. Self-interacting dark matter (SIDM): If dark matter particles scatter off each other, energy transfer thermalizes the inner halo, producing a core
3. Observational systematics: Some analyses may underestimate the inner slope due to limited resolution or assumptions about tracer kinematics

None of these fully resolves the problem across all galaxy masses, which is why it remains an active area of research.

Halo Properties and Structure

Halo Concentration and Mass Function

Concentration quantifies how centrally concentrated a halo's mass is:

$c = \frac{r_{\text{vir}}}{r_s}$

where $r_{\text{vir}}$ is the virial radius (the boundary within which the halo is roughly in gravitational equilibrium). A higher $c$ means more mass is packed toward the center relative to the outskirts.

Concentration correlates with formation history: halos that formed earlier tend to be more concentrated because the universe was denser at earlier times. There's also an inverse relationship with mass: low-mass halos typically have higher concentrations than cluster-scale halos.

The halo mass function tells you how many halos exist per unit volume per unit mass. The Press-Schechter formalism gives an analytic approximation:

$\frac{dn}{dM} \propto M^{-\alpha} \exp\left(-M/M_*\right)$

• The power-law term ($M^{-\alpha}$) means low-mass halos vastly outnumber high-mass ones
• The exponential cutoff at the characteristic mass $M_*$ reflects the rarity of the most massive collapsed structures at any given epoch

More refined versions (Sheth-Tormen, Tinker) improve on Press-Schechter by better matching N-body simulations, particularly at the high-mass end. The mass function is a direct prediction of the cosmological model, so comparing it to observed cluster counts constrains parameters like $\sigma_8$ and $\Omega_m$.

Substructure and Hierarchical Formation

In the $\Lambda$CDM framework, structure forms hierarchically: small halos collapse first, then merge to build larger ones. This process leaves behind subhalos, smaller dark matter clumps orbiting within a larger host halo.

• Subhalos are the remnants of previously independent halos that were accreted
• Some host visible satellite galaxies (the Milky Way's dwarf spheroidals like Sagittarius and Fornax sit inside subhalos)
• The subhalo mass function follows a roughly power-law distribution: $dN/dM \propto M^{-\beta}$ with $\beta \approx 1.9$, meaning far more small subhalos than large ones

This creates the missing satellites problem: simulations predict hundreds to thousands of subhalos around a Milky Way-mass galaxy, but only ~60 satellite galaxies have been observed. Possible explanations include:

• Reionization suppressing gas accretion onto the smallest halos, leaving them dark
• Baryonic physics making many satellites too faint to detect with current surveys
• Ultra-faint dwarfs discovered by DES and other surveys are closing the gap

Tidal stripping by the host halo gradually removes mass from subhalos, and some are fully disrupted over time, contributing to a smooth background of dark matter within the host.

Large-Scale Dark Matter Distribution

Dark Matter Filaments and the Cosmic Web

Zooming out from individual halos, dark matter organizes into the cosmic web, a network that emerged from the gravitational amplification of tiny density perturbations in the early universe.

The cosmic web has four structural elements:

• Filaments: elongated overdense structures that form the "skeleton" of the web, spanning tens to hundreds of megaparsecs. They contain a significant fraction of the universe's total dark matter and channel matter toward denser regions.
• Nodes: where filaments intersect, forming the densest environments. Galaxy clusters sit at these nodes.
• Sheets (walls): flattened structures connecting filaments, representing intermediate-density regions.
• Voids: vast underdense regions that can span 20-50 Mpc or more, containing very few galaxies.

Galaxies preferentially form along filaments and at nodes, and their orientations and spin axes tend to correlate with the filament direction (this is sometimes called cosmic alignment). The anisotropic nature of gravitational collapse, governed by the tidal field, is what produces filaments rather than spherically symmetric structures.

Observational Evidence and Mapping Techniques

Several complementary methods map the dark matter distribution on large scales:

Weak gravitational lensing is the most direct probe. Background galaxies are subtly sheared (distorted in shape) by foreground mass concentrations. By measuring the statistical pattern of these distortions across millions of galaxies, you can reconstruct the projected mass distribution without assuming anything about the mass-to-light ratio. Surveys like the Dark Energy Survey (DES), KiDS, and HSC have produced detailed mass maps this way.

Galaxy redshift surveys trace the luminous matter distribution, which tracks the underlying dark matter (with a bias factor). The 2dF Galaxy Redshift Survey and the Sloan Digital Sky Survey (SDSS) mapped hundreds of thousands of galaxy positions in 3D, clearly revealing the filamentary pattern of the cosmic web.

N-body simulations provide the theoretical counterpart. The Millennium Simulation and its successors (Illustris, IllustrisTNG, EAGLE) start from initial conditions calibrated to the CMB power spectrum and evolve billions of dark matter particles (and, in hydrodynamic runs, baryons) forward in time. The resulting structures reproduce the observed cosmic web with striking fidelity, which is one of the strongest pieces of evidence for the $\Lambda$CDM model.

The agreement between lensing maps, galaxy surveys, and simulations gives us confidence that the large-scale distribution of dark matter is well understood, even as smaller-scale puzzles like the core-cusp problem and missing satellites remain open questions.