Dark Matter Subhalo Parameters

• Dark matter subhalo population parameters are statistical measures describing the mass, abundance, and concentration of subhaloes within larger host haloes.
• They quantify scaling relations that reveal host mass dependence, redshift evolution, and the impacts of tidal stripping and baryonic physics.
• These parameters underpin models for satellite galaxies, dark matter annihilation signals, and strong lensing phenomena by calibrating simulation outcomes.

The dark matter subhalo population parameters describe the statistical distribution, scaling relations, and evolution of gravitationally bound substructures (“subhaloes”) residing within larger host dark matter haloes. These parameters are foundational for modeling the internal structure of dark matter haloes in ΛCDM and alternative dark sector scenarios, influencing not only theoretical predictions for satellite galaxy populations but also prospects for indirect dark matter detection and strong lensing constraints.

1. Subhalo Mass Fraction and Its Host Dependence

The subhalo mass fraction quantifies the fraction of a host halo’s mass bound in subhaloes above a given fractional mass threshold. High-resolution N-body simulations demonstrate that this fraction increases systematically with host halo mass: cluster-scale haloes ([1–3]×10¹⁴ h⁻¹ M_⊙) have on average 7.5% of their mass within $R_{200}$ locked in subhaloes of fractional mass $M_{\text{sub}}/M_{200} \gtrsim 10^{-5}$, which is approximately 25% higher than that found in typical galactic haloes (mass range [1–3]×10¹² h⁻¹ M_⊙), which have ≈6% mass fractions. However, substantial halo-to-halo variance is observed, particularly in the high-mass subhalo regime where a few massive satellites can dominate the fraction; this variance persists even at fixed host mass, implying the governing processes are more complex than a simple universal scaling.

The subhalo mass fraction,

$$f_{\text{sub}} = \frac{\sum_{i} M_{\text{sub},i}}{M_{200}},$$

exhibits significant scatter, and its cumulative distribution (as a function of normalized subhalo mass) displays higher normalization in clusters relative to groups and galaxies. This non-universality must be accounted for in precision models of dark matter substructure (Gao et al., 2010).

2. Subhalo Mass Function and Redshift Evolution

The cumulative subhalo abundance as a function of fractional mass $\mu = M_{\text{sub}}/M_{200}$ follows a nearly universal functional form:

$$N(>\mu) = \eta \left(\frac{\mu}{\mu_0}\right)^{a} \exp\left[-\left(\frac{\mu}{\mu_{\text{cut}}}\right)^b\right],$$

with $a \simeq -0.94$ and $b \simeq 1.2$ for clusters, and best-fit parameters for normalization $\eta$ and cutoff $\mu_{\text{cut}}$ depending on host mass and redshift. For instance, in clusters, $\eta = 0.0110$, $\mu_{\text{cut}}=0.10$, while for groups, $\eta$ and $\mu_{\text{cut}}$ decrease to $\sim0.0092$ and $0.07$ at $z=0$.

Redshift evolution is substantial: for group-scale haloes, the subhalo abundance drops by ≈30% from $z=2$ to $z=0$ (e.g., $\eta=0.0140$ at $z=2$ to $\eta=0.0092$ at $z=0$); for Milky Way–like haloes the decline is somewhat weaker, reflecting earlier formation epochs. This trend is attributed to ongoing tidal disruption, dynamical friction, and deepening of the host potential at late times, which continuously depletes the subhalo population of systems with long infall timescales (Gao et al., 2010).

3. Correlations with Halo Concentration, Formation Redshift, and Intrinsic Scatter

Subhalo abundance exhibits significant but secondary correlations with halo concentration and formation redshift. At fixed host mass, haloes with higher concentration (i.e., higher $V_{\text{max}}/V_{200}$ at $z=0$) or earlier formation epochs possess fewer subhaloes, particularly at high subhalo mass. Quantitative subdivision of a group-scale sample shows that the least concentrated/most recently formed third of host haloes hosts ≈25% more subhaloes (at low mass end) than the most concentrated/earliest forming third, with differences increasing to nearly a factor of 2 above $M_{\text{sub}}/M_{200}>0.01$. Crucially, even after controlling for these parameters, the intrinsic scatter in subhalo abundance remains large (e.g., variance at $\mu\simeq0.01$ is only marginally reduced, from $\sim1.08$ to $\sim1.5$). This suggests that halo concentration and formation epoch are not the dominant variables, highlighting the stochasticity inherent in assembly and merger histories (Gao et al., 2010).

4. Quantitative Impact of Tidal Stripping and Baryonic Physics

Tidal effects play a central role in setting the subhalo population, with consequences for both spatial distribution and internal structure. Upon accretion, subhaloes are truncated and lose mass, with their survival probability and present-day mass retention depending on accretion epoch and radial position:

• Massive infallers ($M_{\text{prog}}/M_{200}\gtrsim10^{-2}$) accreted at higher redshift survive at higher fractions (∼80% for Aquarius, ∼70% for Phoenix), while the global survival fraction from entire accretion histories is 50–55%.
• The fraction of retained mass is small, $\sim$10% on average, and declines closer to the host center, scaling approximately linearly with normalized radius via $f(r/R_{200})=a(r/R_{200})+b$, where $a\simeq0.4$–$0.38$ and $b\simeq0.08$–$0.06$.

Baryonic components drastically affect the number and orbital structure of subhaloes, especially in the innermost tens of kpc. Inclusion of a galactic disk potential leads to enhanced tidal disruption (“disk shocking” and impulsive heating) resulting in local subhalo mass fractions $\lesssim1\%$ near the solar radius, even if global subhalo fractions remain ∼10% as calibrated against dark-matter–only simulations.

In hydrodynamical non-radiative runs, the mass function of surviving subhaloes steepens ($N(>M)\propto M^{-1.1}$ versus $M^{-1.0}$ in DMO), and for subhaloes with $M<10^{10}\,h^{-1}M_\odot$, V$_{\max}$ increases due to higher concentrations pre-infall. This demonstrates the necessity of accounting for baryonic physics even at the non-radiative level (Jia et al., 2020).

5. Subhalo Concentration: Scaling, Environmental Dependence, and Functional Fits

The internal structure of subhaloes is most succinctly captured by a concentration parameter $c_V$, defined not with respect to a specific density profile but as

$$c_V = \frac{\bar{\rho}(R_{\max})}{\rho_c(z)} = 2\left(\frac{V_{\max}}{H(z)R_{\max}}\right)^2,$$

where $R_{\max}$ is the radius of maximum circular velocity. Fitting results (e.g., Uchuu simulations) show:

• $c_V$ increases as subhalo mass or V$_{\max}$ decreases, i.e., lower-mass subhaloes are more concentrated.
• At fixed subhalo mass, $c_V$ is higher the deeper the subhalo resides in the host potential (dependence on $x_{\text{sub}}=r_{\text{sub}}/R_{\text{vir,host}}$), and in more massive hosts.

The combined dependence is fit with

$$c_V(V_{\max}, x_{\text{sub}}) = c_0 \left[1 + \sum_{i=1}^{3}a_i (\log_{10}\frac{V_{\max}}{1~\text{km/s}})^i\right][1 + b\log_{10}x_{\text{sub}}],$$

with empirical $c_0$, $a_i$, and $b$ coefficients provided for both subhaloes and field haloes, and robustly capturing the environmental scaling found in simulations (Moliné et al., 2021, Moliné et al., 2016).

6. Fitting Functions and Model Calibration

State-of-the-art N-body and hydrodynamical simulations provide empirical fitting functions for subhalo abundance and concentration, such as:

• Subhalo mass function (Milky Way–scale):

$$N(>m_{\text{vir}}) = f(x) = \left(\frac{x}{\mu}\right)^{-\alpha}\exp\left[-\left(\frac{x}{\mu_{\rm cut}}\right)^{\beta}\right],\quad x = \frac{m_{\text{vir}}}{M_h}$$

with $\alpha \simeq 1.84$, $\beta \simeq2.91$, $\mu\simeq0.15$, $\mu_{\rm cut}\simeq0.59$.

• Subhalo velocity function:

$$N(>V_{\max}/V_{\max,{\rm host}}) = f(x) = \left(\frac{x}{\mu}\right)^{-\alpha}\exp\left[-\left(\frac{x}{\mu_{\rm cut}}\right)^{\beta}\right]$$

where typical $\alpha\simeq3.91$ and cutoff parameters reflect empirical steepening in the high-V$_{\max}$ end (Moliné et al., 2021).

7. Broader Implications and Applications

The parameterization of dark matter subhalo populations underpins a range of astrophysical applications:

• Calibrating semi-analytic galaxy formation models and predicting satellite galaxy counts.
• Modeling the boost to dark matter annihilation signals (where the annihilation luminosity scales with $\rho^2$ and is heightened in highly concentrated subhaloes).
• Quantifying substructure lensing effects (as subhalo statistics set the expected anomalous flux ratios and image perturbations).
• Informing the expected spatial and orbital distributions of detected Milky Way satellites and substructure-induced gaps in stellar streams.
• Providing a statistical framework for tests of alternative dark matter models (e.g., WDM, SIDM) via departures from the established subhalo mass and concentration functions (Gao et al., 2010, Moliné et al., 2021, Moliné et al., 2016).

The results demonstrate that the subhalo abundance function is close to universal in form, but with moderate systematic variation in amplitude linked to host mass, assembly history, and environment. The normalization, the concentration scaling, and the degree of variance must be treated as quantifiable model uncertainties in interpreting small-scale structure constraints.

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In conclusion, the statistics of the subhalo abundance of dark matter haloes are governed by a combination of mass-dependent and host-dependent scaling laws, secondary but nontrivial correlations with the assembly history and concentration, a universal mass function shape modulated by redshift and environment, and significant residual scatter due to stochastic assembly. High-resolution N-body and hydro simulations provide both the empirical fits and robust statistical interpretation necessary for current theoretical and observational studies of dark matter substructure (Gao et al., 2010).