Time-Dependent Dark Matter Assembly

• Time-dependent dark matter assembly is the process by which dark matter halos grow and evolve through continuous mergers and accretion within the ΛCDM paradigm.
• Analytical models like the extended Press–Schechter formalism and empirical methods such as Diffmah quantitatively describe halo mass growth, concentration evolution, and subhalo accretion.
• Environmental effects and baryonic interactions significantly modulate halo assembly, influencing galaxy formation, merger rates, and observational signatures.

Time-dependent dark matter assembly refers to the continuous accretion, redistribution, and structural evolution of dark matter halos over cosmic time, as governed by hierarchical structure formation in the ΛCDM paradigm. This process directly links the internal mass distribution, dynamical state, baryonic content, and observable galaxy properties to the detailed merger and accretion histories of individual halos and populations. A robust empirical and theoretical framework underpins the quantification of halo mass growth, concentration evolution, substructure accretion, and environmental modulation, revealing assembly bias and providing granular predictions for both dark matter and baryonic observables.

1. Analytical and Empirical Frameworks for Halo Growth

A comprehensive description of halo mass assembly is provided by analytical models such as the extended Press–Schechter (EPS) formalism and its subsequent refinements. The median mass accretion history, $M_\mathrm{med}(z)$, for a halo of given present-day mass $M_0$ at redshift $z_0$, follows the solution to an ODE involving the collapse threshold $\delta_c(z)$, linear density fluctuation amplitude $\sigma(M)$, and empirical corrections for tidal disruption and unbound material (Yang et al., 2011). The full population scatter is well described by a log-normal distribution in $\ln M$, with the standard deviation parameterized as:

$$\sigma_{10} = 0.12 - 0.15\,\log_{10}\bigl[M_\mathrm{med}(z)/M_0\bigr]$$

Time-dependent assembly histories are thus explicitly specified by the cosmological growth factor and the power spectrum shape.

Empirical models such as Diffmah (Hearin et al., 2021) successfully approximate individual halo assembly with a rolling power-law in cosmic time:

$M(t) = M_0 (t/t_0)^{\alpha(t)}$

where the power-law exponent $\alpha(t)$ smoothly transitions, via a sigmoid, from fast-accretion (major merger-dominated) to slow-accretion (smooth, minor accretion-dominated) regimes. This 3-parameter model captures both population mean and variance, and its differentiable implementation is used for population-level applications.

2. Observational Probes of Time-dependent Dark Matter Assembly

Constraining time-dependent halo assembly observationally relies on mass modeling of galaxies (e.g., rotation curves), dynamical tracers, and the connections between stellar population ages and dynamical mass indicators. Recent studies (Kottur et al., 30 Nov 2025) employ detailed three-component decomposition (Hernquist bulge, exponential disk, NFW halo) of rotation curves, with the onset of dark matter dominance identified at the NFW scale radius $r_s$ (where $d\ln\rho/d\ln r=-3$). The total enclosed mass,

$$M_{\rm DM}(r) = 4\pi \rho_s r_s^3 \Big[\ln \big(1 + \tfrac{r}{r_s}\big) - \frac{r/r_s}{1+r/r_s}\Big]$$

is rigorously compared to I-band luminosity via the calibrated Tully–Fisher relation.

Pearson and Spearman correlation analyses (for a sample of 16 spiral galaxies) robustly demonstrate that galactic age strongly correlates with both dark matter mass ($r \simeq 0.91$) and density ($r \simeq 0.91$), as well as the dark-matter-dominated mass-to-light ratio. These trends are consistent with predictions from cosmological simulations where older systems, having collapsed earlier in high-density peaks, exhibit more concentrated and massive halos—a direct observational validation of time-dependent assembly.

3. Subhalo Accretion, Merger Rates, and Hierarchical Structure

Hierarchical assembly naturally implies time-dependent subhalo accretion. The joint rate $d^2 N_a$ at which halos accrete progenitors of mass $m_a$ at redshift $z_a$ is calculated as:

$$d^2 N_a = \mathcal{N}_a(S_a, \delta_a| S_0, \delta_0)\;d\ln m_a\,d\ln(1+z_a)$$

with all variables defined in terms of the EPS framework (Yang et al., 2011). Integrating over redshift provides the unevolved subhalo mass function; subhalos of lower mass are preferentially accreted earlier, while massive host halos delay accretion due to their lower $S_0$.

Major merger rates—where accretion events with $m_a \geq M_a/3$ are counted—show strong redshift dependence, faithfully reproduced in N-body simulations and analytic models. The full distribution of subhalo accretion times and masses ties directly into galaxy satellite abundance, stellar mass buildup, and environmental signatures in galaxies.

4. Environmental Modulation and Assembly Bias

The time-dependent growth of halos is substantially modulated by environment. Assembly bias arises as halos in denser, high-tidal-field regions form earlier and grow more concentrated, while those in low-density environments accrete mass more gradually (Wang et al., 2010). Tidal field amplitude $t_1$, not local overdensity or morphology, drives variations in halo formation epoch at fixed mass. For low-mass halos ($\lesssim 10^{11.5}\,h^{-1}M_\odot$), assembly redshift $z_f$ varies by $\sim$0.4–0.5 as $t_1$ spans two orders of magnitude. In contrast, massive halos' deep potential wells buffer environmental impact, yielding weak assembly-time modulation.

Quantitatively, mass accretion and potential growth histories are tightly correlated (Bosch et al., 2014): the central potential well deepens rapidly at early times, with $V_\mathrm{max}$ reaching half its present value when only 2% of the final mass has assembled, demonstrating "inside-out" growth.

5. Baryonic Coupling, Contraction, and Galaxy Assembly History

The baryonic assembly history critically impacts and is impacted by dark matter halo growth. Simulation analyses with SPH and moving-mesh codes (Artale et al., 2019) find that dark matter density profiles stabilize when the central baryon fraction reaches $\sim$80% of its final value. The inner angular momentum of the halo freezes once $\sim$60% of the stellar mass accumulates. Early-assembling halos reach this configuration at higher redshift ($z \sim 2$), correlating with early, rapid galaxy formation, while later-assembling halos stabilize at lower redshift ($z \sim 1$).

Halo contraction models must therefore be parameterized by both redshift and stellar mass content in the inner regions. The Gnedin et al. (2004) adiabatic contraction model, appropriately calibrated, best reproduces contraction trends for $M_*/M_\mathrm{halo} \gtrsim 0.02$ across redshift. A fitting ansatz for contraction factor is $\Delta r/r_i \simeq A(z)[\mu(z)]^{\beta(z)}$, with future work refining $A(z)$ and $\beta(z)$ via hydro-simulations.

6. Time-dependent Assembly in Alternative Scenarios and Observables

Exotic scenarios involving time-dependent couplings (e.g., dark energy–CDM interaction) produce striking nonlinear structure formation effects (Baldi, 2010). Depending on the coupling form, halo profiles can become either more concentrated (scale-factor couplings) or more diffuse (field-dependent exponential couplings), altering the concentration–mass relation by $\pm$20–40%. These models address small-scale structure tensions by modifying halo assembly histories distinctly from standard constant-coupling models.

Wave-like dark matter models with quadratic couplings to ordinary matter yield time-dependent local field assembly, with the relaxation timescale for the field profile around dense bodies (such as Earth) being $\tau\sim 2/(\pi)mR^2$ (Burrage et al., 30 Oct 2024). For terrestrial experimental applications, the stationary solution for the dark matter density profile is valid except for extremely light masses ($m\lesssim 10^{-20}$ eV), ensuring accurate reinterpretation of detection bounds.

7. Observational Diagnostics and Classification

Determining the assembly state of halos observationally employs combinations of dynamical and morphological indicators: centroid offset, virial ratio, mean radial velocity, sparsity, and ellipticity (Vallés-Pérez et al., 2023). Classification schemes, optimized for redshift and mass dependence, allow robust separation of relaxed, marginally relaxed, and unrelaxed halos, closely tracking merger histories and recent accretion rates. For clusters, these observables are measurable via X-ray, lensing, and spectroscopic data, enabling empirical mapping to assembly history and time-dependent evolution.

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Time-dependent dark matter assembly is a multi-faceted, physically essential process underpinning the formation and evolution of cosmic structure. Quantitative frameworks connect theoretical modeling to observational diagnostics and simulation-based predictions, revealing the granular linkages between cosmic epoch, environment, baryonic content, and halo substructure in both global and local contexts. The strong statistical and physical correspondence between halo age, concentration, and mass accretion histories is now robustly validated across analytic models, simulations, and direct observational probes (Kottur et al., 30 Nov 2025, Yang et al., 2011, Bosch et al., 2014, Wang et al., 2010, Artale et al., 2019, Hearin et al., 2021).