Universal Fitting Formulae for the Peak Concentration of Dark Matter Halos

Abstract: The prediction of the structural properties of dark matter halos is crucial for studies in modern cosmology and galaxy formation. Utilizing a comprehensive suite of N-body simulations spanning diverse cosmologies and box sizes, we derive a universal halo concentration prescription aligned with the excursion set theory framework. The halo peak height parameter was revised to incorporate the linear growth factor at the halo formation epoch, enhancing its physical relevance to assembly history tracking. For fixed halo mass, we fitted the distributions of the halo concentration and revised peak height to lognormal functions to extract their peak values. We find that these peaks follow a universal, tight relation invariant to redshift, box size, initial power spectrum, and cosmology, exhibiting remarkably small scatter. In particular, the relation flattens systematically with increasing revised peak height, approaching an asymptotic value of 2.54, consistent with previous reports of a concentration floor. The fitting formula consistently describes both the mass-concentration and peak height-concentration relations, easily quantifying how these relations depend on redshift and cosmological parameters. This universal framework enables robust prediction of the most probable (peak) halo concentration using our fitting formula or theoretical/semi-analytical mass assembly histories. A software package for calculating peak concentrations is publicly available online.

• The paper introduces a universal analytic framework for predicting dark matter halo peak concentration by incorporating formation history into revised peak height definitions.
• It leverages high-resolution N-body simulations across diverse cosmologies to calibrate lognormal distributions of halo concentration and formation peak height.
• The resulting universal fitting relations outperform earlier models, offering improved tools for semi-analytic galaxy formation and gravitational lensing analyses.

Universal Fitting Formulae for the Peak Concentration of Dark Matter Halos

Overview and Motivation

The paper "Universal Fitting Formulae for the Peak Concentration of Dark Matter Halos" (2606.24071) establishes a universal analytic framework for predicting the peak concentration parameter of dark matter halos, vital for the modeling of nonlinear structure formation and galaxy evolution. The concentration parameter, typically defined through the NFW profile as $c = r_{\mathrm{vir}}/r_s$, encapsulates the structural properties of halos and links halo mass assembly history (MAH) to observable density profiles.

Building on excursion set formalism and extensive N-body simulations across a spectrum of cold and warm dark matter cosmologies, the authors revise halo peak height definitions to include the linear growth factor at the formation epoch, enabling robust tracking of assembly histories. A central result is the derivation and calibration of lognormal probability distribution functions for both halo concentration and revised peak height, facilitating extraction of their most probable (mode) values for fixed halo mass bins.

Methodological Framework

Simulation Suite and Data Handling

The study leverages high-resolution N-body datasets from TNG and MultiDark-Planck projects, spanning box sizes (30–2500 $h^{-1}$ Mpc) and cosmological models (LCDM, OCDM, scale-free, WDM, and wCDM). Halo catalogs are constructed with stringent particle-count thresholds ($\geq$2000) to ensure convergent concentration measurements, with the concentration parameter consistently estimated from direct NFW profile fits, mitigating artifacts induced by force/mass resolution and inner fitting radius.

Revised Peak Height and Lognormal Distributions

Concentration is correlated with halo peak height $v$, traditionally a function of mass and redshift, but the canonical definition excludes explicit assembly history. The authors introduce a formation-time-dependent peak height $v_f(M, z, z_f) = v(M, z) D(z_f)$, crucially incorporating the growth factor at the formation redshift $z_f$ (epoch when the main progenitor attains half its final mass). This definition absorbs cosmological dependencies arising from power spectrum variations and growth histories.

Concentration and formation peak height distributions in mass bins are shown to be lognormal, with mode values preferred over mean or median due to their physical representativeness for population statistics. These are designated as $C_{\mathrm{peak}}$ and $v_{f,\mathrm{peak}}$, respectively.

Universal Fitting Relations

Peak Concentration vs. Effective Formation Peak Height

The principal empirical result is a power-law relation between the concentration mode and a corrected effective peak height:

$$C_{\mathrm{peak}} = 12.03\, v_{\mathrm{eff}}^{-1.09} + 2.54$$

where $$v_{\mathrm{eff}} = v_{f,\mathrm{peak}} (1 + a\, M_{\mathrm{hm}}/M)$$ accounts for WDM suppression via the half-mode mass $h^{-1}$0 ($h^{-1}$1). For standard CDM, $h^{-1}$2. The relation is invariant across redshift, box size, power spectrum shape (including scale-free and WDM), and cosmological parameters, yielding residuals $h^{-1}$3, except for OCDM at low $h^{-1}$4.

Calibration is likewise performed for $h^{-1}$5, yielding a flatter relation with slightly lower amplitude and concentration floor. The analytic form guarantees a minimum concentration value, corresponding to an empirical floor observed in high-mass regimes.

Incorporation of Assembly History

By rewriting the formula in terms of key physical parameters:

$h^{-1}$6

where $h^{-1}$7, the universality across cosmologies is ensured by absorbing model-specific assembly times and power spectrum suppression.

A secondary universal fitting formula for $h^{-1}$8 as a function of mass and redshift is also calibrated:

$h^{-1}$9

with $\geq$0 capturing WDM suppression.

Implications and Comparative Analysis

Robustness and Theoretical Significance

The universal fitting formulae yield several crucial implications:

• Redshift and Mass Anti-correlation: Concentration decreases with increasing redshift and mass due to earlier formation times and diminished fluctuation amplitudes.
• Concentration Floor: All cosmological models exhibit an empirical minimum concentration ($\geq$1), consistent with previous analyses and indicating structural self-similarity at high mass.
• WDM Turnover: In WDM cosmologies, low-mass halos can show a positive slope in $\geq$2-$\geq$3 at masses below a threshold determined by $\geq$4, distinct from CDM predictions and theorized by previous WDM studies.
• Parameter Dependencies: The dependence on cosmological parameters ($\geq$5, $\geq$6, $\geq$7, $\geq$8, $\geq$9) is encapsulated within the growth factors and fluctuation amplitude, enabling theoretical and empirical constraints.

Discrepancies With Prior Models

Comparison with established models (Zhao et al., Bhattacharya et al., Dutton & Macciò, Ludlow et al., Ishiyama et al.) demonstrates that earlier fitting relations (canonical $v$0-$v$1 and $v$2-$v$3) lack universality, often failing at extreme redshifts, mass ranges, or non-standard cosmologies. The revised framework explains the origin of mass-dependent upturns and cosmology-induced scatter, attributing failures to omission of MAH or inadequate treatment of power spectrum suppression.

Practical Applications

The analytic relations are directly applicable for semi-analytic galaxy formation modeling, lensing mass reconstructions, and the interpretation of observed halo populations. The public code provided enables immediate adoption for theoretical predictions and simulation analyses.

Future Perspective

The incorporation of formation peak height unifies halo structure modeling under excursion set theory, providing a pathway for improved cosmological inference—especially relevant in the context of next-generation surveys capable of measuring concentration distributions precisely. Extensions could address non-lognormality, baryonic effects, and extreme dynamical states. Similarly, systematic exploration of residuals in non-flat geometries (OCDM) and refinement for a broader parameter space in WDM cosmologies are warranted.

Conclusion

The study systematically demonstrates that peak halo concentration is a universal, tightly-constrained function of a formation-history-sensitive peak height parameter, independent of redshift, mass, cosmology, and power spectrum details—with particular adaptation for WDM suppression scales. The analytic framework outperforms previous concentration prescriptions both in accuracy and theoretical rigor, yields a physical interpretation for concentration floors and turnovers, and provides a robust foundation for future cosmological modeling and inference (2606.24071).