Universal Halo Concentration Models

• Universal halo concentration prescription models predict dark matter halo concentration by scaling variables like mass fluctuation amplitude and peak height.
• They employ methodologies that reparameterize mass using formation history, power spectrum slope, and assembly variables to achieve near-universality across cosmologies.
• Empirical findings show that true universality emerges only after conditioning on halo definition, baryonic corrections, and modified gravity effects.

Searching arXiv for the key papers and related work on universal halo concentration prescriptions. A universal halo concentration prescription is a model that predicts the concentration parameter of dark matter haloes, usually written as $c_\Delta \equiv R_\Delta/r_s$, across halo mass, redshift, and, in stronger formulations, cosmology. In the literature, “universal” has several distinct meanings: a nearly redshift-invariant relation after replacing mass by $\sigma(M,z)$ or peak height $\nu$; a profile–assembly mapping in which concentration is set by mass accretion history; a two-parameter description involving both concentration and profile shape; or, more recently, an empirical collapse relation for the peak or modal concentration using a formation-dependent peak-height variable. The field therefore does not treat universality as a single settled formula, but as a hierarchy of increasingly successful reparameterizations with explicit domain restrictions (Prada et al., 2011, Diemer et al., 2014, Wang et al., 23 Jun 2026).

1. Definitions, conventions, and the meaning of “universal”

Halo concentration is usually defined as the ratio between an outer spherical-overdensity radius and the profile scale radius, $c_\Delta=R_\Delta/r_s$. The reference overdensity matters. Some models are calibrated natively for $c_{200{\rm c}}$, others for virial quantities $c_{\rm vir}$, and others use $M_{200{\rm m}}$ or $M_{500}$ as the mass argument. This choice is not a technicality: Diemer and Kravtsov argue that concentrations are most nearly universal when masses are defined relative to the critical density, especially $c_{200{\rm c}}$, whereas definitions relative to the mean density or virial overdensity show larger apparent redshift evolution (Diemer et al., 2014).

The statistical target also differs across prescriptions. Some models predict the median concentration relation, some the mean, and some the peak or mode of the concentration distribution. Wang et al. explicitly build a universal relation for the peak concentration $c_{\rm peak}$, emphasizing that for skewed lognormal distributions $\sigma(M,z)$0 and giving examples where the peak is lower than the mean by $\sigma(M,z)$1 at low mass and $\sigma(M,z)$2 (Wang et al., 23 Jun 2026). This distinction is central because two prescriptions can use the same halo definition and yet predict systematically different normalizations without being inconsistent.

A further ambiguity concerns what quantity is meant by “halo concentration.” Some papers calibrate concentration from NFW fits to dark-matter-only host haloes; some infer it from observed total-mass distributions; some predict concentration enhancements relative to a GR baseline; and some provide only a measurement prescription rather than a physical $\sigma(M,z)$3 law. The literature therefore uses the phrase “universal halo concentration prescription” for several related but non-identical objects.

2. Early near-universal reformulations in $\sigma(M,z)$4CDM

A major early step was the re-expression of concentration as a function of the linear rms fluctuation amplitude $\sigma(M,z)$5. Prada et al. show that the complicated $\sigma(M,z)$6 pattern becomes a nearly universal U-shaped relation in $\sigma(M,z)$7, with a minimum at $\sigma(M,z)$8, although they also state that exact universality is not achieved because small dependences on redshift and cosmology remain (Prada et al., 2011). Their prescription is therefore best understood as a rescaled $\sigma(M,z)$9CDM median-host-halo concentration model rather than a strict one-parameter law.

A related reformulation replaces mass by the nonlinear mass scale $\nu$0. Child et al. find that in a WMAP7 $\nu$1CDM cosmology the concentration–mass relation is much closer to redshift-independent in the variable $\nu$2 than in either $\nu$3 or $\nu$4, and is well described by a non-power-law function that transitions from an approximate power law to a constant high-mass plateau $\nu$5 at $\nu$6 (Child et al., 2018). This formulation is practically useful because it compresses redshift evolution into the single scaling variable $\nu$7, but Child et al. also state explicitly that the fit is not universal across cosmologies.

These early models established a durable theme: the mass-only form of the concentration relation is misleadingly non-universal, whereas appropriately scaled variables reveal a much tighter structure. They also showed that universality is approximate even inside collisionless $\nu$8CDM, because halo definition, cosmology, and dynamical selection remain relevant.

3. Peak-height, power-spectrum, and assembly-history frameworks

A second line of development ties concentration to halo assembly and the statistics of the linear density field. Ludlow et al. present an assembly-history-based prescription for relaxed CDM host haloes in which the shape of the spherically averaged mass profile and the shape of the main-progenitor mass accretion history, when written in suitable density variables, are nearly the same and are both well approximated by an NFW form (Ludlow et al., 2013). Their practical conversion from assembly-history concentration to NFW concentration is

$\nu$9

and the model predicts a minimum concentration of about $c_\Delta=R_\Delta/r_s$0 for recently formed, very massive haloes.

Diemer and Kravtsov recast the problem in terms of the peak height $c_\Delta=R_\Delta/r_s$1 and the local slope of the linear matter power spectrum $c_\Delta=R_\Delta/r_s$2. They show that the residual non-universality of $c_\Delta=R_\Delta/r_s$3-$c_\Delta=R_\Delta/r_s$4 can be explained by $c_\Delta=R_\Delta/r_s$5, and introduce a seven-parameter model for $c_\Delta=R_\Delta/r_s$6 that matches their $c_\Delta=R_\Delta/r_s$7CDM results to $c_\Delta=R_\Delta/r_s$8 accuracy up to $c_\Delta=R_\Delta/r_s$9 and scale-free models to $c_{200{\rm c}}$0 (Diemer et al., 2014). A central claim of that model is that there is no well-defined floor in concentration; instead, the minimum concentration depends on redshift because the effective spectral slope becomes steeper at higher $c_{200{\rm c}}$1.

Brown et al. extend the scope from concentration alone to the full Einasto profile. They argue that a one-parameter NFW-like description is insufficient, and derive nearly universal relations for both concentration and the Einasto shape parameter $c_{200{\rm c}}$2 by modifying the filter used in the peak-height calculation (Brown et al., 2021). Their final concentration fit is

$c_{200{\rm c}}$3

while the shape parameter is fitted separately as

$c_{200{\rm c}}$4

In this framework, universality is attached to the joint $c_{200{\rm c}}$5 description rather than to concentration alone.

A more recent formulation by Wang et al. targets the peak concentration and incorporates formation information directly into a revised peak-height variable,

$c_{200{\rm c}}$6

They find that the peak concentrations follow a universal, tight relation invariant to redshift, box size, initial power spectrum, and cosmology over the tested range, with the virial-overdensity fit

$c_{200{\rm c}}$7

and a fitted asymptotic floor $c_{200{\rm c}}$8 (Wang et al., 23 Jun 2026). The same paper is explicit that this universality is empirical and approximate: most normalized residuals remain below $c_{200{\rm c}}$9, OCDM can deviate by $c_{\rm vir}$0–$c_{\rm vir}$1 at low $c_{\rm vir}$2, and the concentration floor is a fitted parameter rather than a first-principles constant.

Framework	Native variable	Stated scope
Prada et al.	$c_{\rm vir}$3	Nearly-universal U-shaped median $c_{\rm vir}$4 in standard $c_{\rm vir}$5CDM
Ludlow et al.	MAH and $c_{\rm vir}$6	Relaxed CDM host haloes, assembly-history algorithm
Diemer & Kravtsov	$c_{\rm vir}$7 and $c_{\rm vir}$8	$c_{\rm vir}$9, $M_{200{\rm m}}$0CDM to $M_{200{\rm m}}$1, scale-free tests
Brown et al.	Modified-filter peak heights	Joint universal model for $M_{200{\rm m}}$2 and Einasto $M_{200{\rm m}}$3
Wang et al.	Formation-dependent $M_{200{\rm m}}$4	Peak concentration across broad CDM/WDM/$M_{200{\rm m}}$5CDM/OCDM/SCDM suite

4. Why strict mass-only universality fails

A recurring result is that concentration cannot be fully captured by mass alone. In observed galaxy-scale strong lenses, Napolitano et al. find that the inferred concentration–virial-mass relation is much steeper and higher in normalization than the corresponding relation recovered from EAGLE haloes analyzed with the same pipeline, with the headline comparison $M_{200{\rm m}}$6 for observed lenses and $M_{200{\rm m}}$7 for simulated haloes (Leier et al., 2021). They therefore argue against a single universal one-parameter $M_{200{\rm m}}$8 law that applies simultaneously to observed strong-lens galaxies and typical simulated haloes, and point instead to assembly history and lensing-related sample selection bias as the main remaining explanations.

Wang et al. provide a physical explanation for part of this failure by decomposing concentration evolution into two primary modes: smooth increase due to pseudo-evolution and intense responses to physical merger events (Wang et al., 2020). They show a universal response pattern to mergers when time is scaled by the halo dynamical time, but also argue that merger events are a major contributor to the uncertainty in halo concentration at fixed halo mass and formation time. Even haloes that appear quiescent in a broad sense still experience multiple minor mergers, which drive small deviations from pseudo-evolution and produce effectively irreducible scatter.

These results sharpen the meaning of universality. They suggest that the strongest available prescriptions are universal only after conditioning on variables that encode rarity, power-spectrum shape, and assembly history. A plausible implication is that any strictly deterministic mass-only concentration law is structurally incomplete, even before baryons are introduced.

5. Baryons, modified gravity, and phenomenological departures

Baryons complicate the connection between observed and simulation-calibrated concentration relations. The INFW model of Schaller et al. is explicitly not a new universal concentration–mass relation; it is a phenomenological baryon-modified profile with inner slope $M_{200{\rm m}}$9 and outer slope $M_{500}$0, together with an approximate mapping from an ordinary NFW concentration $M_{500}$1 to an effective INFW concentration $M_{500}$2,

$M_{500}$3

motivated by baryonic cooling (Er, 2012). Its relevance to universal prescriptions is therefore limited to a simple baryonic correction scheme for galaxy haloes.

A complementary caution comes from the fossil group NGC 6482. Humphrey et al. infer $M_{500}$4 and $M_{500}$5 for an uncontracted NFW model, far above the Dutton–Macciò expectation $M_{500}$6 for a relaxed $M_{500}$7CDM halo of that mass (Buote, 2016). They show that weaker adiabatic-contraction variants reduce the DM-only-equivalent concentration while improving baryon-fraction consistency, and argue that observed concentrations in baryon-rich systems should not be naively identified with the concentration of a DM-only progenitor halo.

Modified gravity introduces a different kind of conditional universality. Mitchell et al. derive a universal model for the halo concentration enhancement in Hu–Sawicki $M_{500}$8 gravity relative to GR by expressing the enhancement as a function of the rescaled mass $M_{500}$9, with

$c_{200{\rm c}}$0

and calibrate a universal enhancement curve over $c_{200{\rm c}}$1 in the dark-matter-only regime (Mitchell et al., 2019). By contrast, Ruan et al. argue that standard fitting functions calibrated near WMAP/Planck $c_{200{\rm c}}$2CDM cannot be simply extended into modified gravity and replace analytic universality assumptions with a gravity-specific emulator for $c_{200{\rm c}}$3 in $c_{200{\rm c}}$4CDM, $c_{200{\rm c}}$5, and DGP (Ruan et al., 2023). The contrast between these two approaches shows that universality in non-GR settings is usually response-based or emulator-based rather than an absolute closed-form law.

6. Measurement prescriptions, practical use, and current interpretation

Not every “universal concentration prescription” is a physical predictor. Zhao et al. propose a universal measurement prescription based on the first moment of the density distribution,

$c_{200{\rm c}}$6

which has a monotonic relation with the NFW concentration and can be inverted with a cubic polynomial fit with error $c_{200{\rm c}}$7 (Wang et al., 2023). In ideal NFW haloes with 100 particles, they report systematic errors below $c_{200{\rm c}}$8 for the $c_{200{\rm c}}$9 method, compared with about $c_{\rm peak}$0 bias for conventional NFW profile fitting and about $c_{\rm peak}$1 for the $c_{\rm peak}$2 method. This is a universal estimator in the geometric sense, not a cosmological $c_{\rm peak}$3 law.

For practical implementation, the literature now converges on several non-interchangeable choices that must be specified before any prescription can be used. These include the overdensity definition ($c_{\rm peak}$4, virial, $c_{\rm peak}$5, $c_{\rm peak}$6); the profile family (NFW or Einasto); whether the target statistic is the mean, median, or peak concentration; whether the calibration sample includes all hosts or only relaxed haloes; and whether the halo population is dark-matter-only, baryon-modified, or modified-gravity. These are not bookkeeping issues: each choice alters the fitted relation and the meaning of “universal.”

The broadest synthesis supported by current work is therefore conditional rather than absolute. The most successful universal formulations are those that replace mass by variables tied to fluctuation rarity, power-spectrum shape, and formation history, such as $c_{\rm peak}$7, modified-filter peak heights, $c_{\rm peak}$8, or revised formation-dependent peak heights (Diemer et al., 2014, Brown et al., 2021, Wang et al., 23 Jun 2026). At the same time, observational selection, baryonic contraction or expansion, merger-driven scatter, and gravity-specific screening effects set clear limits on any attempt to compress halo concentration into a single exact mass-only law (Leier et al., 2021, Wang et al., 2020).