Surviving Subhalo Peak Mass Function

• SPMF is the mass function of surviving subhaloes defined by their peak or infall mass, providing a more reliable proxy for satellite properties.
• It bridges the unevolved progenitor mass function and the present-day subhalo mass function, underpinning robust abundance matching and dark matter inference.
• Analytic models and simulations quantify the survival fraction and stripping effects, emphasizing the roles of resolution, baryonic physics, and alternative dark matter scenarios.

Searching arXiv for recent and foundational papers on the surviving subhalo peak mass function and closely related subhalo peak/infall mass statistics. The surviving subhalo peak mass function (SPMF) is the mass function of subhaloes that remain bound at a specified epoch, expressed in terms of their peak or infall mass rather than their current bound mass. In the modern literature this object often appears implicitly rather than by name: in unified subhalo models it is the unevolved subhalo mass function multiplied by a survival fraction, whereas in simulation catalogues it is the abundance of surviving objects binned by $M_{\rm peak}$ or $M_{\rm acc}$ rather than by present-day $M_{\rm sub}$ (Han et al., 2015, He et al., 2023, Rodriguez-Puebla et al., 2016). The SPMF sits between two more familiar statistics: the progenitor peak-mass function of all accreted haloes, which includes objects later disrupted or merged, and the evolved subhalo mass function in current bound mass, which folds in both survival and stripping. Because galaxy stellar mass, satellite occupation, and many lensing observables correlate more tightly with $M_{\rm peak}$ or $V_{\rm peak}$ than with present-day dark-matter mass, the SPMF is a central quantity for subhalo demographics, abundance matching, and non-cold-dark-matter inference.

1. Definition and relation to adjacent mass functions

A useful starting point is to separate three distinct distributions: the mass function of all accreted progenitors, the mass function of surviving subhaloes expressed in peak mass, and the mass function of surviving subhaloes expressed in present-day bound mass. In the terminology of Han et al. and its WDM extension, these correspond to the unevolved SHMF, the SPMF, and the final SHMF, respectively (Han et al., 2015, He et al., 2023).

Quantity	Objects counted	Mass variable
Unevolved SHMF / progenitor PMF	All accreted progenitors	$m_{\rm acc}$ or $m_{\rm peak}$
SPMF	Surviving subhaloes only	$m_{\rm acc}$ or $m_{\rm peak}$
Final SHMF	Surviving subhaloes only	$m \equiv m(z)$

In the unified subhalo model language, the formal definition is

$M_{\rm acc}$0

where $M_{\rm acc}$1 is the infall mass, defined as “the maximum mass the subhalo ever had before accreted into the current host halo,” and $M_{\rm acc}$2 is the survival fraction to the epoch of interest (He et al., 2023). In Han et al.’s earlier CDM formulation, the same construction appears with $M_{\rm acc}$3 identified with peak mass and a nearly mass- and radius-independent survival fraction, so that the SPMF is simply the unevolved peak-mass function scaled by $M_{\rm acc}$4 (Han et al., 2015).

This distinction is essential because the “subhalo peak mass function” in some recent work refers to all progenitors ever accreted. Jiang et al. explicitly define their PMF as the distribution of all progenitor halos that have ever merged into a given host and include disrupted and merged objects, so their PMF is an accreted peak-mass function rather than an SPMF in the strict sense (Jiang et al., 27 Feb 2025). By contrast, in Planck-cosmology ROCKSTAR/Consistent-Trees catalogues, surviving subhaloes at a snapshot carry fields such as M_acc, M_peak, V_acc, and V_peak; binning the surviving objects by M_peak gives a direct empirical SPMF (Rodriguez-Puebla et al., 2016).

2. CDM formulations and analytic structure

In CDM, the simplest and most influential description of the SPMF is that its shape closely tracks the unevolved SHMF. Han et al. define the accretion or peak mass $M_{\rm acc}$5 as the maximum bound mass attained over the history of a subhalo and write the unevolved SHMF as

$M_{\rm acc}$6

with $M_{\rm acc}$7 and $M_{\rm acc}$8 for AqA1 (Han et al., 2015). In the cluster-calibrated version summarized by He et al., the low-mass limit is

$M_{\rm acc}$9

with $M_{\rm sub}$0 and $M_{\rm sub}$1 (He et al., 2023). In both cases the low-mass slope is close to $M_{\rm sub}$2 in $M_{\rm sub}$3.

The survival factor is then the only additional ingredient needed to turn the unevolved peak-mass function into an SPMF. Han et al. model the conditional stripping/disruption distribution as a mixture of a disrupted component and a lognormal survivor component,

$M_{\rm sub}$4

with $M_{\rm sub}$5 and $M_{\rm sub}$6 in the well-resolved CDM regime (Han et al., 2015). In the later Han et al. formalism summarized by He et al., CDM uses a nearly constant $M_{\rm sub}$7, so

$M_{\rm sub}$8

and the SPMF inherits the unevolved slope almost exactly (He et al., 2023).

A complementary, directly usable parameterization in Planck $M_{\rm sub}$9CDM was provided by Rodríguez-Puebla et al. for surviving subhaloes binned by $M_{\rm peak}$0. Their global peak-mass function is written relative to the distinct-halo mass function as

$M_{\rm peak}$1

where $M_{\rm peak}$2 can be taken as $M_{\rm peak}$3, $M_{\rm peak}$4 is a quartic in scale factor, and

$M_{\rm peak}$5

For the $M_{\rm peak}$6 case they give $M_{\rm peak}$7, $M_{\rm peak}$8, $M_{\rm peak}$9, $V_{\rm peak}$0, $V_{\rm peak}$1, $V_{\rm peak}$2, $V_{\rm peak}$3, and $V_{\rm peak}$4 (Rodriguez-Puebla et al., 2016). The same paper also gives a conditional per-host cumulative form for surviving peak-mass-selected subhaloes,

$V_{\rm peak}$5

with $V_{\rm peak}$6, $V_{\rm peak}$7, $V_{\rm peak}$8, $V_{\rm peak}$9, and $m_{\rm acc}$0 with $m_{\rm acc}$1 for $m_{\rm acc}$2 (Rodriguez-Puebla et al., 2016).

3. Present-day survivor statistics as boundary conditions

Although Gao et al. do not define an SPMF explicitly, they provide the principal endpoint constraints that any viable SPMF must reproduce after stripping and disruption: the abundance, mass fraction, scatter, and host-property dependence of the surviving current-mass subhalo population (Gao et al., 2010). They fit the cumulative number of surviving subhaloes above fractional current mass $m_{\rm acc}$3 with

$m_{\rm acc}$4

fixing $m_{\rm acc}$5 and $m_{\rm acc}$6. At $m_{\rm acc}$7 the fitted amplitudes are $m_{\rm acc}$8 for cluster haloes, $m_{\rm acc}$9 for group haloes, and $m_{\rm peak}$0 for galaxy haloes, implying only a weak host-mass dependence: cluster haloes have about $m_{\rm peak}$1 more surviving subhaloes than galaxy haloes at fixed $m_{\rm peak}$2 (Gao et al., 2010).

The mass-fraction constraint is equally important. Gao et al. find that subhaloes with $m_{\rm peak}$3 contain about $m_{\rm peak}$4 of the mass within $m_{\rm peak}$5 in cluster haloes, while Aquarius Milky-Way-mass haloes contain about $m_{\rm peak}$6 above $m_{\rm peak}$7, with galaxy haloes about $m_{\rm peak}$8 lower than clusters at the same $m_{\rm peak}$9 (Gao et al., 2010). Any SPMF plus stripping kernel must integrate to these survivor mass fractions.

Redshift and assembly history further constrain the SPMF-to-SHMF mapping. At fixed host mass, the surviving subhalo abundance decreases by about $m_{\rm acc}$0 between redshift 2 and 0; at $m_{\rm acc}$1 the abundance is about $m_{\rm acc}$2 lower than at $m_{\rm acc}$3, $m_{\rm acc}$4 lower than at $m_{\rm acc}$5, and $m_{\rm acc}$6 lower than at $m_{\rm acc}$7 for group haloes (Gao et al., 2010). More concentrated and earlier-forming haloes have fewer subhaloes, with differences of about $m_{\rm acc}$8 at low subhalo masses and up to a factor of $m_{\rm acc}$9 at $m_{\rm peak}$0, yet conditioning on concentration or formation redshift does not substantially reduce the residual scatter (Gao et al., 2010). A plausible implication is that the SPMF cannot be specified only by host mass and a single formation-time proxy; detailed assembly history remains relevant.

Semi-analytic evolution models supply a physical interpretation of these endpoint statistics. Jiang and van den Bosch’s semi-analytical framework uses merger trees plus an orbit-averaged mass-loss law and finds that the evolved SHMF has a universal shape while the normalization tracks the host’s dynamical age; they also infer that an average dark matter subhalo loses in excess of $m_{\rm peak}$1 of its infall mass during its first radial orbit within the host halo (Jiang et al., 2014). In the EPS-based framework of Hiroshima et al., different tidal evolution models predict a factor of $m_{\rm peak}$2 difference in the number of subhalos with $m_{\rm peak}$3 mass ratio, while host-mass scatter around its mean does not affect the subhalo mass function when expressed in mass-ratio units (Hiroshima et al., 2022). These results indicate that the low-mass tail of the SPMF is less uncertain as a progenitor statistic than as a survivor statistic.

4. Numerical measurement, tracking, and convergence

The SPMF is only as reliable as the subhalo-tracking methodology used to measure survival. Muldrew et al. showed that overdensity-based halo finders recover strongly radius-dependent subhalo masses: SUBFIND underestimates the mass of a subhalo as it approaches the host centre and can underestimate it by around $m_{\rm peak}$4 even near the virial radius, while neither SUBFIND nor AHF accurately recovers the subhalo very near the host centre (Muldrew et al., 2010). They further showed that the maximum circular velocity is much more stable with radius, but is a poor measure of stripping and is resolution dependent; recovering $m_{\rm peak}$5 within $m_{\rm peak}$6 of the analytic value requires $m_{\rm peak}$7 particles in the halo (Muldrew et al., 2010). For SPMF work this is a direct warning: peak-mass or peak-$m_{\rm peak}$8 statistics are safer than current bound mass, but only if the merger tree retains subhalo identity across dense environments and resolution remains adequate.

The most direct convergence study of the SPMF itself is the Jiutian analysis of Xu, who treats the SPMF as the central abundance statistic for surviving subhaloes binned by $m_{\rm peak}$9 (Xu, 11 Aug 2025). Using HBT+, Xu finds that the raw SPMF converges only for subhaloes with $m \equiv m(z)$0 above $m \equiv m(z)$1 particles; this corresponds to $m \equiv m(z)$2 in Jiutian-300 and $m \equiv m(z)$3 in Jiutian-1G (Xu, 11 Aug 2025). Below this threshold the low-resolution run shows a growing deficit of surviving subhaloes, consistent with artificial numerical disruption.

Xu also shows that including orphan subhaloes according to the merger timescale model of Jiang et al. outperforms other tested prescriptions and recovers the SPMF to $m \equiv m(z)$4 accuracy across a wide range of host masses and $m \equiv m(z)$5 (Xu, 11 Aug 2025). With SPMF-matched orphan selection, the real-space spatial and velocity distributions are recovered to $m \equiv m(z)$6–$m \equiv m(z)$7 accuracy down to $m \equiv m(z)$8–$m \equiv m(z)$9, whereas convergence below $M_{\rm acc}$00 remains challenging and redshift-space multipoles remain more difficult because poorly modeled small projected separations contaminate much larger scales through Fingers-of-God (Xu, 11 Aug 2025). A practical consequence is that any empirical SPMF used for galaxy clustering or redshift-space inference should be accompanied by an orphan treatment and explicit scale cuts.

5. Warm, fuzzy, and baryonic modifications

The WDM extension of the unified subhalo model makes the SPMF modification explicit. In that framework,

$M_{\rm acc}$01

but now both factors differ from CDM: the unevolved SHMF is suppressed below the half-mode mass $M_{\rm acc}$02, and the survival fraction decreases toward low infall mass because WDM subhaloes are less concentrated at accretion and more vulnerable to stripping and disruption (He et al., 2023). He et al. write the WDM unevolved suppression as

$M_{\rm acc}$03

and model the survival fraction as an explicit function of $M_{\rm acc}$04, $M_{\rm acc}$05, and $M_{\rm acc}$06, so the WDM SPMF is doubly suppressed relative to CDM: fewer low-mass progenitors form, and fewer of those progenitors survive (He et al., 2023). The low-$M_{\rm acc}$07 turnover of the SPMF is therefore a direct tracer of dark-matter microphysics.

Stuecker et al. generalize this logic to a broad class of non-cold-dark-matter models by parameterizing the linear-power cutoff through the half-mode scale $M_{\rm acc}$08 and a sharpness parameter $M_{\rm acc}$09, and then fitting the suppression of halo and satellite mass functions relative to CDM (Stücker et al., 2021). For satellite subhaloes they provide simple power-law scalings for the masses where the suppression reaches $M_{\rm acc}$10, $M_{\rm acc}$11, and $M_{\rm acc}$12: $M_{\rm acc}$13 with $M_{\rm acc}$14, $M_{\rm acc}$15, and $M_{\rm acc}$16 (Stücker et al., 2021). They also show that the regime where the suppression is smaller than a factor of 20 is robust to halo-definition ambiguities, while the strongly suppressed regime depends strongly on what one means by a halo or subhalo (Stücker et al., 2021). For SPMF construction this suggests a conservative workflow: take a CDM SPMF and graft onto it a non-CDM suppression only in the regime where the survivor statistics are definition robust.

Fuzzy or ultralight bosonic dark matter can be discussed in the same language. Schutz summarizes semi-analytic FDM results in which the surviving SHMF is suppressed relative to CDM through both the primordial cutoff and soliton-core tidal evolution, and argues that the observed abundance of substructure in stellar streams and strong lensing requires a nearly CDM-like surviving population down to $M_{\rm acc}$17, implying $M_{\rm acc}$18 for FDM comprising all dark matter (Schutz, 2020). A plausible implication for the SPMF is that any viable FDM SPMF must remain close to the CDM SPMF over the mass range currently probed by satellite counts, lensing, and streams.

Baryonic physics perturbs the SPMF in a different way: not primarily by changing the primordial progenitor spectrum, but by altering host masses, central potentials, stripping, and disruption. In hydrodynamical runs Despali and Vegetti find that EAGLE has about $M_{\rm acc}$19 fewer subhaloes than DMO at $M_{\rm acc}$20–$M_{\rm acc}$21, whereas Illustris shows up to $M_{\rm acc}$22 suppression at the low-mass end; the corresponding SHMF slopes are $M_{\rm acc}$23 for EAGLE, $M_{\rm acc}$24 for Illustris, and $M_{\rm acc}$25 for DMO (Despali et al., 2016). They further find that DMO and hydro EAGLE are compatible with observational substructure lensing constraints, while hydro Illustris is not (Despali et al., 2016). Since these are current-mass survivor statistics, the direct inference is that the baryonic SPMF is also suppressed, but less strongly than the present-mass SHMF because some of the hydro–DMO difference comes from enhanced stripping at fixed $M_{\rm acc}$26 rather than from complete loss of survivors.

6. Applications, interpretation, and persistent ambiguities

The main application of the SPMF is to any problem where peak mass is a better latent variable than current bound mass. Rodríguez-Puebla et al. explicitly motivate $M_{\rm acc}$27 and $M_{\rm acc}$28 as the appropriate variables for surviving subhaloes because current $M_{\rm acc}$29 and $M_{\rm acc}$30 are poor proxies for satellite galaxy properties after stripping; their Planck-calibrated global and conditional fitting functions therefore provide a direct route from a surviving peak-mass distribution to abundance matching and HOD/HAM constructions (Rodriguez-Puebla et al., 2016). Han et al. likewise emphasize that selecting subhaloes by peak or infall mass removes the apparent “anti-bias” of current-mass-selected subhaloes, because peak-mass-selected survivors trace the host density profile much more closely (Han et al., 2015).

The SPMF also encodes the hierarchical origin of satellites. Jiang et al. show that, in the accreted progenitor PMF, higher-level subhaloes dominate at progressively lower peak mass and are more likely to originate from major mergers than lower-level ones; at fixed final mass ratio, higher-level and higher-mass-ratio subhaloes tend to be accreted more recently (Jiang et al., 27 Feb 2025). This does not by itself yield an SPMF, because survival has not yet been applied, but it fixes the initial conditions from which a level-resolved SPMF must be derived.

Several recurrent misconceptions follow from conflating these different layers of description. First, the near-universality of the low-mass slope does not imply a universal normalization or negligible scatter. Gao et al. show large halo-to-halo variance in the survivor mass fraction and substantial scatter in abundance even at fixed host mass, concentration, and formation redshift (Gao et al., 2010). Hiroshima et al. similarly find that Poisson fluctuation dominates the number count at mass ratios of order $M_{\rm acc}$31, while different tidal evolution models produce a factor of $M_{\rm acc}$32 difference in low-mass survivor counts below $M_{\rm acc}$33 (Hiroshima et al., 2022). Second, the present-day SHMF cannot be inverted into an SPMF without an explicit stripping and disruption model; Gao et al.’s current-mass endpoint statistics constrain the allowed SPMF but do not uniquely determine it (Gao et al., 2010). Third, the low-mass end of a non-CDM SPMF is not merely a shifted CDM power law once the surviving abundance is suppressed by much more than a factor of 20, because in that regime the very definition of a bound halo or subhalo becomes ambiguous (Stücker et al., 2021).

In this sense the SPMF is best regarded as an intermediate statistical object. It is more directly connected than the present-mass SHMF to galaxy observables such as stellar mass, luminosity, and satellite occupation, yet it still requires a dynamical model of survival, stripping, and numerical completeness to be measured or predicted consistently. The modern literature supplies all of these components separately: direct peak-mass survivor catalogues and fitting functions in Planck $M_{\rm acc}$34CDM (Rodriguez-Puebla et al., 2016), unified analytic models in which the SPMF is the unevolved SHMF multiplied by a survival fraction (Han et al., 2015, He et al., 2023), endpoint constraints from the present-day surviving current-mass population (Gao et al., 2010), and explicit convergence tests showing where the statistic can and cannot be trusted numerically (Xu, 11 Aug 2025).