Brightest group and cluster galaxies as indicators of relaxation

Abstract: Context. Galaxy groups and clusters are widely used to probe the evolution of the cosmic web and cosmology, while assuming that they are relaxed. Aims. We identify the properties of the brightest halo galaxies (BHGs) that can be used to predict the most likely sample of dynamically relaxed host halos. Our work combines thoroughly studied galaxy clusters with less frequently analysed groups. Methods. Our analysis was based on data from the IllustrisTNG simulations. We considered several observationally motivated parameters, including the offset of the BHG from the potential well of the host system ($d_\text{off}$) and from the r-band luminosity centre ($d_\text{lum}$), the distance between the brightest and second-brightest galaxies ($d_{12}$), and the r-band magnitude gap between them ($Δm_{12}$). The primary analysis was performed at redshift $z=0$, with an additional investigation of the redshift evolution of halo relaxation up to $z=1$. The observable proxies were applied to construct a halo mass function (HMF), which was then compared to the HMF of the relaxed sample defined from 3D information commonly used in theoretical approaches. Results. We find that $d_\text{off}$ and $Δm_{12}$ are effective indicators of group and cluster relaxation, particularly when used in combination. The selection criteria of $d_\text{off}<0.05~R_{200}$ and $Δm_{12}>1.6$ mag allowed us to reproduce an HMF that closely matches that of the relaxed halo population. These criteria can be applied to observations up to $z\sim0.2$ within a mass range $\text M_{200}\geq10^{12.5}\text M_\odot$ ($$\text M_\text{*, BHG}\gtrsim10^{10.9}\text M_\odot$$), including groups and clusters in the selection. In this mass range, $15-23\%$ of the systems are considered fully relaxed at $z=0$. The fraction of relaxed haloes decreases with redshift up to $z\sim0.4$, after which the decrease is far slower.

• The paper establishes that combining a brightest-halo-galaxy offset below 0.05R200 with a magnitude gap above 1.6 mag identifies relaxed haloes with 80% efficiency and 70% completeness.
• The paper shows that relaxed haloes host brightest galaxies 1.3–1.5 times more massive than those in the full population, demonstrating that dynamical state biases brightest-galaxy mass estimates.
• The paper finds that unrelaxed systems can bias the halo mass function by up to approximately 40% near group scales, while the proposed observable selection reproduces the relaxed mass function but is reliable only to roughly z≈0.2.

Motivation and scope

Cosmological constraints derived from the abundance of galaxy groups and clusters rest on the assumption that the systems used are dynamically relaxed, i.e. virialised, in equilibrium, and free of strong substructure. The standard theoretical relaxation diagnostics — the subhalo mass fraction $f_\mathrm{sub} < 0.1$, the centre-of-mass displacement $s < 0.07$, and the virial ratio $2T/|U| < 1.35$ (tightened to 1.25 with a surface-pressure correction following Shaw et al.) — are well defined only in simulations, where full 6D phase-space information is available. Observers, in contrast, must rely on projected positions, line-of-sight velocities, and imperfect mass tracers. Kelder and collaborators address this gap by asking which observable properties of brightest halo galaxies (BHGs, covering both BCGs and BGGs) can predict halo relaxation, using the IllustrisTNG suite (TNG100-1, TNG300-1, and TNG-Cluster) across the full halo mass continuum from $10^{12.5}\,\mathrm{M_\odot}$ groups to rich clusters, and from $z=0$ up to $z=1$.

The sample comprises 441 haloes (5838 subhaloes) in TNG100-1, 947 haloes (45658 subhaloes) in TNG300-1, and 349 haloes (93515 subhaloes) in TNG-Cluster at $z=0$, after cuts on stellar mass ($M_* \ge 10^9\,\mathrm{M_\odot}$), richness, virial mass, and box-boundary proximity. Stellar masses in the lower-resolution boxes are corrected using the TNG100-1/TNG100-2 convergence ratio binned in halo mass, following the prescription of Pillepich et al. (2018).

Observable proxies of relaxation at $z=0$

Of the four candidate observables — the BHG offset from the halo potential well ($d_\mathrm{off}$), the offset from the r-band luminosity centre ($s < 0.07$0), the BHG–SBHG separation ($s < 0.07$1), and the magnitude gap ($s < 0.07$2) — two emerge as effective relaxation indicators. The BHG–SBHG separation $s < 0.07$3 shows only a weak anticorrelation with relaxation and is discarded, in contrast to the finding of Casas et al. (2024) that relaxed haloes exhibit large $s < 0.07$4; the authors attribute this discrepancy to their substantially larger sample. The luminosity-centre offset $s < 0.07$5 performs reasonably for lower-mass haloes ($s < 0.07$6), but in massive clusters the relaxed fraction dips or cuts off at small offsets, suggesting that the luminosity centre poorly traces the potential minimum in disrupted massive systems. Consequently, $s < 0.07$7 is also set aside in favour of $s < 0.07$8.

Two fitted relations quantify the diagnostics. The relaxed fraction versus $s < 0.07$9 (in units of $2T/|U| < 1.35$0) follows a stretched exponential,

$2T/|U| < 1.35$1

with $2T/|U| < 1.35$2, $2T/|U| < 1.35$3, and $2T/|U| < 1.35$4. The relaxed fraction versus magnitude gap follows a sigmoid with midpoint at $2T/|U| < 1.35$5 mag, saturating near a relaxed fraction of 1 for large gaps (with the caveat that haloes with $2T/|U| < 1.35$6 mag are so rare that bootstrapping uncertainties are unreliable there).

Combining the two parameters, the authors define a proxy sample using cuts derived from the joint distribution of relaxed and unrelaxed haloes at an 80% relaxed-fraction contour: $2T/|U| < 1.35$7 and $2T/|U| < 1.35$8 mag. This selection yields an efficiency (precision) of 80% and a completeness (recall) of 70%. The cuts remain effective when the 3D offset is replaced by its sky-projected counterpart, confirming observational applicability. Notably, the magnitude-gap threshold is more restrictive than the $2T/|U| < 1.35$9 mag break found by Lopes et al. (2018), while the offset cut is looser than X-ray-based criteria (e.g. $10^{12.5}\,\mathrm{M_\odot}$0), illustrating that combining partially dependent parameters permits more flexible individual thresholds at equal or better purity. A multiplicative 2D probability construction from the two fitted curves is presented but found to be stricter than the directly measured fractions, and it relies on the assumption that $10^{12.5}\,\mathrm{M_\odot}$1 and $10^{12.5}\,\mathrm{M_\odot}$2 are independent, which the authors acknowledge is only a first-order approximation since both co-evolve with the halo.

Dynamical-state dependence of the BHG–halo mass relation

A practically important result is that the BHG stellar mass–halo virial mass relation depends on the dynamical state. Linear fits give $10^{12.5}\,\mathrm{M_\odot}$3 for theoretically relaxed haloes and $10^{12.5}\,\mathrm{M_\odot}$4 for the proxy sample, versus $10^{12.5}\,\mathrm{M_\odot}$5 for the full population. The BHGs of relaxed haloes are more massive by a factor of roughly 1.3–1.5 (median $10^{12.5}\,\mathrm{M_\odot}$6 versus $10^{12.5}\,\mathrm{M_\odot}$7 for the full sample), with the two relaxed subsamples statistically indistinguishable and the difference significant at the 99.7% confidence level. The trends converge at $10^{12.5}\,\mathrm{M_\odot}$8. This confirms, at higher statistical significance than earlier work (e.g. Zenteno et al. 2025; Wen & Han 2013, 2015; Lauer et al. 2014), that ignoring the dynamical state biases BHG-based halo mass estimates, and it motivates the authors' conversion of the mass cut $10^{12.5}\,\mathrm{M_\odot}$9 into a BHG stellar-mass threshold of $z=0$0 for relaxed systems.

Redshift evolution to $z=0$1

The fraction of fully relaxed haloes at $z=0$2 is 15–23%, depending on halo mass. This agrees with the ~20% found by Einasto et al. (2012) but sits at the low end of observational estimates, which range from ~28% (SDSS, Wen & Han 2013) and 31% (eROSITA, Seppi et al. 2023) to ~50% at low redshift (eFEDS, Ghirardini et al. 2022). The authors attribute the discrepancy to observational selection biases — notably cool-core biases in X-ray samples, as demonstrated by Rossetti et al. (2017), who found 29 ± 4% cool-core fractions in SZ-selected versus 59 ± 5% in X-ray-selected clusters — and to differing relaxation definitions.

The relaxation fraction rises toward the present day, with a marked acceleration below $z=0$3, consistent with the drop in the cluster merger rate (Mann & Ebeling 2012) and with clusters reaching their effective radii around this epoch (Chiang et al. 2013). The most massive bins relax far more slowly, consistent with their late assembly: haloes with $z=0$4 attain half their mass near $z=0$5 (Amoura et al. 2021), implying at least one subsequent major merger.

Critically, the observable proxies do not track this evolution faithfully. While the proxy-defined and theoretically defined relaxed fractions match at $z=0$6, they diverge at higher redshift, with the proxy criterion overestimating relaxation in low-mass bins. The median $z=0$7 of relaxed haloes increases and the median $z=0$8 decreases with redshift, so the $z=0$9-calibrated cuts cease to enclose the relaxed population. The authors therefore recommend applying the criteria only up to $z=1$0 and note that the proxies themselves evolve with redshift — an upper limit on their applicability that is a genuine limitation of the method rather than a property of the haloes.

Reconstruction of the relaxed halo mass function

The practical payoff is demonstrated through the halo mass function (HMF). Halo masses are estimated from member galaxies via the virial estimator $z=1$1 using all three velocity components, mimicking observational mass reconstruction. The HMF of theoretically relaxed haloes differs significantly from that of the full population: Mann–Whitney U and Kolmogorov–Smirnov tests at 95% confidence consistently reject the null hypothesis of a common distribution. The relaxation-induced bias in the HMF is mass-dependent and positive throughout, peaking at about 40% (at least 20% including errors) near $z=1$2 for TNG100-1 and just below $z=1$3 for TNG300-1. This quantifies how much unrelaxed systems distort cluster-abundance cosmology if not excluded.

When the HMF is instead constructed from the observable proxy sample ($z=1$4, $z=1$5 mag), it statistically overlaps with the true relaxed HMF in both simulation volumes. The authors conclude that an 80%-efficiency, 70%-completeness selection is sufficient to recover the relaxed-halo mass function, validating the two-parameter cut as a low-cost observational tool.

Limitations and open questions

Several caveats bear directly on these results. First, the criteria have not yet been validated against observational data; the authors plan a test on the GAMA group catalogue, and 4MOST WAVES and 4HS will extend coverage to lower-mass systems. Second, resolution effects may still compromise the lowest-mass haloes, so the reliability floor at $z=1$6 may shift once smaller groups are properly resolved — and the authors speculate that such systems may show stronger relaxation signatures. Third, the independence assumption underlying the multiplicative 2D probability is approximate, and the fitted mean curves slightly underestimate the directly measured relaxation fractions. Fourth, the proxies fail beyond $z=1$7, and no redshift-evolution correction is provided; characterising higher-redshift systems requires further work. Finally, the study leaves open the identification of an accurate general tracer of the potential minimum: the centre of light degrades performance (efficiency 40%, completeness 59% when combined with the magnitude gap), X-ray centres are known to be offset from the potential minimum more often than the BHG (Cui et al. 2016), and the intracluster light is proposed — but not tested here — as a potentially superior tracer of the dark-matter halo shape (Fernandez et al. 2026).

Conclusion

This work establishes a two-parameter observational criterion for halo relaxation — $z=1$8 combined with $z=1$9 mag — calibrated against standard three-criteria theoretical definitions across the full group-to-cluster mass range in IllustrisTNG. The selection achieves 80% efficiency and 70% completeness, suffices to reproduce the relaxed halo mass function, and quantifies a relaxation-induced HMF bias of up to ~40% at group scales. The criteria are applicable to $z=0$0 (or $z=0$1) systems out to $z=0$2. The demonstrated dynamical-state dependence of the BHG stellar mass–halo mass relation, with relaxed-host BHGs more massive by a factor of 1.3–1.5, constitutes an additional systematic that must be accounted for in BHG-based mass estimation.