Group Therapy for Halos: Advancing Halo Mass Estimation for Galaxy Groups

TL;DR

This study tackles the challenge of estimating group-scale dark matter halo masses by developing two calibrated estimators: a virial-theorem-based mass with a Bayesianly calibrated coefficient $A$ that accounts for velocity-dispersion and projected-radius biases, and a SHMR proxy using the sum of the three most massive galaxies’ stellar masses via a double power-law relation. Across three semi-analytic models, the calibrated virial method yields near-zero systematic bias with modest scatter ($\sim$0.20 dex), while the SHMR achieves the highest precision ($\sim$0.12–0.14 dex) but exhibits notable model dependence due to baryonic physics. When applied to the observational SGP group sample, these estimators enable construction of an empirical halo mass function and exploration of quenched fractions in the stellar-mass–halo-mass plane, illustrating practical cosmological and galaxy-evolution applications. The authors provide clear guidance: use the calibrated virial theorem for unbiased HMF work in surveys like GAMA ($i<19.2$, $z<0.1$) and the SHMR for high-precision halo masses across broader catalogues up to $z<0.3$ where reliable stellar masses exist. Together, these calibrated tools prepare upcoming wide-area spectroscopic surveys (e.g., WAVES, DESI, 4MOST) to robustly link galaxies to their dark matter haloes and to constrain cosmology.

Abstract

Accurate estimation of dark matter halo masses for galaxy groups is central to studies of galaxy evolution and for leveraging group catalogues as cosmological probes. We present a calibration and evaluation of two complementary halo mass estimators: a dynamical estimator based on the virial theorem, and an empirical relation between the sum of the stellar masses of the three most massive group galaxies and the halo mass (SHMR). Using state-of-the-art semi-analytic models (SHARK, SAGE, and GAEA) to generate mock light-cone catalogues, we quantify the accuracy, uncertainty, and model dependence of each method. The calibrated virial theorem achieves negligible systematic bias (mean $Δ$ = -0.01 dex) and low scatter (mean $σ$ = 0.20 dex) with no sensitivity to baryonic physics. The calibrated SHMR yields the highest precision (mean $Δ$ = 0.02 dex, mean $σ$ = 0.14 dex) but shows greater model dependence due to sensitivity to baryonic physics across the models. We demonstrate applications to observational catalogues, including the empirical halo mass function and mapping quenched fractions in the stellar mass-halo mass plane. We provide guidance: the virial theorem is recommended for GAMA-like surveys (i < 19.2) at z < 0.1 where minimal model dependence is required, while the SHMR is optimal for high-precision halo mass estimates across diverse catalogues with limits of z < 0.3. These calibrated estimators will aid upcoming wide-area spectroscopic surveys in probing the connection between galaxies and their host dark matter halos.

Figures (9)

• Figure 1: Comparison of the SMF from Shark to well-established observational SMFs and SMFs produced by observational data from VanKempen2024. The figure shows the SMF ($\log \Phi$) versus stellar mass ($\log M_{\star}$) for Shark v2.0 (blue circles) alongside canonical constraints from Baldry2012 (black dashed line) and Driver2022 (grey dashed line). Red triangles, orange diamonds, and green crosses represent our observational data from VanKempen2024 (SGP-2dF, G23-GAMA, and G23-2dF datasets, respectively). Error bars indicate Poisson uncertainties. Shark demonstrates excellent agreement with the established SMFs and across our observational datasets, with only minor deviations. The observational datasets exhibit a slight deficit compared to the SMFs from Baldry2012 and Driver2022 at the low-mass end, likely attributable to WISE's inherent limitations in detecting low-surface brightness and low dust content galaxies that contribute to the low-mass regime. The consistency between Shark v2.0 and observational constraints validates our selection of this semi-analytic model as the foundation for calibrating the baryon-halo mass relation.
• Figure 2: Comparison of the HMF derived from Shark with analytic and observational benchmarks. The blue data points represent the Shark HMF, with error bars indicating Poisson uncertainties in each mass bin. The solid black curve denotes the analytic HMF prediction for the input cosmology of Shark, computed using the hmf Python package Murray2013. The grey dashed line shows the empirical fit from Driver2022, based on GAMA5, SDSS5, and REFLEX II data. The magenta dashed curve corresponds to a Schechter function fit to the Shark HMF. The close agreement between the Shark HMF and the analytic prediction across the full mass range demonstrates the fidelity of Shark in reproducing the expected halo abundance for the adopted cosmology. This concordance underpins the selection of Shark as the calibration basis for our baryon–halo mass relation, ensuring that systematic biases in the halo mass distribution are minimised. Residual deviations at the low-mass end reflect both the intrinsic limitations of group identification in low-multiplicity regimes and the increasing impact of resolution effects. Overall, the robust match between Shark and the analytic HMF validates the use of Shark for calibrating observational mass proxies and for probing the baryon–dark matter connection in group-scale haloes.
• Figure 3: Comparison of halo mass estimates derived from the velocity dispersion relation before and after calibration, across three semi-analytic models: Shark, SAGE, and GAEA. All samples have a manitude limit of $i<19.2$ and a redshift limit of $z<0.1$. Top Row: The uncorrected virial theorem halo mass ($\log M_{\mathrm{halo, \, VT}}$) versus the true halo mass ($\log M_{\mathrm{halo}}$) for each model, with colour indicating the logarithmic number density of halos in each bin. The dashed line denotes the one-to-one relation. Second Row: Corresponding residuals ($\Delta \log M_{\mathrm{halo}} = \log M_{\mathrm{halo, \, VT}} - \log M_{\mathrm{halo}}$) and error bars indicate $16^{th}$ and $84^{th}$ percentiles in each bin. The mean offset and scatter indicated in the legend. Third Row: The MCMC fitted velocity dispersion/group radius calibrated dispersion-based halo mass estimates ($\log$ M$_{\mathrm{halo, \, CVT}}$), compared to the true halo mass ($\log M_{\mathrm{halo}}$). Bottom Row: Corresponding median of residuals ($\Delta \log M_{\mathrm{halo}} = \log M_{\mathrm{halo, \, CVT}} - \log M_{\mathrm{halo}}$) and error bars indicate $16^{th}$ and $84^{th}$ percentiles in each bin. The mean offset and scatter indicated in the legend. Calibration significantly reduces both the systematic offset and scatter, particularly at the low-mass end where uncorrected estimates exhibit substantial bias and increased uncertainty. This improvement is consistent across all three models, demonstrating the robustness of the calibration method in recovering unbiased halo mass estimates, especially for low-multiplicity groups where traditional techniques are less reliable.
• Figure 4: Comparison of baryonic and halo mass relations across three SAMs: Shark, SAGE, and GAEA. The Shark sample has a fainter magnitude limit of $Z<21.2$ and a deeper redshift limit of $z<0.3$ compared to SAGE and GAEA ($i<19.2$ and $z<0.1$) Top Row: The relationship between the sum of the stellar masses of the three most massive group galaxies ($\log \Sigma M_{*, \mathrm{3}}$) and the true halo mass ($\log M_{\mathrm{halo}}$) is shown for each SAM, with the colour scale indicating the logarithm of the number of groups per bin. The black dashed line in each panel represents the MCMC-derived fit to the Shark data, optimised to minimise scatter in halo mass. This same fit is overlaid on the SAGE and GAEA panels to illustrate model dependence. Second Row: The residuals in halo mass ($\Delta \log M_{halo} = \log \Sigma M_{*, \, \mathrm{3}} - \log M_{\mathrm{halo}}$) are shown as a function of the summed stellar mass. Third Row: Estimated halo masses, derived by applying the Shark-calibrated relation to the summed stellar masses, are plotted against the true halo masses from each simulation. The black dashed line denotes the one-to-one relation. The close alignment of the points along this line in all models demonstrates that the Shark-based calibration provides robust halo mass estimates, with low bias and scatter, even when applied to independent SAMs. Bottom Row: The residuals in halo mass are shown as a function of the estimated halo mass. In both residual panels, the colour scale again indicates the logarithm of the number of groups per bin. These residual plots highlight the accuracy and systematic trends of the Shark-calibrated relation across the full range of halo and stellar masses, demonstrating low scatter and minimal bias, and further supporting the robustness of this approach for group-scale halo mass inference across diverse physical models.
• Figure 5: Empirical halo mass function (HMF) for galaxy groups with three or more members, constructed from the group catalogue of VanKempen2024. The HMF is shown separately for groups in the SGP region dominated by 2dF coverage (red triangles), the G23 region with GAMA spectroscopy (orange diamonds), and the G23 region with 2dF spectroscopy (green crosses). Halo masses are estimated using the traditional and calibrated virial theorem method ($\log M_{\mathrm{halo,\,VT}}$ or $\log M_{\mathrm{halo,\,CVT}}$). Number densities are not corrected for survey volume or selection effects, due to the heterogeneous nature of the group sample and the challenges in defining a complete selection function. The solid black curve represents the analytic HMF prediction for the adopted cosmology of Shark, while the grey dashed line shows the empirical fit from Driver2022 based on GAMA5, SDSS5, and REFLEX II data. Error bars reflect Poisson uncertainties.