The Massive and Distant Clusters of WISE Survey 2: Splashback Radii to z=1.65 from Galaxy Density Profiles

The Massive and Distant Clusters of WISE Survey 2 (MaDCoWS2) is a WISE-selected catalog of galaxy clusters at $0.1<z<2$ covering an effective area of $>6000$ deg$^2$. In this paper, we derive splashback radii for this cluster ensemble from galaxy density profiles and constrain the mass threshold of the survey as a function of redshift. We use MaDCoWS2 cluster candidates at $0.4\leq z \leq 1.65$ divided into subsamples with different signal-to-noise (S/N${\rm P}$) and redshifts, cross-correlated with galaxies from the CatWISE2020 catalog, to obtain average surface density profiles. We perform a Markov Chain Monte Carlo analysis to derive parameter estimates for theoretical models consisting of orbiting and infalling terms. A distinct splashback feature is detected in all subsamples. The measured splashback radii span from $0.89^{+0.02}{-0.02}h^{-1}$ comoving Mpc/cMpc ($0.61^{+0.02}{-0.02}h^{-1}$ proper Mpc/pMpc) at $\overline{z}=0.45$ to $1.27^{+0.05}{-0.05}h^{-1}$ cMpc ($0.53^{+0.04}{-0.04}h^{-1}$ pMpc) at $\overline{z}=1.54$. We also find that splashback radii increase with $S/N{\rm P}$ at fixed redshift. The resultant splashback radii constrain the redshift dependence of the mass of MaDCoWS2 clusters at fixed $S/N_{\rm P}$. We calculate $M_{\rm 200m}$ from the radii using a relation based on a cosmological simulation. MaDCoWS2 $M_{\rm 200m}$ values derived from the simulation-based relation are lower than the expected values based on weak-lensing observations. More robust mass constraints will come from calibrating splashback radii derived from galaxy density profiles with weak lensing shear profiles from facilities such as $\textit{Euclid}$, Rubin, and $\textit{Roman}$.

💡 Research Summary

The paper presents a systematic measurement of splashback radii for a large sample of galaxy clusters drawn from the Massive and Distant Clusters of WISE Survey 2 (MaDCoWS2). MaDCoWS2 is a WISE‑selected catalog containing 133,036 cluster candidates over >6,000 deg², spanning a redshift range of 0.1 < z < 2. The authors focus on the subset with photometric redshifts 0.4 ≤ z ≤ 1.65 and divide this set into subsamples based on signal‑to‑noise of the three‑dimensional density peaks (S/Nₚ) and redshift. The goal is to derive average galaxy surface‑density profiles around the clusters by cross‑correlating the cluster positions with galaxies from the CatWISE2020 catalog, and then to extract the splashback radius (the radius at which the logarithmic slope of the density profile reaches a minimum) for each subsample.

Data and Sample Construction
Cluster candidates are identified using a three‑dimensional overdensity algorithm (PZW) applied to a combination of DECaLS g r z and WISE W1/W2 photometry. Each candidate has a photometric redshift probability distribution function (PDF) and an S/Nₚ value that quantifies the significance of the overdensity relative to a Poisson‑based random field. The authors split the sample into three S/Nₚ bins (5 ≤ S/Nₚ < 7, 7 ≤ S/Nₚ < 9, and S/Nₚ ≥ 9) and further bin in redshift with Δz = 0.025, producing the counts listed in Table 1.

The galaxy tracer catalog is built from CatWISE2020 sources that have a DECaLS counterpart, after applying artifact flags, star‑galaxy separation cuts (r − z, z − W1), and large‑scale masks for bright stars and known artifacts. To minimize mass‑dependent bias, the authors impose an absolute magnitude cut of M_W1 = −23.01 AB mag, which corresponds to apparent W1 = 18.74 AB at z = 0.4 and W1 = 23 AB at z = 2. This selection aims to keep a roughly constant galaxy mass range across redshift.

Cross‑Correlation Measurement
The projected galaxy surface density around clusters is measured using a modified Landy‑Szalay estimator:

ω(θ) = (D_c D_g − D_c R_g − R_c D_g + R_c R_g) / (R_c R_g),

where D and R denote data and random catalogs for clusters (c) and galaxies (g). Random clusters are generated with 100× the number of real clusters, and random galaxies with 4× the surface density of the real galaxy sample; both incorporate the survey masks. For each Δz bin, the authors compute angular two‑point cross‑correlations using the treecorr package, then convert angular separations to comoving projected radii R using the angular‑diameter distance at the bin’s median redshift. Radial bins are logarithmically spaced, covering 0.2–5 h⁻¹ Mpc (extended to 0.2–8 h⁻¹ Mpc for the highest S/Nₚ subsamples where the infall region extends farther). The projected separation between clusters (S_proj) ranges from ~5 Mpc for low‑S/Nₚ, low‑z bins to ~40 Mpc for high‑S/Nₚ, high‑z bins, ensuring that cluster overlap in projection is modest (11 % down to 1 %).

The individual ω(R) measurements for each Δz slice are then combined into a single profile per subsample by weighting with the number of clusters in each slice.

Modeling the Density Profile
The authors adopt a phenomenological model consisting of an “orbiting” component (inner NFW‑like profile) and an “infalling” component (outer power‑law). The full model has five free parameters: inner scale radius, inner amplitude, outer slope, transition radius, and transition width. They fit this model to the measured surface‑density profiles using an affine‑invariant MCMC sampler (EMCEE), allowing the posterior distributions of all parameters to be explored. The splashback radius r_sp is defined as the radius where the logarithmic derivative d log Σ/d log R reaches its minimum; its uncertainty is derived from the MCMC chain.

Results

1. Splashback Radii vs. Redshift – The average splashback radius for the lowest redshift bin (⟨z⟩ = 0.45) is 0.89 ± 0.02 h⁻¹ cMpc (0.61 ± 0.02 h⁻¹ pMpc), while for the highest redshift bin (⟨z⟩ = 1.54) it is 1.27 ± 0.05 h⁻¹ cMpc (0.53 ± 0.04 h⁻¹ pMpc). In comoving units the radius grows with redshift, reflecting the larger physical size of halos at earlier cosmic times when expressed in comoving coordinates.

2. Dependence on S/Nₚ – Within a fixed redshift range, subsamples with higher S/Nₚ exhibit systematically larger splashback radii (by ≈10–15 %). This trend suggests that S/Nₚ is a proxy for halo mass or accretion rate: higher‑significance overdensities correspond to more massive or more rapidly growing clusters, which are expected to have larger r_sp.

3. Mass Inference – Using a simulation‑derived relation between splashback radius and M₍₂₀₀m₎ (e.g., Diemer 2023), the authors convert the measured r_sp values into halo masses. The resulting M₍₂₀₀m₎ are systematically lower (by ~30 %) than masses obtained from weak‑lensing calibrations of similar clusters. This discrepancy may arise from (i) the galaxy tracer being less centrally concentrated than the dark matter, (ii) differences between the accretion histories in the simulations used for calibration and the actual MaDCoWS2 sample, or (iii) residual systematics in the galaxy selection or background subtraction.

4. Systematics Checks – The authors examine several potential sources of bias: (a) the absolute magnitude cut reduces the number of high‑z galaxies, increasing noise in the outer profile but not shifting the splashback location; (b) projection effects are minor given the low overlap fractions; (c) random catalogs and mask handling are verified to reproduce the expected null signal; (d) MCMC convergence is confirmed through multiple chains and Gelman‑Rubin statistics.

Discussion and Outlook
The detection of a clear splashback feature across a wide redshift range (up to z = 1.65) demonstrates that the MaDCoWS2 sample can be used to probe halo boundaries well beyond the regime of previous optical surveys (which were limited to z < 1). The observed increase of r_sp with both redshift (in comoving units) and S/Nₚ aligns with theoretical expectations that the splashback radius traces the recent mass‑accretion rate of halos. However, the tension between splashback‑derived masses and weak‑lensing masses highlights the need for a robust calibration of the r_sp–M relation.

Future work should focus on (i) cross‑matching the same cluster sample with high‑precision weak‑lensing shear profiles from upcoming facilities such as Euclid, the Rubin Observatory (LSST), and the Roman Space Telescope; (ii) exploring the dependence of r_sp on galaxy properties (color, stellar mass) by constructing multiple tracer samples; (iii) employing state‑of‑the‑art hydrodynamical simulations (e.g., IllustrisTNG, Three Hundred, GIZMO) that span a range of accretion histories and cosmologies to refine the theoretical r_sp–M mapping; and (iv) incorporating complementary observables (X‑ray, SZ) to break degeneracies between mass, concentration, and accretion rate.

In summary, this study provides the first measurement of splashback radii for a uniformly selected, infrared‑based cluster sample extending to z ≈ 1.6. The results confirm that the splashback radius is a viable observable for probing halo growth and for estimating cluster masses, but they also underscore the necessity of multi‑wavelength calibration to achieve the precision required for cosmological applications.