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

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

Problem and Motivation
Galaxy clusters serve as critical probes for cosmology and astrophysics, with their mass function depending sensitively on matter density ($\Omega_m$), the amplitude of the power spectrum ($\sigma_8$), and dark energy. However, deriving accurate cluster masses is challenging because standard mass definitions based on overdensity thresholds (e.g., $M_{200m}$) are not directly observable and suffer from "pseudo-evolution" as the background matter density changes with redshift. The splashback radius ($r_{sp}$), defined as the apocenter of the first orbit of infalling matter, offers a physically motivated alternative. It marks a distinct boundary in the density profile where the logarithmic slope reaches a minimum, potentially breaking degeneracies between cosmological parameters and providing a cleaner tracer of halo assembly history. While splashback radii have been detected in various surveys (e.g., SDSS, DES, Planck, ACT), previous observational work has been largely limited to redshifts $z < 1$. This paper addresses the need to extend these measurements to higher redshifts ($z \sim 1.65$) using a large, uniform sample to test the evolution of cluster scales and constrain the mass threshold of the survey.

Methodology
The authors utilize the MaDCoWS2 catalog, a WISE-selected sample of 133,036 cluster candidates at $0.1 < z < 2$ covering over 6000 deg$^2$. The analysis focuses on candidates within $0.4 \le z \le 1.65$, divided into subsamples based on photometric redshift ($z$) and signal-to-noise ratio ($S/N_P$), where $S/N_P$ is defined as the significance of the 3D galaxy density peak.

1. Cross-Correlation: The study employs a modified Landy-Szalay estimator to compute the two-point cross-correlation between MaDCoWS2 cluster candidates and galaxies from the CatWISE2020 catalog. Galaxy positions are derived from infrared data, with strict masking and magnitude cuts ($M_{W1} = -23.01$ AB) to ensure a consistent selection function across redshifts.
2. Density Profile Modeling: The measured projected galaxy surface density profiles are fitted using a theoretical model proposed by Diemer (2023, D23). This model decomposes the density profile into orbiting and infalling components:
   • $\rho_{orbit}(r)$: Describes the inner halo, modeled with a generalized Einasto profile.
   • $\rho_{infall}(r)$: Describes the outer infalling matter, modeled with a power-law term.
     The modification removes the mean background density from the infalling term to suit galaxy density profiles rather than weak-lensing matter profiles.
3. Statistical Analysis: A Bayesian Markov Chain Monte Carlo (MCMC) analysis (using the emcee ensemble sampler) is performed to derive posterior distributions for the model parameters. The splashback radius is identified as the radius corresponding to the steepest logarithmic slope of the 3D density profile.
4. Systematics: The authors assess uncertainties arising from cluster position and redshift errors by generating 10 realizations with Gaussian scatter in position (based on offsets from SPT clusters) and redshift (based on MaDCoWS2 photometric redshift scatter). These systematics are combined with jackknife resampling errors to produce final uncertainties.

Key Results

• Detection: A distinct splashback feature is detected in all subsamples, with the logarithmic slope dropping below $-3$. The measured splashback radii span from $0.89^{+0.02}_{-0.02} h^{-1}$ cMpc at $z \approx 0.45$ to $1.27^{+0.05}_{-0.05} h^{-1}$ cMpc at $z \approx 1.54$.
• Redshift Dependence: In comoving units, the splashback radius increases with redshift. In physical (proper) units, it decreases by approximately 13% from $z=0.4$ to $z=1.65$ for the lowest $S/N_P$ bin.
• Signal-to-Noise Dependence: At fixed redshift, splashback radii increase with $S/N_P$. This correlates with the expectation that higher $S/N_P$ clusters are more massive and have larger physical extents.
• Mass Constraints: Using the empirical $r_{sp}$–mass relation from More et al. (2015), the authors derive $M_{200m}$ values for the MaDCoWS2 sample. They find that the derived mass threshold increases by a factor of roughly 5 from $z=0.46$ to $z=1.55$.
• Discrepancy with Weak Lensing: The $M_{200m}$ values derived from the More et al. (2015) simulation-based relation are systematically lower (by factors of 2.4–2.8) than mass estimates inferred from weak-lensing observations in the literature for similar splashback radii. The authors suggest this may be due to galaxy selection effects, baryonic physics (e.g., dynamical friction), or limitations in the simulation-based calibration.

Significance and Claims
The paper claims to present the first detection of splashback radii in a uniform cluster sample out to $z=1.65$, significantly extending the redshift reach of previous studies which were limited to $z < 1$. The large sample size allows for statistical uncertainties as low as 2% in specific redshift and $S/N_P$ bins.

The authors emphasize that while the results successfully constrain the redshift dependence of the MaDCoWS2 mass threshold, the systematic offset between simulation-based mass estimates and weak-lensing expectations highlights the need for more robust calibration. They conclude that future work combining these galaxy density profiles with weak-lensing shear profiles from facilities like Euclid, Rubin, and Roman is essential to calibrate the splashback radius as a precise mass proxy and to fully understand the evolution of cluster assembly at high redshift. The study demonstrates the statistical power of large photometric surveys in probing the structural properties of the cosmic web, even in the absence of direct mass measurements.