Amico Meter

[Brunilda Chestnut]

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Context. Weak gravitational lensing offers a powerful method to investigate the projected matter density distribution within galaxy clusters, granting crucial insights into the broader landscape of dark matter on cluster scales.

Methods. Employing a comprehensive Bayesian analysis, we model the stacked excess surface mass density distribution of the clusters. We adopt a model from recent results on numerical simulations that capture the dynamics of both orbiting and infalling materials, separated by the region where the density profile slope undergoes a pronounced deepening.

Results. We find that the adopted profile successfully characterizes the cluster masses, consistent with previous works, and models the deepening of the slope of the density profiles measured with weak-lensing data up to the outskirts. Moreover, we measure the splashback radius of galaxy clusters and show that its value is close to the radius within which the enclosed overdensity is 200 times the mean matter density of the Universe, while theoretical models predict a larger value consistent with a low accretion rate. This points to a potential bias of optically selected clusters preferentially characterized by a high density at small scales compared to a pure mass-selected cluster sample.

Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In the standard structure formation scenario, dark matter haloes represent the building blocks of galaxy assembly processes (White & Rees 1978; Lacey & Cole 1993; Tormen 1998). Systems assemble via repeated merging events, and haloes hosting galaxy clusters are the last forming structures at the top of this hierarchical pyramid (Springel et al. 2001; van den Bosch 2002; Giocoli et al. 2007). In the last three decades, numerical simulations have clarified this picture, allowing us to define the sizes and boundaries of dark matter haloes hosting hundreds or thousands of galaxies (Springel 2010; Bonafede et al. 2011; Cui et al. 2018).

Recently, a different definition of the halo boundary has been proposed (Diemer & Kravtsov 2014; Adhikari et al. 2014), related to the transition region between the orbiting and infalling mass components. It corresponds to the distance at which satellite galaxies and matter, after their first apocentric passage, splash back toward the halo centre: accreting matter is in the process of reversing direction to fall back into the halo under the influence of gravitational forces. The term splashback radius is used to describe this distance and, in practice, it is close to the turnaround radius of matter that previously fell into the halo when following the nonlinear evolution of collapsing spherical shells (Gunn et al. 1972; Fillmore & Goldreich 1984; Bardeen et al. 1986).

It is worth underlining that the density profiles of gravitationally bound structures mirror their hierarchical growth along the cosmic time as a consequence of repeated merging events. In this respect, at a fixed dark matter halo mass, the location of the splashback radius reflects the recent accretion history via the accretion rate. In particular, the splashback radius is very sensitive to the recent accretion rate over the past crossing time and to the concentration parameter (More et al. 2015; Shin & Diemer 2023).

Beyond the splashback radius, the density profile of the halo steeply declines. Therefore, it serves as a crucial parameter in characterising the spatial extent of dark matter haloes and provides insights into the dynamic processes of matter accretion along the cosmic web. This has been explored and validated through both numerical simulations (Diemer & Kravtsov 2014; More et al. 2015; Xhakaj et al. 2020; Pizzardo et al. 2024) and observational data (Baxter et al. 2017; Chang et al. 2018; Murata et al. 2020; Adhikari et al. 2021). It is worth highlighting that the location could also represent a cosmological test which can be used to look for possible signatures beyond standard Λ-cold dark matter (ΛCDM) model (Adhikari et al. 2018; Contigiani et al. 2019; Despali et al. 2020, 2022). At a fixed accretion rate, the splashback radius depends on the dark energy equation of state parameter w; being related to the expansion history of the Universe, it tends to be larger for lower w. Adhikari et al. (2018) have shown that the location of the splashback radius depends on the gravity model too.

Using galaxy cluster data, different authors have explored the galaxy cross-correlation (Chang et al. 2018; Zrcher & More 2019; Shin et al. 2019, 2021; Murata et al. 2020; Rana et al. 2023; Contigiani et al. 2023), that probes the satellite galaxy distribution to model the projected matter density profiles and constrain the splashback radii. However, it is worth noticing that in these methods, the radius at which the density profile exhibits a sharp steepening of the slope depends on the selection in galaxy magnitudes and colours (Murata et al. 2020). Red and luminous galaxies are typically more centrally concentrated than the blue and faint ones, resulting in different splashback radii for the two groups, even if consistent within measurement uncertainties (Murata et al. 2020).

Following Umetsu & Diemer (2017), Contigiani et al. (2019), Shin et al. (2021), Rana et al. (2023), we base our analysis on weak gravitational lensing of optically selected clusters to model their projected matter density distribution. We believe this method provides an unbiased definition of the radius at which the density profile slope steepens, free from possible systematic uncertainties related to the satellite galaxy selection. Nonetheless, possible systematic uncertainties may arise from selection effects that depend on the specific observables (Wu et al. 2022) that could select systems in particular dynamical states and accretion rates (Shin & Diemer 2023).

This paper is organised as follows. In Sect. 2, we present the photometric properties of the KiDS-DR3 data, and in Sect. 3, we measure the stacked excess surface mass density signal in different redshift and amplitude bins. In Sect. 4, we introduce the density profiles used to model the cluster weak-lensing data, and in Sect. 5, we show our results on the characterization of the splashback radii of our cluster sample, together with the comparison with previous works. Finally, in Sect. 6, we summarize and discuss our results.

In particular, here we use the KiDS-DR3 catalogue (de Jong et al. 2017) and the AMICO cluster sample (Bellagamba et al. 2018). This comprises about 100 000 sources per square degree, resulting in about 50 million sources over the full survey area. KiDS-DR3 has a sky coverage of approximately 447deg2, composed of 440 survey tiles. It includes photometric redshifts and the corresponding probability distribution functions, a globally improved photometric calibration with respect to the previous releases, weak-lensing shear catalogues (Hildebrandt et al. 2017), and lensing-optimised image data. Source detection, positions, and shape parameters used for weak-lensing measurements are all derived from the r-band images, while magnitudes are measured in all filters using forced photometry.

The 440 survey tiles of DR3 mostly cover a small number of large contiguous areas. This enables a refinement of the photometric calibration that exploits both the overlap between observations within a filter, as well as the stellar colours across filters (de Jong et al. 2017)1.

At scales larger than R200m, the shear signal originates from the correlated matter distribution around the galaxy clusters. In practice, the 2-halo term characterises the cumulative effects of the large-scale structures in which galaxy clusters are embedded. The uncorrelated matter distribution along the line-of-sight produces only a modest random contribution to the stacked shear signal, accounted for in the error budget. We model the 2-halo term following the recipe by Oguri & Takada (2011).

In the DK14 model (Diemer & Kravtsov 2014), the total matter density present in collapsed structures is constructed using the Einasto profile (Einasto 1965), which has an extra parameter compared to NFW that captures the slope variation in the inner region. DK14 used the Einasto model to describe the orbiting material in the internal part of the halo and included a new infall term to characterize the matter at larger distances from the centre. The DK14 profile can then be read as:

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