The galaxy-subhalo connection in low-redshift galaxy clusters from weak gravitational lensing

TL;DR

This study measures the connection between satellite galaxy subhalos and their stellar masses in low-redshift clusters using weak gravitational lensing, correcting for observational biases to reveal the subhalo-stellar mass relation.

Contribution

It provides the first direct measurement of the subhalo-to-stellar-mass relation in galaxy clusters, accounting for shape measurement biases and defining subhalo mass based on bound mass.

Findings

Subhalo-to-stellar-mass relation characterized with a power-law fit.

Subhalo masses are about 50% of central galaxy masses at certain stellar masses.

Evidence suggests tidal stripping effects near the cluster scale radius.

Abstract

We measure the gravitational lensing signal around satellite galaxies in a sample of galaxy clusters at $z<0.15$ by combining high-quality imaging data from the Canada-France-Hawaii Telescope with a large sample of spectroscopically-confirmed cluster members. We use extensive image simulations to assess the accuracy of shape measurements of faint, background sources in the vicinity of bright satellite galaxies. We find a small but significant bias, as light from the lenses makes the shapes of background galaxies appear radially aligned with the lens. We account for this bias by applying a correction that depends on both lens size and magnitude. We also correct for contamination of the source sample by cluster members. We use a physically-motivated definition of subhalo mass, namely the mass bound to the subhalo, $m_\mathrm{bg}$, similar to definitions used by common subhalo finders in numerical simulations. Binning the satellites by stellar mass we provide a direct measurement of the subhalo-to-stellar-mass relation, $\log m_\mathrm{bg}/\mathrm{M}_\odot = (11.54\pm0.05) + (0.95\pm0.10)\log[m_\star/(2\times10^{10}\mathrm{M}_\odot)]$. This best-fitting relation implies that, at a stellar mass $m_\star\sim3\times10^{10}\,\mathrm{M}_\odot$, subhalo masses are roughly 50 per cent of those of central galaxies, and this fraction decreases at higher stellar masses. We find some evidence for a sharp change in the total-to-stellar mass ratio around the clusters' scale radius, which could be interpreted as galaxies within the scale radius having suffered more strongly from tidal stripping, but remain cautious regarding this interpretation.

Tables

Table 1. Table 1: Cluster sample. Clusters with a u 𝑢 u and/or i 𝑖 i superscript were observed with INT/WFC in the respective band(s); all other observations were performed with CFHT/Megacam.

Cluster	R.A.	Decl.	Redshift	$r$-band
	(hh:mm:ss.s)	(dd:mm:ss)		seeing (′′)
Abell 7ui	00:11:45.3	32:24:57	0.106	0.60
Abell 21ui	00:20:37.0	28:39:33	0.095	0.63
Abell 85ui	00:41:50.4	$-$09:18:11	0.055	0.62
Abell 119i	00:56:16.1	$-$01:15:19	0.044	0.65
Abell 133ui	01:02:41.7	$-$21:52:55	0.057	0.68
Abell 646ui	08:22:09.5	47:05:53	0.129	0.69
Abell 655	08:25:29.0	47:08:01	0.127	0.65
Abell 754i	09:08:32.4	$-$09:37:47	0.054	0.74
Abell 780ui	09:18:05.7	$-$12:05:44	0.054	0.80
Abell 795i	09:24:05.3	14:10:22	0.136	0.72
Abell 961ui	10:16:22.9	33:38:18	0.124	0.71
Abell 990ui	10:23:39.9	49:08:39	0.144	0.78
Abell 1033ui	10:31:44.3	35:02:29	0.126	0.65
Abell 1068ui	10:40:44.5	39:57:11	0.138	0.61
Abell 1132ui	10:58:23.6	56:47:42	0.136	0.68
Abell 1285ui	11:30:23.8	$-$14:34:52	0.106	0.82
Abell 1361ui	11:43:39.6	46:21:21	0.117	0.61
Abell 1413i	11:55:18.0	23:24:18	0.143	0.66
Abell 1650ui	12:58:41.5	$-$01:45:41	0.084	0.76
Abell 1651ui	12:59:22.5	$-$04:11:46	0.085	0.91
Abell 1781ui	13:44:52.5	29:46:16	0.062	0.73
Abell 1795ui	13:48:52.5	26:35:35	0.062	0.68
Abell 1927ui	14:31:06.8	25:38:02	0.095	0.62
Abell 1991i	14:54:31.5	18:38:33	0.059	0.67
Abell 2029ui	15:10:56.1	05:44:41	0.077	0.65
Abell 2033ui	15:11:26.5	06:20:57	0.082	0.61
Abell 2050ui	15:16:17.9	00:05:21	0.118	0.62
Abell 2055ui	15:18:45.7	06:13:56	0.102	0.61
Abell 2064ui	15:20:52.2	48:39:39	0.108	0.69
Abell 2065ui	15:22:29.2	27:42:28	0.073	0.66
Abell 2069ui	15:24:07.5	29:53:20	0.116	0.62
Abell 2142ui	15:58:20.0	27:14:00	0.091	0.62
Abell 2420ui	22:10:18.8	$-$12:10:14	0.085	0.67
Abell 2426ui	22:14:31.6	$-$10:22:26	0.098	0.73
Abell 2440ui	22:23:56.9	$-$01:35:00	0.091	0.70
Abell 2443ui	22:26:07.9	17:21:24	0.108	0.62
Abell 2495ui	22:50:19.7	10:54:13	0.078	0.61
Abell 2597ui	23:25:19.7	$-$12:07:27	0.085	0.67
Abell 2627ui	23:36:42.1	23:55:29	0.126	0.64
Abell 2670i	23:54:13.7	$-$10:25:08	0.076	0.77
Abell 2703ui	00:05:23.9	16:13:09	0.114	0.60
MKW3Sui	15:21:51.8	07:42:32	0.045	0.65
RX J0736ui	07:36:38.1	39:24:53	0.118	0.70
RX J2344ui	23:44:18.2	$-$04:22:49	0.079	0.70
ZwCl 1023ui	10:25:58.0	12:41:09	0.143	0.72
ZwCl 1215ui	12:17:41.1	03:39:21	0.075	0.86

Table 2. Table 2: Number of galaxies and average properties of stellar mass and cluster-centric distance bins used in Sections   6 and  7 . Sub-columns correspond to the values of the fiducial spectroscopic-plus-red-sequence and the spectroscopic-only samples.

Binning	Bin	Range	$N_{\mathrm{sat}}$		$⟨R_{\mathrm{sat}}/\mathrm{Mpc}⟩$		$\log ⟨m_{\star }/M_{\odot }⟩$
observable	label		spec+RS	spec	spec+RS	spec	spec+RS	spec
$\log \left(m_{\star }/M_{\odot }\right)$	M1	$\left[9.0-9.8\right)$	2144	1010	0.66	0.88	9.51	9.51
	M2	$\left[9.8-10.2\right)$	2017	1315	0.67	0.87	10.01	10.03
	M3	$\left[10.2-10.5\right)$	1387	1146	0.80	0.91	10.36	10.35
	M4	$\left[10.5-10.9\right)$	1178	1052	0.83	0.89	10.67	10.67
	M5	$\left[10.9-11.2\right]$	278	265	0.93	0.98	11.01	11.01
$R_{\mathrm{sat}}\left(\mathrm{Mpc}\right)$	D1	$\left[0.1-0.35\right)$	1346	664	0.23	0.23	9.97	10.20
	D2	$\left[0.35-0.7\right)$	1934	1139	0.52	0.52	10.03	10.20
	D3	$\left[0.7-1.2\right)$	1994	1397	0.90	0.94	10.07	10.22
	D4	$\left[1.2-2.0\right)$	1550	1529	1.55	1.55	10.24	10.25

Table 3. Table 3: Prior ranges and marginalized posterior estimates of best-fitting parameters of the satellite lensing model. Masses are in units of M ⊙ subscript M direct-product \mathrm{M}_{\odot} . Uncertainties correspond to 68 per cent credible intervals. All parameters have flat priors in the quoted ranges. Note that priors are defined in linear, rather than logarithmic, space. The binning schemes are summarized in Table   2 .

Parameter	Prior	$m_{\star }$ bins	$R_{\mathrm{sat}}$ bins
	range	(Section 6)	(Section 7)
$\log ⟨m_{\mathrm{bg},1}⟩$	$[7,14]$	${10.64}_{-0.53}^{+0.39}$	${10.49}_{-0.47}^{+0.35}$
$\log ⟨m_{\mathrm{bg},2}⟩$	$[7,14]$	${11.41}_{-0.21}^{+0.17}$	${11.60}_{-0.15}^{+0.16}$
$\log ⟨m_{\mathrm{bg},3}⟩$	$[7,14]$	${11.71}_{-0.17}^{+0.15}$	${11.55}_{-0.21}^{+0.21}$
$\log ⟨m_{\mathrm{bg},4}⟩$	$[7,14]$	${11.84}_{-0.15}^{+0.15}$	${11.46}_{-0.33}^{+0.25}$
$\log ⟨m_{\mathrm{bg},5}⟩$	$[7,14]$	${12.15}_{-0.20}^{+0.17}$	…
$a_{c,h}$	$[0,10]$	${7.9}_{-1.7}^{+1.4}$	$7,2_{-2.0}^{+2.4}$
$b_{c,h}$	$[-1,1]$	$-{0.74}_{-0.14}^{+0.17}$	$-{0.51}_{-0.31}^{+0.38}$
$\log ⟨M_{h,1}⟩$	$[13,16]$	${15.54}_{-0.51}^{+0.30}$	${15.31}_{-0.34}^{+0.43}$
$\log ⟨M_{h,2}⟩$	$[13,16]$	${15.54}_{-0.37}^{+0.29}$	${15.39}_{-0.32}^{+0.36}$
$\log ⟨M_{h,3}⟩$	$[13,16]$	${15.72}_{-0.43}^{+0.20}$	${15.69}_{-0.76}^{+0.23}$
$\log ⟨M_{h,4}⟩$	$[13,16]$	${15.72}_{-0.49}^{+0.21}$	${15.68}_{-0.55}^{+0.24}$
$\log ⟨M_{h,5}⟩$	$[13,16]$	${15.73}_{-0.37}^{+0.20}$	…

Equations

$$\left(\begin{array}{c}{\gamma }_{\mathrm{t}}\\ {\gamma }_{\times }\end{array}\right)=\left(\begin{array}{cc}-\cos 2\varphi & -\sin 2\varphi \\ \sin 2\varphi & -\cos 2\varphi \end{array}\right)\left(\begin{array}{c}{\gamma }_1\\ {\gamma }_2\end{array}\right),$$

$$\Delta \Sigma (R)\equiv \bar{\Sigma }(<R)-\bar{\Sigma }(R)={\gamma }_{\mathrm{t}}{\Sigma }_{\mathrm{c}},$$

$${\Sigma }_{\mathrm{c}}=\frac{c^2}{4\pi G}\frac{D_{\mathrm{s}}}{D_{\mathrm{l}}{D}_{\mathrm{ls}}},$$

$$\Delta \Sigma =\frac{{\sum }_i{w}_i{\Sigma }_{\mathrm{c},i}{\gamma }_{\mathrm{t},i}}{{\sum }_i{w}_i},$$

$$w_i=\frac{1}{⟨{ϵ}_{\mathrm{int}}^2⟩+({\sigma }_{\gamma ,i}{)}^2}.$$

$$g(R)=\frac{\gamma (R)}{1-\bar{\Sigma }(R)/{\Sigma }_{\mathrm{c}}}=\frac{\Delta \Sigma (R)/{\Sigma }_{\mathrm{c}}}{1-\bar{\Sigma }(R)/{\Sigma }_{\mathrm{c}}}.$$

$${\text{C}}_{mnij}={\Sigma }_{\mathrm{c}}^2⟨{ϵ}^2⟩\frac{{\sum }_s\left(C_{si,m}{C}_{sj,n}+S_{si,m}{S}_{sj,n}\right)}{\left({\sum }_s{Z}_{si,m}\right)\left({\sum }_s{Z}_{sj,n}\right)},$$

$$\begin{array}{rl}{C}_{si}& =-\sum _l{w}_{ls}\cos 2{\varphi }_{ls},\\ S_{si}& =-\sum _l{w}_{ls}\sin 2{\varphi }_{ls},\\ Z_{si}& =\sum _l{w}_{ls},\end{array}$$

$$⟨X⟩={\text{C}}_{\mathrm{tot}}\left[\sum _i{\text{C}}_i^{-1}{\Sigma }_{\mathrm{c},i}{\gamma }_i\right]$$

$${\text{C}}_{\mathrm{tot}}={\left[\sum _i{\text{C}}_i^{-1}\right]}^{-1}$$

$${\gamma }^{\mathrm{obs}}({\theta }_{\mathrm{ls}}|R_{\mathrm{sat}})=\left(1+\mu \right){\gamma }^{\mathrm{true}}({\theta }_{\mathrm{ls}}){\mathcal{B}}^{-1}(R_{\mathrm{sat}})+c,$$

$${\xi }_{\mathrm{s}}\equiv (1/A)\sum _i{w}_i,$$

$$\delta m_{\mathrm{phot}}<-49.04-7.00m_{\mathrm{phot}}+0.333m_{\mathrm{phot}}^2-0.0053m_{\mathrm{phot}}^3,$$

$${\mathcal{F}}_{\mathrm{obsc}}(R_{\mathrm{BCG}})\equiv \frac{{\xi }_{\mathrm{s},\mathrm{cluster}}(R_{\mathrm{BCG}})}{{\xi }_{\mathrm{s},\mathrm{background}}},$$

$$⟨{\xi }_{\mathrm{s},\mathrm{blank}}⟩(⟨r_{\mathrm{h}}^{\star }⟩,\zeta ,A_r)=p_1⟨r_{\mathrm{h}}^{\star }⟩+p_2\zeta +p_3{A}_r+p_4,$$

$$\mathcal{B}(R_{\mathrm{BCG}})=⟨\frac{{\xi }_{\mathrm{s},\mathrm{data}}(R_{\mathrm{BCG}})}{{\xi }_{\mathrm{s},\mathrm{blank}}}⟩{⟨1-{\mathcal{F}}_{\mathrm{obsc}}(R_{\mathrm{BCG}})⟩}^{-1},$$

$$\Delta {\Sigma }_{\mathrm{sat}}(R)=\Delta {\Sigma }_{\star }(R|m_{\star })+\Delta {\Sigma }_{\mathrm{sub}}(R|m_{\mathrm{bg}},c_{\mathrm{sub}})+\Delta {\Sigma }_{\mathrm{host}}(R|M_{\mathrm{h}},c_{\mathrm{h}}),$$

$$\Delta {\Sigma }_{\star }(R|m_{\star })=\frac{m_{\star }}{\pi R^2}.$$

$${\rho }_{\mathrm{NFW}}(r)=\frac{{\delta }_c{\rho }_{\mathrm{m}}}{r/r_{\mathrm{s}}{\left(1+r/r_{\mathrm{s}}\right)}^2},$$

$${\delta }_c=\frac{200}{3}\frac{c^3}{\ln (1+c)-c/(1+c)}.$$

$$c_{\mathrm{h}}(M_{200,\mathrm{h}})=a_{\mathrm{c}}{\left(\frac{M_{200,\mathrm{h}}}{10^{15}{\mathrm{M}}_{\odot }}\right)}^{b_{\mathrm{c}}}$$

$$\begin{array}{rl}{c}_{\mathrm{sub}}(m_{200},x)=& c_0\left(1+\sum _{i=1}^3{\left[a_i\log \left(\frac{m_{200}}{10^8{h}^{-1}{\mathrm{M}}_{\odot }}\right)\right]}^i\right)\\ & \times \left[1+b\log x\right],\end{array}$$

$$⟨r_{\mathrm{sat}}⟩=\frac{{\sum }_{i}n(R_{\mathrm{sat},i})R_{\mathrm{sat},i}}{{\sum }_{i}n(R_{\mathrm{sat},i})},$$

$$\begin{array}{rl}\mathcal{L}=\frac{1}{(2\pi {)}^{k^2/2}}& \prod _{m=1}^k\prod _{n=1}^k\frac{1}{\sqrt{\det ({\text{C}}_{mn})}}\\ & \times \exp \left[-\frac{1}{2}{(\mathbf{O}-\mathbf{E})}_m^T{\text{C}}_{mn}^{-1}{(\mathbf{O}-\mathbf{E})}_n\right],\end{array}$$

$$\frac{m_{\mathrm{bg}}}{{\mathrm{M}}_{\odot }}=10^{11.54\pm 0.05}{\left(\frac{m_{\star }}{2\times 10^{10}{\mathrm{M}}_{\odot }}\right)}^{0.95\pm 0.10}.$$

$$c_{\mathrm{t}}({\theta }_{\mathrm{ls}})=a_{\mathrm{bias}}\exp \left[\frac{-{\theta }^2}{2{\sigma }_{\mathrm{bias}}^2}\right].$$

$$\begin{array}{rl}{a}_{\mathrm{bias}}& =-0.15-0.023(m_{\mathrm{phot}}-16)-0.066\log (s_{\mathrm{eff}}/15\mathrm{pix}),\\ {\sigma }_{\mathrm{bias}}& =6.27-14.01\log (m_{\mathrm{phot}}/16)+7.04\log (s_{\mathrm{eff}}/15\mathrm{pix}).\end{array}$$

$$c(R_{\mathrm{sat}})=A{R}_{\mathrm{sat}}^B,$$

$$\rho (r)=\frac{{\delta }_{\mathrm{c}}{\rho }_{\mathrm{m}}}{r/r_{\mathrm{s}}(1+r/r_{\mathrm{s}}{)}^2}{\left(\frac{r_{\mathrm{t}}}{r^2+r_{\mathrm{t}}^2}\right)}^2,$$

Full text

††pubyear: 2018††pagerange: The galaxy-subhalo connection in low-redshift galaxy clusters from weak gravitational lensing–LABEL:lastpage

1 Introduction

According to the hierarchical structure formation paradigm, galaxy clusters grow by the continuous accretion of smaller galaxy groups and individual galaxies. Initially, each of these systems is hosted by their own dark matter halo, but as a galaxy falls into a larger structure, tidal interactions transfer mass from the infalling galaxy to the new host. The galaxy then becomes a satellite and its dark matter halo, a subhalo.

Detailed studies on the statistics of subhaloes from numerical N-body simulations have revealed that subhaloes are severely affected by their host haloes. Dynamical friction makes more massive subhaloes sink towards the centre faster, while tidal stripping removes mass preferentially from the outskirts of massive subhaloes closer to the centre. These two effects combined destroy the most massive subhaloes soon after infall (e.g., Tormen et al., 1998; Taffoni et al., 2003), a result exaggerated in simulations with limited resolution (e.g., Klypin et al., 1999; Taylor & Babul, 2005; Han et al., 2016). Tidal stripping makes subhaloes more concentrated than field haloes of the same mass (e.g., Ghigna et al., 1998; Springel et al., 2008; Moliné et al., 2017), and counterbalances the spatial segregation induced by dynamical friction (van den Bosch et al., 2016).

One of the most fundamental questions is how these subhaloes are linked to the satellite galaxies they host, which are what we observe in the real Universe. Taking N-body simulations at face value results in serious inconsistencies with observations, the most famous of which are known as the “missing satellites” (Klypin et al., 1999; Moore et al., 1999) and “too big to fail” (Boylan-Kolchin et al., 2011) problems. It has since become clear that these problems may arise because baryonic physics has a strong influence on the small-scale distribution of matter. Energetic feedback from supernovae at the low-mass end, and active galactic nuclei at the high-mass end, of the galaxy population affect the ability of dark matter (sub)haloes to form stars and retain them. In addition, the excess mass in the centre of galaxies (compared to dark matter-only simulations) can modify each subhalo’s susceptibility to tidal stripping (e.g., Zolotov et al., 2012).

Despite these difficulties, given the current technical challenges of generating cosmological high-resolution hydrodynamical simulations (in which galaxies form self-consistently), N-body simulations remain a valuable tool to try to understand the evolution of galaxies and (sub)haloes. In order for them to be applied to real observations, however, one must post-process these simulations in some way that relates subhaloes to galaxies, taking into account the aforementioned complexities (and others). For instance, semi-analytic models contain either physical or phenomenological recipes whether or not to form galaxies in certain dark matter haloes based on the mass and assembly history of haloes (e.g., Bower et al., 2006; Lacey et al., 2016). A different method involves halo occupation distributions (HODs), which assume that the average number of galaxies in a halo depends only on host halo mass. Because they provide an analytical framework to connect galaxies and dark matter haloes, HODs are commonly used to interpret galaxy-galaxy lensing and galaxy clustering measurements through a conditional stellar mass (or luminosity) function (e.g., Seljak, 2000; Peacock & Smith, 2000; Mandelbaum et al., 2006; Cacciato et al., 2009; van den Bosch et al., 2013).

One of the key aspects of these prescriptions is the stellar-to-halo mass relation. While many studies have constrained the stellar-to-halo mass relation of central galaxies (e.g., Hoekstra et al., 2005; Heymans et al., 2006b; Mandelbaum et al., 2006, 2016; More et al., 2011; van Uitert et al., 2011; van Uitert et al., 2016; Leauthaud et al., 2012; Velander et al., 2014; Coupon et al., 2015; Zu & Mandelbaum, 2015), this is not the case for satellite galaxies, whose subhalo-to-stellar mass relation (SHSMR) remains essentially unexplored, and the constraints so far are largely limited to indirect measurements. Rodríguez-Puebla et al. (2012) used abundance matching (the assumption that galaxies rank-ordered by stellar mass can be uniquely mapped to [sub]haloes rank-ordered by total mass) to infer the SHSMR using the satellite galaxy stellar mass function, and Rodríguez-Puebla et al. (2013) extended these results using galaxy clustering measurements. They showed that the SHSMR is significantly different from the central stellar-to-total mass relation, and that assuming an average relation when studying a mixed population can lead to biased results (see also Yang et al., 2009).

Instead, only stellar dynamics and gravitational lensing provide direct ways to probe the total gravitational potential of a galaxy. However, the quantitative connection between stellar velocity dispersion and halo mass is not straightforward (e.g., Li et al., 2013b; Old et al., 2015), and only gravitational lensing provides a direct measurement of the total surface mass density (Fahlman et al., 1994; Clowe et al., 1998). Using deep Hubble Space Telescope (HST) observations, Natarajan et al. (1998); Natarajan et al. (2002); Natarajan et al. (2007); Natarajan et al. (2009) measured the weak (and also sometimes strong) lensing effect of galaxies in six clusters at $z=0.2-0.6$. After fitting a truncated density profile to the ensemble signal using a maximum likelihood approach, they concluded that galaxies in clusters are strongly truncated with respect to field galaxies. Using data for clusters at $z\sim 0.2$ observed with the CFH12k instrument on the Canada-Hawaii-France Telescope (CFHT), Limousin et al. (2007) arrived at a similar conclusion. Halkola et al. (2007) and Suyu & Halkola (2010) used strong lensing measurements of a single cluster and a small galaxy group, respectively, and also found evidence for strong truncation of the density profiles of satellite galaxies, and other strong lensing studies have reached similar conclusions (e.g., Eichner et al., 2013; Monna et al., 2015, 2017). Likewise, Okabe et al. (2014) analyzed the weak lensing signal of subhaloes in the Coma cluster and found that, while group-scale subhaloes show (mild) evidence of sharp truncations at radii $r<200$ kpc, stacked weak lensing measurements of satellite galaxies show no signs of truncation. Similarly, Pastor Mira et al. (2011) found no evidence of truncation of subhaloes in the Millenium simulation (Springel et al., 2005); their density profiles are instead fully consistent with a NFW profile. It is unclear whether the differences between these studies are due to different galaxy (and cluster) samples, different modelling assumptions or, in the case of Pastor Mira et al. (2011), due to the lack of baryonic physics in the simulations.

In addition, recent combinations of large weak lensing surveys with high-purity galaxy group catalogues have allowed direct measurements of the average subhalo masses associated with satellite galaxies using weak galaxy-galaxy lensing (Li et al., 2014, 2016; Sifón et al., 2015a; Niemiec et al., 2017). Like the studies cited above, these studies did not focus on the SHSMR but on the segregation of subhaloes by mass within galaxy groups, by measuring subhalo masses at different group-centric distances. The observational results are consistent, within their large errorbars, with the mild segregation of dark matter subhaloes seen in numerical simulations (Han et al., 2016; van den Bosch et al., 2016). However, it is not clear whether results based on subhaloes in N-body simulations can be directly compared to observations. In fact, van den Bosch (2017) has shown that the statistics of subhaloes inferred from N-body simulations are problematic even to this day, because of severe numerical destruction of subhaloes.

In this work, we present weak gravitational lensing measurements of the total mass of satellite galaxies in 48 massive galaxy clusters at $z<0.15$. Our images were taken with the MegaCam instrument on the Canada-France-Hawaii Telescope (CFHT), which has a field of view of 1 sq. deg., allowing us to focus on very low redshift clusters and take advantage of the $<0.8^{\prime\prime}$ seeing (corresponding to 1.48 kpc at $z=0.1$) typical of our observations. We can therefore probe the lensing signal close to the galaxies themselves, at a physical scale equivalent to what can be probed in a cluster at $z\sim 0.5$ with HST, but out to the clusters’ virial radii. In addition, the low-redshift clusters we use have extensive spectroscopic observations available from various data sets, compiled by Sifón et al. (2015b), so we do not need to rely on uncertain photometric identification of cluster members.

This paper is organized as follows. We summarize the galaxy-galaxy lensing formalism in Section 2. We describe our data set in Section 3, taking a close look at the source catalogue and the shapes of background sources in Section 4. We present our modelling of the satellite lensing signal in Section 5, and discuss the connection between mass and light in satellite galaxies, in the form of the subhalo-to-stellar mass relation and subhalo mass segregation, in Sections 6 and 7, respectively. Finally, we summarize in Section 8.

We adopt a flat $\Lambda$ cold dark matter ($\Lambda$CDM) cosmology with $\Omega_{\mathrm{m}}=0.315$, based on the latest results from cosmic microwave background observations by Planck Collaboration et al. (2016), and $H_{0}=70\,\mathrm{km\,s^{-1}Mpc^{-1}}$. In this cosmology, $10\arcsec=\{9.8,18.4,26.1\}\,{\rm kpc}$ at $z=\{0.05,0.1,0.15\}$. As usual, stellar and (sub)halo masses depend on the Hubble constant as $m_{\star}\sim 1/H_{0}^{2}$ and $m\sim 1/H_{0}$, respectively.

2 Weak galaxy-galaxy lensing

Gravitational lensing distorts the images of background (“source”) galaxies as their light passes near a matter overdensity along the line-of-sight. This produces a distortion in the shape of the background source, called shear, and a magnification effect on the source’s size (and consequently its brightness). The shear field around a massive object aligns the images of background sources around it in the tangential direction. Therefore, starting from a measurement of the shear of an object in a cartesian frame with components $(\gamma_{1},\gamma_{2})$ (see Section 3.3), it is customary to parametrize the shear as

[TABLE]

where $\phi$ is the azimuthal angle of the lens-source vector, $\gamma_{\mathrm{t}}$ measures the ellipticity in the tangential ($\gamma_{\mathrm{t}}>0$) and radial ($\gamma_{\mathrm{t}}<0$) directions and $\gamma_{\times}$ measures the ellipticity in directions $45^{\circ}$ from the tangent. Because of parity symmetry, we expect $\langle\gamma_{\times}\rangle=0$ for an ensemble of lenses (Schneider, 2003) and therefore $\gamma_{\times}$ serves as a test for systematic effects.

The shear is related to the excess surface mass density (ESD), $\Delta\Sigma$, via

[TABLE]

where $\bar{\Sigma}(<R)$ and $\bar{\Sigma}(R)$ are the average surface mass density within a radius111As a convention, we denote three-dimensional distances with lower case $r$ and two-dimensional distances (that is, projected on the sky) with upper case $R$. $R$ and within a thin annulus at distance $R$ from the lens. The critical surface density, $\Sigma_{\mathrm{c}}$, is a geometrical factor that accounts for the lensing efficiency,

[TABLE]

where, $D_{\mathrm{l}}$, $D_{\mathrm{s}}$, and $D_{\mathrm{ls}}$ are the angular diameter distances to the lens, to the source and between the lens and the source, respectively. The ESD for each bin in lens-source separation is then

[TABLE]

where the sums run over all lens-source pairs in a given bin and the weight of each source galaxy is given by

[TABLE]

Here, $\sigma_{\gamma}$ is the measurement uncertainty in $\gamma_{\mathrm{t}}$, which results from the quadrature sum of statistical uncertainties due to shot noise in the images (see Section 3.3) and from uncertainties in the modelling of a measurement bias detailed in Sections 4.2 and A.222In practice, the latter is negligible in most cases. We set the intrinsic root-mean-square galaxy ellipticity, $\langle\epsilon_{\mathrm{int}}^{2}\rangle^{1/2}=(\langle\epsilon^{2}\rangle-\langle\sigma_{\gamma,i}^{2}\rangle)^{1/2}=0.25$ (e.g., Hoekstra et al., 2000; Schrabback et al., 2018), where $\langle\epsilon^{2}\rangle=0.104$ is the observed rms ellipticity. In Equation 4, we use a single value for $\Sigma_{\mathrm{c}}$ for all satellites in each cluster (see Section 4.5).

In fact, the weak lensing observable is the reduced shear, $g\equiv\gamma/(1-\kappa)$ (where $\kappa=\Sigma/\Sigma_{\mathrm{c}}$ is the lensing convergence), but in the weak limit $\kappa\ll 1$ so that $g\approx\gamma$. However, close to the centres of galaxy clusters the convergence becomes significant, so this approximation is not accurate anymore. To account for this, the lensing model presented in Section 5 is corrected using

[TABLE]

Because the gravitational potential of satellites in a cluster is traced by the same background source galaxies, data points in the ESD are correlated. Following equations 13–17 of Viola et al. (2015), we can re-arrange Equation 4 to reflect the contribution from each source galaxy. The data covariance of measurements in a single cluster can then be written as

[TABLE]

where index pairs $m,n$ and $i,j$ run over the observable bins (e.g., stellar mass) and lens-source separation, $R$, respectively, and $C$, $S$ and $Z$ are sums over the lenses:

[TABLE]

where we explicitly allow for the possibility that the source weight, $w$, may be different for each lens-source pair (as opposed to a unique weight per source). This is indeed the case when we consider the corrections to the shape measurements from lens contamination discussed in Sections 4.2 and A, although in practice differences are negligible. As implied by Equation 7, we assign the same $\Sigma_{\mathrm{c}}$ to all galaxies that are part of the same cluster. The total ESD is then the inverse-covariance–weighted sum of the ESDs of individual clusters.

In addition to the data covariance there are, in principle, contributions to the measurement uncertainty from sample variance and from distant large scale structure. By comparing Equation 7 to uncertainties estimated by bootstrap resampling, Sifón et al. (2015a) have shown that the contribution from sample variance is less than 10 per cent for satelite galaxy-galaxy lensing measurements when limited to small lens-source separations ($R\lesssim 2$ Mpc). Since the signal from satellites themselves is limited to $R\lesssim 300\,{\rm kpc}$ (Figure 8; see also Sifón et al., 2015a), in this work we ignore the sample variance contribution to the lensing covariance. Similarly, the distant large scale-structure introduces correlations preferentially on large scales when the signal is averaged around a few positions, as first shown in Hoekstra (2001). This adds noise and is relevant for individual cluster mass estimates (e.g., Okabe et al., 2014). Here, we stack the signal at relatively small scales around many different positions (all the lenses) and the large-scale structure contribution is suppressed accordingly. It is therefore reasonable to ignore this contribution as well.

Measurements are independent between clusters. We therefore combine measurements from different clusters, $X_{i}\equiv\Sigma_{\mathrm{c},i}\mathbf{\gamma}_{i}$, as:

[TABLE]

where $\textbf{{C}}_{i}^{-1}$ is the inverse covariance matrix of measurements of the $i$-th cluster, and

[TABLE]

is the covariance of the average measurements entering our analysis (Section 5.3).

Finally, we note that the formalism described above is valid under spherical symmetry, and for a smooth cluster stack. For an rms cluster ellipticity of roughly 0.3 (e.g., van Uitert et al., 2017), the ellipticity of our stack is $0.3/\sqrt{48}\sim 0.04$, close enough to circular for our purposes. Substructure from individual clusters will naturally smooth out during the stacking process as well. We therefore consider said assumptions of our formalism to be valid within the precision of our measurements.

3 Data set

In this section we describe the lens and source galaxy samples we use in our analysis. In the next section, we make a detailed assessment of the shape measurement and quality cuts on the source sample using extensive image simulations.

3.1 Cluster and lens galaxy samples

The Multi-Epoch Nearby Cluster Survey (MENeaCS, Sand et al., 2012) is a targeted survey of 57 galaxy clusters in the redshift range $0.05\lesssim z\lesssim 0.15$ observed in the $g$ and $r$ bands with MegaCam on CFHT. We only use the 48 clusters affected by $r$-band Galactic extinctions $A_{r}\leq 0.2$ mag, since we find that larger extinctions bias the source number counts and the correction for cluster member contamination (Section 4). The image processing and photometry are described in detail in van der Burg et al. (2013); most images have seeing $<0.8^{\prime\prime}$. We list our sample of 48 clusters in Section 3.1. Sifón et al. (2015b) compiled a large sample of spectroscopic redshift measurements in the direction of 46 of these clusters, identifying a total of 7945 spectroscopic members. Since, Rines et al. (2016) have published additional spectroscopic redshifts for galaxies in 12 MENeaCS clusters, six of which are included in Sifón et al. (2015b) but for which the observations of Rines et al. (2016) represent a significant increase in the number of member galaxies. We select cluster members in these 12 clusters in an identical way as Sifón et al. (2015b). The median dynamical mass of MENeaCS clusters is $M_{200}\sim 6\times 10^{14}\,\mathrm{M}_{\odot}$ (Sifón et al., 2015b).

From the member catalogue of Sifón et al. (2015b), we exclude all brightest cluster galaxies (BCGs), and refer to all other galaxies as satellites. Because the shapes of background galaxies near these members are very likely to be contaminated by light from the BCG, we also exclude all satellite galaxies within 10$\arcsec$ of the BCGs to avoid severe contamination from extended light. Finally, we impose a luminosity limit $L_{\mathrm{sat}}<\min(2L^{\star},0.5L_{\mathrm{BCG}})$ (where $L^{\star}(z)$ is the $r$-band luminosity corresponding to the characteristic magnitude, $m^{\star}_{\mathrm{phot}}(z)$ of the Schechter (1976) function, fit to red satellite galaxies in redMaPPer galaxy clusters over the redshift range $0.05<z<0.7$ (Rykoff et al., 2014)).333Equation 9 of Rykoff et al. (2014) provides a fitting function for the $i$-band $m^{\star}_{\mathrm{phot}}(z)$, which we convert to $r$-band magnitudes assuming a quiescent spectrum, appropriate for the majority of our satellites, using EzGal (http://www.baryons.org/ezgal/, Mancone & Gonzalez, 2012). We use SExtractor’s mag_auto (Bertin & Arnouts, 1996) as our estimates of galaxy magnitudes. We choose the maximum possible luminosity, $2L^{\star}$, because the BCGs in our sample have $L_{\mathrm{BCG}}\gtrsim 3L^{\star}$, so this ensures we do not include central galaxies of massive (sub)structures that could, for instance, have recently merged with the cluster. In addition, we only include satellites within 2 Mpc of the BCG. At larger distances, contamination by fore- and background galaxies becomes an increasingly larger problem. Our final spectroscopic sample consists of 5414 satellites in 45 clusters.

In addition, we include red sequence galaxies in all MENeaCS clusters in low Galactic extinction regions in order to improve our statistics. We measure the red sequence by fitting a straight line to the colour-magnitude relation of red galaxies in each cluster using a maximum likelihood approach, based on the methodology of Hao et al. (2009). Following Sifón et al. (2015b), we include only red sequence galaxies brighter than $M_{\mathrm{r}}=-19$ and within 1 Mpc of the BCG.444Here, $M_{\mathrm{r}}$ is the $k+e$–corrected absolute magnitude in the $r$-band, calculated with EzGal using a passively evolving Charlot & Bruzual (2007, unpublished, see Bruzual & Charlot, 2003) model with formation redshift $z_{\mathrm{f}}=5$. When we include red sequence galaxies, we also use the six clusters without spectroscopic cluster members. Therefore our combined spectroscopic plus red sequence sample includes 7909 cluster members in 48 clusters (including three clusters without spectroscopic data). Throughout, we refer to the spectroscopic and spectroscopic plus red sequence samples as ‘spec’ and ‘spec+RS’, respectively.

For the purpose of estimating stellar masses and photometric redshifts, the original MENeaCS observations in $g$ and $r$ were complemented by $u$- and $i$-band observations with the Wide-Field Camera on the Isaac Newton Telescope in La Palma (except for a few clusters with archival MegaCam data in either of these bands, see van der Burg et al., 2015, for details). Stellar masses were estimated by van der Burg et al. (2015) by fitting each galaxy’s spectral energy distribution using fast (Kriek et al., 2009) assuming a Chabrier (2003) initial mass function.