The kinematics of satellite galaxies reflect the masses of the extended dark matter haloes in which they orbit, and thus shed light on the mass–luminosity relation (MLR) of their corresponding central galaxies. In this paper, we select a large sample of centrals and satellites from the Sloan Digital Sky Survey and measure the kinematics (velocity dispersions) of the satellite galaxies as a function of the r‐band luminosity of the central galaxies. Using the analytical framework presented in More, van den Bosch & Cacciato, we use these data to infer both the mean and the scatter of the MLR of central galaxies, carefully taking account of selection effects and biases introduced by the stacking procedure. As expected, brighter centrals on average reside in more massive haloes. In addition, we find that the scatter in halo masses for centrals of a given luminosity, σlog M, also increases with increasing luminosity. As we demonstrate, this is consistent with σlog L, which reflects the scatter in the conditional probability function P(Lc|M), being independent of halo mass. Our analysis of the satellite kinematics yields σlog L= 0.16  ±  0.04, in excellent agreement with constraints from clustering and group catalogues, and with predictions from a semi‐analytical model of galaxy formation. We thus conclude that the amount of stochasticity in galaxy formation, which is characterized by σlog L, is well constrained, independent of halo mass and in a good agreement with current models of galaxy formation.

Galaxy clustering and galaxy–galaxy lensing probe the connection between galaxies and their dark matter haloes in complementary ways. Since the clustering of dark matter haloes depends on cosmology, the halo occupation statistics inferred from the observed clustering properties of galaxies are degenerate with the adopted cosmology. Consequently, different cosmologies imply different mass‐to‐light ratios for dark matter haloes. Galaxy–galaxy lensing, which yields direct constraints on the actual mass‐to‐light ratios, can therefore be used to break this degeneracy, and thus to constrain cosmological parameters. In this paper, we establish the link between galaxy luminosity and dark matter halo mass using the conditional luminosity function (CLF), Φ(L|M) dL, which gives the number of galaxies with luminosities in the range L± dL/2 that reside in a halo of mass M. We constrain the CLF parameters using the galaxy luminosity function and the luminosity dependence of the correlation lengths of galaxies. The resulting CLF models are used to predict the galaxy–galaxy lensing signal. For a cosmology that agrees with constraints from the cosmic microwave background, i.e. (Ωm, σ8) = (0.238, 0.734), the model accurately fits the galaxy–galaxy lensing data obtained from the Sloan Digital Sky Survey. For a comparison cosmology with (Ωm, σ8) = (0.3, 0.9), however, we can accurately fit the luminosity function and clustering properties of the galaxy population, but the model predicts mass‐to‐light ratios that are too high, resulting in a strong overprediction of the galaxy–galaxy lensing signal. We conclude that the combination of galaxy clustering and galaxy–galaxy lensing is a powerful probe of the galaxy–dark matter connection, with the potential to yield tight constraints on cosmological parameters. Since this method mainly probes the mass distribution on relatively small (non‐linear) scales, it is complementary to constraints obtained from the galaxy power spectrum, which mainly probes the large‐scale (linear) matter distribution.

We use ROSAT All Sky Survey (RASS) broadband X-ray images and the optical clusters identified from SDSS DR7 to estimate the X-ray luminosities around ∼ 65, 000 candidate clusters with masses > ∼ 10 13 h −1 M ⊙ based on an Optical to X-ray (OTX) code we develop. We obtain a catalogue with X-ray luminosity for each cluster. This catalog contains 817 clusters (473 at redshift z ≤ 0.12) with S/N > 3 in X-ray detection. We find about 65% of these X-ray clusters have their most massive member located near the X-ray flux peak; for the rest 35%, the most massive galaxy is separated from the X-ray peak, with the separation following a distribution expected from a NFW profile. We investigate a number of correlations between the optical and X-ray properties of these X-ray clusters, and find that: the cluster X-ray luminosity is correlated with the stellar mass (luminosity) of the clusters, as well as with the stellar mass (luminosity) of the central galaxy and the mass of the halo, but the scatter in these correlations is large. Comparing the properties of X-ray clusters of similar halo masses but having different X-ray luminosities, we find that massive halos with masses > ∼ 10 14 h −1 M ⊙ contain a larger fraction of red satellite galaxies when they are brighter in X-ray. An opposite trend is found in central galaxies in relative low-mass halos with masses < ∼ 10 14 h −1 M ⊙ where X-ray brighter clusters have smaller fraction of red central galaxies. Clusters with masses > ∼ 10 14 h −1 M ⊙ that are strong X-ray emitters contain many more low-mass satellite galaxies than weak X-ray emitters. These results are also confirmed by checking X-ray clusters of similar X-ray luminosities but having different characteristic stellar masses. A cluster catalog containing the optical properties of member galaxies and the X-ray luminosity is available at