# Modeling Galaxy-Galaxy Weak Lensing with SDSS Groups

Ran Li1,2, H.J. Mo2, Zuhui Fan1, Marcello Cacciato3, Frank C. van den Bosch3,
Xiaohu Yang4, Surhud More3
1Department of Astronomy, Peking University, Beijing 100871, China
2Department of Astronomy, University of Massachusetts, Amherst MA 01003, USA
3Max-Planck Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg,
Germany
4Shanghai Astronomical Observatory, the Partner Group of MPA, Nandan Road 80,
Shanghai 200030, China<EMAIL_ADDRESS>
###### Abstract

We use galaxy groups selected from the Sloan Digital Sky Survey (SDSS)
together with mass models for individual groups to study the galaxy-galaxy
lensing signals expected from galaxies of different luminosities and
morphological types. We compare our model predictions with the observational
results obtained from the SDSS by Mandelbaum et al. (2006) for the same
samples of galaxies. The observational results are well reproduced in a
$\Lambda$CDM model based on the WMAP 3-year data, but a $\Lambda$CDM model
with higher $\sigma_{8}$, such as the one based on the WMAP 1-year data,
significantly over-predicts the galaxy-galaxy lensing signal. We model,
separately, the contributions to the galaxy-galaxy lensing signals from
different galaxies: central versus satellite, early-type versus late-type, and
galaxies in haloes of different masses. We also examine how the predicted
galaxy-galaxy lensing signal depends on the shape, density profile, and the
location of the central galaxy with respect to its host halo.

###### keywords:

dark matter - large-scale structure of the universe - galaxies: haloes -
methods: statistical

††pagerange: Modeling Galaxy-Galaxy Weak Lensing with SDSS
Groups–LABEL:lastpage††pubyear: 2008

##  1 Introduction

According to the current paradigm of structure formation, galaxies form and
reside inside extended cold dark haloes. While the formation and evolution of
dark matter haloes in the cosmic density field is mainly determined by
gravitational processes, the formation and evolution of galaxies involves much
more complicated, and poorly understood processes, such as radiative cooling,
star formation, and all kinds of feedback. One important step in understanding
how galaxies form and evolve in the cosmic density field is therefore to
understand how the galaxies of different physical properties occupy dark
matter haloes of different masses. Theoretically, the connection between
galaxies and dark matter haloes can be studied using numerical simulations
(e.g., Katz, Weinberg & Hernquist 1996; Pearce et al. 2000; Springel 2005;
Springel et al. 2005) or semi-analytical models (e.g. White & Frenk 1991;
Kauffmann et al. 1993, 2004; Somerville & Primack 1999; Cole et al. 2000; van
den Bosch 2002; Kang et al. 2005; Croton et al. 2006). These approaches try to
model the process of galaxy formation from first principles. However, since
our understanding of the relevant processes is still poor, the predicted
connection between the properties of galaxies and dark matter haloes needs to
be tested against observations. More recently, the halo occupation model has
opened another avenue to probe the galaxy-dark matter halo connection (e.g.
Jing, Mo & Börner 1998; Peacock & Smith 2000; Berlind & Weinberg 2002; Cooray
& Sheth 2002; Scranton 2003; Yang, Mo & van den Bosch 2003; van den Bosch,
Yang & Mo 2003; Yan, Madgwick & White 2003; Tinker et al. 2005; Zheng et al.
2005; Cooray 2006; Vale & Ostriker 2006; van den Bosch et al. 2007). This
technique uses the observed galaxy luminosity function and clustering
properties to constrain the average number of galaxies of given properties
that occupy a dark matter halo of given mass. Although the method has the
advantage that it can yield much better fits to the data than the semi-
analytical models or numerical simulations, one typically needs to assume a
somewhat ad-hoc functional form to describe the halo occupation model.

A more direct way of studying the galaxy-halo connection is to use galaxy
groups111In this paper, we refer to a system of galaxies as a group regardless
of its richness, including isolated galaxies (i.e., groups with a single
member) and rich clusters of galaxies., provided that they are defined as sets
of galaxies that reside in the same dark matter halo. Recently, Yang et al.
(2005; 2007) have developed a halo-based group finder that is optimized for
grouping galaxies that reside in the same dark matter halo. Using mock galaxy
redshift surveys constructed from the conditional luminosity function model
(e.g. Yang et al. 2003) and a semi-analytical model (Kang et al. 2005), it is
found that this group finder is very successful in associating galaxies with
their common dark matter haloes (see Yang et al. 2007; hereafter Y07). The
group finder also performs reliably for poor systems, including isolated
galaxies in small mass haloes, making it ideally suited for the study of the
relationship between galaxies and dark matter haloes over a wide range of halo
masses. However, in order to interpret the properties of the galaxy systems in
terms of dark matter haloes, one needs to know the halo mass associated with
each of the groups. One approach commonly adopted is to use some halo mass
indicator (such as the total stellar mass or luminosity contained in member
galaxies) to rank the groups. With the assumption that the corresponding halo
masses have the same ranking and that the mass function of the haloes
associated with groups is the same as that given by a model of structure
formation, one can assign a halo mass to each of the observed groups. This
approach was adopted by Y07 for the group catalogue used in this paper. There
are three potential problems with this approach. First, the approach is model-
dependent, in the sense that the assumption of a different model of structure
formation will lead to a different halo mass function, and hence assign
different masses to the groups. Second, even if the assumed model of structure
formation is correct, it is still not guaranteed that the mass assignment
based on the ranking of group stellar mass (or luminosity) is valid. Finally,
even if all groups are assigned with accurate halo masses, the question how
dark matter is distributed within the galaxy groups remains open. Clearly, it
is important to have independent mass measurements of the haloes associated
with galaxy groups to test the validity of the mass estimates based on the
stellar mass (luminosity) ranking.

Gravitational lensing observations, which measure the image distortions of
background galaxies caused by the gravitational field of the matter
distribution in the foreground, provide a promising tool to probe the dark
matter distribution directly. In particular, galaxy-galaxy weak lensing, which
focuses on the image distortions around lensing galaxies, can be used to probe
the distribution of dark matter around galaxies, hence their dark matter
haloes. The galaxy-galaxy lensing signal produced by individual galaxies is
usually very weak, and so one has to stack the signal from many lens galaxies
to have a statistical measurement. The first attempt to detect such galaxy-
galaxy lensing signal was reported by Tyson et al. (1984). More recently, with
the advent of wide and deep surveys, galaxy-galaxy lensing can be studied for
lens galaxies of different luminosities, stellar masses, colors and
morphological types (e.g. Brainerd et al. 1996; Hudson et al. 1998; McKay et
al. 2001; Hoekstra et al. 2003; Hoekstra 2004; Sheldon et al. 2004; Mandelbaum
et al. 2005, 2006; Sheldon et al. 2007a; Johnston et al. 2007; Sheldon et al.
2007b; Mandelbaum et al. 2008). Given that galaxies reside in dark matter
haloes, these results provide important constraints on the mass distribution
associated with galaxies in a statistical way.

In this paper, we use the galaxy groups of Y07 selected from the Sloan Digital
Sky Survey (SDSS), together with mass models for individual groups, to predict
the galaxy-galaxy lensing signal expected from SDSS galaxies. We compare our
model predictions with the observational results obtained by Mandelbaum et al.
(2006) for the same galaxies. Our goal is threefold. First, we want to test
whether the method of halo-mass assignment to groups adopted by Y07 is
reliable. Since the method provides a potentially powerful way to obtain the
halo masses associated with the galaxy groups, the test results have general
implications for the study of the relationship between galaxies and dark
matter haloes. Second, we want to examine in detail the contributions to the
galaxy-galaxy lensing signal from different systems, such as central versus
satellite galaxies, early-type versus late-type galaxies, and groups of
different masses. Such analysis can help us interpreting the observational
results. Finally, we would like to study how the predicted galaxy-galaxy
lensing signal depends on model assumptions, such as the cosmological model
and the density profiles of dark matter haloes. In a companion paper (Cacciato
et al. 2008, hereafter C08), we use the relationship between galaxies and dark
matter haloes obtained from the conditional luminosity function (CLF) modeling
(Yang et al. 2003; van den Bosch et al. 2007) to predict the galaxy-galaxy
cross correlation and to calculate the lensing signal, while here we directly
use the observed galaxy groups and their galaxy memberships.

This paper is organized as follows. In Section 2 we define the statistical
measure that characterizes the galaxy-galaxy lensing effect expected from the
mass distribution associated with the galaxy groups. We provide a brief
description of the galaxy group catalogue and the models of the mass
distribution associated with galaxy groups in Section 3. We present our
results in Section 4 and conclude in Section 5. Unless specified otherwise, we
adopt a $\Lambda$CDM cosmology with parameters given by the WMAP 3-year data
(Spergel et al. 2007, hereafter WMAP3 cosmology) in our analysis: $\Omega_{\rm
m}=0.238$, $\Omega_{\Lambda}=0.762$, and $h\equiv
H_{0}/(100\rm{km\,s^{-1}\,Mpc^{-1}})=0.73$, $\sigma_{8}=0.75$ .

##  2 Galaxy-Galaxy Lensing

Galaxy-galaxy lensing provides a statistical measure of the profile of the
tangential shear, $\gamma_{t}(R)$, averaged over a thin annulus at the
projected radius $R$ around the lens galaxies. This quantity is related to the
excess surface density (hereafter ESD) around the lens galaxy, $\Delta\Sigma$,
as

$\Delta\Sigma(R)=\gamma_{t}(R)\Sigma_{\rm crit}=\bar{\Sigma}(<R)-\Sigma(R)\,,$
(1)

where $\bar{\Sigma}(<R)$ is the average surface mass density within $R$, and
$\Sigma(R)$ is the azimuthally averaged surface density at $R$. Note that,
according to this relation, $\Delta\Sigma(R)$ is independent of a uniform
background. In the above equation,

$\Sigma_{\rm crit}=\frac{c^{2}}{4\pi G}\frac{D_{s}}{D_{l}D_{ls}(1+z_{l})^{2}}$
(2)

is the critical surface density in comoving coordinates, with $D_{s}$ and
$D_{l}$ the angular distances of the lens and source, $D_{ls}$ the angular
distance between the source and the lens, and $z_{l}$ the redshift of the
lens.

By definition, the surface density, $\Sigma(R)$, is related to the projection
of the galaxy-matter cross-correlation function, $\xi_{\rm g,m}(r)$, along the
line-of-sight. In the distant observer approximation

$\Sigma(R)=\bar{\rho}\int^{\infty}_{-\infty}\left[1+\xi_{\rm
g,m}(\sqrt{R^{2}+\chi^{2}})\right]\,d\chi\,,$ (3)

where $\bar{\rho}$ is the mean density of the universe and $\chi$ is the line-
of-sight distance from the lens.

The cross-correlation between galaxies and dark matter can, in general, be
divided into a 1-halo term and a 2-halo term. The 1-halo term measures the
cross-correlation between galaxies and dark matter particles in their own host
haloes, while the 2-halo term measures the cross-correlation between galaxies
and dark matter particles in other haloes. In the present work, we are
interested in the lensing signals on scales $R\leq 2h^{-1}{\rm Mpc}$ where the
observational measurements are the most accurate. As we will show in § 4, on
such scales the signal is mainly dominated by the 1-halo term. Nevertheless,
our model also takes the contribution of the 2-halo term into account. More
importantly, since central galaxies (those residing at the center of a dark
matter halo) and satellite galaxies (those orbiting around a central galaxy)
contribute very different lensing signals (e.g. Natarajan, Kneib & Smail 2002;
Yang et al. 2006; Limousin et al. 2007), it is important to model the
contributions from central and satellite galaxies separately.

As an illustration, in Fig. 1 we show the ESDs expected from a single galaxy
in a host halo of mass $10^{14}h^{-1}{\rm M}_{\odot}$. The solid line
represents the lensing signal expected for the central galaxy of the halo.
While the dotted and dashed lines show the lensing signal of a satellite
galaxy residing in a sub-halo of $10^{11}h^{-1}{\rm M}_{\odot}$ with a
projected halo-centric distance $r_{p}=0.2\,h^{-1}{\rm Mpc}$ and
$r_{p}=0.4\,h^{-1}{\rm Mpc}$ from the center of the host halo, respectively.
In the calculation, the dark matter mass distribution in the host halo is
assumed to follow the Navarro, Frank & White (1997) profile and that in the
sub-haloes is assumed to follow the Hayashi et al. (2003) model. These models
are described in detail in § 3.3. In order to estimate the ESD, we sample
these profiles with mass particles and project the positions of all particles
to a plane perpendicular to the line of sight. The $\Sigma(R)$ is then
estimated by counting the number of dark matter particles in a annulus with
radius $R$ centred on the selected galaxies. Fig. 1 shows clearly that the
lensing signals of the central and the satellite are quite different. The ESD
of the central galaxy follows the mass distribution of the host halo,
decreasing monotonically with $R$. The ESD of a satellite, on the other hand,
consists of two parts: one from the subhalo associated the satellite, which
contributes to the inner part, and the other from the host halo, which
dominates at larger $R$. This simple model demonstrates clearly that, in order
to model the galaxy-galaxy lensing signal produced by a population of lens
galaxies, one needs to model carefully the distribution of matter around both
host haloes and subhaloes. To do this, we need not only to identify the haloes
in which each lens galaxy resides, but also to model the mass and density
profile of each host halo and subhalo. In addition we also need to model the
distribution of dark matter relative to galaxies. In the following section, we
describe our modeling with the use of observed galaxy groups.

Figure 1: The ESD expected for a single galaxy. Here the host halo mass is
assumed to be $10^{14}h^{-1}M_{\odot}$. The solid line represents the lensing
signal for the central galaxy in such a halo. The dotted line represents the
lensing signal of a satellite galaxy residing in a sub-halo of mass
$10^{11}h^{-1}M_{\odot}$ which has a projected distance $r_{\rm
p}=0.2h^{-1}{\rm Mpc}$ from the center of the host halo. The dashed line is
the same as the dotted line, except that the subhalo’s projected distance is
$0.4h^{-1}{\rm Mpc}$ from the center of the host halo.

##  3 Modeling the Mass Distribution Associated with the SDSS Groups

###  3.1 The SDSS Group Catalogue

Our analysis is based on the SDSS galaxy group catalogue constructed by Y07.
The groups are selected with the adaptive halo-based group finder developed by
Yang et al. (2005), from the New York University Value Added Galaxy Catalog
(NYU-VAGC; Blanton et al. 2005) which is based on the SDSS Data Release 4
(Adelman-McCarthy et al. 2006). Only galaxies with redshifts in the range
$0.01\leq z\leq 0.2$, and with redshift completeness $\mathcal{C}>0.7$, are
used in the group identification. The magnitudes and colors of all galaxies
are based on the standard SDSS Petrosian technique (Petrosian 1976; Strauss et
al. 2002), and have been corrected for galactic extinction (Schlegel,
Finkbeiner & Davis 1998). All magnitudes have been K-corrected and evolution-
corrected to $z=0.1$ following the method described in Blanton et al. (2003).
In Y07, three group samples were constructed using galaxy samples of different
sources of galaxy redshifts. Our analysis is based on Sample II, which
includes 362,356 galaxies with redshifts from the SDSS and 7091 galaxies with
redshifts taken from alternative surveys: 2dFGRS (Colless et al. 2001), PSOz
(Saunders et al.2000) or from the RC3 (de Vaucouleurs et al. 1991). There are
in total 301,237 groups, including those with only one member galaxy. The
group finder has been applied to mock catalogue to test the completeness and
purity of the groups in Y07. About $90\%$ of the groups have a completeness
$f_{c}>0.6$ and $80\%$ groups with $f_{c}>0.8$, where $f_{c}$ is defined as
the ratio between the number of true members that are selected as the members
of the group and the number of the total true members of the group.

###  3.2 Halo Mass Assignment

Figure 2: The halo mass $M_{S}$ (estimated using stellar mass), versus
$M_{\rm\ast}$ (lower panels) and $L$ (upper panels) of the galaxies in the
haloes. The left panels are for central galaxies and the right panels are for
satellite galaxies. Figure 3: The distribution of the host halo masses for the
central and satellite galaxies in different luminosity bins, as indicated by
the $r$-band absolute-magnitude range in each panel.

An important aspect of the group catalog construction is the determination of
the halo mass, $M_{\rm vir}$, of each group. In Y07, two estimators are
adopted. The first, $M_{L}$, is estimated using the ranking of the
characteristic luminosity of a group, which is the total luminosity of all
member galaxies in the group with $M_{r}-5\log h\leq-19.5$ (hereafter referred
to as $L_{19.5}$). The second, $M_{S}$, is estimated using the ranking of the
characteristic stellar mass, $M_{\rm stellar}$ which is defined to be the
total stellar mass of group members with $M_{r}-5\log h\leq-19.5$. For each
galaxy the stellar mass is estimated from its absolute magnitude and color
using the fitting formula given by Bell et al. (2003).

The basic assumption of the ranking method is that there is a one-to-one
relation between $M_{\rm stellar}$ (or $L_{19.5}$) and the group mass. Using
the dark matter halo mass function predicted by a model of structure
formation, one can assign a halo mass to each group according to its $M_{\rm
stellar}$ \- ranking (or $L_{19.5}$ \- ranking). In this paper, we use the
mass function obtained by Warren et al. (2006). Note that this one-to-one
mapping is applicable only when the group sample is complete. In Y07, three
complete samples are constructed in three redshift ranges. Only groups in the
complete samples are used in the ranking. The mass of other groups are
estimated by a linear interpolation based on the $M_{\rm stellar}$-$M_{\rm
vir}$ relation (or the $L_{19.5}$ \- $M_{\rm vir}$) obtained from the complete
sample. Detailed tests using mock galaxy redshift samples have shown that the
1-$\sigma$ error of the estimated halo mass is $\sim 0.3$ dex (Y07). In
addition, the two mass estimators, $M_{L}$ and $M_{S}$, agree remarkably well
with each other, with a scatter that decreases from about 0.1 dex at the low-
mass end to about 0.05 dex at the high-mass end. Since the correlation between
$M_{\rm stellar}$ and halo mass is somewhat tighter than that between
$L_{19.5}$ and halo mass, we adopt $M_{S}$ as our fiducial halo mass
throughout. As we demonstrate in § 4.3, using $M_{L}$ instead yields results
that are fairly similar.

Fig. 2 shows the relation between the host halo mass, $M_{S}$, and the galaxy
stellar mass $M_{\ast}$ (the lower two panels) or the galaxy luminosity $L$
(the upper two panels). Results are shown separately for central galaxies
(left panels) and satellite galaxies (right panels). As one can see, the
stellar mass (luminosity) of central galaxies is quite tightly correlated with
their host halo masses. However, for satellite galaxies of a given stellar
mass (or luminosity), their host halo mass covers a very large range,
reflecting the fact that many low-mass galaxies are satellites in massive
haloes. The distributions of host halo masses, $M_{S}$, for central or
satellite galaxies in different luminosity bins are shown in Fig. 3. On
average, brighter central galaxies reside in more massive haloes. For faint
galaxies, the halo-mass distribution is broader, again because many faint
galaxies are satellites in massive systems.

In the group catalogue, the mass assignment described above is used only for
groups where the brightest galaxy is brighter than $M_{r}-5\log h=-19.5$. This
is because the mass ranking used in the group catalog is based on the total
stellar mass (or total luminosity) of the member galaxies that are brighter
than $M_{r}-5\log h=-19.5$. The groups with no galaxies brighter than this
magnitude thus have no assigned rank. As described in Y07, the reason for
choosing this maginitude is a compromise between having a complete sample in a
relatively large volume and having more groups that are represented by a
number of member galaxies. For groups in which all member galaxies have
$M_{r}-5\log h>-19.5$, a different method has to be adopted. In modeling the
luminosity function and stellar mass function of the central galaxies based on
the same SDSS group catalogue as used here, Yang et al. (2008) obtain an
average relation between the luminosity (or stellar mass) of the central
galaxy and the halo mass down to $M_{r}-5\log h\sim-17$. We adopt this
relation to assign halo masses to all groups (including those containing only
one isolated galaxy) represented by centrals with $M_{r}-5\log h>-19.5$. For
convenience, the halo masses obtained in this way are also referred to as
$M_{S}$ (based on the stellar mass of central galaxies) and $M_{L}$ (based on
the $r$-band luminosity of the central galaxies), respectively.

###  3.3 Mass Distribution in Haloes and Subhaloes

With the group catalogue described above, we can model the dark matter
distribution by convolving the halo distribution with the density profiles of
individual haloes. In our modeling of the density profiles, the host halo of a
group is assumed to be centered on the central galaxy. There are two ways to
define a central galaxy: one is to define the central in a group to be the
galaxy with the highest stellar mass, and the other is to define the central
to be the brightest member. For most groups these two definitions give the
same results, but there are very few cases (less than $\sim 2\%$) where
different central galaxies are defined. In our fiducial model, we define the
most massive galaxies (in term of stellar mass) to be the central galaxies.

In a hierarchical model, a dark matter halo forms through a series of merger
events. During the assembly of a halo, most of the mass in the merging
progenitors is expected to be stripped. However, some of them may survive as
subhaloes, although the total mass contained in subhaloes is small, typically
$\sim 10\%$ (van den Bosch, Tormen & Giocoli 2005). Some of the subhaloes are
associated with ‘satellite galaxies’ in a halo. In our modeling of the galaxy-
galaxy weak lensing, we only take into account subhaloes associated with
satellite galaxies, treating other subhaloes as part of the host halo.
Giocoli, Tormen & van den Bosch (2008) provide a fitting function of the
average mass function for subhaloes at the time of their accretion into the
parent halo of a given mass. Using this mass function, we first sample a set
of masses for each group mass. We then set the mass originally associated with
a satellite galaxy according to the stellar mass ranking of the satellites in
the group. Here we implicitly assume that the initial subhalo mass function is
the same as the mass function of the subhalos that host satellites. This
assumption is not proved by any observations, and we have to live with it
since a more realistic model is not currently available. Fortunately, subhalos
only contribute a small fraction to the total lensing signal on small scales.
The uncertainty here will not have a significant impact on any of our
conclusions. To obtain the final mass in the subhalo at the present time, the
evolution of the subhaloes needs to be taken into account. In other words, we
need to know the fraction of the mass that is stripped and how the structure
of a subhalo changes after the stripping. Here it is convenient to introduce a
parameter $f_{m}$ which is the retained mass fraction of the subhalo. Gao et
al. (2004) studied the radial dependence of the retained mass fraction $f_{m}$
from a large sample of subhaloes in a large cosmological simulation. In their
work, $f_{m}$ is considered as a function of $r_{s}/r_{\rm vir,h}$, where
$r_{s}$ is the distance of the subhalo from the center of the host halo and
$r_{\rm vir,h}$ is the virial radius of the host halo. The simulation of Gao
et al. gives

$f_{m}=0.65(r_{s}/r_{\rm vir,h})^{2/3}\,.$ (4)

We will adopt this in our modeling of the masses associated with subhaloes.
However, in the group catalogue, only the projected distance, $r_{p}$, from
the group center is available. The 3D-distance, $r_{s}$, is obtained by
randomly sampling the NFW profile of the host halo with the given projected
radius $r_{p}$.

Thus, the mass assigned to a subhalo is determined by the following three
factors: (1) the stellar mass of the satellite galaxy; (2) the host halo mass;
(3) the distance between the satellite and the center of the host. Here the
host halo mass comes into our calculation in two ways. It not only determines
the subhalo mass function, but also affects the parameter $f_{m}$ in Eqs.4.
Note that the accretion history of the host halo may also affect the value of
$f_{m}$. We have to neglect such effect because it is unclear how to model the
accretion histories for individual groups.

For host haloes, we use the following NFW profile (Navarro, Frank & White
1997) to model the mass distribution:

$\rho(r)=\frac{\delta_{0}\bar{\rho}}{(r/r_{c})(1+r/r_{c})^{2}}\,.$ (5)

where $\bar{\rho}$ is the mean density of the universe, $r_{c}$ is a scale
radius, related to virial radius $r_{\rm vir}$ by the concentration, $c=r_{\rm
vir}/r_{c}$, and $\delta_{0}$ is a characteristic over-density related to the
average over-density of a virialized halo, $\Delta_{\rm vir}$, by

$\delta_{0}=\frac{\Delta_{\rm vir}}{3}\frac{c^{3}}{\ln(1+c)-c/(1+c)}\,.$ (6)

We adopt the value of $\Delta_{\rm vir}$ given by the spherical collapse model
(see Nakamura & Suto 1997; Henry 2000). Numerical simulations show that halo
concentrations are correlated with halo mass, and we use the relations given
by Macciò et al. (2007), converted to our definition of halo mass. Note that
here we use $r_{c}$, instead of the conventional notation $r_{s}$, to denote
the scale radius of the NFW profile, as $r_{s}$ has been used to denote the
distance of a subhalo from the center of its host.

For sub-haloes, we model their density profiles using the results obtained by
Hayashi et al. (2003), who found that the density profiles of stripped sub-
haloes can be approximated as

$\rho_{s}(r)=\frac{f_{t}}{1+(r/r_{t,\rm eff})^{3}}\rho(r)\,,$ (7)

where $f_{t}$ is a dimensionless factor describing the reduction in the
central density, and $r_{t,\rm eff}$ is a cut-off radius imposed by the tidal
force of the host halo. For $f_{t}=1$ and $r_{t,\rm eff}\gg r_{c}$,
$\rho_{s}(r)$ reduces to the standard NFW profile $\rho(r)$. Here $\rho(r)$ is
calculated using the mass of the subhalo at the time of its accretion into the
host halo. Both $f_{t}$ and $r_{t,\rm eff}$ depend on the mass fraction of the
sub-halo that remains bound, $f_{m}$. Based on $N$-body simulations, Hayashi
et al. obtained the following fitting formulae relating $f_{t}$ and $r_{t,\rm
eff}$ to $f_{m}$:

$\log(r_{t,\rm eff}/r_{c})=1.02+1.38\log f_{m}+0.37(\log f_{m})^{2}\,;$ (8)
$\log(f_{t})=-0.007+0.35\log f_{m}+0.39(\log f_{m})^{2}+0.23(\log
f_{m})^{3}\,.$ (9)

It should be pointed out, though, that there are substantial uncertainties in
modeling the mass distribution around individual satellite galaxies. In
particular, many of the results about subhaloes are obtained from $N$-body
simulations, and it is unclear how significant the effect of including
baryonic matter is. Fortunately, the total mass associated with satellite
galaxies is small (see e.g. Weinberg et al. 2008). Furthermore, the
contribution of the subhaloes associated with the satellite galaxies to the
galaxy-galaxy lensing signal is confined to small scales. We therefore expect
that these uncertainties will not change our results significantly.

With the mass distributions described above, we use a Monte-Carlo method to
sample each of the profiles with a random set of mass particles. Note that the
halo mass assigned to a group in the SDSS Group Catalog is $M_{180}$, which is
the mass enclosed in the radius, $r_{180}$, defined such that
$M_{180}={4\pi}r_{180}^{3}(180\bar{\rho})/3$. We therefore sample the particle
distribution within $r_{180}$. After all the particles in each halo are
sampled, we project the positions of all the particles to a plane and
calculate $\Delta\Sigma(R)$ by stacking galaxies in each of the luminosity
bin. Since the mass distribution is isotropic, an arbitrary direction can be
chosen for the stacking. Thus, the projection effect is naturally included in
our calculation. Each of the particles has a mass of $10^{10}h^{-1}{\rm
M}_{\odot}$. Our test using particles of lower masses shows that the mass
resolution adopted here is sufficient for our purpose. Using 2 times more
particles leads to a difference of about $5\%$ at $R\sim 0.02\,h^{-1}{\rm
Mpc}$, and almost no difference at $R\geq 0.1\,h^{-1}{\rm Mpc}$.

Figure 4: Comparison of the lensing signal predicted by the fiducial model
with the observational results. Here the ESD is plotted as the function of the
transverse distance $R$ for lensing galaxies in different luminosity bins.
Data points with error bars are the observational results of Mandelbaum et al.
, while the lines are the model predictions. The dotted, dashed and dot-dashed
lines represent the contributions of the ‘1-halo term’ of central galaxies,
the ‘1-halo term’ of satellite galaxies, and the ‘2-halo term’ (of both
centrals and satellites), respectively. The solid lines show the predicted
total ESD. The $r$-band magnitude range for each case can be found in Table 1.
Figure 5: The contribution to the ESD plotted separately for dark matter
haloes of different masses. In each panel, the dotted line shows the
contribution from haloes with $M_{S}\geq 10^{14}h^{-1}M_{\odot}$. The dashed
line shows the contribution from haloes with $10^{13}h^{-1}M_{\odot}\leq
M_{S}<10^{14}h^{-1}M_{\odot}$, and the dot-dashed line shows the contribution
from haloes with $M_{S}<10^{13}h^{-1}M_{\odot}$. The solid line shows the
total lensing signal predicted by the fiducial model. For comparison, the
observational data are included as data points with error-bars. Figure 6: The
right panels show the ESD of early galaxies in different luminosity bins,
while the left panels show the results for late galaxies. The data points with
error-bars show the observational results. The model predictions of the ESD
using stellar mass as halo mass indicator are shown as the solid lines. For
comparison, the dashed lines show the corresponding model predictions using
$M_{L}$ as the halo masses.  Figure 7: The dependence of the predicted
$\Delta\Sigma(R)$ on various model parameters. For comparison, the
observational data are included as data points with error-bars. The left
column shows the dependence on halo concentration: the halo concentrations in
CL (dotted line) and CH (dashed line) are assumed to be 1/2 and 2 times those
in the fiducial model (solid line), respectively. The middle column of shows
the effects of halo center offset. The dotted line and the dashed line show
the results of models Dev1 and Dev2 model, respectively, while the solid line
is the fiducial model. The right column shows the effect of assuming triaxial
halo density profile. The dotted line shows the result of model TRI, while the
solid line again shows the fiducial model. Both the observation and model
predicted $\Delta\Sigma(R)$ are normalized by the fiducial prediction.  Figure
8: The model prediction of ESD assuming different cosmological models. The
solid and dotted lines show the fiducial model and model using WMAP1
parameters, respectively . The observational data are plotted as data points
with error-bars.

##  4 Results

###  4.1 SDSS Lensing Data

Before presenting our model predictions, we first describe the observational
results that we will use for comparison. The observational results to be used
were obtained by Mandelbaum et al. (2006), who analyzed the galaxy-galaxy
lensing effects using galaxies in a sample constructed from the SDSS DR4
spectroscopic sample. Their sample of lensing galaxies is similar to the
galaxy sample used in Y07 to construct the group catalogue used here. The only
difference is that Mandelbaum et al’s sample includes all galaxies with
redshifts in the range $0.02<z<0.35$, while the galaxies in Y07’s group
catalogue are in $0.01\leq z\leq 0.2$. Since, as to be described below, we are
interested in the lensing signals around galaxies of given luminosity and
morphological type, this difference in redshift range is not expected to have
a significant impact on our results. For the faint luminosity bins, our galaxy
samples should be almost identical to that of Mandelbaum et al. (2006),
because all faint galaxies are at $z\leq 0.2$ in the SDSS catalog. For bright
galaxies we also expect the statistic properties of the two samples to be
similar. Both Mandelbaum et al. (2006) and Y07 applied similar evolution
correction and K correction, so that the evolution in the galaxy population
has been taken into account, albeit in a simple way. As we will see, even for
the the two brightest bins, the lensing signal is dominated by halos with
masses $\sim 10^{14}\,h^{-1}{\rm M}_{\odot}$, and the change in the halo mass
function around this mass is small between $z=0.2$ and $0.35$. Following
Mandelbaum et al. (2006), we split the galaxy sample into 7 subsamples
according to galaxy luminosity. Table 1 shows the properties of these
subsamples: the luminosity range covered by each subsample, the mean redshift,
the mean luminosity, and the fraction of late-type galaxies. As expected the
mean redshifts of brightest bins are different from the corresponding
redshifts in Mandelbaum et al.

Table 1: The properties of galaxy samples. In each case, the absolute-magnitude range, the mean redshift, the mean luminosity, and the fraction of late-types are listed. Note that $L_{*}=1.2\times 10^{10}h^{-2}L_{\odot}$ Sample | $M_{r}$ | $\langle z\rangle$ | $\langle L/L_{*}\rangle$ | $f_{\rm late}$
---|---|---|---|---
L1 | $-18<M_{r}<-17$ | 0.031 | 0.075 | 0.81
L2 | $-19<M_{r}<-18$ | 0.048 | 0.191 | 0.70
L3 | $-20<M_{r}<-19$ | 0.074 | 0.465 | 0.54
L4 | $-21<M_{r}<-20$ | 0.111 | 1.13 | 0.35
L5f | $-21.5<M_{r}<-21$ | 0.145 | 2.09 | 0\. 22
L5b | $-21.5<M_{r}<-22$ | 0.150 | 3.22 | 0.12
L6f | $-22<M_{r}<-22.5$ | 0.152 | 5.01 | 0.04

Also following Mandelbaum et al. (2006), we split each galaxy subsample into
two according to galaxy morphology. The separation is made according to the
parameter $frac\\_{\rm dev}$ generated by the PHOTO pipeline. The value of
$frac\\_{\rm dev}$ is obtained by fitting the galaxy profile, in a given band,
to a model profile given by $frac\\_{\rm dev}\times F_{\rm deV}+(1-frac\\_{\rm
dev})\times F_{\rm exp}$, where $F_{\rm deV}$ and $F_{\rm exp}$ are the de
Vaucouleurs and exponential profiles, respectively. As in Mandelbaum et al.,
we use the average of $frac\\_{\rm dev}$ in the $g$, $r$ and $i$ bands.
Galaxies with $frac\\_{\rm dev}\geq 0.5$ are classified as early-type, while
those with $frac\\_{\rm dev}<0.5$ as late-type.

It should be pointed out that we did not carry a ray-tracing simulation to
predict the galaxy-galaxy lensing results. Instead, we directly calculate the
excess surface density around SDSS galaxies. Thus, our calculation doesn’t
deal with source galaxies. On the other side, in Mandelbaum et al. (2006) the
source galaxies are carefully weighted, and lensing signals are calibrated, to
reduce any bias in the observational measurements (see Mandelbaum et al. 2005
for details). Thus, we assume the observational results are unbiased, and
compare them directly with our model predictions.

The observation data used here was kindly provided by R. Mandelbaum. The data
used in Fig. 6 has been published in Mandelbaum et al. (2006) where lensing
signal is calculated for early and late type galaxies separately. R.
Mandelbaum also provided the lensing data combining early and late type
galaxies for us to make the comparisons presented in all other figures. Note
that here we only show the comparison of the lensing signal for galaxies
divided according to luminosity.

The errorbars on the observational points are $1\sigma$ statistical error. The
systematic error of the galaxy-galaxy lensing has been discussed in detail in
Mandelbaum et al. (2005). The test has been carried out for three source
samples: $r<21$, $r>21$, and high redshift LRGs. The overall systematic error
is found to be comparable to or slightly larger than the statistical error
shown here.

###  4.2 The Fiducial Model

In Fig. 4 we show the lensing signal around galaxies in different luminosity
bins obtained from our fiducial model, which has model parameters as described
in the last section and assumes the WMAP3 cosmology. Here the ESD is plotted
as a function of the projected distance $R$ from galaxies. The solid line
shows the averaged ESD of all galaxies in the corresponding luminosity bin.
The amplitude of the predicted ESD increases with galaxy luminosity,
reflecting the fact that brighter galaxies on average reside in more massive
haloes, as shown in Figs. 2 and 3. These results are to be compared with the
data points which show the observational results obtained by Mandelbaum et al.
(2006) for the same luminosity bins. Overall, our fiducial model reproduces
the observational data reasonably well, especially for bright galaxy bins
where the observational results are the most reliable. The reduced $\chi^{2}$
is $3.2$ combining all the luminosity bins. The best match is for L5f, with a
reduced $\chi^{2}$ of $0.9$. Given that we do not adjust any model parameters,
the $\chi^{2}$ indicates a good agreement. For the three low-luminosity bins,
the predicted ESD is lower than the corresponding observational result. For L1
and L2, the observational data are very uncertain. For L3, if we take the
observational data points at face-value, the discrepancy with the model
prediction is significant. As described in § 3.2, for groups which do not
contain any member galaxies with $M_{r}-5\log h<-19.5$, their halo masses are
not obtained from the ranking of $M_{\rm stellar}$, but from the average
stellar mass-halo mass relation of central galaxies that is required to match
the observed stellar mass function of central galaxies. While all the galaxies
in the bright luminosity bins have their host halo mass assigned by ranking
method, the fraction of the galaxies in halos that have masses assigned
according to the mass-halo mass relation is about $30\%$ in L3 and about
$70\%$ in L2 and L1. It is possible that this relation underestimates the halo
mass. In order to see the effect caused by such uncertainties, we have used a
set of parameters from Yang et al. (2008) that are still allowed by the
observed stellar mass function but give larger halo masses to the hosts of
faint central galaxies. This increases the predicted ESD for L3 by $\sim
20\%$, not sufficient to explain the discrepancy. Indeed, this discrepancy is
not easy to fix. In the observational data, the amplitudes of the ESD for L3
at $R\sim 0.3$ \- $0.9h^{-1}{\rm Mpc}$ are actually slightly higher than for
the brighter sample L4, while in our model the ESD for L3 is always lower than
that for L4. There is an effect that may help to reduce the discrepancy
between our model prediction and the observational results for low-mass
galaxies. Since the group catalogue is only complete down to certain halo mass
limit at different redshift (see Y07), additional assumptions have to be made
in order to model the distribution of the haloes below the mass limit. In the
model described above, we have assumed that the haloes below the mass limit
have a random distribution, so that they do not contribute to the ESD.
However, in reality these low-mass haloes are correlated with the more massive
ones. As a result, our assumption will underestimate the 2-halo term of the
ESD.

In order to understand how the predicted ESD is produced, we also show
separately the contributions from different sources. The dotted lines show the
ESD contributed by the 1-halo term of central galaxies. For all the luminosity
bins, this term dominates the ESD at small $R$. For the brightest two bins,
this term dominates the ESD over the entire range of $R$ studied. This
reflects the fact that almost all the galaxies in these two bins are central
galaxies and the haloes in which they reside are more extended. The dashed
lines show the ESD contributed by the 1-halo term of satellite galaxies. This
term first decreases with $R$ and then increases to a peak value before
declining at large $R$. This owes to the fact that, in the inner part, the
lensing signal produced by satellite galaxies is dominated by the subhaloes
associated with them, while at larger $R$ the lensing signal produced by
satellite galaxies is dominated by their host halos. The value of $R$ at which
the ESD reaches the minimum corresponds roughly to the average halo-centric
distance of the subhaloes in the luminosity bin. Note that the contribution of
satellites to the total ESD is only important at large $R$ in the low-
luminosity bins. This reflects the fact that a significant fraction of the
low-luminosity satellites reside in massive haloes. Note that although the
1-halo satellite term dominates the lensing signal at $R\geq 0.3$$h^{-1}{\rm
Mpc}$ for faint luminosity bins, the discrepancy between the observation and
our prediction cannot be simply solved by boosting up the satellite
contribution. The reason is that the increase of the 1-halo satellite term
requires the increase of the host halo mass of the satellites, which will make
the 1-halo central term of the bright bins increase as well, causing
significant discrepancy for the bright bins. Finally, the dash-dotted lines
represent the contribution of the 2-halo term. As expected, this term is
relevant only on relatively large scales. In our model this term never
dominates at $R\leq 2\,h^{-1}{\rm Mpc}$. However, as discussed before this
term may be underestimated here. In C08, the 2-halo term is found to be
comparable to the 1-halo satellite term even at $R\sim 0.3h^{-1}{\rm Mpc}$.
Unfortunately the 2-halo term in C08 may be overestimated because halo-
exclusion effect is not properly included.

The large fluctuations seen in L6f are due to the small number of galaxies in
this luminosity bin.

Since the halo mass of each lensing galaxy is known in our model, we can also
examine the contributions to the total ESD in terms of the halo mass. Fig. 5
shows the results where the host haloes are split into three bins of $M_{S}$:
$M_{S}\geq 10^{14}h^{-1}{\rm M}_{\odot}$ (dotted lines); $10^{13}h^{-1}{\rm
M}_{\odot}\leq M_{S}<10^{14}h^{-1}{\rm M}_{\odot}$ (dashed lines); and
$M_{S}<10^{13}h^{-1}{\rm M}_{\odot}$ (dashed-dotted lines). As one can see,
the lensing signals in brighter bins are dominated by more massive haloes.
Very massive haloes with $M_{S}\geq 10^{14}h^{-1}{\rm M}_{\odot}$ are not the
dominant contributor, even for the brightest luminosity bin considered here,
because the total number of galaxies hosted by such haloes are relatively
small. For the low-luminosity bins, relatively massive haloes dominate the ESD
at large $R$, because a significant fraction of the low-luminosity satellites
are hosted by massive haloes (see Figs. 2 and 3).

###  4.3 Dependence on Galaxy Type

In Fig. 6 we present the results separately for early-type and late-type
galaxies. For a given luminosity bin, the predicted ESD has a higher amplitude
for early-type galaxies, clearly due to the fact that early-type galaxies are
more likely to reside in massive haloes (see e.g. van den Bosch, Yang & Mo
2003). For the faint samples, L1 and L2, the behavior of the predicted ESD for
early-type galaxies resembles that of satellite galaxies in these luminosity
bins, while the predicted ESD for the late-type galaxies looks like of the
central galaxies in the corresponding luminosity bins. This, again, reflects
the fact that faint early-type galaxies are mostly satellites in massive
haloes, while the faint late-type population is dominated by the central
galaxies in low-mass haloes.

The dashes lines in Fig. 6 show the results obtained using $M_{L}$ as halo
mass, rather than $M_{S}$ (see § 3.2). For early-type galaxies, the results
based on this halo mass estimate are very similar to those based on $M_{S}$.
However, for the late-type samples, the ESDs obtained using $M_{L}$ are
significantly higher than those obtained using $M_{S}$, especially for the
brighter samples. This is mainly due to the fact that late-type galaxies
contain significant amounts of young stars, so that their stellar mass-to-
light ratios are relatively low. Consequently, they are assigned a larger halo
mass based on their luminosity than based on their stellar mass.

The model predictions are compared with the observational results of
Mandelbaum et al. (2006). Here again, the model prediction based on $M_{S}$
matches the observational data for the four bright samples. For the three
faint bins, the model predictions are again lower than the observational
results. As shown in Fig. 6, it seems that our model prediction agrees better
with the observation for the late type galaxies. For example, for L3 our model
prediction matches the observation reasonably well for the late-type subsample
(with the reduced $\chi^{2}$ equal to $1.6$), while the discrepancy is quite
large for the early-type subsample (with reduced $\chi^{2}$ equal to $5.3$).
Since faint, early-type galaxies are preferentially satellite galaxies in
relatively massive halos while faint, late-type galaxies are mostly central
galaxies in relatively low mass halos, the above results seem to indicate that
the discrepancy is due to the underestimate of the 1-halo satellite term in
the model. Unfortunately, there is no simple modification of the model that
can fix the discrepancy. Since the spatial distribution of galaxies is fixed
by the observation, the only change that can be made is in halo properties. As
we will see in the following subsection, increasing the halo concentration
even by a factor of two can only increase the predicted ESD by about 20% for
the L3 sample, insufficient to explain the discrepancy. An increase in the
halo masses assigned to galaxy groups can reduce the discrepancy for faint
galaxies, but it would also significantly over-predict the ESD for bright
galaxies. The other possibility is that the halo masses of isolated, faint
early-type galaxies are significantly underestimated. For example, isolated
early-type galaxies may reside in much more massive halos than that given by
their stellar masses. In order to explain the discrepancy, the halo masses for
these galaxies need to be larger by a factor of at least 3. This will not
affect significantly the prediction for bright galaxies, but can boost the
prediction for L3 by 50% at $R\sim 0.2h^{-1}{\rm Mpc}$. Unfortunately, it is
still unclear if the required halo mass increase is feasible in current models
of galaxy formation.

###  4.4 Dependence on other Model Parameters

The ESD signal predicted by the model outlined in § 4.2 is based on several
assumptions. Beside the underlying cosmology and the halo mass assignment
which are the most crucial model ingredients (see § 4.5 and § 4.6), the model
requires a concentration-halo mass relation and it assumes that central
galaxies reside at rest at the centre of a spherical dark matter halo. In this
subsection, we test how our results are affected by these assumptions.

The concentration parameter is a measure of the amount of dark matter in the
central regions of the haloes. Accordingly, different models for the
concentration-halo mass relations are expected to result in different
predictions for the ESD signal (at least on small scales). The fiducial model
described in § 4.2 uses the model of Macciò et al. (2007). However, other
models are also available in the literature (e.g. Bullock et al. 2001; Eke,
Navarro & Steinmetz 2001), which predict concentration-mass relations that are
slightly different (see C08 for an assessment of the impact on galaxy-galaxy
lensing). Furthermore, the presence of a (central) galaxy in a dark matter
halo may have an impact on its concentration via, for example, adiabatic
contraction (e.g. Blumenthal et al. 1986), which is not accounted for in the
concentration-mass relations obtained from pure $N$-body simulations. Finally,
attempts to measure halo concentrations observationally have thus far given
conflicting results (e.g. van den Bosch & Swaters 2001; Comerford & Natarajan
2007, and referenecs therein). To examine how the lensing predictions depend
on changes in halo concentation we consider two models: model CH, in which the
concentrations are 2 times as high as in our fiducial model, and model CL, in
which the concentrations are 2 times smaller. The left column of Fig. 7 shows
the predictions of models CH (dashed lines) and CL (dotted lines) compared to
our fiducial model (solid lines). Results are only shown for four luminosity
bins, as indicated. Note that the model with higher (lower) halo
concentrations predicts ESDs that are higher (lower). The effect is stronger
on scales where the 1-halo central term dominates (see Fig 4). Accordingly, in
the case of the brightest sample (L6f), models CL and CH differ from the
fiducial model on scales up to $R\simeq 1\,h^{-1}{\rm Mpc}$, while in sample
L4 the differences are only appreciable out to $R\simeq 0.5\,h^{-1}{\rm Mpc}$.
We conclude that the predicted ESD depends quite strongly on the assumed halo
concentrations, indicating that galaxy-galaxy lensing has the potential to
constrain the density profiles of dark matter haloes (see Mandelbaum et al.
2008). In this paper, we have assumed that the concentration of a halo depends
only on its mass, we have ignored the possible halo age-dependence of the
concentration-mass relation (e.g. Wechsler et al. 2002; Zhao et al. 2003; Lu
et al. 2006). If for a given mass older halos have higher concentration, and
if the formation of a galaxy in a halo depends strongly on the formation
history of the halo, then the age-dependence of the halo concentration must be
taken into account. Unfortunately, it is unclear how to connect the halo age
(hence halo concentration) with the properties of galaxies. For a given halo
mass, the dispersion in the concentration is about 0.12 dex (e.g. Jing 2000),
which corresponds to a change of about 20% in the predicted ESD on $R\sim
0.1\,h^{-1}{\rm Mpc}$.

In our fiducial model, central galaxies are assumed to reside at the center of
their dark matter haloes. However, as shown in van den Bosch et al. (2005), in
haloes with masses $M_{h}>10^{13}h^{-1}{\rm M}_{\odot}$ there is evidence to
suggest that central galaxies are offset from their halo centers by $\sim 3$
percent of the virial radius. A similar result was obtained by Berlind et al.
(2003) using SPH simulations of galaxy formation. To examine how such an
offset impacts on the ESDs, we consider two additional models. Following van
den Bosch et al. (2005) and Yang et al. (2006), we assume that central
galaxies in haloes with $M_{S}>10^{13}h^{-1}{\rm M}_{\odot}$ are offset from
their halo centers by an amount that is drawn from a Gaussian distribution
with zero mean. We use two different values for the dispersion of the
distribution: $3\%$ of the virial radius (model ‘Dev1’) and $6\%$ of the
virial radius (model ‘Dev2’). The corresponding lensing predictions are shown
in the middle column of Fig. 7 as the dotted (Dev1) and dashed (Dev2) lines.
Note that the offsets only affect the lensing signal for the brightest sample
(L6f), where a larger offset results in a stronger suppression of the ESD on
small scales ($R<0.1\,h^{-1}{\rm Mpc}$). For fainter samples, no (significant)
differences with respect to the fiducial model are apparent, which owes to the
fact that fainter centrals typically reside in haloes with
$M_{h}<10^{13}h^{-1}{\rm M}_{\odot}$ which do not have an offset (at least in
our models).

As a final test, we examine the impact of halo shapes on the galaxy-galaxy
lensing signal. In our fiducial model, dark matter haloes are assumed to be
spherically symmetric. However, $N$-body simulations show that, in general,
they are triaxial rather than spherical. Jing & Suto (1998) proposed a fitting
formula for triaxial dark matter haloes, which has been applied to both strong
and weak lensing analyses (e.g. Oguri, Lee & Suto 2003; Oguri & Keeton 2004;
Tang & Fan 2005). In order to examine the impact of our assumption of halo
sphericity, we consider an alternative model (model ‘TRI’), in which we assume
that the dark matter density distribution is given by $\rho_{\rm TRI}(R)$,
where $R$ specifies an ellipsoidal surface:

$R^{2}=\left(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}\right)c^{2}\,.$
(10)

Here $a\leq b\leq c$ are the three principal semi-axes of the ellipsoid. We
set $\rho_{\rm TRI}(R)=\rho(R^{\prime})$, where $\rho(R^{\prime})$ is the NFW
profile, so that the total mass within a sphere of radius $R^{\prime}$ in
$\rho(R^{\prime})$ is equal to the mass within the elliptical shell at $R$.
For the axis ratios we adopt the distribution function given by Jing & Suto
(2002):

$\displaystyle p(a/c)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi}\times 0.113}\left(\frac{M_{\rm
vir}}{M_{NL}}\right)^{0.07[\Omega(z)]^{0.7}}$ $\displaystyle\times$
$\displaystyle\exp\left\\{-\frac{\left[(a/c)(M_{\rm
vir}/M_{NL})^{0.07[\Omega(z)]^{0.7}}-0.54\right]^{2}}{2(0.113)^{2}}\right\\}$

and

$\displaystyle p(a/b|a/c)$ $\displaystyle=$
$\displaystyle\frac{3}{2(1-\max(a/c,0.5))}$ (12) $\displaystyle\times$
$\displaystyle\left[1-\left(\frac{2a/b-1-\max(a/c,0.5)}{1-\max(a/b,0.5)}\right)^{2}\right]\,,$

where $M_{NL}$ is the characteristic mass scale, on which the rms of the top-
hat smoothed over-density is equal to $1.68$. In practice, we proceed as
follows: For each dark matter halo we first draw the axis ratio $a/c$ and
$a/b$ using Eqs. (4.4) and (12), respectively. Next we draw a random 3D
orientation of the principal axes, and project the dark matter particles along
the (fiducial) line-of-sight. Next, for each halo, we determine the major axis
of the projected distribution which we align with the major axis of the
central galaxy. This assumption is motivated by observational claims that the
major axis of a central galaxy tends to aligned with that of its host halo
(e.g. Yang et al. 2006; Faltenbacher et al. 2008; Wang et al. 2008). As shown
in the right-hand panels of Fig. 7, changing from spherical haloes (solid
lines) to triaxial haloes (dotted lines) has almost no impact on the predicted
ESDs. This should not come as an entire surprise, since the ESDs are
azimuthally averaged over many haloes, which have random orientations on the
sky. Note that here we assume that the halo is perfectly aligned with the
central galaxy. However, the observational results mentioned above actually
suggest a misalignment. We have tried a model in which the orientation of the
host halo is uncorrelated with that of the central galaxy. The change in the
results is very small compared with the model assuming perfect alignment.

###  4.5 Dependence on Cosmology

Another very important assumption in our model prediction is the cosmological
model in the calculations of the halo mass function and the geometrical
properties of spacetime. The redshifts of galaxies considered here are
restricted to $z\leq 0.2$, and the impact of changing cosmological parameters
on the spacetime geometry is quite small in our analysis. On the other hand,
changing cosmological parameters can change the halo masses assigned to
individual groups, which may have significant impact on the expected lensing
signal. In our fiducial model, we adopt the cosmology parameters obtained from
the WMAP 3-year data, with $\Omega_{\rm m}=0.238$, $\Omega_{\Lambda}=0.762$,
$n=0.951$, and $\sigma_{8}=0.75$ (Spergel et al. 2007). As comparison, we will
show some results obtained assuming another set of cosmological parameters
with $\Omega_{\rm m}=0.3$, $\Omega_{\Lambda}=0.7$, $n=1.0$, and
$\sigma_{8}=0.9$, which is strongly supported by the first year data release
of the WMAP mission (see Spergel et al. 2003) and has been considered in many
previous studies. In what follows we will refer to this second set of
parameters as the WMAP1 cosmology. Note that the cosmological parameters given
by the recent WMAP 5-year data (Komatsu et al. 2008) are in between those of
WMAP1 and WMAP3.

Fig. 8 compares the ESD predicted by the fiducial model using the WMAP3
cosmology (solid lines) and that predicted by the WMAP1 cosmology (dotted
lines). As one can see, the ESD predicted by WMAP1 is significantly higher
than that predicted by WMAP3, especially for bright galaxies. Most of this
increase is due to changes in the halo mass function, which causes (massive)
groups to be assigned a larger halo mass. The changes in the halo
concentrations and the spacetime geometry play only a minor role. A comparison
with the SDSS data clearly favors the WMAP3 cosmology over the WMAP1
cosmology, especially for the brighter luminosity bins. The reduced $\chi^{2}$
for the WMAP1 cosmology is $21.3$, much larger than $3.2$ for the WMAP3
cosmology. This result is in good agreement with that obtained in C08 using
the CLF model.

Thus, we conclude that the galaxy-galaxy lensing data either prefer the WMAP3
cosmology, which has a relatively low $\sigma_{8}$, or our halo mass
assignment is in serious error. In the following subsection we show that the
uncertainties in our halo mass assignments are unlikely to change our results
significantly. We therefore conclude that the galaxy-galaxy lensing data
prefer a $\Lambda$CDM model with a relatively low $\sigma_{8}$. If we use
WMAP5 parameters, the model prediction is in between WMAP1 and WMAP3, which is
still too high to matched the observed ESD of bright galaxies.

###  4.6 Uncertainties in Halo-Mass Assignment

In our model, the masses of groups are assigned according to the stellar-mass
ranking and the halo mass function predicted by the adopted cosmology. The
underlying assumption is that the mass function of the host haloes of groups
is the same as that predicted by the cosmological model. However, even if the
cosmological model adopted is a good approximation to the real universe, the
observed halo density may be different from the model prediction because of
cosmic variance introduced by the finite observational volume. The effect of
such variance is expected to be most important for massive haloes, because the
number density of such systems is small. If, for example, the number density
of massive haloes in the observational sample is, due to cosmic variance,
smaller than the model prediction, the mass assignment with the use of the
theoretical halo mass function would assign a higher halo mass to groups.
Consequently, the ESD of bright galaxies, which are biased toward massive
haloes, would be overestimated. Here we test the importance of such effects by
considering the uncertainties due to Poisson fluctuations. We use the halo
mass function predicted with the WMAP3 cosmology to generate a set of random
halo samples, each of which contains the same number of groups as the
observational sample. The halo masses in each of these samples is then ranked
in descending order and the halo mass of a given rank is then assigned to the
group with the same rank in stellar mass. We find that the scatter in the ESD
obtained in this way is negligibly small, even for the brightest sample. The
reason is that, even for the brightest sample, the ESD is dominated by haloes
with intermediate masses, $10^{13}h^{-1}{\rm
M}_{\odot}<M_{S}<10^{14}h^{-1}{\rm M}_{\odot}$ (see Fig. 5), and the total
number of such groups is quite large.

Another uncertainty in the mass assignment may arise from fiber collisions. In
the SDSS survey, no two fibers on the same SDSS plate can be closer than 55
arcsec. Although this fiber collision constraint is partially alleviated by
the fact that neighboring plates have overlap regions, $\sim 7$ percent of all
galaxies eligible for spectroscopy do not have a measured redshift. Our
analysis here is based on the group catalog constructed from galaxy Sample II
(see Y07), in which many of the galaxies missed due to fiber collisions are
not included. Consequently, the total stellar mass (or the total luminosity)
of some of the groups may be underestimated, thus introducing a bias in the
ranking. This bias is expected to be stronger in richer systems because they
have higher projected galaxy number density and are more likely to suffer from
fiber collisions. However, this effect is not likely to have a big impact on
our results, because it only changes the relative ranking of the groups that
have similar stellar mass (luminosity) and because the galaxy-galaxy lensing
signals are averaged in relatively broad bins of galaxy luminosity.

In order to quantify this effect we carry out a similar analysis using the
group catalog constructed from Sample III, where a galaxy affected by fiber
collisions is assigned the redshift of its nearest neighbor (see Y07 for
details). In this case, the situation is the opposite to that in Sample II,
because here some galaxies may be wrongly assigned to groups in the foreground
or the background due to the wrong redshifts assigned to some of the fiber-
collision galaxies. The stellar mass and the luminosity of some of the groups
will therefore be overestimated. In our test, we use Sample III to construct
the group and to set mass to groups according to their ranking in Sample III,
but we calculate the ESD only around galaxies that are in Sample II. The
results obtained in this way are very similar to those based on Sample II
alone, suggesting that the effect of missing fiber-collision galaxies in the
mass assignment is not important. However, if the fiber-collision galaxies are
themselves included in the calculation of the ESD, i.e. if we use galaxies in
Sample III to calculate the ESD, the amplitude of the ESD is significantly
larger than that obtained from the galaxies in Sample II. The reason for this
is that the galaxies affected by fiber collisions are preferentially located
in high density regions, so that they are more likely associated with a
massive halo. The observational results of Mandelbaum et al. (2006) are based
on galaxies that have spectroscopic redshifts, and so a fair comparison
between the observational results and our model predictions can only be made
with the use of galaxies in Sample II. Our test above therefore demonstrates
that our conclusions about the comparison between the observational data and
the model predictions are robust against the uncertainties due to fiber
collisions.

The scatter in the relation between halo mass and stellar mass (or luminosity)
may also produce some uncertainties in our model prediction. As shown in
Mandelbaum & Seljak (2007), the scatter is in partial degeneracy with
cosmology model. In our investigation, we have fixed cosmological parameters
and allowed no dispersion in the halo mass-total stellar mass relation. If,
for example, we assume a log normal distribution with a dispersion of 0.3
index in halo mass for a given stellar mass, the predicted ESD would be a few
percent larger than that predicted by the fiducial model.

##  5 Summary

In this paper we model the galaxy-galaxy lensing signal expected for SDSS
galaxies, using the galaxy groups selected from the SDSS to represent the dark
matter haloes within which the galaxies reside. We use the properties of the
dark halo population, such as mass function, density profiles and shapes,
expected from the current $\Lambda$CDM cosmogony to model the dark matter
distribution in each of the groups identified in the SDSS volume. The use of
the real galaxy groups allows us to predict the galaxy-galaxy lensing signals
separately for galaxies of different luminosity, morphological types, and in
different environments (e.g. central versus satellite galaxies). We check the
robustness of our model predictions by changing the assumptions about the dark
matter distribution in individual groups (such as the shape, density profile,
and center offset of dark matter haloes), as well as the cosmological model
used to predict the properties of the halo population. We compare our model
predictions with the observational data of Mandelbaum et al. (2006) for
similar samples of lens galaxies. Although there is some discrepancy for lens
galaxies in the low-luminosity bins, the overall observational results can be
well understood in the current $\Lambda$CDM cosmogony. In particular, the
observed results can be well reproduced in a $\Lambda$CDM model with
parameters based on the WMAP3 data, but a $\Lambda$CDM model with a
significantly higher $\sigma_{8}$, such as the one based on the WMAP1 data,
significantly over-predicts the galaxy-galaxy lensing signal. Our results also
suggest that, once a correct model of structure formation is adopted, the halo
masses assigned to galaxy groups based on ranking their stellar masses with
the halo mass function, are statistically reliable. The results obtained imply
that galaxy-galaxy lensing is a powerful tool to constrain both the mass
distribution associated with galaxies and cosmological models. In the future,
when deep imaging surveys provide more sources with high image quality in the
SDSS sky coverage, the galaxy-galaxy lensing signals produced by the SDSS
galaxies can be estimated to much higher accuracy. The same analysis as
presented here is expected to provide stringent constraints on the properties
of the dark matter haloes associated with galaxies and galaxy systems, as well
as on cosmological parameters.

## Acknowledgments

We thank Rachel Mandelbaum for providing the SDSS lensing data in electronic
format. Part of the computation was carried out on the SGI Altix 330 system at
the Department of Astronomy, Peking University. Li Ran is supported by the
National Scholarship from China Scholarship Council. XY is supported by the
One Hundred Talents project of the Chinese Academy of Sciences and grants from
NSFC (Nos. 10533030, 10673023). HJM would like to acknowledge the support of
NSF AST-0607535, NASA AISR-126270 and NSF IIS-0611948. Zuhui Fan would like to
acknowledge the supports from NSFC under grants. 10373001, 10533010, and
10773001, and 973 Program (No. 2007CB815401).

## References

  * Adelman-McCarthy (2006) Adelman-McCarthy J. K., et al., 2006, ApJS, 162, 38
  * Bell (2003) Bell E. F. , McIntosh D. H., Katz N., Weinberg M. D., 2003, ApJS, 149, 289
  * Berlind (2002) Berlind, A. A., Weinberg D. H., 2002, ApJ, 575, 587
  * Berlind (2003) Berlind A. A., et al., 2003, ApJ, 593, 1
  * Blanton (2003) Blanton M. R., et al. , 2003, AJ, 125, 2348
  * Blanton (2005) Blanton M. R., et al., 2005, AJ, 129, 2562
  * Blumenthal (1986) Blumenthal G. R., Faber S. M., Flores R., Primack J. R., 1986, ApJ, 301, 27
  * Brainerd (1996) Brainerd T. G, Blandford R. D., Smail I., 1996, ApJ, 466, 623
  * Bullock (2001) Bullock J. S., Kolatt T. S., Sigad Y., Somerville R. S., Kravtsov A. V., Klypin A. A., Primack J. R., Dekel A., 2001 MNRAS, 321, 559
  * Cacciato (2008) Cacciato M., van den Bosch F. C., More S., Li R., Mo H. J, Yang X., 2008, submitted to MNRAS (C08)
  * cole (2000) Cole S., Lacey C.G., Baugh C.M., Frenk C.S. , 2000, MNRAS, 319, 168
  * coll (2001) Colless M., et al., 2001, MNRAS, 328, 1039
  * Comerford (2007) Comerford J. M., Natarajan P., 2007, MNRAS, 379, 190
  * Cooray (2002) Cooray A., Sheth R., 2002, PhR, 372, 1
  * Cooray (2006) Cooray A., 2006, MNRAS, 365, 842
  * Croton (2006) Croton D. J., et al., 2006, MNRAS, 367, 864
  * deVau (deVau) de Vaucouleurs G., de Vaucouleurs A., Corwin H.G., Buta R.J., Paturel G., Fouque P., 1991, Third Reference Catalogue of Bright Galaxies, (Springel-Verlag Heidelber)
  * Eke (2001) Eke V.R., Navarro J.F., Steinmetz M., 2001, ApJ, 554, 114
  * Faltenbacher (2007) Faltenbacher A. , Li C., Mao S., van den Bosch F. C., Yang X., Jing Y. P., Pasquali A., Mo H. J. , 2007, ApJ, 662, 71
  * Giocoli (2008) Giocoli C., Tormen G., van den Bosch F.C., 2008, MNRAS, 386, 2135
  * Gao (2004) Gao L., White S. D. M., Jenkins A., Stoehr F., Springel V., 2004, MNRAS, 355, 819
  * Hayashi (2003) Hayashi E., Navarro J. F., Taylor J. E., Stadel J., Quinn T., 2003, ApJ, 584, 541
  * Henry (2000) Henry, J. P., 2000, ApJ, 534, 565
  * Hoekstra (2003) Hoekstra H., Franx M., Kuijken K., Carlberg R. G., Yee H. K. C., 2003, MNRAS, 340, 609
  * Hoekstra (2004) Hoekstra H., 2004, MNRAS 347, 1337
  * Hudson (1998) Hudson M. J., Gwyn S. D. J., Dahle H., Kaiser N., 1998, ApJ, 503, 531
  * Jing (1998) Jing Y. P., Suto, Y., 1998, ApJ, 494L, 5
  * Jing (00) Jing Y. P., 2000, ApJ, 535, 30
  * Jing (2002) Jing Y. P., Suto Y., 2002, ApJ, 574, 538
  * JMB (1998) Jing Y. P., Mo H. J., Börner G., 1998, ApJ, 494, 1
  * Johnston (2007) Johnston D. E., et al., 2007, preprint (arXiv: astro-ph/0709.1159)
  * Kang (2005) Kang X., Jing Y. P., Mo H. J., Börner G., 2005, ApJ, 631, 21
  * (33) Kauffmann G., White S. D. M., Guiderdoni B., 1993, MNRAS, 264, 201
  * (34) Kauffmann G. , White S. D. M. , Heckman T. M. , Menard B. , Brinchmann J. , Charlot S. , Tremonti C. , Brinkmann J. 2004, MNRAS, 353, 713
  * (35) Katz N., Weinberg D. H., Hernquist L., 1996, ApJS, 105, 19
  * (36) Komatsu E. , Dunkley J. , 2008, preprint (arXiv: astro-ph/0803.0547)
  * Lin (2007) Limousin M, Kneib J. P., Bardeau S., Natarajan P., Czoske O., Smail I., Ebeling H., Smith G.P., 2007, A&A, 461, 881
  * Lu (2006) Lu Y., Mo H. J., Katz N., Weinberg M. D., 2006 , MNRAS, 368 1931
  * (39) Macciò A. V., Dutton A. A., van den Bosch F. C., Moore B., Potter D., Stadel J., 2007, MNRAS, 378, 55.
  * (40) Mandelbaum R., et al., 2005, MNRAS, 361, 1287
  * (41) Mandelbaum R., Seljak U., Kauffmann G., Hirata C. M., Brinkmann J., 2006 MNRAS, 368, 715
  * (42) Mandelbaum R., Seljak, U., 2007, JACP, 06, 024
  * (43) Mandelbaum R., Seljak, U., Hirata C. M., 2008, preprint (arXiv: astro-ph/0805.2552)
  * (44) McKay T. A., Sheldon E. S., Racusin J., et al. 2001, preprint (arXiv: astro-ph/0108013)
  * (45) McKay T. A., et al., 2002, ApJ, 571, L85
  * (46) Nakamura T. T., Suto Y., 1997, PThPh, 97, 49
  * Natarajan (2002) Natarajan P., Kneib J.P., Smail I., 2002, ApJ, 580, L11
  * (48) Navarro J.F., Frenk C.S., White S.D.M., 1997, ApJ, 490, 493
  * (49) Oguri M., Lee J., Suto Y., 2003, ApJ, 599, 7
  * (50) Oguri M., Keeton C., 2004, ApJ, 610, 663
  * (51) Peacock J. A., Smith R. E., 2000, MNRAS, 318, 1144
  * (52) Pearce F. R., Thomas P. A., Couchman H. M. P., Edge A. C., 2000, MNRAS, 317, 1029
  * (53) Petrosian V., 1976, ApJ, 209, L1
  * (54) Saunders W., et al., 2000, MNRAS, 317, 55
  * (55) Scranton R., 2003, MNRAS 339, 410
  * (56) Schlegel D.J., Finkbeiner D.P., Davis M., 1998, ApJ, 500, 525
  * (57) Sheldon E. S., et al., 2004, AJ, 127, 2544
  * (58) Sheldon E. S., et al., 2007a, preprint (arXiv:astro-ph/0709.1153)
  * (59) Sheldon E. S., et al., 2007b, preprint (arXiv: astro-ph/0709.1162)
  * (60) Somerville R. S., Primack J. R., 1999, MNRAS, 310, 1087
  * (61) Spergel D. N., et al., 2003, ApJS, 148, 175
  * (62) Spergel D. N., et.al., 2007, ApJS, 170, 377
  * (63) Springel V., 2005, MNRAS, 364, 1105
  * (64) Springel V. , et al. , 2005, Nature, 435, 629
  * (65) Strauss M.A., et al., 2002, AJ, 124, 1810
  * (66) Tang J. Y. , Fan Z. H., 2005, ApJ, 635, 60
  * (67) Tinker J.L., Weinberg D.H., Zheng Z., Zehavi I., 2005, ApJ, 631, 41
  * (68) Tyson J. A., Valdes F., Jarvis J. F., Mills A. P. Jr., 1984, ApJ, 281L, 59
  * (69) Vale A., Ostriker J. P., 2006, MNRAS, 371, 1173
  * (70) van den Bosch F. C. , Swaters R. A., 2001, MNRAS, 325, 1017
  * (71) van den Bosch F. C., 2002, MNRAS, 332, 456
  * (72) van den Bosch F. C., Yang X. , Mo H. J. , 2003, MNRAS, 340, 771
  * (73) van den Bosch F. C., Norberg P., Mo H. J., Yang X., 2004, MNRAS, 352, 1302
  * (74) van den Bosch F. C., Weinmann S. M., Yang X., Mo H. J., Li C., Jing Y.P., 2005, MNRAS, 361, 1203
  * (75) van den Bosch F. C., Tormen G., Giocoli C., 2005, MNRAS, 359, 1029
  * (76) van den Bosch F. C., et al., 2007, MNRAS, 376, 841
  * (77) Wang Y., Yang X., Mo H. J., Li C., van den Bosch F. C., Fan Z. H., Chen X., 2008, MNRAS, 385, 1511
  * (78) Warren M. S., Abazajian K., Holz D. E., Teodoro L., 2006, ApJ, 646, 881
  * (79) Wechsler R. H., Bullock J. S., Primack J. R., Kravtsov A. V., Dekel A., 2002, ApJ, 568, 52
  * (80) Weinberg D. H., Colombi S., Davè R., Katz N., 2008, ApJ, 678, 6
  * (81) White S. D. M., Frenk C., 1991, ApJ, 379, 52
  * (82) Yan R., Madgwick D. S., White M. , ApJ, 2003, 598, 848
  * (83) Yang X., Mo H. J., van den Bosch F. C., 2003, MNRAS, 339, 1057
  * (84) Yang X., Mo H. J., van den Bosch F. C., Jing Y. P., 2005, MNRAS, 356, 1293
  * (85) Yang X., Mo H. J., van den Bosch F. C., Jing Y. P., Weinmann S. M., Meneghetti M., 2006, MNRAS, 373, 1159
  * (86) Yang X., Mo H. J., van den Bosch F. C., Pasquali A., Li C., Barden M., 2007, ApJ, 671, 153 (Y07)
  * (87) Yang X., Mo H. J., van den Bosch F. C., 2008, preprint, (arXiv: astro-ph/0808.0539)
  * (88) Zhao D. H., Jing Y. P., Mo H. J., Börner G., 2003, ApJ, 597, 9
  * (89) Zheng Z. , et al. , 2005, ApJ, 633, 791
  * (90) Zheng Z., Weinberg D. H., 2007, ApJ, 659, 1


* Genda & Abe (2005) Genda, H., & Abe, Y. 2005, Nature, 433, 842, doi: 10.1038/nature03360
  * Genda et al. (2017) Genda, H., Iizuka, T., Sasaki, T., Ueno, Y., & Ikoma, M. 2017, EPSL, 470, 87, doi: 10.1016/j.epsl.2017.04.035
  * Genda et al. (2012) Genda, H., Kokubo, E., & Ida, S. 2012, ApJ, 744, 137, doi: 10.1088/0004-637X/744/2/137
  * Gladman (1993) Gladman, B. 1993, Icarus, 106, 247, doi: 10.1006/icar.1993.1169
  * Goldstein et al. (1976) Goldstein, B. E., Phillips, R. J., & Russell, C. T. 1976, Lunar and Planetary Science Conference Proceedings, 3, 3321
  * Gross et al. (2014) Gross, J., Treiman, A. H., & Mercer, C. N. 2014, EPSL, 388, 318, doi: 10.1016/j.epsl.2013.12.006
  * Haghighipour et al. (2018) Haghighipour, N., Maindl, T. I., Schäfer, C. M., & Wandel, O. J. 2018, ApJ, 855, 60, doi: 10.3847/1538-4357/aaa7f3
  * Halliday (2000) Halliday, A. N. 2000, EPSL, 176, 17, doi: 10.1016/S0012-821X(99)00317-9
  * Hallis et al. (2010) Hallis, L. J., Anand, M., Greenwood, R. C., et al. 2010, Geochim. Cosmochim. Acta, 74, 6885, doi: 10.1016/j.gca.2010.09.023
  * Hartmann (2014) Hartmann, W. K. 2014, Philosophical Transactions of the Royal Society of London Series A, 372, 20130249, doi: 10.1098/rsta.2013.0249
  * Hartmann & Davis (1975) Hartmann, W. K., & Davis, D. R. 1975, Icarus, 24, 504, doi: 10.1016/0019-1035(75)90070-6
  * Hauri et al. (2011) Hauri, E. H., Weinreich, T., Saal, A. E., Rutherford, M. C., & Van Orman, J. A. 2011, Science, 333, 213, doi: 10.1126/science.1204626
  * Hosono et al. (2017) Hosono, N., Iwasawa, M., Tanikawa, A., et al. 2017, PASJ, 69, 26, doi: 10.1093/pasj/psw131
  * Hosono et al. (2019) Hosono, N., Karato, S.-i., Makino, J., & Saitoh, T. R. 2019, Nat. Geosci., 12, 418, doi: 10.1038/s41561-019-0354-2
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Ida et al. (1997) Ida, S., Canup, R. M., & Stewart, G. R. 1997, Nature, 389, 353, doi: 10.1038/38669
  * Jackson et al. (2018) Jackson, A. P., Gabriel, T. S. J., & Asphaug, E. I. 2018, MNRAS, 474, 2924, doi: 10.1093/mnras/stx2901
  * Jackson & Wyatt (2012) Jackson, A. P., & Wyatt, M. C. 2012, MNRAS, 425, 657, doi: 10.1111/j.1365-2966.2012.21546.x
  * Jacobsen (2005) Jacobsen, S. B. 2005, AREPS, 33, 531, doi: 10.1146/annurev.earth.33.092203.122614
  * Jacobson et al. (2014) Jacobson, S. A., Morbidelli, A., Raymond, S. N., et al. 2014, Nature, 508, 84, doi: 10.1038/nature13172
  * Jacobson et al. (2017) Jacobson, S. A., Rubie, D. C., Hernlund, J., Morbidelli, A., & Nakajima, M. 2017, EPSL, 474, 375, doi: 10.1016/j.epsl.2017.06.023
  * Johansen et al. (2021) Johansen, A., Ronnet, T., Bizzarro, M., et al. 2021, arXiv e-prints, arXiv:2102.08611.  https://arxiv.org/abs/2102.08611
  * Jutzi (2015) Jutzi, M. 2015, Planet. Space Sci., 107, 3, doi: 10.1016/j.pss.2014.09.012
  * Kaib & Cowan (2015) Kaib, N. A., & Cowan, N. B. 2015, Icarus, 252, 161, doi: 10.1016/j.icarus.2015.01.013
  * Kenyon & Bromley (2006) Kenyon, S. J., & Bromley, B. C. 2006, AJ, 131, 1837, doi: 10.1086/499807
  * Kokubo et al. (2000a) Kokubo, E., Canup, R. M., & Ida, S. 2000a, in Origin of the Earth and Moon, ed. R. M. Canup, K. Righter, & et al., 145–163
  * Kokubo & Genda (2010) Kokubo, E., & Genda, H. 2010, ApJ, 714, L21, doi: 10.1088/2041-8205/714/1/L21
  * Kokubo & Ida (2002) Kokubo, E., & Ida, S. 2002, ApJ, 581, 666, doi: 10.1086/344105
  * Kokubo et al. (2000b) Kokubo, E., Ida, S., & Makino, J. 2000b, Icarus, 148, 419, doi: 10.1006/icar.2000.6496
  * Kortenkamp & Hartmann (2016) Kortenkamp, S. J., & Hartmann, W. K. 2016, Icarus, 275, 239, doi: 10.1016/j.icarus.2016.04.007
  * Kraus et al. (2012) Kraus, R. G., Stewart, S. T., Swift, D. C., et al. 2012, Journal of Geophysical Research (Planets), 117, E09009, doi: 10.1029/2012JE004082
  * Kruijer & Kleine (2017) Kruijer, T. S., & Kleine, T. 2017, EPSL, 475, 15, doi: 10.1016/j.epsl.2017.07.021
  * Lambrechts & Johansen (2012) Lambrechts, M., & Johansen, A. 2012, A&A, 544, A32, doi: 10.1051/0004-6361/201219127
  * Lambrechts et al. (2019) Lambrechts, M., Morbidelli, A., Jacobson, S. A., et al. 2019, A&A, 627, A83, doi: 10.1051/0004-6361/201834229
  * Leinhardt et al. (2010) Leinhardt, Z. M., Marcus, R. A., & Stewart, S. T. 2010, ApJ, 714, 1789, doi: 10.1088/0004-637X/714/2/1789
  * Leinhardt & Stewart (2012) Leinhardt, Z. M., & Stewart, S. T. 2012, ApJ, 745, 79, doi: 10.1088/0004-637X/745/1/79
  * Lock et al. (2020) Lock, S. J., Stewart, S. T., & Ćuk, M. 2020, EPSL, 530, 115885, doi: 10.1016/j.epsl.2019.115885
  * Lock et al. (2018) Lock, S. J., Stewart, S. T., Petaev, M. I., et al. 2018, Journal of Geophysical Research (Planets), 123, 910, doi: 10.1002/2017JE005333
  * Marcus et al. (2009) Marcus, R. A., Stewart, S. T., Sasselov, D., & Hernquist, L. 2009, ApJ, 700, L118, doi: 10.1088/0004-637X/700/2/L118
  * Mastrobuono-Battisti et al. (2015) Mastrobuono-Battisti, A., Perets, H. B., & Raymond, S. N. 2015, Nature, 520, 212, doi: 10.1038/nature14333
  * Matsumura et al. (2017) Matsumura, S., Brasser, R., & Ida, S. 2017, A&A, 607, A67, doi: 10.1051/0004-6361/201731155
  * Maurice et al. (2020) Maurice, M., Tosi, N., Schwinger, S., Breuer, D., & Kleine, T. 2020, Science Advances, 6, eaba8949, doi: 10.1126/sciadv.aba8949
  * Melosh (1989) Melosh, H. J. 1989, Impact cratering : a geologic process (New York: Oxford University Press and Oxford: Clarendon Press)
  * Melosh (2007) —. 2007, Meteoritics and Planetary Science, 42, 2079, doi: 10.1111/j.1945-5100.2007.tb01009.x
  * Melosh (2014) —. 2014, Philosophical Transactions of the Royal Society of London Series A, 372, 20130168, doi: 10.1098/rsta.2013.0168
  * Meyer et al. (2010) Meyer, J., Elkins-Tanton, L., & Wisdom, J. 2010, Icarus, 208, 1, doi: 10.1016/j.icarus.2010.01.029
  * Monaghan (1992) Monaghan, J. J. 1992, ARA&A, 30, 543, doi: 10.1146/annurev.aa.30.090192.002551
  * Morbidelli et al. (2000) Morbidelli, A., Chambers, J., Lunine, J. I., et al. 2000, Meteoritics and Planetary Science, 35, 1309, doi: 10.1111/j.1945-5100.2000.tb01518.x
  * Movshovitz et al. (2016) Movshovitz, N., Nimmo, F., Korycansky, D. G., Asphaug, E., & Owen, J. M. 2016, Icarus, 275, 85, doi: 10.1016/j.icarus.2016.04.018
  * Nakajima & Stevenson (2014) Nakajima, M., & Stevenson, D. J. 2014, Icarus, 233, 259, doi: 10.1016/j.icarus.2014.01.008
  * Nakajima & Stevenson (2015) —. 2015, EPSL, 427, 286, doi: 10.1016/j.epsl.2015.06.023
  * Nakajima & Stevenson (2018) —. 2018, EPSL, 487, 117, doi: 10.1016/j.epsl.2018.01.026
  * Nakajima et al. (2021) Nakajima, M., Golabek, G. J., Wünnemann, K., et al. 2021, Earth and Planetary Science Letters, 568, 116983, doi: 10.1016/j.epsl.2021.116983
  * Nemchin et al. (2009) Nemchin, A., Timms, N., Pidgeon, R., et al. 2009, Nat. Geosci., 2, 133, doi: 10.1038/ngeo417
  * Nie & Dauphas (2019) Nie, N. X., & Dauphas, N. 2019, ApJ, 884, L48, doi: 10.3847/2041-8213/ab4a16
  * O’Brien et al. (2006) O’Brien, D. P., Morbidelli, A., & Levison, H. F. 2006, Icarus, 184, 39, doi: 10.1016/j.icarus.2006.04.005
  * Pahlevan & Stevenson (2007) Pahlevan, K., & Stevenson, D. J. 2007, EPSL, 262, 438, doi: 10.1016/j.epsl.2007.07.055
  * Peplowski et al. (2011) Peplowski, P. N., Evans, L. G., Hauck, S. A., et al. 2011, Science, 333, 1850, doi: 10.1126/science.1211576
  * Perera et al. (2018) Perera, V., Jackson, A. P., Elkins-Tanton, L. T., & Asphaug, E. 2018, Journal of Geophysical Research (Planets), 123, 1168, doi: 10.1029/2017JE005512
  * Pierazzo & Melosh (2000) Pierazzo, E., & Melosh, H. J. 2000, Annual Review of Earth and Planetary Sciences, 28, 141, doi: 10.1146/annurev.earth.28.1.141
  * Qin et al. (2010) Qin, L., Alexander, C. M. O. D., Carlson, R. W., Horan, M. F., & Yokoyama, T. 2010, Geochim. Cosmochim. Acta, 74, 1122, doi: 10.1016/j.gca.2009.11.005
  * Quarles & Lissauer (2015) Quarles, B. L., & Lissauer, J. J. 2015, Icarus, 248, 318, doi: 10.1016/j.icarus.2014.10.044
  * Raskin & Owen (2016) Raskin, C., & Owen, J. M. 2016, ApJ, 831, 26, doi: 10.3847/0004-637X/831/1/26
  * Raymond et al. (2004) Raymond, S. N., Quinn, T., & Lunine, J. I. 2004, Icarus, 168, 1, doi: 10.1016/j.icarus.2003.11.019
  * Reinhardt & Stadel (2017) Reinhardt, C., & Stadel, J. 2017, MNRAS, 467, 4252, doi: 10.1093/mnras/stx322
  * Reufer (2011) Reufer, A. 2011, PhD thesis, University of Bern
  * Reufer et al. (2012) Reufer, A., Meier, M. M. M., Benz, W., & Wieler, R. 2012, Icarus, 221, 296, doi: 10.1016/j.icarus.2012.07.021
  * Rosswog (2009) Rosswog, S. 2009, New A Rev., 53, 78, doi: 10.1016/j.newar.2009.08.007
  * Rufu et al. (2017) Rufu, R., Aharonson, O., & Perets, H. B. 2017, Nat. Geosci., 10, 89, doi: 10.1038/ngeo2866
  * Rufu & Canup (2020) Rufu, R., & Canup, R. M. 2020, Journal of Geophysical Research (Planets), 125, e06312, doi: 10.1029/2019JE006312
  * Safronov (1969) Safronov, V. S. 1969, Evolution of the Protoplanetary Cloud and Formation of the Earth and the Planets (Moscow: Nauka)
  * Salmon & Canup (2012) Salmon, J., & Canup, R. M. 2012, ApJ, 760, 83, doi: 10.1088/0004-637X/760/1/83
  * Sekine & Genda (2012) Sekine, Y., & Genda, H. 2012, Planet. Space Sci., 63, 133, doi: 10.1016/j.pss.2011.05.015
  * Shoemaker (1962) Shoemaker, E. M. 1962, in Physics and Astronomy of the Moon, ed. Z. Kopal (Academic Press, New York), 283–359
  * Solomon (1977) Solomon, S. C. 1977, Physics of the Earth and Planetary Interiors, 15, 135, doi: 10.1016/0031-9201(77)90026-7
  * Spicuzza et al. (2007) Spicuzza, M. J., Day, J. M. D., Taylor, L. A., & Valley, J. W. 2007, EPSL, 253, 254, doi: 10.1016/j.epsl.2006.10.030
  * Sridhar & Tremaine (1992) Sridhar, S., & Tremaine, S. 1992, Icarus, 95, 86, doi: 10.1016/0019-1035(92)90193-B
  * Stevenson (1987) Stevenson, D. J. 1987, AREPS, 15, 271, doi: 10.1146/annurev.ea.15.050187.001415
  * Stewart et al. (2020) Stewart, S., Davies, E., Duncan, M., et al. 2020, in American Institute of Physics Conference Series, Vol. 2272, American Institute of Physics Conference Series, 080003
  * Stewart & Leinhardt (2012) Stewart, S. T., & Leinhardt, Z. M. 2012, ApJ, 751, 32, doi: 10.1088/0004-637X/751/1/32
  * Thompson & Lauson (1972) Thompson, S. L., & Lauson, H. S. 1972, Improvements in the CHART-D Radiation-hydrodynamic code III: Revised analytic equations of state, Tech. Rep. SC-RR-71 0714, Sandia National Laboratories
  * Tian & Wisdom (2020) Tian, Z., & Wisdom, J. 2020, PNAS, 117, 15460, doi: 10.1073/pnas.2003496117
  * Tian et al. (2017) Tian, Z., Wisdom, J., & Elkins-Tanton, L. 2017, Icarus, 281, 90, doi: 10.1016/j.icarus.2016.08.030
  * Timpe et al. (2020) Timpe, M. L., Han Veiga, M., Knabenhans, M., Stadel, J., & Marelli, S. 2020, Computational Astrophysics and Cosmology, 7, 2, doi: 10.1186/s40668-020-00034-6
  * Tonks & Melosh (1993) Tonks, W. B., & Melosh, H. J. 1993, J. Geophys. Res., 98, 5319, doi: 10.1029/92JE02726
  * Touboul et al. (2007) Touboul, M., Kleine, T., Bourdon, B., Palme, H., & Wieler, R. 2007, Nature, 450, 1206, doi: 10.1038/nature06428
  * Touma & Wisdom (1998) Touma, J., & Wisdom, J. 1998, AJ, 115, 1653, doi: 10.1086/300312
  * Visser et al. (2020) Visser, R. G., Ormel, C. W., Dominik, C., & Ida, S. 2020, Icarus, 335, 113380, doi: 10.1016/j.icarus.2019.07.014
  * Wada et al. (2006) Wada, K., Kokubo, E., & Makino, J. 2006, ApJ, 638, 1180, doi: 10.1086/499032
  * Ward et al. (2020) Ward, W. R., Canup, R. M., & Rufu, R. 2020, Journal of Geophysical Research (Planets), 125, e06266, doi: 10.1029/2019JE006266
  * Warren (1985) Warren, P. H. 1985, AREPS, 13, 201, doi: 10.1146/annurev.ea.13.050185.001221
  * Wetherill (1980) Wetherill, G. W. 1980, ARA&A, 18, 77, doi: 10.1146/annurev.aa.18.090180.000453
  * Wiechert et al. (2001) Wiechert, U., Halliday, A. N., Lee, D. C., et al. 2001, Science, 294, 345, doi: 10.1126/science.1063037
  * Wisdom & Tian (2015) Wisdom, J., & Tian, Z. 2015, Icarus, 256, 138, doi: 10.1016/j.icarus.2015.02.025
  * Wissing & Hobbs (2020) Wissing, R., & Hobbs, D. 2020, A&A, 643, A40, doi: 10.1051/0004-6361/201936227
  * Wolf & Anders (1980) Wolf, R., & Anders, E. 1980, Geochim. Cosmochim. Acta, 44, 2111, doi: 10.1016/0016-7037(80)90208-2
  * Wood et al. (1970) Wood, J. A., Dickey, J. S., J., Marvin, U. B., & Powell, B. N. 1970, Science, 167, 602, doi: 10.1126/science.167.3918.602
  * Yang et al. (2007) Yang, J., Goldstein, J. I., & Scott, E. R. D. 2007, Nature, 446, 888, doi: 10.1038/nature05735
  * Youdin (2011) Youdin, A. N. 2011, ApJ, 742, 38, doi: 10.1088/0004-637X/742/1/38
  * Young et al. (2016) Young, E. D., Kohl, I. E., Warren, P. H., et al. 2016, Science, 351, 493, doi: 10.1126/science.aad0525
  * Zahnle et al. (2007) Zahnle, K., Arndt, N., Cockell, C., et al. 2007, Space Sci. Rev., 129, 35, doi: 10.1007/s11214-007-9225-z
  * Zahnle et al. (2015) Zahnle, K. J., Lupu, R., Dobrovolskis, A., & Sleep, N. H. 2015, EPSL, 427, 74, doi: 10.1016/j.epsl.2015.06.058
  * Zhang et al. (2012) Zhang, J., Dauphas, N., Davis, A. M., Leya, I., & Fedkin, A. 2012, Nat. Geosci., 5, 251, doi: 10.1038/ngeo1429