# Galaxy density profiles and shapes – I. simulation pipeline for lensing by realistic galaxy models

###### Abstract

Studies of strong gravitational lensing in current and upcoming wide and deep photometric surveys, and of stellar kinematics from (integral-field) spectroscopy at increasing redshifts, promise to provide valuable constraints on galaxy density profiles and shapes. However, both methods are affected by various selection and modelling biases, whch we aim to investigate in a consistent way. In this first paper in a series we develop a flexible but efficient pipeline to simulate lensing by realistic galaxy models. These galaxy models have separate stellar and dark matter components, each with a range of density profiles and shapes representative of early-type, central galaxies without significant contributions from other nearby galaxies. We use Fourier methods to calculate the lensing properties of galaxies with arbitrary surface density distributions, and Monte Carlo methods to compute lensing statistics such as point-source lensing cross-sections. Incorporating a variety of magnification bias modes lets us examine different survey limitations in image resolution and flux. We rigorously test the numerical methods for systematic errors and sensitivity to basic assumptions. We also determine the minimum number of viewing angles that must be sampled in order to recover accurate orientation-averaged lensing quantities. We find that for a range of non-isothermal stellar and dark matter density profiles typical of elliptical galaxies, the combined density profile and corresponding lensing properties are surprisingly close to isothermal around the Einstein radius. The converse implication is that constraints from strong lensing and/or stellar kinematics, which are indeed consistent with isothermal models near the Einstein radius, cannot trivially be extrapolated to smaller and larger radii.

###### keywords:

gravitational lensing – stellar dynamics – galaxies: photometry – galaxies: kinematics and dynamics – galaxies: structure – methods: numerical## 1 Motivation

### 1.1 Learning from galaxy density profiles and shapes

Observational constraints on galaxy density profiles and shapes can be used to study a great variety of problems, from basic cosmology to the connections between dark matter and baryons. For example, the spherically-averaged form of the dark matter halo density profile, and the distribution of halo shapes, both depend on cosmological parameters to the extent that those parameters affect the process of structure formation (e.g., Bullock et al., 2001; Allgood et al., 2006), and on the physics of galaxy formation to the extent that baryons modify the dark matter distribution (e.g., Blumenthal et al., 1986; Gnedin et al., 2004; Kazantzidis et al., 2004; Naab et al., 2007; Rudd et al., 2008). While there is still some disagreement in the literature about the theoretical predictions, even among those papers cited above, one key result is that there is considerable scatter in both the density profiles and shapes of dark matter halos, which presumably reflects different formation histories (e.g., Wechsler et al., 2002).

Several observational tools have been used to constrain the density profiles and shapes of galaxies, at different length and mass scales, and redshifts. The main tools we consider are gravitational lensing and stellar kinematics. X-ray data are also useful for understanding the density profiles and shapes of massive galaxies and galaxy clusters, but are beyond the scope of our investigation.

Gravitational lensing is the deflection of light from distant sources by the gravitational fields of intervening lens galaxies. Strong lensing occurs in the central regions of galaxies (and clusters) where the light bending is extreme enough to produce multiple images of a background source. It can be used to study the mass distributions of individual galaxies on projected scales of typically several kpc (for a review, see Kochanek, 2006). Weak lensing is a complementary phenomenon in which the deflection of light slightly distorts the shapes of background sources without creating multiple images (for a review, see Bartelmann & Schneider, 2001). Weak lensing probes galaxy mass distributions on scales from tens of kpc out to about 10 Mpc, but only yields constraints on ensemble averages, since currently weak lensing by galaxies can only be detected by stacking many lens galaxies. While galaxy clusters can by studied on an individual basis using weak lensing, our focus is on galaxies.

High-quality (two-dimensional) data on the kinematics of stars as well as gas in the inner parts (a few to ten kpc) of galaxies are now readily available, thanks in particular to strong progress in integral-field spectroscopy in the last decade. Kinematics in the outer parts of late-type galaxies can often be observed from the presence of neutral hydrogen (e.g., Bosma, 1981; van Albada et al., 1985; Persic et al., 1996; Noordermeer et al., 2007). In the outer parts of early-type galaxies, however, cold gas is scarce (but see e.g. Franx et al., 1994; Morganti et al., 1997; Weijmans et al., 2008), so we are left with discrete kinematic tracers such as planetary nebulae and globular cluster (e.g., Côté et al., 2003; Douglas & et al., 2007) out to tens of kpc.

Current and upcoming wide and deep photometric surveys will reveal a vast number of strong lensing events (e.g., Fassnacht et al., 2004; Koopmans et al., 2004; Kuhlen et al., 2004; Marshall et al., 2005), and will also allow for extensive, complementary weak lensing analysis (e.g. Tyson, 2002; Kaiser, 2004). At the same time, there is a rapid increase in the availability of two-dimensional kinematics of galaxies nearby (e.g., Emsellem & et al., 2004; McDermid & et al., 2006), and even at high(er) redshift (e.g., van der Marel & van Dokkum, 2007; Bouché & et al., 2007). These data sets, individually as well as combined, may provide strong constraints on galaxy density profiles and shapes, but the analyses involved are subject to selection and modelling biases.

### 1.2 Selection and modelling biases

If we want to use strong lensing and kinematics to make robust tests of cosmological predictions, we need to answer two questions: Can strong lensing and kinematic analyses yield accurate constraints on galaxy density profiles? Are the galaxies in which we can make the measurements (especially in the case of strong lensing) representative of all galaxies? We refer to these two concerns as modelling biases and selection biases, respectively.

The question of selection bias is particularly important given the diversity in the galaxy population. It is well known that strong lensing favours early-type over late-type galaxies, and massive galaxies over dwarfs (e.g., Turner et al., 1984; Fukugita & Turner, 1991). But even within the population of massive early-type galaxies, to what extent does strong lensing favour galaxies whose dark matter halos have inner slopes that are steeper than average, or concentrations that are higher than average? Also, to what extent does strong lensing favour galaxies with particular shapes and/or orientations with respect to the line-of-sight? To phrase these questions formally, consider some parameter describing the galaxy density profile or shape (e.g., the inner slope of the dark matter density profile). We need to understand how the distribution among strong lens galaxies compares to the underlying distribution for all galaxies. Any analysis that includes strong lensing with some other technique used to study the same systems will suffer from strong lensing-related selection biases, since we never get to choose which systems will be a strong lens. Selection biases are critical when we want to interpret constraints on lens galaxy density profiles and/or shapes from strong lensing, possibly in combination with kinematics (e.g. Rusin & Kochanek, 2005; Koopmans et al., 2006) in comparison with predictions from cosmological models.

On the other hand, modelling biases may occur because of (often
unavoidable) assumptions made when analysing gravitational lensing
and/or kinematic data because of the finite number of constraints
available from the data. Since galaxies are in general non-spherical,
we can only correctly interpret the observations if we know the
viewing direction, and even then the deprojection might not be unique
(e.g. Rybicki, 1987). Moreover, strong lensing is
subject to the so-called mass-sheet degeneracy: part of the deflection
and magnification of the light from the background source can be due
to mass along the line-of-sight that is not associated with the lens
galaxy itself^{1}^{1}1Part of this mass-sheet degeneracy may be
overcome in future surveys through the measurements of time delays
for an adopted Hubble constant.. In addition, the constraints from
strong lensing on the galaxy density are typically limited in radius
to around the Einstein radius, and even then, without secure,
unaffected measurements of the flux of the lensing images, the density
profile is difficult to recover. Kinematics can be obtained over a
larger radial extent, but in particular stellar kinematics suffer from
the so-called mass-anisotropy degeneracy: a change in the measured
line-of-sight velocity dispersion can be due to a change in total
mass, but may also be the result of velocity anisotropy.

In a series of papers, we are developing a pipeline for using realistic galaxy models to simulate strong lensing and kinematic data, and assess how both selection and modelling biases affect typical strong lensing and kinematic analyses. In this first paper, we present the simulation pipeline for point-source lensing. We focus on strong lensing of quasars by early-type, central galaxies at different mass scales, using two-component mass profiles (dark matter plus stellar component) that are consistent with existing photometry and stacked weak lensing data from SDSS (Kauffmann & et al., 2003; Mandelbaum et al., 2006b), -body, and hydrodynamic simulations. In Mandelbaum et al. (2008, hereafter Paper II), we use this pipeline to study selection biases in strong lensing surveys. In future work we will address modelling biases in strong lensing and kinematics (both when studied separately and when combined). Our overall goal is to determine how strong lensing, kinematics, and possibly other probes (weak lensing, and X-ray data for cluster mass scales) can be used to obtain robust, unbiased constraints on galaxy density profiles and shapes.

To make our presentation coherent, and to clarify our notation and terminology, we first review the analytic treatment of ellipsoidal mass distributions with different profiles and shapes (Section 2), and the basic theory of strong lensing (Section 3). We then describe our simulation pipeline for point-source lensing in depth. In Section 4 we present our choices for the masses, profiles, and shapes of the galaxy models. In Section 5 we discuss the numerical methods we use for lensing calculations, including numerous tests. A summary of the simulation pipeline, and some implications for strong lensing analyses are given in Section 6.

## 2 Galaxy density profiles and shapes

In this section we describe our treatment of galaxy density profiles and shapes. While our approach is fairly conventional, it is important to present it carefully to clarify our notation and terminology, collect useful technical results, and provide a firm foundation for the work in this series of papers. This section focuses on the formal framework; the specific galaxy models used in our simulations are discussed in Section 4.

### 2.1 Notation

We compute distances using a flat CDM cosmology with and . We use , , and to denote intrinsic, three-dimensional (3d) coordinates, and and for the projected, two-dimensional (2d) coordinates. Similarly, is the intrinsic radius () and is the projected radius (). For non-spherical galaxies, we use , , and for the major, intermediate, and minor semi-axis lengths of the intrinsic, 3d density profiles; and we use and for the major and minor semi-axis lengths of the projected, 2d surface densities. Subscripts “dm” and “” are used to indicate whether a quantity describes the dark matter or stellar component of the galaxy model. Any gas component that has not cooled to form stars — which is expected to be subdominant relative to the stellar component for early type galaxies (e.g. Read & Trentham, 2005; Morganti & et al., 2006) — is implicitly included in the component that we label dark matter.

### 2.2 Ellipsoidal shapes

We consider mass density distributions that are constant on ellipsoids

(1) |

with . The major semi-axis length is a scale parameter, whereas the intermediate-over-major () and minor-over-major () axis ratios determine the shape. In the oblate or prolate axisymmetric limit we have (pancake-shaped) or (cigar-shaped), respectively, while in the spherical limit (so then ). Note that is a dimensionless ellipsoidal radius.

Under the thin-lens approximation, the gravitational lensing
properties are fully characterised by the mass density projected along
the line-of-sight. We introduce a new Cartesian coordinate system
, with and in the plane of the sky and the
-axis along the line-of-sight. Choosing the -axis in the
-plane of the intrinsic coordinate system (cf. de Zeeuw & Franx 1989 and their Fig. 2), the transformation
between both coordinate systems is known once two viewing angles, the
polar angle and azimuthal angle , are specified.
The intrinsic -axis projects onto the -axis; for an
axisymmetric galaxy model the -axis aligns with the short axis
of the projected mass density,^{2}^{2}2For an oblate galaxy, the
alignment () follows directly from
equation (2) since . For a prolate
galaxy it is most easily seen by exchanging and (),
so that the -axis is again the symmetry axis (instead of the
-axis).
but for a triaxial galaxy model the -axis is misaligned by an
angle such that (cf. equation B9 of
Franx 1988)

(2) |

where is the triaxiality parameter defined as . A rotation through transforms the coordinate system to such that the and axes are aligned with the major and minor axes of the projected mass density (respectively), while is along the line-of-sight (see also Section 4.4.3 below).

Projecting along the line-of-sight yields a surface mass density that is constant on ellipses in the sky-plane,

(3) |

where we have used , and . The sky-plane ellipse is given by

(4) |

The projected major and minor semi-axis lengths, and , depend on the intrinsic semi-axis lengths , , and and the viewing angles and as follows:

(5) |

where and are defined as

(6) | |||||

(7) | |||||

It follows that is proportional to the area of the ellipse.

The flattening of the projected ellipses actually depends on the viewing angles and the intrinsic axis ratios , and is independent of the scale length. Since the intrinsic and projected semi-major axis lengths are directly related via the left equation of (5), the scale length can be set by choosing either or .

Finally, we define the negative logarithmic slope of the mass density and of the surface mass density . Substituting equation (3), the latter follows as

(8) |

which in general has to be evaluated numerically.

### 2.3 Choice of density profiles

We consider two families of density profiles, motivated by both observations and simulations of galaxies. Historically, there has been considerable interest in cusped density profiles that have a shallower power-law at small radii and a steeper power-law at large radii, with a smooth transition in between. This family includes the Hernquist (1990) profile that is commonly used to model the stellar components of early-type galaxies and spiral galaxy bulges, along with (generalised) NFW (Navarro et al., 1997) profiles often used to describe the dark matter profiles of simulated galaxies.

An alternative, observationally motivated family of models is obtained by deprojecting the Sérsic (1968) profile that is known to give a good fit to the surface brightness profiles of early-type galaxies and spiral galaxy bulges. There have been claims that deprojected Sérsic profiles actually provide a better fit to simulated dark matter halos than (generalised) NFW profiles (e.g., Navarro & et al., 2004; Merritt et al., 2005). Several recent studies argue that a Sérsic profile that is not deprojected —also referred to as an Einasto (1965) profile— provides an even better fit (Merritt et al., 2006; Gao, L. and Navarro), although when compared against a generalised NFW or Sérsic profile the improvement is marginal at best.

X-ray (e.g., Allen et al., 2001, 2002; Vikhlinin et al., 2005, 2006) and weak lensing (e.g., Kneib et al., 2003; Limousin et al., 2007) observations of individual galaxy clusters suggest that outside of kpc, the density profiles are consistent with the NFW profiles seen in N-body simulations (no attempts were made to compare with the Sérsic model, however). The mass range for these observations is about a factor two above our higher mass model.

Since there is still no consensus opinion about which models are best, we consider how the choice of density profile affects our conclusions by using both cusped density profiles and deprojected Sérsic density profiles

### 2.4 Cusped density profiles

The cusped density distribution with inner slope and outer slope ,

(9) |

includes the following well-known spherical () density profiles: , the Hernquist profile (Hernquist, 1990); , the Jaffe profile (Jaffe, 1983); and , the NFW profile (Navarro et al., 1997).

The mass enclosed within the ellipsoidal radius (for inner slopes ) is

(10) | |||||

where is the hypergeometric function. For outer slopes , equation (10) reduces to . When , the latter incomplete beta function becomes the complete beta function , and we obtain a finite total mass.

Substituting into equation 10 yields , so that for , we find the total mass of the Hernquist profile, which we adopt for the stellar component when using cusped density profiles. For outer slopes , the total mass is infinite, but for certain half-integer values of and the expression for the enclosed mass simplifies significantly. For example, for , the value which we adopt for the outer slope of the dark matter profile, we find

(11) |

for inner slope values of , respectively.

Although for certain values of and , lengthy analytic expressions for the surface mass density can be derived, we evaluate the integral in equation (3) numerically. The logarithmic slope of the cusped density profile is

(12) |

while of the surface mass density follows from numerical evaluation of equation (8).

### 2.5 Deprojected Sérsic density profiles

It has long been known that the surface brightness profiles of early-type galaxies and of spiral galaxy bulges are well fit by a Sérsic (1968) profile , with the effective radius enclosing half of the total light. A key conceptual difference from the cusped models is that the profile does not converge to a particular inner slope on small scales. The deprojection of the Sérsic profile has to be done numerically. However, the analytic density profile of Prugniel & Simien (1997),

(13) |

provides a good match to the deprojected Sérsic profile when the inner negative slope is given by

(14) |

The enclosed mass for the Prugniel-Simien model is

(15) |

where is the incomplete gamma function, which in the case of the total mass reduces to the complete gamma function .

The expression for the surface mass density is, to high accuracy, the Sérsic profile

(16) |

Given the enclosed projected mass

(17) |

the requirement that the total intrinsic and projected mass have to be equal yields a normalisation

(18) |

The value of depends on the index and the choice for the scale length. The latter is commonly chosen to be the effective radius in the surface brightness profile, which contains half of the total light. We adopt a similar convention requiring that the ellipse contains half of the projected mass. This choice results in the relation , which to high precision can be approximated by (Ciotti & Bertin, 1999)

(19) |

The logarithmic slope of the Prugniel-Simien and Sérsic profile are related by

(20) |

where the logarithmic slope of the surface mass density is

(21) |

## 3 Strong lensing

In this section, we review the strong lensing concepts that are most important for our work. See Schneider et al. (1992) and Kochanek (2006) for more discussion of strong lensing theory. The specific lensing calculations used in our simulations are discussed in Section 5.

### 3.1 Basic theory

The gravitational lensing properties of a galaxy with surface mass density are characterised by the lens potential (in units of length squared) that satisfies the two-dimensional Poisson equation

(22) |

Here is the surface mass density scaled by the critical density for lensing (, see Section 3.2), and is referred to as the “convergence.” The positions and magnifications of lensed images depend on the first and second derivatives of the lens potential, respectively. The image positions are the solutions of the lens equation,

(23) |

where is the deflection angle (in
units of length), while and
are two-dimensional angular
positions on the sky in the lens and source planes, respectively. The
angular coordinates are related to the projected physical coordinates
by and , where is the
angular diameter distance to the lens; and and
, where is the angular diameter distance to
the source.^{3}^{3}3Note that because of this trivial conversion
between angular and physical units, we do not explicitly distinguish
between angles and lengths in the lens plane. For example in the
text we refer to the deflection angle in physical units (e.g., by
stating that ) whereas our plots may show
in angular units. The magnification of an image at
position is given by

(24) |

A typical lens galaxy has two “critical curves” along which the magnification is formally infinite, which map to “caustics” in the source plane. The caustics bound regions with different numbers of images (see Section 5.7 for examples).

If the lens is circularly symmetric, the deflection is radial and has amplitude

(25) |

The outer or “tangential” critical curve corresponds to the ring image that would be produced by a source at the origin. The radius of this ring is the Einstein radius , which is given mathematically by the solution of the equation . This definition is equivalent to saying that the Einstein radius bounds the region within which the average convergence is unity, such that the enclosed mass . By contrast, the inner or “radial” critical curve has radius given by the solution of .

If the lens is non-circular, the conditions for the critical curves
and caustics are more complicated, but we can still retain some
concepts from the circular case. In particular, we can say that the
outer/tangential critical curve is principally determined by the
enclosed mass, whereas the inner/radial critical curve is determined
by the slope of the deflection curve. When we map the critical curves
in the lens plane to the caustics in the source plane, the
arrangement of curves is inverted: the outer critical curve
corresponds to the inner caustic; while the inner critical curve
corresponds to the outer caustic.^{4}^{4}4If the lens is highly
flattened, the tangential caustic can actually pierce the radial
caustic; some specific examples are discussed below (e.g.,
Section 5.7). However, this situation is unusual, and it
does not substantially alter the present discussion. Since the
outer caustic bounds the multiply-imaged region, it determines the
total lensing cross-section. This means the cross-section is sensitive
to the surface mass density (in particular its logarithmic slope) at
small radii in the lens plane, which will be important to bear in
mind when interpreting our results.

While a non-circular lens has non-circular critical curves, it is still occasionally useful to characterise the scale for strong lensing with a single Einstein radius. We generalise the definition of the Einstein radius to a non-circular lens by using the monopole deflection, or the deflection angle produced by the monopole moment of the lens galaxy,

(26) |

where is the average convergence within radius , and are polar coordinates on the sky. The Einstein radius is then given by , or equivalently . This is the definition that emerges naturally from models of non-spherical lenses (e.g., Cohn et al., 2001), and it matches the conventional definition in the circular case.

It is instructive to think of lensing in terms of Fermat’s principle and say that lensed images form at stationary points of the arrival time surface (e.g., Blandford & Narayan, 1986). Since the arrival time is a 2d function of angles on the sky, there are three types of stationary points: minima, maxima, and saddle points. We can classify them using the eigenvalues of the inverse magnification tensor,

(27) |

Here, is the shear^{5}^{5}5In general the shear has two
components conveniently expressed in complex notation
(e.g. Schneider et al., 1992), but as usual in strong lensing
analyses we only use the amplitude (or norm) for which is
the standard symbol. The context should make clear whether we are using
to refer to a lensing shear or to
the negative logarithmic slope of the density profile., which
quantifies how much a resolved image is distorted. The eigenvalues are
both positive for a minimum, both
negative for a maximum, and mixed for a saddle point. Since our
galaxies have central surface densities shallower than , the lensing “odd image theorem” applies: the total number
of images must be odd, and the number of minimum, maximum, and
saddle point images must satisfy the following relation
(Burke, 1981; Schneider et al., 1992):

(28) |

In practice, images at maxima are rarely observed because they form near the centres of lens galaxies and are highly demagnified; the only secure observation of a maximum image in a galaxy-scale lens required a deep and dedicated search (Winn et al., 2004). We do compute maxima because they are useful in checking our numerical methods (see Section 5.2), but we focus our analysis on minimum and saddle point images because they constitute the bulk of lensing astrophysics.

### 3.2 Redshift dependence

The redshifts of the lens ( and source () play an important role in determining what region of a galaxy is relevant for strong lensing. Roughly speaking, strong lensing occurs in the region where the surface mass density exceeds the critical density for lensing,

(29) |

where the values are the angular diameter distances to the lens (), to the source (), and from the lens to the source (). Note that can be computed for a flat universe using the relation

(30) |

and for more general cosmologies, see Hogg (1999). For a fixed source redshift, is lowest when the lens is roughly halfway between the observer and source, and it increases as the lens moves toward the observer or source. For a fixed lens redshift, the critical density is formally infinite for , and it decreases monotonically as the source redshift increases past .

We adopt fiducial redshifts of for the lens and for the source. The resulting critical density, , is such that quasar images generally appear at about one effective radius from the centre of the lens galaxies. At this radius, the stellar component and dark matter halo may both play significant roles in the lensing signal, which is important to keep in mind when interpreting our results in this and subsequent papers.

Our results apply equally well to any other combination of lens and source redshifts that yield . The reason is that if we fix the galaxy density profile and vary and so as to keep fixed, the strong lensing region will always have the same physical scale and hence the same relation to the density profile. If we instead vary and in a way that increases , that will tend to push the lensed images to smaller radii and hence make the stellar component somewhat more dominant. Conversely, varying the redshifts in a way that lowers will tend to make the dark matter component more significant. We examine the effect of varying and on our results explicitly in subsequent papers.

We note that the lens redshift also affects strong lensing in the conversion from physical to angular scales. Even if the physical size of the lensing region stays fixed as we vary the source and lens redshifts, the angular separation between the lensed images will vary with . This may lead to observational selection effects: for example, resolution-limited surveys may miss lenses for which is high and the images cannot be resolved, while spectroscopic surveys may miss lenses for which is low and the images fall outside the spectroscopic slit or fiber. However, our focus in this work is on physical selection biases in strong lensing, which means that we consider the lens and source redshifts only in terms of how they affect .

Our choice for the source redshift is typical of quasar
lenses in the CASTLES^{6}^{6}6CASTLES (see
http://cfa-www.harvard.edu/castles/) is a collection of uniform HST
observations of mostly point-source lenses from several samples with
differing selection criteria, rather than a single,
uniformly-selected survey. sample, but our choice
places the lens galaxy on average at a smaller distance. The latter
is, however, closer to the typical lens redshift in the SLACS sample
of galaxy/galaxy strong lenses (Bolton et al., 2006, 2008a), but those lenses are explicitly selected to
have source redshifts (Bolton et al., 2004).
As a result, our fiducial value of is typically between
that of CASTLES and SLACS lenses. The Einstein radii at in our case appear at larger radii in the CASTLES sample
(; Rusin & et al., 2003), and at smaller radii
in the SLACS sample (; Koopmans et al., 2006). The statistics of the SLACS survey are
much more involved than the statistics of quasar lens surveys, because
SLACS lenses have extended rather than point-like sources, and the
survey relies on fiber spectra from SDSS. SLACS lens statistics have
recently been examined by Dobler et al. (2008), and we do not
consider them explicitly in the current work.

## 4 Construction of Galaxy Models

In this section we construct the galaxy models used in our simulations. We seek models with a variety of density profiles and shapes that not only are realistic but also sample the range of systematic effects in strong lensing. To that end, we must carefully consider the mass and length scales (Sections 4.1–4.2), the profiles (Section 4.3), and the shapes (Section 4.4) of both the stellar and dark matter components. In addition to a main set of simulations with cusped density profiles, we also create a subset of simulations based on deprojected Sérsic density profiles (Section 4.5). After discussing the choice of the model parameters, we briefly describe the part of our pipeline that computes surface mass density maps for these models (Section 4.6), which can then be used for lensing calculations (Section 5).

### 4.1 Mass scales

We use stellar and dark matter halo masses from SDSS for early-type galaxies, where analyses of the spectra give an estimate of the stellar mass (Kauffmann & et al., 2003), and weak lensing analyses from – Mpc yield the dark matter mass (Mandelbaum et al., 2006b). The former are subject to systematic uncertainties due to the initial mass function (IMF) at the – per cent level, whereas the latter is subject to per cent uncertainty in the modelling in addition to – per cent statistical error.

The selection of mass scales begins with the ansatz that we would like to approximately bracket the typical luminosity and mass range of observed lensing systems. Given the range of lens galaxy luminosities in the CASTLES and SLACS strong lens samples, we choose models with total luminosities of and in the (SDSS) -band. We refer to the lower-luminosity model interchangeably as “galaxy scale” or “lower mass scale” model. The higher-luminosity model is essentially a massive galaxy with convergence that is significantly boosted by a group dark matter halo, but we often refer to it as a “group scale” or “higher mass scale” model for brevity. In both cases, the model includes concentric stellar and dark matter components; thus, our current results cannot be applied to satellites in groups/clusters, which will be the subject of future work. We also neglect contributions of nearby (satellite) galaxies to the surface density, focusing only on the host galaxy, and we neglect contributions from other structures along the line of sight (e.g. Keeton & Zabludoff, 2004; Momcheva et al., 2006; Williams et al., 2006). The masses are selected from the Mandelbaum et al. (2006b) SDSS weak lensing analysis using results for early-type galaxies as a function of luminosity, as shown in Table 3 or Fig. 4 of that paper.

The model parameters for these mass scales are summarised in Table 1. We define the virial radius entirely using comoving quantities, with satisfying the relation

(31) |

where is the virial mass and is the mean density of the universe (using ). The definition is in principle arbitrary, but this choice was used to analyse the weak lensing results. In the weak lensing results, the profiles were found to be consistent with NFW for early-type galaxies on all scales studied (see also Mandelbaum et al. 2006a, which focuses on Luminous Red Galaxies). Thus, not only our use of the masses, but also the form of the density profiles, is observationally motivated outside of kpc.

Description | Symbol | Galaxy-scale | Group-scale | Unit |

Dark matter component | ||||

Virial mass | M | |||

M | ||||

Virial radius | comoving kpc | |||

physical kpc | ||||

arcsec | ||||

Concentration | () | – | ||

Scale radius | () | physical kpc | ||

() | arcsec | |||

Stellar component | ||||

Total mass | M | |||

Half-mass radius | kpc | |||

(projected) | arcsec | |||

Scale radius | () | physical kpc | ||

() | arcsec |

### 4.2 Length scales

We choose the dark matter halo concentration from the Bullock et al. (2001) result that , after converting to our mass definition and using inner slope , , and . The nonlinear mass is the mass in a sphere within which the rms linear density fluctuation is equal to , the overdensity threshold for spherical collapse. We obtain for the lower mass scale and for the higher mass scale (see also Table 1). The scale radius of the dark matter component then follows as .

We determine the scale length of the stellar components based on SDSS photometry of galaxies of the appropriate luminosity. Fitting de Vaucouleurs profiles, i.e., Sérsic profiles with index , to the growth curve of these elliptical galaxies yields the half-light (or effective) radius . Assuming a constant stellar mass-to-light ratio, is equal to the projected stellar half-mass radius as given in Table 1. For a Hernquist cusped density profile with as we adopt for the stellar component, the scale radius follows as .

Given the large scatter in dark matter halo concentration values in simulations (0.15 dex; Bullock et al., 2001), we will also redo a subset of the analysis using concentrations lower and higher than the fiducial values used for the main analysis. This work will allow us to quantify biases associated with halo concentration. Realistically, we expect the changes in the concentration to be somewhat degenerate with changes in the inner slope , as both change the amount of dark matter in the inner parts, although there may be some differences because changing affects the inner logarithmic slope while changing does not. In Paper II we quantify the approximate degeneracy between changes in and for the dark matter component, while keeping the stellar component fixed.

### 4.3 Density profiles

For our main set of simulations, we use Hernquist and generalised NFW profiles for stellar and dark matter components, respectively (cf. Sections 2.3 and 2.4). While there is some uncertainty over the true dark matter profile in -body simulations (particularly in the inner regions; see Section 2.5), and the initial dark matter profile may in any case be modified due to the presence of baryons, we use the generalised NFW model because it allows freedom in the form of the cusp. As discussed in more detail below (Section 4.5), we also examine how our results depend on the form of the density profiles by running a set of simulations that use deprojected Sérsic profiles for the intrinsic (3d) densities of both the stellar and dark matter components. Here we focus on the main set of simulations with cusped profiles.

#### 4.3.1 Inner and outer slope

For our main set of simulations with cusped density profiles, we fix the stellar component to have a Hernquist profile, with inner slope and outer slope . Based on cosmological -body simulations, we fix the outer slope of the dark matter simulation to (e.g., Navarro & et al., 2004).

The inner slope of the dark matter profile is more problematic. While -body simulations can suggest forms for dark matter halo profiles (e.g., Navarro & et al., 2004), there is some question whether those profiles are affected by the presence of a stellar component, and if so, in what way. One formalism for analysing how a stellar component affects a dark matter profile is called adiabatic contraction (AC: Young, 1980; Blumenthal et al., 1986; Gnedin et al., 2004; Sellwood & McGaugh, 2005). The physical picture is that as the baryons cool and condense into the centre of the system, they draw some of the dark matter in as well, causing the dark matter halo profile to steepen. Persuasive observational evidence for or against AC in the galaxies we are modelling does not yet exist, though there are claims that the theoretical assumptions behind AC are not valid for these galaxies. Naab et al. (2007) suggest that the opposing effect, growth through dissipationless infall of matter onto the galaxy (which tends to push the dark matter outwards from the centre) may cancel out the steepening due to baryonic cooling. We expect the effects of infalling matter to be even more significant for our higher mass model, which is equivalent to a fairly massive group. Furthermore, hydrodynamic simulations that demonstrate that adiabatic contraction dominates may suffer from over-cooling of the baryonic component, so we cannot use them to determine the extent of the effect either. Thus, we cannot be certain how much the dark matter profiles may be modified due to the different processes involved in galaxy formation.

We consider the range of possibilities in two ways. First, for the main set of simulations with cusped density profiles, we examine three values of the dark matter inner slope: . As we shall see, this allows for a broad range of lensing properties. Second, for a subset of simulations with NFW dark matter profiles, we explicitly include the effects of AC, using the formalisms in Blumenthal et al. (1986) and Gnedin et al. (2004).

#### 4.3.2 Normalisation

When the inner slope of the dark matter is changed, the virial mass also changes. To preserve we must change either the density amplitude or the scale radius of the dark matter (or both at the same time). Preserving in addition the (reduced) shear at the virial radius , would break this “degeneracy” in the normalisation. However, for individual strong lens galaxies, weak lensing and other constraints on the density are very weak if present at all. Nevertheless, the different choices for the normalisation might not result in significantly different lensing cross sections.

To investigate this, we compute the change in total lensing cross-section with respect to the fiducial case with when varying and/or to preserve . The results are summarised in Fig. 1 for both the lower (top) and higher (bottom) mass scale. The curves in the left panels show the relative difference in unbiased cross-sections, while the curves in the right panels are for biased cross-sections with a limiting luminosity of (see Section 5.4 below). The solid curves show the relative difference in cross-section when both and are kept fixed when changing the inner slope away from the fiducial value , i.e., not preserving the virial mass . The dotted and dashed curve show the difference when preserving by varying either (fixing ) or (fixing ), respectively.

The results for the unbiased and biased cross-section are similar. At , the offsets between the solid, dotted and dashed curves corresponding to the three different normalisations are marginal, which is expected since for a shallower slope the contribution of the dark matter component decreases with respect to the stellar component which we keep fixed. However, the dark matter contribution becomes important when , in particular for the high-mass scale as can be seen from the different scaling of the vertical axes. We see that in all cases the solid curve leads to higher cross-sections than the dotted and dashed curve when we preserve , which is expected given the increase in mass of tens of per cent.

The latter two normalisations result in changes in the cross-section of which the relative difference is well within the combined statistical and systematic uncertainty in the cross-sections (i.e., while we have attempted to make these models fairly realistic, there are some systematic uncertainties, so we do not trust them to per cent level precision). This implies that varying and/or to preserve does not change the lensing properties in a way that is significantly different. In what follows we choose to fix (and hence the concentration ), because it is simpler and in line with preserving the (weak) concentration-mass relation for dark matter halos (e.g. Bullock et al., 2001) and the size-luminosity relation for early-type galaxies (e.g. Shen et al., 2003; Bernardi et al., 2007), as discussed further in Section 4.5.2 below.

#### 4.3.3 Intrinsic profiles

For the spherical density profiles, we present a set of plots showing various quantities for both mass scales. We begin with Fig. 2, which shows the intrinsic, 3d mass density profile for the cusped density profiles (NFW plus Hernquist), for both mass scales. It is apparent that for and density profiles (stellar plus dark matter), the dark matter component is negligible for scales below kpc for both mass scales, whereas the models have significant contributions from dark matter for all scales shown. The difference between these cases is somewhat reduced when considering the profiles in projection along the line-of-sight, due to the dominant dark matter contributions at large intrinsic radius. Nonetheless, we expect a significant break in the strong lensing properties going from to .

The colour curves in Fig. 2 show the effects of AC on the cusped, density profiles. We can see (bottom-left panel) that for the lower mass scale, which has a higher ratio of stellar to halo mass, AC tends to increase the density on scales below kpc by per cent depending on the model used, at the expense of a slightly decreased density ( per cent) for larger scales. For the higher mass scale (bottom-right panel), the density below 10 kpc tends to increase by per cent for the Gnedin et al. (2004) AC prescription, again at the expense of a percent-level decrease on much larger scales.

Fig. 3 shows the negative logarithmic slope of the intrinsic mass density for both mass scales. We can see that for the individual components of the models, the logarithmic slopes behave smoothly according to equation (12), but for the full model, there can be complex, non-monotonic behaviour. The latter “wiggle” in Fig. 3 is stronger for the cusped than the Sérsic density profiles, reflecting the break in the cusped density profile while the Sérsic density profile turns over smoothly.

The intrinsic profiles of our galaxy models are consistent with the total mass density profile derived by Gavazzi et al. (2007), based on a joint strong-lensing and weak-lensing analysis of 22 early-type galaxies from the SLACS sample. The inferred intrinsic density profile (their Fig. 8b) has an amplitude which falls in between our lower-mass galaxy and higher-mass group scale (for an adopted Hubble constant ). Moreover, the corresponding slope matches very well the wiggle in the logarithmic slope of our models in Fig. 3. Since this non-monotonic variation is around a value of (indicated by the dashed horizontal line in Fig. 3), our models as well as the total mass density inferred by Gavazzi et al. (2007) are rather close to isothermal over a large radial range around the Einstein radius. However, the density overall is not consistent with isothermal, with clear deviations at small and large radii, which happen well within the two-decade radial range for which Gavazzi et al. (2007) claim consistency with isothermal. We discuss this “isothermal conspiracy” further when studying the profiles in projection next.

#### 4.3.4 Projected profiles

In addition to the plots of the intrinsic, 3d profiles, we also show projected, 2d quantities, which are observationally more relevant. Fig. 4 shows the surface mass density as a function of the projected radius , in physical units (kpc) and at the top in angular units (arcsec) for a lens galaxy at the fiducial redshift of . The total (dark matter plus stars) for and are nearly identical and star-dominated for kpc. As anticipated, this is a smaller transverse scale than that for which they are identical in 3d.

The colour curves in Fig. 4 show the effect of AC on the projected cusped, density profiles. As expected, the projection effects have brought in larger scales where AC is less significant, so the projected profiles with AC tend to be higher by (lower mass scale) and (higher mass scale) per cent on the scales of interest (a few kpc for the lower mass scale, kpc for the higher mass scale).

Fig. 5 shows the negative logarithmic slope of the surface mass density, . If the density profile were a pure power-law, the 2d and 3d negative logarithmic slopes would be related by . Comparing Fig. 5 to Fig. 3, we see that this simple relation does hold at small and large radii for our cusped models, since these models are asymptotically power-laws. However, it is not valid at intermediate radii, because our models — especially the full models containing both stellar and dark matter components — are far from being simple power-laws. This is particularly notable in the vicinity of the Einstein radius, where the 3d slope varies much stronger with radius than the 2d slope.

Fig. 6 shows the projected dark matter fraction within projected radius . As shown, by varying the dark matter inner slope in the cusped models, we are able to explore models with widely varying fractions of dark matter. The dark matter fractions in our models are consistent with the range of values inferred from attempts to compare stellar and dark matter masses in observed lens systems (Rusin et al., 2003; Rusin & Kochanek, 2005; Ferreras et al., 2005; Bolton et al., 2008b).

Finally, Fig. 7 shows the deflection angle as function of the projected radius . The Einstein radii follow from the solution of , which corresponds to the intersections of the dashed one-to-one line with the thick curves, indicated by the filled circles. The corresponding Einstein radii appear at about the half-mass radius , which in case of a constant stellar mass-to-light ratio corresponds to the effective radius .

Figs. 5 and 7 contain an important point related to attempts to combine strong lensing and stellar dynamics to measure the density profile in the vicinity of the Einstein radius (e.g., Treu & Koopmans, 2004; Koopmans et al., 2006). The slope of the composite surface mass density for models with of , and are all close to unity near the Einstein radius. For the lower-mass scale models (top-left panel) there is a little more spread around a somewhat higher value than for the higher-mass scale (top-right panel) with . However, all are close to the slope of a projected isothermal density as indicated by the horizontal dashed line in Fig. 5. In Fig. 7, the deflection angle for the higher-mass scale (top-right panel) increases slowly with radius around the Eintein radius, but for the lower-mass scale the curves are nearly flat around (and beyond) , similar to a constant deflection angle in case of an isothermal density. Even though the stellar density and dark matter density with different inner slopes are clearly non-isothermal, the combined density has a projected slope and corresponding deflection angle consistent with isothermal around the Einstein radius. This finding implies that it will be challenging to use strong lensing to constrain the dark matter inner slope for a two-component model without bringing in information from significantly different scales.

We note that the apparent isothermality of our composite models is a completely unintended consequence of the mass models we have chosen, not something we have imposed by design. However, it provides support for the realism of our models, when compared against observations. There is much evidence from strong lensing (especially in combination with kinematics) that early-type galaxies have total density profiles that are approximately isothermal in the vicinity of the Einstein radius (e.g., Rusin et al., 2003; Treu & Koopmans, 2004; Rusin & Kochanek, 2005; Koopmans et al., 2006). The fact that a superposition of non-isothermal stellar and dark matter profiles can produce a total profile that is approximately isothermal over a wide range of scales — known as the “isothermal conspiracy” — is related to the tendency towards flat rotation curves originally used to argue for a dark matter component. Finally, the same set of SLACS lenses can be fit with a single power-law profile that is consistent with isothermal (Koopmans et al., 2006), or by a more physically intuitive combination of a stellar Hernquist profile and a dark matter halo (Jiang & Kochanek, 2007). All in all, the quasi-isothermal nature of our models appears to be consistent with interpretations of strong lensing by real galaxies.

### 4.4 Density shape models

We consider seven different models for the shape of the galaxy density: spherical, moderately oblate, very oblate, moderately prolate, very prolate, triaxial, and a mixed model described below. In all but the mixed model, the axis ratios used for the dark matter and stellar components are related in a simple way (see the full description in Section 4.4.1). In all cases the axes of the two components are intrinsically aligned in three dimensions.

#### 4.4.1 Axis ratios

We adopt the mean values and from Jing & Suto (2002) for the axis ratios of the triaxial density of the dark matter component. These ratios were determined from cosmological -body simulations, without any stellar component. We set for the oblate shapes, with the same for the moderately oblate case, and for the very oblate case. We set for the prolate shapes, with for the moderately prolate case and for the very prolate case. Note that in the very oblate and very prolate case these choices preserve the product of the triaxial case, and hence the enclosed ellipsoidal mass.

Next, we use results from the cosmological gasdynamics simulations in Kazantzidis et al. (2004) to make a (crude) conversion from the dark matter shape to the rounder shape of the stars: , and similarly for . In practice, in that paper, the axis ratios were a function of separation from the centre of the halo; we neglect those effects in this work. Also, in that work the dark matter halos were themselves made more round due to the presence of baryons, with up to per cent effects out to the virial radius. However, those simulations may suffer from baryonic overcooling, which would tend to accentuate these effects, so we do not incorporate the rounding of the dark matter component at this time.

The final model, the mixed shape model, is constructed using the very oblate stellar shape combined with the triaxial dark matter halo shape. Part of the motivation for doing so is based on the results given by Lambas et al. (1992) for the axis ratio distributions and 3d shapes of early-type galaxies as inferred from the distributions of their projected shapes. In that paper, the elliptical sample containing galaxies was found to have a projected shape distribution consistent with a (weakly) triaxial intrinsic shape distribution. Fitting Gaussian distributions in both axis ratios, they found a best-fitting mean and standard deviation of and in , and and in . The S0 galaxies in their sample had a projected shape distribution with best-fit mean value consistent with oblate axisymmetry, and with best-fitting mean and standard deviation of and in .

Further motivation is provided by detailed dynamical models of nearby elliptical and lenticular galaxies that accurately fit their observed surface brightness and (two-dimensional) stellar kinematics. These models show that the inner parts of lenticular as well as many elliptical galaxies, collectively called fast-rotators (Emsellem & et al., 2007), are consistent with an oblate axisymmetric shape (Cappellari & et al., 2007). The slow-rotator ellipticals, on the other hand, might be rather close to oblate axisymmetric in the centre but rapidly become truly triaxial at larger radii (van den Bosch et al. 2008, 2008).

We thus consider this mixed model, with a triaxial dark matter shape as in cosmological -body simulations plus an oblate stellar shape as inferred from observations, as being potentially consistent with reality for the lower mass scale. The triaxial shape for both dark matter and stars may be more appropriate for the higher mass scale.

Shape | ||||
---|---|---|---|---|

Spherical | 1.00 | 1.00 | 1.00 | 1.00 |

Moderately oblate | 1.00 | 0.50 | 1.00 | 0.70 |

Very oblate | 1.00 | 0.36 | 1.00 | 0.62 |

Moderately prolate | 0.71 | 0.71 | 0.83 | 0.83 |

Very prolate | 0.60 | 0.60 | 0.76 | 0.76 |

Triaxial | 0.71 | 0.50 | 0.83 | 0.70 |

Mixed | 0.71 | 0.50 | 1.00 | 0.62 |

The resulting axis ratios for each shape model, both for the dark matter and stellar component, are given in Table 2. While we do not explore a broad range of axis ratios due to the prohibitive numbers of simulations involved, we can still get some sense of the trend with axis ratio by looking at the changes from spherical to moderately and very oblate or prolate shapes.

#### 4.4.2 Scale lengths for non-spherical density profiles

When constructing the non-spherical density profiles, we fix the concentration to be the same as in the spherical case, but set the scale length such that the mass within the ellipsoidal virial radius is equal to the virial mass . This is done using

(32) |

In this case, the length scale of the halo is rescaled while maintaining its concentration .

#### 4.4.3 (Mis)alignment of projected axes

We assume that the intrinsic axes of the stellar and dark matter component are aligned. For the axisymmetric models this means that the minor and major axes of the surface mass densities of the stars and dark matter are aligned as well. However, in case of a triaxial dark matter component the projected axes are misaligned with respect to those of the intrinsically rounder stellar component.

The misalignment follows from equation (2) for the different intrinsic axis ratios of the dark matter and stellar component (Table 2) as a function of the polar and azimuthal viewing angles . The left panels of Fig. 8 show the resulting for the triaxial (top) and mixed (bottom) shape model. Depending on the viewing direction, the projected axis ratios of the stars and dark matter can become completely orthogonal (). At the same time, however, the projected ellipticity of the stars goes to zero, as shown in the right panels of Fig. 8. This means that the stars appear round on the plane of the sky, so that their position angle on the sky is very hard to establish.

Even ignoring the stellar ellipticity, the probability of a
significant misalignment is rather small in the case of random viewing
directions^{7}^{7}7We show in Paper II that there is a selection bias
in orientation for non-spherical lens galaxies.. For the triaxial
shape model only % of the area on the viewing sphere leads
to misalignments . This number increases to
% for the mixed shape model, but at the same time the area
on the viewing sphere that leads to a round appearance of the stars is
also larger: % of the area has ,
while this is only % for the triaxial shape model.

Our use of models that lead to almost no situations with significant projected ellipticity and misalignment of DM and stellar components is consistent with studies of relatively isolated lens systems, for which the position angles of the light and of the projected total mass agree to within (Keeton et al., 1998; Kochanek, 2002a; Koopmans et al., 2006; Bolton et al., 2008b). Moreover, both the observed surface brightness and stellar kinematics of early-type galaxies are well-fitted by dynamical models with the intrinsic density parametrised by multiple (Gaussian) components that have different shapes but are all co-axial (e.g. Emsellem et al., 1994; Cappellari & et al., 2006). This includes the photometric and kinematic twists observed in particularly the giant ellipticals (van den Bosch et al. 2008, 2008).

The above is all empirical evidence in favour of an initial guess involving a high degree of alignment. While there are many reasons why these assumptions are not perfect, here we do not explore the effects of intrinsic misalignment between the shapes, because of the enormous amount of parameter space involved. Future work should explore this issue further, as well as the issue of warps in the intrinsic and/or projected profiles.

### 4.5 Deprojected Sérsic simulations

In addition to the simulations with cusped density profiles, we also create a set of simulations with deprojected Sérsic density profiles (Section 2.5), which also provide a good description of both stars and dark matter over a range of mass scales (Section 2.3). As before, we choose the different parameters for the dark matter and stellar density to create observationally realistic galaxy models.

#### 4.5.1 Density parameters

For the total mass of the stellar component we use the same values given in Table 1 for the lower and higher mass scales. Even though the total mass of the dark matter component is also finite in this case, we adopt for consistency the same virial mass scale by setting the mass within the virial radius equal to as given in Table 1. In spherical geometry, the scale radius of the Sérsic density is the half-mass radius, which for the stellar component is given in Table 1. This, in turn, is equal to the effective radius in the case of a constant stellar mass-to-light ratio. For the dark matter component we numerically derive the radius that encloses a projected mass that is half of the virial mass of the spherical NFW dark matter halo, and use the same half-mass radius for the deprojected Sérsic profiles. This procedure results in kpc and kpc for the lower and higher mass scale, respectively

The only parameter left to choose is the Sérsic index . To derive the Sérsic indices of the stellar component, we use the relation for Virgo early-type galaxies as derived by Ferrarese & et al. (2006). A luminosity of corresponds to an absolute Johnson -band and SDSS -band magnitude of about (Faber & et al., 2007) and (Blanton & et al., 2003), respectively. With as a typical colour for elliptical galaxies (Fukugita et al., 1995), this means that our choice for the lower and higher mass scales of and in the -band translate to and in the -band, or about and . We thus obtain Sérsic indices and for the stellar component of the lower and higher mass scale, respectively. For the dark matter component, we use the Sérsic fits to simulated CDM halo profiles presented in Merritt et al. (2005). Taking into account the difference in virial mass definition, we obtain and for the lower and higher mass scales, respectively.

In addition to these fiducial Sérsic indices, we also consider values consistent with the observed scatter. For a given absolute luminosity , the scatter in the observed in Ferrarese & et al. (2006) is approximately dex. This scatter results in a range in of and for the lower and higher mass scales, respectively. Similarly, for a given virial mass , the range of fitted in Merritt et al. (2005) is approximately covered by dex. Since these findings are based on a low number (19) of simulations, we consider indices dex around the fiducial values. This scatter implies a range in of and for the lower and higher mass scales, respectively.

#### 4.5.2 Normalisation

For the cusped density profiles in Section 4.3.2, changing the dark matter inner slope from the fiducial value only affects the inner parts of the density profile. As a result, the corresponding change in virial mass (about % and % when changing to respectively and ), could easily be compensated by changing the density amplitude and/or the scale radius . For the deprojected Sérsic density profiles, changing the index affects the whole density profile rather than just the inner parts, so that the corresponding changes in the mass are much larger. Changing the fiducial dark matter index to the limits of the adopted range (while keeping fixed), results in a factor change in the virial mass for both mass scales. Changing the fiducial stellar index to the limits of the adopted range, results in a factor change in the total stellar mass . Preserving the mass thus requires very large variations in the central density and/or the half-mass radius .

Certainly for the stellar component, changing is not an option, since early-type galaxies obey a size-luminosity relation (e.g. Shen et al., 2003; Bernardi et al., 2007). Although there is some freedom due to scatter in this relation and the stellar mass-to-light conversion, this relation excludes larger variations in when preserving . A similar conclusion is reached based on the Kormendy (1977) relation , where is the mean surface brightness within the effective radius . In case of spherical symmetry and a constant mass-to-light ratio and , where the latter dependence on the Sérsic index follows from equation (15). Therefore, obeying the Kormendy relation means , while preserving the total stellar mass implies . Since even at high(er) redshift (e.g. La Barbera et al., 2003; di Serego Alighieri & et al., 2005), these two relations together imply that remains to be approximately constant, while can be varied to compensate for the change in .

A similar scaling relation seems to hold for NFW dark matter halos between their concentration (or scale radius in case of a constant virial radius), and virial mass of the form (Bullock et al., 2001). Since the dependence on is weak and the scatter in the relation is rather large, we could not use it before in Section 4.3.2 when changing the inner slope of the cusped density profile to break the degeneracy between the different normalisation options (which we showed in case of preserving was not necessary, since the effect of the different normalisations on the lensing cross-sections is similar). Salvador-Solé et al. (2007) show that when an Einasto profile is used instead of a NFW profile, a similar concentration-mass relation exists for dark matter halos. Because deprojected Sérsic profiles fit simulated dark matter halos equally well as Einasto profiles (Section 2.3), we expect also in that case a correlation between and . As mentioned above, for the Sérsic density the effect of changing is much stronger than changing for the cusped density, so that varying (while fixing ) to preserve would violate this concentration-mass relation. As a result, for both the stellar and dark matter component we vary to preserve the (total and virial) mass, while we keep the half-mass radius fixed.

#### 4.5.3 Density profiles

The profile of the spherical deprojected Sérsic mass density is shown in Fig. 2 in the middle row and compared to the cusped mass density in the bottom row, for the lower and higher mass scales. The thin curves indicate the stellar and dark matter profiles which combined yield the thick curves. The solid curves are based on the fiducial Sérsic indices for both components, whereas the two other curves indicate the variation in the mass density when adopting the above limits in and . In a similar way, we show in Fig. 3 the corresponding negative logarithmic slope , in Fig. 4 the Sérsic surface mass density , in Fig. 5 the corresponding negative logarithmic projected slope , in Fig. 6 the projected dark matter fraction , and finally in Fig. 7 the deflection angle.

As for the cusped density in Section 4.3.4, we find that both the slope of the composite surface mass density and the deflection angle (Fig. 5 and Fig. 7) are consistent with an isothermal density around the Einstein radius. Especially for the lower-mass scale, the projected slope for all cases converges to around , and the curves of the deflection angle are extremely flat at and beyond . As for cusped models, this finding implies that constraints from strong lensing cannot trivially be extrapolated to smaller and larger radii.

### 4.6 Computation of surface mass density maps

The lensing properties of a galaxy follow from its surface mass density . For the above galaxy models this is the sum of the surface mass density of the stellar and dark matter components, each of which can be computed as follows. For a given intrinsic shape seen from a viewing direction , the projected semi-axis lengths are given by equations (5)–(7). At each position on the plane of the sky, equation (4) then yields the elliptic radius . The surface mass density follows by evaluation of equation (3) for a cusped density given in equation (9), or in the case of a Sérsic density directly as given in equation (16).

However, rather than doing the numerical evaluation for each position and viewing direction, we instead compute on a fine one-dimensional grid in , once for each intrinsic density profile and shape. We then derive by linearly interpolating onto this grid at the value that (according to equations 4–7) corresponds to the position and chosen viewing direction . In particular, in the case of the cusped density, this procedure means that we can avoid doing the numerical evaluation of equation (3) on a two-dimensional grid repeatedly for each viewing direction.

Along with the construction of the cusped and Sérsic galaxy models
described above, we implemented and tested this efficient computation
of the surface mass density in
idl.^{8}^{8}8http://www.ittvis.com/ProductServices/IDL.aspx We use
a linear grid in , except in the inner parts where we sample
logarithmically in to accurately resolve a central peak in the
density. Even so, when we compute the surface mass density on a
regular square grid in with a given dimension (box size) and
sampling (resolution), the resulting fixed pixel size might not be
small enough to account for a strong central cusp. This is not the
result of the interpolation but is due to the discreteness of the
map-based approach. The effects of finite resolution and box size on
the surface mass density and corresponding lensing properties are
further tested and discussed in Sections 5.7
and 5.8.

We choose the grid in such that the -axis is aligned with the major axis of the surface mass density of the dark matter component. Even though we assume that the intrinsic axes of the stellar and dark matter component are aligned, in case of a triaxial dark matter component the projected axes are misaligned with respect to those of the intrinsically rounder stellar component. As discussed in Section 4.4.3 above, the resulting misalignment angle follows from equation (2), and depends on the viewing direction . To take this misalignment of the stellar component with respect to the dark matter component into account, we first rotate over the misalignment angle

and instead of equation (4), we then use to define the elliptic radius . In case of the axisymmetric models , so that the stellar and dark matter component are always aligned. Finally, we add the maps of the stellar and dark matter components together to arrive at the total surface mass density of the galaxy model.

In Fig. 9, we show examples of derived surface mass density maps for lower, galaxy-scale mass models with cusped density profiles. The figure includes, from top to bottom, the very oblate mass model viewed close to face-on and close to edge-on, and the mixed shape model. The latter is seen close to the viewing direction that results in the maximum combination of ellipticity of the stars (left column) and misalignment with respect to the dark matter (middle column), resulting in a twist in the combined surface mass density (right column). In Section 5.9 we address the technical issue of how many viewing angles we need to sample in order to accurately characterise how the lensing behaviour changes with viewing direction.

## 5 Lensing calculations

In this section, we describe in detail how we do the lensing
calculations necessary for our analysis. We present numerical methods
designed to balance efficiency with accuracy
(Sections 5.1–5.5), together with
extensive tests to validate those methods
(Sections 5.6–5.9). All of the
calculations described here can be done with gravlens version 1.99k,
which is a version of the original software by
Keeton (2001b) that has been extended for this
project^{9}^{9}9The features that are new in gravlens v1.99k are
identified in the text. All the new features have been tested, and
have instructions available through the gravlens help
command, but are not fully documented in the manual. Version 1.99k
can be accessed using the link for latest updates at
http://redfive.rutgers.edu/~keeton/gravlens..

### 5.1 Lens potential, deflection, and magnification

Given the two-dimensional convergence distributions for our models, we need to solve the Poisson equation (22) to find the lens potential. In the present work, we actually need the first and second derivatives of the lens potential (for the deflection and magnification, respectively), and do not work with the potential directly. We use different approaches to obtain the derivatives, depending on the model properties. Note that we apply the methods discussed here to the stellar and dark matter components separately, and construct the composite lens model from a simple linear superposition of the mass components.

If the model has circular symmetry and the convergence is known analytically, the deflection can be obtained from the one-dimensional integral over given in equation (25). Often this integral can be evaluated analytically, so the lens model is fully analytic. This procedure is used for circular deprojected Sérsic models. (For more detail about lensing by circular Sérsic models, see Cardone 2004 and Baltz et al. 2009.)

If the model has elliptical symmetry and the convergence is known
analytically, the first and second derivatives of the lens potential
can be obtained from a set of one-dimensional integrals over
and its derivative
(Schramm, 1990; Keeton, 2001a). For elliptical
deprojected Sérsic models, we evaluate the integrals
numerically,^{10}^{10}10We note that the steep central profile of
Sérsic models can lead to numerical precision requirements that
are more stringent than usual. With inadequate precision, there is
a systematic and cumulative error that effectively changes the
density profile. We achieve the necessary precision by decreasing
the gravlens integration tolerance parameter inttol from
its default value of down to . yielding what
we call numerical lens models. Circular and elliptical Sérsic models
are new features in gravlens that have been added for this project.

For some models, not even the convergence is known analytically, so the projection integral in equation (3) must be evaluated numerically. In such cases we compute the convergence on a map and then solve the Poisson equation using Fourier methods. Here, is a two-dimensional angular position on the sky in the lens plane, with its counterpart in Fourier space. We use FFTs to find the Fourier transform of the convergence. We then rewrite the real-space Poisson equation (22) in Fourier space as