**B.4.1. Common Models for the Monopole**

Most attention in modeling lenses focuses on the monopole or radial mass distribution of the lenses. Unfortunately, much of the lensing literature uses an almost impenetrable array of ghastly non-standard acronyms to describe the mass models even though many of them are identical to well-known families of density distributions used in stellar dynamics. Here we summarize the radial mass distributions which are most commonly used and will keep reappearing in the remainder of Part 2.

The simplest possible choice for the mass distribution is to simply trace the light. The standard model for early-type galaxies or the bulges of spiral galaxies is the de Vaucouleurs ([1948]) profile with surface density

(B.53) |

where the effective radius *R*_{e} encompasses half the
total mass (or light) of the profile. Although the central density of a
de Vaucouleurs model is finite,
it actually acts like a rather cuspy density distribution and will generally
fit the early-type lens data with no risk of producing a detectable central
image (e.g. Lehár et al.
[2000],
Keeton
[2003a]).
The simplest model for a disk galaxy is an exponential disk,

(B.54) |

where *R*_{d} is the disk scale length. An exponential disk
by itself is rarely a viable lens model because it has so little density
contrast between the center and the typical radii of images that
detectable central images are almost always predicted but not
observed. Some additional component, either a de Vaucouleurs bulge or a
cuspy dark matter halo, is always required. This makes spiral galaxy
lens models difficult because they generically require two stellar
components (a bulge and a disk)
and a dark matter halo, while the photometric data are rarely good enough to
constrain the two stellar components (e.g. Maller, Flores & Primack
[1997],
Koopmans et al.
[1998],
Maller et al.
[2000],
Trott & Webster
[2002],
Winn, Hall & Schechter
[2003]).
Since spiral lenses are already relatively rare, and spiral lens
galaxies with good photometry are rarer still, less attention
has been given to these systems. The de Vaucouleurs and exponential disk
models are examples of Sersic
([1968])
profiles

(B.55) |

where the effective radius *R*_{e}(*n*) is defined to
encompass half the light and *n* = 4 is a de Vaucouleurs model and
*n* = 1 is an exponential disk. These profiles
have not been used as yet for the study of lenses except for some quasar
host galaxy models (Section B.10).
The de Vaucouleurs model can be approximated (or the reverse) by
the Hernquist
([1990])
model with the 3D density distribution

(B.56) |

and *a*
0.55*R*_{e} if matched to a de Vaucouleurs model.
For lensing purposes, the Hernquist model has one major problem. Its
1 / *r* central
density cusp is shallower than the effective cusp of a de Vaucouleurs
model, so Hernquist models tend to predict detectable
central images even when the
matching de Vaucouleurs model would not. As a result, the Hernquist model
is more often used as a surrogate for dynamical normalization of the de
Vaucouleurs model than as an actual lens model (see below).

Theoretical models for lenses started with simple, softened power laws of the form

(B.57) |

in the limit where there is no core radius. We are using these simple
power law lenses in all our examples (see
Section B.3). These models include many
well known stellar dynamical models such as the singular isothermal
sphere (SIS, *n* = 2,
*s* = 0), the modified Hubble profile (*n* = 3) and the
Plummer model (*n* = 5).
Since we only see the projected mass, these power laws are also related
to common models for infinitely thin disks. The Mestel
([1963])
disk (*n* = 2, *s* = 0) is the disk that produces a flat
rotation curve, and the Kuzmin
([1956])
disk (*n* = 3) can be used to mimic the rising and
then falling rotation curve of an exponential disk.
The softened power-law models have generally fallen out of favor other
than as
simple models for some of the visible components of lenses because the
strong evidence for stellar and dark matter cusps makes models with core
radii physically unrealistic. While ellipsoidal versions of these models
are not available in useful form, there are fast series expansion methods
for numerical models (Chae, Khersonsky & Turnshek
[1998],
Barkana
[1998]).

Most "modern" discussions of galaxy density distributions are based on sub-cases of the density distribution

(B.58) |

which has a central density cusp with
*r*^{-n}, asymptotically declines as
*r*^{-m}
and has a break in the profile near *r*
*a*
whose shape depends on
(e.g. Zhao
[1997]).
The most common cases are the Hernquist model (*n* = 1, *m* = 4,
= 1) mentioned above,
the Jaffe
([1983])
model (*n* = 2, *m* = 4,
= 1), the NFW (Navarro,
Frenk & White
[1996])
model (*n* = 1, *m* = 3,
= 1) and the Moore
([1998]) model
(*n* = 3/2, *m* = 3,
= 1). We can view the
power-law models either as the limit
*n* 0 and
= 2, or we could
generalize the *r*^{-n} term to
(*r*^{2} + *s*^{2})^{-n/2} and consider
only regions with *r* and *s* << *a*. Projections
of these models are similar to surface density distributions of the form

(B.59) |

(although the definition of the break radius *a* changes) with the
exception of the limit
*n* 1 where the
projection of a 3D density cusp
1 / *r* produces
surface density terms
ln *R* that
cannot be reproduced by the broken surface
density power law. This surface density model is sometimes called the
Nuker law (e.g. Byun et al.
[1996]).
A particularly useful case for lensing is the pseudo-Jaffe model with
*n* = 2, *m* = 4 and
= 2 (where the normal
Jaffe model has
= 1) as the only
example of a broken power law with simple analytic
deflections even when ellipsoidal because the density distribution is
the difference between two isothermal ellipsoids (see Eqn. B.41).
These cuspy models also allow fast approximate solutions for their
ellipsoidal counterparts (see
Chae 2002).

The most theoretically important of these cusped profiles is the NFW
profile
(Navarro et al. 1996)
because it is the standard model for dark matter halos. Since it
is such a common model, it is worth discussing it in a little more detail,
particularly its peculiar normalization. The NFW profile is normalized by
the mass *M*_{vir} inside the virial radius
*r*_{vir}, with

(B.60) |

where *f* (*c*) = ln(1 + *c*) - *c* / (1 + *c*) and
the concentration *c* = *r*_{vir} / *a* ~ 5 for
clusters and *c* ~ 10 for galaxies. The concentration is a function
of mass whose scaling is determined from N-body simulations. A typical
scaling for a halo at redshift *z* in an
_{M} = 0.3
flat cosmological models is
(Bullock et al. 2001)

(B.61) |

with a dispersion in log *c* of
_{log c}
0.18 dex. Because
gravitational lensing is very sensitive to the central density of the lens,
including the scatter in the concentration is quantitatively important for
lensing by NFW halos (Keeton
[2001b]).
The virial mass and radius are related and determined by the overdensity
_{vir}(*z*) required
for a halo to collapse given the cosmological model and the
redshift. This can be approximated by

(B.62) |

where _{u}(*z*) = 3*H*_{0}^{2}
_{M}(1 +
*z*)^{3} /
8*G* is the mean matter
density when the halo forms and
_{vir}
(18^{2} + 82*x* -
39*x*^{2}) /
(*z*) with
*x* = - 1 is
the overdensity needed for a halo to collapse. There
are differences in normalizations between authors and with changes in the
central cusp exponent
, but
models of this type are what we presently
expect for the structure of dark matter halos around galaxies.

For most lenses, HST imaging allows us to measure the spatial distribution of the stars, thereby providing us with a model for the distribution of stellar mass with only the stellar mass-to-light ratio as a parameter. For present purposes, gradients in the stellar mass-to-light ratio are unimportant compared to the uncertainties arising from the dark matter. Unless we are prepared to abandon the entire paradigm for modern cosmology, the luminous galaxy is embedded in a dark matter halo and we must decide how to model the overall mass distribution. The most common approach, as suggested by the rich variety of mass profiles we introduced in Section B.4.1, is to assume a parametric form for the total mass distribution rather than attempting to decompose it into luminous and dark components. The alternative is to try to embed the stellar component in a dark matter halo. Operationally, doing so is trivial - the lens is simply modeled as the sum of two mass components. However, there are theoretical models for how CDM halos should be combined with the stellar component.

Most non-gravitational lensing applications focus
on embedding disk galaxies in halos because angular momentum conservation
provides a means of estimating a baryonic scale length (e.g. Mo, Mao
& White
[1998]).
The spin parameter of the halo sets the
angular momentum of the baryons, and the final disk galaxy is defined
by the exponential disk with the same angular momentum. As the baryons
become more centrally concentrated, they pull the dark matter inwards
as well through a process known as adiabatic contraction
(Blumenthal et al.
[1986]).
The advantage of this approach, which in lensing has been used only by
Kochanek & White
([2001]),
is that it allows a full
*ab initio* calculation of lens statistical properties when
combined with a model for the cooling of the baryons (see
Section B.7).
It has the major disadvantage that most lens galaxies are
early-type galaxies rather than spirals, and that there is no analog
of the spin parameter and angular momentum conservation to set the
scale length of the stellar component in a model for an early-type galaxy.

Models of early-type galaxies embedded in CDM halos have to start with an empirical estimate of the stellar effective radius. In models of individual lenses this is a measured property of the lens galaxy (e.g. Rusin et al. [2003], [2004], Koopmans & Treu [2002], Kochanek [2003a]). Statistical models must use a model for the scaling of the effective radius with luminosity or other observable parameters of early-type galaxies (e.g. Keeton [2001b]). From the luminosity, a mass-to-light ratio is used to estimate the stellar mass. If all baryons have cooled and been turned into stars, then the stellar mass provides the total baryonic mass of the halo, otherwise the stellar mass sets a lower bound on the baryonic mass. Combining the baryonic mass with an estimate of the baryonic mass fraction yields the total halo mass to be fed into the model for the CDM halo.

In general, there is no convincing evidence favoring either approach - for the regions over which the mass distributions are constrained by the data, both approaches will agree on the overall mass distribution. However, there can be broad degeneracies in how the total mass distribution is decomposed into luminous and dark components (see Section B.4.6).