B.6.2. The Lens Population

The probability that a source has an intervening lens requires a model for the distribution of the lens galaxies. In almost all cases these are based on the luminosity function of local galaxies combined with the assumption that the comoving density of galaxies does not evolve with redshift. Of course luminosity is not mass, so a model for converting the luminosity of a local galaxy into its deflection scale as a lens is a critical part of the process. For our purposes, the distributions of galaxies in luminosity are well-described by a Schechter ([1976]) function,

 (B.100)

The Schechter function has three parameters: a characteristic luminosity L* (or absolute magnitude M*), an exponent describing the rise at low luminosity, and a comoving density scale n*. All these parameters depend on the type of galaxy being described and the wavelength of the observations. In general, lens calculations have divided the galaxy population into two broad classes: late-type (spiral) galaxies and early-type (E/S0) galaxies. Over the period lens statistics developed, most luminosity functions were measured in the blue, where early and late-type galaxies showed similar characteristic luminosities. The definition of a galaxy type is a slippery problem - it may be defined by the morphology of the surface brightness (the traditional method), spectral classifications (the modern method since it is easy to do in redshift surveys), colors (closely related to spectra but not identical), and stellar kinematics (ordered rotational motions versus random motions). Each approach has advantages and disadvantages, but it is important to realize that the kinematic definition is the one most closely related to gravitational lensing and the one never supplied by local surveys. Fig. B.39 shows an example of a luminosity function, in this case K-band infrared luminosity function by Kochanek et al. ([2001], also Cole et al. [2001]) where MK*e = - 23.53 ± 0.06 mag, n*e = (0.45 ± 0.06) × 10-2 h3 Mpc-3, and e = - 0.87 ± 0.09 for galaxies which were morphologically early-type galaxies and MK*l = - 22.98 ± 0.06 mag, n*l = (1.01 ± 0.13) × 10-2 h3 Mpc-3, and l = - 0.92 ± 0.10 for galaxies which were morphologically late-type galaxies. Early-type galaxies are less common but brighter than late-type galaxies at K-band. It is important to realize that the parameter estimates of the Schechter function are correlated, as shown in Fig. B.40, and that it is dangerous to simply extrapolate them to fainter luminosities than were actually included in the survey.

 Figure B.39. Example of a local galaxy luminosity function. These are the K-band luminosity functions for either all galaxies or by morphological type from Kochanek et al. ([2001]). The curves show the best fit Schechter models for the luminosity functions while the points with error bars show a non-parametric reconstruction.

 Figure B.40. Schechter parameters and M* for the 2MASS luminosity functions shown in Fig. B.39. Note there is a significant correlations not only between and M* but also with the comoving density scale n* that should be included in lens statistical calculations but generally are not.

However, light is not mass, and it is mass which determines lensing properties. One approach would simply be to assign a mass-to-light ratio to the galaxies and to the expected properties of the lenses. This was attempted only in Maoz & Rix ([1993]) who found that for normal stellar mass to light ratios it was impossible to reproduce the data (although it is possible if you adjust the mass-to-light ratio to fit the data, also see Kochanek [1996a]). Instead, most studies convert the luminosity functions dn / dL into a velocity functions dn / dv using the local kinematic properties of galaxies and then relate the stellar kinematics to the properties of the lens model. As Fig. B.41 shows (for the same K-band magnitudes of our luminosity function example), both early-type and late-type galaxies show correlations between luminosity and velocity. For late-type galaxies there is a tight correlation known as the Tully-Fisher ([1977]) relation between luminosity L and circular velocity vc and for early-type galaxies there is a loose correlation known as the Faber-Jackson ([1976]) relation between luminosity and central velocity dispersion v. Early-type galaxies do show a much tighter correlation known as the fundamental plane (Dressler et al. [1987], Djorgovski & Davis [1987]) but it is a three-variable correlation between the velocity dispersion, effective radius and surface brightness (or luminosity) that we will discuss in Section B.9. While there is probably some effect of the FP correlation on lens statistics, it has yet to be found. For lens calculations, the circular velocity of late-type galaxies is usually converted into an equivalent (isotropic) velocity dispersion using vc = 21/2 v. We can derive the kinematic relations for the same K-band-selected galaxies used in the Kochanek et al. ([2001]) luminosity function, finding the Faber-Jackson relation

 (B.101)

and the Tully-Fisher relation

 (B.102)

These correlations, when combined with the K-band luminosity function have the advantage that the magnitude systems for the luminosity function and the kinematic relations are identical, since magnitude conversions have caused problems for a number of lens statistical studies using older photographic luminosity functions and kinematic relations. For these relations, the characteristic velocity dispersion of an L* early-type galaxy is *e 209 km/s while that of an L* late-type galaxy is *l 143 km/s. These are fairly typical values even if derived from a completely independent set of photometric data.

 Figure B.41. K-band kinematic relations for 2MASS galaxies. The top panels show the Faber-Jackson relation and the bottom panels show the Tully-Fisher relations for 2MASS galaxies with velocity dispersions and circular velocities drawn from the literature. The left hand panels show the individual galaxies, while the right hand galaxies show the mean relations. Note the far larger scatter of the Faber-Jackson relation compared to the Tully-Fisher relation.

 Figure B.42. The resulting velocity functions from combining the K-band luminosity functions (Fig. B.39) and kinematic relations (Fig. B.41) for early-type (top), late-type (middle) and all (bottom) galaxies. The points show partially non-parametric estimates of the velocity function based on the binned estimates in the right hand panels of Fig. B.41 rather than power-law fits. Note that early-type galaxies dominate for high circular velocity.

Both the Faber-Jackson and Tully-Fisher relations are power-law relations between luminosity and velocity, L / L* (v / *)FJ. This allows a simple variable transformation to convert the luminosity function into a velocity function,

 (B.103)

There are three caveats to keep in mind about this variable change. First, we have converted to the distribution in stellar velocities, not some underlying velocity characterizing the dark matter distribution. Many early studies assumed a fixed transformation between the characteristic velocity of the stars and the lens model. In particular, Turner, Ostriker & Gott ([1984]) introduced the assumption dark = (3/2)1/2 stars for an isothermal mass model based on the stellar dynamics (Jeans equation, Eqn. B.90 and Section B.4.9) of a r-3 stellar density distribution in a r-2 isothermal mass distribution. Kochanek ([1993b], [1994]) showed that this oversimplified the dynamics and that if you embed a real stellar luminosity distribution in an isothermal mass distribution you actually find that the central stellar velocity dispersion is close to the velocity dispersion characterizing the dark matter halo. Fig. B.43 compares the stellar velocity dispersion to the dark matter halo dispersion for a Hernquist distribution of stars in an isothermal mass distribution. Such a normalization calculation is required for any attempt to match the observed velocity functions with a particular mass model for the lenses. Second, in an ideal world, the luminosity function and the kinematic relations should be derived from a consistent set of photometric data, while in practice they rarely are. As we will see shortly, the cross section for lensing scales roughly as *4, so small errors in estimates of the characteristic velocity have enormous impacts on the resulting cosmological results - a 5% velocity calibration error leads to a 20% error in the lens cross section. Since luminosity functions and kinematic relations are rarely derived consistently (the exception is Sheth et al. [2003]), the resulting systematic errors creep into cosmological estimates. Finally, for the early-type galaxies where the Faber-Jackson kinematic relation has significant scatter, transforming the luminosity function using the mean relation as we did in Eqn. B.103 while ignoring the scatter underestimates the number of high velocity dispersion galaxies (Kochanek [1994], Sheth et al. [2003]). This leads to underestimates of both the image separations and the cross sections. The fundamental lesson of all these issues is that the mass scale of the lenses should be "self-calibrated" from the observed separation distribution of the lenses rather than imposed using local observations (as we discuss below in Section B.6.7).

 Figure B.43. Stellar velocity dispersions vlos for a Hernquist distribution of stars in an isothermal halo of dispersion DM. The solid curves show the local value vlos and the dashed curves show the mean interior to the radius . Local velocity dispersions are typically measured on scales similar to Re / 8 where the stellar and dark matter dispersions are nearly equal rather than matching the viral theorem limit which would be reached in an infinite aperture. The upper, lower and middle curves are for stars with isotropies of = 0.2 (somewhat radial), = 0 (isotropic) and = - 0.2 (somewhat tangential).

Most lens calculations have assumed that the comoving density of the lenses does not evolve with redshift. For moderate redshift sources this only requires little evolution for zl < 1 (mostly zl < 0.5), but for higher redshift sources it is important to think about evolution as well. The exact degree of evolution is the subject of some debate, but a standard theoretical prediction for the change between now and redshift unity is shown in Fig. B.44 (see Mitchell et al. [2004] and references therein). Because lower mass systems merge to form higher mass systems as the universe evolves, low mass systems are expected to be more abundant at higher redshifts while higher mass systems become less abundant. For the v ~ * ~ 200 km/s galaxies which dominate lens statistics, the evolution in the number of galaxies is actually quite modest out to redshift unity, so we would expect galaxy evolution to have little effect on lens statistics. Higher mass systems evolve rapidly and are far less abundant at redshift unity, but these systems will tend to be group and cluster halos rather than galaxies and the failure of the baryons to cool in these systems is of greater importance to their lensing effects than their number evolution (see Section B.7). There have been a number of studies examining lens statistics with number evolution (e.g. Mao [1991], Mao & Kochanek [1994], Rix et al. [1994]) and several attempts to use the lens data to constrain the evolution (Ofek, Rix & Maoz [2003], Chae & Mao [2003], Davis, Huterer & Krauss [2003]).

 Figure B.44. The ratio of the velocity function of halos at z = 1 to that at z = 0 from Mitchell et al. ([2004]). The solid curve shows the expectation for an 0.78 flat cosmological model. The points show results from an N-body simulation with 0.7 and the dashed curve shows the theoretical expectation. For comparison, the dotted curve shows the evolution model used by Chae & Mao ([2003]).