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The existence of dark matter halos surrounding large, bright galaxies is well established (e.g., [1], [2], [3], [4] and references therein), and in the standard cold dark matter (CDM) paradigm, the halos of large field galaxies are expected to extend to virial radii of ~ 100h-1 kpc to ~ 200h-1 kpc and have masses of ~ 1012 h-1 Modot (e.g., [5], [6], [7]). Until very recently, however, direct observational constraints on the nature of the dark matter halos of field galaxies have not been especially strong. In particular, it has been challenging to address the question as to whether the halos of observed galaxies are consistent with the halos that one would expect in a CDM universe.

The lack of a Keplerian fall-off in the rotation curves of the disks of most spiral galaxies (e.g., [2]) indicates that the dark matter halos extend far beyond the visible radii of the galaxies. Therefore, in order to place constraints on the total mass distribution, it is necessary to use tracers of the halo potential that exist at large projected radii (gtapprox 100h-1 kpc). Two such tracers of the large-scale potential are satellite galaxies that are in orbit about isolated host galaxies, and photons emitted by distant galaxies that, on their way to the observer, happen to pass through the potential wells of more nearby galaxies at small impact parameters (i.e., gravitational lensing). "Strong" gravitational lensing, in which multiple and highly-distorted images of a source occur, is a rare phenomenon because it requires nearly perfect alignment of the lens and source galaxy (e.g., [8]). "Weak" gravitational lensing, in which multiple images and significant image distortion do not occur, is, however, commonplace in the universe (e.g., [9], [10], [11], [12]), and it is on this extremely mild regime of gravitational lensing that I will focus for this discussion.

Weak lensing of background galaxies by foreground galaxies ("galaxy-galaxy" lensing) and the motions of satellite galaxies about host galaxies are phenomena that can only lead to constraints on halo potentials through ensemble averages over statistically large samples. That is, for any given foreground galaxy, the distortion that it induces in the images of background galaxies due to weak lensing is so small that the signal cannot be detected convincingly for any one foreground lens galaxy. Similarly, isolated host galaxies are typically found to have 1 to 2 satellite galaxies on average and, so, the potential of any one host galaxy cannot be constrained at all well by the motions of its own satellites. Both galaxy-galaxy lensing and satellite dynamics, therefore, lead to statistical constraints on the halo population as a whole and by their very nature they require large samples of galaxies in order to obtain such constraints. Until several years ago, galaxy-galaxy lensing and satellite dynamics were both tantalizingly close (or frustratingly close, depending on one's point of view) to being able to fulfill their theoretical promise to map out the gravitational potentials of the halos of field galaxies. With the advent of routine availability of wide-field imagers and the completion (or near completion) of large redshift surveys, however, both galaxy-galaxy lensing and satellite dynamics are now yielding sufficiently strong constraints on the dark matter halos of galaxies that the observations can be used to test the theoretical predictions (i.e., CDM) at a substantive level.

There are distinct advantages and disadvantages of galaxy-galaxy lensing versus satellite dynamics when it comes to constraining halo potentials. A clear advantage of galaxy-galaxy lensing is that it can be applied to all foreground galaxies and, since gravitational lensing is affected only by the total mass along the line of sight and not its dynamical state, the halos of the foreground lens galaxies need not be virialized. A complicating factor in galaxy-galaxy lensing is that it is not correct to assume that each background galaxy has been lensed solely by one foreground galaxy (e.g., [13], [14], [15]). Instead, photons emitted by the distant galaxies are deflected by all mass along the line of sight, including individual galaxies, groups, and clusters (e.g., Figure 1). That is, galaxy-galaxy lensing is inherently a multiple-deflection problem and care must be taken when using observations of galaxy-galaxy lensing to constrain the halos of a given subset of lens galaxies (i.e., the halos of early-type galaxies versus late-type galaxies, or the halos of high-luminosity galaxies versus low-luminosity galaxies). Therefore, a computation of the weak lensing signal about the white lenses in Figure 1 above is not identical to a measurement of weak lensing signal produced by the white lenses since the black lenses also contribute to the net shape of the final image. That is not to say that galaxy-galaxy lensing cannot be used to probe the potentials of halos surrounding lenses of differing types; it most certainly can, but the presence of multiple deflections in the data must be taken into account when modeling the observed signal. In the case of relatively shallow data (zlens ~ 0.15), most sources will have been lensed by only one foreground galaxy (e.g., [16]), but in deep data sets (zlens ~ 0.5) most source galaxies will have been lensed at a significant and comparable level by two or more foreground galaxies (e.g., [13], [15]). A further disadvantage of galaxy-galaxy lensing is that the signal is very small (systematic image distortions of ltapprox 1% or in the image ellipticities), so the images of millions of background galaxies must be obtained and, in general, be meticulously corrected for the presence of anisotropic, spatially-varying point spread functions. Finally, it is possible that Newtonian tidal distortions of genuine satellites of the lens galaxies could masquerade as a weak lensing signal. Happily, such distortions appear to be at most a very small contributor to the observed weak lensing signal (e.g., [13], [17]. [18], [19]).

Figure 1

Figure 1. Schematic representation of multiple lenses along the line of sight to a given source galaxy.

An advantage to using dynamics of satellite galaxies to probe the potentials of the halos of isolated host galaxies is that, unlike deep weak lensing data, the only important potential well in the problem is that of the host galaxy. In principle, this is a "cleaner", more straightforward probe of the halo potential which is intentionally restricted to the physical scales that one would expect to characterize the halos of individual large galaxies. However, there are a number of arguments against the use of satellite dynamics to probe the mass distributions of host galaxies: (1) satellites must be found at large projected radii in order to probe the halo potential on the very largest scales, (2) noise is introduced by the presence of "interlopers" (i.e., galaxies that are selected as satellites but which are, in fact, not associated dynamically with the host galaxy), and (3) the relaxation times of these systems are large compared to the age of the universe. The first argument is much less compelling now than it was in the past simply because of the availability of large redshift surveys (i.e., the data bases are now sufficiently big that although it is rare to find satellites at, say, a projected radius of 500 h-1 kpc, the large number of redshifts that are now available makes it possible to compile statistically significant samples). The second argument has also become much less compelling with the realization that it is straightforward to account for the effects of interloper galaxies on the determination of the velocity dispersion (see below). The third argument can still be compelling, since it makes little sense to apply a virial-type mass estimator to systems which are not relaxed. However, an assumption of virialization is not necessary a priori, and the use of secondary infall models can be used to bypass this assumption (e.g., [20]).

For the sake of a certain amount of brevity and, at the very least, an attempt at providing some level of coherent argument, I will focus here on only the most recent results that are directly relevant to the halos of field galaxies and which have been obtained from four large surveys: the COMBO-17 Survey, the Red-Sequence Cluster Survey (RCS), the Sloan Digital Sky Survey (SDSS), and the Two Degree Field Galaxy Redshift Survey (2dFGRS). Even with this restriction, it is simply not possible to discuss all of the most recent results from these studies in great detail, and the reader should consult the source literature for further information. Finally, I should hasten to add that all errors or omissions in this article in regards to my colleagues' work are entirely unintentional and entirely my own fault. I can only hope that my colleagues will be kind enough to forgive me.

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