4.3. The central concentration degeneracy

Equations 10 and 11 tell us that the interpretation of the time delay for PG1115+080 depends crucially upon the internal structure of the galaxy. An error in the radial exponent of plus or minus 0.1 produces a 10% change in the predicted time delay as compared to an isothermal model. Fortunately we can test the isothermality hypothesis. For an isothermal sphere, the angular radius of the Einstein ring, b, is a directly proportional to the square of the velocity dispersion, ²:

(4.5)

With an Einstein ring radius b = 1."15, a lens redshift z_L = 0.31 and a source redshift z_S = 1.71 we predict _SIS = 232 km/s. This must be reduced by a factor (1 - _group)^1/2 which amounts to a 5% correction in the present case. We can test our prediction, but the measurement is a difficult one. The lensing galaxy is crowded by four very much brighter images. [Tonry (1998)] measures a velocity dispersion of 281 ± 25 km/s, very much larger than predicted. [Treu & Koopmans (2002)] use Tonry's measurement in modeling PG1115+080 and derive a value for H₀ very much larger than under the isothermal hypothesis. Their value is the one plotted in figure 1.

Why are we so reluctant to abandon the isothermal hypothesis for PG1115+080? First, because velocity dispersion estimates from equation 12 for an ensemble of lensing galaxies are consistent with the fundamental plane relation for non-lensing ellipticals. PG1115+080 is in no way unusual [(Kochanek et al. 2000)], but would be if we adopted the direct measurement.

A second argument is that the lenses for which we can measure the radial exponent are very nearly isothermal. Lenses with multiple sources, rings or central images all break the central concentration degeneracy and permit measurement of the radial exponent . The results for six systems are shown in table 3 (see also WAYTH'S contribution to the present proceedings). While there are differences in the way was calculated for each of these systems, the results are consistent with isothermal. Unfortunately only one of these systems, JVAS0218+357, is also a system that has a measured time delay.

Table 3. Exponents for power law density profiles: ~ r^-

lensing galaxy		reference
JVAS0218+357	1.96+0.02-0.02	[Wucknitz, Biggs, & Browne (2004)]
Q0957+561	1.84	[Barkana et al. (1999)]
MG1131+0456	2.40+0.2-0.2	[Chen, Kochanek, & Hewitt (1995)]
PMN1632-0033	1.91+0.02-0.02	[Winn, Rusin, & Kochanek (2004)]
MG1654+1346	1.90+0.16-0.01	[Kochanek (1995)]
CLASS1933+503	1.86+0.17-0.11	[Cohn, Kochanek, McLeod, & Keeton (2001)]
<      >	1.98

A third argument is that a great many nearby galaxies have been studied, and their potentials are consistent with isothermals. This is nicely shown in a figure published by [Romanowsky & Kochanek (1999)]. For a sample of twenty bright ellipticals, the circular velocity inferred from the velocity dispersion declines only slightly over a factor of 30 in radius. The corresponding decline in PG1115+080, computed from its measured central dispersion and the Einstein ring radius, is very much larger. It is nonetheless true that the potentials for nearby ellipticals are only very-nearly isothermal and not perfectly isothermal. The central concentration degeneracy qualifies as another major difficulty associated with time delay estimates of H₀.