**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 *H*_{0} 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.

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 *H*_{0}.