The observations of gravitationally lensed quasars are best understood in light of Fermat's principle (e.g., Blandford & Narayan 1986). Intervening mass between a source and an observer introduces an effective index of refraction, thereby increasing the light-travel time. The competition between this Shapiro delay from the gravitational field and the geometric delay due to bending the ray paths leads to the formation of multiple images at the stationary points (minima, maxima, and saddle points) of the travel time (for more complete reviews, see Narayan & Bartelmann 1999 or Schneider, Ehlers, & Falco 1992).
As with glass optics, there is a thin-lens approximation that applies
when the optics are small compared to the distances to the source
and the observer. In this approximation, we need only the effective
potential,
(
) = (2 /
c2)(Dls / Ds)
dz
, found
by integrating the 3D potential
along the line
of sight. The light-travel time is
![]() |
(1) |
where = (x,
y) = R(cos
,
sin
) and
are the angular positions of the image and the source,
(
) is the effective
potential, (
-
)2 / 2 is the geometric delay in the small-angle
approximation, zl is the lens redshift,
and Dl, Ds, and
Dls are angular-diameter distances
to the lens, to the source, and from the lens to the source,
respectively. The only dimensioned quantity in the travel time
is a factor of H0-1
10h-1
Gyr arising from the
H0-1 scaling of the angular-diameter distances.
We observe the images at the extrema of the time delay function, which
we find by setting the gradients with respect to the image positions
equal to zero,
x
= 0, and finding all the
stationary points
(
A,
B,
...) associated with a given source position
.
The local magnification of an
image is determined by the magnification tensor Mij, whose
inverse is determined by the second derivatives of the time
delay function,
![]() |
(2) |
where the convergence =
/
c is the
local surface density in units of the critical surface density
c =
c2 Ds /
4
G Dl
Dls,
and
and
define
the local shear field and its
orientation. The determinant of the magnification tensor is the net
magnification of the image, but it is a signed quantity depending on
whether the image has positive (maxima, minima) or negative (saddle points)
parity.
A simple but surprisingly realistic starting point for modeling lens potentials is the singular isothermal sphere (the SIS model) in which the lens potential is simply
![]() |
(3) |
is a deflection scale determined by geometry and
is the 1D velocity
dispersion of the lens galaxy. For
|
|
< b, the SIS lens produces two colinear images at radii
RA =
|
|
+ b and RB = b -
|
|
on opposite sides of the lens galaxy (as in
Fig. 1 but with
AB = 180°).
(1)
The A image is a minimum of the time delay and leads the
saddle point, B, with a time delay difference of
![]() |
(4) |
Typical time delay differences of months or years are the consequence of
multiplying the ~ 10h-1 Gyr total
propagation times by the square of a very small angle (b
3 ×
10-6 radians so, RA2
10-11).
The SIS model suggests that lens time delay measurements reduce the
determination of the Hubble constant to a problem of differential
astrometry. This is almost correct, but we have made two idealizations
in using the SIS model.
The first idealization was to ignore deviations of the radial
(monopole) density profile from that of an SIS with density
r-2,
surface density
R-1, and a flat rotation curve.
The SIS is a special case of a power-law monopole with lens potential
![]() |
(5) |
corresponding to a (3D) density distribution with density
r-
, surface density
R1-
, and rotation curve
c
r(2-
)
/ 2. For
= 2 we recover the SIS model,
and the normalization is chosen so that the scale b is always the
Einstein ring radius. Models with smaller (larger)
have less (more)
centrally concentrated mass distributions and have rising (falling)
rotation curves. The limit
3 approaches the
potential of a point
mass. By adjusting the scale b and the source position
|
|,
we can fit the observed positions of two images at radii
RA and RB on opposite sides
(
AB = 180°)
of the lens for any value of
.
(2)
The expression for the time delay difference can be well approximated by
(Witt, Mao, & Keeton
2000;
Kochanek 2002)
![]() |
(6) |
where <R> = (RA + RB) / 2
b and
R =
RA - RB (see
Fig. 1). While the expansion assumes
R /
<R> (or
|
|)
is small, we can usually ignore the higher-order terms.
There are two important lessons from this model.
Image astrometry of simple two-image and four-image lenses generally cannot constrain the radial mass distribution of the lens.
More centrally concentrated mass distributions
(larger )
predict longer time delays, resulting in a larger Hubble constant
for a given time delay measurement.
These problems, which we will address from a different perspective in Section 3, are the cause of the uncertainties in estimates of H0 from time delays.
The second idealization was to ignore deviations from circular symmetry
due to either the ellipticity of the lens galaxy or the local tidal
gravity field from nearby objects. A very nice analytic example of a
lens with angular structure is a singular isothermal model with
arbitrary angular structure, where the effective potential is
=
bRF(
),
and F(
) is an
arbitrary function. The model family includes
the most common lens model, the singular isothermal ellipsoid (SIE).
The time delays for this model family are simply
tSIS,
independent of the angular structure of the lens
(Witt et al. 2000)!
This result, while attractive, does not hold in
general, and we will require the results of
Section 3 to
understand the effects of angular structure in the potential.
1 The deflections produced by the SIS
lens are constant,
| -
|
= b, so the total image separation is always 2b. The outer
image is brighter than the inner image, with signed magnifications
MA-1 = 1 - b / RA
> 0 (a positive parity minimum) and
MB-1 = 1 - b / RB
< 0 (a negative parity saddle point). The model parameters,
b = (RA + RB) / 2 =
<R> and
|
|
= (RA - RB) / 2 =
R / 2, can be
determined uniquely from the image positions.
Back.
2 In theory we have one additional constraint
because the image flux ratio measures the magnification ratio,
fA / fB = | MA| / |
MB|, and the magnification ratio depends on
.
Unfortunately, the systematic errors created by milli- and
microlensing make it difficult to use flux ratios as
model constraints (see Section 5).
Back.