B.6.6. Magnification Bias
The optical depth calculation suggests that the likelihood of finding that
a zs 2
quasar is lensed is very small
(
~ 10-4) ,
while observational surveys of bright quasars typically find that of
order 1% of bright quasars are lensed. The origin of the discrepancy
is the effect known as "magnification bias" (Turner
[1980]),
which is really the correction needed to account for the selection of
survey targets from flux limited samples. Multiple imaging
always magnifies the source, so lensed sources are brighter than the
population from which they are drawn. For example, the mean
magnification of all multiply imaged systems is simply the area over
which we observe the lensed images divided
by the area inside the caustic producing multiple images because the
magnification is the Jacobean relating area on the image and source
planes, d2
= |µ|-1 d2
.
For example, an SIS lens with Einstein radius b produces multiple
images over a region
of radius b on the source plane (i.e. the cross section is
b2),
and these images are observed over a region of radius 2b on the image
plane, so the mean multiple-image magnification is
<µ> = (4
b2) / (
b2) = 4.
Since fainter sources are almost always more
numerous than brighter sources, magnification bias almost always
increases your
chances of finding a lens. The simplest example is to imagine a lens which
always produces the same magnification µ applied to a
population with number counts N(F) with flux F. The
number counts of the lensed population are then
Nlens(F) =
µ-1
N(F / µ), so the fraction
lensed objects (at flux F) is larger than the number expected
from the optical depth if fainter objects are more numerous than the
magnification times the density of brighter objects. Where did the extra
factor of magnification come
from? It has to be there to conserve the total number of sources or
equivalently the area on the source and lens planes - you
can always check your expression for the magnification bias by
computing the number counts of lenses and checking to make sure that the
total number of lenses equals the total number of sources if the optical
depth is unity.
Real lenses do not produce unique magnifications, so it is necessary to work
out the magnification probability distribution P( >
µ) (the probability of a
magnification larger than µ) or its differential dP /
dµ and then convolve
it with the source counts. Equivalently we can
define a magnification dependent cross section,
d / d
µ =
dP / d µ
where
is the total
cross section. We can do this
easily only for the SIS lens, where a source at
produces two
images with a total magnification of µ = 2 /
with
µ > 2 in the multiple image region (Eqns B.21, B.22),
to find that P( > µ) = (2 /
µ)2 and
dP / dµ = 8 / µ3. The
structure at low magnification depends on the lens model, but all sensible
lens models have P( > µ)
µ-2 at high magnification because this
is generic to the statistics of fold caustics (Part 1,
Blandford & Narayan
[1986]).
Usually people have defined a magnification bias factor
B(F) for sources of flux F so that the probability
p(F) of finding a lens with flux F is related to the
optical depth by p(F) =
B(F). The
magnification bias factor is
![]() |
(B.115) |
for a source with flux F, or
![]() |
(B.116) |
for a source of magnitude m. Note the vanishing of the extra
1 / µ factor when using logarithmic number counts
N(m) for the sources rather than
the flux counts N(F). Most standard models have
magnification probability distributions similar to the SIS model, with
P( > µ)
(µ0 / µ)2 for
µ > µ0,
in which case the magnification bias factor for sources with
power law number counts N(F) = dN / dF
F-
is
![]() |
(B.117) |
provided the number counts are sufficiently shallow
( < 3). For
number counts as a function of magnitude
N(m) = dN / dm
10am (where
a = 0.4(
- 1))
the bias factor is
![]() |
(B.118) |
The steeper the number counts and the brighter the source is relative to
any break between a steep slope and a shallow slope, the greater the
magnification bias. For radio sources a simple power law model
suffices, with
2.07 ± 0.11 for
the CLASS survey (Rusin & Tegmark
[2001]),
leading to a magnification bias factor of B
5. For quasars,
however, the bright quasars have number counts steeper than this critical
slope, so the location of the break from the steep slope of the bright
quasars to the shallower slope for fainter quasars near B ~ 19
mag is critical to determining the
magnification bias. Fig. B.45 shows an example
of a typical
quasar number counts distribution as compared to several (old) models for
the distribution of lensed quasars. The changes in the magnification bias
with magnitude are visible as the varying ratio between the lensed and
unlensed counts, with a much smaller ratio for bright quasars (high
magnification bias) than for faint quasars (low magnification bias) and
a smooth shift between the two limits as you approach the break in the
slope of the counts at B ~ 19 mag.
For optically-selected lenses, magnification bias is "undone" by extinction
in the lens galaxy because extinction provides an effect that makes lensed
quasars dimmer than their unlensed counterparts. Since the
quasar samples were typically selected at blue wavelengths, the rest
wavelength corresponding to the quasar selection band at the redshift
of the lens galaxy where it encounters the dust is similar to the U-band.
If we use a standard color excess E(B - V) for the
amount of dust, then the images become fainter by of order
AU E(B - V) magnitudes where
AU
4.9. Thus, if lenses had an average extinction of only
E(B - V)
0.05 mag, the net
magnification of the lensed images
would be reduced by about 25%. If all lenses had the same
demagnification factor f < 1 then the modifications to the
magnification bias would be straight forward. For power-law number counts
N(F)
F-
,
the magnification bias is reduced by the factor
f
and a
E(B - V) = 0.05 extinction leads to
a 50% reduction in the magnification bias for objects with a
slope
2 (faint quasars) and
to still larger reductions for bright quasars. Some examples of the
changes with the addition of a simple mean extinction
are shown in the right panel of Fig. B.45,
although the
levels of extinction shown there are larger than observed in typical
lenses as we discuss in Section B.9.1.
Comparisons between the
statistics of optically-selected and radio-selected samples can be
used to estimate the magnitude of the correction. The only
such comparison found estimated extinctions consistent with the
direct measurements of Section B.9.1
(Falco, Kochanek & Muñoz
[1998]).
However, the ISM
of real lenses is presumably far more complicated, with a distribution
of extinctions and different extinctions for different images which
may be a function of orientation and impact parameter relative to the
lens galaxy, for which we have no good theoretical model.
The flux of the lens galaxy also can modify the magnification bias for faint quasars, although the actual sense of the effect is complex. The left panel Fig. B.45 shows the effect of dropping lenses in which the lens galaxy represents some fraction of the total flux of the lensed images. The correction is unimportant for bright quasars because lens galaxies with B < 19 mag are rare. In this picture, the flux from the lens galaxy leads to the loss of lenses because the added flux from the lens galaxy makes the colors of faint lensed quasars differ from those of unlensed quasars so they are never selected as quasars to begin with. Alternatively, if one need not worry about color contamination, then the lens galaxy increases the magnification bias by supplying extra flux that makes lensed quasars brighter.
Any other selection effect, such as the dynamic range allowed for flux ratios between images as a function of their separation will also have an effect on the magnification bias. Exactly how the effect enters depends on the particular class of images being considered. For example, in the SIS lens (or more generally for two-image lenses), a limitation on the detectable flux ratio 0 < fmin < 1 sets a minimum detectable magnification µmin = 2(1 + fmin) / (1 - fmin) > µ0 = 2. Since most lens samples have significant magnification bias, which means that most lenses are significantly magnified, such flux limits have only modest effects. The other limit, which cannot be captured in the SIS model, is that almost all bright images are merging pairs on folds (or triplets on cusps) so the image separation decreases as the magnification increases. The contrast between the merging images and any other images also increases with increasing magnification - combined with limits on the detectability of images, these lead to selection effects against highly magnified images. This is also usually a modest effect - while magnification bias is important, the statistics are dominated by modestly magnified systems rather than very highly magnified images. In fact, there are have been few attempts at complete studies of the complicated interactions between finding quasars, finding lenses, selection effects and magnification bias. There is an early general study by Kochanek ([1991b]) and a detailed practical application of many of these issues to the SDSS survey by Pindor et al. ([2003]). Unfortunately, Pindor et al. ([2003]) seem to arrive at a completeness estimate from their selection model that is too high given the number of lenses they found in practice. Some of this may be due to underestimating the luminosity of lens galaxies, the effects of the lens galaxy or extinction on the selection of quasars or the treatment of extended, multicomponent lenses compared to normal quasars in the photometric pipeline. These difficulties, as well as the larger size of the present radio-selected lens samples, are the reason that almost all recent statistical studies have focused exclusively on radio lenses.
The standard magnification bias expressions (Eqns. B.115 and B.116)
are not always appropriate. They are correct for the statistics of
lenses selected from source populations
for which the total flux of the source (including all images of a lensed
source) is defining F (or m). This is true of most
existing surveys - for example
the CLASS radio survey sources were originally selected from single dish
observations with very poor resolution compared to typical image
separations (see Browne et al.
[2003]).
If, however, the separation of the images is large compared to the
resolution of the observations
and the fluxes of the images are considered separately, then the bias
must be computed in terms of the bright image used to select sources to
search for additional images. This typically reduces the bias. More
subtle effects can also appear. For example, the SDSS survey selects
quasar candidates based on the best fit point-source magnitudes, which
will tend to be an underestimate of
the flux of a resolved lens. Hence the magnification bias for lenses found
in the SDSS survey will be less than in the standard theory. Samples
selected based on more than one frequency can have more complicated
magnification biases depending on the structure of the multidimensional
number counts (Borgeest, von Linde & Refsdal
[1991],
Wyithe, Winn & Rusin
[2003]).
The exact behavior is complex, but
the magnification bias can be tremendously increased if the fluxes in the
bands are completely uncorrelated or tightly but nonlinearly correlated.
For example, if the luminosities in bands A and B are related by
tight, nonlinear correlation of the form
LA
LB1/2,
then the lensed examples of these objects will
lie off the correlation. At present, there are too few deep, wide-area
multiwavelength catalogs to make good use of this idea, but this is
changing rapidly.
In general, the ellipticity of the
lenses has little effect on the expected number of lenses, allowing the use
of circular lens models for statistical studies that are uninterested in the
morphologies of the images (e.g. Keeton, Kochanek & Seljak
[1997],
Rusin & Tegmark
[2001],
Chae
[2003]).
However, the effects of ellipticity are trivially observable in the
relative numbers of two-image and four-image lenses.
We noted earlier that the expectation from the cross section
is that four-image lenses should represent order
2
~
2
~ 0.01 of lenses
where
is the
ellipticity of the lens potential. Yet in
Section B.2 we saw that four-image lenses
represent roughly one third
of the observed population. The high abundance of four-image lenses
is a consequence of the different magnification biases of the two-image
multiplicities - the four-image lenses are more highly magnified than
the two-image lenses so they have a larger magnification bias factor.
Fig. B.46
shows the image magnification contours for an SIS lens in an external shear
on both the image and source planes. The highly magnified regions are
confined to lie near the critical line. If we Taylor expand the inverse
magnification radially, then µ-1 =
x|
dµ-1 / dx| where
x is the distance
from the critical line, so
the magnification drops inversely with the distance from the critical line.
If we Taylor expand the lens equations, then we find that the change in
source plane coordinates is related to the change in image plane coordinates
by
=
µ-1
x
µ-2. Thus, if L is the
length of the astroid curve, the probability of a magnification larger than
µ scales as P( > µ)
µ-2L / | dµ-1
/ dx|. This applies only to the
four image region, because the only way to get a high magnification in the
two-image region is for the source to lie just outside the tip of a cusp.
The algebra is overly complex to present, but the generic result is that
the region producing magnification µ extends
µ-2 from the cusp tip but has a width that scales
as µ-1/2, leading to an
overall scaling that the asymptotic cross section declines as
P( > µ)
µ-7/2 rather than P( > µ)
µ-2. This can all be done formally (see
Blandford & Narayan
[1986])
so that asymptotic cross sections can be derived for any model
(e.g. Kochanek & Blandford
[1987],
Finch et al.
[2002]), but
a reasonable approximation for the four-image region is to compute the
magnification, µ0, for the cruciform lens formed
when the source is
directly behind the lens and then use the estimate that
P( > µ) = (µ0 /
µ)2. Unfortunately,
such simple estimates are not feasible for the two-image region. These
distributions are relatively easy to compute numerically, as in the
example shown in Fig. B.47.
![]() |
Figure B.47. The integral magnification
probability distributions for a singular isothermal ellipsoid
with an axis ratio of q = 0.7 normalized by the total cross
section for finding
two images. Note that the total four-image cross section is only of order
|
Because the minimum magnification of a four-image lens
increases µ0
-1
even as the cross section decreases as
4
2,
the expected number
of four-image lenses in a sample varies much more slowly with ellipticity
than expected from the cross section. The product
4
B(F)
2
µ0
-1, of the four-image cross section,
4,
and the magnification bias, B(F), scales as
3-
for the
CLASS survey (
2), which is a much
more gentle dependence on ellipticity
than the quadratic variation expected from the cross section. There is
a limit, however, to the fraction of four-image lenses. If the
potential becomes too flat, the astroid caustic extends outside the
radial caustic (Fig. B.18), to
produce three-image
systems in the "disk" geometry rather than additional four-image lenses.
In the limit that the axis ratio goes to zero (the lens becomes a line),
only the disk geometry is produced. The existence of a maximum four-image
lens fraction, and its location at an axis ratio inconsistent with the
observed axis ratios of the dominant early-type lenses has made it
difficult to explain the observed fraction of four image lenses
(King & Browne
[1996],
Kochanek
[1996b],
Keeton, Kochanek & Seljak
[1997],
Keeton & Kochanek
[1998],
Rusin & Tegmark
[2001]).
Recently, Cohn & Kochanek
([2001])
argued that satellite galaxies of the lenses provide the explanation by
somewhat boosting the fraction of four-image lenses while at the same
time explaining the existence of the more complex lenses like B1359+154 (Myers et al.
[1999],
Rusin et al.
[2001])
and PMNJ0134-0931 (Winn et al.
[2002c],
Keeton & Winn
[2003])
formed by having multiple lens galaxies with more complex caustic
structures. It is not, however, clear in the existing
data that four-image systems are more likely to have satellites to the lens
galaxy than two-image systems as one would expect for this explanation.
Gravitational lenses can produce highly magnified images without multiple
images only if they are highly elliptical or have a low central density.
The SIS lens has a single-image magnification probability distribution of
dP /
dµ =
2
b2 /
(µ - 1)3 with µ < 2 compared to
dP /
dµ = 2
b2 / µ3 with
µ
2 for the
multiply imaged region,
so single images are never magnified by more than a factor of 2. For
galaxies, where we always expect high central densities, the only way
to get highly magnified single images is when the astroid caustic
extends outside
the radial caustic (Fig. B.18).
A source just outside an exposed
cusp tip can be highly magnified with a magnification probability
distribution dP / dµ
µ-7/2. Such single image magnifications
have recently been a concern for the luminosity function of high redshift
quasars (e.g. Wyithe
[2004],
Keeton, Kuhlen & Haiman
[2004])
and will be the high magnification tail of any magnification
perturbations to supernova fluxes (e.g. Dalal et al.
[2003]).
As a general rule for galaxies, the probability of a single image being
magnified by more than a factor of two is comparable to the probability
of being multiply imaged.