B.8.1. Low Mass Dark Halos
When we are examining a particular lens, almost all the substructure
will consist of
satellites associated with the lens, with only a ~ 10% contamination from
other small halos along the line-of-sight to the source
(Chen, Kravtsov & Keeton
[2003]).
However, the excess of low mass halos in CDM mass functions relative to
visible galaxies is a much more
general problem because the low mass CDM satellites should exist everywhere,
not just as satellites of massive galaxies. Crudely, luminosity
functions diverge as dn / dL ~ 1 / L ~ 1 / M
while CDM mass functions diverge as
dn / dM ~ M-1.8 so the fraction of low
mass halos that must be dark increases ~ M-0.8 at low
masses.
Fig. B.51 illustrates this
assuming that all low mass halos have baryons which have cooled
(e.g. Gonzalez et al.
[2000],
Kochanek
[2003c]).
In the context of CDM, the solution to this general problem is
presumably the same as for the satellites responsible
for anomalous flux ratio - they exist but lost their baryons before they
could form stars. Such processes are implicit in semianalytic models which
can reproduce galaxy luminosity function (e.g. Benson et al.
[2003])
but can be modeled empirically in much the same way was employed for the
break between galaxies in clusters in
Section B.7 (e.g. Kochanek
[2003c]).
In any model, the probability of the
baryons cooling to form a galaxy has to drop rapidly for halo masses below
~ 1011
M just
as it has to drop rapidly for halo masses above ~ 1013
M
.
Unlike groups and clusters, where we still expect to
be able to detect the halos from either their member galaxies or X-ray
emission from the hot baryons trapped in the halo, these low mass halos
almost certainly cannot be detected in emission.
We can only detect isolated, low-mass dark halos if they multiply image
background sources. For SIS lenses the distribution of image separations
for small separations (
/
*
<< 1, Eqn. B.112) scales as
![]() |
(B.126) |
where describes the
divergence of the mass/luminosity function at low mass and
FJ
is the conversion from mass to velocity dispersion
(see Section B.6.2). For the standard
parameters of galaxies,
- 1 and
FJ
4, the separation
distribution is
d
SIS /
d
. In practice we do
not observe this distribution because the surveys have angular selection
effects that prevent the detection of small image separations (below
0."25 for the radio surveys), so the observed distributions show a
much sharper cutoff (Fig. B.1).
Even without a cutoff, there
would be few lenses to find - the CLASS survey found 9 lenses between
0."3
1."0 in which case we
expect only one lens with
< 0."3 even in
the absence of any angular
selection effects. A VLBI survey of 3% of the CLASS sources
with milli-arcsecond resolution found no lenses (Wilkinson et al.
[2001]),
nor would it be expected to for normal galaxy populations. Our
non-parametric reconstruction of the velocity function including
selection effects confirms
that the existing lens samples are consistent with this standard model
(Fig. B.50).
The result is very different if we extrapolate to low mass with the
- 1.8
slope of the CDM halo mass function. The separation distribution becomes
integrably divergent,
d
SIS /
d
-0.6, and we
would expect 15 lenses with
< 0."3 given 9
between 0."3
1."0. Unfortunately, the
Wilkinson et al.
([2001])
VLBI survey is too small
to rule out such a model. A larger VLBI survey could easily do so,
allowing the lenses to confirm the galaxy counting argument for the
existence of second break in the density structure of halos at low
mass (Kochanek
[2003c],
Ma
[2003])
similar to the one between galaxies and high mass halos
(Section B.7).
If the baryons in the low mass halos either fail to cool, or cool and
are then ejected by feedback, then their density distributions should
revert to those of their CDM halos. If they are standard NFW halos, Ma
([2003])
shows that such low mass dark lenses will be very difficult to detect
even in far larger surveys than are presently possible. Nonetheless,
improving the scale of searches for
very small separations from the initial attempt by Wilkinson et al.
([2001])
would provide valuable limits on their existence.
The resulting small, dark lenses would be the same as the dark lenses we discussed in Section B.7.2 for binary quasars and explored by Rusin ([2002]). They will also create the same problems about proving or disproving the lens hypothesis as was raised by the binary quasars with the added difficulty that they will be far more difficult to resolve. Time delays, while short enough to be easily measured, will also be on time scales where quasars show little variability. Confirmation of any small dark lens will probably requires systems with three or four images, rather than two images, and the presence of resolvable (VLBI) structures.