We will now talk about gravitational 
lensing, which is a very powerful 
technique to find about distribution of 
mass in the universe, whether it's 
visible or not. 
As you will recall from the general 
relativity, it predicts that the presence 
of the mass, again whether you can see 
it, invisible light or not, will distort 
the paths of the light rays passing by. 
Therefore we have a method of learning 
about mass distribution, regardless of 
whether we can see it in any way. 
this was originally considered, 
considered by Chowlson and Einstein, and 
also Zwicky, but it took until the 1979 
to find the first real example. 
This was measurement of a quasar, which 
we have to have 2 images which are just 
mirages. 
They're both the same quasar and after 
that hundreds if not thousands of lenses 
have been found. 
So, the geometry shown here, if we have a 
perfect alignment of the observer, the 
background source and the lens. 
And distributional symmetric, then each 
azimuth will be equal and, and so the 
image of the background object will be 
now distorted in to a ring, called the 
Einstein ring. 
If we deviate from that symmetry, either 
by moving the lens a little bit in the 
lens plane, or by changing the mass 
distribution to be asymmetric, the ring 
will break up, 
and we'll see multiple images of the lens 
source. 
and this is indeed what's seen in a case 
of multiple image gravitationalized 
quasar. 
So Zwicky already in 1937 predicted that 
gravitational lensing should not only be 
detected, but it can be used to probe 
mass distribution in galaxies in the 
universe, as well as to use clusters or 
galaxies as telescopes to magnify more 
even distant ones and after all, it can 
also be used to test the theory of 
gravity, all of those predictions have 
come true. 
As far as measuring masses using 
gravitational lensing, that took until 
the 1990's, and Hubble Space Telescope, 
in order to be able to really do this 
well. 
These are Hubble Space Telescope images 
of a particular cluster, that serves as a 
gravitational lens. 
All of these blue arches, arcs and 
arclets are images of background galaxies 
that have been lensed by this cluster, 
split and distorted, and thus the cluster 
itself is essentially a gigantic lens in 
space that magnifies and distorts their 
images. 
From the distribution and shape of these 
arcs and arclets, one can reconstruct the 
mass distribution in the cluster 
regardless of where the galaxies are. 
So the gravitational bending of light in 
general of theory is twice what you heard 
from the utonian gravity. 
Most importantly if it's achromatic, 
protons in any wavelength are deflected 
in exactly same way. 
And this is unlike say, plans that were 
made out of box. 
This adds to the power of the method 
because it's been used not just in 
invisible light but in radio and also 
x-rays. 
Obviously the effects would be the 
strongest for the strongest gravitational 
field. 
and the strongest you can get is if you 
compress the lens mass into a black hole. 
Then you get maximum effect right at 
Schwarzschild radius, 
but in reality that doesn't happen. 
Most of the physical objects in galixies 
are clusters are vastly larger than their 
Schwarzschild radii, so the effect isn't 
quite as strong. 
Here is a simple schematic geometry 
shown. 
It is fairly obvious there is observer, 
the lengths and the source in the 
background and destortion of the light 
path shown here that straight lines must 
really curve can be respresented as the 
angle by which, between the direction to 
the lens and the lens image. 
All these angles are small, usually arc 
seconds, or even fractions thereof, and 
so small angle approximations apply. 
There is some fairly straightforward 
geometry, that then, you can derive from 
here, and in the end, you have a 
quadratic equation that describes the 
deflection of light rays, in this 
particular geometrical setting. 
We can simplify that expression a little 
bit, by introducing Acquainted he called 
the Einstein radius, it's really an 
angle, angular radius if you will and 
it's given by this formula. 
Essentially this is the characteristic 
radius in projection away from the lens 
where distortions will be strongest and. 
It may not be a perfect circle, depending 
on the mass distribution, so the, the 
line that connects points to satisfy this 
criterion, regardless of whatever the 
mass distribution is, is called critical 
line. 
Generally speaking, for deflections 
within Einstein radius or thereabouts, 
the, you'll see multiple images, that's 
called strong lens. 
for deflections further out you'll see 
only distortions but generally not 
multi-split images, that's called the 
weak lensing regime. 
So quadratic equation is not shown very 
simply and it has 2 obvious solutions 
that, this is high school math. 
And so, in the cases where a lens isn't 
perfectly aligned, there are at least two 
solutions, corresponds to, to different 
images. 
One inside, and one outside the Einstein 
radius. 
I mentioned different lensing regimes. 
If the deflection is strong, hence the 
image splitting, that is called strong 
lensing regime. 
however, deflections will occur even 
further out, and obviously, the further 
out you go, the weaker the effect. 
But, nevertheless, they'll be detectable 
in the form of image distortion. 
And that's called the weak lensing 
urgings, that turns out to be a very 
important for studies of cluster mass 
distribution. 
If you can map all of the distorted 
images, you can, in principle, 
reconstruct the distribution of mass that 
led to the lensing This has been done on 
galaxies as well as cluster with galaxies 
we're talking about something in the of 
the order of an arc second. 
Observations with Hubel telescopes were 
essential to do this. 
A particular group called SLACS 
collaboration is one of them. 
they studied fairly large sample of 
galaxies acting as lenses to some 
background galaxies that appearing as are 
thin lenses or as segments thereof, and 
with some reasonable assumptions they can 
infer the mass distributions in 
projection for the lensing galaxies. 
The results are sh, shown schematically 
here for a handful of cases. 
The lines on the plots describe density 
distribution as a function of radius, for 
visible matter, which is just measured 
stars, or other massive stars, for, the 
whole thing that causes the distortion, 
and the difference which represents 
contribution of the dark matter halo. 
as you can see, a smaller radii, visible 
material dominates, something we already 
learned from kinematical studies. 
As you go further out in radius, the dark 
matter component becomes inevitable, and 
dominates the geometry. 
We can simulate the effects of 
gravitationalizing, by taking an image of 
the sky, placing fictitious lens of given 
mass distribution in front, and seeing 
what it will do to the picture behind it. 
From that, we can then learn how to 
invert actual observations to derive 
distribution of mass That is doing the 
lensing. 
These are simulations so you can see that 
clearly that the effects are the 
strongest in the middle and get weaker 
and weaker the further out you go. 
This is another simulatio used to 
illustrate this. 
Imagine that somehow there is a gigantic 
graph paper in the sky. 
and then you put a cluster of galaxies in 
front of it, then it will distort the 
geometry of the background as shown here. 
It is complicated because it's not a 
simple, simple mass distribution. 
There are many different galaxies. 
Each of them acts like little lensing, 
in, you know, of itself, and together 
they have a collective effect. 
So inverting these geometries can be a 
very complicated business. 
Nevertheless, the metal is well 
understood now and a lot of good data 
have been obtained. 
Many clusters have been studied using 
this technique. 
Here is an example of a particular 
cluster, where different contours show, 
say, average distribution of galaxy 
light, 
or the number density, or the inferred 
galaxy mass, or mass distribution for the 
whole cluster. 
This was done for quite a number of 
clusters. 
A small subset is shown here, 
and lensing gives a new independent 
estimate of the lensing mass in a 
cluster, regardless of whether it's 
baryons, or dark baryons, or dark matter 
or anything. 
As long as it's some matter of any kind 
that exerts gravitational pull, it will 
contribute. 
And remarkably enough, the inferred mass 
to light ratios for clusters, are in 
excellent agreement with those we 
inferred from using very old theorem 
arguments, from either X-ray gas or 
Galaxy velocity dispersion. 
This is completely different physics and 
different kind of measurement, 
and this is why we believe, that indeed, 
there is some kind of dark mass there. 
We see its effects not just in kinematics 
of test particles like galaxies or 
protons or electrons but also in 
manifestations of general activity. 
The geometry of the images seen as the 
lensing mass. 
For this reason, people really believe 
that we are, indeed, dealing with dark 
matter, 
and it's not something like the 
distortion of gravity, 
because that will spoil gravitational 
lensing, 
and you never get the same result as you 
would for measurements using variable 
care. 
This is one thing, for example, that 
Monde cannot possibly reproduce. 
This kind of study is, is now done not 
just for big clusters of galaxies, but 
just in general deep field surveys, where 
we measure a lot of faint galaxies, and 
sometimes get measurements of their 
redshifts. 
The, what's shown here is a snapshot from 
one such survey called COSMOS, and what's 
show on the left is the smoother 
distribution of light that's observing 
galaxies. 
What's shown on the right is the 
reconstructed distribution of mass in 
projection that is determined from the 
distortions of the images of galaxies in 
the background. 
Because, they measured distance's to 
galaxies, then they can actually 
constructed 3-dimensional dark matter in 
space, and the resolution isn't very good 
still, but here it is. 
It's actually remarkable that we can map 
mass distribution in space, because the 
physiological distances in full 
3-dimension's, even though we don't see 
the mass. 
This is likely to become a very powerful 
tool in our underst, our understanding of 
structure formation. 
Next we will talk about Gravitational 
Microlensing. 
That's lensing of stars by stars and it 
was used to constrain contributions of 
matches to the dark mass.