We now turn to the concept of the dark 
matter which is where the most of the 
matter in the universe is. 
The original discovery of the dark matter 
was due to Fritz Zwicky in 1930's who did 
a very simple experiment. 
He looked at galaxies in Coma cluster, 
some of the velocities were measured. 
From that, he could derive the kinetic 
energy in the coma cluster. 
From virial equilibrium that told him how 
much total mass there has to be in order 
to bind galaxies together. 
And that turned out to be several hundred 
times more than the actual amount of mass 
he could see in stars. 
Nobody quite believed it at the time. 
Never the less he was completely right. 
And the more modern versions of same 
measurement bore him correct. 
The reason why we believe that dark 
matter must be there is not just this one 
thing. 
There are many different methods that 
leads to more or less the same result. 
So here is how this works. 
In a system in a gravitational 
equilibrium or virial equilibrium, the 
kinetic energy should be 1/2 of the 
potential energy to it inside and for 
galaxies as test particles probing 
cluster potential, we can look at 
velocity dispersion as means of their 
kinetic energy per unit mass that has to 
be balanced by the potential energy due 
to the gravitational traction. 
And that's how we can estimate mass of 
clusters of galaxies. 
We measure the velocity dispersion of 
galaxies through the red shifts. 
Assuming that there is a tropic so there 
is just, the radial motion is represented 
to other 2 outer components. 
Correct for that, measure the mean 
projected separation of galaxies, plug it 
in this formula and derive cluster mass. 
Typical cluster velocity dispersion are 
several hundred kilometers per second, 
even a couple thousand kilometers per 
second, for galaxies moving within the 
cluster potential well. 
Typical mean separation cluster radii are 
of the, between galaxies, are of the 
order of a few megaparsecs. 
Plugging these two, in, numbers, into the 
equation above, gives us cluster masses 
of the order of 10^14 or 10^15 solar 
masses. 
At the same time, we can simply add up 
luminosity in galaxies, and find out that 
the net mass to light ratio, that is mass 
of stars plus the mass of dark matter, 
derived by the luminosity from stars, is 
of the order of several hundred, exactly 
what Zwicky originally predicted. 
So clusters must be dominated by huge 
amounts of dark Dark matter. 
A different way to measure that, would be 
from the x-ray gas they contain. 
In addition to galaxies and dark matter, 
cluster's also contain copious amount's 
of intergalactic gas. 
This is gas that was partly ejected from 
galaxies, due to the super nova 
treatments stellar winds, and partly 
simply leftover gas from the original 
formation of structure fell into the 
potential well. 
In any case, any test particle whether 
it's a proton or electron or entire 
galaxy has to move with same average 
velocities to balance the gravitational 
potential. 
So instead of hundreds of galaxies, we 
have untold numbers of protons and 
electrons serving as test particles. 
The kinetic energy then translates into a 
temperature which we can measure from 
extra observations. 
And thus we can find out what the 
velocity dispersion of particles is in 
the cluster. 
Turns out to be the same as in the case 
of galaxies. 
If we assume the clusters are an average 
in hydrostatic equilibrium, then that 
gives us a means of, the, the body think 
their masses directly, from, and actually 
their temperature and their density 
profiles. 
And the result is the same, the typical 
masses of clusters are in the range of 
10^14, or 10^15 solar masses, only about 
5% of which is luminous mass, that we see 
in stars and x-ray gas. 
Only about, 1/6 is the stuff in Baryons 
and everything else must be the dark 
matter. 
A slightly different approach to this is 
to measure the Baryonic fraction, in 
clusters of galaxies. 
So the fraction of the total mass in 
Baryons is the sum of fraction in stars, 
frac-, in gas and in dark Baryonic which 
as we already learned are actually form 
of extra gas. 
So if we assume that, 
that's a fair mix, that mix in clusters, 
which are the largest bound structures we 
see, is the same as universe at large, 
then gives us an opportunity of actually 
estimating omega variance or omega dark 
matter. 
We do have to assume Omega variance from 
nuclear synthesis. 
And, that is a fairly reliable 
measurement and from micro background and 
from cosmic nucleus synthesis. 
The physics of that is well understood. 
It's a fairly well established number. 
So roughly speaking, 15 or 20% of mass in 
clusters is in form of x-ray gas, and 
that gives us limits. 
Same time, we can say what is the lower 
limit to be a fraction of Baryons and 
also what is the upper limit on omega 
matter. 
A more detailed analysis which also 
includes mass and galaxies then yields to 
the typical Baryonic fraction clusters 
of, of the order of 15%. 
and the omega matter total for clusters 
of galaxies, which we can then assume is 
not different from universe as a whole on 
the order of 0.25. 
This is in excellent agreement with 
completely different measurements 
obtained by other means that we covered 
in cosmological tests The second major 
piece of evidence for the existence of 
dark matter are the rotation curves of 
these galaxies, spiral galaxies. 
And they all tend to look more or less 
the same. 
There is a rapid rise in the middle and 
then the curve becomes more or less flat, 
all the way we can measure it. 
It's only in the more modern days that 
people manage to measure them so far that 
actually see them decline. 
But for the, most of the extent of 
visible galaxies, by visible, I mean both 
stars and gas, the rotation curves remain 
fairly flat. 
This has a very important consequences. 
We can decompose rotation curves to the 
contributions from different components. 
The gas is moving a circular orbit, 
roughly speaking, and centrifugal forces 
equal centripetal force. 
So, we can derive the total amount of 
mass inside the given radius. 
We could simply add up the mass that we 
see inside that radius in stars or gas. 
And from that, we can see that, up to 
radius of several kiloparsecs, like solar 
radius and milky way, visible stuff does 
not mix, the total irritational 
potential. 
But further out you go, it's contribution 
drops now. 
And something else must be contributing 
the mass which requires high circular 
speeds. 
If you think in terms of say, solar 
system, the velocities of outer planets 
are much slower than the velocities of 
the inner planets, for the reason, that 
the rotational potential in outer regions 
is much weaker. 
And if there were no dark matter, 
something similar would apply to 
galaxies. 
So the argument is really very simple. 
This is the most elementary Newtonian 
dynamics. 
If you have test particles orbiting in 
circles, which is more or less correct, 
we can immediately say, what is the mass 
inside of that circle. 
Now, if we only take luminous mass, we 
can come up with a model that predicts 
what circular velocity should be, as a 
function of radius. 
And, in the inner parts of disks, that's 
not so bad. 
But then in the outer parts the model 
predicts that velocity should be 
declining and it doesn't. 
That means that it has to be, there is a 
missing component in the model. 
So measuring the circular velocity as a 
function of radius gives us a direct 
estimation of the mass inside the radius 
and therefore what is the denisity is a 
function of radius. 
If we assume that it is a power-law, we 
can, and this is a reasonable assumption, 
it turns out, flat rotation curve has 
shown a simple evaluation here, so it's 
very easy to derive that a flat rotation 
curve implies the density law that is 
behaving as r^-2. 
That is called singular isothermal 
sphere. 
It's singular because mass that goes to 
infinity in the middle. 
This is obviously not the case in reality 
but small rounding off in the middle make 
no difference here, in size of thermal 
because the velocity is the same 
throughout. 
If you integrate density distribution 
like this in the sphere, 
you get mass than increases linearly with 
the radius. 
Recall, if the density was constant, the 
mass would increase as a cube of the 
radius, but because that the radius, 
because the density declines as a square 
of the radius then whats left is just one 
power, which means that as far as we can 
trace the rotation curve and spirals. 
And they stay flat. 
The mass keeps increasing and until 
recently we didn't really know where it 
stops. 
You can do very similar thing for 
elliptical galaxies which is essentially 
same thing we've done for clusters. 
There, the orbits of stars are fairly 
random and we cannot make circular orbit 
assumption, we can follow velocity 
dispersion of stars, and that will do 
more or less the same thing. 
But we can also use gas because 
elliptical galaxies do contain copious 
amounts of x-ray gas, and treatment there 
is exactly the same as we did for 
clusters. 
And the answer is that the inner visible 
parts of the elliptical galaxies, 
visible material dominates the 
gravitational potential, but as you go to 
ever larger radii, the contribution of 
the dark matter does increase. 
Instead of the x-ray gas, you can use 
other test particles in case of 
elliptical galaxies, those could be 
globular clusters or dwarf galaxies 
around them. 
And even though their numbers are much 
smaller, the, the, they produce exactly 
the same result. 
There is one kind of galaxy that is 
actually completely dominated by dark 
matter and those are so called dwarf 
spheroidals. 
These are dwarf galaxies, Milky Way has 
on the order of 20 of them orbiting 
around, they do not contain much in terms 
of Stellar mass, indeed, millions to 100 
millions of solar luminosities. 
But, they do seem to contain a lot of 
dark matter, and we evaluate that in 
exactly same way as before. 
We use their stars as test particles, 
probing the gravitational potential, we 
measure their velocity of dispersion, 
function radius and we derive the amount 
of mass needed in order to bind these 
stars to a galaxy. 
The mass to light ratios reached levels 
of 100. 
Recall that for a pure stellar population 
the mass to light ration would be on the 
order of few. 
And so therefore there could be a large 
amount of dark matter in these galaxies. 
There is a reasonable physical model how 
that can happen, that the superrnova 
driven Winds can expel a lot of gas from 
our galaxy, but they do not expel the 
dark matter, because it does not react to 
the pressure it's just non-gravitational. 
And there is one more way in which we can 
do this, and this is by using now 
velocities all the galaxies and 
large-scale structure, overall measuring 
the peculiar velocities of clusters and 
groups inferring what the density field 
is from them to have such velocities and 
then averaging to find our what is the 
overall matter density corresponding to 
such a velocity field. 
The answer to that, again, brings that. 
Omega matter is less than 0.3 but it's 
probably more than 0.2.