Finally, let us address the question of, 
is the universe really expanding? 
The reason we think its expanding is the 
existence of Hubble's Law, and this is 
generally accepted to be true. 
However, there are other possibilities, 
hence there is a so-called Tired Light 
theory, which states that for some reason 
that as yet is completely unknown, 
photons coming from very far away lose 
their energy on the way here in a way 
that's proportional to their travel 
distance. 
There is no physics behind it, but it's a 
possibility. 
So, several tests have been actually 
designed to demonstrate that universe 
actually is expanding. 
The first of those is so called Tolman 
Test, which Tolman and Hubble came up 
with in 1930s, and it uses behavior as 
surface brightness as a function of 
distance. 
And here is how it works. 
If the Universe wasn't expanding at all, 
nothing else funny was going on. 
Surface brightness will be constant, not 
depending on the distance because 
remember, the luminosity of the clients 
was the square of the distance, 
but so does the area of the angular areal 
over which its distributed so the ratio 
of the two remains constant. 
In Tired Light Theory, the brightness 
will decline with redshift as first power 
of one plus redshift. 
And finally, in an expanding Universe, it 
will decline as the fourth power of 
stretch factor one plus redshift. 
The second method uses time dilation of 
Supernova light curves. 
Recall that those can be standardized for 
type 1A Supernova to the same shape. 
Now, these Supernova or rather their host 
galaxies are receiving from us with 
relativistic speeds, and so clocks are 
ticking slower there. 
So, we have to compensate for the time 
dilation. So, in addition to the stretch 
factor that brings them all together, 
light curves have to be compensated for 
time dilation that's proportional to the 
first power of, of the stretch factor 1 + 
z And finally, somewhat indirect 
argument, is the black body nature and 
temperature of the cosmic 
micro-background. 
In an expanding Universe, the shape of 
the black body curve is preserved and the 
energy density has to scale as the fourth 
power of temperature. 
If the Universe wasn't expanding, that 
relationship would not be exactly power 
of four. 
But we do see essentially perfect black 
body radiation, 
which is perfectly consistent with an 
expanding Universe. 
So, here's how Tolman Test works. In non 
expanding Euclidian space, surface 
brightness is constant and does not 
depend distance. Because luminosity 
declines a second power distance and so 
does the angular area over which we're 
dividing it to get surface brightness. 
However, in an expanding Universe, we 
have to deal with angular diameter 
distance and luminosity distance. 
As you recall, the angular diameter 
distance is equal to the physical 
distance divided by one plus redshift 
because of the objects fixed and proper 
coordinates do not expand with co-moving 
coordinates. And luminosity distance is 
bigger than co-moving distance by factor 
one plus redshift because the photons 
lose energy and the rate of the photon 
emission is also direct. 
So the upshot of this is that surface 
brightness will decline as one plus 
redshift to the 4th power relative to 
what it would be if the Universe wasn't 
expanding. 
Note, that this has nothing to do with 
curvature of space or anything else. 
It simply tests whether universe is 
expanding or not, and assumes that 
special relativity is valid and nobody's 
doubting that. 
So, in some sense, it's completely 
independent of cosmology of Hubble 
constant or all cosmological paramaters. 
In order to perform this test, we need 
something that can be seen far away that 
has a constant surface brightness, what 
we may call standard fuzz. 
One good choice is the surface brightness 
intercept of the fundamental plane 
correlations. 
Remember, they connect things like radii, 
velocity dispersion, 
and means surface brightness of galaxies 
in what's essentially perfect correlation 
module of the Earth, module of measument 
of the Earth. 
So, we can reproject it in such a way 
that one axis has surface brightness on 
it, and the intecept on that axis will be 
defining the standard fuzz. 
So, if we have two clusters of galaxies, 
one larger distance than the other, 
and we compare the intercepts for their 
own fundamental plane solutions, 
they should shift according to the 
expansion law. 
This test was done, 
and here is the result. 
This is a log log plot so the power law 
is a straight line, 
and the line that's drawn through the 
points here is exactly one plus redshift 
to the -4 power. 
So, the Universe does really seem to be 
expanding exactly as Tolman Test would 
say it would. 
Now, the time dilation of Supernova light 
curves. 
On, shown on the left here is a set of 
light curves as observed for the 
normalized to the same peak brightness 
for Supernova of type 1A, a different 
redshift. 
On the right, we apply the stretch factor 
to, that's normally used to standardize 
them. 
However, no correction was made for 
realistic time dilation. 
In the lower left now, we see what 
happens when we apply the relativistic 
time dilation correction, the scatter 
goes way down. 
And then, if we, of course, bend the 
points, it becomes very obvious. 
Thus, Supernova light curves a sense of 
giant clocks, do behave exactly in the 
way that should if the Universe was 
expanding. 
Another way to show this result is to 
plot this characteristic width of 
Supernova light curves before and after 
relativistic correction and before you 
can see there is a residual trend, 
after the distribution is flat. 
Which means that is the way it should be. 
So, next week, we'll start talking about 
cosmological tests. 
How do we actually figure out in what 
kind of Universe do we live?