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?