We'll talk now about source counts, which is an old cosmological test which never
really did much for cosmology itself. But it's very useful for studies of
galaxy evolution. And the idea here is that this is really
volume-redshift test. Normally we're thinking about comparing
distance to redshift but we may as well compare volumes.
And we could in principle measure change in volume as a function of redshift if
there was a population of objects uniformly filling the space.
If we could measure distances to them then that would be all we needed, however
obtaining redshifts is very expensive and instead of that we use number counts
versus flux. Note.
That there is an assumption here again that the source population isn't evolving
which, of course, is not really true for anything that we know.
The test has been applied in radio astronomy as well as in optical
astronomy. Actually in the radio astronomy, it did
see its first cosmological use, which is to try to distinguish the steady state
cosmology from the big bang models. But that too, was a subject of
evolutionary effects because populations of radial sources have all been tried.
Nowadays, we know that there are way too many evolutionary effects for this to
really be useful as a cosmological tool, but it does provide some hints.
Because it is a useful tool for studies of galaxy evolution, let's find out what
cosmological background of it is. Now, assume non-expanding simple
Euclidean space, populated uniformly with some kind of sources.
Their number will increase as the cube of the distance.
But the fluxes will decrease as the inverse square of the distance.
Thus, the number will scale as a flux to -3/2 power, and that is what we call
Euclidean source counts. Since the exact same scaling would apply
to sources of any intrinsic luminous bundles, even mix of them say Galaxies of
different brightness. If the mix is the same all the time that
too will behave in a similar fashion. But in practice what we do a differential
counts, how many per unit flux or other per unit magnitude.
And that is obtained thoroughly from original formula.
So for magnitudes of the coefficient will be different.
But nevertheless the principle is the same.
Now in relativistic cosmology, things get a little more complicated, because there
is the dependence of both volume element and fluxes on cosmological parameters in
a non-trivial fashion. And this is written here.
In general case this will have to be integrated numerically.
The point is, however, that both numbers of sources And their fluxes depend on
cosmological, but as it turns out for all reasonable cosmological models the slopes
can only deviate in one way from the Euclidean.
Let's see if we can illustrate that. So here is a schematic outline, what
source counts might look like, say as a function of magnitude.
Very near us. Things are close to Euclidean and
therefore the counts will be a asymptotically going to the straight line
with the slope we just derived. But then the further out we go the
relativistic effects become more important and the line peels off from the
Euclidean asymptote. There are two reasons why sources go
fainter, the first one is the luminosity distances the one plus z factor if you
recall because of that alarm, regardless more or less regardless of cosmology the,
The sources will be fainter in an expanding universe then they would be in
a equivalent euclidean space. In addition to that there will be a
effective K-correction and if you recall for galaxies in visible light more or
less, those tend to be positive. Galaxies tend to look dimmer.
Further away they are because that's the shape of their spectrum.
In some special cases like in sub-millimeter, K-corrections can be
negative and can overcome some of the geometrical effects.
Now let's look at the effect of cosmology.
Generally speaking, models with more volume would tend to have more counts.
And models with more volume are those that have low densities and/or positive
cosmological constant. But again, low distances from us makes us
still nearly Euclidean. So the lines will be deviating.
Away from the, Euclideas slope to further up we go.
Now, let's look what galaxy evolution does.
It turns out, does something very similar.
Both luminosity evolution, meaning say galaxies were brighter in the past
because there were more younger stars or density evolution meaning there were more
smaller pieces then merged Tend to bend the line in similar fashion.
But in both cases the counts will be higher than for the relativistic case
when there is no evolution. And you can see why this is.
If the sources were brighter in the past, than those which were really faint by
coming closer the words the Euclidean slope line, and so that will tend to push
it above the no evolution cosmological line, and you can catch the density
evolution, where there is simply more faint pieces further out, and therefore
that will also be above. The cosmological line with no evolution
and the only way we can tell is to actually measure redshifts and to find
out as a function of redshift itself, how does the density change.
So here is some actual galaxy counts, these are from Hubble space telescopes
and they are more or less, the deepest we ever got, about 29th magnitude.
Here they're shown in four different filters.
The behavior is fairly similar. But the axes above the new evolution is
always the highest in the bluer bands. And that's because the bluer bands are
more sus, more susceptible. To presence of young stars and so your
galaxies were evolving so stellar populations were aging.
Then you expect that there'll be more effect in the ultraviolet to then say in
the red. A simple extrapolation of these counts
through the entire sky suggest that there are about hundred billion galaxies within
the observable universe. That probably doesn't count a whole
number of little dwarf galaxies. Here is an example of galaxy counts in
blue filter from several different groups.
There is a good mutual agreement and the counts are above the new evolution
prediction. The amount of deviation is again an
indicator or of galaxy evolution. The question then is can we minimize the
effects of galaxy evolution by going to say mid inferred where stellar
populations do not evolve very much. And that has been done, the counts then
favor models with low density and or high, and or positive cosmological
constant. But then again because the evolutionary
effects could not be entirely discounted this was not seen as a direct evidence
for presence of dark energy. Or low density universe.
A sum of different source counts involve galaxy clusters.
Galaxy clusters form in time in fact in fact they are forming today and so
counting them as a function of red shift. Is sensitive to cosmology in two
different ways. First the same way the galaxies are, but
second the longer time they had to form the more clusters we will see.
And this turns out to be actually far more sensitive to cosmology then just
simple galaxy counts. So if there are more massive clusters at
high redshifts, that means they had. More time to form and that favors low
density or positive ethological constant universes.
So next time, we will talk about the cosmic concordance.
How all different measurements converge to tell us about parameters, the
cosmological parameters of the universe today.