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We'll talk now about source counts, which 
is an old cosmological test which never 

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really did much for cosmology itself. 
But it's very useful for studies of 

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galaxy evolution. 
And the idea here is that this is really 

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volume-redshift test. 
Normally we're thinking about comparing 

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distance to redshift but we may as well 
compare volumes. 

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And we could in principle measure change 
in volume as a function of redshift if 

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there was a population of objects 
uniformly filling the space. 

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If we could measure distances to them 
then that would be all we needed, however 

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obtaining redshifts is very expensive and 
instead of that we use number counts 

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versus flux. 
Note. 

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That there is an assumption here again 
that the source population isn't evolving 

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which, of course, is not really true for 
anything that we know. 

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The test has been applied in radio 
astronomy as well as in optical 

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astronomy. 
Actually in the radio astronomy, it did 

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see its first cosmological use, which is 
to try to distinguish the steady state 

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cosmology from the big bang models. 
But that too, was a subject of 

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evolutionary effects because populations 
of radial sources have all been tried. 

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Nowadays, we know that there are way too 
many evolutionary effects for this to 

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really be useful as a cosmological tool, 
but it does provide some hints. 

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Because it is a useful tool for studies 
of galaxy evolution, let's find out what 

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cosmological background of it is. 
Now, assume non-expanding simple 

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Euclidean space, populated uniformly with 
some kind of sources. 

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Their number will increase as the cube of 
the distance. 

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But the fluxes will decrease as the 
inverse square of the distance. 

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Thus, the number will scale as a flux to 
-3/2 power, and that is what we call 

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Euclidean source counts. 
Since the exact same scaling would apply 

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to sources of any intrinsic luminous 
bundles, even mix of them say Galaxies of 

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different brightness. 
If the mix is the same all the time that 

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too will behave in a similar fashion. 
But in practice what we do a differential 

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counts, how many per unit flux or other 
per unit magnitude. 

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And that is obtained thoroughly from 
original formula. 

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So for magnitudes of the coefficient will 
be different. 

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But nevertheless the principle is the 
same. 

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Now in relativistic cosmology, things get 
a little more complicated, because there 

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is the dependence of both volume element 
and fluxes on cosmological parameters in 

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a non-trivial fashion. 
And this is written here. 

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In general case this will have to be 
integrated numerically. 

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The point is, however, that both numbers 
of sources And their fluxes depend on 

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cosmological, but as it turns out for all 
reasonable cosmological models the slopes 

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can only deviate in one way from the 
Euclidean. 

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Let's see if we can illustrate that. 
So here is a schematic outline, what 

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source counts might look like, say as a 
function of magnitude. 

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Very near us. 
Things are close to Euclidean and 

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therefore the counts will be a 
asymptotically going to the straight line 

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with the slope we just derived. 
But then the further out we go the 

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relativistic effects become more 
important and the line peels off from the 

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Euclidean asymptote. 
There are two reasons why sources go 

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fainter, the first one is the luminosity 
distances the one plus z factor if you 

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recall because of that alarm, regardless 
more or less regardless of cosmology the, 

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The sources will be fainter in an 
expanding universe then they would be in 

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a equivalent euclidean space. 
In addition to that there will be a 

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effective K-correction and if you recall 
for galaxies in visible light more or 

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less, those tend to be positive. 
Galaxies tend to look dimmer. 

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Further away they are because that's the 
shape of their spectrum. 

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In some special cases like in 
sub-millimeter, K-corrections can be 

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negative and can overcome some of the 
geometrical effects. 

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Now let's look at the effect of 
cosmology. 

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Generally speaking, models with more 
volume would tend to have more counts. 

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And models with more volume are those 
that have low densities and/or positive 

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cosmological constant. 
But again, low distances from us makes us 

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still nearly Euclidean. 
So the lines will be deviating. 

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Away from the, Euclideas slope to further 
up we go. 

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Now, let's look what galaxy evolution 
does. 

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It turns out, does something very 
similar. 

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Both luminosity evolution, meaning say 
galaxies were brighter in the past 

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because there were more younger stars or 
density evolution meaning there were more 

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smaller pieces then merged Tend to bend 
the line in similar fashion. 

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But in both cases the counts will be 
higher than for the relativistic case 

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when there is no evolution. 
And you can see why this is. 

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If the sources were brighter in the past, 
than those which were really faint by 

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coming closer the words the Euclidean 
slope line, and so that will tend to push 

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it above the no evolution cosmological 
line, and you can catch the density 

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evolution, where there is simply more 
faint pieces further out, and therefore 

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that will also be above. 
The cosmological line with no evolution 

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and the only way we can tell is to 
actually measure redshifts and to find 

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out as a function of redshift itself, how 
does the density change. 

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So here is some actual galaxy counts, 
these are from Hubble space telescopes 

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and they are more or less, the deepest we 
ever got, about 29th magnitude. 

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Here they're shown in four different 
filters. 

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The behavior is fairly similar. 
But the axes above the new evolution is 

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always the highest in the bluer bands. 
And that's because the bluer bands are 

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more sus, more susceptible. 
To presence of young stars and so your 

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galaxies were evolving so stellar 
populations were aging. 

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Then you expect that there'll be more 
effect in the ultraviolet to then say in 

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the red. 
A simple extrapolation of these counts 

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through the entire sky suggest that there 
are about hundred billion galaxies within 

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the observable universe. 
That probably doesn't count a whole 

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number of little dwarf galaxies. 
Here is an example of galaxy counts in 

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blue filter from several different 
groups. 

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There is a good mutual agreement and the 
counts are above the new evolution 

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prediction. 
The amount of deviation is again an 

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indicator or of galaxy evolution. 
The question then is can we minimize the 

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effects of galaxy evolution by going to 
say mid inferred where stellar 

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populations do not evolve very much. 
And that has been done, the counts then 

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favor models with low density and or 
high, and or positive cosmological 

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constant. 
But then again because the evolutionary 

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effects could not be entirely discounted 
this was not seen as a direct evidence 

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for presence of dark energy. 
Or low density universe. 

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A sum of different source counts involve 
galaxy clusters. 

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Galaxy clusters form in time in fact in 
fact they are forming today and so 

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counting them as a function of red shift. 
Is sensitive to cosmology in two 

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different ways. 
First the same way the galaxies are, but 

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second the longer time they had to form 
the more clusters we will see. 

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And this turns out to be actually far 
more sensitive to cosmology then just 

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simple galaxy counts. 
So if there are more massive clusters at 

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high redshifts, that means they had. 
More time to form and that favors low 

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density or positive ethological constant 
universes. 

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So next time, we will talk about the 
cosmic concordance. 

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How all different measurements converge 
to tell us about parameters, the 

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cosmological parameters of the universe 
today.