>> We have seen how redshift surveys can
be used to describe large scale structure,
at least in the cosmographic sense.
But how do we quantify the distributional
galaxies such that we can compare with
theoretical models?
The first way in which people have done
this is to use the so called two-point
correlation function.
Nowadays, power spectrum is more often
used and the two are actually related in a
very simple fashion.
We will go about that next time.
So, let's talk about the galaxy two-point
correlation function.
What it means is that, if galaxies are
clustered together, they're correlated.
Each galaxy is somehow more likely to be
found next to another galaxy.
And one way to quantify this is to ask the
question.
Assume that galaxies are actually
uniformly, randomly distributed in space.
Then, at a distance from any given galaxy,
there will be a certain probability of
finding another galaxy.
If you actually measure this, you'll find
out, there is an axis.
There are more galaxies near other
galaxies and you'd expect from purely
random distribution.
And that access above the random is what
correlation function is.
So, one simply does the counting of galaxy
pairs for each galaxy and then normalizes
by what the random distribution with the
same number of data points would be.
As it turns out, the two-point correlation
function is well-represented by power-law,
and it's usually written in this form.
That, radius, divided by some scaling
radius, to some power which is close to
minus 1.8.
And typically, for normal galaxies in this
neck of the universe, the scaling length
is about 5 megaparsec.
This, however, is not universal.
Different kinds of galaxies have different
clustering properties, as you'll see
shortly.
So, here is an example of a modern, well
measured, two-point correlation function
from, from the galaxies from, to the F
redshift survey.
It does look pretty close to the
power-law.
Although, if you subtract the best fit
power-law, you'll see that there are some
significant deviations from it.
Now, if you don't have redshifts, you can
measure the angular correlation fraction
just projected in the sky.
And that this two-dimensional projected
correlation function usually [unknown]
looks like little w, not to be confused
with the equation of state parameter, is
related to the three-dimensional
correlation function, xi of r, in a fairly
simple fashion.
The parallax--point is differed exactly by
one because we reduced the dimensionality
of problem from 3 to 2.
Another important point is that if
galaxies are more likely to be found near
other galaxies, there is that axis
probability.
Then, in order to keep the average
constant, it has to turn negative at some
point, and it does.
Scales that are roughly corresponding to
those of voids seen in galaxy
distribution.
If there is a void in distribution, in
some sense, that's anti-correlation.
It's less likely to find galaxy there,
then if it would be, if the space was
uniformly populated with galaxies.
So, how do we do this in practice?
Suppose we have a catalog of galaxies from
some survey.
Then, you can create an, a random catalog
of galaxies.
Same number of galaxies, but randomly
distributed in a Poissonian fashion.
Then, you can do the simple counts.
Count galaxy pairs, one against the other,
and count the fake galaxy pairs, the
random catalog, divide the two and
subtract to one because its the axis
probability.
For large extensive catalog of galaxies
with uniform borders, this will work
fairly well.
But in reality catalogs do have some
incompleteness or uneven borders, and so
on.
So, there is a better estimator, which is
called the Landy-Szalay estimator, and its
formula is given here.
We make a correction by counting galaxy
versus random catalog pairs.
This takes care of the boundary
conditions.
Now, if you're not counting individual
galaxies, but have, essentially, a galaxy
density field where you can divide
galaxies in, in boxes or, or pixels, then
numerical galaxy density can be used in
the same fashion.
Just subtract the expected average from
the actual count n, and divide by the
average.
So, if you do this, for any kinds of pairs
or density pairs, then xi of r is really
the expectation value of this
probabilistic distribution.
Now, strictly speaking, because we are
correlating galaxies with themselves, this
should be called the autocorrelation
function.
But common usage is just to call it
correlation function.
Note, also, that you can correlate sample
of one kind of objects versus the others.
So, for example, you could ask, are
galaxies clustered around quasars?
And, and we can evaluate the
cross-correlation function between
galaxies and quasars, and so on.
By analogy, we can expand this to
higher-order terms.
The three-point correlation function, now
asks, given a galaxy, given a probability
of finding another galaxy at certain
distance, what is now probability of
finding a third galaxy at some other
distance?
This obviously gets lot more complicated
and numerically tedious very fast.
However, there is some useful information
in these high-order correlation functions,
and sometimes they're evaluated.
I mentioned that different kinds of
galaxies can cluster differently, and here
is a simple thing you can do.
You can ask, how are galaxies clustered,
say, bright ones versus faint ones?
And it turns out the bright ones are
clustered more strongly.
The amplitude is higher and the slope is
steeper.
The steepness of the slope obviously
correlates with how strongly they are
clustered.
If the correlation function was perfectly
flat, there would be no correlation.
Understanding effects like this contains
some useful information about galaxy
formation mechanisms.
And we'll talk more about those later in
the class.
Or, you can divide galaxies by
morphological type, say, ellipticals
versus spirals.
It turns out that elliptical galaxies or
redder galaxies are clustered more
strongly than the blue ones, the disk
galaxy.
That, too, has an interesting clue about
galaxy formation and evolution.
So, it's a power-law.
Does that mean it's a fractal?
Remember, for any distribution of points,
the probability of finding another one
from the same set increases the sum
dimensionality of the space.
In normal tedious space, this would be
like cube of radius.
If that number, is not an integer, the set
is called fractal.
So, we can write this formula, which is
resembling correlation function.
And if, indeed, it was pure power law,
then you could say, universe was fractal,
if it was a pure power-law extending to
infinitely large distances.
But in reality, that's not the case.
There are significant deviations from
power-law, it's slightly bent.
And, therefore, universe is not fractal or
large-scale structure is not fractal,
although pretty close to it.
Next time, we'll talk about power spectrum
of galaxy clustering, which can be
directly related to theoretical
predictions.