We now turn to the subject of Evolution of
Galaxy Clustering.
But before we get into that subject
proper, we have to introduce another
concept.
This is the concept of galaxy biasing
which we did mention before.
And the way it works is like this.
In principle, visible material, galaxies,
stars in them, may not be distributed in
exactly the same way as the dark matter.
It turns out that by and large they are,
but at some scales that may not be the
case.
And so we simply Introduce the possibility
that density contrast in variance is some
factor b times the density contrast in the
dark matter.
And, when expressed through the two point
correlation function, then the correlation
function of visible material, is factor of
b squared times the correlation function
of the dark matter.
Note that There is no assumption that one
is stronger than the other.
If B is greater than 1 than baryons, or
visible light are plus or more strongly
than dark matter, it's less than 1 and the
other way.
And so, it is possible that the visible
material is not entirely representative of
the underlying mass distribution, but it
gives you a biases answer and does, does
the name of Galaxy bias.
Now, one, why could that be?
And so one possibility is that as the
structure forms, it forms in density peaks
of the primordial density field and it may
start forming faster or go further in the
densest spots.
So those would be the high peaks of the
density field.
Instead of average or one sigma deviations
or two sigma deviations.
This could be five sigma deviations.
And that turns out that for just about any
type of noise it is true that the highest
density peaks are clustered more strongly.
So here is an example of this.
Lets take just, exaggerated density cut
through a density field in your universe
and you can think of it as small scale
fluctuations riding on top of large scale
fluctuations.
Or large scale fluctuations lifting the
small ones even further.
So if you impose a threshold and saying it
has to be at least this dense for galaxies
to really form, to ignite star formation
and so on, then it's those fluctuations
that are lifted by the large waves.
That will start doing it first.
And, by the construction of this, they
will be clustered strongly together.
And so you'd expect, then, that the first
structure galaxies form in the highest
spot of the density field.
Deepest potential wells.
And that you might see a formation of
proto-clusters on the other hand.
On the hand there could be proto-voids
which are negative fluctuations.
Things that are less dense than the other
places.
Let me give you a metaphor for this.
Imagine asking that one the planet earth,
what are the highest points.
Above the sea level.
And, if you are to put that cut at few
kilometers elevation, you'll find out that
all of them are concentrated in areas like
Rocky Mountains or Himalayas or
Cordilleras and so on.
And then as you slowly lower down the
threshold.
More and more lands enclosed and
eventually the entire surface of the
planet is enclosed.
Same thing here, the highest points will
be by nature clustered together because
they correspond to smaller fluctuations.
A top of larger ones.
Here is a direct illustration of this, not
from real universe, but from simulated one
done by Ray Carlberg.
So, he followed formation of the structure
and then asked, there is an average value
of the density and their deviations of it.
So what is the distribution of all data
points and just those that are higher than
1 sigma above the mean?
2 sigma, 3 sigma and you can see as you
increase the threshold that they get to be
clustered more and more strongly.
They'll pile up in just 1 part of, of the
slice.
So now we can estimate this from redshift
surveys.
The redshift surveys as you recall will
give us actual measurement of the mass
distribution because.
By the coupling Hubble expansion peculiar
velocities, we can find out where the mass
really is that causes galaxies to move.
And, we can also see where the galaxies
are.
So we can directly compare the
distribution inferred from the redshift
surveys with those of galaxies themselves.
And so here is a plot from slong retro sky
survey, and he plots the value of the, of
the bias as a function of galaxy
luminosity.
And you find out that, past certain
luminosity, clustering grows dramatically,
more luminous galaxies are Fostered more
strongly.
And this makes sense in the previously
described scenario because the earliest
forming ones will eventually grow to be
the largest ones and they did form at, at
the highest peaks of the denisty field and
they will be clustered more strongly.
As it turns out, this is also the case
with all redshifts and it's even stronger
at high redshifts.
So now let's look at evolution of
clustering itself.
Generally speaking, you expect clustering
to grow in time as the structures collapse
to density contrast increases.
Remember, it starts as a few parts in a
million at the time of the recombination
and now grows to factor of a million in,
in galaxies themselves or few 100 in a
large scale structure.
So.
The general expectation is that clustering
has to get stronger in time or conversely,
it'll be weaker at higher redshifts.
A way to quantify this is to follow the 2
point correlation function.
And a simple model is shown here, it
changes the aptitude of correlation
function or some power of the expansion
factor 1 plus z.
And it's written in the following form so
different values of this parameter epsilon
correspond to different types of
clustering evolution as.
Indicated here.
If epsilon is equal to minus 1.2, then the
plus string is fixed in comoving
coordinates.
It doesn't change.
It just expands.
If epsilon is 0, then clustering is fixed
in proper coordinates.
The universe expands but the structures
stay just as they are.
And, pushing further if epsilon is
positive, that means the clustering will
grow in time, in proper coordinates
themselves.
And the observation indicate that, that's
in, indeed the case.
But, any one single value of epsilon
describes this process at all different
scales.
So let's look at some data.
What's plotted here is the amplitude of
projected angular two point correlation
function, as a function of depth of the
serving.
Now the deeper you look the fainter
galaxies you see.
And so on average, fainter galaxies are
further away.
At higher edges.
And you can see that this it's exactly
what you expect.
As you go deeper, meaning further, the
strength of the clustering diminishes.
But because we actually have redshifts for
those, we can follow the, follow the
strength of the clustering as a function
of redshift.
And here it's plotted.
As clustering length are not, which you
may recall, is work correlation function
has amplitude of 1 and it's about 5
megaparsec divided by little h, near us.
And, then as you go deeper redshift, it
goes down.
The lower clustering length means weaker
clustering.
And so, here are the redshifts of about
unity or slightly beyond this is exactly
what we expect, but then something
interesting happens.
Going to larger redshifts, past redshift
of 1, the clustering start increasing
again with redshift meaning it's been
decreasing in time, which is exactly the
opposite of what you expect.
So, even in the deepest redshift survey
measurements over very small fields going
very deep, pencil being surveys, we see
spikes, as you may recall by having pencil
being survey going through large scale
structure, spikes correspond to
intersections with filaments or void in
sun.
And here you see them occurring of
redshifts of three or four and even
beyond.
Turns out, the strength of clustering back
then when the universe was only a couple
gill, couple giga-years old is about the
same as it is today.
So the clustering got weaker, and then got
stronger again.
And it's not just galaxies.
You could use quazars, which we can see
very far away, and they tell the same
story.
Beyond redshifts of one, going to higher
redshifts, clustering strength increases.
Exactly the opposite from your What you
expect from naiive picture of structure
formation.
So how can this possibly be?
And the answer is it's due to the
evolution of bias itself.
And here is how it works.
Consider the fate of fluctuations of
different amplitude, here shown with thin
black lines and dashed red lines the
highest fluctuations say 5 sigma lines
will evolve fastest, they will reach
higher values sooner, the lower contract
fluctuations will get there eventually but
takes longer time Now if the structures
form according to the biased contrast,
then as you look back is you'll be looking
at different type of fluctuations.
Nearest you might be looking at 1 or 2
signal fluctuations to see which.
Clustered however they are.
If you work, be, if you could follow them
to higher redshifts, you would see that
they, their clustering indeed was lower at
larger redshifts.
But you don't.
You'll see structures that are
corresponding to higher contrast as you go
to deeper redshifts.
Against two sigma peaks, you may be
looking at five sigma peaks because that's
where galaxies are back then.
And those peaks were clustered more
strongly, in other words, at different
redshifts, you're looking at a different
sample of objects and this is again why
this is called biased galaxy clustering.
So this is almost certainly what's
happening.
By working at high redshifts, we're
looking at the highest peaks of the
density field and they're strongly
clustered and then as you look at lower
ratchets, then you see lower contrast
peaks and they're clustered less.
And this is why there is an apparent
effect of clustering getting weaker in
time.
It's not.
It's just that you are changing the sample
to which you're looking at, from say 5
sigma peaks to 3, 2 sigma peaks and so on,
but all of them increase.
In time.
Each of these density contreras, get ho,
stronger in time, it's just that the
highest ones get there first.
And indeed, this can be proved with large
deep regid surveys and here is a result
from one of them from European Sultan
Observatory deep survey.
It shows the value of the effective, or
average, biased parameter for galaxies,
starting from retch of unity to about 1
and a half.
And you can see that it was higher in the
past.
So galaxies near us are almost an unbiased
tracer of underlying mass.
Near us when we look at large scale
structures, as painted by galaxies we see
exactly how the dark matter is distributed
as well.
But as we go deeper in the past, higher
ridges, then we see a.
A more biased set of tracers.
Galaxies are more clustered than the
underlying dark matter back then because
it's those that are higher density
contrast that lighted up and so these are
what you see.
So to summarize the evolution of
clustering and bias, the generic
expectation is that clustering grows in
time, and it does.
But the rate at which it does depends on
the mass or the density contrast of the
initial conditions.
The type of the objects that you're
following.
And it's fastest for the highest-contrast
ones.
We can quantify this through the evolution
of the two point correlation function, as
shown here, and observationally, we can
follow clustering to the highest reaches
we can probe.
What we observe is always light, not mass.
And so this is my introduce concept of
biasing to allow for a possibility that
light, visible parts of galaxies, are
clustered in a different way from the
underlying mass distribution.
The simplest way of quantifying this is
saying that they're proportional and the
factor of proportionality is B, and it's
greater than 1.
If light is clustered more strongly than
mass, and it turns out usually is.
Now B is not really a constant, it is a
function of all manner of things, not just
time, but also the density contrast itself
and so on.
And there is no antalytical theory for its
change but it is something that we can
reproduce for numerical simulations and
then compare to the observations.
And it turns out that the low redshifts,
galaxies are pretty unbiased tracer of the
underlying mass.
Galaxies have formed fully by and large.
Of course they keep merging and so on, but
as you look at higher redshifts, galaxies
become an increasingly more biased tracer
of the underlying mass distribution.
That is, the deeper in the past you go,
the more biased sample of the higher
density peaks you see.
Next, we'll talk about clusters of
galaxies.