Finally let us address the subject of the power spectrum of large scale structure.
You may recall that to introduce this concept when we talked about premordial
density field. And now just like then some familiarity
with free analysis is absolutely essential.
So if you are familiar with Fourier Analysis, the main subject of today's
lecture, will be almost trivial. And if you are not, it will make no sense
whatsoever. So this would be like really good time to
refresh your knowledge of Fourier Analysis.
Done? Okay, well that was fast.
So let's get going. So we can characterize the density field
of large scale structure in the same way we did before.
So lets take this Fourier transform here is the formula.
It's essentially almost the and the inverse transform is given by this.
So in this way spacial scales, a wave number scale are connected and the power
spectrum of that defluctuation is obviously a complex product of spectrum by
itself. Now recall what was the value of two point
correlation function expressed from density field of the expectation value.
In other words, this is exactly what it is.
So, correlation function is a transform of the power spectrum.
Power spectrum and correlation function are a Fourier pair.
They are completely equivalent to each other mathematically.
And here is an example for what they look like from Las Campanas Redshift Survey
that you may recall, what is shown here is both 3 dimensional 2 point correlations
functions and power spectrum corresponding to the same large scale structure.
You may recall that 2 point new point correlation function was fairly easy to
evaluate in any one of several ways. And, but power spectrum is really what
theory produces. So, one can be translated into the other.
Now, we usually express the power spectrum on a log lock axis, and theoretical
models, fitted against observations, give us the shape of it.
However, the amplitude has not been defined.
There are two ways in which that can be done.
One is to measure fluctuations so that the density field at the particular scale.
And typically, sphere of radius of H, H to the minus 1 megaparsecs is used because
that gives the RMS of fluctuations close to unit.
Turns out it's a little less than that for modern measurements but traditionally 8, 8
megaparsecs is used. So, whatever the value of that is gives us
unit normalization of the power spectrum at that spatial scale.
Essentially what's been done here is you're convolving what's technically
called a spherical top-hat filter but it's really nothing but a sphere with a power
spectrum itself. An alternative way is to simply use
observations of cosmic microwave background which give us power spectrum on
very large scales. So if you have a model like cold dark
matter model, you can fit it on those scales, evolve for the appropriate linear
growth of fluctuations, and that works just as well.
So, here is one of the modern renderings of the power spectrum of large scale
structures. Spanning the range all the way from super
horizon sizes, down to the scales of galaxies.
The plot assembles several different kind of measurements.
They all brought to the same redshift by scaling in linear regime.
At large scales cosmic microwave background is what's done.
That overlaps with measurements of clustering from large scale structure from
rigid surveys and such. And then going to smaller scales, we used
things like abundance of clusters of galaxies, that's essentially, statistics
of high density peaks and clustering of Lyman alpha clouds which are sub-galactic
fragments. Weak gravitational lensing reflects the
nasty distribution of the dark matter and so that too can be used to probe the
distribution of CDM on intermediate scales.
And the line going through at points is the standard cold dark matter model.
As you can see, it's a remarkable good fit.
And also an excellent agreement between these completely different ways of
observing it, in the regions where they overlap.
So this gives us a real confidence that we actually, you know at some level of
certainty what's going on and one of the major reasons why people believe CDM model
is correct one. Just as an aside numerical called baryonic
acoustic oscillations. And this is actually how they were
detected. By looking at the two point correlation
function or power spectrum at scales in the vicinity of hundred megaparsecs.
So this an absolutely essential use of this uh,, concept as a cosmological test
that can be, be used to improve our knowledge of cosmological parameters.
So far we talked about powerless spectrum, that is the distribution of amplitudes of
dense defluctuations. But in that process, as in computing power
spectrum, we'll completely neglect phase information.
Now here is a dramatic illustration of why phase is information is important,
produced by [unknown] And it's a toy model.
The density model that is really [unknown] form kind of mimicking the filamentary
structure that we see in large scale structure.
What was done here is free transform was taken and then the phases were randomly
scrambled, leaving the power spectrum identical.
Then transformed back, with scrambled phases, and you see the plot on the
bottom. Looks like a complete noise.
The coherence is completely gone. So whereas the distribution of
fluctuations is the same. Their arrangement is completely different.
The phase coherence is been lost in this computation, and obviously we have to
somehow learn how to use statistics of the phases of the density field power spectrum
or rather the Fourier spectrum in order to characterize it fully, because when you
look at large scale structure, you don't see just lumpy density field.
You do see a measure of elements and voids and what not.
Amazingly enough, so far nobody has come up with a really simple, elegant, complete
way of describing this so it might be a little.
So we discussed clustering of galaxies in some detail but what about clustering of
clusters of galaxies? Or in fact you can discuss clustering of
any elements of large scale structure galaxies, groups, clusters, super
clusters, and so on. As it turns out clusters also cluster and
their clustering is stronger than that of galaxies.
Actually there is an excellent correlation between the strength of clustering, say
measured through clustering length, and the richness of the system at which you're
looking. The richer clusters are clustered more
strongly. This is actually a very interesting
result, and at first it was puzzling why should that be so.
Until Nick Keiser came up with the explanation that this is due to phenomenon
now called bias. You can consider density field, and so
there are some high peaks and low peaks. Remember it can be also be composed in
waves of large wavelengths and small wavelengths fluctuations riding on top of
large wavelength. So if you impose a threshold for a
formation of certain amount, a kind of object say five sigma cut.
Then the highest peaks will be strongly clustered because the smaller fluctuations
are carried together by the large waves. And those that happen to be a crest of a
big wave will be all clumped together. This is in same sense as if you were to
ask what are the highest points on the surface of planet Earth.
And if you put a cap a few kilometers elevation above the sea level, you'll see
they are very strongly clustered where the high mountains are.
In this case say Himalayas and[UNKNOWN] and so on.
Whereas if you lower down the threshold. To the ground level, then everything will
be more or less uniformly distributed. So this is a simple explenation.
At the highest density peaks, always cluster more strongly than the lower
density peaks And this applies to galaxies as well.
We find out the denser galaxies, elliptical galaxies say, plus their more
strongly than spiral galaxies. And I've shown you part of that we've seen
earlier. This is the explanation why.
It also has some implications. The highest density peaks will be the
first ones to form, at small scales. This we'll come back to this phenomenon
when we talk about galaxy formation and evolution.
So let us recap everything we learned about large scale structure.
So we see a range of scales from galaxies which are scales of a kiloparsecs through
groups and clusters in scales of megaparsecs to superclusters on scales of
maybe hundred parsecs. We measure it, largely, by using Redshift
series of galaxies, and today we have Redshift for couple million galaxies,
which gives us really excellent insights into what large scale structure looks like
as a function of Redshift. But clustering itself is not enough.
There's also lot of interesting topology going on.
Coherent structures like filaments and voids and sheets that separate them in We
understand why this happens. You may recall that aspherical fluctuation
will first turn around and collapse along one axis while expanding in the other two,
making things like sheets or, or walls, then will turn around on an intermediate
axis, collapse into a filament-like structure, and eventually all drain into,
again, to. So this is the origin of the large scale
topology, although we still don't have good ways of characterizing it.
Clustering itself is customarily characterized through 2-point correlation
function or equivalently the power spectrum.
The 2-point correlation function has a fairly robust functional form.
It is well represented. Not perfectly, but well over large range
of scales as a parallel with slope of minus 1.8 in three dimensional space.
For galaxies, the normalization given through clustering length is of the order
of five Megaparsec or so. For clusters, it's larger, responding to
larger amplitude of clustering. And whereas we can easily measure a 2
point correlation function for any kind of objects.
To compare them with theory, we actually have to turn 2 point correlation function
into a power spectrum. And fortunately the two are equivalent,
and therefore a pair, so we know how to compute that.
Overall, the cold dark matter model, fits the data remarkably well over a full range
of scales that we can measure through a variety of different methods.
Which gives us confidence that cold dark matter model is, in fact, correct.
And I should know to this point, that would be very difficult to achieve this if
there were, were no dark matter at all. Generally speaking, objects of different
kinds have different clustering strengths that applies to different kinds of
galaxies. Luminous versus less luminous.
Early versus late types, and so on. Carrying over to the clusters of different
regions, and so on. Next we will talk about large-scale
peculiar velocity field, which is a direct consequence of the existence of
large-scale density field.