Module 9.1.1: Large Scale Structure Observations and Redshift Surveys
So, we have seen how large-scale structure is supposed to form theoretically, but now, let us
take a look at what observations have to say.
[Slide 2] The basic paradigm, of course, is that density fluctuations from the early universe
collapse into structures that we see today and that formation continues through hierarchical
merging of smaller pieces into ever larger ones. So, clearly, we distinguish between galaxies
which are obviously seen as unit things in space and are dissipatively formed structures
achieving density contrast of the order of a million relative to the average density, which
clearly puts them apart from large-scale structure itself, consisting of assemblies of really
dark matter with galaxies used as tracers. So by large-scale structure, we talk about galaxy
groups, filaments, clusters, sheets, superclusters, and that is what we will be addressing now.
Note that the basic assumption here is that we do know that most of the mass is in form of
non-baryonic dark matter and that most of the baryonic matter we don't really see directly. So
we hope, and this turns out to be actually a pretty good assumption, that galaxies, visible
ones, trace the large-scale structure that's actually there in an invisible material.
The basic way of quantifying the large-scale structure is to first map it out, find out where all
the galaxies are and see what their density field is like. To do this, we obviously measure two
coordinates in the sky and we need a third, radial coordinate. This is obtained through
redshifts, assuming of course that Hubble expansion works. So, we need to do large redshift
surveys, measuring redshifts for as many galaxies as possible in order to trace the large-scale
structure.
It was recognized already in the early 20th century that galaxies are not randomly distributed
in space, but rather, they seem to congregate. However, it took until about 1970s to really
start quantifying this.
[Slide 3] So in 1930s, Shapley and Zwicky, who started the first systematic sky surveying,
noted that galaxies do seem to form large structures. Shown here on the left is one of the
contour maps of projected galaxy densities on the sky from one of Zwicky's atlases.
The quantitative analysis of galaxy clustering was started in 1950s by Shane and Wirtanen,
astronomers who counted galaxies per unit area on photographic sky plates taken at Lick
Observatory. Those so-called Lick galaxy counts turn out to be a basic material for a lot of
the early theoretical studies of large-scale structure. From 1950s to 1970s, more and more
attention was paid to it. Gerard de Vaucouleur in particular noticed what we now call the
Local Supercluster, and more and more galaxy redshift surveys were conducted. The redshift
surveys really came into prominence in the 1980s and beyond, and today we have redshifts
for maybe 2 million galaxies or so.
[Slide 4] Here is a simple projection on the sky of the 6,000 brightest galaxies. Two things
are immediately apparent. There is a belt where galaxies are absent and that is called the
zone of avoidance. We now understand that this is due to the interstellar dust absorption in
our own galaxy that we simply cannot see very much through the dusty interstellar medium
in the Milky Way's disk. Outside of that, you see that there is some structure.
Now, if you were to make a plot of the 6,000 brightest stars in the sky, it would be exactly the
opposite. They are all concentrated in what is the zone of avoidance of the galactic world, the
angular disk of the Milky Way, and otherwise, they'd be fairly randomly distributed, on the
rest of the sky.
[Slide 5] So here is the 3D picture of our immediate extragalactic neighborhood, the so-called
Local Group of galaxies. Milky Way and Andromeda are the two dominant galaxies in the
group and there is a large number of their dwarf galaxy satellites. Groups of galaxies are
indeed where most of the galaxies are. So they are the most common building block of larger
scale structures.
[Slide 6] We mentioned the Local Supercluster earlier. This was noted as an elongated
structure in the sky and that we are kind of in its plane. The center is on Virgo cluster, the
nearest cluster of galaxies to us and it's roughly 60 megaparsecs in scale. As superclusters go,
this is not a very large supercluster, but it is the one in which we belong.
In addition to the commonly used equatorial or ecliptic or galactic coordinates from
astronomy, we now define so-called supergalactic coordinates. The XY plane of the
supergalactic coordinates is the plane of the Local Supercluster, and we are at its origin, and
the vertical Z direction is orthogonal to it. Later on, we will see plots of some of the
distributions projected onto this XY plane of the supergalactic coordinates.
[Slide 7] So here is the rough 3D sketch of what the Local Supercluster might look like. This
plot is centered on the Local Group of galaxies, but really the center of the Local Supercluster
is at Virgo. The rest of the structures are various galaxy groups or individual galaxies.
Actually, there are very few isolated galaxies, most of them are in groups. It is not a
perfectly flat system like the disk of the Milky Way, but it is flat. Two other clusters that are
sort of peripheral to the Local Supercluster are Fornax and Eridanus. They are on sort of the
other side of the sky from Virgo.
[Slide 8] Now this becomes a little more apparent if we plot not 6,000 but 15,000 brightest
galaxies in the sky. This was typical of galaxy catalogs until say 1990s, where those are
galaxies selected by eye or on photographic sky survey plates. Here we can see that there is a
great circle shown here through these arches that does go through a lot of galaxies, and that is
indeed the plane of the Local Supercluster. To the upper left, sort of, you see a structure
called Centaurus or Hydra-Centaurus, this is another supercluster of galaxies, and in fact, it's
bigger than the Local Supercluster and we're falling into it. We'll talk more about that when
we talk about peculiar velocities. On the other side is Perseus-Pisces Supercluster, roughly
the same distance. So, superclusters themselves tend to form slightly more coherent
structures, and this is not surprising from everything that we know about large-scale
structure.
[Slide 9] So the mapping of large-scale structure begins with catalogs of galaxies on the sky
and those were counted by eye, well, originally by Messier and Herschel and so on. But,
really, the more modern ones begin with Shapley and Ames in 1930s and then the Lick
galaxy counts in 1950s, and there are famous catalogs like Uppsala General Catalog of
Galaxies and the Southern Extension. Still, the Shane-Wirtanen Lick counts were the largest
up until 1980s. They counted close to 1 million galaxies in the sky, which is why this was a
good statistical sample for the early studies. The next step came with the digitization of sky
survey plates, and that was done for both northern and southern skies, starting sometime in
the late 1980s. And APM is for Automated Plate Measuring machine in Cambridge, England
and people who did this work produced a catalog of roughly 2 million galaxies. The northern
version of it, from the Second Palomar Sky Survey, the so-called Digital Palomar
Observatory Sky Survey or DPOSS, counted about 50 million galaxies.
By the mid 2000s, Sloan Digital Sky Survey obtained much higher quality data for a couple
hundred million galaxies at first. Nowadays, they all have almost a billion, both stars and
galaxies. I n the future, we'll look towards even larger sky surveys and Large Synoptic
Survey Telescope or LSST, which certainly will not be completed in 2015, this is an old
slide, more like 2020, we'll probably map billions of galaxies.
But that's just pictures in the sky. In order to get the third dimension, redshifts are needed.
And the first modern redshift survey was probably the one done at the Harvard-Smithsonian
Center for Astrophysics in the mid 1980s. They obtained redshifts of the order of thousand
nearby galaxies and started to quantify large-scale structure near us. Along with its Southern
Extension, they covered in the order of 2,500 galaxies, this was already good enough to begin
first modern quantitative structures of galaxy clustering. Now you may recall the zone of
avoidance was placing this artificial belt of missing galaxies in the sky, simply due to the
extinction in our own Milk Way. The way to bypass this was to look at infrared, as infrared
light goes through the interstellar dust relatively unobscured.
By the mid 1990s there was a good catalog of infrared selected galaxies from the All Sky
Survey by IRAS satellite. That produced a fairly unbiased set of galaxies that enabled
astronomers to go deep into the zone of avoidance. The resulting redshift surveys now went
up to 9,000 galaxies. Encouraged by their initial success, the Center for Astrophysics
conducted a second generation redshift survey which compiled the redshifts of about 25,000
galaxies. John Huchra was the leading astronomer in that enterprise, and also, his
collaborators including Margaret Geller and others. Roughly at the same time, Carnegie
Observatories did Las Campanas Redshift Survey in the southern hemisphere using one of
the first modern multi-object spectrographs, and they obtained redshifts for a comparable
number, about 23,000 galaxies. These surveys of 1990s started revealing more interesting
features in the galaxy distribution, but really, the state of the art was achieved by two large
surveys, one in Australia, the two-degree field (2dF) redshift survey, and the Sloan Digital
Sky Survey, we'll talk a little more about them later, which together, now mapped well over a
million galaxies.
[Slide 10] Let's now look at the sky in the successive redshift shells as we step out. The
closest one would be shown here. These are galaxies with the recession velocities due to the
Hubble Expansion up to 3,000 kilometers per second, and the biggest feature you see is the
Local Supercluster. Most of the galaxies are being centered on Virgo, but with some pieces
in Fornax. Stepping further out between 3,000 and 6,000 kilometers per second, new
structures come in, the most notable one is the Perseus-Pisces supercluster opposite from
Virgo. This was largely done by Giovanelli and Haynes at Cornell and their collaborators,
who conducted their work in parallel with the CfA group. And again, sort of opposite to
Perseus-Pisces and almost hidden by the Galactic plane, is the Hydra-Centaurus Supercluster,
which later made an appearance under the name of Great Attractor, but we'll talk about that
later. And now, even further out between 6,000 and 9,000 kilometers per second, the biggest
feature is the Coma - Abell 1367 supercluster, composed largely of these two rich clusters.
But there are also some vestiges of others, especially the Perseus-Pisces.
[Slide 11] And so here is a 3D sketch of what our local supergalactic neighbourhood might
look like, with the Local Supercluster and nearby superclusters like Perseus-Pisces and
Hydra-Centaurus. You have probably noticed that superclusters, at least those near us, are
given names by the constellations in which most of the galaxies are placed. Constellations of
course have no physical meaning, but they're still used as a traditional convenience to
designate where in the sky some things might be.
Module 9.1.2: Large Scale Structure Observations and Redshift Surveys, cont.
[Slides 1-2] So, once you get a 3-D distribution, you can project it onto the supergalactic
plane. And here is a density landscape, high peaks corresponding to high projected density
from one of the IRAS-based redshift surveys. So the big set of mountains off from the center
is the Local Supercluster. We are always in the middle, so we're sort of on the slopes of it.
[Slide 3] But why project on the supergalactic plane if you have 3-D picture? And so, here is
a highly smoothed density distribution of galaxies in 3-D from the IRAS Redshift Survey.
Now you see that there are really all these blobs that really are in complex relation to each
other.
[Slide 4] I mentioned a second Center for Astrophysics survey by Huchra and collaborators,
and what they did is they chose a slice in the sky, very thin in declination, long in right
ascension, fan-shaped like that. They did this simply in order to have manageable numbers
of galaxies to observe, they went deeper than the previous survey, and went past the Coma
Cluster of galaxies, and obtained this famous Stickman Diagram. The structure in the middle
is the Coma cluster, and it actually doesn't look like that, it is elongated in radial direction by
the so-called finger of God effect. Namely, remember, that measured velocities of galaxies
are the vector sum of the Hubble expansion velocity and a peculiar velocity the galaxy may
have on its own. Now, in rich clusters of galaxies, galaxies have high velocity dispersions.
They have to move fast in order to balance the gravitational potential of the dark matter. So,
if you look through a cluster of galaxies, suddenly some galaxies will appear to be closer to
us because they're moving towards us relative to Hubble expansion. Some are moving away
from us, and there will be an elongation in what we call the redshift space, as opposed to
actual 3-D space.
So, this is a two-dimensional slice through a three-dimensional structure. And you can see
that galaxies actually form a kind of mesh of filaments, and there are big voids that had been
noticed previous to this. Because this is a two-dimensional cut, then it's reasonable to assume
that, in fact, voids must be really more of a spherical nature, or at any rate, not flat, and that
sheets intersecting through this plane may actually look like filaments. In reality, there are
both sheets and filaments out in the large-scale structure.
[Slide 5] So, there are actually two important effects about the redshift space. And remember,
redshift space has the radial coordinate that is associated with total measure velocity of the
galaxy. And a priori, we do not know which part is due to Hubble velocity, which would be
the real radial coordinate, and which part is due to peculiar motion. Statistically, we can
decompose that later. Then what happens, in rich clusters of galaxies, as I already
mentioned, there will be an elongation towards the observer, due to the radial velocity
dispersion of galaxies moving fast inside a cluster. On the other hand, if we're looking not at
a dense structure like cluster but, say, an intersection with one of the sheets or walls or
filaments, galaxies will be falling towards it. It's not yet a virialized structure, but it's a
density excess, and so galaxies fall to it. So, those which are on our side are falling towards
it, acquiring velocity that makes them look like they're a little further out. Those on the other
side are falling back towards that filament and that will make them look like they're a little
closer. So then, this will squish the structure along the line of sight, the opposite of the finger
of God effect. It all depends on how dense or how virialized is the density that we see. In
cases of higher densities like clusters of galaxies, we see the elongation, the finger of God
effect. In cases of low density enhancements, like filaments, we see the opposite. So, these
effects are well-understood and they have to be deconvolved out of the observations when
performing the full analysis.
[Slide 6] So, the full compilation of CfA2 redshift survey by Huchra et al. went well past
10,000 kilometers per second and they noticed that there is an even larger structure they
called the Great Wall, which contains, in part, the Coma - Abell 1367 cluster, or maybe
supercluster. And it was basically filling up the space available to their survey. Up until
then, every time a redshift survey was done, a structure was seen that was as large as it can be
fit in the survey volume. Which, on face value of it, doesn't bode well for the assumptions of
homogeneity. The resolution of this came with Las Campanas redshift survey, but before I
tell you about that, let me just give you some idea of how these measurements are done.
[Slide 7] In the early days, spectrographs were doing one galaxy at a time. This was a tedious
work and that's what CfA survey was. But then, astronomers designed multi-object
spectrographs, and they come in two varieties. First, we have to know, of course, where the
galaxies are in the sky. In one type, you have a metal plate that's positioned in the focal plane
of a telescope, and a little opening or slit was made where each galaxy should be. Then,
instead of having one spectrum with lot of blank space on each side, you have a whole lot of
little short spectra with object in the middle. The other way is to use optical fibers. Again,
the fiber head is positioned to where the galaxy should be, using some kind of device, usually
a robotic arm. And then, these fibers are grouped together to enter the slit of the spectograph,
and then the spectra are taken together. Both approaches worked fairly well. And this is what
really moved the redshift surveys to industrial strength of hundreds of thousands and now
millions.
[Slide 8] Recall that the second CfA survey was a slice in the sky. A fan-shaped slice, thin
declination, long in right ascension, and they saw all these voids and filaments and stuff. Las
Campanas Redshift Survey did the same thing but they did three slices, and they also went
further out. So, in their survey, they could see structures of same size up to a hundred
megaparsec or so, that were seen by the CfA Redshift Survey as well. But they also went
further, and no larger structures were seen. In other words, homogeneity does seem to apply
on scales larger than hundreds of megaparsecs. And for cosmological purposes, that's
perfectly good enough.
[Slide 9] Two huge redshift surveys really transformed this field. The first one was done by
Anglo-Australian observatory, an Australian-UK Consortium, who used one of those robotic
fiber spectrographs on the 4 meter Anglo-Australian telescope in Siding Spring in Australia.
They collected about a quarter million redshifts and they did the job first. They also observed
a lot of quasars, and so on. Their analysis really started revealing interesting new features of
large-scale structure. At the same time and continuing beyond, was the Sloan Digital Sky
Survey. They used a 2.5 meter telescope in New Mexico, at Apache Point Observatory. And
they did both imaging survey and spectroscopic survey. They did imaging of large areas on
the sky first, selected galaxies, then they used multifiber spectrographs to obtain redshifts of
galaxies, as well as quasar,s and so on. This was spectacularly successful, an earlier release
of the Sloan Digital Sky Survey had something like 800,000 galaxy redshifts, but they did
continue since then. And now, they have of the order of 1 and a half million galaxy redshifts,
and a comparable number of velocity measurements for stars, as well as a couple hundred
thousand quasars. This is truly a fundamental data set on which many of the modern studies
of the large-scale structure are based.
[Slide 10] So, here is the redshift slice projection of the 2dF survey. And it looks sort of like
Las Campanas Redshift Survey, only with much better resolution, because they had an order
of magnitude more galaxies. You can see the mesh of filaments and voids and so on, but you
can also see that on scales larger than about 100 megaparsecs, we do not see super voids or
super filaments or anything like that.
[Slide 11] Here is a picture showing sky coverage from the latest incarnation of Sloan Digital
Sky Survey. They have these strange belts in the sky because they operate in drift scanning
mode. Telescope moves in large circles in the sky as they collect the data. And then they
obtain redshifts along the same areas. This may look a little strange at first. But if you know
exactly where you are pointing, you can certainly take this into account in your analysis.
[Slide 12] And here is the projection of the Sloan Sky Survey. Now again, this squishes 3D
information into a plane that cuts across the sky. And we see the same qualitative features as
before: a frothy kind of large-scale structure, composed of filaments or walls or sheets
intersecting and more or less quasi-spherical voids inside of them, sort of like sponge-like
topology.
[Slide 13] There is always a tradeoff, of covering a large area on the sky, and being complete,
versus going deep. If you have a finite amount of observing time, there are only so many
galaxies you can observe. And so, you can trade one for the other. These large scale surveys
that we described covered large areas, but, they did not go very deep on cosmological scales.
Complementary to them were surveys done over small areas in the sky, but going very deep,
so-called pencil beam surveys, because of the narrow cone. The first of those were done in
1990s, and they saw presence of large-scale structure out of high redshifts, spikes in the
redshift distribution. At first, it seemed like those are periodic but that turn out not to be the
case and now it's believed that the characteristic scale that they saw there was really due to
the baryonic acoustic oscillations at scales of 100 megaparsecs or a little more.
[Slide 14] Here is the little redshift slice distribution from one of these modern survey. This
one is called DEEP and it was done at Keck telescope reaching out to redshift 1.5, or several
times deeper than the large area surveys from before. Now, if you make a histogram of
redshifts along the line of sight, you see these spikes. They correspond to intersections of
clusters and filaments and walls along your line of sight. And the interesting thing is that
structure keeps persisting out to the largest distances we can measure. This leads to an
interesting new phenomenon called biasing, which we'll discuss in a little more detail.
[Slide 15] So, here is a comparison of some of the redshift surveys. There are many ways in
which you can compare redshift surveys. In this particular case, it is the number of objects
measured versus the volume of space that's been covered. Obviously, the deeper you go, the
more volume you get at the expense of taking more spectra, having more objects. Now, as the
SDSS-Main is the actual Sloan Redshift Survey, they had few others. For one of them, they
didn't really actually use spectroscopic redshifts. They used multicolor photometry to derive
statistical estimates of redshifts, the so-called photo-z survey. This turns out to work
reasonably well if you have good photometry and at least several filters. And also they can
look at spectra of quasars, very far away, and look at absorption line clouds along the line of
sight. Those are very numerous and they provide means of probing large-scale structure out
to the highest redshifts.
[Slide 16] Another way to compare them is to look at area coverage versus number density of
objects that are being covered. And so again, there is a trade-off. Now, in this plane, diagonal
lines show the total numbers of objects, as you can see progressing first from thousands to
now millions.
[Slide 17] Next time, we will talk about how we actually quantify galaxy clustering using the
two-point correlation functions.
Module 9.2: Galaxy Clustering: The Two-Point Correlation Function
We have seen how redshift surveys can be used to describe large-scale structure, at least in a
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.
[slide2] 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 this 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 excess: there are more galaxies near
other galaxies than you'd expect from purely [uniform] random distribution. And that excess
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, gamma, which is
close to -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.
[slide 3] So, here is an example of a modern, well-measured, two-point correlation function
from the galaxies from 2dF 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.
[slide 4] Now, if you don't have redshifts, you can measure the angular correlation function
just projected in the sky. And that this two-dimensional projected correlation function,
usually denoted with a 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
power-law exponents differ 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 excess probability. Then, in order to keep the average constant, it has to turn
negative at some point, and it does, at the 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 anticorrelation. It's less likely to find galaxy there, than if it would be, if the space was uniformly
populated with galaxies.
[slide 5] So, how do we do this in practice? Suppose we have a catalog of galaxies from some
survey. Then, you can create a random catalog of galaxies, with the 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 galaxies pairs, from the
random catalog, divide the two and subtract one because it is the excess probability. For
large extensive catalogs 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.
[slide 6] Now, if you're not counting the individual galaxies, but have, essentially, a galaxy
density field where you can divide galaxies in boxes 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 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
a 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.
[slide 7] 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 that 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.
[slide 8] 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 galaxies. That, too, has some interesting clues about galaxy formation and
evolution.
[slide 9] 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 3-D 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 a power-law, it is slightly
bent. And, therefore, universe is not fractal or large-scale structure is not fractal, although it
is pretty close to it.
[slide 10] Next time, we'll talk about power spectrum of galaxy clustering, which can be
directly related to theoretical predictions.
Module 9.3: Large Scale Structure: Power Spectrum
[slide 1] Finally let us address the subject of the power spectrum of large-scale structure. You
may recall that we introduced this concept when we talked about primordial density field.
And now, just like then, some familiarity with Fourier 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.
[slide 2] So we can characterize the density field of large-scale structure in the same ways we
did before. So let's take this Fourier transform. Here is the formula. It's essentially almost
the definition of Fourier transforms. And, the inverse transform is given by this. So in this
way spatial scales and wave number scale are connected, and the power spectrum of density
fluctuations is obviously a complex product of spectrum by itself.
Now recall what was the value of 2-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.
[slide 3] 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 three-dimensional 2-point correlations
function and power spectrum corresponding to the same large scale structure. You may
recall that 2-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.
[slide 4] Now, we usually express the power spectrum on a log-log 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 is at a particular scale. And typically, sphere of radius of 8 h-1
megaparsecs is used because that gives the RMS of fluctuations close to unity. Turns out it's
a little less than that for modern measurements, but traditionally the scale of 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.
[slide 5] 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 kinds of measurements. They are all brought
to the same redshift by scaling in the linear regime. At large scales, cosmic microwave
background is what's being used. That overlaps with measurements of clustering from large
scale structure from redshift 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 density 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 remarkably 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 know at some level of certainty what's going on, and one of the
major reasons why people believe CDM model is correct one.
[slide 6] Just as an aside, you may recall the baryonic acoustic oscillations. And, this is
actually how they were detected. By looking at the 2-point correlation function or power
spectrum at scales in the vicinity of hundred megaparsecs. So this is an absolutely essential
use of this concept as a cosmological test that can be used to improve our knowledge of
cosmological parameters.
[slide 7] So far we talked about power spectrum, that is the distribution of amplitudes of
density fluctuations. But in that process, as in computing power spectrum, we completely
neglect phase information.
Now here is a dramatic illustration of why phase information is important, produced by Alex
Szalay. And it's a toy model, the density model that is really Voronoi foam, kind of
mimicking the filamentary structure that we see in the large-scale structure. What was done
here, is that its Fourier transform was taken, and then the phases were randomly scrambled,
leaving the power spectrum identical, then transformed back, with the 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 has 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 mesh of
filaments and voids, and what not. Amazingly enough, so far nobody has come up with a
really simple, elegant, complete way of describing this, so maybe you will.
[slide 8] So we discussed clustering of galaxies at 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.
[slide 9] 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 wavelength and small wavelength
fluctuations riding on top of large waves. So, if you impose a threshold for a formation of
certain kind of object, say, 5-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 cut at few kilometers elevation above the sea level, you'll see they're
very strongly clustered where the high mountains are, in this case, say, Himalayas and
Cordilleras 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 explanation. 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, cluster more strongly than spiral galaxies. And I've shown you plot 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.
[slide 10] So let us recap everything we learned about large scale structure. So we see a range
of scales from galaxies, which are scales of kiloparsecs, through groups and clusters, on
scales of megaparsecs, to superclusters on scales of maybe hundred megaparsecs. We
measure it, largely, by using redshift surveys of galaxies, and today we have redshifts for
couple of 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 is also lot of interesting topology going on.
Coherent structures like filaments and voids and sheets that separate them, and so on. We
understand why this happens. You may recall that an aspherical fluctuation will first turn
around and collapse along one axis while expanding in the other two, making things like
sheets or walls, then will turn around on intermediate axis, collapse into a filament-like
structure, and eventually all drain into, again, to quasi-spherical one. o 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
power law with a slope of -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 2point 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 note at this point, that would be very
difficult to achieve this if there 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.
[slide 11] Next, we will talk about large-scale peculiar velocity field, which is a direct
consequence of the existence of large-scale density field.