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So far we've discussed how equal ops of 
primordial dense defluctuations would 

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lead to formations of structures in the 
universe, but the fluctuations will have 

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a spectrum of different sizes or masses. 
And so the question is how do we 

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characterize that? The way we do this is 
in the language of Fourier analysis. 

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If you are not familiar or you forgot 
about it, this would be a really good 

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time to refresh your knowledge of it. 
Otherwise, you may find this module a 

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little difficult to follow. 
We will represent the Density Field in 

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the early universe as a superposition of 
waves of different frequencies and 

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different phases. 
In other words, we're decomposing it into 

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a Fourier spectrum in three dimensions. 
The power spectrum of this density field 

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and characterizes the distribution of one 
of different size. 

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It is customary expressed as a function 
of the wave number, the inverse of the 

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special wavelength. 
So we start with the fluctuation density 

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field, the relative fluctuation delta 
that we introduced earlier. 

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We take the Fourier spectrum of this 
field and it's power spectrum is of 

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course, the square of the applicants. 
In this way we can completely 

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characterize the distribution of lumps of 
different size, not of their shapes which 

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is where phasing for mission comes in. 
But instead of using spacial frequency, 

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we can also express it as a function of 
mass. 

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The larger ones obviously having larger 
physical sizes so a larger wavelength. 

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The formula shown here is what's 
customarily used. 

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It is usually assumed that all 
fluctuations have about the same 

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amplitude, of the order of 10 to the -4 
when they enter the horizon. 

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This is more than what you see in cosmic 
micro background. 

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But recall the presence of the dark 
matter is what helps. The dark matter 

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fluctuations have grown already relative 
to those in baryons and photons. 

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So, if all of the fluctuations of 
different frequencies have same 

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amplitude. 
That's called a scale-free spectrum, or 

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Harrison-Zeldovich according to the 
cosmologist who first introduced it. 

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It so happens that this is false so what 
was predicted in inflation theory. 

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More generally, the power spectrum can be 
represented as a power-law of the wave 

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number exponent of n. 
If that is equal to 1, that is the 

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Harrison-Zeldovich spectrum. 
In any case, the fluctuations will always 

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grow, because they attract material near 
them and so the amplitudes will always 

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grow. 
They will grow same way at all sizes 

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during the linear regime, and therefore 
the whole thing just shifts in 

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logarithmic space. 
But the important point is that 

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fluctuation growth by itself does not 
change the shape of the spectrum. 

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Let's look at some illustrative examples. 
These are two-dimensional images that 

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correspond to different spacial 
frequencies. 

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And, our spectra shown in the box to the 
right. They're almost delta functions 

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favoring waves, spacial waves of 
frequencies. 

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And the pictures show that representation 
using random phases and clearly the 

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higher the wave number, the lower the 
wavelength, the smaller the lumps you 

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see. 
In reality, it will be a broader 

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distribution. 
And so, if we allow for a broader 

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spectrum, but with a cut-off at at some 
frequency or some wavelength, then the 

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pictures might look like something like 
this. 

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We talk about several different types of 
density fluctuations. 

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The most commonly used ones are 
so-called, Adiabatic fluctuations. 

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[SOUND] They are the same for photons and 
invariance in terms of number dynasties. 

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But because of the expansion, their 
energy densities change differently. 

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This is also commonly known as one of the 
F noise, or Harrison-Zeldovich spectrum, 

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and it has equal power on all masses. 
It also happens to be exactly what we 

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observe in the cosmic micro background. 
The other kind of perturbations are 

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Isocurvature perturbations. 
Where there are perturbations in the 

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numbers of photons, as well as baryons, 
but in the opposite directions. 

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So they composite each other and so the 
curvature, spatial curvature that 

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corresponds to the mass density is 
unperturbed. 

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And finally there are Isothermal 
fluctuations where the photons are 

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unperturbed does the equal temperature, 
but there are underlying fluctuations in 

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the regular matter. 
This corresponds to so-called white noise 

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and these are no longer used. 
Now the important point here is the 

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different types of dark matter would lead 
to different structure formation 

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scenarios. 
And we can see that through behavior of 

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their power spectrum. 
Recall also that initially it's just the 

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fluctuations in the dark matter that 
matter because baryons are coupled to 

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photons and they only participate in 
growth of fluctuations after the 

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recombination. 
Perturbations can be raised by sound 

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waves. 
Obviously, the density waves will carry 

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the density of particles from more dense 
regions to less dense regions. 

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And thus, they would start the raising 
the fluctuations. 

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This is also known as the Meszaros 
effect. 

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This turns out to be important for 
circled cold dark matter. 

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Massive particles that don't move very 
fast. 

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If on the other hand, we have hot dark 
matter composed of low mass, low 

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relativistically moving particles, like 
say neutrinos, they would be far more 

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effective in raising filtrations on small 
scales because they move faster after 

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all. 
This is also known as Silk damping. 

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Because of this reason, the different 
mechanisms dominating erasure of small 

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scale fluctuations with different 
efficiency. 

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The two types of dark matter make very 
different predictions, 

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in terms of large scale structure as it 
how it develops from these initial 

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conditions. 
But do remember, that in either case, the 

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smallest fluctuations are erased first, 
and erasure progresses towards ever 

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larger ones as the universe expands. 
So here is a schematic diagram that shows 

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the damping process. 
The original power law, shown here as a 

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straight line is been modified by the 
damping process. 

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In the case of the cold dark matter, the 
damping removes lower mass fluctuations, 

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high special frequency ones, and does so 
in a more gradual fashion. 

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In the case of hot dark matter, the 
erasure of small scale fluctuations is 

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very effective, making a sharp cut-off in 
the wave number or spatial frequency Now 

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expressed against mass instead of wave 
number, this becomes perhaps a little 

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more intuitive. 
Note that in the case of the cold dark 

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matter, there's still some power even at 
very small masses. 

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Whereas in the case of hot dark matter, 
there's essentially no fluctuations left 

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below certain critical mass. 
The characteristic mass that corresponds 

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to the peak, just before cutoff, is 
called silk mass. 

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It's about mass of clusters of galaxies. 
So, if the universe were dominated by hot 

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dark matter extremely relativistic 
particles like sigmas and neutrinos would 

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erase very effectively all small scale 
fluctuations. 

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They'll very effectively dissolve all 
fluctuations smaller than roughly silk 

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mass before the recombination and does 
what will be the left at the time of 

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recombination will be just very large 
scale fluctuations of high amplitude. 

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The expectation then is that cosmic 
microwave background will have large, 

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large scale fluctuations with fairly high 
amplitudes maybe 10 to the -3. 

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The idea is that smaller structures will 
form through fragmentation, of the larger 

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ones, 
like galaxies forming after the clusters. 

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Neither the prediction of the cosmic 
micro background sky, nor formation of 

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galaxies essentially today are is what's 
observed. 

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Therefore, this scenario was rejected by 
observations. 

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Now, let's look at the case of the cold 
dark matter. 

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The small scale fluctuations are damp, 
but not nearly as efficiently. 

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So, some power does exist. 
And essentially, structure forms at all 

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scales at once. 
The small ones will collapse first, 

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if you recall the discussion of the free 
fall time. 

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Then they can start merging making ever 
larger ones. 

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Note, that this is a different process 
from fluctuation growth by slow 

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accretion. 
Nevertheless, it results in ever larger 

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clumps being made. 
So since the structure that it forms from 

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smaller fluctuations going up, this is 
called the bottom up or hierarchical 

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formation scenario, 
and this is exactly what we actually 

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observe. 
It also predicts exactly the right kind 

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of cosmic micro background fluctuations, 
as they are observed on micro background 

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sky. 
So even though there are some 

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cosmological neutrinos. 
Because of this, we know that their 

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contribution to the dark matter must be 
very small. 

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Here is just an illustrative simulation 
of what two different scenarios would 

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predict in terms of large scale 
structure. 

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The picture on the left is for cold dark 
matter and you can see that there are 

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structures on all different scales. 
Whereas the picture on the right shows 

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only the large lumps with no small 
struct, scale structure at all. 

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Next time we will talk more how the 
fluctuations grow in time.