So far we've discussed how equal ops of primordial dense defluctuations would
lead to formations of structures in the universe, but the fluctuations will have
a spectrum of different sizes or masses. And so the question is how do we
characterize that? The way we do this is in the language of Fourier analysis.
If you are not familiar or you forgot about it, this would be a really good
time to refresh your knowledge of it. Otherwise, you may find this module a
little difficult to follow. We will represent the Density Field in
the early universe as a superposition of waves of different frequencies and
different phases. In other words, we're decomposing it into
a Fourier spectrum in three dimensions. The power spectrum of this density field
and characterizes the distribution of one of different size.
It is customary expressed as a function of the wave number, the inverse of the
special wavelength. So we start with the fluctuation density
field, the relative fluctuation delta that we introduced earlier.
We take the Fourier spectrum of this field and it's power spectrum is of
course, the square of the applicants. In this way we can completely
characterize the distribution of lumps of different size, not of their shapes which
is where phasing for mission comes in. But instead of using spacial frequency,
we can also express it as a function of mass.
The larger ones obviously having larger physical sizes so a larger wavelength.
The formula shown here is what's customarily used.
It is usually assumed that all fluctuations have about the same
amplitude, of the order of 10 to the -4 when they enter the horizon.
This is more than what you see in cosmic micro background.
But recall the presence of the dark matter is what helps. The dark matter
fluctuations have grown already relative to those in baryons and photons.
So, if all of the fluctuations of different frequencies have same
amplitude. That's called a scale-free spectrum, or
Harrison-Zeldovich according to the cosmologist who first introduced it.
It so happens that this is false so what was predicted in inflation theory.
More generally, the power spectrum can be represented as a power-law of the wave
number exponent of n. If that is equal to 1, that is the
Harrison-Zeldovich spectrum. In any case, the fluctuations will always
grow, because they attract material near them and so the amplitudes will always
grow. They will grow same way at all sizes
during the linear regime, and therefore the whole thing just shifts in
logarithmic space. But the important point is that
fluctuation growth by itself does not change the shape of the spectrum.
Let's look at some illustrative examples. These are two-dimensional images that
correspond to different spacial frequencies.
And, our spectra shown in the box to the right. They're almost delta functions
favoring waves, spacial waves of frequencies.
And the pictures show that representation using random phases and clearly the
higher the wave number, the lower the wavelength, the smaller the lumps you
see. In reality, it will be a broader
distribution. And so, if we allow for a broader
spectrum, but with a cut-off at at some frequency or some wavelength, then the
pictures might look like something like this.
We talk about several different types of density fluctuations.
The most commonly used ones are so-called, Adiabatic fluctuations.
[SOUND] They are the same for photons and invariance in terms of number dynasties.
But because of the expansion, their energy densities change differently.
This is also commonly known as one of the F noise, or Harrison-Zeldovich spectrum,
and it has equal power on all masses. It also happens to be exactly what we
observe in the cosmic micro background. The other kind of perturbations are
Isocurvature perturbations. Where there are perturbations in the
numbers of photons, as well as baryons, but in the opposite directions.
So they composite each other and so the curvature, spatial curvature that
corresponds to the mass density is unperturbed.
And finally there are Isothermal fluctuations where the photons are
unperturbed does the equal temperature, but there are underlying fluctuations in
the regular matter. This corresponds to so-called white noise
and these are no longer used. Now the important point here is the
different types of dark matter would lead to different structure formation
scenarios. And we can see that through behavior of
their power spectrum. Recall also that initially it's just the
fluctuations in the dark matter that matter because baryons are coupled to
photons and they only participate in growth of fluctuations after the
recombination. Perturbations can be raised by sound
waves. Obviously, the density waves will carry
the density of particles from more dense regions to less dense regions.
And thus, they would start the raising the fluctuations.
This is also known as the Meszaros effect.
This turns out to be important for circled cold dark matter.
Massive particles that don't move very fast.
If on the other hand, we have hot dark matter composed of low mass, low
relativistically moving particles, like say neutrinos, they would be far more
effective in raising filtrations on small scales because they move faster after
all. This is also known as Silk damping.
Because of this reason, the different mechanisms dominating erasure of small
scale fluctuations with different efficiency.
The two types of dark matter make very different predictions,
in terms of large scale structure as it how it develops from these initial
conditions. But do remember, that in either case, the
smallest fluctuations are erased first, and erasure progresses towards ever
larger ones as the universe expands. So here is a schematic diagram that shows
the damping process. The original power law, shown here as a
straight line is been modified by the damping process.
In the case of the cold dark matter, the damping removes lower mass fluctuations,
high special frequency ones, and does so in a more gradual fashion.
In the case of hot dark matter, the erasure of small scale fluctuations is
very effective, making a sharp cut-off in the wave number or spatial frequency Now
expressed against mass instead of wave number, this becomes perhaps a little
more intuitive. Note that in the case of the cold dark
matter, there's still some power even at very small masses.
Whereas in the case of hot dark matter, there's essentially no fluctuations left
below certain critical mass. The characteristic mass that corresponds
to the peak, just before cutoff, is called silk mass.
It's about mass of clusters of galaxies. So, if the universe were dominated by hot
dark matter extremely relativistic particles like sigmas and neutrinos would
erase very effectively all small scale fluctuations.
They'll very effectively dissolve all fluctuations smaller than roughly silk
mass before the recombination and does what will be the left at the time of
recombination will be just very large scale fluctuations of high amplitude.
The expectation then is that cosmic microwave background will have large,
large scale fluctuations with fairly high amplitudes maybe 10 to the -3.
The idea is that smaller structures will form through fragmentation, of the larger
ones, like galaxies forming after the clusters.
Neither the prediction of the cosmic micro background sky, nor formation of
galaxies essentially today are is what's observed.
Therefore, this scenario was rejected by observations.
Now, let's look at the case of the cold dark matter.
The small scale fluctuations are damp, but not nearly as efficiently.
So, some power does exist. And essentially, structure forms at all
scales at once. The small ones will collapse first,
if you recall the discussion of the free fall time.
Then they can start merging making ever larger ones.
Note, that this is a different process from fluctuation growth by slow
accretion. Nevertheless, it results in ever larger
clumps being made. So since the structure that it forms from
smaller fluctuations going up, this is called the bottom up or hierarchical
formation scenario, and this is exactly what we actually
observe. It also predicts exactly the right kind
of cosmic micro background fluctuations, as they are observed on micro background
sky. So even though there are some
cosmological neutrinos. Because of this, we know that their
contribution to the dark matter must be very small.
Here is just an illustrative simulation of what two different scenarios would
predict in terms of large scale structure.
The picture on the left is for cold dark matter and you can see that there are
structures on all different scales. Whereas the picture on the right shows
only the large lumps with no small struct, scale structure at all.
Next time we will talk more how the fluctuations grow in time.