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.