So far, we have seen how the nature of the initial conditions, as specified by the
density perturbation spectrum, and the nature of the dark matter that determines
the damping mechanisms, govern the formation of the large scale structure.
Now let us look in a little more detail what happens to those density fluctuations
in the early universe. During the radiation dominated era, fluctuations do
not grow. And their, the density field is dominated by the radiation, and the matter
is tightly coupled to it. However, after the matter domination begin, begins, the
fluctuations could grow. And the crucial quantity here is the Jeans Length, which
you may recall from learning about star formation. Essentially, the pressure
suppresses collapse of fluctuations at all scales smaller than genes length. This
formula is given here. Physically, the way you can think of it is, the genes length
is the size of the collapsing cloud, where the sound waves can cross before it
collapses. In a radiation dominated universe, Jeans Length is very close to
the horizon size itself. However, after the matter domination begins, it peels off
and becomes smaller. The physical reason for this is the speed of sound drops,
because it's dominated by matter, not radiation, and fluctuation can start to
grow. This is schematically shown here. The diagonal line shows the mass enclosed
within the horizon. It grows in time, because the horizon grows in time. And the
Jeans Mass is very close to it. At the time of radiation matter equality, the
curves begin to split, and Jeans Mass stays almost constant until the
recombination, whereas the horizon mass keep growing. At the time of the
recombination, the Jeans Mass drops precipitously by many orders of magnitude,
roughly from super clusters of galaxies to scale of globular clusters. And so the key
points about the growth and fluctuation is that always the horizon scale determines
the characteristic length of the fluctuations. After the radiation matter
equality, barriers are still in teracting with radiation field and interacting with
it through Thompson scattering. The density perturbations show us the baryonic
acoustic oscillations that we discussed earlier. Those are of course responsible
for the Doppler peaks in the power spectrum with a cosmic micro-background.
However, after recombination fluctuations can grow and essentially variance fall
into the potential wells defined by the dark matter fluctuations. Schematically,
this is shown here. The density contrast does not grow until, roughly, matter and
radiation equality. At that point the dark matter fluctuations can begin to grow in
contrast, whereas the baryons stay coupled to the radiation. After the recombination,
the baryons are free to fall into the potential wells defined by the dark matter
perturbations. And soon enough, they become equal. A more detailed computation
is shown here. This shows the actual Baryonic Acoustic Oscillation as they
would appear, and those will map into the peaks of the upper peaks in the cosmic
micro-background. So here is the power spectrum of advanced defluctuations. At
the very large scales, it is normalized by using the cosmic micro background. Those
are fluctuations that go beyond the horizon, and here's the spectrum.
Then there is the rollover due to the damping, as predicted by the cold dark
matter theory. And there, of course, the baryonic acoustic oscillations. The data
points here show the galaxy clustering power spectrum scaled to the apropriate
size. It does not follow exactly the theoretical prediction, and this is due to
the non-linear effects of merging hierchical assembly of galaxies. As you
recall, the observations of micro-background confirm this theoretical
picture to exquisite precision. So we know that this is roughly what happened in the
early universe. So what happens after the recombination? The density perturbations
continue to grow. They're already imprinted in the dark matter. The baryons
soon follow. Baryons falls in those potential wells, and as t hey do so, they
will start dissipating energy. Essentially, gas clouds keep colliding,
that will inevitably cause some shock heating and energy dissipation. Later on,
there will be things like start formation and energy input by stars. Or by coals in
form of active galactic nuclear. Simulating these in detail is at the
cutting edge of our ability to model universe in computers. But we can still
make some progress on that. Now the key point here, is as you recall, the
transition from simply gravitationally bound fluctuation to the one that's fully
virialized, meaning matter is just rearanged due to the gravity. There is no
energy dissipation per se. Leads to collapse by factor of 2, because the ratio
of kinetic and potential energy changes by a factor of 2 and potential energy is
inversely proportional to the characteristic size. So the density
contrast that can be achieved due to dissipationless collapse is on the order
of magnitude. Cube of 2, 8 or close to 10. In order for density fluctuation to
achieve higher density contrasts, energy must be dissipated in some way. This
process is sometimes called cooling. A good physical mechanism for this to happen
is inverse Compton cooling of hot gas on cosmic micro background photons. This
really only becomes effective at z less than 100. Because before then, the photons
are too hot for them to act as an effective coolant for the gas. To quantify
this, we will introduce the concept of the cooling time, and that's simply the energy
that needs to be dissipated divided by the energy dissipation rate. So, it has
dimensions of seconds. Plasma physics gives us the formula for this which is
shown here. And there is a function lambda which is not to be confused with
cosmological constants called the, the cooling function and its value depends on
the density and temperature and composition of the material. So the key
question here is the relationship between characteristic cooling time. The
gravitational free fall time and the Hubble time. If the cooling ti me is
shorter than the free fall time which we defined earlier, then objects will
condense faster than they would just through gravitational collapse alone,
reaching higher densities. If, on the other hand, both cooling time and
free-fall time are greater than Hubble time, objects simply cannot form. Here is
what cooling curves for plasmas of appropriate chemical composition look
like. To show as a function of temperature. The relevant range of.
Chemical compositions is from pure hydrogen and helium which is primordial
gas, through all the way to solar composition gas, which is produced by
stellar evolution. So, somewhere in between is all the relevant range. So,
objects inside the cooling curve can cool faster than they can fall together, due to
gravitation alone, and outside of cooling curve they cannot cool as fast and on the
gravitational collapse of matter. Now we can recast the cooling curve to be shown
in the temperature density diagram. Like most thermodynamical quantities, it can be
expressed as a function of these two variables. And since the Jeans mass is
also a function of density and temperature, the lines of equal Jeans mass
correspond to diagonal lines in this log, log block shown here. You may notice that
characteristic mass galaxies, 10 to the 12 solar masses, goes right through the
region where cooling would be dominant, whereas characteristic masses of clusters
of galaxies, something like 10 to the 15 solar masses, are outside of its region.
So, to bring this point again. As a density fluctuation collapses, 2 things
can happen. If it cannot cool faster than collapses, then it'll be just pure
gravitational collapse on a free fall time scale. If however it does cool faster than
it collapses, then it will inevitably fragment into smaller pieces, which then
later may merge. So here is the cooling diagram again. And this time, shaded
regions indicate where objects of different kinds are. Remarkably enough,
galaxies of different travel types, occupy a region inside the cooling curve. And
groups and clusters of galaxies occupy region outside. You may recall that
galaxies are over dense by a factor of a million, whereas a venial collapse in an
expanding universe.can only achieve contrast in the order of 100. And indeed,
we do believe the galaxies formed through dissipative fra-, processes, and where as
groups and clusters and large scale structure are product of dissipationless,
purely gravity driven collapse. So, let us recap the key ideas of structure
formation. Structure formation grows out of dense defluxuations in the early
universe, which can grow under their own gravity, accreting more material, and
through merging. The initial conditions are often described using the power
spectrum of the density field. And the one that is often used is power law dependent
power spectrum, with an exponent of one. Also called, Harrison-Zeldovich spectrum.
Spectrum. The role of dark matter is essential. Dark
matter fluctuations can grow before baryons can actually follow. Which is why
it's possible to actually have the structure that we can see. In some sense,
dark matter seeds this large scale structure prior to recombination. And
after the recombination, baryons, the visible material, falls into those
potential wells. There is damping mechanisms, say photon diffusion or sound
waves. They raise small scale fluctuations. The nature of dark matter
determines which mechanism will be at play, and therefore how many of small
fluctuations will be left. The observations show that cold dark matter,
composed of massive particles. Does fit all of the data, whereas the hot dark
matter composed of light relativistically moving particles, say as massive
neutrinos, simply does not reproduce the observed picture. And the topology of the
density fluctuations will always go from there are actual blobs which first
collapse into the sheets, which then collapse into filaments, which then drain
into an quasi-spherical blobs like clusters. An important thing to remember
is that, as shown throug h spherical top-hat model. The simple virialization,
dissipationless collapse of a fluctuation in an expanding universe, can reach a
density contrast of 200. This is more than what will happen in non-expanding universe
where only a factor of eight will be achieved because the surrounding
background also expands while the perturbation collapses. Free-fall time
scales, that is, in fall just due to the gravity with nothing to oppose it,
indicate that for galaxy scale fluctuations, the formation will be on the
scale of hundreds of millions of years. Where some scales of clusters of galaxies
and large scale structure formation would be on scales of billions of years, and
that scale responds to observations. Just as in the case of star formation here in
galaxy formation, genus length or genus mass are key concept to determine the
scale of fluctuations that can actually grow or fragment in the smaller one. The
Jeans mass is essentially within, mass within horizon prior to the
radiation-matter equality. Becomes constant after that, and then drops
precipitously at the time of the recombination. Cooling is the key concept
because, in order to achieve higher density contrast, energy must be
dissipated. Which is known as cooling. And galaxies form largely through dissipate
processes where as large scale structure is formed dissipation less leak, purely
through gravitational collapse and assembly. Next we will see how the
nonlinear processes of structure formation are simulated using computers.