Let us now take a closer look at the collapse of density fluctuations.
The toy model that's often used is called the spherical top-hat model.
What it means is simply that there is a uniform overdense sphere embedded in the
constant density background. As you recall from the last time, you can
consider the background universe expanding according to the critical
universe line. And then the fluctuation itself, which is
overdense, acts like a little closed universe.
So, it splits from its background. And if it were perfect model that can
recollapse to a point, then it will be simply shown as the curve here.
In reality, it does not collapse back in a point, it collapses to a certain
radius, finite radius in which system is in a variable equilibrium.
But qualitatively, the density contrast, which is the difference between that line
and the background line, grows as the universe expands.
There is a turnover radius at the turnover time.
And then, there is a collapse that caught, that takes about same amount of
time as it took to get to the turn-around time, and the system ends up at some
finite radius of equilibrium state. A simple argument to understand the final
virial radius is that the system is just bound, so kinetic energy and potential
energy are equal at the beginning. When it's in the equilibrium state, the
Virial theorem state, tells you kinetic energy will be 1/2 the potential
[UNKNOWN]. Now, if you'll look at simple formulas
for combining energy [UNKNOWN] gravitational system,
you can see the potential energy is inversely proportional to the radius.
And so, if we're going to achieve a factor of 2 difference, that means that
the radius of the final system will be 1/2 exactly of the initial one and
therefore, the density will increase by cube of that or a factor of 8.
Now, let's go back to the Friedmann equation.
We can break it down as follows. And it has the well-familiar solution of
universe expanding at time, it's time to the third, to the third power.
Now, for the overdense perturbation, we can express the solution in a parametric
form shown here. The derivation of those is beyond this
class, but there are places we can look it up such as Barbara Ryden's book.
For now, assume that this is a view of what's happening.
So, we can find out the turnaround time or radius by asking where is the maximum
of this curve, which means that its first [UNKNOWN] is zero.
Obviously, since this is a periodic solution, even though universe will not
necessarily bounce back, it tells us that for the first turn around, it will be at
the value of pi. And so, the solution for this is given
here. The turn-around time is given by this
formula. Now, if we look at the background
universe, which continues expanding, its density go, keeps going down.
And the ratio of the densities of the spheres at the turnaround time and
surrounding the universe, is given as the cube of the expansion factors that
correspond to them, which is a simple numerical factor of 9pi^2/16, for a very
idealized case that we are talking. But remember,
that by the time the universe collapses, again, to the Virial equilibrium, its
physical scale will go down by a factor of two, and that means the density will
go by the factor of eight. In the meantime, the background will get
further diluted by a factor of 2^3, to the cube, and that means four.
So, there is an additional gain of a factor of 32 in density contrast.
Multiplying these two together, the final virialized state of structured
equilibrium, would be 180 times denser than surrounding background.
Remarkably, that's exactly what we see in large scale structure today.
This is a treatment of single idealized density fluctuation.
Now, of course, what happens later, such density fluctuations, what happen for
that matter halos, keep merging, building larger ones and that is called a
hierarchy confirmation setup Or or the bottom up structure information because
you build bigger ones starting from small ones.
Now in reality, the density fluctuations are unlikely to be perfect spheres.
More likely, they can be approximated, say, as triaxial ellipsoids.
They might have some very irregular shapes, but the triaxial ellipsoid is a
better approximation than a sphere. Now, look what happens.
At first, the perturbation expands within surrounding areas.
But then, it turns around and it will turn around first at its shortest axis
and it'll start collapsing along the shortest axis while still expanding in
the other two. So, a three-dimensional blob will flatten
itself into a pancake like structure. And then, we'll turn around on the second
shortest axis and it will collapse along two axis and expanding a third one.
The structure will turn into a filament. Finally, the third axis turns around and
the, that's really [UNKNOWN] cluster collapses along that.
and it will become, again, a quasi-spherical object in the end.
This is the origin of the topology of the large scale structure that we see walls,
and filaments, and the intersecting things which really are clusters, and
this is the simple reason why it happens. So, let's figure out how long it will
take for this to happen. Now, let's divide initial density
perturbation, that's collapsing into a lot of thin layers.
They will be all falling towards the middle.
Let's say, two adjacent ones are, are essentially subject to the same
acceleration, the inner one slightly higher acceleration, so it would be
falling a little faster than the one just behind it.
Therefore, the layers will never cross, the inner parts falling the fastest and
the outer one is the slowest. So, if we take the point of the outermost
shell and ask how long before a point collapses to the middle, that is
basically the time for the collapse of the fluctuation, that's called a free
fall time. And since all of the mass is enclosed to
that radius at all times for all practical purposes, it is a free fall in
the gravitational field of a point mass in the middle.
And the solution for this is very simple and is shown here.
You can also immediately see that intuitively, it is also true, not
formally, that the lowest density fluctuations would collapse slowest and
the highest density ones fastest. Now, also the largest ones would take
longest time, the smaller ones would take less.
Now, if we plug in typical numbers, say, for extent of dark halos of galaxies and
sizes of clusters of galaxies, we find out that the relative collapse times are
a few hundred million years for galaxies and few billion years for clusters.
Which means that galaxies will collapse first in the early universe, and large
constructions will be collapsing even today, and that's exactly what's
observed. Next, we will talk about primordial
density spectrum of fluctuations, which is a descriptor of how many fluctuations,
of what size they are.