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.