We now turn our attention to structure 
formation in the universe. 
Actually since the original 
disappointment of the failure of 
classical cosmological tests due to 
galaxy evolution in 1970s until the 
revival of precision cosmology in late 
1990s. 
This was the main subject of cosmological 
research. 
So here is the problem in the nutshell. 
We see the dance defluxuations in the 
early universe imprinted on the cosmic 
microbe background photosphere. 
They're roughly parts in a million. 
Today, we see a large-scale structure, 
clusters, filaments, voids with density 
contracts of about 100. 
And then, within them there are galaxies 
whose average density within say half 
life radius is about million times larger 
than the surrounding So how do go from 
fluctuation, that there a part in a 
million, to large scare structure, that 
is factor of 100 over dense to galaxy 
that are, factor million over dense and 
that is the job. 
So we believe that the original 
fluctuations came from quantum 
fluctuations in the early universe that 
you know due to fluctuations in the 
number of density of particles and then 
they're inflated through the inflationary 
era to the large scale structures or are 
the seeds of large scale structures. 
The matter then falls into them, and the 
density contrast grows. 
So the observed thing is, fluctuations in 
micro background, is radiation and 
variations that are coupled before the 
recombination, which are parts in a 
million. 
But we also observe, galaxy-like 
structures at the [UNKNOWN] greater than 
6. 
Those are luminous quasars and Even some 
real galaxies, and so we know that at 
least some little pockets of the universe 
have achieved the density contrast which 
are not the part in a million, but a 
million for orders of magnitude, in the 
time elapsed from the microbackground 
release until then is of the order of 
half a billion years, so this is not an 
easy thing to achieve. 
And the question then is, how do we go 
from such small seeds of density 
fluctuations to the large density 
contrast that is observed in the objects 
that we see. 
Well here is a schematic flow chart of 
what has to happen. 
It begins with inflationary physics or 
very early [UNKNOWN] physics, which is 
not directly observable except as 
consequences of inflation. 
But then those fluctuations will grow and 
we see their imprint in microbackground 
even after microbackground is released, 
they do grow through gravitation and 
stability. 
The dense parts will attract more matter 
and become denser. 
That part is fairly easily understood and 
even modeled. 
It's Newtonian gravity because it's weak 
field. 
But, then gas falls into those potential 
wells, gets condensed, dissipates energy 
as it falls in. 
It makes stars, stars dump energy back, 
and those are very messy processes which 
have to be modeled hydrodynamically, and 
through radiative transfer. 
So, there is transitions from relatively 
simple, elegant, theoretical [UNKNOWN]. 
Geometry and gravitation into the really 
messy hydrodynamics radiation physics of 
what actually happens to the variance 
after the recombination. 
so let us set the stage for us. 
In the early universe, we know that it's 
very close to the critical matter 
dominated universe. 
Matter or radiation. 
And so, it's a very simple overall 
Friedmann equation. 
Regardless of what universe will evolve 
to later. 
Early on, its really close to a omega 
matter equal 1. 
And let's look at an over-dense region. 
Some were inside that exactly critical 
universe. 
It is essentially like little closed 
universe of its own. 
And initially, of course it expands, but, 
as you know, since its densities are 
greater than critical and surrounding 
universes are created also, overdense 
must be. 
that means that at some point, it'll have 
to turn around and fall back upon itself. 
So let's just subtract those two 
equations. 
And so that's, that's Friedmann equation 
for the difference in density, which we 
can simply rewrite as follows. 
So that's the difference of densities, 
but really we're looking at the density 
contrast. 
So, let's just divide it by the mean 
density, and so we'll call that as the 
overdense [UNKNOWN] delta, and its 
equation is shown here. 
Here. 
Now, since the density is inversely 
proportional to the scale factor. 
This is density of the matter, remember? 
So it just gets diluted as the cube of 
the radius. 
We see that the density contrast, the 
delta, will grow as the first power of 
the expansion factor, r of t. 
And we call that the era of the linear 
growth of perturbations. 
So between any two different redshifts, 
the change in the contrast will be given 
by the ratio of 1 plus redshift or rather 
the inverse thereof. 
So thet's try this from micro background 
roughly redshift 1000 to say first quasar 
roughly at redshift 5 or so. 
and we find out that the density contrast 
should have grown by a couple parts in 
thousands, whereas it should have been 
millions. 
So we were off by 9 orders of magnitude 
and clearly linear treatment does not 
apply. 
The solution to this is in part that, at 
the time of the micro background, the 
fluctuations that we see are those in 
varions and photons. 
But, underlying them are much deeper 
fluctuations in dark matter, and those 
have been growing since the universe 
became matter-dominated. 
So this is another reason why we think 
that dark matter must exist, because 
without it, we would simply not obtain 
the large-scale structure that we see. 
And that's a key idea. 
That the density fluctuations are 
dominated with the dark matter which, as 
you recall is several times more by 
density of regular bariums and those 
fluctuations grow even before the 
micro-background is released. 
So the fluctuations we see in 
micro-background are just little icing on 
the cake and after that Those variables 
can fall into those already preexisting 
potential wells built by the dark matter. 
Next time we will address this process in 
a little more detail.