As we know the cosmological principle asserts the Universe is
homogeneous and isotropic. And in fact the extra-ordinary smoothness
of the distribution of radio galaxies and of the microwave background
indicates that this isn't very far from the truth. But when we look at
the Universe today it is not at all smooth and homogeneous in
detail. The density in galaxies is 
times greater than the
average density in the Universe. The Universe is smooth but only
on very large scales, in excess of 100 megaparsecs (1 megaparsec = 1
million parsecs). But the temperature fluctuations in the microwave
background are very low, only around one part in which
translates to density fluctuations at the level of one part in
. How did such minute density fluctuations grow to the degree of
contrast we observe today? The short answer is gravity. The tiny
fluctuations of early times gravitationally grew to the observed
density contrast today. The study of the formation of structure has
been and continues to be a gradual understanding of this process.
The basic idea of structure formation in the Universe is to grow density fluctuations from the very small fluctuations in the past to the present day through gravity. Gravity because any other way of growing the contrast involves too much energy that would be obvious in other observations. To study this process we have to start with some initial conditions, that describe the condition of the Universe at some early epoch when the fluctuations were still small. Then use these initial conditions to evolve the Universe using Einstein's gravity. And then compare with the Universe of today to provisionally accept the model for further study or to reject it. We could of course in principle just take the Universe of today and evolve it backwards in time using the equations of physics and find out what the initial conditions were. But this proves to be impossible in practise. This has to do with the extremely complicated astrophysics in the very recent past of the Universe. Things like star formation, supernova explosions, hot gas in clusters of galaxies are very poorly understood phenomenon that are of substantial importance to the structure formation today. This makes it impossible to decipher the past from today directly. It is however easier to evolve a given set of initial conditions to some time just before the present when the astrophysics was still relatively simple. And the hope is that we will be able to narrow the range of allowed models from the gross features of the large scale structure in the Universe today that must be reproduced by the models regardless of astrophysical complications.
The information we need to do this are:
One of the first complications that show up has to do with the
collapsing gas. Two moments in the Universe's history have special
significance in the structure formation process. The first is epoch of
equalization. At early times the Universe is dominated by radiation
and later by matter. The transition point is the equalization
time. When the Universe is radiation dominated the density
fluctuations hardly grow, the radiation keeping the matter in its
place. When the Universe becomes matter dominated the density
fluctuations can grow increasing their contrast. But even then the
baryonic matter is restrained by radiation. The photons interact
strongly with the electrons restricting them in their movement and so
keeping the baryonic gas from collapsing very fast. The dark matter
however hardly interact with photons, and so can get a head start in
their gravitational collapse. Then after the recombination time the
electrons in the Universe attach themselves to nuclei to form neutral
atoms. They are confined to a small region and the photon can travel
freely without being assailed by the electrons. The Universe becomes
transparent. And the baryonic gas can collapse freely into the
gravitational wells that the collapsed dark matter halos provide. The
first gas clouds that form are about times as heavy as the
Sun. These clouds will presumably form the first stars. However what
kind of stars they will form remains entirely unclear. They could form
one massive star each or fragment into many tiny stars. And the
consequence of one is drastically different from the other. If a tiny
number of stars are indeed massive then they immediately heat up the
collapsing baryon gas sufficiently so no cloud can less than 
times as heavy as the Sun can collapse. These stars can also produce
metals that can pre-enrich the gas forming the galaxy and thus making
sure that even the oldest stars we see today would have some metals in
them. The clouds with over the mass of Sun can collapse to form
galaxies. It does so by cooling, getting rid of its heat through
radiation. The efficiency of this cooling mechanism is what sets the
timescale of formation of the galaxy. If the galaxy gets too big,
heavier than 
Suns, then cooling isn't efficient enough to
for such a large gas cloud to collapse. The cloud fragments into
smaller pieces each of which form a galaxy. This restricts the masses
of galaxies to between and the mass of Sun, which is
what is observed. However it is almost impossible to predict the
distribution of the galaxies in different masses. That is, how many
galaxies have a mass equal to say, Suns, as opposed to
Suns. This depends on two pieces of information, both
unknown to us, due to the crudeness of our simulations. The first is
the mass distribution of the dark halos themselves. The second is the
relation between the size of the dark halo and the amount of gas the
halo can capture. The latter issue is closely connected with star
formation in the galaxy and how the energy produced by stars might
feedback to the collapsing gas and hinder the collapse.
There are still some constraints that we can impose from observations, that restrict the range of allowed models. The first of course is the cosmic microwave background. This restricts the amplitudes of initial fluctuations as well as their nature. The second is the structure on the largest scales. On these scales astrophysics hasn't become very significant and we should be able to connect model predictions to the observations relatively easily. The third observations are the dwarf galaxies. Now one of the problems with ordinary galaxies like our Milky Way, is that while we can tell from the rotation speeds that there is dark matter in the Galaxy it is hard to say how much of it there is. In dwarf galaxies there are some indications that the gas in them may be only just bound to the galaxies. The supernovae going off in them are on the verge of blowing them apart. This allows us to deduce the total amount of dark matter in these galaxies. This could help us draw conclusions between size of dark matter halo and the gas content in the galaxy. And finally there are the searches for the dark matter. By definition dark matter particles interact weakly except by gravitation, hence a search for them is problematic. Secondly any observations have to be done locally which means we must make assumptions about the universality of the observations. But still this would be the only direct way of detecting the dark matter particles and so is essential.
One of the assertions of the Cosmological principle that make
cosmology simpler is the assumption of isotropy. The best evidence for
it is the extreme smoothness of the microwave background
radiation. But this gives rise to a conundrum. One of the most
important features of the big bang Universe is that it has
horizons. The spatial extent of the Universe may be infinite, as is
the case for models with 
or finite, as with models with
, but the relevant concept is that of horizons. Since the
Universe has a finite age, and because light has a finite travel time,
there is a finite distance at any given moment of the Universe' life
that light will have had time to reach us from. This distance defines
our horizon at that age of the Universe. Obviously as the Universe
ages, light has more time to travel larger distances and so our
horizon grows. But when we look at large distances, we are looking
back in time and so to times when the horizons were small. Now about
each point the horizon defines the region over which the point can
communicate any changes happening to it. Regions outside its horizon
are inaccessible to it. If we look at the microwave sky we are looking
back to the recombination era when the Universe went transparent. The
horizon size then was far smaller. So two points at opposite ends of
the sky had horizons that couldn't communicate. And yet they have
nearly identical temperatures of the microwave background. How did
they manage to arrange that? This is called the Horizon problem in
cosmology.
One solution to the horizon problem is simply to say that is how the Universe was born. As we have only one Universe it is impossible to make any statements about probabilities based on it and so it isn't a problem to have the isotropy as an initial condition for the Universe. However this answer is unsatisfying to most physicists. They would prefer to have the isotropy drop out as a natural consequence of the theory. One that does provide an answer to this and several other problems is the Inflationary model. In this model at early times the Universe is filled with some substance that behaves like coiled spring, i.e., possesses substantial amounts of potential energy. At some point the force holding the spring coiled is lifted and spring is released. The substance releases its potential energy which causes the Universe to expand at a phenomenal rate. A piece of the Universe that was very small early grows incredibly large very fast. The piece of the Universe that was the size of the horizon early when equilibrium existed and so that patch of the Universe was isotropic now becomes many times larger than what it was. The horizon however increases at its usual pace and is now trying to catch up with the original little homogeneous piece of the Universe that has grown so much bigger. This model beside solving the horizon problem also forces the density to be equal to the critical density. Equal aside from small density fluctuations very close to the amplitude and nature required to produce the large scale structure today! So inflationary models solve all these problems at once.
However this is also its problem. All dynamical observational measurements of the galaxies and their motions appear to indicate that the density of the Universe is only 20% the critical density. This means either the inflationary model is wrong, or somehow the galaxies don't tell the full story of mass in the Universe. That is the galaxies are biased estimators of mass. Imagine trying to sample the population of a city using the telephone book. It will be representative sample if all the people in the city do own telephones. If many people in the cities are homeless or simply do not own a telephone the sample will be very biased. Most likely against the poorer people. Biasing in galaxies is essentially asserting the same thing. If for some reason galaxies only form in the biggest dark matter halos there will be many smaller dark matter halos that aren't counted in the evaluation of the density of the Universe because we simply don't know of their existence. There are theoretical reasons why such biasing may occur. The heaviest dark matter halos form the earliest. So gas will collect there first and form stars. These first generation stars may be able to heat up gas yet to collapse and so prevent galaxies from forming in the smaller dark matter halos that form later. And so these smaller dark matter halos may never fill up with gas and form stars and thus be visible. This may have even been observed. Galaxies in clusters of galaxies appear to move too fast to be bound to the cluster if all the mass was that associated with galaxies (including dark matter in galaxies). Indicating that the clusters have extra mass in halos not associated with galaxies, unless all clusters are on the verge of exploding today. Any firm theoretical statements are very difficult because of the poor dynamic range of modern computer simulations. As we must simulate large chunks of the Universe, restricted computer power means we are limited in the detail we can resolve. And to measure biasing we would need tremendous amount of detail. However the point is that biasing is natural and should be expected regardless of what the density of the Universe is. However whether there is enough biasing to give us a closed universe is a difficult question as yet without an answer.
The observational constraints and theoretical efforts described before have narrowed our choice of models to two.
. The
horizon problem becomes a feature the Universe was born with. The dark
matter that is observed in rotation of galaxies and clusters of
galaxies is baryonic dark matter, viz., white dwarf stars, neutron
stars, black holes and brown dwarf stars. The last being stars that
are very light and are too dim to be observed. This model has received
some boost of late from observations of what might be brown dwarf
stars. Even though they are too dim to see they can be observed
through their gravitational effects. When they pass in front of other
stars they bend the light from the stars making them twinkle (like our
atmosphere does). Recent large studies have found these twinkling of
stars in our and neighboring galaxies.
, the
horizon problem is solved by inflation. And the dark matter is some
exotic material that is between ten to a hundred times as abundant as
the baryonic matter.
In either case some problem remain difficult to solve. The very large scale structure that is observed, like the filaments of superclusters of galaxies extending to hundreds of megaparsecs, or the bulk motion of all galaxies and clusters at about 600 km/s within the nearby 100 megaparsecs. This field remains under intensive study and one with the most important and outstanding questions in astronomy. We seem as far away from understanding the origin of the Universe as the Greeks were, but perhaps we have started asking the right questions.