One of the greatest achievements of Einstein's theory of gravitation is to provide a coherent description of the Universe. As you may have noticed this has been the driving aim of astronomy. From the earliest times the effort has been to be able describe the functioning of the Universe. Whether the Universe was the Solar System, the Milky Way or what we today think of as the Universe. It has an incredible number of galaxies like our Milky Way separated by unimaginable distances each of which have immensely large numbers of stars in them. We've come some distance from the seven crystal spheres. The extra-ordinary thing about this evolution has been that our theories have got simpler as our understanding of the Universe has gone deeper and our observations have become more detailed and broader. Thus, cosmology is the culmination of this process. A simple and elegant theory describes the extra-ordinary wealth of detail that we now know about the Universe.
The starting point of cosmology is the Cosmological Principle. It asserts that the Universe is homogeneous, i.e., would appear the same for any observer anywhere in the Universe, and is also isotropic, i.e., appears the same to a observer looking in any direction. What evidence do we have for such an assertion? It is clearly wrong in detail. Different directions in the sky certainly look different. Also the Universe is anything but homogeneous in detail. The galaxies are many times more dense than regions in between galaxies. The stars are far more dense that the region between stars. So it would appear that the ``principle'' is not only a presumption, but also incorrect. The trick is to think of it in terms of homogenized milk available in cartons in supermarkets. The milk, of course, isn't homogeneous in detail, being made up of many different molecules of fat and protein. But each serving of the milk has roughly the same amount of saturated fats, sodium, proteins and other essential ingredients. Similarly we must talk about the homogenized Universe in appropriate serving sized chunks. The evidence for isotropy comes from two major sources. One is very distant, extremely bright galaxies, called radio galaxies because they are unusually bright in radio waves. Their distribution in the sky is extremely uniform. Another is the microwave background radiation. This is believed, as we shall see later, to be the relic light from the original explosive formation of the Universe, the big bang. This is also extremely smooth in the sky, to better than one part in a hundred thousand. This gives us some confidence in the presumption of isotropy. Homogeneity is more difficult to justify. However there is some evidence for it. The microwave background sometimes passes through material at large distances before it reaches us. As the light passes through these materials it is affected by it and affects it. By noting these interactions we can tell how these distant material sees the microwave background. And they appear to be the same as how we see it. Also abundances of the different elements, the temperatures and densities that can be measured in galaxies at large distances appear to be similar to our Galaxy, which suggests they formed in similar circumstances. But aside from such indirect evidence there is no hard reason for the cosmological principle. At least there are no evidence to suggest otherwise.
The convenience of accepting the cosmological principle is that then the description of the geometry of the Universe becomes quite simple. This was done in 1922 by Aleksandr A. Friedmann, a Russian mathematician. In the Friedmann Universe the geometry of space can be curved or flat depending on the value of a parameter called curvature, k. The Universe isn't static, but can expand or contract. This change of scale is described by the function a(t) called the scale factor. Curiously because this change of scale is experienced by all observers in the Universe, they all measure the same time. Of course in practise the time measured by an observer is affected by the local gravitational field. But in the smoothed out Universe of the large serving size, all observers would measure the same time, returning us to some form of Newtonian absolute time! Although this just a shared sense of time for observers in this Universe, and not at all an absolute time in principle.
There is one obvious consequence of the change of scaling of the Universe. Light has a finite speed and so takes time to travel from one point to another. So when we see a distant galaxy we are infact seeing it as it was in the past. As long ago as it takes light to reach us from that galaxy. Now if the Universe is expanding, the distance scale then was shorter than it is now. So an atom making a specific transition in that galaxy then would produce light of the appropriate wavelength. But by the time that photon reaches us the Universe has expanded by some amount. The wavelength that was appropriate then now becomes longer than that is radiated by an atom here undergoing the same transition (the wavelength having been stretched out with the expanding space). This is infact what was observed by Edwin Hubble. He discovered the relation between the distance of a galaxy and its redshift that goes by his name. The further the galaxy the larger its redshift. Which means that the Universe has been expanding as far back as we can see. Because of the enormous implications of this assertion we should examine a little the evidence behind it.
The most important part of establishing the Hubble's law is the measurement of distances to the distant galaxies. As you have learnt this simply cannot be done directly because of the immense distances. Consequently a long ladder of arguments have to be raised that connect the distant galaxies to the nearest stars whose distances we can measure directly. One of the features of the ladder is that there are more than one ways to go from one rung to another. This is important because this is what adds confidence to what might be otherwise a complex and unstable hierarchy of arguments.
At the bottom is the solar system. The distances here are measured to exquisite accuracy, using laser reflection off the moon and radar reflection off Venus and most importantly Newtonian gravity. The next step is trigonometric parallax. This is the apparent motion of stars because of the motion of the Earth around the Sun. This allows the direct measurement of the distances to the nearest stars. Then we have several possibilities open to us. One is fitting the main sequence of clusters of stars to the main sequence of field stars (i.e., stars not in clusters) of known brightness. The other is the period-brightness relations of variable stars like RR Lyrae or Cepheids. Both of these are bright enough to allow us to leap frog over to the nearby galaxies. Then we can use the bright standard candles like supernovae, globular clusters which have nearly the same brightness or are bright and have characteristics that all objects of the same class share. This takes us to distant galaxies. Then we can apply the standard candle argument to the galaxies themselves. Methods like surface brightness fluctuations described before, or the brightness-rotation speed relationship of galaxies can be used to measure distances to very distant galaxies, because of the great brightness of the galaxies. This multi-pronged approach to the measurement of distances of galaxies means that even if each individual step in the distance scale isn't completely reliable, the fact that many independent approaches yield similar results allows us to have confidence in the distance scale of the Universe. And therefore in the Hubble law asserting that more distant the galaxy larger its redshift.
The Hubble's law of the expansion of the Universe states that the
further the galaxy the larger its redshift. If we forget about the
expansion about the galaxy and continue our charade of pretending the
redshift was because of a recession velocity of the galaxies, then the
velocity of recession, v, of a galaxy at a distance d is given by,
, where 
is called the Hubble constant. Current
observations constrain to being roughly within 50 km/s/Mpc and
80 km/s/Mpc.
If the Universe is expanding, and has been expanding for ever it was smaller before. So if we go backwards in time, the Universe will keep getting smaller, in the sense that everything will be a lot closer to each other. Presumably if we continue this we will end up with everything on top of each other and a vanishingly small Universe. If we take this moment to be the birth (more on this later) of the Universe then Universe is not infinitely old. This is a momentous assertion, that calls for some time to be spent on it. Because the space time must also have been born with the Universe, there was no ``before'' this birth because time was born with it. It wasn't born into any place because space was born with the Universe. As far as empirical evidence is concerned, there is no before to this at all that one can sensibly ask about. And what is the age of the Universe? The age that we get by rewinding the Universe backwards is around 12-18 billion years. There are other ways in which we can constrain the age of the Universe. Radio active dating, achieved by comparing the abundance of daughter products to parent products of nuclear transmutations, puts the age of the Earth around 5 billion years. The oldest stars in our galaxy are around 16 billion years. Remember white dwarf stars gradually cool and get dimmer and redder. If they've only had a fixed amount of time to cool then the oldest dwarf will be a certain brightness and color if they all follow the same track in the HR diagram. This cut-off in the HR diagram for white dwarf stars fixes the age of the galaxy around 10 billion years. So all these completely independent measures of the age of the old objects in the Universe are within shouting range of the possible age of the Universe itself. This is a remarkably strong argument for the start of the Universe as very small a few tens of billions of years ago.
If the Universe was infact very small before and is expanding now,
everything is moving away from each other. This means that the density
of material in the Universe (the mass per unit volume) is
decreasing. The temperature is decreasing as well in the same way an
gas that is expanding will cool. Thus as we go back to the past, the
Universe was smaller, denser and hotter. So at very early stages of
the Universe, it was intensely hot, dense and small. This is referred
to as the Hot Big Bang. In the same way as the piston in an car
cylinder is violently pushed out by the exploding petrol fumes after
the spark plug fires, the Universe is violently dispersed out in a
gigantic explosion that is still continuing in its expansion. An
extremely strong evidence for this was the discovery of the cosmic
microwave background. Like any explosion the birth of the Universe was
accompanied by a flash. The photons expanded with the gradually
cooling Universe, their wavelength lengthening until today they are
only detectable as microwave radiation a mere 2.7 degrees above
absolute zero 
. Nobel laureates Arno
Penzians and Robert Wilson were characterizing the noise that radio
transmitters need to be aware of, when they realized that there was
one noise whose source they couldn't identify that came from all
directions of the sky. They later realized that theoreticians working
in Princeton expected this radio noise from the hot big bang model of
the Universe and was looking for it. The discovery of the microwave
background at the temperature expected from a cooling Universe born in
an explosion was a very positive step for the hot big bang model.
Contrary to other suggestions as far as most cosmologists are
concerned the hot big bang model doesn't start with a point like
Universe. It starts 
seconds after the ``explosion'' when
the Universe though incredibly small is of a finite size. The reason
this restriction is insisted on is that present physics is expected to
only work upto these conditions. Physics appropriate to the conditions
before this point is yet unknown and hence outside the purview of
scientific speculation. The density of the Universe was a
mind-boggling 
grams per cubic cm and the temperature
degrees Celsius! From this the Universe starts to expand and
cool as well as get less dense. Electrons, neutrinos and photons fill
the Universe. After 
seconds the Universe cools to 
Celsius at densities of 
grams per cubic cm. Nuclear
particles like neutrons and protons begin to appear. Two processes
drive the evolution of the Universe. One is the rate of interactions
of these elementary particles. But as the Universe is expanding and
these particles are moving apart their chances of meeting and thus
interacting is going down. So the competing process is the expansion
of the Universe. When a particular elementary particle moves apart
faster than they can interact it stops interacting and freezes
out. One of the first elementary particle to freeze out are the
neutrinos. From now on they just expand with the Universe, unaware of
anything else in the Universe, as everything else is unaware of
them. When the Universe is about 1 second old, the first nuclear
cooking in the Universe starts. This is called big bang
nucleosynthesis. First deuterium 
and light helium 
is
produced. Then in the following 1 to 3 minutes lots of ordinary helium
and a little lithium 
is produced. By now the
temperature and densities fall so far that no further nuclear
processing can occur. Any further nucleosynthesis happens in
stars. All through this the background radiation leftover from the big
bang is energetic enough to constantly produce electron and positron
pairs, which promptly annihilate to produce the radiation back. But
after the big bang nucleosynthesis the temperature of the background
radiation falls below the threshold needed to produce electron
positron pairs and the remaining electron-positron pair annihilate to
add to the background radiation one last time. At early times the
radiation dominates the Universe, that is what drives the expansion
rate. Gradually it loses its dominance to matter and after 14,000
years after the birth of the Universe it becomes dominated by matter
as it is today. The next major event in the Universe is
recombination. Up until now the electrons and photons were closely
coupled. No photon could travel very far before being scattered by an
electron. Eventually though the Universe becomes sufficiently empty as
the electrons move away from each other, so that after around hundred
thousand years after its birth the photons can travel large distances
unassailed by electrons and the Universe becomes transparent. This
sets the distance that we can see to. The distance which corresponds
to the time of recombination is when all the background radiation we
see is coming from. This is called the surface of last scatter.
The hot big bang model makes specific predictions of the abundance of helium, lithium, deuterium and light helium that we should be left with after the initial nucleosynthesis. Depending on how much neutrons and protons we started off with we get specific amounts of the light elements in the end of the nucleosynthesis. However observationally confirming these predictions are not easy. Deuterium for example is extremely difficult to produce and very easy to destroy. Which means any measurement of deuterium, regardless of how old we think the region is, will only be a lower limit on the amount of deuterium produced in the big bang nucleosynthesis. However by comparing the amount of deuterium and light helium (which is produced by the destruction of deuterium and is itself not easily destroyed) we can put a limit on the amount of big bang nucleosynthesis. Although only trace amounts lithium is produced in big bang nucleosynthesis it turns out to be of great significance in constraining the big bang model. Once this has been done we can fix the amount of neutrons and protons we started of with. Then we have a hard prediction on the amount of helium made in the big bang model. Now of course the primary nuclear process in stars is the burning of hydrogen to produce helium. Which confuses any measurement of primordial helium. However we can plot the fraction of helium measured against the age of the region of observation. Then older the object the closer the measured fraction of helium approaches the primordial abundance. This asymptotic fraction of helium (between 23.5% and 25.4%) sits comfortably with the amount of protons and neutrons at the start with that allowed by the observations of lithium, deuterium and light helium. This was the other significant success of the big bang model.
The big bang nucleosynthesis and the observed abundance of light
elements leads to a measurement of the amount of neutrons and protons
in the Universe. This is a very significant measurement. The big bang
model of course has an expanding Universe. But the geometry of space
time in this expanding Universe is decided by the amount of matter in
the Universe. It will also decide the final fate of the Universe. This
lets us define a critical density. If the Universe has this
density then the space behaves as if it were flat (i.e., satisfies
Euclid's geometry), and is asymptotically closed. What the latter
statements means has to do with the fact that the explosion in the
Universe is slowing down because of the gravitational pull of the
matter inside the Universe. If this matter is of critical density it
will eventually stop the expansion albeit after infinite time. If the
Universe has density greater than critical density it has positive
curvature (like the surface of a sphere) and is closed, i.e., will
stop expanding after finite time turn around and collapse in on itself
in finite time. If the density of the Universe is less than critical
it has negative curvature (like the surface of water in a rotating
bucket) and open, i.e., will never stop expanding. Everything will
continue to move away from each and the Universe will become barren
and very cold. This is why it is customary to express the density of
the Universe as a ratio to the critical density, 
. It ought to be realized that the critical
density is actually an incredibly small density, only about 
grams per cubic cm.
The density of baryonic matter (i.e., ordinary matter like neutrons,
protons, electrons etc.) as predicted by big bang nucleosynthesis is
far less than critical, 
is between 0.012 and 0.037. This
would then lead us to presume that the Universe is open. But we know
there is evidence for dark matter that is different in nature from
baryonic matter. The problem of measuring the density of the dark
matter is getting a representative sample of the Universe. We couldn't
possibly measure the rotation velocity of all galaxies, and in any
case it doesn't tell us about the matter, if any, between
galaxies. This is why using large concentrations of matter, like
clusters of galaxies, are hoped to yield values of density that are
more representative of the Universe. These measurement appear to
indicate an 
of about 0.2 to 0.3, nearly ten times as much as
the density in baryonic matter! The dark matter, which we have no idea
what it is, threatens to dominate the matter. There are indications
from measurements on an even larger scale that may be as high
as 1. Along with the nature of dark matter, the question of how much
of it is there in the Universe remains one that has most astronomers
working for today.