Basic assumptions
The fundamental laws of physics do not change with time. We do not have any observational evidence that supports a variation in the basic laws of physics or the fundamental physical constants such as electron or proton mass, charge, spin, fine structure constant etc. We can and do test this assumption by looking back in time when we see objects that are far away.
The perfect cosmological principle wherein the universe itself does not evolve, that is, looks the same at all times, is no longer accepted by most astronomers. This assumption leads to the steady state cosmology and is at odds with the big bang cosmology. (The universe can expand, and as it does the density is kept constant through the creation of new matter. Why assume everything was created at a single instant?)
Einstein's theory of relativity (gravitation) is correct. Newtonian cosmology is quite an important thing to understand and agrees with general relativity (GR) in the limit of no gravity (flat space). But discrepancies between Newtonian and GR are obvious in the solar system and GR wins hands-down. To date no non geometric theory of gravitation has been able to compete with GR. A number of more complicated theories have been proposed (Dicke's scalar-tensor) but GR has so far agreed with all experiments.
The early universe was filled with an almost perfectly uniform intensely hot plasma composed of elementary (quarks and leptons?) particles in thermodynamic equilibrium. This is really the basis of the standard big bang theory of the universe. The models that start with this assumption after -say- a femptosecond are quite succesful. The real trick is getting this condition. This is where the standard big bang is very vulnerable.
The gas filled all available space. OK.
The changes in state of matter and radiation are smooth. That is, there are no phase transitions that cause an abrupt evolutionary change. Not unlike water freezing or thawing, it is assumed if there exist such things as "symmetry breaking" transitions they don't do anything. Relaxation of this assumption leads to the difference between the standard and inflationary big bang theories.
The standard big bang cosmology predicts
The 2.7K background radiation. No contest. This was predicted before Penzias and Wilson detected this "cosmic background radiation", a remnant left over from the time that the expanding hot plasma fell below 3000K and became transparent (so this corresponds to a redshift z of about 1000).
The composition of the early universe. After a few minutes the expanding plasma cools to the point that it looks just like the interior of a star, conditions well understood by astronomers. A straightforeward application of nuclear astrophysics results in a composition at the end of this expansion interval that agree well with observation. The early composition is intimately related to the current density of the universe.
Three families of quarks and leptons. One aspect of the formation of the elements in the early universe depends on the number of quark/lepton families there were since the number governs the ratio of protons to neutrons at the time of helium synthesis. four families gives too many neutrons, two families too few. Three is just right. And now the particle physicists by watching the decay of the Lo find there are three families. Astronomers were there first. By the way, for there to be more matter than antimatter you need at least three families of quarks and leptons.
You cannot discount these results. But Big bang cosmology does have some problems that people love to jump on. Most recent is a problem with the age of the universe obtained from Hubble's constant being less than the age of globular clusters or the oldest white dwarf stars. It is always the big bang that is jumped on, never the ages which somehow are reported as gospel. But there are some real problems that are rather subtle. Fortunately, nobody worries about something they cannot really understand as being a problem!
The horizon problem. The 2.7K cosmic background radiation (CBR) has been mapped with low resolution by the COBE and found it to be, apart from the expected dipole asymmetry due to the motion of the sun through the cosmos, remarkably uniform. Too uniform in fact. Big bang cosmology always predicts a horizon for any observer, to which light signals can propagate over the lifetime of the universe, that is much smaller than the present observable radius of the universe. That is, if you look at the CBR in one direction then look at a reginon in the opposite direction (the most extreme case) the two regions have never been in thermal contact. Now this is certainly "the perfectly uniform" assumption verified but completely unexplained. There is just no reason for this remarkable uniformity. In fact, it is almost as if there were no quantum fluctuations allowed until the first femptosecond and that violates our "no law change" assumption!
The smoothness problem. Sure, the background radiation is smooth yet we do have structure on the size of galaxies, clusters and superclusters. Now the thing about the universally attractive force of gravity is that in any model of the universe any small amount of uniformity must grow very fast. We can "run the reel backwards", that is, take the present structure and work backwards to estimate the initial roughness. It turns out to be incredibly small, even at z=1000. If you go back to log(t)=-40 the universe has to be fantastically but not exactly uniform. Coupled with the fact that one part of the universe has never seen the other part is a real dilemma. In fact, many astronomers at this point resort to the anthromorphic principle, "it had to be this way or we just would not be around here to argue about this"...
The flatness problem. If you pick a cube of space having sides ten megaparsec or so you will probably find a observable matter density of about r=6´ 10-28 kg/m3. Dark matter must increase this by a factor of ten, perhaps even more. Depending on the initial conditions, big bang models can predict a universe that expands forever (a low density or "open" universe) or expands to a maximum radius, halts, then contracts to a "big crunch" (high density or "closed" universe). There is a critical model that expands asymptotically to a zero expansion rate H(infinity)=0 and requires very special initial conditions (a critical density or "flat" universe). The flat model requires a density that is within an order of magnitude of the 10% observed plus 90% dark matter universe we live in, and it is not impossible that there is enough dark matter to close the universe. Nobody ever comes to me and says "it must be flat because it just has to be that way". I like the idea of a critical universe but I can also appreciate that there is no reason in the world for the observed density to be within a hundred orders of magnitude of critical. It is all really strange. Note that just like smoothness the flatness (often described by "omega", the ratio of observed density to the density required to just close the universe Wo=ro/rc about 0.1 at present) rapidly divirges from W=1 so if it as within a factor of ten now than at t=ten to the -35 seconds it would have to be almost exactly one.
Odd Physics
Everybody knows that Maxwell unified the forces of electricity and magnetism around 1850 and we now refer to the electromagnetic or simply electric force. Fewer know that the electric and weak forces were sucsessfully "united" in 1960. This is a bit different because the electric force only acts on charged particles while the weak force acts on just about everything. But on a small enough scale the forces are essentially the same and they were indistinguishable in the very early universe. At t=10-12 seconds the forces "froze out", a process sometimes called spontaneous symmetry breaking.[Some examples of spontaneous symmetry breaking follow:
Balance a pencil on its point. The symmetry will soon be broken.
Twist a rope ladder. This destroys one form of symmetry but there is still order from rung to rung. When you twist too much the rope between two of the rungs will "kink", breaking the order.
Put a bead on a vertical hoop, it sits on the bottom. Spin the hoop and it still sits on the bottom. Above some critical angular speed the bead will hop up one side or the other breaking the symmetry.]
A recent theory known as the Grand Unified Theory (GUT) unifies the electroweak and strong nuclear forces that natrally predicts this phase change at about t=10-34 second. The GUT has an added feature that in the broken symmetry phase that we enjoy now a certain physical law known as "conservation of baryon number" is obeyed but not before the phase transition. This could explain the observed fact that matter predominates over antimatter. It also predicts that protons decay (half life greater than 10+35 years) and an experiment (kamiokande, the same one used to detect neutrinos from the sun and supernovae) is underway to verify this. A real drag if true since after 1040 years the universe would be a pretty dull place. Unless we have a big crunch.
Now for something a bit different. The GUT requires a set of two special "Higgs fields" which account for spontaneous symmetry breaking. At very high temperature like existed prior to t=10-34 second both fields are "zero" and the vacuum state is symmetric but nonzero. When the temperature falls low enough one of the fields becomes nonzero, the symmetry is broken and the vacuum makes a transition to the "true vacuum" state with enormous release of energy. The symmetry breaking causes the universe to increase or inflate over fifty orders of magnitude in 10-33 second. Thereafter the universe is identical to the standard big bang. But the inflation very neatly solves the horizon, flatness and smoothness problems.
The horizon problem is avoided since just prior to inflation (when inflation and standard theory are in accord) the universe evolves from a region fifty orders of magnitude smaller than the horizon distance and so has time to thermalize. The horizon distance for the inflationary model is many orders of magnitude greater than the radius of the observable universe while in the standard model it is always less. So the isotropy of the cosmic microwave background radiation is explained.
The flatness problem is eliminated because whatever W was before inflation it is naturally driven to unity during the inflationary phase. Recall that in the standard model W could be anything at all. the fact that 0.1>W>2 is very compelling. The figure shows how the horizon prior to log(t)=-35 was much greater than the universe (everything was in touch so to speak) while after inflation it is less. Motion faster than the speed of light? No. Nothing moves, space just gets bigger really fast!
The smoothness problem might be explained by inflation since it predicts a spectrum of inhomogenieties that are sensitive to GUT details and will be reflected in the slight roughness that is now being found on the cosmic background radiation. (It is smooth, but not completely smooth.) Unfortunately, the simplest GUT predicts fluctuations that are far to large. (It also predicts a proton half life that is far too short.) Astronomers are looking for the spectrum of inhomogenieties in the cosmic background and particle physicists are working on the GUT. And even trickier theories that unite gravity, strong and electroweak such as supersymmetry, supergravity, and superstring theories.