Notes on General Relativity and Cosmology

 (updated 8/30/04)

What do we observer?

• The sky is dark (Olber's paradox: If the universe were infinite then every line of sight would end on a star, eventually so the sky should be bright, not dark).
  
• The universe is isotropic (looks the same in every direction, at least on a large scale > 200Mpc).
• The universe is homogeneous (looks the same at every location; we are not at a privileged location).
• There is a cosmic microwave background radiation which is very uniform corresponding to a black body curve for an object at 2.725K + or - 0.001K.
• Objects (stars, galaxies etc.) are moving faster the further away from us they are in a nearly linear relation. We also can measure how this rate changes over time since when we look out into space we are looking backward in time (it isn't exactly linear- see below).
  
• There are no magnetic monopoles in the universe.
• The matter that we can detect in the universe is about 3/4 ths hydrogen, 1/4 helium and a tiny fraction consisting of all the other baryonic material. (i.e. normal every day stuff; protons neutrons, etc.).
• Galaxies and superclusters of galaxies do not fly apart (even though it looks like they don't have enough mass to stay together, if we only count matter that we can detect directly).
  

What is the current model (called the Benchmark Universe) of how the universe is constructed?

• The General theory of Relativity is the correct set of equations which describe the large scale structure of the universe.
• The universe started with a hot big bang.
• There was a period early in the universe where there was a sudden expansion (called inflation) before settling down to a more steady expansion.
• The expansion rate of the universe is now accelerating and has been for about the past 4 billion years.
• Currently about 0.001% of the universe is made of photons.
  
• Currently about 4% + or - 1% of the universe is made of baryonic material (the stuff we can detect).
• Currently about 26% of the universe is made of dark matter (which keeps galaxies and clusters from flying apart). This could be neutrinos, black holes, WIMPS (weakly interacting massive particles) or some other unknown mass.
• Currently about 70% of the universe is dark energy. This could be vacume energy or something else we don't yet know about.
  

How do the observations support the current model?

Evidence for Einstein's equations:

• Newton's equations told us: Mass tells gravity how to exert a force (F=GMm/r²) and F=ma tells a mass how to move as a result of a force. He has no explanation for why m is the same in both equations. Einstein's equations tell us: Mass/energy tells space time how to curve and curved space time tells mass/energy how to move.There is no difference between inertia mass and gravitational mass as there is in Newton's theory.
• The three major tests we have made of Einstein's equations (the shift in the perihelion of Mercury, the red shift of light traveling away from a massive object, the bending of light by the sun) turn out correct. Global Position Satellite positioning and airplane navigation makes use of the theory to give correct positions.
  
• Einstein's equations tell us there should be black holes and we detect them (rather, we see evidence that they are there such as gravitational lensing).
• Einstein's equations tell us the universe should be either expanding or contracting; it is expanding.
• Once the big bang has started, Einstein's equations give the correct amount and time for galaxy and star formation (clumping due to gravitation).
  

Evidence for the hot big bang:

• The Doppler shift (red shift) of stars tells us how fast they are moving. From this Hubble found that most stars and galaxies are moving away from us and the ones further away are moving away faster. This tells us the universe is expanding. (No, we are not at the center, think about dots on an expanding balloon, they all move away from each other and each dot will see the dots further away as moving faster.). The rate of expansion has been approximately constant over time but not quite (this small change is significant).
  
• An expanding universe solves Olber's paradox. The universe is not infinite in time so light has not had enough time to reach us from stars that are further than a certain distance (the horizon). Current calculations of the horizon distance and the age of the universe are consistent with each other.
  
• If we extrapolate backwards in time to a point when the matter and energy density of the universe is very high (the first few minutes of the universe) we can apply the equations of nuclear physics since the universe at that time is a plasma (similar to what goes on in a hydrogen bomb when it explodes). At first the energy density is so high that quarks cannot come together to form protons and neutrons. At a later time the energy density is low enough to form protons and neutrons but not atoms because the photons present have enough energy to keep all atoms completely ionized. At a still later time the energy density drops to a point where photons decouple from matter and atoms can form. All of this happens in the first three minutes of the existence of the universe. When the equations of nuclear physics are solved for this transition period we get a prediction that 3/4ths of the matter in the universe should be hydrogen, 1/4 helium and small amounts of isotopes of hydrogen (i.e. deuterium), lithium and beryllium. These predictions are very precise and they give the exactly quantities we see in the universe today, with the exception of heavier elements which are created in stellar processes (fusion in stars and nova). Nuclear physics (physics of the tiny) at the big bang determined the makeup of the universe we see today.
• The energy of the photons when they decouple from matter in the first three minutes continue to travel through the universe but they are very red shifted because of the gravity of the 'stuff' in the universe. These photons are what we see as the cosmic microwave background. Saying the photons are red shifted as they travel through the universe is exactly equivalent to saying the universe cools over time and it's black body radiation shifts to lower frequency. The current temperature of the cosmic microwave background is exactly what is predicted from the temperature at the time of the photon decoupling and subsequent cooling since then.
  
• Einstein's equations predict either an expanding or contracting universe, not a static one. All tests of the equations have been successful so an expanding universe is consistent with Einstein's equations.

Evidence for inflation:

• A sudden expansion of the early universe solves the flatness problem: The universe appears flat today (i.e. not much acceleration- the rate of expansion is fairly constant). In order to be this flat it had to be pretty flat to start with. Imagine an ant crawling on a balloon. How could we make the balloon appear more flat to the ant? Make it bigger. A sudden inflation of the early universe does this for us.
  
• A sudden expansion of the early universe solves the horizon problem: How did the universe become isotropic? For the microwave background to be that smooth, the universe had to be well mixed yet we know that some parts of the universe have not had time to communicate (mix) with other parts. So how did the universe become isotropic? If the universe was smaller than we thought at the beginning the various parts could mix before losing contact with other parts in a sudden expansion.
• A sudden expansion solves the monopole problem: We don't see any monopoles although theory says they should have existed in large numbers at the beginning. A sudden expansion means the density of monopoles is now very low, so low we do not expect to find any.
• The very small variations in the current cosmic microwave background tell us the amount of 'clumpyness' of the early universe if we extrapolate backwards in time. This clumpyness fits nicely with the idea that the universe at the very beginning was smooth but quantum fluctuations (due to Heisenberg's uncertainty principle) caused it to be just a little lumpy. The lumps got stretched out into the current observed microwave variations by inflation. This same lumpiness also made star and galaxy formation possible (a perfectly smooth universe would not form stars). So due to inflation, the quantum fluctuations of the early universe led to galaxy formation and variations in the microwave background which are observed.
  

Evidence for 'dark matter':

• Without a halo of dark matter (something which has mass/energy) there is not enough gravity to keep galaxies and super clusters of galaxies together. Based only on the material we can see, galaxies should fly apart but they don't. Gravity gives us a specific value(26% of the universe) for how much dark matter there must be to keep things together.

Evidence the universe is accelerating and there is 'dark energy':

• We can get the rate of expansion of the universe from red shift data and also the rate of change of expansion (the acceleration). The acceleration is determined by how much 'stuff' there is in the universe (much like if you were to shoot a rocket straight upward; a bigger earth would pull it down, a smaller earth would not because it would cause less acceleration). The current acceleration is very small (i.e. the expansion rate is fairly constant- but see the next bullet).  This means two things: the universe has very little curvature (is flat) and there ought to be a lot more 'stuff' (70% more!) than what we detect as either baryons or 'dark matter' to account for the small acceleration we do measure.
  
• In 1998 two separate groups using different techniques measured the red shift for 'standard candles' (astronomical objects with known light emitting properties) for galaxies very far away. These were Type Ia supernova and are the furthest away of anything we can measure as a standard candle. Because of the expansion of the universe we expect objects which are further way to be moving faster (and therefore have a higher red shift). These groups expected to measure a slowing of the expansion (a deceleration) and get a better value for the missing 'stuff' mentioned in the last bullet. But these Type Ia supernova are moving a little TOO fast for their distance. This means the universe has to be accelerating, not decelerating. 
• In order to get an expanding universe from Einstein's equations we have to add a term corresponding to negative energy (Einstein originally tried to put his term in to get a static universe but then took it out when he found out the universe was expanding). Positive mass/energy tends to slow (decelerate) the expansion (this was in fact what the astronomers were trying to measure when the found the acceleration). So not only is 70% of the universe stuff that we can't detect and have no theory for, it has to have negative energy!
  
• Fluctuations in the cosmic microwave background are also very consistent with a flat universe where 70% of the 'stuff' is missing. Notice this is two totally different sets of measurements (supernova and microwaves) giving the same result (70% of the universe is missing).
• In quantum mechanics we will learn that because of the Heisenberg uncertainty principle, what we think of as a vacume can have mass/energy fluctuating into existence and then out of existence for short times. On a small scale this makes very little difference (although it has been measured- see the Casmir effect). On the scale of the universe, however, this is a lot of 'stuff'. Unfortunately the current calculations give a number that is way TOO BIG (more than the missing 70% by a factor of millions!!!). Keep in mind, however, that General Relativity and quantum mechanics are not compatible. So it is possible that at some point in the future a quantum version of General Relativity will correctly predict the amount of missing stuff.