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VBHT18: Big Bang Cosmologies
The basic framework for the physics of the universe on the largest scales is the family of homogeneous, isotropic cosmological models, the so-called Big Bang models.
This part will be the first of a subseries talking about important solutions to the Einstein field equations. This time around I'll explain the physics of homogeneous, isotropic cosmologies, the next part will consider the evidence for the Big Bang, and then I'll move on to discuss non-rotating, uncharged black holes and maybe some other exact solutions too.
The most basic element of Big Bang models is the idea of "cosmological observers" who ride along with the expansion (and possible later contraction). The time coordinate of an event in the universe is then the proper time that elapses between the initial singularity and that event for the cosmological observer whose worldline passes through the event. Each cosmological observer is given a set of spatial coordinates which remain constant along that observer's worldline, so that the spatial coordinates of an event are just the coordinates carried by the cosmological observer who passes through that event. It's important to note that these coordinates only serve to label events and not to measure distances between them. To convert differences of coordinates into proper distances (in other words, the minimum number of standard length rulers you can lay end to end between the two locations) we need to know the metric for the universe, and to find that we need to solve the Einstein equations.
It's an observational fact that as we look at the universe on larger and larger scales it looks more and more homogeneous and isotropic. Homogenenous means that all places look the same and isotropic means that all directions look the same. This means that to a good approximation we can describe the dynamics of the universe by considering homogeneous, isotropic world models. In more careful terms what this means is that if you pick a given amount of time and then see how the universe looks for each cosmological observer when that much time has elapsed since the initial singularity then you'll find that each observer will see the same local properties of spacetime and local matter content, and will see the same things further out in space when looking in any direction. Cosmological models that obey these conditions are called Friedmann-Robertson-Walker models. If you pick a given amount of time and then look at the spacelike surface given by the events that much time along the worldlines of all the observers then there are only three possible intrinsic geometries for the slice (i.e. the intrinisic geometry is the set of geometrical properties that can be determined using only measurements within the slice): it can be spherical, flat or hyperbolic. Note carefully that even if the spacelike surfaces are all flat it's still possible for the spacetime to be curved.
If the universe is homogeneous and isotropic then the only quantity associated with the geometry that can vary with time is the "scale factor". The spacelike parts of the metric, which enable the conversion of coordinate differences to proper distances at a given cosmological time, only vary with time through this scale factor: as the scale factor varies, the distances vary in the same proportion. If the scale factor is increasing with time (as it is now and probably will be forever) then "space is expanding" and if the scale factor is decreasing then "space is contracting". An expanding universe isn't expanding into anything, or even expanding to fill some pre-existing void; it's just that all the distances between cosmological observers are increasing. All those graphics that are shown on television shows that make the Big Bang look like a really impressive supernova fling matter into space are just plain wrong.
The FRW models are actually a whole family of universes. To get one specific universe, you have to decide what matter to put into it. The really neat thing about FRW cosmologies is that we can deduce the dynamics from purely locally considerations. Recall the Einstein equations as described in the last part. We consider an initially comoving sphere of test particles and look at how the shape and volume of the sphere changes as time advances. The Einstein equations directly relate the change in volume of the sphere to the energy density and pressure inside the sphere (that is, to the matter content of the universe). If the universe is isotropic then the shape of the sphere cannot change with time but only its radius, and the radius is directly proportional to the scale factor. It's conventional to talk about universes filled with "dust", which is a technical term for matter whose pressure is very, very small compared with its energy density, so that the pressure can be neglected. The fact that particles of dust are conserved (as each particle of "dust" is really a whole galaxy) means that that we know how the energy density varies with scale factor, because the energy density is really the same thing as the total mass of galaxies inside our sphere divided by the volume of the sphere. Combining this fact with our nice simple form of the Einstein equation gives us the equation that determines how the scale factor varies with cosmological time.
We can also consider universes filled with different types of energy. For example, a radiation filled universe has an energy density that falls as the inverse fourth power of the scale factor because the number density of photons falls as the inverse cube and each photon also gets redshifted to lower energy as the inverse of the scale factor. Another example is a universe with a cosmological constant, which is an energy density associated with space itself that stays constant as the universe expands. If we're feeling especially ambitious, we can even work out the dynamics of a universe containing a mixture of stuff. Such a universe will start out with dynamics dominated by its radiation content, then pass through a stage dominated by matter and finally end up dominated by the vacuum itself.
A quite remarkable thing about the equation for the variation of the scale factor in a matter-dominated universe is that it looks just like the equation for the motion of a test particle in the field of a massive body under Newtonian gravitation. Imagine throwing a ball upwards from the surface of the Earth. If you throw it too slowly then it will just fall down again, but if you throw it at the escape velocity then it will just get to infinite distance, but moving ever more slowly, and if you throw it even faster it will get to an infinite distance moving at a finite speed (of course, it will take an infinite time to do this, but you can use the equations to work out what the speed would be if you waited an infinite time). The escape velocity depends on the mass of the body. Similarly, in the case of cosmology there are three classes of universe. Given a certain amount of matter content, there are universes that are expanding more slowly than a critical rate and which will fall back in on themselves to a "Big Crunch", there are universes expanding exactly at the critical rate which will go on expanding forever but ever more slowly, and universes expanding at greater than the critical rate, which will never slow down towards zero expansion. Even more remarkably, these three classes correspond to the three classes of intrinsic spatial geometry: universes with a Big Crunch have a spherical spatial geometry, those expanding at the critical rate have flat spatial geometry, and those which are expanding at greater than the critical rate have hyperbolic spatial geometry.
(As I mentioned above, a "flat" universe can [and will] still have a curved spacetime. Just imagine a simple three-dimensional model for the four-dimensional universe. Each spacelike slice will be a flat sheet with some galaxies marked on it, and in each successive sheet the galaxies will be in the same relative positions, but with a larger distance scale. Now imagine the worldlines of cosmological observers, each of which threads through all the images of a particular galaxy. The way things work for a flat universe is that high in the stack [i.e. at late times] each of the worldlines is pretty much vertical, but as you go back towards the Big Bang they curve towards each other, and eventually intersect at the initial singularity. The fact that worldlines that are parallel at late times intersect as you follow them back in time shows that spacetime isn't flat even if the spacelike slices are, because each of the worldlines is a geodesic of the spacetime and parallel geodesics can only intersect in the presence of curvature.)