The existence of three or more such observers places very strong constraints on the possible structure of space-time. For a start, there is a "natural" time coordinate: the time as measured by each co-moving observer, equipped with a standard clock. Notice that their clocks all measure the same time because they are all controlled by the same rules of physics, and the observers can synchronize their clocks by agreeing to define t=0 to be some special moment in the life of the universe, for instance its beginning (if it had one). The existence of a cosmic time is an enormous simplification; without it, there is no sensible way of separating space from time in the 4-D continuum of space-time. Next one can show that in such a universe there could be a co-moving observer at every point, all of whom see the universe as isotropic, i.e. 3 co-moving observers imply an infinite number of co-moving observers.1 Next, you can show that at a given time, the universe is homogeneous (the same at all points). Finally, there is also a "natural" way of fixing space coordinates: we can choose co-moving co-ordinates which move along with the co-moving observers (this gives us a way of defining what we mean by the "same" point at different times).
In such a universe, the interval (space-time separation) between events ("points" in space-time) can be described by the Robertson-Walker metric. By fixing the distances between all points, the metric also defines the geometry of space-time, and, because there is a meaningful cosmic time, the geometry of space at a given time. In fact, there are only three possibilities for the local geometry of space, because the curvature of space must be the same at all points (homogeneity) and not pick out any particular direction (isotropy):
In this equation the evolution of the Universe is governed by the scale-factor R(t): the term in square brackets is the square of the co-moving distance, dl ², and the physical distance between two points is just R dl. At the moment R is increasing with time, which is another way of saying that the Universe is expanding.
The geometry of space at a given time is defined by the function Sk(chi), which is just a neat way of writing the 3 different functions needed for the 3 values of k:
Notice that the geometry of space cannot change with time in a homogeneous
universe: if the curvature is negative to start with, it stays that way.
The same applies to topology: a fundamental observer's neighbours cannot
suddenly (or gradually) change with time.
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1 This does not mean the universe seems isotropic to every possible observer. Although SR emphasises that all observers should be treated equally, in cosmology the co-moving observers are more equal than others: anyone who presumes to move relative to them will get a distorted view of the universe in which the observer's direction of motion relative to the co-movers will pick out a special direction in space.
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