It is sometimes said that the universe is homogeneous and isotropic. What is meant by each of these descriptions? Are they mutually exclusive, or does one require the other? And what implications rise because of this?
Answer
A homogeneous cosmology is one in which there are no "special" places in the universe: at a given instant in time, the universe appears the same at every location (on large enough spatial scales).
An isotropic cosmology is one in which there are no "special" directions: at a given instant in time, the universe appears the same in every direction (again, on sufficiently large spatial scales).
Together, they form the Cosmological principle.
As pointed out by Brian Hooper, these symmetries (when applied to physical laws) give rise to the conservations of linear and angular momentum as a result of Noether's theorem.
In addition, the cosmological principal is important for the physical interpretation of observational data, and not only because it is a generally unspoken assumption when using physical laws tested on Earth to model distant objects (galaxies, quasars, etc.) For example, it supports the interpretation of the Hubble diagram as the result of the expansion of the universe as opposed to evidence that the Earth (or someplace "nearby") was at the center of a very big conventional explosion. After all, in a conventional explosion, the fragments that travel the farthest are those that had the highest velocity, so some time after the explosion, faster moving fragments are further away from the center. If, however, observers on all fragments see the same density of galaxies and relationship between velocity and distance in all directions, then this model doesn't work.
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