Red shifting light is a feature of special relativity.
In classical mechanics, when light is emitted from an object that is moving away from you, it will appear to be moving towards you slower than when emitted from a stationary observer.
This does not fit with observed facts.
In Special Relativity, the speed of light is constant for all reference frames. In relativistic mechanics, the wave length of the light shifts. For an emitter moving away from you, the light will appear red and for an emitter moving towards you, the light will appear blue. This is just Doppler shifting. This does fit empirical facts.
So the biggest supporting factor for it is the doppler shifting of everything outside our 'local cluster' and the fact that special relativity tells us that this means that every point in the universe is moving away from us.
Another huge supporting bit of evidence is the Cosmic Microwave Background Radiation. This is coming at us from all direction in the night sky and we interpret it as a record of the big bang.
Another bit of evidence is that the ratios of elements in the universe are consistent with a universe being around 13 billion years old. We can count the generations of stars that could occur in that timespan and since they manufacture the elements, the number of generations of stars affects the ratio of hydrogen to everything else.
The big shortcoming in it is that, as the theory stands today, one has to have a non-zero gravitational constant in Einsteins field equations for it to work (this is general relativity now). So the question is, "where does this come from?" In Quantum Field Theories, one gets a vacuum effect where one constantly has particle/anti-particle virtual pairs popping into existence and annihilating each other. This is one popular candidate but all the QFD's give a value for it of around 1 and our observations indicate that it should be slightly greater than 0. Another way of going about it is to postulate the existence of a 'quintessence' field. This is not well understood.
At any rate, postulating a non-zero graviational constant is supported by the evidence and clears up a couple of other questions, for example "why is the expansion of the universe accelerating as opposed to slowing down as GR would have us expect?"
Another possible solution is to propose that gravity tapers off to mM/r at great distances as opposed to mM/r^2. This is supposed to fit the facts almost perfectly but nobody's figured out how to make it work.
All in all, it's a pretty good theory with some problems.
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