Observational evidence for the Big Bang includes the analysis of the spectrum of light from galaxies which reveal a shift towards longer wavelengths proportional to each galaxy's distance in a relationship described by Hubble's law. If this observation that galaxies are moving away from us is not at any special position in the universe (the Copernican principle), then this suggests that space itself is expanding. Extrapolation of this expansion back in time yields an initial state in the distant past in which all matter and energy was at an immense temperature and density. This initial state is the key premise of Big Bang theory.
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