The evidence for the Big Bang comes from many pieces of observational data that are consistent with the Big Bang. None of these prove the Big Bang, since scientific theories are not proven. Many of these facts are consistent with the Big Bang and some other cosmological models, but taken together these observations show that the Big Bang is the best current model for the Universe. These observations include:
- The darkness of the night sky -Olbers'
paradox.
- The Hubble Law - the linear distance vs redshift law. The data is
now very good.
- Homogeneity - fair data showing that our location in the Universe is
not special.
- Isotropy - very strong data showing that the sky looks the same in
all directions to 1 part in 100,000.
- Time dilation in supernova light curves.
The observations listed above are consistent with the Big Bang or with the Steady State model, but many observations support the Big Bang over the Steady State:
- Radio source and quasar counts vs. flux. These show that the
Universe has evolved.
- Existence of the blackbody This shows that the Universe has evolved
from a dense, isothermal state.
- Variation of TCMB with redshift. This is a direct
observation of the evolution of the Universe.
- Deuterium, 3He, 4He, and 7Li abundances. These light isotopes are all well fit by predicted reactions occurring in the First Three Minutes.
Finally, the angular power spectrum of the CMB anisotropy that does exist at the several parts per million level is consistent with a dark matter dominated Big Bang model that went through the inflationary scenario.
http://www.astro.ucla.edu/~wright/cosmology_faq.html#BBevidence
http://www.astro.ucla.edu/~wright/cosmolog.htm