How does nucleosynthesis provide evidence of the big bang?
Big bang nucleosynthesis explains the abundance of Helium in the universe.
If matter is being produced in a relatively cool universe, it would be mainly Hydrogen. In fact much of the material in the universe consists of 74% Hydrogen and 26% Helium. This asks the question "Where did all the Helium come from?".
The big bang can answer this question. The early universe had a very high temperature. There were no atoms. It was filled with photons, electrons, protons and neutrons.
Protons are effectively stable. Neutrons are unstable and decay into protons with a half life of about 10 minutes. Given enough energy protons can become neutrons.
In the early universe the number of protons and neutrons were about the same. As the universe cooled, neutrons started to decay faster than they were being produced. Another consequence of cooling was that protons and neutrons were able to fuse to become isotopes of Hydrogen and helium.
This fusion process is called big bang nucleosynthesis. It explains the relative abundance of Helium in the universe. It is seen as evidence providing evidence the big bang.