Xyooj said:
when do a person earn that title?
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When you say title it sounds like you are talking about degrees, like PhD, and professorship ranks. It has institutional connotations.
Some universities have a separate department of Astronomy, or a department of Astrophysics, which is separate from the main Physics department. In other universities Astronomy is included in the Physics department.
So in some cases you would just get a PhD in Physics and you would then do your research in cosmology. Cosmology would be your specialization. there is not always a separate TITLE for a research specialty. People would get to know that your special interest was cosmology after you had published some papers in that field.
If you got to be promoted to a professorship, then you would be Professor of Physics, or Professor of Astronomy, depending on how the departments were organized. But you would not normally have a title like "Professor of Cosmology".
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The people I think of as cosmologists are people whose work I know, the ones I see being invited to give the featured talks about cosmology at conferences, and being invited to write the major survey papers reviewing the progress in their specialty. They are usually the ones whose papers are highly cited by other researchers in that field.
I don't think there is an official certificate. It is just a
type of astronomer.