Cosmology is the study of the origin, the evolution,
and the eventual fate of the universe.
The term physical cosmology can be used to
specify that physical science is the basis of your study.
The term cosmos is a synonym for universe.
Current physical cosmology is based on
the cosmological principle that sufficiently basic natural
laws that apply here also apply to the far reaches of the
universe, i.e., objects observed at a distance are assumed
to follow such laws. General relativity plays a big role in current
cosmological theory (relativistic cosmology).
Current thought places the beginning of the universe at about 13.8
billion years ago, the term for this beginning being the Big Bang.
The Lambda-CDM model is the current favored detailed model, much
contemporary cosmological research effort being to pin down its
details through observation, and to explore its problems, such
discrepancies with observation, and to explore possible improvements.
The adjective cosmological is often used to mean "involving
distances or times when cosmology is relevant", i.e.,
the early universe and distances at which only the
early universe is visible because of the time
required for the electromagnetic radiation to reach us
(cosmological distances).
The adjective cosmic has the same meaning (at least roughly)
and often which is used seems mere preference,
or habit regarding a particular phrase, e.g., cosmological distance
versus cosmic distance scale.
A cosmogony is a theory or model of the origin of the universe.