My preprint "The Beginning to the End of the Universe and Eternal Space-time " is available at:
https://www.preprints.org/manuscript/202207.0121/v3
In this manuscript I explained the brief history of the universe from the beginning to the end of the Universe. My research suggests that the universe will not continue to expand forever, no need however, for dark energy. The theory of eternal Space-time and it consists of unique properties, which enables us to describe a sequence of events from the Big Bang to the Big Crunch. It appears that we now have two disparate possibilities: It could be that our universe is cyclic and no beginning; there may have been Big Bangs before ours, and a universe with a definite beginning. The ultimate arbiter will be Nature.
Currently, this model is best understood by pictures rather than a large number of equations.