My question specifically, how has the curvature of the universe remained just about flat topologically as the universe has expanded and thus if the expansion of the universe is accelerating then suppose in a few billion years will there be a decrease in energy density of matter that causes the curvature to change( e.g. to being more hyperbolic) ?


You are mixing different concepts and things. First, the universe is flat is its space dimensions, and will always be so. Spacetime is curved but it is the time dimension that contributes to it, while you can consider the constant cosmological time spatial slices as flat. Easy to get confused.

The dark energy density already is larger than the matter density, 67% of total energy density vs 33%. As we go to the future the universe expands diluting the matter density but the energy density remains the same, and eventually is all there is (yes, billions of more years). Since there is more space as the universe expands, a constant dark energy density means more total energy. But this energy has negative pressure and causes acceleration of the expansion. This is all part of the standard cosmology model, the so called Lambda-CDM model (lambda for the dark energy labeled that way, CDM is cold dark matter which is about five time more than normal matter)

So as we go further to the future the spatial slices continue to be flat (in the large scale), and as it expands faster it becomes more and more homogeneous, with all matter diluting. The scale factor (defined as 1 now, meaning the ratio of the spatial dimensions sizes to today's) as we come closer to 100% dark energy, starts growing exponentially with an e-doubling time the inverse of the Hubble parameter, and the universe expansion rate and acceleration grow with the same exponential e-doubling time. This spacetime at late stages is so called deSitter spacetime, and can be sliced as in cosmology with flat spatial slices. See the cosmology model at


and the deSitter model at https://en.m.wikipedia.org/wiki/De_Sitter_universe

And deSitter spacetimes at https://en.m.wikipedia.org/wiki/De_Sitter_space


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