# De Sitter universe

A de Sitter universe is a solution to Einstein's field equations of General Relativity which is named after Willem de Sitter. It is common to describe a patch of this solution as an expanding universe of the FLRW form where the scale factor is given by:

[itex] a(t) = e^{Ht} [itex],

where the constant H is the Hubble expansion rate and t is time. As in all FLRW spaces, [itex]a(t)[itex], the scale factor, describes the expansion of physical spatial distances.

A de Sitter universe is one with no ordinary matter content but with a positive cosmological constant which sets the expansion rate, H. A larger cosmological constant leads to a larger expansion rate:

[itex]H \propto \frac{\Lambda}{M_{pl}}[itex],

where the constants of proportionality depend on conventions. The cosmological constant is [itex]\Lambda[itex] and [itex]M_{pl}[itex] is the Planck mass.

Our universe may be approaching a de Sitter universe in the infinite future. If the current acceleration of our universe is due to a cosmological constant then as the universe continues to expand all of the matter and radiation will be diluted. Eventually there will be nothing left but the cosmological constant, and our universe will have become a de Sitter universe.

The exponential expansion of the scale factor means that the physical distance between any two non-accelerating observers will eventually be growing faster than the speed of light. At this point those two observers will no longer be able to make contact. Therefore any observer in a de Sitter universe would see horizons beyond which that observer can not see. If our universe is approaching a de Sitter universe then eventually we will not be able to observe any galaxies other than our own Milky Way.

Another application of de Sitter space, is in the early universe during cosmic inflation. Many inflationary models are approximately de Sitter space and can be modeled by giving the Hubble parameter a mild time dependence. For simplicity, some calculations involving inflation in the early universe can be performed in de Sitter space rather than a more realistic inflationary universe. By using the de Sitter universe instead, where the expansion is truly exponential, there are many simplifications.

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