A de Sitter universe is a solution to Einstein's field equations of General Relativity which is named after Willem de Sitter. It models the universe as spatially flat and neglects ordinary matter, so the dynamics of the universe are dominated by the cosmological constant, thought to correspond to dark energy.

Mathematical Expression

A de Sitter universe has 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:

H \propto \frac{\sqrt{\Lambda}}{M_{pl}},

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

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:

a(t) = e^{Ht} \,,

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

Our universe is becoming like de Sitter universe

Because our Universe has entered the Dark Energy Dominated Era a few billion years ago, our universe is probably 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 almost nothing left but the cosmological constant, and our universe will have become a de Sitter universe.

Acceleration becomes faster than light

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 event horizons beyond which that observer can never see nor learn any information. 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 and a few others in the gravitationally bound Local Group.

Modelling cosmic inflation

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.

See also

• Cosmic inflation
• Deceleration parameter
• De Sitter space for more mathematical properties

References

1. An Introduction to General Relativity by Ronald Adler, Maurice Bazin, and Menahem Schiffer

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