De Sitter universe

A de Sitter universe is a cosmological solution to the Einstein field equations of general relativity, 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 in our universe or the inflaton field in the early universe. According to the models of inflation and current observations of the accelerating universe, the concordance models of physical cosmology are converging on a consistent model where our universe was best described as a de Sitter universe at about a time

t

{\displaystyle t}

= 10−33 s after the fiducial Big Bang singularity, and far into the future.

Mathematical expression

A de Sitter universe has no ordinary matter content but with a positive cosmological constant (

Λ

{\displaystyle \Lambda }

) that sets the expansion rate,

H

{\displaystyle H}

. A larger cosmological constant leads to a larger expansion rate:

H

∝

Λ

,

{\displaystyle H\propto {\sqrt {\Lambda }},}

where the constants of proportionality depend on conventions.

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

H

t

,

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

where the constant

H

{\displaystyle H}

is the Hubble expansion rate and

t

{\displaystyle t}

is time. As in all FLRW spaces,

a

(

t

)

{\displaystyle a(t)}

, the scale factor, describes the expansion of physical spatial distances.

Unique to universes described by the FLRW metric, a de Sitter universe has a Hubble Law that is not only consistent through all space, but also through all time (since the deceleration parameter is

q

=

−

1

{\displaystyle q=-1}

), thus satisfying the perfect cosmological principle that assumes isotropy and homogeneity throughout space and time. There are ways to cast de Sitter space with static coordinates (see de Sitter space), so unlike other FLRW models, de Sitter space can be thought of as a static solution to Einstein's equations even though the geodesics followed by observers necessarily diverge as expected from the expansion of physical spatial dimensions. As a model for the universe, de Sitter's solution was not considered viable for the observed universe until models for inflation and dark energy were developed.