The final fate of the universe can be computed. The de Sitter spacetime cloaks every region with a cosmological horizon, which is similar in some ways to a Rindler horizon. $10^{100} years from now there will be nothing going on: Black holes will have decayed away and largely the universe is a de Sitter vacuum. There may be a smattering of neutron stars around, which as I recall these exist for a very long time. This horizon does emit quanta, though the flux of this radiation is exceedingly small. The universe will then eventually decay as the vacuum decays and the cosmological horizon retreats off “to infinity.” Nothing else much will be going on.
We consider the decay of the de Sitter vacuum by quantum means and the prospect for this as a mechanism for the production of nascent cosmologies or baby universes. The observable universe, under eternal inflation from dark energy, will asymptotically evolve to a de Sitter spacetime. This spacetime is a vacuum configuration with a cosmological constant Λ. The stationary metric for this spacetime is
ds2 = Adt2 – A(r)−1dr2 − r2dΩ2, A(r) = (1 − Λr2/3)
A radial null geodesic with
ds2 = 0 and
dΩ2 = 0 gives the velocity
˙r = dr/dt = A(r), where this pertains to both out and in going geodesics near the cosmological horizon
r = √3/Λ as measured from
r = 0. The total action for the motion of a particle is
S = ∫prdr − ∫Hdt. Consider the bare action of massless particles, using methods found in [1], across the horizon from
r to
r′,
S = ∫r′rprdr = ∫r′r∫pr0dprdr.
The radial velocity of a particle is
˙r = dr/dt = dH/dpr, which enters into the action as,
S =∫r′r∫H0dH′˙rdr.
The field defines
H′ = ℏω′. The integration over frequencies is from
E to
E − ω, for the ADM energy. The action is properly written as
S = −ℏ∫r′r∫E−ωEdω′˙rdr,
where the negative sign indicates the quanta is tunneling across the horizon to escape the Hubble region with radius The radial velocity
˙r = √Λ/3r
defines the action
S = −ℏ∫r′r∫ω0dωdr±1 − √Λr2/3 = √3/Λtanh−1(√Λ/3r)
The action is then the delay coordinate evaluated as
r∗ = ∫dr1 – Λr2/3 = √3/Λtanh−1(√λ/3r).
The domain
[0, √3/Λ) defines a real valued action. Since,
tanh−1(x) = 12ln((1 + x)/(1 − x)) for
r > √3/Λ the argument of the logarithm is negative. In this case the action is
S = √3/Λln(√Λ/3r + 1√Λ/3r − 1) + iπ√3/Λ.
The imaginary part represents the action for the quantum field emission as
r → ∞. The delay coordinate is defined on
[0, ∞) which assures an S-matrix is defined on an unbounded causal domain, and this holds in general as well.
This action does describe the emission of photons by the cosmological horizon. A tiny production of bosons occurs which causes the horizon to slowly retreat away. Eventually the de Sitter spacetime decays away into a Minkowski spacetime as t → ∞.
This post imported from StackExchange Physics at 2016-01-10 17:44 (UTC), posted by SE-user Lawrence B. Crowell