Bousso mentions the decay of the vacuum as needed to prevent Boltzmann brains in the distant future. If the universe were to reach a final state as a deSitter vacuum this eternal state would permit Boltzmann brains. The small cosmological constant Λ = 10^{-123} would permit statistical fluctuations as a finite (though very small temperature) to give rise to any configuration, including a brain or whole person --- even a whole planet or solar system with people and so forth. However, the decay of the vacuum state which would otherwise permit this can be seen from rather elementary grounds.
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
ds^2 = Adt^2 - A(r)^{-1}dr^2 - r^2dΩ^2, A(r) = (1 - Λr^2/3)
A radial null geodesic with ds^2 = 0 and dΩ^2 = 0 gives the velocity v = dr/dt = A(r), where this pertains to both out and in going geodesics near the cosmological horizon r = sqrt{3/Λ} as measured from r = 0. The total action for the motion of a particle is S = ∫ p_r dr - ∫ Hdt. Consider the bare action of massless particles, across the horizon from r to r',
S = ∫_r^{r'}p_rdr = ∫_r^{r'}∫_0^{p_r}dp_rdr.
The radial velocity of a particle is dr/dt = v = dH/dp_r, which enters into the action as,
S =∫_r^{r'}∫_0^H drdH'/v.
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^{E-ω}drdω'/v,
where the negative sign indicates the quanta is tunneling across the horizon to escape the Hubble region with radius The radial velocity
v = sqrt{Λ/3}r
defines the action
S = -ħ∫_r^{r'}∫_0^ω dωdr/(1 - sqrt{Λr^2/3}) = sqrt{3/Λ}tanh^{-1}sqrt{Λ/3}r
We now perform a similar analysis above, but instead consider the radiation production according to the transition Λ --- > Λ δΛ. The transition is considered according to the metric back reaction of the de Sitter vacuum. This tunneling defines the imaginary part of the action
S = ∫_{Λ _0}^Λ pdr.
The velocity term here is the computed from the Hamilton equation {v} = \partial H/\partial p which permits this to be written as
S = ∫_{Λ _0}^Λ ∫_0^RdHdr/v.
The Hamiltonian used is the H = G^{00} computed here, with v = (1 - Λr^2/3) with
S = ∫_{Λ _0}^Λ∫_{r_i}^{r_f}dr(4r/3)^2∫_{Λ _0}^Λ(Λ/(1-Λr^2/3)^2}}(1Λ r^2/3)/(1 - Λr^2/3)dΛ
This complicated expression is evaluated for Λ = Λ _0 - δ Λ with
S = 8(sqrt{Λ/3}arctanh({Λ /3}r) r/(1-3/(Λr^2)) 2/(Λ-3/r^2))|_{r_i}^{r_f}δΛ
Of course this solution exhibits a singularity for the radius extended across sqrt{3/ Λ }. We then consider the integration with respect to the small change in the cosmological constant and the calculus of residues and Cauchy integral formula gives this result in a very simple form for z^2 = Λ r^2 and we then left with the result
δS = (π sqrt{3}/4)δΛ.
The cosmological constant will decrease to zero as r_cc = sqrt{Λ/3} --- > ∞. The decay of this vacuum will be extremely slow, on the order of 10^{10^{100}} years for a half life (cosmological horizon doubles), which should then be still fast enough to prevent the formation of Boltzmann brains. This is something which would have to be verified explicitly.
LC
