Preamble

Assumptions

$Assumptions->{τ,s,λ}∈Reals

$Assumptions->λ>0

Definitions of (asymptotically de Sitter) scale factor, trajectory, and the two point function (σ^2)(τ,s)

σ[τ_,s_,λ_]:=(Sqrt[-a[τ,λ] a[τ-s,λ](((η[τ,λ]-η[τ-s,λ]))^2)])

a[τ_,λ_]:=Cosh[λ τ]

η[τ_,λ_]:=2(λ^(-1)) ArcTan[Tanh[λ τ/2]]

λ is a constant related to the field temperature ~ (Subscript[T, ds])=H/2π

Definition of transition rate

NRateComoving0[τ_,ω_,λ_]:=(1/(2(π^2)))NIntegrate[(((Cos[ω s])/((σ[τ,s,λ])^2))+(1/(s^2))),{s,0,τ},WorkingPrecision->40,AccuracyGoal->20,MaxRecursion->Infinity]+(1/(2(π^2) τ))-(ω/(4π))

Conformal metric

FRWc:=DiagonalMatrix[a[τ,λ]^2{-1,1,1,1}]

Four-velocity (comoving)

u:={D[η[τ,λ],τ],0,0,0}

This shows that u is timelike

FRWc.u.u//Simplify

where

u//FullSimplify

Results

Transition rate over detection time

Here is a plot of the transition rate for a variety of choices of energy gap. The blue curve has the value of ω=1/10 and appears to be monotonically decreasing. As the energy gap increases, the magnitude decreases and a dip emerges for early times (see yellow and green curve). If we further increase the energy gap, this dip gives way to oscillations, best exemplified by the choice of ω=20 for the purple curve. We interpret these oscillations (and the associated ‘dips’) as clear signals of transience associated with turning on the detector sharply. For longer detection times, the transition rate settles down to an almost constant value, where the value of this constant is inversely proportional to the magnitude of the energy gap in the region of positive energy gap

ListLinePlot[{Table[{τ,NRateComoving0[τ,1/10`64,1`64]},{τ,.1`64,5`64,.1`64}],Table[{τ,NRateComoving0[τ,1/2`64,1`64]},{τ,.1`64,5`64,.1`64}],Table[{τ,NRateComoving0[τ,1`64,1`64]},{τ,.1`64,5`64,.1`64}],Table[{τ,NRateComoving0[τ,2`64,1`64]},{τ,.1`64,5`64,.1`64}],Table[{τ,NRateComoving0[τ,20`64,1`64]},{τ,.1`64,5`64,.1`64}]},InterpolationOrder->3,AxesLabel-> {Style[Δτ,Larger],Style["(OverDot[ℱ])",Larger]}]

Temperature

Temperature Estimator

TEDRcomoving[τ_,ω_,λ_]:=((-ω)/(Log[((NRateComoving0[τ,ω,λ])/(NRateComoving0[τ,-ω,λ]))]))

NRateComoving0[τ_,ω_,λ_]:=(1/(2(π^2)))NIntegrate[(((Cos[ω s])/((σ[τ,s,λ])^2))+(1/(s^2))),{s,0,τ},WorkingPrecision->460,AccuracyGoal->300,MaxRecursion->Infinity]+(1/(2(π^2) τ))-(ω/(4π))

Note: we’ve had to raise (considerably!) the WP and AG.

ListLinePlot[{Table[{τ,TEDRcomoving[τ,2`464,1`464]},{τ,10.4`464,40`464,.4`464}]},InterpolationOrder->3,AxesLabel-> {Style[Δτ,Larger],Style["(Subscript[T, EDR])",Larger]}]

The Hayward-Kodama temperature for the given spacetime is worked out to be

THK[τ_,λ_]:=(1/(4π)) (λ Coth[λ τ]+λ Tanh[λ τ])

Then

ListLinePlot[{Table[{τ,TEDRcomoving[τ,2`464,1.0`464]},{τ,15`464,40`464,.4`464}],Table[{τ,THK[τ,1.0`464]},{τ,15`464,40`464,.4`464}],Table[{τ,Teff[τ,1.0`464]},{τ,15`464,40`464,.4`464}]},InterpolationOrder->3,AxesLabel-> {Style[Δτ,Larger],Style["(Subscript[T, EDR])",Larger]},PlotRange->{0.1591544,0.1591555},PlotStyle->{,,Dashed}]

We see that the temperature of the detector (as modeled by (Subscript[T, EDR])) thermalises to the local dS temperature after a suitable length of time.