Geometric Setup

2 Geometric Setup

We work in closed FRW minisuperspace with cosmological constant $Λ = 1$ and a scalar field $ϕ$ , Wick-rotated to Euclidean time $τ ∈ [0,τmax]$ . The Euclidean action is:

∫ τmax[ ˙a2 1 2 1 3 2 Λ 3] SE = − a + κ-a-+ 2a ˙ϕ + 2-a ϕ + 3a dτ + 2a(τmax), 0

with $κ = 0.15$ . The classical Hartle-Hawking no-boundary instanton satisfies $a(0) = 0$ and $a˙(τmax) = 0$ , and is given by:

∗ 1- ∘ ---- π-- a (τ) = ω sin (ω τ), ω = Λ∕3, τmax = 2ω,

with $ϕ(τ) = 0$ as the classical scalar background.

We generate large ensembles of random histories that respect these boundary conditions and compute both the Euclidean action $S E$ and the spectral complexity $C s$ of each history.

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