Manuscript XX-A@1 · XX-A.tex · section · 3a586110

Action, conventions, and background

An explicit on-shell background and the perturbative freeze › Action, conventions, and background · lines 234–282

XX-A@1#eq:action, XX-A@1#eq:background-data, XX-A@1#eq:potential, XX-A@1#eq:einstein, XX-A@1#eq:scalar

Use signature $(-, +, +, +)$ , a reduced Planck mass $M > 0$ , and the minimally coupled action density

S [g, ϕ] = \int - g (\frac{M ^{2}}{2} R [g] - \frac{1}{2} g^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - V (ϕ)) d^{4} x .

On noncompact spatial slices the background action need not be finite. We use its local Euler equations and compactly supported variations, not an asserted finite total background action.

Choose $p > 1$ , $t_{*} > 0$ , $ϕ_{*} \in R$ , and set

A = 2 p M, a (t) = (\frac{t}{t _{*}})^{p}, ϕ ˉ (t) = ϕ_{*} + A lo g (\frac{t}{t _{*}}),

g ˉ = - d t^{2} + a (t)^{2} δ_{ij} d x^{i} d x^{j}, V (ϕ) = \frac{M ^{2} p ( 3 p - 1 )}{t _{*}^{2}} exp (- \frac{2 ( ϕ - ϕ _{*} )}{A}) .

Work on

M = I \times R^{3}, I = (t_{-}, t_{+}), 0 < t_{-} < t_{+} < \infty.

No limit toward the cosmological singularity $t = 0$ is taken.

The Euler equations are

M^{2} G_{μν} □_{g} ϕ - V^{'} (ϕ) = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} g_{μν} (g^{ρ σ} \partial_{ρ} ϕ \partial_{σ} ϕ + 2 V (ϕ)), = 0.

Displayed formulas in this passage

XX-A@1#eq:action · line 238
$S [g, ϕ] = \int - g (\frac{M ^{2}}{2} R [g] - \frac{1}{2} g^{μν} \partial_{μ} ϕ \partial_{ν} ϕ - V (ϕ)) d^{4} x .$
XX-A@1#eq:background-data · line 251
$A = 2 p M, a (t) = (\frac{t}{t _{*}})^{p}, ϕ ˉ (t) = ϕ_{*} + A lo g (\frac{t}{t _{*}}),$
XX-A@1#eq:potential · line 257
$g ˉ = - d t^{2} + a (t)^{2} δ_{ij} d x^{i} d x^{j}, V (ϕ) = \frac{M ^{2} p ( 3 p - 1 )}{t _{*}^{2}} exp (- \frac{2 ( ϕ - ϕ _{*} )}{A}) .$
XX-A@1#eq:einstein · line 272
$M^{2} G_{μν} □_{g} ϕ - V^{'} (ϕ) = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} g_{μν} (g^{ρ σ} \partial_{ρ} ϕ \partial_{σ} ϕ + 2 V (ϕ)), = 0.$
XX-A@1#eq:scalar · line 272
$M^{2} G_{μν} □_{g} ϕ - V^{'} (ϕ) = \partial_{μ} ϕ \partial_{ν} ϕ - \frac{1}{2} g_{μν} (g^{ρ σ} \partial_{ρ} ϕ \partial_{σ} ϕ + 2 V (ϕ)), = 0.$
unlabeled · line 265
$M = I \times R^{3}, I = (t_{-}, t_{+}), 0 < t_{-} < t_{+} < \infty.$

Nearby in this section

lemma: Lemma: On-shell power-law background