Manuscript Em02-E@1 · Em02-E-v1.tex · section · f3795050

Fixed background

Einstein–scalar theory and an explicit inflationary background › Fixed background · lines 290–323

Em02-E@1#eq:background, Em02-E@1#eq:potential, Em02-E@1#eq:vH

Fix

p > 1, a_{*} > 0, t_{*} > 0, ϕ_{*} \in R,

and define

a (t) V (ϕ) = a_{*} (t / t_{*})^{p}, = \frac{p ( 3 p - 1 ) M _{Pl} ^{2}}{t _{*}^{2}} exp [- \frac{2}{p} \frac{ϕ - ϕ _{*}}{M _{Pl}}] . ϕ_{0} (t) = ϕ_{*} + 2 p M_{Pl} lo g (t / t_{*}),

Work on

M = I \times R^{3}, I = (t_{*} /2, 2 t_{*}), g_{0} = - d t^{2} + a (t)^{2} δ_{ij} d x^{i} d x^{j} .

A dot denotes cosmic-time differentiation. Put

H = \frac{a ˙}{a} = \frac{p}{t}, v = ϕ ˙_{0} = \frac{2 p M _{Pl}}{t} > 0.

There is no limit toward $t = 0$ in this paper.

Displayed formulas in this passage

Em02-E@1#eq:background · line 297
$a (t) V (ϕ) = a_{*} (t / t_{*})^{p}, = \frac{p ( 3 p - 1 ) M _{Pl} ^{2}}{t _{*}^{2}} exp [- \frac{2}{p} \frac{ϕ - ϕ _{*}}{M _{Pl}}] . ϕ_{0} (t) = ϕ_{*} + 2 p M_{Pl} lo g (t / t_{*}),$
Em02-E@1#eq:potential · line 297
$a (t) V (ϕ) = a_{*} (t / t_{*})^{p}, = \frac{p ( 3 p - 1 ) M _{Pl} ^{2}}{t _{*}^{2}} exp [- \frac{2}{p} \frac{ϕ - ϕ _{*}}{M _{Pl}}] . ϕ_{0} (t) = ϕ_{*} + 2 p M_{Pl} lo g (t / t_{*}),$
Em02-E@1#eq:vH · line 317
$H = \frac{a ˙}{a} = \frac{p}{t}, v = ϕ ˙_{0} = \frac{2 p M _{Pl}}{t} > 0.$
unlabeled · line 293
$p > 1, a_{*} > 0, t_{*} > 0, ϕ_{*} \in R,$
unlabeled · line 311
$M = I \times R^{3}, I = (t_{*} /2, 2 t_{*}), g_{0} = - d t^{2} + a (t)^{2} δ_{ij} d x^{i} d x^{j} .$