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

Proposition: Background and clock

Einstein–scalar theory and an explicit inflationary background › Fixed background · lines 324–359 · proof included or linked

Em02-E@1#prop:background

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The pair $(g_{0}, ϕ_{0})$ solves (eq:Einstein)–(eq:KG). It is expanding and accelerating, and its scalar clock is nondegenerate on every compact subinterval of $I$ .

Proof. For a spatially flat homogeneous background, the equations obtained from (eq:Einstein)–(eq:KG) are

3 M_{Pl}^{2} H^{2} = \frac{1}{2} v^{2} + V (ϕ_{0}), - 2 M_{Pl}^{2} H ˙ = v^{2}, v ˙ + 3 H v + V^{'} (ϕ_{0}) = 0.

Substitution gives

V (ϕ_{0} (t)) = \frac{p ( 3 p - 1 ) M _{Pl} ^{2}}{t ^{2}}, \frac{1}{2} v^{2} = \frac{p M _{Pl}^{2}}{t ^{2}},

which proves the first equation. The second follows from $H ˙ = - p / t^{2}$ . Writing $c = 2 p M_{Pl}$ , the third equation has terms

- c / t^{2}, 3 p c / t^{2}, - (3 p - 1) c / t^{2},

whose sum is zero.

Finally, $a ˙ > 0$ , $a ¨ / a = p (p - 1) / t^{2} > 0$ , and $v > 0$ has a positive lower bound on each compact time interval.

Displayed formulas in this passage

unlabeled · line 335
$3 M_{Pl}^{2} H^{2} = \frac{1}{2} v^{2} + V (ϕ_{0}), - 2 M_{Pl}^{2} H ˙ = v^{2}, v ˙ + 3 H v + V^{'} (ϕ_{0}) = 0.$
unlabeled · line 343
$V (ϕ_{0} (t)) = \frac{p ( 3 p - 1 ) M _{Pl} ^{2}}{t ^{2}}, \frac{1}{2} v^{2} = \frac{p M _{Pl}^{2}}{t ^{2}},$
unlabeled · line 351
$- c / t^{2}, 3 p c / t^{2}, - (3 p - 1) c / t^{2},$

Nearby in this section

section: Fixed background