9D (Cosmology)

The Friedmann equation immediately gives

Ω - 1 = \frac{k c ^{2}}{H ^{2}} .

Using

P = wρ

in the acceleration equation and eliminating

ρ a^{2}

with Friedmann gives

2 H^{'} + (1 + 3 w) (H^{2} + k c^{2}) = 0.

Differentiate

Ω - 1 = k c^{2} H^{- 2}

and substitute this equation:

Ω^{'} = - 2 (Ω - 1) \frac{H ^{'}}{H} = (1 + 3 w) H Ω (Ω - 1),

where

H^{2} + k c^{2} = Ω H^{2}

was used in the last step.

b

For an expanding universe with

0 \leq w \leq 1

(1 + 3 w) H

is positive. Thus

Ω = 1

is unstable toward the future: an initial

0 < Ω < 1

decreases away from one toward zero, whereas an initial

Ω > 1

increases and eventually corresponds to recollapse. In radiation domination,

w = 1/3

and

Ω^{'} = 2 H Ω (Ω - 1),

so a phase-line sketch has arrows away from the fixed point

1

on both sides. Consequently the observed near-flatness today requires extremely fine tuning of the early value of

Ω

w ≃ - 1

, so

1 + 3 w ≃ - 2

. For expansion,

Ω^{'} ≃ - 2 H Ω (Ω - 1),

and the arrows point toward

Ω = 1

. Equivalently, accelerated expansion makes the curvature contribution

∣ k c^{2} ∣/ H^{2}

rapidly decay. A sufficiently long inflationary epoch therefore drives a broad range of initial conditions extremely close to spatial flatness.

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