15D (Cosmology)

The conservation equation with

P = wρ

gives

d ρ / ρ = - 3 (1 + w) d a / a

, hence

ρ = ρ_{0} a^{- 3 (1 + w)} .

ρ_{crit} = \frac{3 c ^{2} H ^{2}}{8 π G}, Ω_{i} = \frac{ρ _{i}}{ρ _{crit}} .

b

w = 1/3

w = 0

, and a cosmological constant has constant density. With negligible curvature and present-day normalization,

H^{2} = H_{0}^{2} (\frac{Ω _{R 0}}{a ^{4}} + \frac{Ω _{M 0}}{a ^{3}} + Ω_{Λ0}),

w = - 4/3

has

ρ_{P} \propto a

, so flatness gives

\overset{a}{˙} = H_{0} Ω a 1 + β^{2} a, β^{2} = \frac{1 - Ω}{Ω} .

For

b^{2} = 1 + β^{2} a

, integration yields

lo g \frac{b - 1}{b + 1} = H_{0} Ω (t - t_{0}) + lo g κ, κ = \frac{1 - Ω}{1 + Ω} .

Solving gives the displayed scale factor

a (t) = \frac{4Ω e ^{H_{0} Ω (t - t_{0})}}{[ ( 1 + Ω ) - ( 1 - Ω ) e ^{H_{0} Ω (t - t_{0})} ] ^{2}} .

For

1 - Ω ≪ 1

and times near

t_{0}

, this approaches de Sitter growth

e^{H_{0} Ω (t - t_{0})}

. The denominator vanishes at

t_{BR} - t_{0} = \frac{1}{H _{0} Ω} lo g \frac{1 + Ω}{1 - Ω} .

Even a tiny phantom fraction therefore ends expansion in finite time; its divergent density ultimately disrupts loosely bound systems.

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