Solution

Take a fixed comoving region, whose physical volume is

V = V_{0} a^{3}

. Its energy is

E = ρ V

, so the first-law relation

d E = - P d V

gives

d (ρ a^{3}) = - P d (a^{3}) .

Differentiating with respect to cosmic time and using the Hubble parameter

H = \overset{a}{˙} / a

a^{3} \overset{ρ}{˙} + 3 a^{2} \overset{a}{˙} ρ = - 3 a^{2} \overset{a}{˙} P, \overset{ρ}{˙} = - 3 H (ρ + P) .

P = wρ

ρ = ρ_{0} (\frac{a}{a _{0}})^{- 3 (1 + w)} .

Substituting this into the flat Friedmann equation and integrating the expanding branch gives, for

w \neq = - 1

a (t) \propto (t - t_{*})^{2/ [3 (1 + w)]} .

For

w = - 1

, the density and Hubble parameter are constant and instead

a (t) \propto e^{H t}

d τ = \frac{d t}{a ( t )}, \frac{d a}{d τ} = a \overset{a}{˙} = a^{2} H .

At radiation–string equality, let each component have density

ρ_{eq}

a = a_{eq}

w = 1/3

w = - 1/3

ρ_{r} = ρ_{eq} (\frac{a _{eq}}{a})^{4}, ρ_{s} = ρ_{eq} (\frac{a _{eq}}{a})^{2} .

Consequently

(\frac{d a}{d τ})^{2} = a^{4} \frac{8 π G}{3 c ^{2}} (ρ_{r} + ρ_{s}) = \frac{8 π G a _{eq}^{2}}{3 c ^{2}} ρ_{eq} (a^{2} + a_{eq}^{2}) .

Thus

B = \frac{8 π G a _{eq}^{2}}{3 c ^{2}} .

On the expanding branch, integration gives

arsinh (\frac{a}{a _{eq}}) = B ρ_{eq} (τ - τ_{0}) .

Choosing the Big Bang to occur at

τ = 0

a (τ) = a_{eq} sinh (B ρ_{eq} τ) .

 Ancestors (10)

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