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 Incoming links: Conformal time during de Sitter expansion

With an approximately constant Hubble parameter

H

a (t) = e^{H t}

after a choice of normalization. Integrate

d τ = d t / a (t)

and choose the integration constant so that the future endpoint is

τ = 0

τ = - \frac{e ^{- H t}}{H}, a (τ) = - \frac{1}{H τ}, - \infty < τ < 0.

Substituting

d t = a d τ

in the FLRW metric gives

d s^{2} = a^{2} (τ) (- d τ^{2} + d x^{2})

d τ = \frac{d t}{a ( t )} .

For

a = a_{0} e^{H_{inf} t}

, choose

τ \to 0

t \to + \infty

. Integration gives

τ = - \frac{e ^{- H_{inf} t}}{a _{0} H _{inf}} = - \frac{1}{a H _{inf}}, - \infty < τ < 0,

Writing a prime for

d / d τ

, the relations

\frac{d}{d t} = \frac{1}{a} \frac{d}{d τ}, \frac{a ^{'}}{a} = - \frac{1}{τ}

ϕ_{k}^{''} + 2 \frac{a ^{'}}{a} ϕ_{k}^{'} + c^{2} k^{2} ϕ_{k} = 0.

ϕ_{k} = a ϕ_{k} = - \frac{1}{H _{inf} τ} ϕ_{k} .

A direct substitution gives

ϕ_{k}^{''} + (c^{2} k^{2} - \frac{a ^{''}}{a}) ϕ_{k} = 0.

Here

a^{''} / a = 2/ τ^{2}

\frac{d ^{2} ϕ _{k}}{d τ ^{2}} + (c^{2} k^{2} - \frac{2}{τ ^{2}}) ϕ_{k} = 0 .