User:Dendropithecus/Sandbox

(Work on the Lambda-CDM metric)

The FLRW metric with two spatial dimensions suppressed is

$ds^{2}=c^{2}dt^{2}-a(t)^{2}dx^{2}$

Ignoring the effects of radiation in the early universe and assuming k = 0 and w = −1, the Lambda-CDM scale factor is

$a(t)=\left[{\frac {\Omega _{m}}{\Omega _{v}}}\sinh ^{2}\left({\frac {3}{2}}{\sqrt {\Omega _{v}}}H_{0}t\right)\right]^{\frac {1}{3}}$

and the WMAP five-year report gives

${\begin{aligned}\Omega _{m}&\approx 0.279\\\Omega _{v}&\approx 0.721\\H_{0}&\approx 70.1\ {\text{km}}\ {\text{s}}^{-1}\ {\text{Mp}}^{-1}\approx 0.0717\ {\text{Ga}}^{-1}\end{aligned}}$

(Mp = megaparsec, Ga = gigayear).

Putting

$A=\left[{\frac {\Omega _{m}}{4\Omega _{v}}}\right]^{\frac {1}{3}}\approx$ and

$\alpha ={\sqrt {\Omega _{v}}}H_{0}\approx$,

a(t) can be rewritten as

$a(t)=A\left[2\sinh \left({\frac {3}{2}}\alpha t\right)\right]^{\frac {2}{3}}$

$a(t)=A\left[\exp \left({\frac {3}{2}}\alpha t\right)-\exp \left(-{\frac {3}{2}}\alpha t\right)\right]^{\frac {2}{3}}$

$a(t)=A\exp \left(\alpha t\right)\left[1-\exp \left(-3\alpha t\right)\right]^{\frac {2}{3}}$

The path of the light ray satisfies $dx/dt=c/a(t)$.