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

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

Putting (for reasons that will emerge later)

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

and

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

the Lambda-CDM scale factor may be rewritten as

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

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

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

Formally expanding the binomial and simplifying gives

$a(t)=Ae^{\alpha t}\left[1-{\frac {2}{3}}e^{-3\alpha t}-{\frac {1}{9}}e^{-6\alpha t}-{\frac {4}{81}}e^{-9\alpha t}-{\frac {7}{243}}e^{-12\alpha t}...+{\frac {\text{-2.1.4.7....(3n-5)}}{3^{n}n!}}e^{-3n\alpha t}-...\right]$

Best Current Numerical Values

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).

These give

$A=\left[{\frac {\Omega _{m}}{4\Omega _{v}}}\right]^{\frac {1}{3}}\approx 1.15_{7}$

and

$\alpha ={\sqrt {\Omega _{v}}}H_{0}\approx 0.0609\ {\text{Ga}}^{-1}$

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