Summary

Mathematical details

The FLRW metric with two spatial dimensions suppressed is

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

where ${\displaystyle a(t_{\text{now}})=1}$. If we flip the sign of the dx term, making the metric Euclidean, it can be embedded isometrically in Euclidean 3-space with cylindrical coordinates ${\displaystyle (r,\phi ,z)}$ by

${\displaystyle {\begin{aligned}r&=a(t)R\\\phi &=x/R\\z&=\int {\sqrt {c^{2}-a'(t)^{2}R^{2}}}\,dt\end{aligned}}}$

where R is a free parameter. z is only defined when ${\displaystyle |a'(t)|\leq c/R}$, and ${\displaystyle a'(t)}$ goes to infinity for both small and large t in ΛCDM, so a smaller R allows us to embed a larger fraction of the universe's history. On the other hand, with a large R we can embed larger spatial distances, since the embedding curves around on itself at a comoving distance of 2πR.

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

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

and the WMAP five-year report gives

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

(Mpc = megaparsec, Gyr = gigayear). For the embedding above I chose ${\displaystyle R=c/a'(0.7\ {\text{Gyr}})\approx 9\ {\text{Gly}}}$ and a time range of 0.7 Gyr to 18 Gyr. I deliberately cut off the embedding short of a full circle to emphasize that space does not loop back on itself (or, if it does, not at a distance governed by the arbitrary parameter R).

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

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