The FLRW metric with two spatial dimensions suppressed is

where . 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 by

where R is a free parameter. z is only defined when , and 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

and the WMAP five-year report gives

(Mpc = megaparsec, Gyr = gigayear). For the embedding above I chose 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 .

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