File:Embedded LambdaCDM geometry.png

Embedded_LambdaCDM_geometry.png (640 × 544 pixels, file size: 38 KB, MIME type: image/png)

Summary

Description Euclidean embedding of a part of the Lambda-CDM spacetime geometry, showing the Milky Way (brown), a quasar at redshift z = 6.4 (yellow), light from the quasar reaching the Earth after approximately 12 billion years (red), and the present-era metric distance to the quasar of approximately 28 billion light years (orange). Lines of latitude (purple) are lines of constant cosmological time, spaced by 1 billion years; lines of longitude (cyan) are world lines of objects moving with the Hubble flow, spaced by 1 billion light years in the present era (less in the past and more in the future).
Date
Source Own work (see mathematical details below)
Author Ben Rudiak-Gould
Permission
(Reusing this file)
Public domain (an attribution would be appreciated but isn't required)
Other versions Image:Embedded LambdaCDM geometry (alt view).png

Mathematical details

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 .

Licensing

Public domain I, the copyright holder of this work, release this work into the public domain. This applies worldwide.
In some countries this may not be legally possible; if so:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

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18 March 2008

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Date/TimeThumbnailDimensionsUserComment
current00:37, 19 July 2012Thumbnail for version as of 00:37, 19 July 2012640 × 544 (38 KB)LucasVBLess saturation on the blue makes it easier on the eye.
21:09, 18 March 2008Thumbnail for version as of 21:09, 18 March 2008640 × 544 (30 KB)BenRGEuclidean embedding of a part of the Lambda-CDM spacetime geometry

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