Image:Embedded LambdaCDM geometry.png

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[edit] 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 worldlines 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).
Source

Own work (see mathematical details below)
Date

2008 March 18
Author

Ben Rudiak-Gould
Permission
(Reusing this image)

Public domain (an attribution would be appreciated but isn't required)
Other versions Image:Embedded LambdaCDM geometry (alt view).png

[edit] Mathematical details

The FLRW metric with two spatial dimensions suppressed is

ds2 = c2dt2 − a(t)2dx2

where a(tnow) = 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 (r,φ,z) by

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

where R is a free parameter. z is only defined when |a'(t)| \le c/R, and 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

a(t) = \left[ \frac{\Omega_m}{\Omega_v} \sinh^2 \left( \frac32 \sqrt{\Omega_v} H_0 t \right) \right]^{\frac13}

and the WMAP five-year report gives

\begin{align} \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{align}

(Mpc = megaparsec, Gyr = gigayear). For the embedding above I chose 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 dx / dt = c / a(t).

[edit] Licensing:

Public domain I, the copyright holder of this work, hereby release it into the public domain. This applies worldwide.

In case this is not legally possible:
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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current21:09, 18 March 2008640×544 (30 KB)BenRG (Euclidean embedding of a part of the Lambda-CDM spacetime geometry)
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