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I am trying to find out why the Hubble radius, defined as the distance at which cosmological objects recede from the observer at the speed of light because of the expansion of the universe, is equal to $cH_0^{-1}$. How can I prove it?

I'm confused about which type of distance we are considering here: physical distance, comoving distance, luminosity distance...? I thought about using the Hubble-Lemaître law, $v=H_0d$, which would give the correct result, but I am not sure of whether it applies to this context or not.

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    $\begingroup$ You can find an answer to your question at: physics.stackexchange.com/a/518704/193705 $\endgroup$
    – Albert
    Commented Apr 11, 2023 at 6:44
  • $\begingroup$ @Albert I don't see any explanation in that answer, they seem to be just assuming the formula. And the expression $v=H_0d$ is supposed to only be valid for small values of the redshift, which is what confuses me the most. $\endgroup$ Commented Apr 11, 2023 at 8:49

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The fact that the Hubble radius equals c/H can easily be shown if you assume the universe has the shape of the surface of a 4-sphere, with a current radius of 13.80 billion light-years that is increasing at the speed of light. Using that model, a simple formula for Hubble’s parameter, and, also the Hubble radius, can be easily derived. Since Hubble’s parameter is defined as a VELOCITY divided by a relevant DISTANCE, from the model, since the radius of the 4-sphere is increasing with a velocity of c, it means the velocity at which its circumference is increasing is 2pic when the length of the circumference is 2piR, so H=(2pic)/(2piR), which simplifies to H=c/R. Rearranging, you see that R=c/H, which is what you wanted to ‘prove’. In so far as the model from which this formula is derived is correct, that formula is correct. But about the only way to check the model is to check the predictions of the formula. The observational values of H vary between 68 and 74 km/s/MPc. Plugging in 13800 MLY for the current radius of the 4-sphere (current radius of the 4-sphere equals the age of the universe – 13.80 billion years - times c) one finds:

H=c/R = 299,792,458 m/s /13800 MLY = 21,724 m/s/MLY= 70.85 km/s/MPc

for the 4-sphere’s current surface expansion rate, which is very close to the WMAP probe’s value of 71.0 km/s/MPc, and in the middle of other determinations of H (68 to 74 km/s/Mpc), so the formula, and therefore the model on which the formula was based may be correct.

The formula also shows that the length of the Hubble radius is always the same as the length of the radius of the universe’s 4-sphere, and both are increasing at the speed of light, therefore, both are increasing in length by one light-year per year. (The two are not the same, however. The 4-sphere’s radius is in 4D space, whereas the Hubble radius is in 3D space.) Lastly, R is a physical distance.

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