# When will the Hubble volume coincide with the volume of the observable Universe?

The Hubble volume is the volume that corresponds to objects so far from the Earth that the space between us and them is expanding faster than the speed of light. (I.e. objects outside this volume could never again be visible to us, even in principle.)

The volume of the observable Universe extends from us to the maximum distance light could have travelled since the Universe became transparent; when It was roughly 380,000 years old.

Since c1998 we have known the expansion of the Universe is accelerating, implying that the number of galaxies within the Hubble volume is decreasing. Since the Big Bang (well infinitesimally close to it at least) we know that time has been going forwards, and thus that the observable Universe is expanding.

When do these two volumes coincide with one another and what will the corresponding maximum volume of the observable Universe be at that time? The associated calculation or link to a suitable reference would also be very much appreciated.

Supplementarily/

• very interesting article - btw i wonder if that observation happens to be also a significant validation of dark energy being a universal constant? i.e: cosmological constant. At the end it says "If dark energy is not uniform as is usually assumed, then one could see this in anisotropies of the CMB in a similar way as in the void models,", which is not completely free of any ambiguity Jul 26 '11 at 17:25
• @lurscher: Whilst I don't think it answers you query there was another independent varification of DE published recent. Here is a sciam article about it and here is the paper on which it is based. Jul 26 '11 at 17:59

The question can be reposed as asking when the Hubble distance, which is not $c t_0$ but rather $c/H = c a/\dot{a}$ at $t_0$, is equal to the horizon distance $\int_0^{t_0} a^{-1} c dt$ where $a(t)$ is the scale factor normalized as $a(t_0) = 1$. For a power law $a(t) = (t/t_0)^n$ this gives $c t_0/n = c t_0 \int_0^1 x^{-n} dx = c t_0/(1-n)$. Thus $n = 1/2$ which is the case for a radiation dominated Universe.
Since the Universe is expanding, $a(t) < 1$ for $t < t_0$, so the horizon distance is always bigger than $c t_0$.
• Oops, thanks for pointing out my mistake in construing the question as asking about $ct_0$ rather than $c/H$.