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/
I have often wondered about this; ever since reading Professor Sir Roger Penrose's Cycle's of Time a year or two ago.
I thought about it again today after reading a somewhat unassociated article about how recent results had shown the accelerating expansion of the Universe cannot be explained by the "Hubble Bubble" hypothesis.
Before asking this question I did of course search the site for an existing answer: this question is extremely similar but does not appear to include an exact answer (in fact the answers appear to somewhat contradict each other. Moreover, it does not appear to address the subtleties involved in making the estimate; such as the variation in inter-galactic recession speeds, due to the change in gravitational force between them, since the Big Bang.
 A: 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 $c t_0$.
If you take last scattering for the lower limit of the integral then there will some more complicated equation.  But that's a photist approach.  Neutrinos and gravitational waves don't care about last scattering at 380,000 years.
A: Never if space keeps accelerating. When the universe accelerates, the Hubble volume decreases. When the universe decelerates it will increase. The only way the whole universe's size can be measured is when it neither decelerates or accelerates, i.e. remains constant, which is essentially impossible.
