How much new 3d space does the expansion of the universe create? Assuming the current understanding of our universes expansion velocity, ((73.24 ±1.74 kms/Mpc) Supernovae results), is correct throughout the observable universe, how many cubic light seconds of volume are added to the universe every second?
Please assume the universe is an extremely large sphere and that it expands 3 dimensionally.
 A: The current radius of the observable universe is estimated to be 14.25 gigaparsecs or 14,250 megaparsecs. Expansion will add (73.24 km/s)*(14,250) = 1,043,670 km/s or 3.38230412e-17 gigaparsec/s to the radius of the universe. So for that second $$\frac {4\pi}{3} ((14.25 + 3.38e-17)^3 - (14.25)^3) Gpc^3$$ was added to the volume of the universe as a result of expansion.
A: In our best cosmological model, $\Lambda$CDM, the universe is modelled as a manifold with a metric,
$$\text{d}s^2 = - \text{d}t^2 + a^2(t) (\text{d}x^2 + \text{d}y^2 + \text{d}z^2)
$$
(I'm neglecting spatial curvature for simplicity).
Metrics allow us to measure distances and volumes; any fixed coordinate volume $V_c$ (something like "$x^2 + y^2 + z^2 \leq 1$") will expand, scaling like $a^3$, and the corresponding physical volume will be $a^3 V_c = V_p$.
We will assume we keep the coordinate (or, as it is often called, "comoving") volume fixed, so we can differentiate this relation with respect to time to figure out how fast the physical volume changes:
$$ \frac{\text{d}V_p}{\text{d}t} = 3 V_c a^2 \dot{a} = 3 V_c a^3 H = 3 H V_p
$$
where dots denote derivatives with respect to time, and $H= \dot{a} / a$ is the Hubble parameter.
So, the relative change in any (cosmological) volume per unit of time will be $\dot{V_p} / V_p = 3H$; we can look at this at the current time, so using $H_0 \approx 70 \text{km/s/Mpc} \approx 2 \times 10^{-18} \text{Hz}$.
In other words, any cosmological volume increases by roughly $7 \times 10^{-16}$ percent every second (2 point something times 3 rounds up to 7).
That's great, but what volume should we refer this to?
One option is to look at the volume, as measured today, of the sphere defined by the particle horizon, which is the furthest light has had time to reach us.
This is roughly $r_p = 14.2 \text{Gpc}$ away (as measured now), and the corresponding volume is simply* $4 \pi r_p^3 / 3 \approx 12000 \text{Gpc}^3$.
Then, the increase in volume is readily computed to be roughly $\dot{V_p} \approx 80\ 000 \text{kpc}^3 / \text{s}$; for reference, the volume of the region in the Milky Way galaxy in which there are stars is roughly $15 \ 000 \text{kpc}^3$, so the rate is about 5 Milky Ways per second.**
I should stress, though, that picking this volume is kind of arbitrary; it's not the "volume of the universe", as we don't even know whether that is finite.
The main result to keep in mind is the formula for the rate of change of any finite volume, as that is a well-defined consequence of our cosmological models.
*: In this case the calculation is simple, as we are always considering volumes as measured today (fixing the time coordinate), which makes the metric basically flat for the purposes of the calculation.
**: Note that this is refers to the volume of the stellar disk, while the dark matter halo is hundreds of times larger.
