# Forum > Discussion > Mad Science and Grumpy Technology >  What precisely is meant when it is said that universal expansion is accelerating

## Bohandas

Specifically, does it merely mean that the total additional volume gained per unit time is increasing, or does it mean that the additional volume gained _per unit volume_ per unit time is increasing?

Or to put it another way, imagine I have a sphere whose radius is increasing at a constant rate - and whose volume is therefore increasing at an ever growing rate - is that the kind of thing they're talking about? Or would it be more equivalent to sphere whose radius is increasing at an ever increasing rate?

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## NichG

It means that the radius is increasing at an increasing rate.

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## McGarnagle

> Specifically, does it merely mean that the total additional volume gained per unit time is increasing, or does it mean that the additional volume gained _per unit volume_ per unit time is increasing?


This isn't a meaningful distinction because if we have the former, the latter necessarily follows.

Take r to be the radius of the universe; r-dot is the expansion rate, r-double-dot is the change in the expansion rate over time.  Also, take V to be the volume of the universe; V-dot is the volume rate of expansion, and V-double-dot is the change in the volume rate of expansion over time.

In the case of accelerating expansion, r, r-dot, and r-double-dot are all greater than zero.

The volume of the universe, V, is proportional to the cube of its radius.

V-dot is proportional to r2*(r-dot).

V-double-dot is proportional to r*(r-dot)2 + r2*(r-double-dot), which will be greater than zero.

The change in the volume rate of expansion per unit volume over time is (V-double-dot)/V.

(V-double-dot)/V is proportional to [(1/r)*(r-dot)]2 + (r-double-dot)/r, which will also be greater than zero.

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## NichG

> This isn't a meaningful distinction because if we have the former, the latter necessarily follows.
> 
> Take r to be the radius of the universe; r-dot is the expansion rate, r-double-dot is the change in the expansion rate over time.  Also, take V to be the volume of the universe; V-dot is the volume rate of expansion, and V-double-dot is the change in the volume rate of expansion over time.
> 
> In the case of accelerating expansion, r, r-dot, and r-double-dot are all greater than zero.
> 
> The volume of the universe, V, is proportional to the cube of its radius.
> 
> V-dot is proportional to r2*(r-dot).
> ...


Er, r''=0, r'=C would still have V''>0

V''= 4 pi ( r^2 r'' + 3r r'^2 )

So it is an important distinction. In fact, you could have r'' = -3 C r'^2 / r, for 0<C<1 (e.g. strictly negative), and V'' would still be positive.

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## Chronos

Take two widely-separated galaxies, and measure the distance between them as a function of time.  Take the first derivative of that function, and you get the speed at which they're separating.  Take the second derivative, and that's the acceleration.  If all that were going on were the sort of gravity that we're familiar with, then that acceleration would be negative (and you'd get different long-term effect depending on just how negative it is).  But it's not:  It's positive.  That's what we mean when we say that the Universe is accelerating.

In the simplest models, the speed divided by the distance (Hubble's "constant") truly is a constant, or at least asymptotically approaches a constant as the age of the Universe increases.  There are also other models where it's not constant, but those models are hard to distinguish from the simple model, with the data that we have, and so the simple model is usually preferred.

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## Rockphed

To answer the question with less math: we can only measure linear distances and velocities away from us; we technically cannot tell how far things in space are from each other* when they are interesting  distances apart. So it is the derivatives of that linear distance we speak of. Since everything past Andromeda is red shifted we know it is moving away. We expected to see the redshift describe velocities with a constantish 2nd derivative. No such luck. Velocities seem to have decreased over time for the first few billion years (as you would expect from objects moving away from each other) but have started increasing again (like a cubic function). Since we don't have a good explanation for why we call the cause of this annually "dark energy".

*Occasionally we see reflections of things like the reflection of a supernova off a nearby cloud of gas telling us that the nebula we assumed was 400 years old was actually 400 years old.

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## Quizatzhaderac

The rate they are referring to is that rate of space expanding, not the rate of the universe as a whole expanding.

Currently, two objects 1 megaparsec apart are expanding away from each other at a rate of 70 km/s. Greater distance means a greater velocity of expansion.

As the rate increases this will mean a rate of 71 km/s/parsec.

In either case, any positive rate of space expanding means the observable universe is expanding at an exponential rate.

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