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I have been reading about Hubble's constant and trying to make 'sense' of the theory of the expanding Universe. Is is possible that space in the universe expands uniformly? If so, absent of other forces (ie gravity), how long does it take for the distance between any to dimensionless points in the universe to double in length?

I've tried to work the math as follows:

$\frac{74.2 \text{ km}}{\text {s Mpc}}\times\frac{\text{Mpc}}{3,261,564\text{ ly}}\times\frac{\text{ly}}{9,460,730,472,581 \text{ km}}\times\frac{31,557,600\text{ s}}{\text{yr}} = \frac{1}{13,177,793,645 \text{ yr}}$

Using the continuously compounding interest formula

$FV= Pe^{rt}$

$2 = 1\times e^{(1/13,177,793,646)t}$

$\ln(2) = \frac{1}{13,177,793,646}\times t$

$t = 9,134,150,511 \text{ yr}$

So it would take 9 billion years for the distance between any two points in space to double in length?

If this is so, when two points in 3D space double in distance apart, the space itself increased by $2^3 = 8$ so the time it would take for space itself to double in size would be $t = 1,141,768,813 \text{ yr}$

Since the Universe is only about 15 billion years old and started from a singularity of volume $0$,

I would have to assume that the rate of expansion of space isn't constant over time?

Does the time for the distance between two points to double in length vary based on the original distance between those to points?

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The significant figures in your calculation are really, really, really silly. –  Ben Crowell Mar 31 '13 at 14:20

3 Answers 3

up vote 1 down vote accepted

The way to answer your question is to take the Friedman equation and put in the components you want to. In the Standard Model of Cosmology you'd put in radiation, matter and lamdba. You then solve the equation for the scale factor $a(t)$. (This can be automated with a program like Mathematica.) You'll get an explicit $a(t)$ function that you can plot and see how the Universe expands.

A very nice pedagogical introduction is in Barbara Ryden's "Introduction to Cosmology", there's a PDF version online:

http://www.astro.caltech.edu/~george/ay21/readings/Ryden_IntroCosmo.pdf

Pg. 119 Figure 6.5 is just what you're looking for.

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It's going to take me some time to digest that PDF document. –  Doug Coburn Nov 10 '11 at 16:41

You are correct that the expansion of the universe hasn't been constant over time. A period of the very early universe is considered to have undergone exponential inflation doubling in size continuously to result in a $10^{78}$ increase in size between $10^{-36}$ to $10^{-32}$ seconds after the big bang.

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The increase of the size of the Universe isn't quite exponential yet but it's getting close to it. It will become (nearly) exponential when the dark energy (cosmological constant) constitutes (nearly) 100% of the energy density. So far, it is only 73 percent.

Because the expansion hasn't been quite exponential so far, the answer to the question "how much time is needed for doubling of the linear dimensions" depends on where you start to measure it. When we're controlled fully by the cosmological constant, the doubling time will converge to a constant inversely proportional to the cosmological constant, with a natural coefficient. It won't be far from those 9 billion years. Well, it will be a bit shorter, I guess, because the acceleration will jump relatively to the present one (which you used in your calculation) as the percentage of the dark energy increases.

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