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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

4 Answers 4

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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

good strategy!
Hubble Constant = Ho = [d/s]/[d/1] in basic variables, thus it reduces to 1/s which is time inversed, that is, the inverse of the age of the universe at whatever Ho constant value used and further adjustments from the Standard Model equations. The big adjustment is U. acceleration [ and significant deceleration during first 2 billion years]. Thus, for the U to double again, it needs another full universe age ... at least at this instant of time and U size or stretch.

Every part in the U expands, thus any part of it expands like another. Like a loaf of raisin bread, everything expands, and all raisins move apart. Raisins separated by more distance will move apart further, because more expanding bread is between them – expanding at every point. The big bang is not points traveling from a center, but all space however small expanding. Where matter coalesced to stars and proto-galaxies, gravity overcomes matter moving apart, to preserve coalescence, while total U in all parts continues to expand - ?a space fabric expansion, whatever that might be.

So, the Ho needs to be velocity per some chosen distance to define the rate of expansion everywhere. Two points, as you put your question, are distance between, and not related to volume of U, but to linear separation. Linear is related to diameter doubling, also linear.

So, how much time did it take for the U to ‘double diameters’ in the past? Present U age [ = U] in a series of ½ U + ¼ U + 1/8 U + 1/16 U + 1/32 U + 1/64 U etc. as a total is good. Smaller the diameters, the faster it doubles in relation to what it was. Choose any part of the history of the universe and slice it to find how much time was needed to double space between two points, that is, double the size of the observable universe.

We do not know how big is the U. We observe that part of the U that communicates to us with light. Space can expand faster than light. Matter that has a speed limit in relation to itself, but not space. [So what does that mean for matter - galaxies in the part of space moving apart faster than light, one part beyond our view and us in the other galaxy, like 2 super-light fast raisins, and we are one raisin? Does their length disappear to each other, and if so, to what if anything? Better said that they cannot communicate. So, what does that say about relativity and communication?] We are a part of the loaf of bread, but do not know which part, or if or where a center is. We must know some edge to know a center. The U is bigger than we can see and know. U for unknown.

On this topic, have you heard of a writer, Gerald Schroeder, who relates Bible and Genesis creation by a growth-decay formula that uses Bible creation days 1 – 6 as numerical data in the formula to divide the 13.8 billion year history of the universe into six related parts, using diameter doubling for measure? Days can be eras in Hebrew used in the Bible. The six divisions are not equal, but as these fractions given above. The formula is a well accepted growth-decay function with a natural log for universal expansion, and integrated. Data put into the formula is from both nature and Bible, to bridge both with accountability. The nature side relates the age of the U, using a ratio of composed of the Microwave Background Radiation temperature at first matter, that is at quark confinement becoming neutrons [and a little later, protons overlapping], divided by the MBR now. That sets two data points for expansion, and starts the ‘beginning’ in the Bible at first matter. That ratio is related to diameter doublings of the universe. And ‘t’ in the exponent slices all U time into 6 parts, that data number taken from the Bible for 6 creation day-eras. It traces the length of each day-era or ‘slice’ of the whole. An other events fit well on the function graph, such as ‘light of last scattering’ etc. Bible descriptions and nature agree for those slices, especially if using Hebrew correctly to understand Bible words, of which there is much misuse in western Bible translations, and yes, even the Jewish Torah. We must use word roots. The ‘Science of God’ is the book, and his website is his name.

Take care, Steve Huffey

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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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