Is the Universe's Size Determined by Hubble's Rate of Expansion?

[TheSicilianSa]

Gentlemen:

I am a rank amateur, so I would appreciate it if you would critique the following logic:

If the Universe has been expanding at a maximum rate = H (m/s) for a period = Y (years), then the current “radius” = R of the 4D-hypersphere in which we assume to exist is:

(1) R (light years) = HY/c, where c = the speed of light

Since the Hubble Rate of Expansion (H) is estimated to be:

(2) 6.7E4 < H < 7.5E4 (m/s)

And the estimated Age of the Universe (Y) is estimated to be

(3) 1.361E10 < Y < 1.385E10 (years)

Then the estimated Radius of the Hypersphere (R) is:

(4) 3.04E6 < R < 3.47E6 (light years)

Thus since the Volume of the Surface of a 4D-hypersphere = 2π^2R^3, then

The Volume of the Universe (V):

(5) 5.545E20 < V < 8.247E20 (cubic light years)

Of course, if the Hubble constant is (and has been) increasing, then this estimated is a maximum, and the size of the universe must be considerably smaller.

Where am I going wrong?

 

[marcus]

If the Universe has been expanding at a maximum rate = H (m/s) for a period = Y (years), then the current “radius” = R of the 4D-hypersphere in which we assume to exist is:

...
Where am I going wrong?

Thanks for laying out things clearly, with simple math equations and units.
This makes it easier to understand what you are saying, much easier than when people just use words (and one doesn't know what they think they are saying in real math terms.)

Your problem is you think that the Hubble rate H is a speed. Measured in meters per second.

It is not a speed.
So all your reasoning in that post is wrong and you must start with a better picture and the right sort of quantities---the real H, not a speed in m/s.
But that is good! Because you are thinking clearly and quantitatively from the start, not just talking words.

The cleanest way to say what the Hubble rate is is to express it as a percentage rate of increase of distance. Large scale (intergalactic) distances increase 1/140 percent every million years.

This rate has always been declining and it will continue to decline, but more and more gradually because of the accelerating effect of dark energy.
When people hear about "accelerating expansion" they think it means the Hubble rate H is increasing. That is not what is meant. H is not a speed.
H will continue to decline but with an asymptotic value of around 1/160 percent per million years.

What would you like to calculate with this rate? You think of things and say, maybe some of us can help.

One thing you should know about is the Hubble time. It is 1/H.
The reciprocal of the Hubble rate. See if you can calculate it.

H is 1/140 per million years
So it is an increase rate of 1 percent every 140 million years.
Which (if it would stay constant for such a long time) would be 100 percent in
140 x 100 million years----which is 14 billion years.

That is the Hubble time. The time required for distances to double, i.e. increase 100 percent, if the rate would stay the same indefinitely.

The Hubble distance is c times 1/H, that is c times the Hubble time. It is 14 billion lightyears. This is the distance that is growing at exactly the rate c.
If we observe a galaxy which is today at a distance of 14 billion lightyears from us, then the distance to that galaxy is increasing at the rate c.

We can observe galaxies which are far more distant than that. Recession rate is not the same as a real motion speed and is not governed by special relativity.

See what questions you can come up with! There are plenty of people around here who can respond.

 

[marcus]

Then the estimated Radius of the Hypersphere (R) is:

(4) 3.04E6 < R < 3.47E6 (light years)

Thus since the Volume of the Surface of a 4D-hypersphere = 2π^2R^3, then

The Volume of the Universe (V):

(5) 5.545E20 < V < 8.247E20 (cubic light years)

You have the right formula for the hypersphere volume! Excellent!
The most recent WMAP report gives a 95% confidence lowerbound on the radius of curvature R
that you can plug into this!
This is an authoritative 2008 paper by the WMAP team. There is currently no more reliable source of cosmo data.
Their lowerbound is 101 billion lightyears. Let's round it to 100 billion LY.
It could be much larger but with 95% confidence they say it is at least that.However your figure for R is not good because it derives from a bad picture and a misunderstanding of what H is. Not your fault. A respectable mistake.
They say how they got the figure of R = 100 billion LY. See Table 2 on page 4 of the
WMAP report. If you stubbornly ask questions you can probably get people to explain how they got this interesting figure for R.

It involves H all right, but not the way you thought. Different formula.
http://arxiv.org/abs/0803.0547
Exciting stuff! A lower bound on the volume of the whole universe!

You can always get this paper with google if you google "cosmology WMAP komatsu"
Komatsu is the lead author. Another maybe easier to remember is Joanna Dunkley (an attractive woman and ace cosmologist). You could google
"cosmology WMAP dunkley" and probably also get it.

 

[Chalnoth]

Since the Hubble Rate of Expansion (H) is estimated to be:

(2) 6.7E4 < H < 7.5E4 (m/s)

This isn't right. The units of Hubble expansion are inverse time, not distance over time. It looks like the units you use here are m/s/Mpc.

The Volume of the Universe (V):

(5) 5.545E20 < V < 8.247E20 (cubic light years)

I haven't paid too close attention to your calculations here, but if accurate, this would only make for the volume of the visible portion of the universe. The universe as a whole is vastly, vastly larger than the visible portion, as evidenced by the fact that the universe is smooth out to as far as we can see: we would expect some tapering off if it ended close to our horizon.

Of course, if the Hubble constant is (and has been) increasing, then this estimated is a maximum, and the size of the universe must be considerably smaller.

Actually it's decreasing. The Hubble constant will continue to fall until the universe is empty of all matter and radiation. If dark energy turns out to be a cosmological constant, or just act like one at late times, then the Hubble constant will approach a constant value dependent upon the energy density of the dark energy.

 

[TheSicilianSa]

Thus, as Marcus suggests (that R > 1.0E11) then V > 1.97E34 (cubic light years) ?
And, by (1), that the minimum rate of expansion of the universe (if constant) = 2.16E9 (m/s)?

Thank you all for your critique; I hope that all my future queries are likewise edifying.

 

[marcus]

Thus, as Marcus suggests (that R > 1.0E11) then V > 1.97E34 (cubic light years) ?
And, by (1), that the minimum rate of expansion of the universe (if constant) = 2.16E9 (m/s)?

Thank you all for your critique; I hope that all my future queries are likewise edifying.

I think that's right! You have the correct formula for the volume of a hypersphere namely
V = 2 pi² R ³ ; and 2 pi² is about equal to 20.

And the WMAP team (cosmo greats like Ned Wright, Joanna Dunkley, David Spergel!) tell us that with 95% confidence R > 100 billion LY, or as you say E11 LY
So R ³ > E33 cubic LY
And we just have to multiply that by 20 to get the present volume of space (lower bound assuming finite at all). So that's 2 E34 cubic LY.

Now I think you are thinking clearly and constructively, as a selfdesignated amateur, and you go on to calculate the minimum growth rate of R. Which I'm not familiar with people doing. Let me see if you get the right answer.

The presentday Hubble rate is 1/(14 billion years). That is, if that rate were to continue a distance would double (increase by 100 percent) in 14 billion years.
So to find the rate of increase of a distance R = 100 billion LY we have to multiply that by the Hubble rate. And that is 100/14 lightyears per year. Seven times the speed of light.

Lets see if that agrees with what you calculated. You put it in m/s and the speed of light is 3E8 m/s so that 7c equals 21E8 m/s. Yes! That is the answer you got.

Now keep in mind that the universe might already be infinite in size---both in extent and volume. I think you realize that we are just considering the case where it is finite volume, the positive curved case. This is just to get a 95% lower bound.

Also it is not usual to do what you did, namely to call the radius of curvature R "the size of the universe". And people don't usually want to calculate the speed that R is growing. But indeed the WMAP lowerbound is 100 billion LY and it is growing at the rate you say, namely at 7c. (In this one finite volume case that we are using.)

I should say it is growing at least 7c, because it might be more than 100 billion LY and then its rate of increase would be proportionately larger. I think you see this clearly enough, I just want to be deliberate and explicit.

Neat.
====================================
BTW there seems to have been a tiny bit of confusion. Chalnoth apparently thought you were calculating the volume of the currently visible portion:

...
I haven't paid too close attention to your calculations here, but if accurate, this would only make for the volume of the visible portion of the universe...

I assumed you were trying to get a handle on the volume of the whole thing, in the case it is finite (pos. curv.) and that is why you used the hypersphere volume formula. If you were just after the currently visible volume, then I'm confused. But from your last post it sounds like we are clear about what we're doing.

You might also want to know the volume of the currently visible, say out to the surface of last scattering where the source material for the current CMB is----redshift z = 1090. That would be a different radius r, and a different volume formula, v = (4 pi/3) r³.
If you want some discussion of those things, just ask. That's another idea of size.

 

[TheSicilianSa]

Thank you for verifying my figures; but the fact that the universe is expanding faster than the speed of light brings up a second question:

I realize that our metrics are unaffected by a "faster than light" expansion, but wouldn't the intensity of the light reaching us from stars/galaxies (viz. # of photons/second received) be less than that actually emitted?

If this were to be the case, then how would our current estimates of "distance" (to stars/galaxies) be affected? - and, by measuring the fluctuation in intensity over time, would we not be able to estimate the acceleration rate (if any) of the expansion?

 

[Chalnoth]

Thank you for verifying my figures; but the fact that the universe is expanding faster than the speed of light brings up a second question:

It's not. As I said, expansion does not have units of speed. So the statement that the universe is expanding faster than any speed is frankly nonsensical. You might as well say that my car drives faster than 30Hz. It's ridiculous and makes no sense.

I realize that our metrics are unaffected by a "faster than light" expansion, but wouldn't the intensity of the light reaching us from stars/galaxies (viz. # of photons/second received) be less than that actually emitted?

This is the nature of the expansion of the universe, yes. As the redshift of light between source and receiver reduces the energy of each individual photon that reaches the receiver, this is seen as an extra drop in brightness.

If this were to be the case, then how would our current estimates of "distance" (to stars/galaxies) be affected? - and, by measuring the fluctuation in intensity over time, would we not be able to estimate the acceleration rate (if any) of the expansion?

This effect is accounted for in the standard measure of distance used for brightness-measured distances: the luminosity distance.

 

[marcus]

Thank you for verifying my figures; but the fact that the universe is expanding faster than the speed of light brings up a second question:

I realize that our metrics are unaffected by a "faster than light" expansion, but wouldn't the intensity of the light reaching us from stars/galaxies (viz. # of photons/second received) be less than that actually emitted?

If this were to be the case, then how would our current estimates of "distance" (to stars/galaxies) be affected? - and, by measuring the fluctuation in intensity over time, would we not be able to estimate the acceleration rate (if any) of the expansion?

You are welcome! I have to say that for a self-declared rank amateur you come up with congenial and interesting questions.

I understand exactly what you mean by the universe (in the particular finite volume case you have chosen to consider) having a size. This size was estimated in the 2008 WMAP cosmology report that I linked to earlier. And this size is increasing at a rate of at least 7c with 95% confidence (WMAP's table 2 page 4).

A less model-specific way to say this would be to say that the current distances to many of the galaxies we observe are increasing faster than c. And indeed in many cases were already increasing faster than c when the light was emitted.

How you say it is not the main thing. Let's look at your questions. Wouldn't the light be affected? Yes! Fewer photons per second and also each photon's energy diminished.
This is directly related to the factor by which distances have expanded while the light was in transit. In the case of CMB photons, distance has expanded by a factor of 1090 while they were in transit, and therefore their wavelengths have been extended by a factor of 1090, from a range around 2 microns to around 2 millimeters. Longer wavelength means lower energy. The CMB photons have each lost about 1089/1090 of their original energy that they had when they were emitted by the hot early universe medium (partially ionized gas).

And we get fewer per second, just as you surmised! If you think of them traveling to us in a kind of train, the expansion of distances has lengthened the interval between the cars.

In cosmology courses one is taught not to analyze the cosmo redshift as a Doppler effect* but instead to use a formula based on a measure of the universe's expansion called the scalefactor a(t).
The redshift z is the fractional increase in wavelength, so if the length doubles z = 1.
If the length triples, z = 2. It's just a convention. So z+1 is the ratio. This ratio is given by the formula:
z+1 = a(now)/a(then)
The universe's size or scale now divided by its size then (when the light started out on its way.)

In the case you chose to concentrate on, with finite volume 2 pi² R³,
and R being defined as in the WMAP report, you could just as well use R as a measure of size. The choice of a(t) is in some sense arbitrary. Cosmologists use it as a handle on size and they normalize it so that a(present) = 1. It is a relative scale parameter. It doesn't have units like meters or lightyears. Your measure R is in some sense more intuitive and less technical. You could write the formula for redshift this way:
z+1 = R(now)/R(then)
Wavelengths are stretched by the same factor that distances lengthened during the light's transit.
============================

Now we come to the interesting part. You suggest that we might DETECT stuff by keeping track of how the cosmological redshift increases over time!
This is an apt suggestion. Just a couple of weeks ago a professional astronomer Matt 0. here at PF was talking about doing just this. There are planned instruments with such fine accuracy that they may be able to detect the increase in redshift of a galaxy over a practical timeperiod like 10 or 20 years.

This wouldn't be detecting the acceleration people talk about so much. That is the second time-derivative of the scalefactor and it is very slight. If you do calculus and use prime notation the acceleration is a"(t).

The detectable effect would be just the fact that as the galaxy gets farther away the light takes longer to get here, and the universe expands more during transit, so there is more redshift. Over 10 years, even over 100 years, it is a small effect but it is calculable and Matt says that planned instruments may be able to measure it. Assuming it is measured as predicted, this will be another very strong support for the basic expansion model.

*except as a rough approximation or for short distances as appropriate.

 

[TheSicilianSa]

Again, I thank you - most informative.

I have a question on the Schwarzschild equation; should I continue to post here, post a new question under "cosmology" or post a new question under "black holes"?

 

[Carid]

marcus

The CMB photons have each lost about 1089/1090 of their original energy that they had when they were emitted by the hot early universe medium (partially ionized gas).

Can you please explain where that energy went?

 

[marcus]

Can you please explain where that energy went?

Thanks for asking, it's an interesting question. No. I cannot explain where it went, if it went anywhere at all. I don't think anyone can at present, at least I haven't heard a convincing explananation.

BTW a newcomer to PF just posted these quotes:

...
Misner, Thorne, Wheeler (Gravitation, section 29.2): “A detailed analysis focuses attention on three processes: emission, propagation, and absorption. Emission and absorption occur in the proper reference frames (orthonormal tetrads) of the emitter and receiver; they are special-relativistic phenomena. Propagation, by contrast, is a general-relativistic process: it is governed by the law of geodesic motion in curved spacetime.”

Sten Odenwald and Rick Fienberg (Sky and Telescope, February 1993, ...): “...The frequency of light is also affected by the gravitational field of the universe, and it is neither useful nor strictly correct to interpret the frequency shifts of light...in terms of the special relativistic Doppler effect.” "In fact, general relativity allows the Conservation of Energy to be suspended so that matter and energy may be created quite literally from the nothingness of curved spacetime.”

Peebles (Principles of Physical Cosmology, 1995, p. 139): “Where does the lost energy go? ... The resolution of this apparent paradox is that while energy conservation is a good local concept ... and can be defined more generally in the special case of an isolated system in asymptotically flat space; there is not a general global energy conservation law in general relativity theory.”
...

I've speculated about what might have happened to that energy. It might have somehow been imparted to the gravitational field. We are in a period during which the gravitational field is being studied using new conceptual tools, and being quantized. Maybe we will understand more a few years from now.

A lot of what happens in cosmology is outside the range of special relativity. It doesn't all fit onto one rigid non-expanding SR coordinate frame. So for now we have to use General Relativity. GR is our theory of spacetime geometry and the best theory of gravity we have. But if you are depending on GR then you don't have a global conservation law. You can't expect energy to be conserved over large reaches of dynamically changing geometry.
At least in any simple well-understood way.

Yeah! It's a good question. Maybe someone else wants to speculate.

 

[Carid]

marcus,

Thank you for your answer.
My cosy little world view is in pieces.

 

[marcus]

Again, I thank you - most informative.

I have a question on the Schwarzschild equation; should I continue to post here, post a new question under "cosmology" or post a new question under "black holes"?

Probably starting a new thread in astrophysics would be best.
But people do start black hole threads in cosmology forum as well.
When you have been around a while the Mentors may want you to use the search tool (top menu bar, advanced version) to find out if earlier threads have answered the question. But I often find that a fresh question, even if a partial repeat of something discussed earlier, is more fun. There are downsides to digging up old threads.

Here is some ridiculous pedantry for you, Sicilian: the German word Schild means a sign like what you hang up in front of a store or tavern. Maybe it is a cognate with the shield of a knight, with identifying heraldic emblems. Schwarz Schild means "black sign". Roth Schild means "red sign".
Schwarzschild has two esses.