Effort to get us all on the same page (balloon analogy)

[Jorrie]

...
H(then) = (1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5
I paste that in, and it tells me H(then) = 3.9 x 10^-18 Hz
in other words a number per second. But I want a number per million years so I multiply the answer by a million years:
(1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5*million years
and it tells me H(then) = 0.000123231935 per million years
So I tell it 1/0.000123231935 and it says 8115
The answer therefore is H(then) = 1/8115 per million years. The Hubble growth rate, which is now 1/144% per million years WAS back then 1/81% per million years.
And the Hubble radius, which is now 14.4 Gly, was 8.1 Gly.

I know you are fond of the % per million years growth rate. My question is, with everyone around here being used to think billions of years (Gy) in large scale cosmology, why not stick to it. One then uses the Hubble radii as we talk about them, i.e. your paragraph "paraphrased":

"H(then) = (1/17.3^2 per (billion years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5
I paste that in, and it tells me H(then) = 3.9 x 10^-18 Hz,
in other words a number per second. But I want a number per billion years so I multiply the answer by a billion years:
(1/17.3^2 per (billion years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5*(billion years)
and it tells me H(then) = 0.123231935 per billion years
So I tell it 1/0.123231935 and it says 81.15
The answer therefore is H(then) = 1/81.15 per billion years. The Hubble growth rate, which is now 1/14.4 per billion years WAS back then 1/81.15 per billion years."

I think this may avoid any confusion about the units used.

 

[marcus]

Hi Jorrie, thanks for the comment! Please keep me apprised of other things you notice. I'll think about switching to a coarser timescale and play around with it, but I probably won't shift over at least right away at this point. Very used to the 1/144 % per million year format, now. It has become a habit. But as you show it wouldn't be difficult to edit over to the coarser timescale format--just by moving the decimal point at strategic places. I'll try a kind of compromise edit in this post to see how it goes (but am not promising to shift permanently.)

Here's another example. Suppose you want to know the rate distances were growing at a time in the past when distances (between unmoving points) were HALF what they are today. Well then volumes were 1/8 of present size and the matter density then ρ(then) = 8*0.239 nanopascal.

We can use Friedman equation again

H(then)² = H_∞² + (1/Φ)8*0.239 nanopascal

H(then) = (1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5

I paste that in, and it tells me H(then) = 3.9 x 10^-18 Hz

in other words a number per second. But I want a number per billion years so I multiply the answer by a billion years. That means I paste in:
(1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5 * billion years
and it tells me H(then) = 0.12323... per billion years
Now 0.123… is about one eighth. Actually a bit less than 1/8, more like 1/8.1
The answer therefore is H(then) = 1/8.1 per billion years, and taking it in smaller steps that is 1/81 of a percent growth per million years.
The Hubble growth rate, which is now 1/144% per million years WAS back then 1/81% per million years.
And the Hubble radius, which is now 14.4 Gly, was 8.1 Gly.

Actually I like that kind of hybrid explanation. Let's try "take two" of the hybrid:

in other words a number per second. I want to know the Hubble time back then in billion years so I multiply the answer by a billion years. That means I paste in:(1/17300^2 per (million years)^2 + 8*pi*G/(3c^2)*8*0.239 nanopascal)^0.5 * billion years
and it tells me H(then) = 0.12323... per billion years
That 0.123… is about one eighth--actually a bit less than 1/8, more like 1/8.1
So the Hubble time back then was 1/H(then) = 8.1 billion years. As we know from regularly converting Hubble times and Hubble radii to percentage growth rates, this corresponds to H(then) being a distance growth rate of 1/81 of a percent per million years.
The Hubble growth rate H(t), which is now 1/144% per million years was 1/81% per million years, back then, and the Hubble radius, which is now 14.4 Gly, was 8.1 Gly.


You might be right. It might be better to completely switch over to billion years.
Then one gets fractions with a decimal point in the denominator: e.g. 1/14.4
and 1/17.3 but and one loses touch with the language of percentage growth rates. The percentages would be
1/0.144 percent per billion years and 1/0.173 % per billion years.
But to compensate, some calculations like this one would be considerably trimmer! We can keep this open and continue considering it.

 

[marcus]

The first occurrence of the cosmological constant Lambda in the GR equation, around 1920 (actually as early as 1917), is of interest in this connection. A source:
Einstein, A. 1917. Kosmologischege Betrachtungen zur allgemeinen Relativitätstheorie. Sitzungsb. König. Preuss. Akad. 142-152, reprinted and translated in The Principle of Relativity (Dover, 1952) 175-188
This recent historical paper quotes a recently translated original source that is later (1931):
http://arxiv.org/abs/1402.0132

I found the first occurrence of Lambda (cosmological constant) I know of in a 1917 paper Cosmological Considerations on the General Theory of Relativity translated in the Dover book on page 179, equations 2 and 3.
the book is available online at Internet Archive (archive.org) so it doesn't require a trip to the library.

 

[marcus]

The issue of entropy gets raised from time to time in connection with bounce cosmologies. People who think of entropy as an ABSOLUTE physical quantitity, rather than an observer-dependent one, occasionally ask how it apparently got to be so low at the start of expansion (if the start was a rebound from prior contracting phase.) I responded in the context of separate discussion so, for convenience, I'll save the reply here where I can refer to it easily.

If you look at how entropy is defined you see it is observer-dependent because it depends on the observer's coarse-graining---the macrovariable versus microvariable distinction. Entropy is the logarithm of the number of microstates (based on degrees of freedom irrelevant to the observer) comprising one grand macrostate (based on d.o.f that he actually interacts with and which affect him).

Any observer has a coarse-graining map corresponding to the lumping together of microstates into macrostates (consolidating all those which don't make any difference to the observer). Entropy measures the "size" in the particular macrostate we're in. The amount of information in it, that we ignore.

There's a group of people who think of entropy as absolute, who don't think that when you talk about it you have to specify a coarse-graining map. It is difficult for them to accept bounce cosmology because it looks to them as if "the entropy" (an absolute quantity) was reset to zero at the bounce. And there are other people who don't have that problem.

If you think of entropy as defined for a particular coarse-graining, then you don't encounter that mental obstacle. There is a pre-bounce guy and according to his coarsegraining the entropy increases astronomically as you go into the bounce, and it never thereafter declines! Because everything post-bounce is irrelevant to him, like it was inside the horizon of a black hole, the whole universe.
The post-bounce guy has a DIFFERENT coarsegraining and he sees the entropy initially low, everything about the bounce matters to him, is of vital importance, affects him thru variables he interacts with. Then as the U expands and diversifies regions of phase space become indifferent and irrelevant to him and entropy (for the post-bounce guy) increases.

The second law holds for any particular guy's entropy---defined based on his coarse-graining of the world.

This has been pointed out by various people. I think probably it would have come up in your Abhay&Ivan interview documentary video. As I recall Thanu Padmanabhan stated it clearly. Entropy is observer-dependent, or words to that effect. I've lost track of all the people who have made that point. Recently it came up here:
http://arxiv.org/abs/1407.3384
Why do we remember the past and not the future? The 'time oriented coarse graining' hypothesis
Carlo Rovelli
(Submitted on 12 Jul 2014)
Phenomenological arrows of time can be traced to a past low-entropy state. Does this imply the universe was in an improbable state in the past? I suggest a different possibility: past low-entropy depends on the coarse-graining implicit in our definition of entropy. This, in turn depends on our physical coupling to the rest of the world. …

Some more reading, if curious:
http://arxiv.org/abs/gr-qc/9901033
http://arxiv.org/abs/hep-th/0310022
http://arxiv.org/abs/hep-th/0410168

To give a bit of the flavor I'll quote a passage from Don Marolf's 2004 paper

==quote http://arxiv.org/abs/hep-th/0410168 from conclusions==
the realization that observers remaining outside a black hole associate a different (and, at least in interesting cases, smaller) flux of entropy across the horizon with a given physical process than do observers who themselves cross the horizon during the process. In particular, this second mechanism was explored using both analytic and numerical techniques in a simple toy model. We note that similar effects have been reported³⁵ for calculations involving quantum teleportation experiments in non-inertial frames. Our observations are also in accord with general remarks^36,37 that, in analogy with energy, entropy should be a subtle concept in General Relativity.
We have concentrated here on this new observer-dependence in the concept of entropy. It is tempting to speculate that this observation will have further interesting implications for the thermodynamics of black holes. For example, the point here that the two classes of observers assign different values to the entropy flux across the horizon seems to be in tune with the point of view (see, e.g., Refs. 38,39,40,41,42) that the Bekenstein-Hawking entropy of a black hole does not count the number of black hole microstates, but rather refers to some property of these states relative to observers who…
==endquote==
For context see: https://www.physicsforums.com/showthread.php?p=4810929#post4810929

 

[marcus]

I got a PM letter yesterday from one of our members asking basic questions about the declining Hubble expansion rate, the increasing growth *speed* of the scale factor and related matters, so decided to respond here in case it could be useful for anybody else.
In words: H(t) is a fractional growth rate, conveniently expressed as percentage growth per million years, rate is different from *speed*. The speed a distance grows is proportional to its size, larger distances grow faster.
So the percentage RATE can be DECLINING even though if you watch a particular distance it's growth SPEED can be increasing.
Like if you have a savings account at the bank, the percent INTEREST on it can be constant or even slowly declining and yet, because your principal is growing your savings can still be increasing by a greater amount each year, in gross dollar terms.

I think it's good to look at a concrete example. A purely verbal description like the above leaves something missing. Let's look at some numbers, using Jorrie's Lightcone calculator:
This table runs from year 67 million, when distances were 1/40 their present size, out to year 28.6 billion when distances will be 2.5 times their present size. The present is year 13.787 billion where you see scale factor a(t) = 1 and its reciprocal S(t) = 1 indicating distances are exactly their present size.
The way to read the percentage growth rate is to mentally multiply the R column number by TEN and take ONE OVER THAT. So in year 67 million, the growth rate was one percent per million years
Is that clear? You multiply 0.1 by ten and get 1, and one over one is one.

I would advise getting used to reading the R column that way. Another example: in year 135 million distances were growing 1/2 percent per million years , you take 0.2, multiply by ten to get 2 and take one over two to get 1/2.

You can see from the table that at present, year 13.787 billion, distances are growing 1/144 percent per million years. The percentage rate has come down a lot over time and it is continuing to decline towards a limit of 1/173 %. The table extends into the future far enough to show it getting down to 1/171 %, which is getting close to where it is expected to end up.

$${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&a'R_{0} (c) \\ \hline 0.025&40.000&0.067&0.1&3.53\\ \hline 0.031&31.764&0.096&0.1&3.14\\ \hline 0.040&25.223&0.135&0.2&2.79\\ \hline 0.050&20.030&0.192&0.3&2.49\\ \hline 0.063&15.905&0.271&0.4&2.22\\ \hline 0.079&12.630&0.384&0.6&1.97\\ \hline 0.100&10.030&0.543&0.8&1.76\\ \hline 0.126&7.965&0.768&1.2&1.57\\ \hline 0.158&6.325&1.085&1.6&1.40\\ \hline 0.199&5.022&1.531&2.3&1.25\\ \hline 0.251&3.988&2.159&3.2&1.13\\ \hline 0.316&3.167&3.035&4.5&1.02\\ \hline 0.398&2.515&4.243&6.1&0.94\\ \hline 0.501&1.997&5.876&8.1&0.89\\ \hline 0.631&1.586&8.009&10.4&0.87\\ \hline 0.794&1.259&10.665&12.6&0.91\\ \hline 1.000&1.000&13.787&14.4&1.00\\ \hline 1.259&0.794&17.262&15.6&1.16\\ \hline 1.495&0.669&20.000&16.3&1.32\\ \hline 1.774&0.564&22.823&16.7&1.53\\ \hline 2.106&0.475&25.701&16.9&1.79\\ \hline 2.500&0.400&28.613&17.1&2.11\\ \hline \end{array}}$$

This table is an implementation of the FRIEDMANN EQUATION as a table of numbers instead of as an equation. It's good to study the equation and understand it, but I think it also helps to mull over the actual numbers of the history of the universe which the equation generates when you plug in the observed values of the parameters and run it.

The rightmost column is the growth *SPEED* of a chosen sample distance whose present size is 14.4 billion light years. You can see it starts out (way back in year 67 million) at 1/40 of its present size and growing at 3.53 times the speed of light.
And that speed declines until around year 8 billion.
And then it starts to increase.
And by now, in year 13.787 billion, it is increasing at exactly the speed of light.
So ever since year 8 billion it has been, in a manner of speaking, "accelerating".
But the word is not quite apt. Distance growth is not like ordinary motion. Nobody GETS anywhere by it, everything just becomes farther apart. So the word "accelerating" is just slightly misleading and can give a false mental image. It just means that the speed of distance growth is increasing.
Although of course as we noted earlier the percentage RATE of distance growth is declining.

 

[thedaybefore]

I said I would try to avoid abbreviations, but I need another one: CMB for cosmic microwave background.

The balloon analogy teaches various things, but sometimes you have to concentrate in order to learn them.

One thing it teaches is what it means to be not moving with respect to CMB.

the balloon is a spherical surface and as it gradually expands a point that always stays at the same longitude and latitude is stationary with respect to CMB.

Distances between stationary points can increase, and in fact they do. They increase at a regular percentage rate (larger distances increase more). In our 3D reality this is called Hubble Law. It is about distances between points which are at rest wrt CMB.

In our 3D reality you know you are at rest wrt CMB if you point your antenna in all directions and get roughly the same temperature or peak wavelength. There is no doppler hotspot or coldspot in the CMB sky. That means you are not moving with respect to the universe.

In cosmology being at rest is a very fundamental idea, we had it even before the 1960s when the CMB was discovered. Then it was defined as being at rest with respect to the process of expansion---you could tell you were at rest with respect to the universe if the expansion around you was approximately the same in all directions---not faster one one side of the sky and slower on the other, but balanced. It is the same idea but now we use the CMB to define it because it is much more accurate. Sun and planets are traveling about 380 km/s with respect to CMB in a direction marked by the constellation Leo in the sky. It is not very fast but astronomical observations sometimes need to be corrected for that motion so as to correspond to what an observer at CMB rest would see.

Now let's take another look at the balloon and see what else it can tell us.

The CMB is electromagnetic radiation and all non-accelerating reference frames are non-moving to the observer. Accordingly, how can a non-accelerating observer not be at rest with respect to the CMB? All CMB will be moving at C to every observer.

 

[marcus]

At rest wrt CMB MEANS temperature essentially the same in all directions.

Solar system we know is not at rest, for reason given in what you quote. There is a hot spot in constellation Leo. And a cold spot in the opposite direction.
What you quoted says 380 but a better figure is the solar system is moving about 370 km/s in the Leo direction, relative to the soup of ancient light. A recent report says 369 km/s
That is about 0.123 of a percent of the speed of light.
Therefore the temperature in that direction in the sky is 0.123 of a percent WARMER than the average CMB sky temperature. Something like 0.003 kelvin warmer than the average 2.725 kelvin

Another way to say observer "at rest" is to say ISOTROPIC observer. Isotropic means "universe looks the same in all directions" In particular the CMB temperature is the same in all directions.

An observer riding with the solar system is not an isotropic observer because there is a measurable temperature "dipole", a hotspot coldspot axis.

Hubble already discovered this motion, or dipole, before the CMB was known. The galaxies in the Leo direction are receding on average a LITTLE SLOWER than the overall Hubble rule predicts. This is because the solar system is not at rest wrt universal expansion process. And galaxies in the opposite direction are receding a little faster .

Discovering the CMB and the temperature hotspot only made this more accurate, but it was already known that the universe has a criterion of rest.

It does not depend on the electromagnetic field, it depends on the approximately uniform distribution of the ancient matter, the primordial gas.

 

[marcus]

A good recent report is
http://arxiv.org/abs/0803.0732
Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Data Processing, Sky Maps, and Basic Results
G. Hinshaw, J. L. Weiland, R. S. Hill, N. Odegard, D. Larson, C. L. Bennett, J. Dunkley, B. Gold, M. R. Greason, N. Jarosik, E. Komatsu, M. R. Nolta, L. Page, D. N. Spergel, E. Wollack, M. Halpern, A. Kogut, M. Limon, S. S. Meyer, G. S. Tucker, E. L. Wright
(Submitted on 5 Mar 2008 (v1), last revised 17 Oct 2008 (this version, v2))
We present new full-sky temperature and polarization maps in five frequency bands from 23 to 94 GHz, based on data from the first five years of the WMAP sky survey. The five-year maps incorporate several improvements in data processing made possible by the additional years of data and by a more complete analysis of the ...
==quote==
samples from both methods to produce the conservative estimate shown in the bottom row. This approach, which enlarges the uncertainty to emcompass both estimates, gives

(d, l, b) = (3.355 ± 0.008 mK, 263.99◦ ± 0.14◦, 48.26◦ ± 0.03◦), (1)

where the amplitude estimate includes the 0.2% absolute calibration uncertainty. Given the CMB monopole temperature of 2.725 K (Mather et al. 1999), this amplitude implies a Solar System peculiar velocity of 369.0 ± 0.9 km s−1 with respect to the CMB rest frame.
==endquote==

See also this one:
http://arxiv.org/abs/1303.5087
Planck 2013 results. XXVII. Doppler boosting of the CMB: Eppur si muove
Planck Collaboration: N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, A. J. Banday, R. B. Barreiro, J. G. Bartlett, K. Benabed, A. Benoit-Lévy, J.-P. Bernard, M. Bersanelli, P. Bielewicz, J. Bobin, J. J. Bock, J. R. Bond, J. Borrill, F. R. Bouchet, M. Bridges, C. Burigana, R. C. Butler, J.-F. Cardoso, A. Catalano, A. Challinor, A. Chamballu, L.-Y Chiang, H. C. Chiang, P. R. Christensen, D. L. Clements, L. P. L. Colombo, F. Couchot, B. P. Crill, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, J. M. Diego, S. Donzelli, O. Doré, X. Dupac, G. Efstathiou, T. A. Enßlin, H. K. Eriksen, F. Finelli, O. Forni, M. Frailis, E. Franceschi, S. Galeotta, K. Ganga, M. Giard, G. Giardino, J. González-Nuevo, v1), last revised 10 Nov 2014 (this version, v2))
Our velocity relative to the rest frame of the cosmic microwave background (CMB) generates a dipole temperature anisotropy on the sky which has been well measured for more than 30 years, and has an accepted amplitude of v/c = 0.00123, or v = 369km/s. In addition to this signal generated by Doppler boosting of the CMB monopole, our motion also modulates and aberrates the CMB...
...gnificant confirmation of the expected velocity.

 

[thedaybefore]

At rest wrt CMB MEANS temperature essentially the same in all directions.

Solar system we know is not at rest, for reason given in what you quote. There is a hot spot in constellation Leo. And a cold spot in the opposite direction.
What you quoted says 380 but a better figure is the solar system is moving about 370 km/s in the Leo direction, relative to the soup of ancient light. A recent report says 369 km/s
That is about 0.123 of a percent of the speed of light.
Therefore the temperature in that direction in the sky is 0.123 of a percent WARMER than the average CMB sky temperature. Something like 0.003 kelvin warmer than the average 2.725 kelvin

Another way to say observer "at rest" is to say ISOTROPIC observer. Isotropic means "universe looks the same in all directions" In particular the CMB temperature is the same in all directions.

An observer riding with the solar system is not an isotropic observer because there is a measurable temperature "dipole", a hotspot coldspot axis.

Hubble already discovered this motion, or dipole, before the CMB was known. The galaxies in the Leo direction are receding on average a LITTLE SLOWER than the overall Hubble rule predicts. This is because the solar system is not at rest wrt universal expansion process. And galaxies in the opposite direction are receding a little faster .

Discovering the CMB and the temperature hotspot only made this more accurate, but it was already known that the universe has a criterion of rest.

It does not depend on the electromagnetic field, it depends on the approximately uniform distribution of the ancient matter, the primordial gas.

What concerns me here is that it sounds suspicously like you are creating a preferred reference frame forhe Universe. It may be that there are differences in temperature of the microwave background and that the objects in the Universe have relative velocity to each other, I don't believe we can say that the CMB has some ultimate zero velocity. At best, I think we can say there is relative velocity between the objects that emitted the CMB 14 billion years ago and us.

 

[marcus]

the U represents a particular solution of GR. In the mother theory (GR) there is no preferred ref frame. But individual solutions to GR equation can have preferred frame specific to that solution.
So in cosmology we have a preferred frame.

It depends on initial conditions---eg a particular configuration of initial matter. even.

The solution is basically the Friedmann solution, the Friedmann metric. It has a preferred time called universe time or Friedmann time.

AFAIK everybody who does cosmology knows there is a preferred time, and a criterion of rest with respect to the universe (aka wrt CMB).

If you don't believe me, there is nothing I can say. This is basic.

 

[thedaybefore]

the U represents a particular solution of GR. In the mother theory (GR) there is no preferred ref frame. But individual solutions to GR equation can have preferred frame specific to that solution.
So in cosmology we have a preferred frame.

Is this preferred frame based upon an analytical solution to Einsteins's equations that is solved for high field assymetric conditions? However, I guess as long as you guys don't posit some type of aether, I shouldn't be too concerned.

 

[GeorgeDishman]

I think it is also important to emphasise that the "preferred frame" isn't a frame in the usual special relativity sense, two objects, both at rest with respect to the CMB will not be at rest with respect to each other as Marcus said. Another way to look at it is that today's galaxies formed out of the gas that emitted the CMB so really we are just measuring the speed of individual items relative the average speed of all those in the neighbourhood.

 

[marcus]

Practice with a HANDS-ON approach to cosmology might help get us all on the same page. I'd like to try it with some volunteers who'd be willing to work some concrete exercises (with self-calculating formulas) and see how much it improved their comprehension.
The thought here is that purely verbal explanations tend to lead to confusion. As can too much reliance on equations with abstract symbols.

It's possible to put the equations to work in simple calculations and that can raise one's level of understanding quite a lot.

Anyway comments and reactions are very welcome. What I'm thinking of doing is carry this effort to get us on the same page, cosmology-wise, a step beyond the balloon analogy and the CMB rest frame, and see how it goes.

 

[Drakkith]

I'd be willing to give it a go.

 

[marcus]

Great! I hope two or three others will join the project. I'd like us to try using zeit (17.3 billion years) as a time unit and lightzeit (17.3 billion lightyears) for distance. It makes the formulas very simple, so they can be effectively self-calculating.
The present age is 0.8 zeit (more precisely 0.797 but 0.8 is close enough).

Fact 1 is at any time t the size of distances that are expanding at the speed of light is tanh(1.5t).

The answer comes out in lightzeits and it's especially convenient because google calculator knows the function "tanh". So you can say what size of distance is growing at speed c right now today. You just type in tanh(1.5*0.8) and press "enter", the * is for multiplication. Let me know if you have any trouble with google.

Exercise 1.1 what size distance WILL be growing at the speed of light in the future 0.1 zeit from now, i.e. when the age t = 0.9.
Exercise 1.2 what size distances WERE expanding at speed c in the past, 0.1 zeit ago, i.e at age t = 0.7.

Drakkith please let me know if this is grossly too simple or too hard. I have very little notion of what the right level is to start with. If this is OK, the focus at first will be on simple hands-on calculation of the universe, getting actual numbers so it is on more than just a verbal level.

 

[Drakkith]

Exercise 1.1 what size distance WILL be growing at the speed of light 0.1 zeit from now in the future, i.e. when the age t = 0.9.
Exercise 1.2 what size distances WERE expanding at speed c

1.1: 0.87 lightzeit, or 15.1 billion light years.

1.2: 0.78 lightzeit.
(Note that if you use another calculator than google, which I did, you have to use radians, not degrees)

 

[marcus]

Yay! BTW I got called away from computer while I was typing in Exercise 1.2, and only finished it later.

 

[Drakkith]

Yay! BTW I got called away from computer while I was typing in Exercise 1.2, and only finished it later.

I've corrected my post.

 

[Drakkith]

Drakkith please let me know if this is grossly too simple or too hard. I have very little notion of what the right level is to start with. If this is OK, the focus at first will be on simple hands-on calculation of the universe, getting actual numbers so it is on more than just a verbal level

I think it's pretty simple. You literally just plug in numbers or type it into google.

 

[marcus]

I see it's too elementary, I'll go to something a bit more complicated in a few minutes.

But what we just did was essentially equivalent to Hubble's law v(t) = H(t)D(t)
because the speed a distance is growing is proportional to its size.

So if you know what size is growing at c, and you have another distance that is HALF that then you know it is growing at half c.

In our units we can say H(t) = 1/tanh(1.5*t) and we can write Hubble law
v(t) = D(t)/tanh(1.5*t)

Exercise 1.3 So thinking back to t = 0.234, how fast was a distance growing that was size 0.337 lightzeit?

 

[marcus]

Exercise 1.4 A distance is 3/4 lightzeit in size at time 0.44, what speed is it growing?

I think you are good with these tanh(1.5t) exercises. I should move on.

How about the inverse, you very quickly got the knack of going from time t to the distance that grows at speed c. Call that R(t).
maybe we can go back from the distance R to t

Can your calculator do the natural logarithm "ln"? If so then you can calculate the inverse of the R(t) function and go back from R --> t

If at some time t, the distances growing at speed c are R of a lightzeit, then the time is
t = ln((1+R)/(1-R))/3

I think you already found that for time t=0.8 that R=0.83, so we could check by working back from R=0.83
Does ln(1.83/0.17)/3 = 0.8?

Imagine you find yourself back in a time when distances sized 0.71 lightzeit are growing at speed c.
Has the Earth formed yet? If it has, are there single-celled living organisms?

I'm hesitant about presenting the formula for the size of a generic distance growing over time, because of the 2/3 power. Does your calculator do sinh(1.5t)^(2/3)?

 

[marcus]

Great! I hope two or three others will join the project. I'd like us to try using zeit (17.3 billion years) as a time unit and lightzeit (17.3 billion lightyears) for distance. It makes the formulas very simple, so they can be effectively self-calculating.
The present age is 0.8 zeit (more precisely 0.797 but 0.8 is close enough).

Fact 1 is at any time t the size of distances that are expanding at the speed of light is tanh(1.5t).

The answer comes out in lightzeits and it's especially convenient because google calculator knows the function "tanh". So you can say what size of distance is growing at speed c right now today. You just type in tanh(1.5*0.8) and press "enter", the * is for multiplication. ..

So far we have one self-computing formula (and its inverse). t→R (and R → t)
It is simply tanh(1.5*t), and it relates the size and growth speed of distances at any given time t.
It gives the size of distances that are growing at speed c at time t. From that you can figure out the speeds that OTHER distances are growing, because speed is proportional to size.

I'm thinking we might be able to make do which just three basic formulas. The next one is more complicated and it calculates the ratio of distance size (or wavelength size) at any two times, call them "then" and "now".
Substitute numbers for the words then and now.
sinh(1.5*then)^(2/3)/sinh(1.5*now)^(2/3)

$$\frac {size\ then}{size\ now} = \frac{sinh(1.5*then)^{2/3}}{sinh(1.5*now)^{2/3}}$$

Fact 2: This compares wavelengths and distances then to what they are now. or actually between any two times you choose, including times in the future.

An example would be to take 0.44 zeit as the age when acceleration began and to take 0.797 as the present.
sinh(1.5*0.44)^(2/3)/sinh(1.5*0.797)^(2/3)
That calculates the size of distances back then compared to what they are today. And if some light arrives here today that some galaxy emitted back at that time its wavelengths will arrive stretched out by the the reciprocal of that number, the factor by which distances have expanded between then and now. You can use it to predict future size of distances compared with their size now---or the wave stretch of some signal we send today that is received some time in the future.

 

[marcus]

This post is a review.
We're looking for a way to present the standard model of expanding universe that can get the maximum number of people on board, the broadest possible hands-on understanding.

the google calculator knows the "tanh" function and it happens that the cosmic expansion rate is well-described by it. that was Fact 1.

So let's get to know this function better. As I see it, at least, it's the simplest function you can build with e^x which starts out at zero at x = 0 and increases steadily leveling off at 1.$$\tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}$$ Readers may find that picturing the graph of the basic exponential function e^x---how it tends to zero at -∞, equals 1 at x=0 and rises steeply for x---helps understand how the "tanh" function defined this way equals zero at x = 0 and rises steadily leveling off at 1.

The simplicity of this function, I would say, is shown by its having a simple inverse---you can UNDO it easily. If you plug x into tanh, and get tanh x, you can always get x back again.

If$$R = \tanh x = \frac{e^x - e^{-x}}{e^x + e^{-x}}$$ then $$2x = \ln\frac{1+R}{1-R}$$ Checking this just involves simplifying a fraction by multiplying numerator and denominator by the same thing, namely e^x+e^-x
$$\frac{1+\frac{e^x - e^{-x}}{e^x + e^{-x}}}{1 - \frac{e^x - e^{-x}}{e^x + e^{-x}}} = \frac{e^x}{e^{-x}}= e^{2x}$$
and the natural logarithm of e^2x is 2x.

In the case of our universe, the R = tanh 1.5t, instead of x we have 1.5t, and 2x = 3t, so $$3t = \ln\frac{1+R}{1-R}$$ and $$t = \frac{1}{3}\ln\frac{1+R}{1-R}$$

 

[marcus]

So if you want to know the SPEED any given distance D is expanding, at some point t in history, you just find R(t)=tanh(1.5t) which is the size of distances expanding at c, and compare D to that. At any given moment, expansion speed is proportional to size.

If D is twice R, then the distance is growing at twice the speed of light.
If D is half R, then the distance is growing at half the speed of light.

Its expansion speed is the distance's size divided by R. Also notice that at any time t, R(t) is the critical size that distinguishes distances growing faster than c from those growing slower than c. That's one of several ways R(t) can give us a handle on the expansion rate history: It is also reciprocal to the "percentage" or fractional growth rate. The larger R is, the slower any given-sized distance is growing (think: larger denominator---the farther out you have to go to get to where distances are expanding at the speed c.)

So R(t) is the critical distance size at time t and it changes over time like this (the blue curve in this picture)

In professional circles this critical distance size R(t) is called the "Hubble radius". One way to picture its role is to think of a bit of light coming from far away aimed in our direction. As long as the light is farther than R(t) (outside the "Hubble radius") it will lose ground. It can't make headway because the distance to us is expanding too fast. So it is swept back. You can see that by looking at the red curve in the picture in the interval from t=0 to t=0.2 where the distance to the light (red) is greater than critical (blue) and the light is swept back--gets farther from us even though traveling towards us thru its surrounding space.

But the critical distance increases until it finally takes the light in (actually around time t = 0.234 is when this happens). At that moment the speed of the light is exactly canceled and you can see the slope of the red curve is flat, zero progress. Then after the moment t = 0.234 has passed the light begins to make progress and narrow the distance to us. It arrives here at t = 0.8 which is the present time in history.

You could say (figuratively of course) that our universe has "chosen", for this critical size, the simplest function you can build from basic exponential e^x that starts at zero, levels off at one, and has a simple inverse. I didn't mention earlier that it has a nice flip symmetry too: tanh(-x) = -tanh(x). Flip it left-to-right and then flip it head-to-toe and you get back what you started with. Maybe that doesn't matter. : ^) To me it suggests a prior contracting phase. The negative branch of the tanh(x) function could describe contraction, mirroring the expansion. Here the red curve is the size history of a sample distance, which is contracting and expanding in accordance with the blue (tanh) curve.

For the sample distance to plot, I chose one which is just slightly over 1.3 lightzeits at the present. That is at t = 0.8. That has the simplest growth history to plot: D(t) = sinh(1.5t)^2/3

 

[marcus]

The third formula (those three may be all we need to do basic cosmology) is going to be the wavestretch-distance relation. The wavestretch is the factor by which wavelengths and distances are enlarged while the light is on its way to us. It equals the conventional "redshift" number plus one.
It's more convenient to work with than the conventional redshift, which when you use in equations you usually have to add one to, so that you are actually working with the wavestretch. I'll denote it with the letter S.

We want to know, if some light comes in stretched by a factor S, how far the source is NOW. Let's call that distance D(S). It turns out that this is an integral that is easy and quick to get evaluated online. There is a website that does this and it remembers what you keyed in the last time you visited, so once you have gone once and done the distance integral D(S) you have very little to do the next time except change one of the limits of integration.

$$D(S) = \int_1^S R(s)ds = \int_1^S ((s/1.3)^3 + 1)^{-1/2}ds$$

Here's a good website for online integration
http://www.numberempire.com/definiteintegralcalculator.php

 

[marcus]

Wavestretch and current distance to the source are the two most important things we more or less directly observe. The first is measured from the incoming light itself, the second (the current distance to the source) is told from the dimness of standard brightness sources (called "standard candles").

Fortunately the critical distance size R(t), for which we already have the formula relating it to time R(t) = tanh(1.5t) is also simply related to the wavestretch of incoming light that was emitted at the time in question. R(S) = ((S/1.3)³ +1)^-1/2

For example suppose some light comes in stretched by a factor of 3, so its wavelengths are 3 times as long as they were in the original light. You want to find the distance to the source galaxy. You go here
http://www.numberempire.com/definiteintegralcalculator.php
put 1 and 3 in for lower and upper limits, change the variable from x to s, put
((s/1.3)^3 + 1)^(-1/2) in the main box and press "calculate"

If greater precision were needed one could use 1.3115 instead of 1.3. I'll have more to say in the morning and hopefully will be able to explain this third formula somewhat. Using this wavestretch-distance formula is basically how the positive cosmological curvature constant was discovered, in 1998, and it's how the longterm value of 17.3 billion lightyears has been determined (by fitting wavestretch-distance data).

 

[Drakkith]

Exercise 1.3 So thinking back to t = 0.234, how fast was a distance growing that was size 0.337 lightzeit?

Just now saw this and your next few posts. I'll have to get back to you, tomorrow hopefully.

 

[marcus]

No rush! Your reactions are a good guide. Help me see which explanations work, which don't. I could review where we are at this point.
The aim is a widely accessible basic introduction to cosmology which goes beyond merely verbal description. The reader should be enabled to calculate a few things (if he or she wishes to) like expansion speeds, expansion ratios, distance to source, or how long ago light with a given stretch factor was emitted...

I'm thinking now that we might narrow it down essentially to three self-calculating formulas. "google-ready" formulas (ones that you can paste into google)

A. tanh(1.5t) gives the "Hubble radius" at time t---the critical size that separates distances growing faster than c from those growing slower. This is the size of those distances which are growing at speed c and by proportionality you can use it to find the expansion speeds of other distances. Reminder: use an asterisk for multiplication as in tanh(1.5*0.8)

B. sinh(1.5"now")^(2/3)/sinh(1.5"then")^(2/3) where you replace "now" and "then" by two different times. It gives the ratio of distance size between those two times, and the ratio of wavelength size. So it gives the expansion factor between those two times.
I would like to call this ratio "stretch" and denote it S, in the special case where "now" is 0.8 or more exactly 0.797.
The stretch factor S compares present day size to that of the same distance or wavelength at another time which can be either in past or future.
That's a special case though. The formula can be used with any two times---it doesn't need for one of the two to be the present.

C. ((s/1.3)^3 + 1)^(-1/2) computes R(S) the "Hubble radius" at a time in the past (or future) corresponding to a given stretch factor S. Pasting that in for the integrand at numberempire gets us
D(S) = ∫₁^S((s/1.3)^3 + 1)^(-1/2)ds, the distance from its source of light arriving today with stretch S.

Example: People are often interested in knowing the radius (today's distance) of the currently observable region. We are currently getting CMB light stretched by roughly a factor of 1000, so we can use that D(S) formula to find out the distance to that matter---roughly the same as the radius of the currently observable region. The latter is only a bit larger to allow for other possible signals from even earlier and more distant matter.
So numberempire integrator can be used to find D(1000), a reasonably good estimate of the radius of the observable.
To improve the accuracy some, we use 1.3115 instead of 1.3.

 

[Drakkith]

Exercise 1.3 So thinking back to t = 0.234, how fast was a distance growing that was size 0.337 lightzeit?

I get 0.215 c.

Exercise 1.4 A distance is 3/4 lightzeit in size at time 0.44, what speed is it growing?

I get 0.416 c.

 

[Drakkith]

I'm hesitant about presenting the formula for the size of a generic distance growing over time, because of the 2/3 power. Does your calculator do sinh(1.5t)^(2/3)?

Plugging in 0.5 for t, I get 0.8778 using this calculator online. Is that right?

 

[marcus]

Plugging in 0.5 for t, I get 0.8778 using this calculator online. Is that right?

Yes! Google calculator agrees with your answer.
When I put this into the google search window
sinh(1.5*0.5)^(2/3)
and press enter, I get 0.8777... hardly any difference at all!
======
tanh(1.5*0.234) ≈ 0.337

.75/tanh(1.5*0.44) ≈ 1.3

 

[marcus]

Since we just turned a page, I'll bring forward this summary of the essentials.

...
The aim is a widely accessible basic introduction to cosmology which goes beyond merely verbal description. The reader should be enabled to calculate a few things (if he or she wishes to) like expansion speeds, expansion ratios, distance to source, or how long ago light with a given stretch factor was emitted...

==quote==
I'm thinking now that we might narrow it down essentially to three self-calculating formulas. "google-ready" formulas (ones that you can paste into google)

A. tanh(1.5t) gives the "Hubble radius" at time t---the critical size that separates distances growing faster than c from those growing slower. This is the size of those distances which are growing at speed c, and by proportionality you can use it to find the expansion speeds of other distances. Reminder: use an asterisk for multiplication as in tanh(1.5*0.8)

B. sinh(1.5"now")^(2/3)/sinh(1.5"then")^(2/3) where you replace "now" and "then" by two different times. It gives the ratio of distance size between those two times, and the ratio of wavelength size. So it gives the expansion factor between those two times.
In the special case where "now" is 0.8 or more exactly 0.797.I would like to call this ratio "stretch" and denote it S. This agrees with the notation in Lightcone calculator. The stretch factor S compares present day size to that of the same distance or wavelength at another time which can be either in past or future.
That's a special case though. The formula here can be used with any two times---it doesn't need for one of the two to be the present.

C. ((s/1.3)^3 + 1)^(-1/2) computes R(S) the "Hubble radius" at a time in the past (or future) corresponding to a given stretch factor S. Pasting that in for the integrand at numberempire gets us
D(S) = ∫₁^S((s/1.3)^3 + 1)^(-1/2)ds, the distance from its source of light arriving today with stretch S.

Example: People are often interested in knowing the radius (today's distance) of the currently observable region. We are currently getting CMB light stretched by roughly a factor of 1000, so we can use that D(S) formula to find out the distance to that matter---roughly the same as the radius of the currently observable region. The latter is only a bit larger to allow for other possible signals from even earlier and more distant matter.
So numberempire integrator can be used to find D(1000), a reasonably good estimate of the radius of the observable.
To improve the accuracy some, we use 1.3115 instead of 1.3.
==endquote==

 

[marcus]

I'll add just one more formula, to close the circle, and see how that looks. Is it an adequate set of tools for basic cosmology?

D. Given a figure for the Hubble radius R > 1, you can find the time in history t when that was the R(t). Namely t = ln((1+R)/(1-R))/3

As a check try putting in 0.83 for R because at present the R is about 0.83 lightzeit. You should get t = 0.8

The reason I say to close the circle is because formulas B, C, and D, enable one to compute t→S→R→t

B. t→S
C. S→R
D. R→t
We already saw formula D, it being the inverse of A. Back in posts#511 and #513.$$t = \frac{1}{3}\ln\frac{1+R}{1-R}$$======================
How can we visualize the fractional growth rate that goes along with a given Hubble radius R? This has to involve taking the RECIPROCAL of R because the larger R is the slower the growth of any given size distance. Remember that you DIVIDE the size by R to get the speed. Let's take the largest R that's in the cards for our universe (according to standard cosmic model), namely R = 1 lightzeit = 17.3 billion lightyears.
Suppose we've reached that point and that is the distance that is growing at the rate of one lightyear per year.
Imagine a microzeit, a millionth of a zeit---that is 17,300 years (longer than human civilization has existed so far, but there were hunt&gather folks that looked like us that long ago.)
What fraction of itself does a distance sized 1 lightzeit grow in a millionth of a zeit? It expands by a millionth of itself.
It expands by 1 ppm---by one part per million---per microzeit.

That works for all. Right now the Hubble radius R(now) = R(0.8) is about 0.833 and the reciprocal of that is about 1.2. To picture the corresponding fractional rate of distance growth, we can think 1.2 ppm per microzeit. 1.2 parts per million in 17,300 years.
Maybe you or somebody else can think of a better way to visualize it. Cosmic expansion is really really slow, in human terms.
Does this way of imagining the expansion rate work?

 

[Drakkith]

Does this way of imagining the expansion rate work?

It's a little abstract, being all numbers, but I think so.

 

[marcus]

======================
How can we visualize the fractional growth rate that goes along with a given Hubble radius R? This has to involve taking the RECIPROCAL of R because the larger R is the slower the growth of any given size distance. Remember that you DIVIDE the size by R to get the speed.
Let's take the largest R that's in the cards for our universe (according to standard cosmic model), namely R = 1 lightzeit = 17.3 billion lightyears.
Suppose we've reached that point and that is the [size of distance that is growing at the rate c.]
Imagine a microzeit, a millionth of a zeit---that is 17,300 years (longer than human civilization has existed so far, but there were hunt&gather folks that looked like us that long ago.)
What fraction of itself does a distance sized 1 lightzeit grow in a millionth of a zeit? It expands by a millionth of itself.
It expands by 1 ppm---by one part per million---per microzeit.

...Right now the Hubble radius R(now) = R(0.8) is about 0.833 and the reciprocal of that is about 1.2. To picture the corresponding fractional rate of distance growth, we can think 1.2 ppm per microzeit. 1.2 parts per million in 17,300 years.
Maybe you or somebody else can think of a better way to visualize it. Cosmic expansion is really really slow, in human terms.
Does this way of imagining the expansion rate work?

It's a little abstract, being all numbers, but I think so.

I'm glad it seems OK, also you put your finger on what I think is the weakness. The fractional growth rate is abstract and harder to picture, because it is too numerical. By contrast, the Hubble radius is more concrete and easier to visualize---the size of a distance that is growing at speed c.