From Aeon to Zeon to Zeit, simplifying the standard cosmic model

[marcus]

This is take 2.0 of the earlier thread, which got a lot of help from George J., Jorrie, Wabbit, Ken G, and others. I'm exploring this simplification of the flat matter-dominated ΛCDM model (basically anything after year 1 million) to see if there is a presentation that would be suitable for PF Insights.
And if not that's fine too. It's a Cosmology forum thread.

The point is the universe's own behavior defines a natural time scale for us. The rate of distance growth H is declining and leveling out so that it approaches a constant longterm rate H_∞. That rate defines a natural unit of time (let's call it a "zeon" to rhyme with "aeon").

If we use zeons instead of (billions of) Earth years the equations get simpler and the numbers get easier to handle.

How I want to present this is first with examples without any theory. Imagine you go to visit some astronomer friends at an observatory. They have several distant galaxies in view and are sampling incoming light to measure its redshift z (or for our purposes the stretch factor s = z+1 by which the wavelengths have been enlarged.) s = 2 means the incoming wavelengths from that galaxy are twice what they were when the light was emitted.

Suppose you have a hand calculator or a cell-phone app that can do the log and square root functions---basic math. Your friend tells you the wave stretch factor s = 2, or 3, or 4.

You can tell from that some things about conditions back when the light was emitted. You can tell what size distances were back then compared with what they are now, just by how much the light was stretched in transit. You can also tell what the expansion rate was back when the light started on its way. $$ H = \sqrt{.443s^3 +1}$$

And you can tell the time when the light was emitted, what the expansion age was back then, how long it had been since the start of expansion.
$$T = \ln(\frac{H+1}{H-1})/3$$

These aren't difficult formulas, as equations go. the light itself is telling you how long it has been traveling, when it started, what size distances were compared to present , how rapid expansion was back then. And to top it off, if your mobile device can google "definite integral calculator" it can find for you the distance D that the light has covered, aided by expansion. It can tell you how far away the source galaxy is now (and dividing D by s tells how far away it was back when the light started out.)
$$D = \int_1^s \frac{ds}{\sqrt{.443s^3 + 1}}$$

 

[marcus]

The thing is, though, that these simple formulas tell you things in terms of an unfamiliar unit. For example the expansion rate H is expressed as a multiple of the longterm rate H_∞.
The current rate, for instance, is 20% larger than the eventual constant rate H_∞.

And they tell you time in zeons instead of billions of Earth years. Figuratively speaking, the way you get simple formulas is by using the units the universe is comfortable with and likes to use.

Let's use the above formulas to find the present-day age of the universe. Light emitted and received the same day has stretch factor s = 1. That represents the present moment. So what is TODAY's rate of expansion? Well 1³=1
$$ H = \sqrt{.443s^3 +1} = \sqrt{.443 +1} = \sqrt{1.443} = 1.201$$
That was mentioned earlier, today's expansion rate is about 20% larger than the eventual one.
Now what about the age, today, the time since the start of expansion?
$$T = \ln(\frac{H+1}{H-1})/3 = \ln(\frac{2.201}{0.201})/3 = \ln(11)/3 = 0.8$$
The present age of the universe (since start of expansion) is 0.8 zeons.

If you want to tell the age in billions of Earth years, you can multiply that 0.8 by 17.3 billion years. That is the equivalent of the zeon in terms that are special to our species and planet. And when you multiply you get the familiar figure of 13.8 billion years.

 

[wabbit]

Zeons sounds great, I wish a long and prosperous life to the new word :)

 

[marcus]

Now I know the secret of Spock's pointed ears, he was part Wabbit. I am greatly encouraged that you like zeons!

Maybe it is time for a picture. Mathematically speaking, the most basic fact about this simplified version of the cosmos (which gives a remarkably good approximation as long as one does not go back into the early hot radiation-dominated days) is that the expansion rate is tracked by the hyperbolic cotangent function "coth".
$$H(x) = \coth(\frac{3}{2}x)$$
Google graphs hyperbolic trig functions. You can say "graph this" and it will plot a graph. Here is a screen shot of the universe's distance growth rate, with a dot at the present.

The present time is 0.8 zeon.
The eventual growth rate is 1 per zeon. What that means is that, for example, in a thousandth of a zeon each distance grows by a thousandth of its size. By a tenth of a percent in other words. Because the growth rate changes we should think in terms of small intervals of time over which it is approximately constant. Unit growth rate means that in a millionth of a zeon a distance will grow by a millionth of its current size.
(If you like referring back to years, a millionth of a zeon is 17,300 years.)
Anyway that is the unit growth rate we are using and it is also, as you can see from the graph, the eventual expansion rate that the universe is approaching. It is interesting how astronomers measure this. We could discuss this later on.

You can see that at the present time, 0.8 zeon, the growth rate is 1.2 per zeon so it is still some 20% larger than the longterm limit and has a ways to come down.

For comparison, if you go back in time, say to time 0.1 zeon, the curve rises sharply. At a tenth zeon, the growth rate was 6.7 per zeon, or seven times the longterm limiting Hubble rate (which we call H_∞).

 

[marcus]

coth = 1/tanh
Just as a "technical" note, to get google to plot coth(1.5x) I had to type in
graph 1/tanh(1.5*x)
A nice thing about the google grapher is that it let's you zoom selectively in the vertical and/or in the horizontal.
A not nice thing is that it does not have coth. So you have to say cosh/sinh or 1/tanh

Let's play a game. Imagine you are an astronomer examining the light from various galaxies in the field of view of your telescope and I've come to visit. The galaxies are at various unknown distances---you isolate their light and run samples thru the spectrometer to determine wavelength stretch. You tell me stretch factors and I try to figure out stuff about what it was like back at the old galaxy when it emitted the light we are getting in now.
You say "stretch factor two"
I say: s=2, distances were half their present size back then, two cubed is eight and 8x.443 is 3.5 plus one is 4.5, square root is 2.1

So expansion was happening back then at a little over TWICE the longterm rate!
And let's see at what point in universe history the light was emitted---*calculates, pokes at cell phone* log of 3.1/1.1 is 1.02 divided by 3 is 0.34, about a third of a zeon!
So that light comes from a time when expansion was a third of a zeon old.

Now you fiddle with the optics and examine the light from another galaxy and you say "stretch factor three".
I say: s=3, distances were a third of their present size back then, three cubed is 27 and 27x.443 is twelve, plus one is thirteen, square root---*pokes at cell phone*---is 3.6.
That light comes from a time when expansion was happening at 3.6 times the longterm rate.
Let's see when that was. *pokes phone* Log of 4.6/2.6 is about 0.6 divided by three is 0.2.
Expansion was a fifth of a zeon old

 

[marcus]

There are parts of Biblical scripture affectionately known as "the Begats": "And A begat B, who begat C. And C lived for 140 years and begat..."
Reading a discussion of theory can be a little like reading the Begats. Cosmology is based on the Einstein GR equation, which if you simplify by assuming matter evenly distributed becomes the Friedmann equation, and since space seems to have large-scale curvature zero or nearly zero, it works very well if we simplify it to the spatial flat case of the Friedmann equation...

Anyway maybe it's time to touch on the foundations. Here's the Einstein GR equation from the Wikipedia article.

This includes no "dark energy". It simply has a curvature constant Λ on the lefthand (curvature side) where Einstein put it in 1917. So far there is no evidence of any actual "energy" associated with it. The evidence is simply of a small intrinsic curvature in the geometry. So we keep it that way here, as in the Wikipedia article. The Einstein equation has geometry (curvature) on the left and matter (energy &momentum) on the right---related by a central constant $\frac{8\pi G}{c^4}$ that indicates something about the stiffness of geometry (what intensity of matter it takes to bend it).

Here's the Friedmann equation which it begat $$H^2 - Λ/3 = \frac{8\pi G}{3c^2}ρ^*$$
The * is a reminder that ρ* the energy density (of radiation and matter) has no "dark energy" component. The central constant $\frac{8\pi G}{3c^2}$ has what it takes to convert the energy density into the square of a growth rate. I'll abbreviate it [const] for short.
Λ/3 is the longterm steady growth rate we are calling H_∞²
A curvature constant built into spacetime geomtry like that can be expressed either as a reciprocal area (number per unit area) or as the square of reciprocal time (number per unit time² or the square of a distance growth rate) In this case we are treating it as the square of a growth rate. $$H^2 - H_\infty^2 = [const]ρ^*$$
That form of the Friedmann is valid in any system of units. Now we make H_∞ our unit growth rate, and its reciprocal (the zeon) our unit of time. Expressed in terms of that unit the Friedmann then reads:
$$H^2 - 1 = [const]ρ^*$$
It turns out, as long as radiation represents only a small part of the overall energy density, that
$$[const]ρ^* = 0.443s^3$$ So we have
$$H^2 - 1 = 0.443s^3$$
$$H = \sqrt{0.443s^3+1}$$
...which begat...

 

[wabbit]

Squared, screamed the rabbit ! H infinity squared !

 

[marcus]

Hi Wabbit, thanks for catching that. I forgot to type the squared on something. Fixed now. Did you see anything else?

BTW in post #1 of this thread I mentioned that if your cell phone can google "definite integral calculator" you can tell, given the light's stretch factor, how far the light is now from its source. Let's do that for stretch factors s=2, 3, 4, and 5, and find the distances in light zeons
$$D = \int_1^s \frac{ds}{\sqrt{.443s^3 + 1}}$$
It's really easy. If you google "definite integral calculator" the first hit is:
http://www.wolframalpha.com/widgets/view.jsp?id=8ab70731b1553f17c11a3bbc87e0b605
This is announced to be a "widget" which you can take back to your own website and install there, and embed in your own HTML code. But you can also use it right there immediately at the WolframAlpha site, which is what I did.

There is a box for the integrand and boxes for the variable and the lower and upper limits.
Paste ( .443s^3+1)^(-1/2) in for the integrand, type in s for the variable and 1 to 2 for the limits. Press submit, then change 2→3→4→5 and repeat.

s=2 gave D = 0.64
s=3 gave D = 1.00
s=4 gave D = 1.23
s=5 gave D = 1.38

Another good online "definite integral calculator" that comes up is the "number empire" one:
http://www.numberempire.com/definiteintegralcalculator.php
It agrees to 3 decimal places, rounded.

So light that arrives to us with its wavelengths enlarged by a factor of three is now almost exactly one light zeon from its source galaxy.

 

[wabbit]

Hi Wabbit, thanks for catching that. I forgot to type the squared on something. Fixed now. Did you see anything else?

Nah, no other typos : )
Well, one thing maybe - You are using c=1, G=1 units I think, so the Einstein equation might be better quoted as $G+\Lambda g =8\pi T$ (with or without indices) then in the Wikipedia form.

 

[marcus]

Nah, no other typos : )
Well, one thing maybe - You are using c=1, G=1 units I think, so the Einstein equation might be better quoted as $G+\Lambda g =8\pi T$ (with or without indices) then in the Wikipedia form.

Interesting idea, but for now at least I'm thinking maybe we better keep the Einstein equation in the form it is.
In discussing the Friedmann equation the units are not specialized. The equations are stated in form that would be valid in metric units or whatever. That's for the first dozen or so lines after the Einstein equation is stated.

[const] = 8πG/3c²

It is a non-technical discussion so I was inclined to try to avoid bewildering readers by spelling out the constant. But I went back and spelled it out in the first statement of the Friedmann equation just now to emphasize that at that point the equation is valid in any system of units.

 

[marcus]

What about ending the main part of the article at the end of post#6 above, and just having one or two technical appendices for explanations that some readers might be interested in, but might not appeal to all?

Post #6 concludes like this:
==quote==
It turns out, as long as radiation represents only a small part of the overall energy density, that
$$[const]ρ^* = 0.443s^3$$ So we have
$$H^2 - 1 = 0.443s^3$$
$$H = \sqrt{0.443s^3+1}$$
...which begat...
==endquote==
We can go on in this thread and work out one or two appendices with extra detail. But maybe an article for "Insights" could be made out of the first 6 posts. Keep it short. I still don't know if it is suitable or if it would prove satisfactory.

But let's elaborate some.

To me it seems interesting that once you know the growth RATE of the universe is coth(1.5x) you can DERIVE a curve that shows the growth history of a generic distance. It's integrating a simple differential equation. Maybe that could be put in an appendix.

The logarithmic derivative of sinh^2/3(1.5x) is coth(1.5x)
So any growth history curve f(x) is determined up to a multiplicative factor by stipulating that the growth rate f'(x)/f(x) = coth(1.5x)

Another technical detail to append might be to show where the number 0.443 comes from (we already sort of did in post #2).
Two things which one can measure by fitting data are:
H_now and H_∞
and the crucial fact is that:
H_now² - H_∞² = 0.443 per zeon²

 

[Jorrie]

What about ending the main part of the article at the end of post#6 above, and just having one or two technical appendices for explanations that some readers might be interested in, but might not appeal to all?

I am following a similar approach in a Blog post for http://cr4.globalspec.com/ (being prepared). Engineers work with equations, but they do not like reading them, so they want lots of explanation and graphs/diagrams. And they are by nature a 'skeptical' and 'traditional' bunch, so I have to carefully draw most of them into even reading something with a 'new idea' look and feel.

BTW, after trying other names and acronyms for your 'zeon', I have decided it is the best name for the unit after all - and without acronyms and abbreviations. Even distance is easily enough indicated as lzeon (not lz, because of possible redshift confusion).

 

[marcus]

I am following a similar approach in a Blog post for http://cr4.globalspec.com/ (being prepared)...

Great news! Encouraging to hear.

 

[marcus]

This might belong in footnotes or endnotes. That number 0.443 is unitless. It's defined as
$$(\frac{H_{now}}{H_\infty})^2 - 1$$
This is a pure number, serving to characterize the present era in universe history---observable in the same sense that H_now and H_∞ are observable, but independent of whatever units you happen to use to measure the two expansion rates.
I want to emphasize that this number is not special to our time scale or way of simplifying the standard model. It pops up as soon as we have measured the present expansion rate and the longterm limit towards which expansion rates are declining.

Here's a version of the Friedmann equation which is also independent of one's choice of units.
$$(\frac{H}{H_\infty})^2 - 1 = 0.443s^3$$
This version gives a good approximation applicable any time after the initial hot radiation-dominated era.
Here's how to derive it from the regular flat-case Friedmann:
$$H^2 - H_\infty^2 = [const]\rho^*$$
$$H^2 - H_\infty^2 = [const]\rho^*_{now} s^3$$
$$H^2 - H_\infty^2 = ( H_{now}^2 - H_\infty^2 ) s^3$$
$$(\frac{H}{H_\infty})^2 - 1 = 0.443s^3$$
None of these steps involves preferred scales or units. At the end one just divides through by one of the quantities, H_∞, and whatever units were in use are canceled out.

 

[marcus]

When you see this version of the Friedmann equation (the equation basic to all cosmology)
$$(\frac{H}{H_\infty})^2 - 1 = 0.443s^3$$ you may notice that H/H_∞ is the same thing as H measured in H_∞ units and the idea of measuring the growth rate H in terms of the eventual one could occur to you. And that amounts to making the zeon your time unit.
Growth rate is reciprocal time, making H_∞ your rate unit is equivalent to making 1/H_∞ your time unit.
When you do that, the Friedmann equation becomes even simpler:
$$H^2 - 1 = 0.443s^3$$At this point it's not a big leap to notice that, if you know the factor s by which wavelengths of some incoming light are enlarged, deducing the conditions where and when the light was emitted only involves simple arithmetic. What time was it? How far was it? How swift was expansion then? And we arrive at the equations at the start of the thread.
$$ H = \sqrt{.443s^3 +1}$$ $$T = \ln(\frac{H+1}{H-1})/3$$ $$D = \int_1^s \frac{ds}{\sqrt{.443s^3 + 1}}$$

 

[marcus]

There is more that happens when one uses the natural time unit that arises from the standard Friedmann equation model. For one thing you get the expansion rate H expressed as a simple hyperbolic trig function. $$H(x) = \coth(\frac{3}{2}x)$$ We saw a graph of that back in post#4.
Another thing that happens is that we can solve for a typical distance growth history u(x).
Whatever the curve u(x) is, it has to have its fractional growth rate u'(x)/u(x) equal to H(x). This determines u(x) uniquely up to a multiplicative constant. So we get $$u(x) = \sinh^{2/3}(\frac{3}{2}x)$$
This gives the basic shape of all distance growth histories, but it contains no information about the present. We can multiply it by the cube root of 0.443 to normalize it, so that it will equal 1 at present. This then becomes the normalized scale factor a(x) familiar to many of us. 1/a(x) = s(x) shows how the stretch factor evolves as a function of time. Here's a plot of a(x).
The slope of a(x) is an indicator of expansion speed as distinct from growth rate. The speed the size of some particular distance was increasing.
Speed is not a good handle on expansion because it is somewhat arbitrary, depending not only on the RATE H(x) but also on the size the particular distance being studied happens to be at the moment. Also distance expansion is not like ordinary motion--nobody gets anywhere by it, everybody just becomes further apart. So it is confusing to picture it as a conventional motion with the usual sort of speed. However some people are interested in distance expansion speeds, and we can see how they change over time from this plot. The expansion speed of our sample distance DECREASES until around time x = 0.45 zeon. And then a gradual acceleration begins---the slope of the a(x) curve begins to steepen. The shape turns from convex to concave around that time.

In plotting this curve I divided by 1.311 to normalize the hyperbolic trig function u(x). This amounts to the same thing as multiplying by the cube root of 0.443 but avoids some round-off error. The changeover from deceleration to acceleration is subtle but you will see that it happens around time 0.45 zeon.
It happens when the decline in H(x) levels out enough. As long as the downwards slope of the H(x) = coth(1.5x) curve (shown in post #4) is steep enough the speed that a(x) is growing lessens. Around x = 0.45 zeon the H(x) growth rate curve is still declining of course---it always declines---but the slope is sufficiently less steep, and then a(x) switches over to acceleration.
You can figure out when that changeover occurred in Earthy terms, if you like. Recall that according to latest measurements a zeon is 17.3 billion Earth years.

 

[marcus]

In post#15, I mentioned a couple of versions of the (spatial flat) Friedmann equation, applicable any time after radiation becomes a negligible part of the energy density. The first version is independent of choice of units. Each side of the equation is simply a pure number, the units having canceled out.
$$(\frac{H}{H_\infty})^2 - 1 = 0.443s^3$$
The other version is where we make H_∞ = 1 by measuring time in zeons.
$$H^2 - 1 = 0.443s^3$$
To extend this equation back into the radiation era we would have to make the exponent 3 gradually change to 4

Matter density increases as s³ as you go back in time. Volumes shrink as the cube of distance, so the number of matter particles per unit volume grows as the cube of s.
But radiation density grows as the fourth power of s. One can think of this as happening because besides the number of particles per unit volume rising as the cube, the wavelengths shorten, increasing the energy per particle.
So if the universe were filled with nothing but radiation, we would have been using$$H^2 - 1 = 0.443s^4$$ throughout.

At what point, as we go back in time, would we start thinking about shifting over from s³ to s⁴? I don't expect us to make that transition in this thread, we will simply not use the simplified model before, say, s = 100.
But Lightcone calculator makes the transition based on the fairly standard assumption that matter and radiation are in balance at s= 3400. That would mean that radiation is about 1/34 of matter at s=100, which is enough to throw things off. So let's not push the model back in time too far--no earlier than s=100.

The first stars probably formed around s = 10 or 11, so s=100 is going pretty far back in history. Let's see how far. We can use those simple formulas from the start of the thread.
$$ H = \sqrt{.443*100^3 +1} = 666\ per\ zeon$$
$$T = \ln(\frac{667}{665})/3 = 0.001\ zeon$$
$$D = \int_1^{100} \frac{ds}{\sqrt{.443s^3 + 1}} = 2.425\ lightzeon$$

Being cautious and not pushing the model too far back into the past still gets us to 0.001, a thousandth of a zeon, which in Earthy terms would be year 17.3 million. I was curious what the distance would be, so multiplied 2.425 by 17.3 and got 42 billion lightyears.
So we don't make it as far as the last scattering surface of the CMB, but still include a respectable chunk of the observable universe.

 

[marcus]

I think all of us are used to the idea that s=1 (wavelengths unchanged) indicates the present and the stretch factor increases as you go back into the past. s=10 indicates a time in the past when distances were 1/10 their present size and light emitted then would have its wavelengths enlarged by a factor of 10 by the time it gets to us.
Some may also be familiar with Jorrie's implementation of the standard cosmic model in Lightcone and the way s is also used to parametrize the future.
How would you describe the time in the future that s=0.1 stands for?
Well s=10 was a time when the matter density was --- (s³ = 1000)--- a thousand-fold what it is today. So by analogy
s=0.1 should be a time when the average density of matter will be (0.1³ = 0.001)---a thousandth of what it is today.

If we send a light signal that is received in some other galaxy at s = 0.1, we expect distances and wavelengths at our end to be 1/10 what they will be at the receiving end. In fact, s can always be interpreted as the size of something at OUR end compared with its size at the OTHER end (whether past or future, emitter or receiver). It is the ratio of size now to size then, regardless of whether then is in the past or the future.

Let's figure out when that s=0.1 reception will occur, what the expansion rate H will be that far in the future, and how far the target galaxy is from us now.

$$ H = \sqrt{.443*0.1^3 +1} = \sqrt{.443*0.001 +1} =\sqrt{1.000443} = 1.0002215...\ per\ zeon$$
$$T = \ln(\frac{2.0002215}{0.0002215})/3 = 3.036 \ zeon$$
$$D = \int_{.1}^1 \frac{ds}{\sqrt{.443s^3 + 1}} = 0.853 \ lightzeon$$

And let's compare these results with Lightcone
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html
Lightcone says the galaxy that receives our flash of light at s=0.1 does so in year 52.52 billion Our arithmetic says 3.036, which in Earth years is 52.52 billion. Smack on.

Lightcone says that galaxy is now 14.77 billion LY from us and our formula gives the distance now as 0.853 lightzeon which translates into 14.76 billion LY. Pretty close.

Of course the galaxy will be ten-fold farther from us when they finally receive the signal. As before we divide the distance now by the stretch factor s=0.1 to get the distance then.
Notice the distance to the target galaxy is currently increasing slightly faster than c (it is beyond the Hubble radius of 14.4 billion LY) but nevertheless the light can still make it.

 

[Ken G]

Being cautious and not pushing the model too far back into the past still gets us to 0.001, a thousandth of a zeon, which in Earthy terms would be year 17.3 million. I was curious what the distance would be, so multiplied 2.425 by 17.3 and got 42

So the answer to life, the universe, and everything, is indeed 42! I knew it.

The above is great stuff, I did not realize the expressions come out in convenient closed forms until I saw your thread. A few practical comments: above you characterize H as the "rate of expansion", but H is actually the fractional rate of expansion, i.e., the fractional change in 1/s per unit time. The words are tricky here because H is indeed like the interest rate on a bank account, which we might think of as the expansion rate of the bank account, but it is more consistent use of language to regard the expansion rate of a bank account to be the amount of money per year that it grows, not the interest rate. This is because what is normally referred to as "constant expansion" corresponds to a constant da/dT for scale parameter a=1/s, so that's if H ~ s. So the expansion rate is actually H/s, which is also da/dT, and that is the thing that is growing with T here, i.e., "accelerating."

Also, I really think you should replace .433 s³ with (s/1.32)³, because the form is not much different, and the number 1.32 has a physical interpretation more along the lines of your "zeon" concept (which I now like since it was suggested not to abbreviate it, just leave it zeon and lzeon). You are saying that there is a natural timescale, and if you write it the way I'm suggesting, you see there is also a natural "stretch" scale, the stretch where matter and the cosmological constant contribute equally in the dynamics. I wouldn't recommend rescaling s to reflect that, because unlike time, s is already unitless, and has a direct connection to redshifts as you say. But it's nice if the equation can tell you what the natural scale for s is, and indeed, remembering that number let's you recreate the equation pretty easily (though 0.433 does too). This alternative form is particularly convenient for stretch factors that are integer multiples of 1.32, obviously! For example, we immediately see that if we look back over a stretch of 2.64, we are talking about an H that is 3 times its asymptotic value.

By the way, are you sure you want to use the stretch factor rather than scale parameter a=1/s? I see where you are coming from, the stretch factor is the amount the wavelength is stretched by the amount the universe has stretched since the light was emitted. But when using stretch factor as the independent variable, we are tracking a quantity that drops with time, rather than the scale parameter which expands with time. In other words, stretch makes for a nice way to talk about times in the past relative to the present, but it's not as good of an independent variable for telling the tale of the expansion of the universe. It's a tough call, because although scale parameter a is more familiar to most, stretch factor s is more like the wavelengths we actually see, and would save us that awkward need to always take 1/a whenever we are talking about the wavelengths we see. Of course, if we were used to talking about frequency instead...

 

[marcus]

Ken G, thanks for your comments! It's encouraging and stimulating to get thoughtful response and suggestions of alternatives. It also gets me to explain more clearly about details like basic parameters, and concerns about roundoff error.
ln((173/144 +1)/(173/144-1))/3 = 0.7972... = x_now
(sinh(ln((173/144 +1)/(173/144-1))/2))^(2/3) = sinh^2/3(1.5 x_now) = 1.31146...

I want to be able to conveniently compare results of this (really primitive ) version of standard ΛCDM with the same numbers calculated using the fuller version embodied in Jorrie's Lightcone calculator.
That uses the exact values 14.4 and 17.3 billion years for the two main parameters, namely the two Hubble times. To be able to check ours easily, we should base things on the same exact values.

That's why when I refer to the present expansion age I usually say 0.8 zeon but if more precision is needed I will say 0.797, or 0.7972 zeon. It's almost always perfectly fine to say 0.8 but as I'm sure you understand the real x_now is that logarithm expression above

The above is great stuff, ...
...
Also, I really think you should replace .433 s³ with (s/1.32)³, because the form is not much different, and the number 1.32 has a physical interpretation more along the lines of your "zeon" concept (which I now like since it was suggested not to abbreviate it,...

I'm glad you are OK with the term "zeon" (as long as not abbreviated)! I think we are talking about the same numbers. Indeed it would be possible to replace 0.443s³
by (s/1.311)³
and s/1.311 has a physical interpretation as the reciprocal of the unnormalized scale factor
u(x) = 1.311a(x) = sinh^2/3(1.5x)
I guess you could say that 1.311 is how big the universe thinks it is at the present moment ,
namely u(x_now). I think that may be what you were referring to when you mentioned a "natural stretch scale". u(x) is suggestive of a "natural scale factor". And stretch and scale-factor are reciprocals.

 

[marcus]

Ken G, I can definitely see a point to using the cube of 1.311 or the cube of its reciprocal, or re-scaling the stretch factor as you suggest, but using this number in our equations also has things going for it:
$$(\frac{H_{now}}{H_\infty})^2 - 1 = (\frac{17.3}{14.4})^2 - 1 = 0.443...$$
It depends in an immediate and transparent way on our two primary model parameters. It arises in the derivation of a simplified version of the Friedmann equation which we use:

This might belong in footnotes or endnotes. That number 0.443 ...
...pops up as soon as we have measured the present expansion rate and the longterm limit towards which expansion rates are declining.

Here's a version of the Friedmann equation which is also independent of one's choice of units.
$$(\frac{H}{H_\infty})^2 - 1 = 0.443s^3$$
This version gives a good approximation applicable any time after the initial hot radiation-dominated era.
Here's how to derive it from the regular flat-case Friedmann:
$$H^2 - H_\infty^2 = [const]\rho^*$$
$$H^2 - H_\infty^2 = [const]\rho^*_{now} s^3$$
$$H^2 - H_\infty^2 = ( H_{now}^2 - H_\infty^2 ) s^3$$
$$(\frac{H}{H_\infty})^2 - 1 = 0.443s^3$$
None of these steps involves preferred scales or units. At the end one just divides through by one of the quantities, H_∞, and whatever units were in use are canceled out.

 

[marcus]

===trying alternate notation===
Let's introduce a notation and see if it helps, or is convenient. If not we can drop it. The two principal things we can measure, to get started, are H_now and H_∞, both are measured by fitting to observed redshift-distance, or as some prefer, wavestretch-distance data. And having measured them, we can readily calculate the difference of their squares.
The basic spatial-flat Friedmann equation relates that difference of squared rates to the overall energy density of matter and radiation.$$H^2 - H_\infty^2 = \frac{8\pi G}{3c^2}\rho^*$$ Its presentday value, H_now² - H_∞², keeps coming up in all sorts of calculations. It is a kind of handle on the present. It often appears as the number 0.443, and is, in a way, how our model gets hold of the information it needs about where we are in expansion history.
Let's call it something. Let's call it ω_now, thought of as a mark of the observer and the present moment. It is related to the presentday energy density ρ*_now$$H_{now}^2 - H_\infty^2 = \omega_{now}$$$$\omega_{now} = \frac{8\pi G}{3c^2}\rho_{now}^*$$As long as we are past the early universe's hot radiation stage, energy density varies approximately as the cube of the stretch factor. It's a close approximation, typically within a percent or so. When distances were half, density was 8-fold what it is today. When they were a tenth, density was 1000-fold today's.$$\rho^* = \rho_{now}^*s^3$$So we can write the Friedmann equation in a simpler form:
$$H^2 - H_\infty^2 = \omega_{now} s^3$$This form of the equation makes sense in any system of units. But now we can simplify still more by making H_∞ our unit growth rate, zeon our unit of time, or simply specifying that H_∞ = 1. In that case H_now=1.2014... and H_now²= 1.443.
The Friedmann equation becomes:$$H^2 - 1 = \omega_{now}s^3$$

 

[marcus]

==trying ω_now as possible new notation==
Since we just turned a page, I bring some reminders forward.
Growth rate is reciprocal time, making H_∞ your rate unit (1 per zeon) is equivalent to making 1/H_∞ your time unit (1 zeon).
When you do that, the Friedmann equation simplifies:
$$H^2 - 1 = 0.443s^3$$ Let's continue using the ω_now symbol for that number.$$H^2 - 1 = \omega_{now}s^3$$If you know the factor s by which wavelengths of some incoming light have been enlarged, deducing the conditions where and when the light was emitted involves fairly simple arithmetic. What time was it? How far was it? How swift was expansion then? And we arrive at the equations at the start of the thread.
$$ H = \sqrt{\omega_{now}s^3 +1}$$ $$T = \ln(\frac{H+1}{H-1})/3$$ $$D = \int_1^s \frac{ds}{\sqrt{\omega_{now}s^3 + 1}}$$
The first equation is the Friedmann in a different guise. The second can be inverted to give H as function of time. I'll switch over to using x for time. $$H(x) = \coth(\frac{3}{2}x)$$ We can use the new ω_now symbol to write the scale and stretch factors as functions of time.
$$a(x) = (\frac{\omega_{now}}{H^2-1})^{1/3}$$$$s(x) = (\frac{H^2-1}{\omega_{now}})^{1/3}$$
Both of them are normalized to equal one at present, so they need the information ω_now provides as to where in time the present is.

 

[marcus]

For plotting curves showing how cosmological quantities evolve over time, probably the most useful equation is $$H(x)=\coth(\frac{3}{2}x)$$ The dot for the present at about 0.8 zeon, shows a growth rate of about 1.2 per zeon.

Applying a little algebra to the previous scale factor equation gives an alternative form: $$a(x) = (\frac{\omega_{now}}{\coth^2(\frac{3}{2}x)-1})^{1/3} = \omega_{now}^{1/3}sinh^{2/3}(\frac{3}{2}x)$$This let's us plot the scale factor showing our universe's expansion history (over the course of the first 0.8 zeon and into the future, normalized to equal one at present.)

 

[Jorrie]

Marcus, I like the idea of a symbol there, but is your $w_{now}$ not likely to be confused with the symbol used for equation of state?

Essentially $\frac{8\pi G}{3c^2}\rho_{now}^*$ represents a curvature, so maybe a more 'k-like' symbol could be more intuitive?

What about $k^*_{now} =\frac{8\pi G}{3c^2}\rho_{now}^*$?

 

[marcus]

Hi Jorrie, I just saw your post. That's a good point. "k-like" makes sense too. Another asterisk might be visually distracting. It's nearly midnight so I'll think about alternatives in the morning. I wonder how κ_now would do.
Need to get some sleep.

 

[Jorrie]

Yes, but then k is normally associated with the conventional curvature constant of Friedman, which means k~0 today. Remember, we started using $\rho^*$ for exactly the same reason.
$$H^2 - H_\infty^2 = k^*_{now}s^3$$
does not look too distracting...

 

[marcus]

Hi Jorrie,
one asterisk is fine, but I hesitate to have two, esp in the same equation
I made some alternatives to compare for general feel. back in post#23 and #22
The alt versions are at the head of the post and I kept the old ones down below for comparison.
I think you are basically right, we need some alternative to the w_now symbol.
It shouldn't overlap with symbols constantly in use in the same Friedmann equation cosmology context.
That could also be a problem with Greek lowercase kappa, because relativists use it for 8πG sometimes.
But I tried out kappa anyway. Also omega.

 

[wabbit]

Hi there, just passing by to throw a vote for $\omega$, it looks better than $w$ and I don't think it would bring confusion with other notations.
Or $\omega^2$ so that $\omega$ is a frequency as befits this letter : )

 

[marcus]

Hi there, just passing by to throw a vote for $\omega$, it looks better than $w$ and I don't think it would bring confusion with other notations.
Or $\omega^2$ so that $\omega$ is a frequency as befits this letter : )

Thanks Wabbit, provisionally at least I want to go with your preference for ω. I'll go back and erase the other parts of posts#22 and 23 which used w and kappa. Because I'm trying to see how to present this at a basic beginner level I won't go for ω² which would be elegantly traditional and would implicitly define a third frequency/growth rate in pythagorean relation to H_now and H_∞. Maybe some other time in a different essay someone else could do that. It could be nice. Anyway I'll adopt plain ω and go back and edit #22 and 23.

 

[wabbit]

I agree, $\omega^2$ wasn't really a serious suggestion, it would make sense only if as you say you were then to interpret it and explore "what it really means" as a growth rate etc. Here it would just beg the question, why the square?

Is the final form going to be an Insight post? I think you mentionned that before, not sure.

 

[marcus]

Would you like to co-author an Insight post? You may be more comfortable than I am with what I think of as magazine/journal format.
I would like this material to be presented in PF Insight in some form. But I'm a bit diffident about actually producing a finished piece. Informal conversation like this---as in this thread---is where I'm not inconvenienced by "writer's bog".
If you like the idea of co-authoring, patch and edit together any of these posts, any of this material as you see fit. If you do, I'm sure to be content with the result.

 

[marcus]

Here's a brief table showing sample numbers calculated with this model. The table goes in time steps. As review, if anyone is new to the thread, x is the usual time divided by 17.3 billion years (so the present 13.787 Gy becomes x_now = 13.787/17.3 = 0.797. I've also tabulated the Hubble time 1/H(x).
a(x) is the normalized scale factor at time x: sinh^2/3(1.5x)/1.311
s(x) is the wavelength and distance stretch, the "now/then" ratio. 1/a(x) = z+1
HubT is the reciprocal growth rate at time x, namely tanh(1.5x)

x-time  (Gy)    a(x)    s       HubT    (Gy)     Dnow      (Gly)
.1      1.73    .216    4.632   .149    2.58    1.712       23.03
.2      3.46    .345    2.896   .291    5.04     .971       16.80
.3      5.19    .458    2.183   .422    7.30     .721       12.47
.4      6.92    .565    1.771   .537    9.29     .525        9.08
.5      8.65    .670    1.494   .635   10.99     .362        6.26
.6     10.38    .776    1.288   .716   12.39     .224        3.87
.7     12.11    .887    1.127   .782   13.53     .103        1.78
.797   13.787  1.000    1.000   .832   14.40    0            0

 

[wabbit]

I don't think a co-author would make a lot of sense, you already have the piece pretty much written already, which is the reason I asked: it seems to be getting close to a final form that I think would make a nice posting in Insights - not that I know its editorial policy, just my feeling as a reader.

I'd be glad to contribute as editor/proofreader though it this could help finalize it.

Just an aside : you're lucky you get writer block only once you've almost finished the writing

 

[marcus]

It could help finalize it. That's a nice offer. I thought of another subtopic, plotting the expansion speed of a selected distance.
The distance chosen for this example (here as in the Lightcone calculator column menu) is one whose current size coincides with the present-day Hubble distance. Since H_now = 1.2014 per zeon, the Hubble distance is currently 1/1.2014 lightzeon.
Here is the sample speed history:

You can see that it is expanding exactly at the speed of light at the present moment (x = 0.8) which is right because that defines the present-day Hubble distance.
And it has its speed minimum around the time x = 0.45. That is where the inflection point comes in the a(x) scale factor curve, when deceleration changes to acceleration. All distances' speed curves look the same, they just differ by a multiplicative factor--they are proportionate to the size of the distance.
Using this model, the size history of that particular distance size(x) = a(x)/1.201 lightzeon
and have the formula for the normalized scale factor a(x) = sinh^2/3(1.5x)/1.311
Then its speed history is simply the size multiplied by H(x) = coth(1.5x).
So the expansion speed of that particular distance is:
speed(x) = H(x)size(x) = coth(1.5x)sinh^2/3(1.5x)/(1.311*1.201)
So that is what is plotted here.

I see that by time 2 zeon that particular distance will be growing at 3 times speed of light. It was also growing at over the speed of light until 0.2 zeon, but between that time and the present 0.8 zeon its expansiopn speed was less than c.