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

[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.

 

[wabbit]

Hmm I'm confused. You mean, the history of the velocity of a galaxy that is currently at that distance, right ?

But yes if I read it correctly this is a weird and interesting chart, showing in concrete terms the deceleration from gravity then acceleration caused by the cosmological constant, looking at just one observer and one distant galaxy - this is nice, a break from the "all encompassing majestic view" one is easily tempted into when talking cosmology.(*)

It just hit me that that one galaxy not only approaches us faster, but at asymptotically infinite speed as one approaches the big bang. Quite obvious really, its distance scales as a which is vertical at the origin, but somehow I didn't quite see it, was lost in comoving coordinates and scale factors I guess, leaving the poor galaxy alone. Good chart indeed.

But.. Everything is superluminal early on ? Come on, you must be joking ! And yet, it is, every galaxy (or what remains of it) eventually exits our Hubble radius as we approach t=0. Hmmm is that right ?

Another nice chart would be this : the history of the Hubble radius, with the paths of a few selected galaxies, entering it, spending some time visiting, then eventually exiting when aunt lambda calls time to go home (he he, a hint of time symmetry here )

(*) You might add the distance chart as well, d0*a/a0, on the same or nearby plot, as it tells that same story in a complementary way.

 

[marcus]

Your interpretation at the start of your post is what was intended, W. Interesting suggestions for graphs illustrating ideas. I'll think about them. For now all I have is a very simple triple graph of that speed(x) and the Hubble-time(x) and the scale factor a(x)

The minimum on the speed(x) curve should come at the same time x as the inflection point of the a(x) scale factor curve. It is when Aunt Lambda tells the children they must hurry up.
The fact that speed(x) intersects a(x) at x = 0.8 is just a formal consequence of the definitions, a(x) is normalized to equal 1 there and the the sample distance was chosen to make its speed equal 1 at present.
The Hubble time or Hubble distance curve has to approach the limit 1 for large x again for a formal reason, the time unit was chosen so that H(x) → 1. So I think the figure has very little content. I made it as an exercise--learning to use the Mac utility called "Grapher".
It would be excellent if it could graph curves defined by definite integrals, like D_now and D_then. You mentioned a figure in which one of those played a part. I wonder if it is possible.

 

[Jorrie]

It would be excellent if it could graph curves defined by definite integrals, like D_now and D_then. You mentioned a figure in which one of those played a part. I wonder if it is possible.

LightCone 7, of course, does all these graphs and more, but in standard units. The problem is that the scales of some of the values in standard units are so different that you cannot graph many on the same graph. So I was wondering if it would be useful to bring out a variant (LightConeZeon), with normalized parameters...

I will look into the feasibility and the effort required.

 

[marcus]

Wow! If that were feasible it would be so neat!

 

[wabbit]

One stray comment here inspired by the zeon and related natural GR unit of length $\Lambda^{-1/2}$
In any flat FRW universe, $\frac{\dot a}{a}\geq\sqrt{8\pi\Lambda}$ which implies $\forall u<v, \frac{a(u)}{a(v)}\leq e^{-(v-u)\sqrt{8\pi\Lambda}}$ so for the radius of the observable universe at u, seen at v, $R(u,v)=\int_u^v \frac{a(u)}{a(t)}dt \leq \frac{1-e^{-(v-u)\sqrt{8\pi\Lambda}}}{\sqrt{8\pi\Lambda}}\leq\frac{1}{\sqrt{8\pi\Lambda}}$, and equality can be approached as closely as one wants in the late exponential era.
So the largest possible measurable distance is $D= \frac{1}{\sqrt{2\pi\Lambda}}$ ~ one zeon.
Or more directly, the maximum area of the boundary of the observable universe, and hence the maximum area of any sphere, is $A_{max}=\pi D^2=\frac{1}{2\Lambda}$ ~ one square zeon.
And the maximum volume of any observable spatial region at fixed comoving time is some constant times $\Lambda^{-3/2}$ ~ one cubic zeon.

This expresses in the flat case the zeon as kind of counterpart to the Planck length, or the inverse of the cosmological constant as the maximum area in units of the minimum area, the Planck area - which I also find interesting from an information viewpoint.

So it seems the zeon (or the zeon squared) may have something fundamental about it :)

 

[marcus]

One stray comment here inspired by the zeon and related natural GR unit of length $\Lambda^{-1/2}$
...
This expresses in the flat case the zeon as kind of counterpart to the Planck length, or the inverse of the cosmological constant as the maximum area in units of the minimum area, the Planck area - which I also find interesting from an information viewpoint.

So it seems the zeon (or the zeon squared) may have something fundamental about it :)

The naturalness of the zeon length.
This is interesting, and it could be a separate topic
http://inspirehep.net/record/899089?ln=en
http://inspirehep.net/record/899089/citations
http://inspirehep.net/search?ln=en&p=refersto:recid:899089
"Smallest measurable angle" call it a "planckian" angle, might be a Planck length or Planck area held out at the "fundamental large distance".
Or perhaps the distance scale is a consequence of the limitation on angle measurement instead of the other way round.
I will try to start a separate thread.
Maybe it should be in BtSM forum.
We could also keep discussing it in this thread, although this thread so far is more about a nice way to think about quantitative cosmology and the various curves, the friedmann equation and so forth. it is very concrete, "counting the cosmos on one's fingers" almost. primitive.

 

[wabbit]

No I agree this thread isn't the place to discuss it, I was discreetly trying to hijack it but you called me out in time

I thought it could be worth a mention in introducing the zeon scale, but it might be hard to "just mention" without delving much more into it than is appropriate in this context.

 

[marcus]

Yes maybe it's only appropriate as a passing mention in this thread, but it's really interesting. I initiated a thread in BtSM.
https://www.physicsforums.com/threads/the-naturalness-of-the-zeon-length-scale.813144/#post-5104511
I hope the title is OK---in your post you referred to the "related natural GR unit of length Λ^−1/2..."
It's possible that geometry in the large is telling us its own built-in preferred length scale. The idea is astonishing.
Since we just turned a page I think I'll bring forward some essentials of this thread, to have handy for reference. Clearly these, by contrast, are very basic hands-on-the-cosmos, straight-forward things. How to interpret the stretch factor you measure in incoming light.

 

[marcus]

Here's a plot of some of the functions or curves we are using. The x-axis is time in zeons. The y-axis has multiple uses. It can serve to show time in zeons (when we plot the Hubble expansion time) or distance size in lightzeons, or growth rate in per zeon units, like for example 1.6 per zeon or 1.2 per zeon.

The Hubble expansion rate is the declining curve H(x) = coth(1.5x) that is about 1.6 at time x=0.5, then equals 1.2 at the present x = 0.8, and levels out at one, longterm.
That reflects the fact that the distance expansion rate is currently 1.2 per zeon, and is tending to the longterm unit rate 1 per zeon.

The Hubble time, 1/H(x) and the Hubble radius, c/H(x) have the same curve. You can see it is steadily rising, starting from zero at time x = 0, and leveling off at 1 longterm. As the reciprocal of the first curve we mentioned, it has the value 0.83 or 1/1.2 at the present. The longterm value is 1 lightzeon if you are thinking of it as the Hubble radius and 1 zeon if you think of the curve as showing Hubble time.

Besides the expansion rate H(x) and the expansion time 1/H(x) the figure has two other curves. One is the normalized scale factor a(x) which shows the expansion history of a typical distance, normalized to equal one at present time. It starts out at zero at time x=0, equals 1 at the present x= 0.8, and continues rising.
The slope of the a(x) curve behaves in a subtle way, at first gradually becoming less steep, until around x≈0.44, after which the curve gradually gets steeper. This is the "acceleration" in distance growth that one hears about.

Here are formulas for the three curves mentioned so far:
H(x) = coth(1.5x)
1/H(x) = tanh(1.5x)
a(x) = sinh^2/3(1.5x)/1.311

The fourth curve is the expansion speed history of a distance whose size at the present moment is 0.83 lightzeon. Since its present size is 0.83, its size at other times must be 0.83a(x). That is what the scale factor function a(x) does for us. Multiply it by the present size of any distance and it gives you the whole growth history of that distance. Notice that 0.83 lightzeon is today's Hubble radius (the boundary between slower than c and faster than c expansion), so at present the size of our sample distance must be increasing at exactly c, the speed of light.
The formula for the fourth curve is H(x)a(x)0.83. Multiply the size at any given time (namely 0.83a(x)) by the Hubble parameter H(x), and you get the speed.
The speed of light is 1, in these units---one lightzeon per zeon. This makes it easy to plot a curve showing how a sample distance's expansion speed varies, along with the rest.

The speed curve starts high, dips down to a minimum, and rises thereafter. The minimum on the speed(x) curve comes at the same time x=0.44 as the inflection point of the a(x) scale factor curve, where the slope of a(x) starts getting steeper.
The fact that speed(x) intersects a(x) at x = 0.8 is a formal consequence of the definitions, a(x) is normalized to equal 1 there and the sample distance was chosen to make its speed equal 1 there.

 

[marcus]

We don't have a curve for D_then but if we did, we'd see the max comes at time x = 0.234 zeon and corresponds to a scale factor of s = 2.6.
I'm thinking about the pear-shaped past lightcone.
The Hubble time 1/H(0.234) = tanh(1.5*0.234) = 0.337 zeon and accordingly the Hubble radius is 0.337 lightzeon.
The integral for D_now divided by 2.6, which defines D_then, is also 0.337 in those units.

We need an expanded scale graph for this because the past lightcone only goes from x=0 to x=0.8,
so the range on the x-axis should be from 0 to 1, and the vertical range should be from 0 to 1. So this should be smaller square cut out of the larger four-curve graph I posted earlier, back a couple of posts, blown up.
...
...
If you are visiting your friends at the observatory, and learn they are studying a galaxy at stretch s = 2.6 then you might remark that the light being collected and measured was emitted at the greatest distance from us of any light we have ever received or will ever receive.
Since we weren't alive when it was emitted maybe I should say it was "emitted at the greatest distance from here", or from the Milkyway matter that became the solar system and us.

To avoid such issues, let's say that if the light comes in with stretch s = 2.6 then it was emitted at the greatest distance of any light we ever have or ever will receive.

Light coming in with stretch greater than 2.6 was emitted closer to us and earlier (before time x = 0.234). It was outside the Hubble radius at that time and was at first "swept back" as the intervening distance expanded, so that it took longer to reach us even though emitted from comparatively nearby.

 

[marcus]

What would be nice would be a graph like this one (made in standard units by Jorrie's Lightcone calculator) but with the x-axis relabeled
0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 0.7. 0.8

or more simply:
0, 0.2, 0.4, 0.6, 0.8

and the pear-shaped light cone curve would be zero at zero and zero at 0.8
It's max would come at x=0.234, and the height there would be y = 0.337
So the max where the two curves cross would be a little ways above the 0.2 mark, between 0.2 and 0.3.

There would also be a section of the tanh(1.5*x) curve which is also equal to 0.337 at x=0.234.
That is the Hubble radius, and a good partner to the lightcone radius.

 

[Jorrie]

Not difficult, but LightCone 7z not release-ready yet...
As you can see, the horizontal scale units are still wrong and so is are the Latex tables-makers.

In LightCone Google-Charts decides on the scale markings, so one does not have much control over them.

 

[marcus]

It's great! It automatically chooses to make the spacing 0.19 zeon, which by good fortune just happens to put a line (at x=0.23) right through the max and and the place where the curves cross.

At the bottom where it now says "Time (Gy)" it would eventually say "Time (zeons)" or something like that I guess, whatever you decide
and similar change to the labeling of the red and blue curves in the upper right corner. But that is non-essential at the moment. I really like the look of the x-in-zeons, y-in-light-zeons graph! Thanks for giving us a foretaste.

 

[marcus]

I'm learning how to use Jorrie's Lightcone7z, which uses zeon units. I want to see how to make the simplest graph pictures of cosmic expansion with it.
One thing I want to do is see how one might address the initial confusions about expansion people get from trying to understand words like "Hubble radius". This is not the "radius of the universe" and it is not even a physical distance in the usual sense, so much as it is a handle on the expansion rate. The Hubble radius is not expanding like, e.g., the distance between two galaxies. It is is growing as the expansion rate H declines because it is the reciprocal of the rate. But its growth is leveling off just as the decline of the Hubble rate is leveling off. They mirror each other and both converge to one, in our picture.

In Lightcone7z pictures with time on the x-axis, now is marked by the point x = 0.8. I'm thinking this picture, simple as it is, could save newcomers a lot of confusion. The blue curve, a(x) "scale factor", is what shows the actual expansion---the expanding scale of physical distances between stuff. there is no "radius of the universe" as such,that we know of,but a(x) is a good way to track expansion. It is mathematically generated by the orange curve which is the Hubble expansion rate H(x) which is in concept very much like a percentage growth rate, not a speed. AND H(x) is declining.
Newcomers who have heard about "acceleration" often get confused and expect H(x) the expansion rate to be increasing, but it has been declining all along since the start at x=0, and is on track to continue declining as the graph shows.

The red curve can represent either of the two reciprocals of the expansion rate: the Hubble time 1/H(x) or its length counterpart c/H(x).
In zeon units the speed of light is one: one lightzeon per zeon. The Hubble time shows how long it takes for any small fractional increase to occur.
Both the Hubble time and the Hubble radius are converging longterm to one---the unit time and unit distance.
They mirror the fact that the Hubble rate is declining and converging to the unit expansion rate of one per zeon.

(The expansion rate of one per zeon is where a distance grows by one thousandth of its length in a thousandth of a zeon, and so on for small fractions like that. It is good to think of small fractions because the rate is always changing so one wants to imagine a small interval over which it is nearly constant.)

You can see the expansion we have heard so much about in the blue curve, the scale factor a(x), which changes from decreasing slope to getting steeper in the time interval x=0.4 to 0.6. The inflection point, where slope stops declining and starts to increase, is actually at time x = 0.44 zeon, but it is hard to see precisely by eye.

In the picture you can see that at present, x=0.8, the Hubble growth rate is 1.2. That means it is still some 20% higher than the longterm rate to which it's expected to decline. And accordingly the Hubble radiius, as of now, is 0.83 lightzeon. That means, for instance, that physical distances, like the distance between two galaxies, which happen to be 0.83 lightzeon at the moment, are growing at the speed of light. This is where the two galaxies, for instance, are not though of as moving significantly in the space around them. Their imdvidual random motions are considered as negligible and the expansion of the distance between them and the patches of space around them.

 

[marcus]

Again, I'm learning how to use Jorrie's Lightcone7z, which employs zeon units. One thing I want to do is see how to make the simplest possible pictures of cosmic expansion with it--that can address some frequent confusions newcomers have.
Recall that the OLDEST light we are getting now is stretch s=1090. That is the ancient light of the cosmic microwave background. Its wavelengths have been stretched out by a factor of 1090 while it was traveling.

Paradoxically it comes from fairly close by. Expansion was so rapid that the light (even though aimed in our direction, or in the direction of the matter that eventually became us, since everybody was hot gas at the time) was initially swept back by the expansion. So the light lost ground at first and ended up taking a very long time to get here.

What about the light that we are getting now that started towards us FROM THE LARGEST DISTANCE? If you were reading some earlier posts in this thread you may recall our discussing that. That is light that arrives here with a stretch of s=2.6.

In the picture it is light that was emitted around time x=0.23, where the two curves intersect, and whose distance from us as it gradually narrowed down, exactly followed the red curve called "distance then" or D_then

This shows one of the ways the Hubble radius curve (blue) can be useful. The intersection around time x=0.23 means that at the moment the light was emitted and started toward us the distance to the space the light was traveling through was increasing at the speed of light. That is what the blue Hubble radius curve tells us. It is a kind of threshold--all distances below the blue curve in size are at that time increasing slower than light--all those above that threshold size are expanding faster. So when it was emitted around time 0.23 zeon, the forward motion of the light was exactly canceled and it made no progress at all.
The humpback part of that red curve is level where the two curves intersect. It shows the light making no progress, just staying at the same distance.

The red D_then is in two parts, a pearshaped past, before the nowmark 0.8, and a flaring hornshaped future. It is actually the PROPER DISTANCE RADIUS OF THE LIGHT CONES past and future.
The distance from us light we are receiving now WAS at times in the past, and the distance the light we are emitting now WILL BE at times in the future. The future lightcone flares because the the light is aided by expansion. It gets away from us faster than it would on its own without expansion.

Proper distance is what you would measure if you could pause the expansion process at that moment long enough to measure it in some conventional way. I often don't bother to say "proper" because it's understood. In many if not most of these threads we don't use any other measure of distance (light "travel time" is not very useful as a measure of distance.)

If you are new to the subject you may think it odd that a flash of light sent by us today could reach another galaxy the distance to which is expanding faster than light when it receives our signal. But that is what the picture shows can happen to all light now being emitted here which is received after time x = 1.5 zeon.
At any time after x = 1.5 the red curve is above the threshold blue curve, which means it is a part of space that is receding faster than light.

But the light can clearly get there all right. The red curve (the radius of the future lightcone) tracks its progress. The secret is it's aided in its travels by expansion, by the expansion of the ground it has already covered.

Earlier in the thread we described how light heading our way in the past can actually have been initially swept back, as long as it was on the part of the red curve that is above the Hubble radius. There is a slice of the fat part of the pear showing that. Just another example of the usefulness of the Hubble radius (although it is certainly not the "radius of the universe"

 

[marcus]

Here's another Lc7z picture that can help us get familiar with the way the cosmos operates.
Recall from post #49 that the Hubble radius is an abstract length that helps us keep track of the expansion rate. It varies over time in a different manner from an ordinary cosmic-scale physical distance like the separation between two galaxies. In fact the Hubble radius is converging to a constant value: one light zeon. It is not expected to increase indefinitely.
The Hubble radius is currently 0.83 light zeon. And it is increasing at a speed which is a little less than half the speed of light, 0.4575 c to be precise. It is increasing not along with physical distances but in a way the reflects the declining rate H(x).
A physical distance like the separation between two galaxies, if it currently equals the Hubble radius in size and is therefore equal to 0.83 light zeon, would be increasing at speed c. That's by definition of the Hubble radius. It's the threshold size for superluminal expansion: At any given time it's supposed to tell us the size of distances which are growing at speed c.

What we see in this figure is the speed history of a sample physical distance which happens to coincide with the Hubble radius in size at one point in its existence, it is 0.83 lightzeon at the present era. This is the curve that swoops down to a minimum around time x = 0.44 and then rises back up.

The other curve is the scalefactor a(x) which shows exactly how distance grows over time (normalized to equal one at present, so multiply the distance's current size by the scale factor and you have it whole growth history.) This applies to physical distance, say between two galaxies, neither of which is moving significantly in the space around it. Such objects are said to be "comoving" or at rest with respect to the expansion process and the background of ancient light.

With regard to the speed history, notice that all other speed histories look the same just with different vertical scales. If a distance is twice the size of our sample one, its speed is scaled up by a factor of two, or if half the size, its speed is taken down by half. the minimum point always comes at the same place in history. Around 0.44 zeon.

You can find 0..44 zeon approximately by eye, on the graph. Go halfway between the 0.4 and the 0.6 mark, that would be 0.5, and then about half way between that and 0.4. That should be where the minimum of the speed curve comes, and also it should be where the inflection point of the scalefactor a(x) curve comes---that is, where the a(x) curve changes from convex upwards to concave.

You can see where there is a time period six tenths of a zeon long during which the sample curve is growing slower than the speed of light. Its speed dips below c around x = 0.2 zeon and finally gets back up to c right at the present x = 0.8 zeon.

EDITED after Jorrie's post #52, where he pointed out that the earlier version of this post wasn't clear enough about the difference between the Hubble radius and an ordinary expanding distance that just happens to coincide it at one point in time. This version is an attempt to avoid any possible confusion about that.

 

[Jorrie]

The other is the speed history of a particular distance, watched over a long span of time.

This particular distance was chosen to be the size of the current Hubble radius. So that at present it is growing at speed c.

I think 'this distance' should be qualified a little better to avoid confusion. This recession rate history is for a comoving object presently 'moving through the Hubble sphere', radius R_now ~ 0.83 zeon. The Hubble sphere itself presently grows at a different rate.

 

[marcus]

Good point. thanks for the suggestion. I'll edit in the morning to make it clearer.

EDIT: I went back and emended post #51 for clarity.

 

[Jorrie]

Lc7z now sport variable cropping of curves (under "Open Chart Options") for better customization of specific graphs.
The defaults are still min=0 and max=2.

Here is the 'quintet' of the default curve selection. Basically all the curves that Marcus has posted above on one chart.

 

[marcus]

This "quintet" of curves picture is rich with insights about the standard cosmic model. I wanted to see if I could reproduce it---just duplicate what Jorrie has in post#54---and I expected it would be a lot of work.

I found out it's easy! Lc7z is set up with the defaults to make it simple. All you do is:
1. open Lc7z
2. change S_lower to 0.3 and S_steps to 100.
3. Tick the "chart" button in the row of "Display Options"
and press calculate.

 

[marcus]

I'm trying to imagine a more interactive less abstract introduction to cosmology--based on concrete (even though imagined) situations.
An example of this sort was discussed early in this thread, someone tells you the stretch of some galaxy's light (the redshift-plus-one) and asks when the light was emitted (so how long has it been traveling) and what the expansion rate was back then.
We had some formulas for those.

Here's another situation. You fall asleep--deep suspended animation--and wake on an uninhabited planet. Or maybe there are lots of nice people but they don't have any notion of cosmology. You wonder what time is it? how long was I asleep? what is the expansion age of the universe now?

Fortunately you discover a sensitive device for measuring temperature and are able to measure the temperature of the background of ancient light. It is 1.3625 kelvin.

 

[marcus]

$$ H = \sqrt{.4433s^3 +1}$$ $$t = \ln(\frac{H+1}{H-1})/3$$
s=1/a is always the size of distance or length NOW compared with some other referenced time either in past or future. In the case of the above the CMB temperature is half, so distances must have doubled. s = 0.5 (those now are half what they will be at the designated future time).
Let's figure out what the expansion rate H will be that far in the future, and knowing H will tell us the time.

$$ H = \sqrt{.4433*0.5^3 +1} = \sqrt{.4433*0.125 +1} = 1.02733...$$
$$t = \ln(\frac{2.02733}{0.02733})/3 = 1.4355$$
times 17.3 if you like billions of years, is 24.83 billion years

And let's compare these results with Lightcone
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html

 

[wabbit]

Maybe I missed it, but I didn't see in this thread an introduction to the CMB temperature decay law - If not I think it deserves a brief explanation.

Not sure this is completely correct, but maybe the simplest way would be to define that temperature as the average kinetic energy of CMB photons, and since each photon's energy scales as $h\nu\propto 1/a$ this temperarure must also scale as $T\propto 1/a$ .

 

[marcus]

Yes! Thank you Wabbit. Each photon's wavelength is doubled, each photon's energy is cut in half. As you indicate, the temperature goes down by a factor of 2. We're used to that in another context: that because the redshift+1 of the CMB is estimated at 1090, the temperature of the hot gas at last scattering, that emitted the ancient background light, was 2.725*1090 ≈ 3000 kelvin

 

[marcus]

We're often using those two equations together, so for convenience I combined the calculation into one expression ready to paste into the google calculator. This is for an example where you might want to find the time corresponding to s = 0.8. That is, a time when distances are 25% larger than today. 1/0.8 = 5/4

Here is what you's paste into google to get the answer in H_∞ units:
ln(((.4433*.8^3+1)^(1/2)+1)/((.4433*.8^3+1)^(1/2)-1))/3

Or, if you want it in terms of billions of years, you would multiply by 17.3, or simply paste in:
17.3*ln(((.4433*.8^3+1)^(1/2)+1)/((.4433*.8^3+1)^(1/2)-1))/3

When you paste this in you'll presumably change both occurrences of the number 0.8 to whatever is appropriate. For example if you want to know the time when distances were 2/3 what they are now, that means s = 1.5 (s is always the size now compared with that at the designated time). So you would paste in:
ln(((.4433*1.5^3+1)^(1/2)+1)/((.4433*1.5^3+1)^(1/2)-1))/3

Of the two equations, one is simply a version of the Friedmann equation itself, showing how H² relates to density, tracked by present density and scale factor---under the square root.
The other is the solution of the Friedmann equation, namely the hyperbolic function coth(1.5t), that relates the expansion rate H(t) to time--with the equation H(t) = coth(1.5t) solved for t, to give t(H) as a function of H. So these two equations are "part and parcel" of the Friedmann.