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

I should summarize where I see this going as a result of the work and discussion in this thread. We have two resources shaping up. On one hand there are Jorrie's calculator(s) which AFAICS deliver professional-grade accuracy and implement the standard cosmic model. One reason they are interesting is that they let you input different model parameters and they output quite a variety of information about the universe: distances to galaxies then and now, distance expansion speeds, percentage expansion rates, and so forth.

On the other hand we've got a simple "do it yourself" cosmic model that delivers reasonably good precision back over the past 12 or 13 billion years. It is more trouble to use because it is essentially just based on a single formula. This is the formula for the expansion history of a generic distance: the scalefactor a(t) as a function of time.

This is arbitrarily pegged at the present value of 1, so a(now) = 1, and it shows how any cosmological distance has grown---it also extends into the future to show expected future growth.

This simple DIY cosmic model is is more trouble to use because essentially given the growth history you have to figure everything else from that, for yourself. But it's not as bad as it sounds and one picks up some understanding along the way.

My posts on the last several pages can be summed up streamlined as follows.
If you go to the online calculator http://web2.0calc.com/ (the first hit if you do a search for "online scientific calculator")
and paste this formula in:

(((16.3/13.9)^2 - 1)/((tanh(1.5*t/16.3))^-2-1))^(1/3)

you will see that as soon as it is pasted in, the calculator displays a neater, easier-to-read form.$$\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{1/3}$$What this computes, given a time t (in billion years) is the scalefactor a(t)---in other words it tells you the expansion history of a generic distance over time.
To use it, just replace the "x" by a time expressed in billions of years (such as 1, or 2, or 13.759 which is the current expansion age) and press the equals sign.
The answer will appear in the window and you can copy it to clipboard if you want. Then to repeat the calculation, click on the neat version of the formula as it appears above the window, and you can substitute something else in for the variable.
========================
1--- 0.1471433... (when the universe was 1 billion years old, distances were 14.7% what they are now)
2--- 0.2342347... (at 2 billion years, distances were about 23% what they are now)
...
...
13.759--- 0.9999836... (at the present age of 13.759 billion years they are of course 100% of their present lengths.:-)
...
20--- 1.5235746... (at age 20 billion years, distances will be 52% bigger than they are today.)
==================
A modified version of this same formula gives you the reciprocal scalefactor, 1/a(t), which turns out to be quite useful
The model's formula for 1/a(t) is
$$\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{-1/3}$$
and the single-file version, that you paste into the calculator if you are working with it that way is:
(((16.3/13.9)^2 - 1)/((tanh(1.5*t/16.3))^-2-1))^(-1/3)

Adding up successive values of 1/a as you work back in time is actually the way the present distance to a source is calculated!
========================
Some examples of the reciprocal scalefactor 1/a(t) for various times.
1---6.80... (distances and wavelengths of traveling light have expanded by a factor of 6.8 since the universe was 1 billion years old
2--- 4.27... (distances and wavelengths of traveling light have expanded by a factor of 4.27 since the universe was 1 billion years old)
...
...
13.759--- 1.000... (this is the present, distances and waves are their present lengths :-)
...
==================
It's conceivable you might sometime want to find the time t that gives a particular scalefactor, IOW invert the above formula for a(t). In that case paste this in
(16.3/1.5)atanh((((16.3/13.9)^2 -1)/a^3+1)^-.5)
which the calculator will display as $$\frac{16.3}{1.5}atanh\left(\left(\frac{(\frac{16.3}{13.9})^2-1}{a^3}+1\right)^{-.5}\right)$$
==================
The main equation in this model is this one. It gives the scalefactor a(t) at each time, going back pretty far into the early history of expansion where it gets a bit off track (because when you get back to the first few 100 million years much of the density in the universe was radiation rather than particles of matter, and radiation behaves differently in expansion , so the physics is not as simple. Anyway the main model equation is this:

$$\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{1/3}$$

The coefficient 1.5 and the exponent 1/3 both reflect the fact that we're in a matter dominated era, and have been since the first few 100 million years, and matter density falls off as volume increases---as the cube of distance. That is where the 3 and the 1.5 come from. In a radiation dominated world they would be 4 and 2. I derived some equations earlier in this thread and can go back to that later if there's interest.

But the most significant parameters in that "expansion history equation" are the Hubbletime parameters 13.9 and 16.3 billion years.

THOSE TWO TIME QUANTITIES *SHAPE* THE GROWTH CURVE.

If you change them the scalefactor curve a(t) showing the growth of a generic distance will change.

These two quantities are worth understanding. They express the CURRENT percentage growth rate of distance and the eventual longterm LIMIT growth rate that the present one is slowly tending towards.

I'll pick up and continue from here in my next post. This is enough for now.

 nobody replied to me ___________________ http://www.astro.ucla.edu/~wright/CM...Oct08clean.pdf (last pic) #do people on galaxies which are 13bly away see "first galaxies" when they look towards milky way?# and do they see that milky way and andromeda are moving away from each other and/or away from observer's direction (like observers on earth see how galaxies are moving away in the distant space(like in picture)) ....................................................................... ............. here it looks like implosion(inside of a sphere), because dark age can be observed around the universe(in each direction) and this looks like surface of a sphere model is reality in between these models? then it would be like a donut model. if there is no centerpoint which serves as center of universe then it would support what i put into "# #" ....................................................................... ............. this doesnt help stuff because if universe would be this way then there one could also look back(away from the center): http://en.wikipedia.org/wiki/File:Em...M_geometry.png ....................................................................... ......... was explosion(big bang) this enormous that space expands with 70mpc / s? is inflation responsible for this? and is speed the same as it was billions of years ago or does it become faster(expansion)? and does it expand now with speed of 2.16 trilliards(number with 21 characters) km/s? ....................................................................... ........... does somebody understands it trully? i follow scientifical concept to understand something and if i take away one step before another i cant precede, understanding starts from simple and gets more complicated once something is understood but i read stuff where people start from the middle and then its complicated, then i must ask things(puzzle parts) and if they dont contradict each other, i get the picture from answers did knowledge came(to cosmologists) from observing mathematical equations which dont contradict each other and was it then applied to understanding(imagination) which then trully understood how universe works/looks like?
 The are different ways that try to show the same thing. The first diagram is a two dimensional diagram in *time*. What it shows is what you see if you look back in time. The second diagram is a three dimensional diagram in both space and time. To get from the second diagram to the first, imagine a cone that ends at now, and shows the path of light that is arriving at you at this very moment. The intersection between the second diagram and the "light cone" will get you the first diagram.

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Hi Twofish! Glad to see you! Let's see if we can get F.F. to start his own thread(s) with these questions.He has so much to ask and learn about I fear it would overload this thread.
 Quote by fat f... http://www.astro.ucla.edu/~wright/CM...Oct08clean.pdf (last pic) #do people on galaxies which are 13bly away see "first galaxies" when they look towards milky way?# ... ... does somebody understands it trully? i follow scientifical concept to understand something and if i take away one step before another i cant precede, understanding starts from simple and gets more complicated once something is understood but i read stuff where people start from the middle and then its complicated, then i must ask things(puzzle parts) and if they dont contradict each other, i get the picture from answers did knowledge came(to cosmologists) from observing mathematical equations which dont contradict each other and was it then applied to understanding(imagination) which then trully understood how universe works/looks like?
You have many questions--many things that you don't understand and want to talk about.
Too many for this thread. You should start your own thread. Start off with one clear question. Don't ask everything all at once.

Like start a thread with this question (it is a good one)
"#do people on galaxies which are 13bly away see "first galaxies" when they look towards milky way?#"

That is a really good question. If you start a thread with just that, I would certainly answer. Other people would also. You might get several hours of people's time discussing that. Clarifying confusions about how distance is measured in cosmology.

BTW personally I think the first cosmological knowledge did not come from equations.

One of the first bits of knowledge came to a man named Anaxagoras in 250 BC (before there was equation-solving as we know it) by carefully reasoning about the distances to things.. He figured out that the sun is more than 10 times farther than the moon.
This enabled him to deduce that the width of earth shadow (at the distance of the moon) was nearly as wide as the earth itself. Then he observed that the earth shadow when cast on moon during eclipse, was about 3 times greater than the width of the moon.
So he could estimate that the earth itself (being slightly wider than its shadow at that distance) is slightly more than 3 times wider, maybe something around 4 times wider, than the moon. Figuring that out from scratch is no small achievement!

Since the time of Anaxagoras, most new knowledge about cosmos has evolved by careful reasoning about distances. What Hubble did was not so different from Anaxagoras. He learned a new way to estimate distances to galaxies, and when he surveyed them he found that distances to most galaxies were increasing a certain tiny percentage each year. Discovering expansion this way came, for him, before understanding and believing the Einstein equation--although that equation has geometric expansion as one of its most likely solutions.

From Anaxagoras to Hubble cosmologists have gained knowledge primarily by reasoning carefully about and devising smart ways to estimate the distances to things. So if you want to understand, a good way is to begin at the beginning and ask yourself how do you know that the sun is more distant than the moon.

Or start a thread with that ONE question you asked. Get people to explain the answer to that one question. But not in this thread, it would get too far off the current topic.

 1) Does somebody understands it trully? As far as the parts for the diagram that you are showing, people understand it pretty well. Part of the reason is that there is very little "weird physics"
 Recognitions: Gold Member Science Advisor About the general question "how do we know", I'll add to what I said a couple of posts back: From Anaxagoras to Hubble cosmologists have gained knowledge primarily by reasoning carefully about distances and angles (often as seen from another observer's viewpoint) and by devising ways to estimate distances to things. It's not a bad idea to go over some of the steps in that long human history of accumulated insight, and in effect re-experience. For instance Anaxagoras (the name means "kings market") had the idea to visualize the angle between earth and sun that someone on moon would see (when we see a half-moon) and realize that it was a right angle. So he could sketch a right triangle, with the square corner at the moon, and realize that the sun was much farther from us than the moon (because of the near-right angle between them that WE see from earth, at half moon time.) Now remember from that "much farther" and watching an eclipse he could tell that earth is something like 4 times wider than moon. But the angle the moon makes in the sky is only 1/120 of a SIXTH of a circle! (Greeks learned from Babs that it's sometimes smart to judge angles as fractions of a SIXTH of a circle rather than of a whole circle.) Which means its distance from us is 120 TIMES ITS WIDTH!!! That would mean, if earth is 4 times wider than moon, that distance to moon is 30 earth diameters. These complicated chains of reasoning about distances and angles are still at the heart of cosmology. If you practice on Anaxagoras it might make it easier to overcome confusion about the temperature map of the microwave background (the most ancient light we can see.) The angular sizes of its fluctuations are the analogs of the angles and proportions Anaxagoras perceived in the sky.
 Recognitions: Gold Member Science Advisor To get back to discussing the simple one-formula model cosmos, here is what I was saying a few posts back: ================== The main equation in this model is this one. It gives the scalefactor a(t) at each time, going back pretty far into the early history of expansion where it gets a bit off track (because when you get back to the first few million years much of the density in the universe was radiation rather than particles of matter, and radiation behaves differently in expansion , so the physics is not as simple. Anyway the main model equation is this: $$\left(\frac{(\frac{16.3}{13.9})^2-1}{(tanh(\frac{1.5}{16.3}t))^{-2}-1}\right)^{1/3}$$ The coefficient 1.5 and the exponent 1/3 both reflect the fact that we're in a matter dominated era, and have been since the first few million years. Matter density falls off as volume increases---as the cube of distance---and that is where the 3 and the 1.5 come from. In a radiation dominated world they would be 4 and 2. I derived some equations earlier in this thread and can go back to that later if there's interest. But the most significant parameters in that "expansion history equation" are the Hubbletime parameters 13.9 and 16.3 billion years. THOSE TWO TIME QUANTITIES *SHAPE* THE GROWTH CURVE. If you change them the scalefactor curve a(t) showing the growth of a generic distance will change. These two quantities are worth understanding. They express the CURRENT percentage growth rate of distance and the eventual longterm LIMIT growth rate that the present one is slowly tending towards. Each of the (rather long) intervals of time is the reciprocal of a (rather slow) instantaneous distance expansion rate. For example the presentday Hubbletime 13.9 billion years can be understood intuitively by thinking that ONE PERCENT of it, namely 139 million years, is the time a distance would take to increase by one percent. (Continuing steadily at its present speed of growth.) The two hubbletimes 13.9 and 16.3 billion years are convenient to handle and rememeber, and we use them as the two main parameters in the model, but the actual growth rates we are concerned with are their RECIPROCALS which you can write as 1/13.9 per billion years and 1/16.3 per billion years---fractional rates of growth. These are ridiculously (I should say "astronomically") slow rates of fractional growth. So it's easier to work with the times than with the rates. Going back to the earlier post, here are some sample outputs from our model's main formula: scalefactors a(t) calculated for various times t; ================== 1--- 0.1471433... (when the universe was 1 billion years old, distances were 14.7% what they are now) 2--- 0.2342347... (at 2 billion years, distances were about 23% what they are now) ... ... 13.759--- 0.9999836... (at the present age of 13.759 billion years distances are of course 100% of their present lengths.:-) ... 20--- 1.5235746... (at age 20 billion years, distances will be 52% bigger than they are today.) ================== What I want to explain is how we can use the output from this one formula to find out other things: DISTANCES to sources which emitted the light we're getting from them at various times in the past. DISTANCES to sources whose light comes to us with wavelengths expanded by some factor SPEEDS that the distances computed as above are now increasing, and were increasing when the light was emitted. I'll try to get to that in the next post.
 Recognitions: Gold Member Science Advisor To begin to deal with distances in terms of this simple model I need to add up this cumulative sum of terms ts where s is the reciprocal scalefactor 1/a and run thru some range like [1,2] in steps of 0.1, say. Let's start it at s = 1.1 and go 1.2, 1.3,.... and at each value of s we will evaluate this formula atanh((.375136* s^3+1)^-.5) using http://web2.0calc.com to get ts the time (expressed in billions of years) when the light was emitted that we now see wave-lengthened by a factor of s. 1.14678............(starting with 1.1) 2.1893904284755 3.14075638766957 4.01193171154709 4.81239653369235 5.55030772362728 6.2327011396146 6.86566003274671 7.45445670103674 8.00367220130405 (this was for 2.0) 8.51729768096363 8.99882016696824 9.4512951759886 9.87740815857389 10.2795265021777 10.65974356945189 11.0199160299271 11.36169555160973 11.68655575226624 11.99581516649566 (the last one was for 3.0) So how to use this cumulative sum to get distances? Well suppose a galaxy's light comes in with a scaleup s=2.0 (wavelengths twice as long as when emitted.) The number from the list we use is the one for 1.9 namely 7.45445670103674, and that gets multiplied by the Δs, the step size, which is 0.1 In addition there are two other things to do: add (1+Δ/2)*t1 which is 1.05*1.2661864372681=1.329495759131505 and subtract (2-Δ/2)*t2 = 1.95*0.54921550026731=1.0709702255212545 The difference is 0.2585255336102505, so that's what gets added: 0.745445670103674+0.2585255336102505 = 1.0039712037139245 Finally multiply that by 16.3/1.5 and get 10.9098... Gly. Ned Wright's calculator says 10.901 Gly (with equivalent model parameters). So the accuracy is not so bad. that's the current distance: 10.91 billion lightyears. All those extra decimals are ridiculous but it is too much trouble to be rounding off all the time so I just take what the calculator gives and finally round it off to something sensible at the end. the corresponding thing for s=3 1.168655575226624+1.329495759131505-0.91231527197679=1.585836062381339 and then again finally multiply by 16.3/1.5, to get 17.23275...billion lightyears Ned Wright says 17.220, so again we are off by 1 in the fourth digit.
 Recognitions: Gold Member Science Advisor I didn't get around to editing the previous post until after the deadline for changes and it needs some clarification. The key formula in the toy version cosmic model I'm working with is (16.3/1.5)*atanh((.375136* s^3+1)^-.5) This gives the time (expansion age in billions of years) at which the reciprocal scalefactor was a particular value s. Another way to say it: ts is the time (expressed in billions of years) when the light was emitted that we now see wave-lengthened by a factor of s. s can be thought of, if you like, as the "scale-up factor": because since the time ts, distances and wavelengths have been scaled up by a factor of s. Back at time t2 distances were 1/2 their present size, back at t9 distances were 1/9 their present size, t1 is the universe's present expansion age, and so on. I won't be using the traditional notation for s, which is "1+z", since it makes the formulas even messier than they are already. The Hubbletime parameter 16.3 billion years represents the cosmological constant (asymptotic distance growth rate.) The other Hubbletime parameter, giving present growth rate, is 13.9 Gy and since that most often appears in combination as (16.3/13.9)2-1=0.375136... I have, for simplicity, packaged it in that number. When I have to do a lot of calculating, e.g. numerical integration, I leave off the coefficient (16.3/1.5) and factor it in only at the end. It turns out that we can do a pretty good job of estimating the distances to sources at various scaleups by essentially just adding up a long string of arctanh values, with s advancing from 1 to s in steps of some smallish stepsize Δ. If we take Δ = 0.1, this amounts to: 1.05t1+ 0.1( t1.1+ t1.2+...+ ts-0.1) - (s-0.05)ts This is the bare bones of a numerical integration for c∫ s dt. The idea is that at each interval dt of time in the past, the light from an object travels a distance cdt and this gets scaled up by a the appropriate factor s. So at present the distance to the object is the sum of all those scaled-up segments and equals c∫ s dt. It looks messy but seems to work out all right. Here's a cumulative sum of atanh((.375136* s^3+1)^-.5) using http://web2.0calc.com 1.14678............(starting with 1.1) 2.1893904284755 3.14075638766957 4.01193171154709 4.81239653369235 5.55030772362728 6.2327011396146 6.86566003274671 7.45445670103674 8.00367220130405 (this was for 2.0) 8.51729768096363 8.99882016696824 9.4512951759886 9.87740815857389 10.2795265021777 10.65974356945189 11.0199160299271 11.36169555160973 11.68655575226624 11.99581516649566 (this was for 3.0) 12.29065686210409 12.57214523554962 12.84124042899438 13.09881073851359 13.34564332215593 13.58245346592667 13.80989262370986 14.02855541222707 14.23898571313266 14.44168201025018 (for 4.0) 14.63710206991032 14.82566705565418 15.00776515463345 15.18375478139281 15.3539674149637 15.51871011700947 15.67826777187171 15.83290508355766 15.98286835979816 16.12838710914517 (for 5.0) So suppose we evaluate 1.05t1+ 0.1( t1.1+ t1.2+...+ ts-0.1) - (s-0.05)ts to find the distance now to a source with s = 4 (its light comes in wavelengthened by a factor of 4). As long as we are using the stepsize Δ=0.1, the first terms is always 1.329495759131505, and the sum, multiplied by the stepsize, can be read off that list: 1.423898571313266. The term at the end, that gets subtracted, is 3.95*0.202696297=0.800650373 1.329495759 +1.423898571 - 0.800650373 = 1.95274396 And then at the end the whole thing gets multiplied by the cosmological constant term 16.3/1.5, to give 21.2198≈ 21.22 billion light years. Sorry about all the meaningless extra digits but it is too much trouble to be rounding off every time I take a result from the calculator, so I just round off at the end. Let's compare this with Wright's calculator. Well, Wright's says 21.204 Gly. So as usual we are OK for three significant figures and off in the 4th place. Notice that since this numerical summing procedure gives us the NOW distance to the source, all we need to do is multiply by the scalefactor a, or alternatively divide by the scaleup s, and we get the THEN distance---how far from our matter or galaxy the thing was when it emitted the light. So this primitive model already does quite a bit that one expects from serious cosmology calculators. Given a scalefactor a (or the reciprocal 1/a = s) it can give the corresponding expansion age---the time when the source galaxy emitted the light. And it can give the Now and Then distances to the source (proper distance, as if you could halt expansion at the given moment and measure directly). The THEN distance is essentially the ANGULAR SIZE distance (our model is spatial flat) so that's taken care of. It still might be nice to be able to calculate the HUBBLETIME corresponding to a given scalefactor a, or its reciprocal s. That is hour handle on the rate of expansion going on at the time the light was emitted. Notice that the light itself, when it arrives, tells us the scalefactor, or equivalently its reciprocal s, which we focus on here, so the other things we want to know should be calculated from s.
 Recognitions: Gold Member Science Advisor It looks as if the Hubbletime Ys corresponding to a given scaleup s should be given in billions of years by: 16.3(0.375136s3+1)-.5=16.3/sqrt(.375136*s^3 + 1) so using the calculator let's try that for s=3 It gives Y3 = 4.8861403 ≈ 4.886 billion years. Let's see if I've made a mistake. Apparently not, Jorrie's calculator (with the corresponding parameters) gives 4.885 billion years. Remember that the two parameters we're using in the model's formulas, namely 16.3 and 0.375136, are just an equivalent form of the two Hubbletimes which determine two key expansion rates, now and in distant future. Ynow = 13.9 billion years Y∞ = 16.3 billion years The number .375136 is simply what (Y∞/Ynow)2 - 1 = (16.3/13.9)2 - 1 works out to be. You can think of 1.375136 as a ratio of two expansion rates, squared. It is simply (Hnow/H∞)2 so it tells you how much more the percentagewise expansion rate is now than it will be in the longterm future. When people talk about "acceleration" what they mean is what you see when the H expansion rate is declining only very slowly or is steady at some given value. As long as H is not declining too rapidly, if you watch a particular distance it will grow by increasing annual amounts as the principal grows. Not terribly dramatic, given the very low "interest rate" but there is acceleration in a literal sense. In the previous post we calculated that a galaxy we see with scaleup factor s = 4 (wavelengths quadrupled) is now at a distance of 21.22 Gly. How fast is that distance now growing? That is very simple to calculate. We just divide 21.22 Gly by 13.9 Gy. Dnow/Ynow = 21.22/13.9 = 1.53c. That means it is growing at 1.53 times the speed of light. Calculation easy with these quantities For comparison and a bit more practice, back in post #433 we found that the distance NOW for s=3 was 17.23275 billion lightyears, which means distance THEN was 17.23275/3=5.744 billion lightyears. However we just found that also for s=3 we have Hubbletime Ythen = 4.886 Gy. So for a s=3 galaxy, whose distance THEN at time light was emitted was 5.744 Gly, how fast was that distance then growing? Well obviously Dthen/Ythen = 5.744 Gly/4.886 Gy = 1.18 ly/y = 1.18c. The notation is still far from perfect, but I hope some of this is comprehensible ============== referring back to post #434 the last increment was 0.14551874934701 and to find the now distance to an s=5 source D5(now) one would take 4.95*0.14551874934701=0.720317809 off at the end, so it looks like 1.329495759 +1.598286836 - 0.720317809 = 2.207464786 which then gets multiplied finally by 16.3/1.5 to give 23.987784 billion lightyears. So unless I've made a mistake that's the distance now to an s=5 source. I'll compare with what Wright's says. It says 23.970 Gly. So we are still OK for three digits. D5(now) = 23.99 Gly D5(then) = 23.987784/5 ≈ 4.798 Gly Y1 = 13.9 Gy Y5 = 16.3/sqrt(.375136*5^3 + 1)= 2.35535 ≈ 2.355 Gy So we can say that for an s=5 galaxy, when the light we are now getting was emitted, the distance was expanding at a speed 4.798 Gly/2.355 Gy = 2.037 c, over twice the speed of light. The then and now speeds of expansion are given by: D5(then)/Y5 and D5(now)/Y1

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 Quote by marcus It looks as if the Hubbletime Ys corresponding to a given scaleup s should be given in billions of years by: 16.3(0.375136s3+1)-.5=16.3/sqrt(.375136*s^3 + 1)
It appears to me that your 'scaleup factor' (s = 1/a = z+1) is a useful one, since it does not go negative for future times, but just goes smaller than 1. I would caution against the terminology though, as it is too close to the conventional 'scalefactor' and may cause confusion. Maybe something like 'upscale ratio' or 'expansion ratio'? I would prefer 'upscale ratio of distances' rather than 'scaleup factor of wavelengths', so as to not also cause potential confusion with Doppler effects.

I'm working on a variant of the cosmo-calculator that will give a table for a range of z (or s?), with some useful values in the columns. Not quite there yet, but it looks practical.

 Recognitions: Gold Member Science Advisor How about calling it "stretch factor"? Or "extension ratio"? Another idea, similar to something you suggested is to call the s number the "enlargement" because it is the ratio by which distances are enlarged during the time the light is on its way somewhat reminiscent of photographs being blown up. I'd welcome more suggestions. I'm glad to hear your new calculator is looking practical! I think the idea of generating tables for a range of z (or for s !!!) is a good one. Even fairly short tables with 5 to 10 lines can give someone extra perspective and intuition about how things are evolving. Just being able to compare two or three lines can be informative. After trying different things I do agree that the reciprocal scalefactor (whatever you call it and whether or not you subtract 1 from it) is the most useful handle on the situation.
 Recognitions: Gold Member Science Advisor Here's one way to think about it: say we number the stages of expansion history according to how much distances have been enlarged since then. It means that earlier slices of spacetime have larger s numbers, which at first seems turned around, but in fact it's cleaner formula-wise to do the numbering backwards that way. To illustrate, suppose we are watching a galaxy as it was when distances were 1/4 of present size. We can denote that stage of expansion history by saying s = 4. While the light was on its way, distances and wavelengths have been enlarged by that factor. So that galaxy, as we see it, is in "slice 4" of expansion history. In that way of denoting stages of expansion, the present is s=1, because enlarging by a factor of 1 is the identity. In the simplified toy model, the expansion age ts associated with a stage s is given (in billions of years) by: $$t_s = \frac{16.3}{1.5}arctanh \left( \left( 0.375136 s^3 + 1\right)^{-1/2}\right)$$ And the corresponding Hubbletime at stage s, also in billions of years, is: $$Y_s = 16.3 \left( 0.375136 s^3 + 1\right)^{-1/2}$$ These are the two basic equations of the model--the other usual quantities such as distances and expansion speeds can be derived from these two. There is one peculiar thing to notice, which is that with expansion stages numbered this way, not only do we have the present tagged s = 1 but also future infinity is s = 0. So the eventual, or longterm value of the Hubbletime (a key parameter in the model) is: $$Y_0 = 16.3 \left( 0.375136 \times 0^3 + 1\right)^{-1/2} = 16.3 Gy$$ while the present Hubbletime is: $$Y_1 = 16.3 \left( 0.375136 \times 1^3 + 1\right)^{-1/2} = 13.9 Gy$$ I keep having to write this number 0.375136, which is kind of like a parameter of the system being the square ratio of our two Hubbletimes, less one. (Y0/Y1)2 - 1. So I will call that number capital Theta Θ. The two basic equations of the model are then: $$t_s = \frac{2}{3}Y_0 arctanh \left( \left( \Theta s^3 + 1\right)^{-1/2}\right)$$ $$Y_s = Y_0 \left( \Theta s^3 + 1\right)^{-1/2}$$ People who don't like greek letters should just remember it is a shorthand for an ordinary number ≈ 0.375 that essentially says something about the amount bigger current expansion rate is than the eventual longterm rate. (their ratio is about sqrt(1.375)
 Recognitions: Gold Member Science Advisor I want to try out some terminology in part suggested to me by Jorrie's comments and which he might be puttng to use in another project.But I can try several ideas out, tentatively, in connection with this simple cosmic model. The main variable could be called the "stretch" because it is the factor by which distances from a past slice of spacetime are enlarged (and wavelengths too) between then and now. The idea is that if you start back to some earlier stage in expansion history and the enlargement of distances (and wavelengths) from then to now is a stretch factor of four (say S = 4) then the scale back then, relative to now is 1/4, or 0.25. So the stetch and scalefactor are reciprocals, like 4 and 1/4. It just turns out that the stretch is a convenient variable to run the model, or the calculator, on. You get the simplest formulas that way, of the various things I've tried. So I'm using S to stand for the stretch and the conventional letter a (= 1/S) to stand for the scale factor. The lineup of numerical information could (tentatively) go like this: Stretch---Scale factor---Expansion age---Hubble time---Distance now---Distance then and then again, this time showing the symbols that might be used to denote these quantities: Stretch (S=1/a)---Scale factor (a)---Expansion age (tS)---Hubble time (YS)---Distance now (DS[now])---Distance then (DS[then]) The idea is, we observe a galaxy and its light tells us the stretch factor S, say it is 4. Wavelengths 4 times what they were at the start of the trip. The galaxy is living back when distances were 1/4 present size. Then D4[now] tells us proper distance to the galaxy NOW, and D4[then] tells us distance back then, when light was emitted, from our matter (that became us) to the galaxy. If we want to know the SPEED of distance growth, you simply divide the distance by the Hubble time belonging to that slice. Back then when the distance was D4[then], it was growing at speed D4[then]/Y4. The present is denoted S=1 and the present Hubbletime is Y1 = 13.9 billion years. So the present distance is expanding at speed D4[now]/Y1. Today's distances, if you want to know what speed they are expanding, you just divide them by 13.9 billion years. So that's a provisional idea for a list of 6 related numbers that the model, or a calculator, can give you, that seems like enough to work with and get a picture of the expansion history from.
 Recognitions: Gold Member Science Advisor I don't want to forget that simple model, although Jorrie has now put an excellent online tabulator on line. Here is the single-line formula calculator we were using, see post #434 http://web2.0calc.com Here is the calculator formula to compute the time given the stretch S: (16.3/1.5)atanh((.375136* S^3+1)^-.5) There's also a more complicated version for it given the scalefactor, but we probably won't use it. (16.3/1.5)atanh((((16.3/13.9)^2 -1)/a^3+1)^-.5) I just had a kind of exciting look into the future. I went to web2.0calc and put in exactly what I mentioned, namely (16.3/1.5)atanh((.375136* S^3+1)^-.5) And decided to see when distances would be 100 times what they are today which means scalefactor a=100 and reciprocal S = 1/a = 0.1 So I put 0.1 in place of S, in the formula and pressed = and it said that would happen in year 87.92 billion. So that is kind of cool. When expansion has been going on for about 88 billion years distances can be expected to be about 100 times what they are today. Let's try another. when will it be that distances are FIFTY times what they are today? Put in S=0.02 to the web2.0calc. Bingo. It says that will happen in year 76.625 billion. And just as a side comment with continuous compounding the Hubbletime 16.3 billion years corresponds to a doubling time of 11.3 billion years. That is the natural log(2) times 16.3. So it seems right that you go from scale 50 to scale 100 in something a little over 11 billion years.
 Recognitions: Gold Member Science Advisor In post#434 I used a crude numerical integration of Sdt to find the distance now to a galaxy in the past in era S It turned out that when you rearrange an Sdt integration to make it easy to add up you get what LOOKS like a tdS integration (with extra terms at either end). this is just algebraic rearrangement. Then the steps can be of S rather than time and we can use the formula tS (16.3/1.5)*atanh((.375136* S^3+1)^-.5) which gives the time (expansion age in billions of years) when the reciprocal scalefactor (stretch) was a particular value S In the earlier post we had S advancing from 1 to S in steps of some smallish stepsize Δ. If we take Δ = 0.1, this amounts to: 1.05t1+ 0.1( t1.1+ t1.2+...+ ts-0.1) - (s-0.05)ts This is what the numerical integration for c∫ S dt boiled down to. The idea was that at each interval dt of time in the past, the light from an object travels a distance cdt and this gets scaled up by the appropriate factor S. So at present the distance to the object is the sum of all those scaled-up segments and equals c∫ S dt. ==================== So I decided to look into the future with the same technique and I found that if a galaxy is going to pass thru your forward lightcone at S=0.5, that is when distances are TWICE what they are today, then the distance NOW to it is 7.5 Gly. Where are the galaxies NOW which you could hit with a flash of light you send today and which arrives wavestretched to double length? They are 7.5 billion lightyears from here. It's like a time reverse image of the earlier game when we asked things about a galaxy whose light comes in wavestretched by a factor of 2, where was it when it emitted the light, where is it now etc. The numerical integration boiled down to: .55t.5+ 0.1( t.6+ t.7+...+ t.9) - .95t1.0 And I evaluated that and got 7.5 billion lightyears. So then we can say that when the signal we send arrives the distance to the target galaxy will be 15.0 billion light years. Because we know the expansion of scale between now and that time in the future.
 Recognitions: Gold Member Science Advisor In the previous post I did a rough numerical integration based on toy model and got the estimate that a galaxy we can send a message to which will arrive when distances are TWICE today (namely an S=0.5 galaxy) is currently at distance 7.5 Gly and when the message arrives it will be at S=15 Gly. Now I can confirm that with the A20 calculator, which sees the shape of expansion history in the future as well as the past http://www.einsteins-theory-of-relat...oLean_A20.html I just make a small table running from present S=1 out to S=.1 in future, in steps of 0.1. So it covers the S=0.5 case but also gives me a little context (to help grow intuition/feel for the expansion process.) ===quote=== Hubble time now (Ynow) 13.9 Gy Change as desired (9 to 16 Gy) Hubble time at infinity (Yinf) 16.3 Gy Change as desired (larger than Ynow) Radiation and matter crossover (S_eq) 3350 Radiation influence (inverse: larger means less influence) Upper limit of Stretch range (S_upper) 1.0 S value at the top row of the table (equal or larger than 1) Lower limit of Stretch range (S_lower) 0.1 S value at the bottom row of table (S_lower smaller than S_upper) Step size (S_step) 0.1 Step size for output display (equal or larger than 0.01) Stretch (S) Scale (a) Time (Gy) T_Hubble (Gy) D_now (Gly) D_then (Gly) 1.000 1.000 13.756 13.900 0.000 0.000 0.900 1.111 15.250 14.444 -1.417 -1.575 0.800 1.250 16.981 14.929 -2.887 -3.608 0.700 1.429 19.004 15.342 -4.401 -6.287 0.600 1.667 21.396 15.677 -5.952 -9.921 0.500 2.000 24.279 15.930 -7.533 -15.066 0.400 2.500 27.856 16.108 -9.135 -22.839 0.300 3.333 32.507 16.218 -10.752 -35.840 0.200 5.000 39.097 16.275 -12.377 -61.886 0.100 10.000 50.388 16.297 -14.006 -140.059 ===endquote=== So the quick and dirty estimate I did earlier worked OK. For an S=.5 galaxy (where our message reaches when distances are TWICE) the present distance really is 7.5 and the distance then when message arrives really is 15 Gly. the minus signs have to do with the direction the light is going, from us to them. whereas in the past the distances have positive sign because the light is coming from them to us----itself a kind of nice feature. Also as an extra bonus the A20 tells me that the message that we send today (expansion age 13.75 billion years) will arrive when expansion age is 24.3 billion years. So it will take around 11 billion years to get there. That makes sense: when it arrives at destination the message will be 15 billion lightyears from us, and will have been traveling 11 billion years---you have to allow for some expansion of distances so naturally 15 > 11.