# Look 88 billion years into future with the A20 tabular calculator

by marcus
Tags: billion, calculator, future, tabular
 PF Patron Sci Advisor P: 22,351 The new A20 tabular calculator lets you look at changing geometry out to about 88 billion years according to the standard LCDM cosmic model (with usual estimates for the parameters.). http://www.einsteins-theory-of-relat...oLean_A20.html It's pretty neat. Here is one sample tabulation. Red stuff is just the three standard parameters, estimated based on observation. No reason to change them, although in this calculator you CAN change them and play around to see the effects. The blue stuff is what I put in to give bounds and step size for the table I wanted it to generate From the present (S=1) to the distant future (S=0.01) when distances are 100 times what they are today. In steps of ΔS = 0.09. those are just what I chose. If you choose a smaller step size like ΔS = 0.01 you get a table with more rows, like around 100 rows instead of only 12 rows. I won't bother to align the columns. It's probably legible as is. ===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) 0.01 S value at the bottom row of table (S_lower smaller than S_upper) Step size (S_step) 0.09 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 13.769 13.896 0.000 0.000 0.910 1.099 15.104 14.387 -1.219 -1.339 0.820 1.220 16.630 14.829 -2.536 -3.093 0.730 1.370 18.374 15.221 -3.884 -5.320 0.640 1.563 20.402 15.545 -5.270 -8.234 0.550 1.818 22.772 15.812 -6.676 -12.138 0.460 2.174 25.618 16.006 -8.108 -17.627 0.370 2.703 29.120 16.143 -9.555 -25.825 0.280 3.571 33.629 16.233 -11.010 -39.323 0.190 5.263 39.934 16.278 -12.474 -65.650 0.100 10.000 50.390 16.296 -13.939 -139.393 0.010 100.000 87.919 16.300 -15.406 -1540.607 For the model used, see this thread on Physicsforums. =====endqquote===== what this tells you, among other things, is which of the galaxies out there you can reach if you flash a signal to them today. It says ANYTHING THAT IS TODAY NEARER THAN 15.4 BILLION LY is a target you can reach if you flash a message today, and it will get there WITHIN 88 BILLION YEARS. It also says that 88 billion years from now is when distances will be 100 times what they are today (cosmological distances, not dimensions of bound structures like a rock or solar system) So if you select a galaxy which is today 15.4 billion LY and you flash a message today, when the message finally gets there the distance to the galaxy (and the message arriving at it) will be 1540 billion LY. You can read that off the table too. Is there anyone to whom this does NOT make sense. This is a great calculator and an interactive version of the standard cosmic model that is in professional use (LCDM) and there must be plenty of people who can explain if you find anything obscure about the table. Everybody should get so they understand the table outputs of this calculator both of past history and of the future, IMHO. They are basic.
Mentor
P: 9,636
 It says ANYTHING THAT IS TODAY NEARER THAN 15.4 BILLION LY is a target you can reach if you flash a message today, and it will get there WITHIN 88 BILLION YEARS.
And anything beyond ~16 billion ly is unreachable?
If I understand this correctly, the range we can contact at all shrinks with ~1 ly / year (using the current distance). This reduces the reachable volume by ~3*10^21 ly^3 per year, more than the volume of our local group (according to WolframAlpha). Claustrophobia anyone? ;)
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P: 22,351
 Quote by mfb And anything beyond ~16 billion ly is unreachable? If I understand this correctly, the range we can contact at all shrinks with ~1 ly / year (using the current distance). This reduces the reachable volume by ~3*10^21 ly^3 per year, more than the volume of our local group (according to WolframAlpha). Claustrophobia anyone? ;)
Hi Mfb, I was glad to see your comment! In terms of what is called proper distance the reachable volume is growing. Its current radius is about 15.6 billion ly.
Its radius is expected to plateau at 16.3 billion ly.
So the reachable volume (just the volume of sphere with that radius) is growing and will plateau accordingly.

Note that the CEH is different from the Hubble radius. The Hubble radius is the distance that is growing at rate c. It is currently 13.9 Gly and the CEH (the reachable radius) is 15.6 Gly.
I think you know this but I'll say it just in case others read this.

There is the tricky idea of COMOVING distance, where everything and every galaxy is permanently assigned its present distance and keeps that like a tattoo for all its past and future life.
Comoving volume corresponds intuitively to "amount of matter".
Now because of expansion the number of galaxies within our CEH range---our 15.6 or eventual 16.3---is declining. If you keep a volume at a fixed proper distance size then stuff will leak out of it. So the amount of matter in our reachable sphere-shaped volume is declining. Even though in proper distance terms the volume is not.

So the reachable "comoving volume" (essentially referring to amount of reachable matter) is slated to decline almost to zero. Just an agglomeration of Milky and Andromeda surrounded by a big 16 Gly radius ball with not much in it.

Mentor
P: 9,636

## Look 88 billion years into future with the A20 tabular calculator

 If you keep a volume at a fixed proper distance size then stuff will leak out of it. So the amount of matter in our reachable sphere-shaped volume is declining.
That was my point. There are co-moving objects reachable today, but not tomorrow.
 PF Patron Sci Advisor P: 22,351 To see the Cosmic Event Horizon actually emerge in the output of this calculator we have it make a longer table with small step. What I showed in post#1 was a shortened form: http://www.einsteins-theory-of-relat...oLean_A20.html ===quote=== ... Red stuff is just the three standard parameters, estimated based on observation. No reason to change them, although in this calculator you CAN change them and play around to see the effects. The blue stuff is what I put in to give bounds and step size for the table I wanted it to generate From the present (S=1) to the distant future (S=0.01) when distances are 100 times what they are today. In steps of ΔS = 0.09. those are just what I chose. If you choose a smaller step size like ΔS = 0.01 you get a table with more rows, like around 100 rows instead of only 12 rows.... 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) 0.01 S value at the bottom row of table (S_lower smaller than S_upper) Step size (S_step) 0.09 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 13.769 13.896 0.000 0.000 0.910 1.099 15.104 14.387 -1.219 -1.339 0.820 1.220 16.630 14.829 -2.536 -3.093 0.730 1.370 18.374 15.221 -3.884 -5.320 0.640 1.563 20.402 15.545 -5.270 -8.234 0.550 1.818 22.772 15.812 -6.676 -12.138 0.460 2.174 25.618 16.006 -8.108 -17.627 0.370 2.703 29.120 16.143 -9.555 -25.825 0.280 3.571 33.629 16.233 -11.010 -39.323 0.190 5.263 39.934 16.278 -12.474 -65.650 0.100 10.000 50.390 16.296 -13.939 -139.393 0.010 100.000 87.919 16.300 -15.406 -1540.607 ===endquote=== One thing this tells us is that if we send a message to a galaxy expecting it to arrive when distances are 100 times what they are now then the galaxy has to be only 15.4 billion ly away. What is happening is that this distance is CONVERGING to about 15.6 billion ly, as we allow time to run to infinity and the expansion factor 100 to grow without bound. I have to go out for the evening so can't copy in the table. But try making the table yourself. Put in upper bound S = 1 lower bound S = 0.01 step size = 0.01 You will see the NOW distance plateau, level out, coverge towards what is around 15.6. the amount it changes decreases with each step. that is because the CEH is at 15.6. If we want to send a message today to a galaxy and have it get there NO MATTER HOW LONG IT TAKES then the galaxy can not be farther than 15.6 at this time. Have to go so must leave the post unedited but hope it's clear and someone will make the table and see the convergence beginning to happen. You can see that kind of thing in a table (with small steps) when you can't see it with a one-shot.
 PF Patron P: 700 I have done an experimental version of A20 with the distance of the cosmic event horizon added. It required increasing the number of integration step significantly, because it has to calculate far into the future, so the calculator becomes slower. It is not all that well tested, so I want to get reaction before finalizing and uploading it for direct access. Here is a screenshot for discussion. Due to the increased number of integration steps, some of the values are marginally different. Still evaluating for possible errors. I think we must carefully consider the descriptions of the (live) info popups so as to not cause confusion with the generalized meaning of distance columns. Edit: my proposal D_now "If positive: present proper distance of a source from which we now receive light with a wavelength stretch S. If negative: present proper distance of a target which will receive our present signals with a wavelength stretch 1/S. Proper distance is like measuring cosmic distances on a hypothetical 'freeze-frame' (no expansion) by means of radar, measuring rods, or similar." D_then "If positive: past proper distance (at emission) of a source from which we now receive light with a wavelength stretch S. If negative: future proper distance of a target which will receive our present signals with a wavelength stretch 1/S. See D_now info for definition of proper distance." D_hor "The cosmic event horizon, which is the largest distance (at time of emission) between an emitter and receiver that light can ever bridge. At larger distances, accelerating expansion prevents light from reaching the receiver." I will appreciate suggestions for improvement of clarity. Attached Thumbnails
 PF Patron Sci Advisor P: 22,351 It's great to have a Cosmic Event Horizon column! the CEH must surely be one of the most interesting dimensions of cosmology. I had assumed that the computational cost would be too great to include a calculation of it---that it would prohibitively slow the generation of the table. Is it really OK to include that much numerical integration? I guess it must be OK! Jorrie may have found a way to make the whole thing more efficient and thus practical. Right now I can't think of suggestions about wording, maybe someone else can suggest some clear concise phrasing to put in the "info" popups. AFAICS they are already pretty clear and helpful.
 PF Patron Sci Advisor P: 22,351 BTW the title should have said look 74 billion years into future, not 88 billion. The calculator goes out to year 88 billion of the expansion, but we are already nearly at year 14 billion. Careless error. Mainly here I want to copy a couple of Jorrie's posts that provide technical background to the A25 calculator. That way we will have them in this thread for reference and aren't so likely to lose track: ===quote Jorrie post#4066605=== The latest version (as in Marcus's signature) is CosmoLean_A20, which adds an 'Introduction' button with some hints for usage. It is supposed to be fairly stable now and it is perhaps time to give an idea of the underlying formulas and conventions. It follows the development of the 13.9/16.3 simplified model proposed by Marcus, but with inclusion of the early stage radiation energy density. The basic input parameters are: present Hubble time $Y_{now}$, long term Hubble radius $Y_{inf}$ and the redshift for radiation/matter equality $z_{eq}$. Since the factor $z + 1$ occurs so often, an extra parameter $S = z + 1$ is defined. From these, the Friedman equation terms for the cosmological constant, radiation and matter can respectively be found for a perfectly flat LCDM model. $$\Omega_\Lambda = \left(\frac{Y_{now}}{Y_{inf}}\right)^2, \ \ \, \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \, \Omega_m = S_{eq}\Omega_r$$ The 'heart' of any simple cosmological calculator is the time variable Hubble constant $H$, which comes from the Friedman equation as: $$H = H_0 \sqrt{\Omega_\Lambda + \Omega_r S^4 + \Omega_m S^3}$$ For perfect flatness, it can be expressed as $$H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}$$ It can be interpreted in terms of the "13.9/16.3 factors" as follows: $\Omega_\Lambda = 0.7272$ and $\Omega_m (1+S/S_{eq})= 0.2728$, which of course sum to 1 (required for perfect flatness). It also shows at a glance how the influence of the various energy densities changes with S. Since S_eq ~ 3350, radiation dominated when S > 3350 and matter dominated for S < 3350, until such time as $\Omega_m (1+S/S_{eq}) < 0.7272$, when the cosmological constant started to dominate the equation. From H, the following calculator outputs are readily available: Hubble time $$Y(a) = 1/H$$ Cosmic time $$T(S) = \int_{S}^{\infty}{\frac{dS}{S H}}$$ Proper distances to a source at stretch S, "now" and "then" respectively, $$D_{now} = \int_{1}^{S}{\frac{dS}{H}},\ \ \ \ D_{then} = \frac{D_{now}}{S}$$ The integration for T(S) to S_infinity is problematic, but is usually stopped at a suitably high S (effectively close enough to time zero). In principle, the equations can be used for projecting into the future as well. This has been "secretly" sneaked into version A20. If you want to try it out, enter 1 into S_upper and 0.1 into both S_lower and S_Step. Note the time going to some 50 Gy, T_Hubble to around 16.3 Gy and the distances to negative values. As Marcus has pointed out before, D_now for this scenario is the present distance to a target that will receive our signals with a wavelength stretch S at future time T(a). D_then means the proper distance of the target when they eventually receive our signal, obviously 1/S times farther. This 'trial feature' can go down to S = 0.01 in steps of 0.01, but not lower at this time. ==endquote== This link will get into the midst of this series posts: http://physicsforums.com/showthread....63#post4072363 They were written during successive stages of development, e.g. A20, A22,..A25, and describe useful features as they were added. The series of posts extends beyond the two quoted here. ==quote Jorrie post#4072363== Hand-in-hand with the 'future option' goes the cosmic event horizon. It has been included in CosmoLean_A22. For completeness, I'll repeat the prior post's equations together with D_CEH. Given present Hubble time $Y_{now}$, long term Hubble time $Y_{inf}$ and the redshift for radiation/matter equality $z_{eq}$. Since the factor $z + 1$ occurs so often, an extra parameter $S = z + 1 = 1/a$ is defined, making the equations neater. $$\Omega_\Lambda = \left(\frac{Y_{now}}{Y_{inf}}\right)^2, \ \ \, \Omega_r = \frac{1-\Omega_\Lambda}{1+S_{eq}}, \ \ \, \Omega_m = S_{eq}\Omega_r$$ Hubble parameter $$H = H_0 \sqrt{\Omega_\Lambda + \Omega_m S^3 (1+S/S_{eq})}$$ Hubble time, Cosmic time $$Y = 1/H, \ \ \, T = \int_{S}^{\infty}{\frac{dS}{S H}}$$ Proper distance 'now', 'then' and cosmic event horizon $$D_{now} = \int_{1}^{S}{\frac{dS}{H}}, \ \ \, D_{then} = \frac{D_{now}}{S}, \ \ \, D_{CEH} = \frac{1}{S} \int_{0}^{S}{\frac{dS}{H}}$$ This essentially means integration for S from zero to infinity, but practically it has been limited to $10^{-7} < S < 10^{7}$ with quasi-logarithmic step sizes, e.g. a small % increase between integration steps. ==enduote==
 PF Patron Sci Advisor P: 22,351 We are now up to version A27 of Jorrie's calculator. It has had a lot of improvements and maybe is nearing final version. I want to use it to answer a question. It is now year 13.7 billion of the expansion and the stretch factor of the CMB is about 1090---the redshift is always 1 less than the stretch ratio so if you like redshifts just subtract 1. What will the CMB stretch be in year 17 billion? What will the CMB stretch factor be in year 19 billion? And what will the radius of the SOURCE SHELL--the spherical surface of last scattering--be at those future times. Now according to Jorrie's calculator the present radius is 45.9 Gly. And when the light was emitted it was 42.1 Mly. I don't have time to explain but using the calculator I found that in year 17 billion the stretch will be 1362 and in year 19 billion it will be 1557. Now what is really interesting to me is the DISTANCE to the hot gas source matter back when the light was emitted. Demit Jorrie calculator says that distance for the PRESENT CMB source matter was 42.1 Mly. Not very far only 42 million light years. What about in the FUTURE? From the calculator I get that in year 17 billion Demit will be 44.8 million ly. That is, then we will be getting CMB light NOT as now from stuff that was 42.1 Mly from here, but from stuff that was 44.8 Mly. And in year 19 billion I see that Demit will be 46.1 Mly. I'll go thru the arithmetic of this later, have to do something else. But anyone interested can click on http://www.einsteins-theory-of-relat...oLean_A27.html and check it out. Put in 1 and .6 for the upper and lower limits of the stretch factor. Put in 0.1 for the step and you will see part of what I just said.
 PF Patron P: 5,505 Thank you for the chart and explanation in post #9. That gets my vote for the best post of all time....
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P: 22,351
 Quote by Naty1 Thank you for the chart and explanation in post #9. That gets my vote for the best post of all time....
I'm delighted you found it good! It's really thanks to Jorrie--I guess everybody realizes that by now. I don't know how he gets the calculator to generate charts like that, with different color curves. I have a vague feeling that he explained to me at one point how to get charts, but I didn't take it in (a limited capacity for new information )

The calculator is the "A27" link in the signature at the end of this post.

It's a really nice calculator.

I should explain the arithmetic used in my preceding post. The question was as follows.

Now (Year 13.7 billion) we get CMB stretched 1090-fold from matter that was 42 Mly away.
The question was what about in future, like in Year 17 billion or 19 billion?
The answers calculated with the Jorrie online device were:

In Year 17 billion we'll be getting CMB stretched 1362-fold from matter that was 44.8 Mly away.

In Year 19 billion we'll be getting CMB stretched 1557-fold from matter that was 46.1 Mly away.

The source shell for the CMB gradually inches out--the distance of that matter from our matter at the epoch of the brief flash has to be larger. Because we GOT the flash from today's source shell and tomorrow we will be getting the flash from a slightly more distant(on average) shell which of course took a day longer to get here.

So how were those numbers calculated?
===========================

first off you put upper limit = 1 and lower limit say .6, and step .1
that tells you that stretch = 1 corresponds to NOW i.e. Year 13.755 billion.
and stretch = .8 corresponds to Year 17 billion
while stretch = .7 corresonds to Year 19 billion.
Then you just say 1090/.8 = 1362
and 1090/.7 = 1557
Does that make sense? If anybody is reading who wants that explained please say.
===========================

And the other thing is to find the distance to the matter back then at time of emission. We can find Dnow and then divide by 1090. By Dnow I mean D13.755.
Think of us as a waystation. The light from the flash has already traveled 45.9 when it passes us and to get to the people in Year 17 billion it still has to how far? In Dnow terms.
If you did that tabulation with step = .1 I suggested earlier, you know that the two distances we need are 2.9 and 4.4
those are the addition distances the light has to travel to get to the people in Years 17 and 19 billion. But it already traveled 45.9 by the time it got to us. (In Dnow terms.)

(45.9+2.9) billion/1090 = 44.8 million
(45.9+4.4) billion/1090 = 46.1 million

That's where the numbers in the earlier part of the post came from.
Again, this may not be a sufficient explanation so if puzzled by anything please ask.

I think the main thing is to click on the calculator and actually put in upper=1, lower=.6, step=.1 and look at the resulting tabulation and think about what the numbers you see in the table mean. that is, the stretch factors .8 and .7, and the times 17 and 19 billion, and the now-distances 2.9 and 4.4

Plus earlier if you were unfamiliar with the now distance 45.9 Gly to the flash source matter, you might have put in upper=1090, step = 0 (a quick way to get a table with only one row)
 P: 694 Hi Marcus, thanks for this very interesting calculator. I just wanted to ask about the Demit number is this the Dthen parameter in the calculator? Also would you be able to set up a version of this with all the necessary parameters in place for the year 88 Billion? It is just that I am not sure which and how many parameters to change. I am just curious what distance the radiation first was that we will receive it then. It would be interesting to see a 20 point chart of how this distance changes from say year 0 to 88 Billlion. Thanks again, great job Marcus.
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P: 22,351
 Quote by Tanelorn Hi Marcus, thanks for this very interesting calculator. I just wanted to ask about the Demit number is this the Dthen parameter in the calculator? Also would you be able to set up a version of this with all the necessary parameters in place for the year 88 Billion? It is just that I am not sure which and how many parameters to change. I am just curious what distance the radiation first was that we will receive it then. It would be interesting to see a 20 point chart of how this distance changes from say year 0 to 88 Billlion. Thanks again, great job Marcus.
It's great you like it! It's really an extraordinarily cool online device IMHO. Jorrie gets all the credit. Actually it's a credit to Physicsforums that the forum provides an environment for stuff like this to happen. Just to be clear, I personally had no hand in the building of the calculator. I'd like to encourage it's wider use.

The calculator is the "A27" link in my signature at the end of this post.

If you want a table that goes from NOW (stretch factor = 1) to Year 88 billion (stretch factor 100) and which has, say 21 rows,
then you just have to type in upper=1, lower=.01, and step = -20

There are popups that will appear to help interpret the numbers in the table. Jorrie put them in, as he did with everything.

I'll say in my own words Dnow is the distance between us and the other matter, today (if you could stop expansion to allow for it to be measured.)
If the Dnow number is positive, think of a signal from the other matter coming towards us
If the Dnow number is negative, think of the signal from us to the other stuff, imagine creatures there to receive it

Dthen is the distance between us and the other matter on the day when the light was emitted (if distance positive) or received (if distance negative).

If Dthen is positive that means the signal is coming towards us from the other matter, so "then" means when it was emitted by the other matter.
If Dthen is negative it means the signal is going away from us towards the other matter, so "then" means when the other "people" receive it.

Far in the future they will be be very far away when they finally recieve our message, because of all that expansion while the signal is en route to them. You can see that if you just put in the suggested numbers upper=1 lower=.01 step=-20.
(or however many steps in the table you want, 10, 20, 30...whatever).

The HORIZON distance in the last column of the table is something I find fascinating. If someone checks out the other stuff and then gets to wondering about that column, it might be fun to discuss some.
 P: 694 Thanks Marcus, Sorry I am still having trouble. I was looking for a column which shows the distance then from us to where the CMBR light was emitted and how this distance changes with time from t=0 to t=88 Billion years. None of the columns seem to have the 41 Million light years distance in there.
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 Quote by marcus If you did that tabulation with step = .1 I suggested earlier, you know that the two distances we need are 2.9 and 4.4 those are the addition distances the light has to travel to get to the people in Years 17 and 19 billion. But it already traveled 45.9 by the time it got to us. (In Dnow terms.) (45.9+2.9) billion/1090 = 44.8 million (45.9+4.4) billion/1090 = 46.1 million That's where the numbers in the earlier part of the post came from. Again, this may not be a sufficient explanation so if puzzled by anything please ask.
Marcus, the latter two values are correct as defined for the calculator, but they may need a little more explanation. When we speak about distances "now", we measure with our local yardstick, which is essentially the local speed of light multiplied by the time it would take light to traverse the distance, provided that we could 'freeze' the expansion of space now (as you have said).

When we speak about distances "then" we use the same yardstick (our present one), but we 'freeze' the expansion at the time of emission. For the past, that was obviously "then" and not "now", because we are not the emitting party. For the future, we are the emitting party, so we freeze the expansion "now" and and D_now is how far light has to travel to reach the other civilization in a static cosmos.

Tanelorn's question may arise from the fact that for the future D_then, we have to freeze the expansion when the light reaches the other civilization and find out how long the light had to travel in an expanding cosmos. So, it is important to note that one cannot always equate D_emit with D_then. This is a price we pay for having the past and the future together in one table.
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 Quote by Jorrie Marcus, the latter two values are correct as defined for the calculator, but they may need a little more explanation...
I am just so pleased with this calculator! It's versatile like a swiss jackknife, you can do a lot of things with.

And that leads to opportunities to explain more stuff, like what you just did. For instance the idea of proper distance (distance at a certain moment with expansion stopped).

Also I guess I should be restrained and not try to bring in TOO MUCH. Introduce concepts slowly. But it's tempting to also mention the idea of the COMOVING DISTANCE which is simply Dnow. The comov. distance of some matter is a way of tagging the matter with a serial number that never changes. So it can be useful. By convention it is the now distance (as of today).

When I did those simple calculations before, conceptually I was working in comoving distance, and then at the end, converting to proper distance at "recombination time" simply by dividing by 1090. That is always how one converts comoving distance to distance at the moment the CMB flash was emitted.

==quote==
... suggested earlier, you know that the two distances we need are 2.9 and 4.4
those are the addition distances the light has to travel to get to the people in Years 17 and 19 billion who are going to receive it. But it already traveled 45.9 by the time it got to us. (In Dnow terms.)

(45.9+2.9) billion/1090 = 44.8 million
(45.9+4.4) billion/1090 = 46.1 million
==endquote==

I guess I am actually talking to people who might be reading (not to you Jorrie). You have to click on the "A27" link and put in something like
upper=1
lower=.6
step=.1
and actually see that for those Years the Dnow distances 2.9 and 4.4

So after the light has covered a comov. distance of 45.9, and has passed the way station (us) and is racing on towards the people who are to receive it in Year 17 billion,
then how far, in comov. terms, does it have to travel?

Well it is just as if we sent them a message today--a flash of a different color to travel neck and neck all the way to them. And the comov. distance from us to them is 2.9

So to get the total comov. diet you just add 45.9 and 2.9. That is the comov. dist between them and the source matter of the CMB which they will see in Year 17 billion.

Now since it is the comov. distance you just have to divide by 1090, and bingo.

Jorrie forgive me if I'm revealing too much enthusiasm, but AFAIK this is unique. I don't know of any online cosmo calculator that has future as well as past, and none of the others give the Event Horizon. In fact they don't do half the things this one does. There's only one other I've seen that tabulates. (A recent one by someone at Oxford in the UK, as I recall.)
The trouble is we have a distribution bottleneck. More beginning astronomy students should get to use it. I don't know how to get the word out. Maybe there are some active academics here at PF who would be willing to pass the link along to colleagues in the astronomy department.

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