Hubble's Deep Field View in the long long Future

[PhanthomJay]

I don't know why I let these things bug me, so I figure I'll ask the experts here.

We've all seen those Hubble photos of the galaxies in their early formations some 12 billion years ago or so.

If Hubble is somehow still around in the year 12 billion AD, and indeed, we assume man is still alive, what will its deep field photo show to those humanoids?

Probably a simple answer, but I don't find Cosmology simple.

Thanks.

 

[marcus]

...If Hubble is somehow still around in the year 12 billion AD, and indeed, we assume man is still alive, what will its deep field photo show to those humanoids?
...

It's an interesting question. To get a realistic idea you have to take account of the redshifting of light from distant galaxies which eventually (I'm thinking way beyond 12 billion AD) could make their light difficult if not impossible to detect and focus into images. Other people here will hopefully contribute more details that can help us imagine.

Actually 12 billion AD is not the really longterm future that people like to write about and discuss. I'll check, but my immediate guess is that there still will be a deep field for the humanoids you mentioned to look at and admire. Jorrie's calculator makes tables of past and future (redshift and distance) conditions. So we can easily consult that specifically about 12 billion AD.

But also it will tell us about times much farther out like say year 50 billion of the expansion (sort of 36 billion AD) in case you are curious about that.

 

[PhanthomJay]

Marcus, thanks for the reply.

Whether it is 12 or 36 or 50 billion AD, when the humanoids look at and admire those photos, they won't be able to see any of the baby formative galaxies that we see today, correct? ...because they will have long since developed into into mature galaxies, and the light from their infant stages will have disappeared beyond their observable universe, right? Or wrong?

 

[phinds]

Marcus, thanks for the reply.

Whether it is 12 or 36 or 50 billion AD, when the humanoids look at and admire those photos, they won't be able to see any of the baby formative galaxies that we see today, correct? ...because they will have long since developed into into mature galaxies, and the light from their infant stages will have disappeared beyond their observable universe, right? Or wrong?

There are not likely to be any forming galaxies in the distant future but you can go out VERY far and there will still be solar systems forming and you can go out a very long way before the Milky Way becomes an island galaxy (perhaps still with some of the Local Group) but eventually, there will be NO galaxies in the observable universe (again, with the possible exception of the Local Group).

 

[marcus]

Marcus, thanks for the reply.

Whether it is 12 or 36 or 50 billion AD, when the humanoids look at and admire those photos, they won't be able to see any of the baby formative galaxies that we see today, correct? ...because they will have long since developed into into mature galaxies, and the light from their infant stages will have disappeared beyond their observable universe, right? Or wrong?

It's a very good question. Type of question where you don't want to jump to conclusions too soon and probably should approach it gradually.

When we look at baby formative galaxies today, we are already in many cases using instruments that can "see" in the infrared that is invisible to our human eyes. Because their starlight has been redshifted so much that it is longer wavelength heat glow, not ordinary visible star-light.

So let's IGNORE that and talk as if the redshift lengthening of wavelengths is not a problem!

What deep field stuff could they be seeing if they have good instruments, or highly evolved eyes?

I think if you are just talking about 36 billion AD or year 50 billion then they STILL get to see baby galaxies in formation. Just ones that were FARTHER AWAY when they emitted the light that will be arriving here to those humanoids of year 50 billion.

I would like to show you Jorrie's "Lightcone" calculator that makes tables spanning standard model past and future. See if you can teach yourself how to use it.
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html

When you go there you get a short table that runs from the origin of the CMB (before there were any stars or galaxies, just hot gas) out to very distant future. What I want to do is explain how to CHANGE the table so that it has MORE STEPS and also does not go back so far into the past.
I want it to be more near-present focused and not take such big steps.

One way to do that is by clicking on the button that says "sample chart range" and then "calculate". That gets you a new table that has 60 steps and runs from the time when distances were 1/40 of their present size, to the present (labeled S = 1) and then out to a future time when distances will be 2.5 times present size.

Give yourself time, go at it gradually. Do you see in the S column how it goes from 1/40 to 2.5?
Do you see in the T column how it goes from year 67.4 million up to present year 13.8 billion and then out into future, to year 28.6 billion?

When you have absorbed the basics about that table, what some of the columns mean (it really is pretty basic but there is a lot of information there, so ask questions) I want to show how to CHANGE the table so that it shows the past history from the viewpoint of a humanoid in year 50 billion, or some future year like that.

The "Lightcone" calculator is somewhat neat in that it does not merely make tables of cosmic history from our present perspective. It can be used to make tables showing how things would have stacked up seen from some ancient animal's perspective back in past, or how things will look from some future being's perspective.

Let me know if you give Jorrie's calculator a try and if some of the columns make sense to you, and if you want to proceed.

The reason for my response is that I think the question you asked is a very good one and looking at a cosmic history table (what we can see, at what distance, at what distance stretch factor, as it was in what year when it emitted the light...) is, I think, the best way to understand the answer to that question.
better than trying to think about it in merely verbal generalities.

Hey! I just noticed you are a Sci Advisor and HW helper! I was talking like you were a Noob. :rofl: This is all easy stuff for you! We don't have to take a gradual approach to this.
Duh. I should read who the people are that are asking questions and not give such a generic response.

 

[marcus]

So let's take a shortcut. Open the calculator
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html
Change "upper row S" to 10
and "lower row S" to 0.5
and "calculate"

That gives you a brief compact history from when distances were 1/10 present size out to when they will be 2 times present size. (you can read those ratios in the scale factor "a" column)

Notice that this table now runs from year 545 million out to year 24.8 billion.

Let's look at cosmic history from standpoint of humanoid in year 24.8 billion, that is, somebody 11 billion years in future.

The thing to do is to jot down his R, his Hubbleradius, which is 16.84 billion lightyears. This is the main key to seeing things from his perspective.
But as a finepoint we also should double the 3400 entry to make it 6800. (This is the scale ratio that takes one back to when radiation and matter energy densities were equal. Because he is out in future when distances are twice ours, he has to refer back by twice the factor to reach the epoch of radiation-matter balance.)
But that is a fine point. The main thing is CHANGE THE 14.4 TO 16.84. His Hubbleradius will be 16.84 while ours, the present one, is only 14.4.

Those two changes, up at the top, put us in his world, or timeframe.

Then we can just press calculate, or first make whatever changes we want in the RANGE and number of steps in the table, and the press calculate. I choose to set "lower row S" to equal 1, so that we are looking at his past. S=1 now means his present. So the table will be purely retrospective. But you could select different ranges. and have more steps.

 

[marcus]

This is what I get when I make the two changes just mentioned and also let the range be from S=20 to S=1 in 17 steps. That means from year 545 million (approx the same baby galaxy stage) to his present of year about 24.8 billion. That is, 11 billion years from now. Some roundoff error because I used only approximate figure for his hubbleradius R.$${\small\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 16.84&17.3&6800&58.1&0.948&0.052\\ \hline \end{array}}$$ $${\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.050&20.000&0.5455&0.8199&77.144&3.857&4.716&4.58&4.70\\ \hline 0.060&16.769&0.7109&1.0673&74.124&4.420&5.445&4.40&4.14\\ \hline 0.071&14.059&0.9260&1.3887&70.829&5.038&6.260&4.21&3.63\\ \hline 0.085&11.788&1.2059&1.8051&67.235&5.704&7.161&3.99&3.16\\ \hline 0.101&9.883&1.5693&2.3427&63.322&6.407&8.146&3.76&2.73\\ \hline 0.121&8.287&2.0405&3.0325&59.069&7.128&9.202&3.51&2.35\\ \hline 0.144&6.948&2.6491&3.9088&54.463&7.839&10.312&3.23&2.01\\ \hline 0.172&5.825&3.4310&5.0036&49.500&8.498&11.447&2.94&1.70\\ \hline 0.205&4.884&4.4266&6.3359&44.201&9.050&12.568&2.62&1.43\\ \hline 0.244&4.095&5.6773&7.8927&38.619&9.431&13.627&2.29&1.19\\ \hline 0.291&3.433&7.2177&9.6081&32.852&9.568&14.573&1.95&1.00\\ \hline 0.347&2.879&9.0654&11.3547&27.050&9.397&15.366&1.61&0.83\\ \hline 0.414&2.414&11.2118&12.9715&21.396&8.865&15.984&1.27&0.68\\ \hline 0.494&2.024&13.6213&14.3227&16.072&7.942&16.434&0.95&0.55\\ \hline 0.589&1.697&16.2404&15.3483&11.218&6.612&16.739&0.67&0.43\\ \hline 0.703&1.423&19.0126&16.0665&6.908&4.856&16.935&0.41&0.30\\ \hline 0.838&1.193&21.8886&16.5402&3.159&2.648&17.055&0.19&0.16\\ \hline 1.000&1.000&24.8318&16.8399&0.000&0.000&17.124&0.00&0.00\\ \hline \end{array}}$$

The printable version is obtained after you make any desired changes, like to the range and number of steps, by ticking the button that says Tex small, and pressing "calculate" again. That gives you Tex code for the table that you can immediately copy and paste into a post.

You can examine the table without prompts but in case others are reading the thread I want to mention that he sees baby galaxies back in year 545 million just like we do BUT the ones HE sees are forming from matter which was at a distance of 3.86 Gly from solar system and Milkyway matter when it emitted the arriving light. Whereas the baby galaxies which WE see (from the same historical era) are forming from matter which was 3.07 Gly from solar system and Milkyway matter when it emitted the light

 

[PhanthomJay]

Marcus, this is too weird for me sorry. If you think Science Advisors know everything there is to know abut anything, I am an exception, and still proud enough to admit it.
The humanoid of the year 12 billion sees forming galaxies that we in the year 2013 do not see because its light has not yet had time to reach us, I gather. That is to say, the light from that infant galaxy is presently beyond our observable universe, but given a few billion more years, it will be part of the observable universe. The observable Universe will have thus grown in size.

But i now must focus my attention on phinds response, which I quote:

(snip) ...eventually, there will be NO galaxies in the observable universe (again, with the possible exception of the Local Group).

Impying, again I humbly presume, that in the year zillion zillion and 25, if man/woman is still alive, he or she, with his/her Hubble eyes, will peer out far and see well er...nothing... but blackness, thus the observable universe will have reduced in size.

Please explain this contradiction.

Thank you.

 

[phinds]

Impying, again I humbly presume, that in the year zillion zillion and 25, if man/woman is still alive, he or she, with his/her Hubble eyes, will peer out far and see well er...nothing ... but blackness

I did not think I was IMPLYING that, I thought I was saying it directly and specifically, since it is the case.

thus the observable universe will have reduced in size.

No, the observable universe will have increased slightly in size, there just won't be anything in it to observe.

 

[PhanthomJay]

I did not think I was IMPLYING that, I thought I was saying it directly and specifically, since it is the case.

I just wanted to be sure that I understood this correctly, thanks.

No, the observable universe will have increased slightly in size, there just won't be anything in it to observe.

If there is nothing to observe, then how do you know the observable universe has increased in size? I mean like heck all you will see is a lone Milky Way star or two that presumably has not burnt out, and, being gravitationally bound, is only a mere few thousand light years away, and no red shifted or whatever stuff beyond that to 'measure' . Now wait a sec...will you still be able to see (or rather, detect) the CMBR?? This will be key to my understanding, thank you.

 

[phinds]

If there is nothing to observe, then how do you know the observable universe has increased in size?

I take the definition of "observable universe" to be that space for which we can observe anything that is emitting radiation. The fact that there isn't anything emitting radiation doesn't change the distance at which we could see it if it WAS there.

I mean like heck all you will see is a lone Milky Way star or two that presumably has not burnt out, and, being gravitationally bound, is only a mere few thousand light years away, and no red shifted or whatever stuff beyond that to 'measure' . Now wait a sec...will you still be able to see (or rather, detect) the CMBR?? This will be key to my understanding, thank you.

Actually, the other galaxies in the local group are MILLIONS of LYs away, not thousands and everything in the Milky Way and the local group will still be there.

My understanding, and if I'm wrong I hope more knowledgeable members will jump in here, is that there will be no CMBR in the far distant future. If there were to be a collapse and then re-emergence of civilization, astronomers would have no way of knowing that they are in an expanding universe or that there is anything other than the local group in all of space.

 

[PhanthomJay]

It is starting to make some sense now, thank you. As I anxiously await any additional response, I've got to get back to my job now because Dilbert and I are already months behind our deadline and guess what...over budget.

 

[marcus]

Hope you are kidding about the project status (behind sched. and over budg.)
Sorry to hear that the standard cosmic model stuff strikes you as weird.
In case other people are reading who want to get into the issue of the longterm size of the observable, at quantitative level, I'll briefly sketch.

In terms of comoving distance (present distance of objects so effect of expansion ignored) the size of observable is expected to increase towards a limit of about 63 billion ly.

The radius of the observable universe is called the "particle horizon" and it is currently about 46 billion ly. this is increasing and is expected to approach a longterm limit of 63 (in present distance terms)

So all the galaxies which are at this moment less than 46 Gly from us are in our observable region. Those that are 47 Gly from us are NOT (yet) but they will be. In fact any galaxy that is now at this moment less than 63 Gly from us eventually WILL be in our observable universe.

Of course the farther away matter is the earlier stages we see it in, so matter very far away is seen before it coalesced into stars and galaxies. But that matter stays in our observable region and eventually we will be getting light the galaxies that form out of it. BUT

The kicker is redshift. As light from the far outlying matter gets redshifted it becomes harder to detect! So there is a PRACTICAL limit. In the very long term we will only have the galaxies of the Local Group that we can see with ordinary visible light.

We will still have as many galaxies in our observable universe as we do now, in fact more, but their light will be way redshifted into the infrared, and eventually even longer wavelengths. It is hard to put a limit on what will be detectable with future instruments.

If anyone is quantitative-minded and wants to see where that figure of 63 Gly comes from, just go to http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html and click on "column selection and definition".
You will see that the "D_par" box has not been checked. It has an X instead of a check mark. Click on the X and the Particle Horizon will be selected, marked with a √ checkmark.
Then press "calculate".

You will see that in future when distances are 100 times what they are today the radius of the observable will be 6300 billion ly. Divide by 100 to get the presentday proper distance from us of that same matter. That is its so-called "comoving" distance, distance-adjusted-for-expansion, namely 63 Gly.

 

[marcus]

Here's what you get. Just so it would print properly in PF, I deselected D_hor to make room for the particle horizon column D_particle. But if you aren't planning to paste the table into PF then you don't have to even do that.
Just go to "Lightcone" calculator, click on "column selection and definition", check the box for "D_particle" and press "calculate".

What you see is that in far future when distances are 100 times present the matter that will then be in our observable region (and that we can in principle be getting some kind of signal from) will be at a maximum of about 6300 Gly from here. So right now that matter is at a maximum distance of around 63 Gly.

Light from that matter is already today within the so-called "cosmic event horizon" range (currently 16.5 Gly) so although it may take a very long time to get here (because of expansion) it will eventually reach us. That is why that matter, although it is not YET in our observable universe, eventually will be included in it.

$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{par}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.001&1090.000&0.0004&0.0006&45.332&0.042&0.001&3.15&66.18\\ \hline 0.003&339.773&0.0025&0.0040&44.184&0.130&0.006&3.07&32.87\\ \hline 0.009&105.913&0.0153&0.0235&42.012&0.397&0.040&2.92&16.90\\ \hline 0.030&33.015&0.0902&0.1363&38.052&1.153&0.249&2.64&8.45\\ \hline 0.097&10.291&0.5223&0.7851&30.918&3.004&1.491&2.15&3.83\\ \hline 0.312&3.208&2.9777&4.3736&18.248&5.688&8.733&1.27&1.30\\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&46.279&0.00&0.00\\ \hline 3.208&0.312&32.8849&17.1849&11.118&35.666&184.083&0.77&2.08\\ \hline 7.580&0.132&47.7251&17.2911&14.219&107.786&458.476&0.99&6.23\\ \hline 17.911&0.056&62.5981&17.2993&15.536&278.256&1106.893&1.08&16.08\\ \hline 42.321&0.024&77.4737&17.2998&16.093&681.061&2639.026&1.12&39.37\\ \hline 100.000&0.010&92.3494&17.2999&16.328&1632.838&6259.262&1.13&94.38\\ \hline \end{array}}$$

 

[PhanthomJay]

Well OK Marcus you are genius in this stuff so pardon my ignorance but here goes:
You know those infant galaxies we see in the Hubble photo...the ones we are talking about that formed a few hundred millions years after the big bang , the light from which is reaching us today 12 billion years later or so... what I want to know is this:

Since that 'baby' is now 12 billion years old, where is this 'ancient' galaxy NOW, as measured by my clock. ? Is it part of the unobservable universe, such that we don't know whether it still exists or has been eaten up by a space dragon or whatever...and when we determine the mass of the universe, well, this baby has none it is only light waves...so it's not counted as part of the total mass of the Universe, right? Does anyone understand my question?? How much of the universe we 'see' is really both 'stuff' and light from same, and how much is just light waves or radiation from the distant past. No need to get into dark matter and dark energy, please.
I know the sun is 'stuff, because 9 minutes after I see its dawn's early light, it is still shining so it must be still there. But what about say Alpha Centauri... its only 4 light years away..but nonetheless, what definitive proof can you give me that say 2 years from now, it is still there as 'stuff' and not as light?

 

[TumblingDice]

I know the sun is 'stuff, because 9 minutes after I see its dawn's early light, it is still shining so it must be still there. But what about say Alpha Centauri... its only 4 light years away..but nonetheless, what definitive proof can you give me that say 2 years from now, it is still there as 'stuff' and not as light?

There isn't any difference between the Sun and Alpha Centauri in your example except distance. We have no definitive proof of the status of the Sun for nine minutes. Longer distances just increase the time required for light and gravity to traverse.

 

[Drakkith]

Jay, while I don't know the math, it is trivial for someone who does to say whether or not the galaxy in question is inside or outside of the observable universe. I don't understand your question about the mass. We can observe it, which means we can estimate its mass. It does not exist merely as light waves any more than the Sun does. The galaxy does not simply disappear.

 

[PhanthomJay]

Jay, while I don't know the math, it is trivial for someone who does to say whether or not the galaxy in question is inside or outside of the observable universe. I don't understand your question about the mass. We can observe it, which means we can estimate its mass. It does not exist merely as light waves any more than the Sun does. The galaxy does not simply disappear.

that was a real poorly worded question and example,sorry.
I didn't mean to talk about 'disappearing ' galaxies but rather 'unobservable' galaxies . I have been incorrectly assuming that ' faster than light' space time expansion would ultimately mean that galaxies that are presently observable , where 'observable ' has been heretofore defined , would one day become part of the unobservable universe. I now realize that quite the opposite is true: galaxies presently unobservable will one day become observable, and galaxies presently observable will always be observable in the future. Thanks all for your help.

 

[phinds]

To Marcus in particular but to anyone who can answer me, here's a question arising out of this thread:

Taking the definition of "observable universe" as being that volume of space from which we COULD, theoretically at least, see radiation from objects present in that space, all the way out to the edge of it. The current size of the observable universe is about 47 billion light years in radius. It will increase somewhat in the future, to about 65 billion light years in radius.

Objects at the edge of the current observable universe are receding from us at about 3c and will continue to recede from us faster and faster, so that at some arbitrary (but calculable) time in the future, those object will be, say, 1,000,000 BILLION light years away from us.

SO ... my conclusion was that eventually there will be no objects in our observable universe because it will have a radius of 65 billion light years and everything but the local group will have left it.

Do I now understand correctly, based on this thread that everything above is correct up to but not including, my conclusion and that in fact the correct conclusion is that the radius of the observable universe will continue to grow and at the arbitrary time mentioned above, it will have a radius of 1,000,000 billion light years and will continue to grow further as time goes on ?

EDIT: and I should add, it is now my understanding that the 65 billion light years is NOT the future size of the observable universe but rather is the radius of the "now" universe (and yes, I understand that "now" is tricky) that will be included in the future observable universe, which itself will have an arbitrarily large "then" radius depending on how far in the future "then" is.

 

[marcus]

EDIT: and I should add, it is now my understanding that the 65 billion light years is NOT the future size of the observable universe...

That is right. What you call 65 Gly is the distance NOW of matter which we will eventually be getting light from (matter which will eventually, at least in principle, be observable by its CMB emission or neutrinos or whatever).

But by the time the news from that matter comes in, by the time the "ancient light" from it arrives, it will be much farther away. Hundreds of times. In the standard cosmic model we assume that expansion will continue indefinitely---we know of no mechanism that would end or reverse the process of expansion. So in proper distance terms the size of the ultimately observable region (matter which is NOW 65 Gly from us) will grow indefinitely.

When distances are 100 times what they are today, the radius we are talking about will be 6500 Gly, and so on.

I've been calling it 63 Gly so I would be saying 6300 Gly but it doesn't matter. It's only known approximately. You know what I mean and I know what you mean.

At any moment in history, the radius of the observable (proper distance, i.e. if you could stop expansion at that moment to allow measurement) is called particle horizon. So it is what gets tabulated as "D_par" in the Lightcone calculator.$${\small\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly) \\ \hline 0.001&1090.000&0.0004&0.0006&45.332&0.042&0.057&0.001\\ \hline 0.003&339.773&0.0025&0.0040&44.184&0.130&0.179&0.006\\ \hline 0.009&105.913&0.0153&0.0235&42.012&0.397&0.552&0.040\\ \hline 0.030&33.015&0.0902&0.1363&38.052&1.153&1.652&0.249\\ \hline 0.097&10.291&0.5223&0.7851&30.918&3.004&4.606&1.491\\ \hline 0.312&3.208&2.9777&4.3736&18.248&5.688&10.827&8.733\\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&16.472&46.279\\ \hline 3.208&0.312&32.8849&17.1849&11.118&35.666&17.225&184.083\\ \hline 7.580&0.132&47.7251&17.2911&14.219&107.786&17.291&458.476\\ \hline 17.911&0.056&62.5981&17.2993&15.536&278.256&17.299&1106.893\\ \hline 42.321&0.024&77.4737&17.2998&16.093&681.061&17.300&2639.026\\ \hline 100.000&0.010&92.3494&17.2999&16.328&1632.838&17.300&6259.262\\ \hline \end{array}}$$

 

[phinds]

Thanks for that clarification, Marcus. I have been making incorrect statements on this forum because of my misunderstanding so I particularly appreciate your setting me straight so I won't do that any more (at least on THIS subject )

 

[marcus]

No problem, you make a whole lot of very helpful correct statements about cosmology in particular. I didn't notice what you are talking about and anyway we are clearing it up. I don't know how it goes with you but I find the Lightcone (history table calculator) enormously useful. Also it uses parameters approximately those from Planck mission reported this year, so up to date. I just used it to kind of put a magnifying glass on the last two rows of the previous table. Let's take a close look at the D_par behavior far in the future:

$${\small\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly) \\ \hline 41.667&0.024&77.2040&17.2998&16.086&670.258&17.300&2597.934\\ \hline 45.479&0.022&78.7184&17.2999&16.121&733.165&17.300&2837.211\\ \hline 49.640&0.020&80.2330&17.2999&16.153&801.827&17.300&3098.380\\ \hline 54.182&0.018&81.7476&17.2999&16.182&876.772&17.300&3383.444\\ \hline 59.139&0.017&83.2622&17.2998&16.209&958.574&17.300&3694.590\\ \hline 64.550&0.015&84.7766&17.2999&16.233&1047.860&17.300&4034.203\\ \hline 70.456&0.014&86.2912&17.2999&16.256&1145.315&17.300&4404.889\\ \hline 76.902&0.013&87.8058&17.2999&16.276&1251.686&17.300&4809.491\\ \hline 83.938&0.012&89.3202&17.3000&16.295&1367.790&17.300&5251.110\\ \hline 91.618&0.011&90.8348&17.2999&16.313&1494.517&17.300&5733.135\\ \hline 100.000&0.010&92.3494&17.2999&16.328&1632.838&17.300&6259.262\\ \hline \end{array}}$$

I am zooming in. There will be a little jitter from roundoff error as we encounter the calculator's limits of precision:$${\small\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\small\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&D_{par}(Gly) \\ \hline 90.909&0.011000&90.7006&17.30&16.311&1482.826&17.300&5688.665\\ \hline 91.780&0.010896&90.8654&17.30&16.313&1497.192&17.300&5743.309\\ \hline 92.659&0.010792&91.0303&17.30&16.315&1511.695&17.300&5798.476\\ \hline 93.546&0.010690&91.1952&17.30&16.316&1526.338&17.300&5854.171\\ \hline 94.442&0.010589&91.3600&17.30&16.318&1541.121&17.300&5910.400\\ \hline 95.346&0.010488&91.5249&17.30&16.320&1556.045&17.300&5967.167\\ \hline 96.259&0.010389&91.6899&17.30&16.322&1571.112&17.300&6024.478\\ \hline 97.181&0.010290&91.8548&17.30&16.323&1586.324&17.300&6082.337\\ \hline 98.112&0.010192&92.0197&17.30&16.325&1601.681&17.300&6140.751\\ \hline 99.051&0.010096&92.1845&17.30&16.327&1617.185&17.300&6199.724\\ \hline 100.000&0.010000&92.3494&17.30&16.328&1632.838&17.300&6259.262\\ \hline \end{array}}$$

 

[marcus]

what is 6199.724 divided by 99.051? That would be the distance NOW of matter which will be on the limit of the observable region for those people living in year 92.1845 billion when distances are 99.051 what they are today.
Heh heh we are pushing the limits of Jorrie's numerical programming way past any reasonable expectation. The calculator will probably scream I will do the division:
6199.724/99.051 = 62.591...

Well well! There seems to be convergence!

The distance NOW of matter which will eventually be on the limit of the observable for people far in future does not change very much when you look way ahead using standard cosmic model.
It is always around 62.59 billion lightyears.

There is convergence to something around that figure. Jorrie's calculator behaved pretty darn well in that experiment.

I would guess if you divide 6140.751/98.112 you will also get something around 62.59.

I've been calling this quantity 63 billion ly, but maybe it would be OK to push the precision a little and say 62.6 billion ly (if we are assuming the March 2013 Planck mission report parameters or something like that---what you see in the top boxes of the calculator printout are the cosmic model parameters used.)