# Travelling against expansion

First question. Suppose you send a rocket with initial speed of c/10 and no further propulsion or gravity to a distant galaxy moving away due to space expansion at a speed of c/3. Will the rocket reach the galaxy?
Temptative answer. If we call f(t) the fraction of the distance covered at time t. Then f(0)=0 and f will be positive soon after take off. So initially f grows. I guess f will grow for ever. So either f will reach a limit somewhere and the rocket will hang at say 1/3 between earth and the galaxy. Or the rocket will smash its goal. which of the two will happen?

Second question. The speed due to expansion is proportional to the distance in between. This suggests that distance and speed increase exponential. So even an initial speed of c/2 doesn't guarantee the rocket in the first question to reach the galaxy because the galaxy might be running even faster before the rocket comes close. Is this reasoning correct?

aboro

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If I send a car going 30 mph towards a car going away from it at 50 mph will it ever catch up?

marcus
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First question. Suppose you send a rocket with initial speed of c/10 and no further propulsion or gravity to a distant galaxy moving away due to space expansion at a speed of c/3. Will the rocket reach the galaxy?
Temptative answer. If we call f(t) the fraction of the distance covered at time t. Then f(0)=0 and f will be positive soon after take off. So initially f grows. I guess f will grow for ever. So either f will reach a limit somewhere and the rocket will hang at say 1/3 between earth and the galaxy. Or the rocket will smash its goal. which of the two will happen?

Second question. The speed due to expansion is proportional to the distance in between. This suggests that distance and speed increase exponential. So even an initial speed of c/2 doesn't guarantee the rocket in the first question to reach the galaxy because the galaxy might be running even faster before the rocket comes close. Is this reasoning correct?

You might want to fix the numbers, maybe swap c/10 and c/3 around. The way you have it now the rocket is going slower (c/10) than the distance is expanding (c/3).
But more generally, Arthur0, you might benefit from learning to use this table-making calculator, where you get to set the number of steps and the distance growth range between S-upper and S-lower.
http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html
I would suggest setting upper S to 10 and lower S to 1 (which is the present era, distances equal to their present size) and letting the number of steps be 40, say, or 50.
Then press calculate and see what you get.

If the percentage rate of distance increase were CONSTANT then you would be absolutely right about exponential. But for most of the history of expansion the distance growth has NOT been exponential because the percentage rate has been declining too rapidly for that.
(you can see that in the tables that Lightcone makes, R is the reciprocal of the percentage growth rate, R increases as the growth rate declines. R=14.4 billion LY corresponds to growth by 1/144 of one percent per million years. R=1.0 billion LY would correspond to growth rate of 1/10 of a percent per million years.)

Because the percentage rate has declined, there can be light which started towards us but was swept BACK at first and actually lost ground, until it reached say 5.8 billion LY from us and was just barely holding its own and then slowly began to narrow the distance and make headway towards us. This is the sort of thing your title "traveling against expansion" suggests. It suggests swimming upstream against the current, and maybe losing ground at first but keeping at it and eventually the current slows and one begins to close the distance to the goal.

You can refine the table to show that. The turning point comes around S = 2.6, that is where "distance-then", the radius of the past lightcone, is about 5.8 billion LY.
If you specify enough steps, or rows for the table, you can see that figure 5.8 in the Dthen column. It is the max distance of any object we can see, if the distance were measured at the time it emitted the light we are now getting from it.

Likewise 5.8 is the max distance any packet of light that is currently reaching us can have been swept back to, by expansion, before it began to make progress towards us. If you have trouble using the calculator and getting a table that shows that, ask. Someone will talk you through. My guess is you aren't likely to need help but no harm if you do and ask.

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phinds
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Second question. The speed due to expansion is proportional to the distance in between. This suggests that distance and speed increase exponential. So even an initial speed of c/2 doesn't guarantee the rocket in the first question to reach the galaxy because the galaxy might be running even faster before the rocket comes close. Is this reasoning correct?
yes, but "speed" turns out to be a poor choice of words here. "Recession velocity" is what it is normally called because we think of "speed" as being due to actual movement but recession is different than that. The proper motion of two galaxies that are 10 billion light years apart (their speed relative to each other) will be close to zero even though the recession velocity is an appreciable percent of c.

Matterwave
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If I send a car going 30 mph towards a car going away from it at 50 mph will it ever catch up?

But this is not the correct analogy, since it is the space between the start and end points that is expanding and not the other object moving away.
There's a thread in the math section which talks about a snail moving from one end of a very long rubber band to the other end. Even though the rubber band stretches considerably at each step, the snail will reach the other end in finite time (although it is a ridiculously large time for the parameters given in that thread). I think the same could be said here, except, for the presence of the dark energy. If there were no dark energy, and the universe was not accelerating in its expansion, then I am quite confident that the spaceship will reach its destination within finite time (in fact, if the big crunch were the ultimate fate of the universe, all this rocket ship has to do is just wait and the other galaxy will come back to it eventually). I do not think this is still the case once we account for the acceleration of the expansion of the universe.

EDIT: I took a look at the thread in the mathematics section, and the stretching of the rubber band was not quite analogous to the expansion of the universe...I'm not sure... so my conclusions here might be invalid.

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Orodruin
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But this is not the correct analogy, since it is the space between the start and end points that is expanding and not the other object moving away.
There's a thread in the math section which talks about a snail moving from one end of a very long rubber band to the other end. Even though the rubber band stretches considerably at each step, the snail will reach the other end in finite time (although it is a ridiculously large time for the parameters given in that thread). I think the same could be said here, except, for the presence of the dark energy. If there were no dark energy, and the universe was not accelerating in its expansion, then I am quite confident that the spaceship will reach its destination within finite time (in fact, if the big crunch were the ultimate fate of the universe, all this rocket ship has to do is just wait and the other galaxy will come back to it eventually). I do not think this is still the case once we account for the acceleration of the expansion of the universe.

EDIT: I took a look at the thread in the mathematics section, and the stretching of the rubber band was not quite analogous to the expansion of the universe...I'm not sure... so my conclusions here might be invalid.

It is a fairly good analogy. The question is how the rubber band is being stretched (i.e., constant, accelerating, or decelerating expansion). The problem usually quotes a rubber band that is extended a certain amount each second, for which case the snail (or ant or whatever small creature you consider) will reach the end. Depending on the universe, this may be a good or bad approximation. It certainly is not true in a universe dominated by a cosmological constant. So in the end, the answer will depend on how the universe expands. The calculator marcus linked will take this into account.

Regardless, it is a better analogy than two objects moving with constant speed relative to each other.

Edit: Let me also add that in this case, the rocket will also lose speed as the universe expands since it has been stated that there is no propulsion rather than constant speed (I am assuming speed is measured relative to a comoving frame).

aboro and marcus
marcus
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Hi Matterwave, I basically agree with what you said here and with Oroduin's comment. It is a fairly good analogy, given the clarifications he supplied. Uniform percentage-rate distance growth is a bit different from ordinary motion we are used to because nobody gets anywhere by it, relative positions don't change, everybody just becomes farther apart. So what you said (it's the distance expanding, not the other object moving away) seems like an OK way to think about it.I took the liberty of removing the line-out to make it easier to read what you said at first. It seems OK. I want to post a PICTURE of "the snail's progress"as soon as I have quoted you and Oro. It shows how some light eventually gets to us (now, in year 13.8 billion) even though in year 4 billion it is making zero progress--it's forward speed towards us is exactly canceled by the growth of the distance between it and us.
...it is the space between the start and end points that is expanding and not the other object moving away.
There's a thread in the math section which talks about a snail moving from one end of a very long rubber band to the other end. Even though the rubber band stretches considerably at each step, the snail will reach the other end in finite time (although it is a ridiculously large time for the parameters given in that thread). I think the same could be said here, except, for the presence of the dark energy. If there were no dark energy, and the universe was not accelerating in its expansion, then I am quite confident that the spaceship will reach its destination within finite time (in fact, if the big crunch were the ultimate fate of the universe, all this rocket ship has to do is just wait and the other galaxy will come back to it eventually). I do not think this is still the case once we account for the acceleration of the expansion of the universe.

EDIT: I took a look at the thread in the mathematics section, and the stretching of the rubber band was not quite analogous to the expansion of the universe...I'm not sure... so my conclusions here might be invalid.
I think they are valid, but of course it depends on how the percentage distance growth rate changes over time. It has been declining (its reciprocal, the Hubble time, has been increasing, as the picture will show. Oroduin comment seems appropriate to quote.

It is a fairly good analogy. The question is how the rubber band is being stretched (i.e., constant, accelerating, or decelerating expansion). The problem usually quotes a rubber band that is extended a certain amount each second, for which case the snail (or ant or whatever small creature you consider) will reach the end. Depending on the universe, this may be a good or bad approximation. It certainly is not true in a universe dominated by a cosmological constant. So in the end, the answer will depend on how the universe expands. The calculator marcus linked will take this into account...
About the picture, in the year 4 billion (or thereabouts) the distance growth rate was 1/58 of a percent per million years. That is to say that a distance of 5.8 billion lightyears was growing at exactly the speed of light. So a "snail-like" flash of light that was crawling towards us and was then 5.8 billion LY from us makes zero progress. The tear-shape lightcone profile is FLAT at that point.

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marcus
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Mwave, I think you probably understand what's being discussed here very well but just to give a concrete example, if you take a picture of a galaxy today and it is what astronomers call redshift z = 1.6, that means that the wavelengths of the hot hydrogen and sodium and stuff have all been stretched by a factor of S = 2.6 (for historical accident reasons, the "redshift" is always one less than the actual stretch factor.)

And that wavelength enlargement, or stretch, factor means that the light was emitted in year 4 billion (because 2.6 is the factor by which distances have been enlarged since then).

And the nice thing, I think, is that when it was emitted the light made zero progress for a while because distance growth exactly canceled the speed towards us of the light. The light just hung there at a distance of 5.8 billion LY, not gaining ground, just holding its own. And then very gradually the percentage rate which was 1/58 % per million years, declined slightly, to like 1/59% per million years, and then 1/60% per million years. And the light began to inch ahead.
You can see in the picture how the Hubble radius, starting at year 4 billion is 5.8 billion LY and then increases, e.g. to 5.9 billion LY and 6.0 billion LY and so on. It is a convenient way to keep track of the declining distance growth rate. It is not a physical distance itself but a tool for keeping track of the growth rate.

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Orodruin
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The light just hung there at a distance of 5.8 billion LY, not gaining ground, just holding its own.

While I agree with most of what you say, I would not consider the light just hanging there. I know that you know this but I just want to make it clear for any readers: While the proper distance may be constant, the light is certainly changing comoving coordinates and to a local observer it will fly by at the speed of light.

marcus
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Absolutely right! It is definitely traveling at the usual speed c in its surroundings. It just isn't getting any closer to us. It is staying the same distance of 5.8 billion LY from us.

Just checked https://www.physicsforums.com/members/orodruin.510075/
Great research interest (neutrino and DM pheno) key area now. Probably know Maxim Markevitch who co-authored Bulbul et al paper.

Some readers may be interested in how you get the Lightcone tabulator to plot graphs (like above) instead of making tables. First, for example, go to Lightcone, set upper and lower S factors at 40 and 1, and the number of steps to 50. That will cover a convenient span of time from around year 67 million up to the present. with 50 steps the resolution of the table will be fine enough that you actually see the Hubble radius and Dthen cross at 5.8 billion LY in year 4 billion.
When you generate the table you will be generating the same numbers that are plotted in that figure with the two curves, a few posts back. Then there's a way to eliminate the distracting other numbers that we're not interested in, at the moment. But for starters, setting steps=50 and S to run from 40 down to 1 here is what you get.
Look down the T column to about 12 rows from the bottom where it says year 4.0233 billion (i.e. approximately year 4 billion):
$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\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_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.025&40.000&0.0674&0.1021&38.875&0.972&1.384&2.70&9.52\\ \hline 0.027&37.155&0.0754&0.1141&38.568&1.038&1.482&2.68&9.10\\ \hline 0.029&34.513&0.0843&0.1275&38.249&1.108&1.586&2.66&8.69\\ \hline 0.031&32.058&0.0943&0.1425&37.918&1.183&1.697&2.63&8.30\\ \hline 0.034&29.778&0.1054&0.1592&37.575&1.262&1.815&2.61&7.93\\ \hline 0.036&27.660&0.1178&0.1779&37.219&1.346&1.942&2.58&7.56\\ \hline 0.039&25.693&0.1317&0.1988&36.849&1.434&2.076&2.56&7.22\\ \hline 0.042&23.866&0.1472&0.2221&36.465&1.528&2.219&2.53&6.88\\ \hline 0.045&22.168&0.1645&0.2481&36.066&1.627&2.371&2.50&6.56\\ \hline 0.049&20.592&0.1839&0.2772&35.653&1.731&2.532&2.48&6.25\\ \hline 0.052&19.127&0.2055&0.3097&35.224&1.842&2.704&2.45&5.95\\ \hline 0.056&17.767&0.2297&0.3460&34.779&1.958&2.885&2.42&5.66\\ \hline 0.061&16.503&0.2567&0.3866&34.317&2.079&3.078&2.38&5.38\\ \hline 0.065&15.329&0.2868&0.4319&33.837&2.207&3.283&2.35&5.11\\ \hline 0.070&14.239&0.3205&0.4824&33.339&2.341&3.499&2.32&4.85\\ \hline 0.076&13.226&0.3582&0.5389&32.823&2.482&3.728&2.28&4.60\\ \hline 0.081&12.286&0.4002&0.6020&32.287&2.628&3.970&2.24&4.37\\ \hline 0.088&11.412&0.4472&0.6724&31.732&2.781&4.225&2.20&4.14\\ \hline 0.094&10.600&0.4996&0.7511&31.155&2.939&4.494&2.16&3.91\\ \hline 0.102&9.846&0.5582&0.8389&30.557&3.103&4.778&2.12&3.70\\ \hline 0.109&9.146&0.6237&0.9369&29.936&3.273&5.076&2.08&3.49\\ \hline 0.118&8.496&0.6967&1.0462&29.292&3.448&5.389&2.03&3.30\\ \hline 0.127&7.891&0.7783&1.1682&28.624&3.627&5.717&1.99&3.10\\ \hline 0.136&7.330&0.8695&1.3043&27.931&3.810&6.060&1.94&2.92\\ \hline 0.147&6.809&0.9712&1.4560&27.213&3.997&6.418&1.89&2.75\\ \hline 0.158&6.325&1.0847&1.6251&26.468&4.185&6.792&1.84&2.58\\ \hline 0.170&5.875&1.2115&1.8134&25.696&4.374&7.180&1.78&2.41\\ \hline 0.183&5.457&1.3528&2.0229&24.896&4.562&7.584&1.73&2.26\\ \hline 0.197&5.069&1.5105&2.2560&24.067&4.748&8.001&1.67&2.10\\ \hline 0.212&4.708&1.6863&2.5148&23.209&4.929&8.431&1.61&1.96\\ \hline 0.229&4.373&1.8823&2.8019&22.320&5.103&8.873&1.55&1.82\\ \hline 0.246&4.062&2.1006&3.1198&21.400&5.268&9.326&1.49&1.69\\ \hline 0.265&3.773&2.3435&3.4711&20.450&5.419&9.789&1.42&1.56\\ \hline 0.285&3.505&2.6136&3.8583&19.468&5.554&10.258&1.35&1.44\\ \hline 0.307&3.256&2.9137&4.2836&18.454&5.668&10.732&1.28&1.32\\ \hline 0.331&3.024&3.2467&4.7491&17.410&5.757&11.208&1.21&1.21\\ \hline 0.356&2.809&3.6155&5.2562&16.335&5.815&11.684&1.13&1.11\\ \hline 0.383&2.609&4.0233&5.8054&15.232&5.837&12.156&1.06&1.01\\ \hline 0.413&2.424&4.4731&6.3963&14.101&5.818&12.620&0.98&0.91\\ \hline 0.444&2.251&4.9681&7.0270&12.945&5.750&13.073&0.90&0.82\\ \hline 0.478&2.091&5.5109&7.6938&11.768&5.627&13.511&0.82&0.73\\ \hline 0.515&1.943&6.1040&8.3911&10.573&5.443&13.930&0.73&0.65\\ \hline 0.554&1.804&6.7496&9.1115&9.364&5.190&14.327&0.65&0.57\\ \hline 0.597&1.676&7.4488&9.8452&8.149&4.862&14.699&0.57&0.49\\ \hline 0.642&1.557&8.2024&10.5811&6.932&4.452&15.042&0.48&0.42\\ \hline 0.692&1.446&9.0099&11.3071&5.720&3.955&15.356&0.40&0.35\\ \hline 0.744&1.343&9.8702&12.0108&4.521&3.366&15.640&0.31&0.28\\ \hline 0.801&1.248&10.7813&12.6807&3.342&2.678&15.892&0.23&0.21\\ \hline 0.863&1.159&11.7403&13.3067&2.189&1.889&16.114&0.15&0.14\\ \hline 0.929&1.077&12.7436&13.8814&1.068&0.992&16.307&0.07&0.07\\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline \end{array}}$$
You will see that the Hubble radius R and the light's distance from us Dthen are both about 5.8 billion LY. That is where the curves cross in the plotted figure.
You can click on "column definition and selection" to open the column menu to deselect the columns you don't want.
You can for example get rid of scale factor a, and stretch S, and keep only time T and Hubble radius R and Dthen

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marcus
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I'm hoping additional people will like to use this online tabulator.
It's easy to learn (and also easy to make switch over to plotting CURVES instead of making tables, whenever you want).

So here is what you get from Lightcone by setting steps to 50 and stretch factor S to run from 40 down to 1 (the present day),
and by opening the column menu and deselecting everything but T, R, and Dthen.
$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline T (Gy)&R (Gly)&D_{then}(Gly) \\ \hline 0.0674&0.1021&0.972\\ \hline 0.0754&0.1141&1.038\\ \hline 0.0843&0.1275&1.108\\ \hline 0.0943&0.1425&1.183\\ \hline 0.1054&0.1592&1.262\\ \hline 0.1178&0.1779&1.346\\ \hline 0.1317&0.1988&1.434\\ \hline 0.1472&0.2221&1.528\\ \hline 0.1645&0.2481&1.627\\ \hline 0.1839&0.2772&1.731\\ \hline 0.2055&0.3097&1.842\\ \hline 0.2297&0.3460&1.958\\ \hline 0.2567&0.3866&2.079\\ \hline 0.2868&0.4319&2.207\\ \hline 0.3205&0.4824&2.341\\ \hline 0.3582&0.5389&2.482\\ \hline 0.4002&0.6020&2.628\\ \hline 0.4472&0.6724&2.781\\ \hline 0.4996&0.7511&2.939\\ \hline 0.5582&0.8389&3.103\\ \hline 0.6237&0.9369&3.273\\ \hline 0.6967&1.0462&3.448\\ \hline 0.7783&1.1682&3.627\\ \hline 0.8695&1.3043&3.810\\ \hline 0.9712&1.4560&3.997\\ \hline 1.0847&1.6251&4.185\\ \hline 1.2115&1.8134&4.374\\ \hline 1.3528&2.0229&4.562\\ \hline 1.5105&2.2560&4.748\\ \hline 1.6863&2.5148&4.929\\ \hline 1.8823&2.8019&5.103\\ \hline 2.1006&3.1198&5.268\\ \hline 2.3435&3.4711&5.419\\ \hline 2.6136&3.8583&5.554\\ \hline 2.9137&4.2836&5.668\\ \hline 3.2467&4.7491&5.757\\ \hline 3.6155&5.2562&5.815\\ \hline 4.0233&5.8054&5.837\\ \hline 4.4731&6.3963&5.818\\ \hline 4.9681&7.0270&5.750\\ \hline 5.5109&7.6938&5.627\\ \hline 6.1040&8.3911&5.443\\ \hline 6.7496&9.1115&5.190\\ \hline 7.4488&9.8452&4.862\\ \hline 8.2024&10.5811&4.452\\ \hline 9.0099&11.3071&3.955\\ \hline 9.8702&12.0108&3.366\\ \hline 10.7813&12.6807&2.678\\ \hline 11.7403&13.3067&1.889\\ \hline 12.7436&13.8814&0.992\\ \hline 13.7872&14.3999&0.000\\ \hline \end{array}}$$

At that point, if you look at the "display as" control buttons, and tick the button that says "chart"
and click on "calculate" again, you will get the plot of curves that I posted earlier. It can be dragged off the Lightcone screen onto your desktop, and uploaded to PF if you want to make a point about some cosmic history topic.

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Matterwave
Gold Member
Right, I wasn't sure for which situation was the snail on the rubber band sheet a valid analogy, since the rubber band sheet stretched by a certain distance at every step, and not a certain percentage. Anyways, certainly, if we had a closed universe without dark energy, any place in the universe can be reached within finite time since the universe will go to a big crunch - worst case scenario you just wait around for the big crunch to happen and for the distant galaxy to come to you. For an open universe, it seems intuitively clear, and mathematically clear, that the total proper distance traveled diverges to infinity as your proper time (or coordinate time) diverges to infinity IF you didn't lose energy to the expansion. If the energy lost due to expansion were taken into account (which I had not considered before Oro brought it up), I am no longer sure that even the proper distance traveled will diverge. The pertinent question to this thread; however, is not the total proper distance traveled, but is in what scenarios will your co-moving distance traveled (coordinate distance traveled in the FLRW coordinates) diverge to infinity? This is the measure by which you might reach a distant co-moving observer.

marcus
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Mwave, suppose you are a regular PF member who visits Cosmo forum sporadically, how would you react if I just show you this plot and ask
"In year 10 billion, what was the percentage growth rate of distances (going by this figure)?"
View attachment 77324

Do you think it's clear enough how to read a percent growth rate from the figure like that? It is the same figure with two curves I included a few posts back. Does it need more explanation or is it clear enough already? I'd like it accessible to fairly wide audience of people interested enough in cosmology to dip into threads here.

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Thx for all the answers. I meant the numbers as I typed them. Vanadium easily shows that my first question is solved easily in a non relativistic setting. I believe now that his reply is also correct in relativistic settings. Unless Matterwave initial counter argument is correct. I now guess it is not. But his thought was mine when I formulated my first question. I need to study Marcus' answer in detail. Thanks for the homework ;) Probably thereafter I'll be in a postion to better understand Phinds' and Oroduin's replies.

Matterwave
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Mwave, suppose you are a regular PF member who visits Cosmo forum sporadically, how would you react if I just show you this plot and ask
"In year 10 billion, what was the percentage growth rate of distances (going by this figure)?"
View attachment 77324

Do you think it's clear enough how to read a percent growth rate from the figure like that? It is the same figure with two curves I included a few posts back. Does it need more explanation or is it clear enough already? I'd like it accessible to fairly wide audience of people interested enough in cosmology to dip into threads here.

I think I would need some further explanation. Which curve, the red or the blue, corresponds to proper distance traveled, and which curve corresponds to comoving distance traveled? Is the red the comoving distance traveled and the Blue the proper distance? Or neither? Does the red-line curving backwards means this light ray will never reach us if it originated from more than 5.8 Gly away?

marcus
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Mwave thanks for the response. I have been wanting someone to discuss this picture with to see what is or is not understood and what kind/amount of explanation is needed!
I think I would need some further explanation. Which curve, the red or the blue, corresponds to proper distance traveled, and which curve corresponds to comoving distance traveled? Is the red the comoving distance traveled and the Blue the proper distance? Or neither? Does the red-line curving backwards means this light ray will never reach us if it originated from more than 5.8 Gly away?

To answer one of your questions, the RED curve, labeled Dthen, is the proper distance the light was from us at some moment (indicated by position on the time axis. This is for a flash of light that we receive today. You can see that if the light was emitted in year 1 billion it will initially be dragged back, away from us (the distance from us increases, the red curve rises). But after year 4 billion, you can see that the light begins to make headway towards us, and its proper distance from us decreases. And the red curve slopes down.
Are you OK with this?

The blue curve is labeled R, like the red, the units are billions of LY. The blue curve is the HUBBLE RADIUS, it is a way of keeping track of the distance growth RATE.
It is defined as the size of distance which, at that moment in time, is growing at speed c. The other largescale distances are growing proportionately. One which is twice R would be growing at speed 2c.

What I think is the important thing to learn and understand here is how the Hubble radius at some particular moment (e.g. year 10 billion) encodes the percentage growth rate. Once the coin drops and you understand, you will be able to simply look at the figure, notice that at that moment, year 10 billion, the Hubble radius is 12 billion LY, and then immediately convert that mentally into the percentage growth rate that prevailed at that time.

It's actually really simple: in a million years a distance that size, growing that rate, would increase by a million LY. So what percentage is that?
what is the fraction 1 million/12 billion ? Well it is 1/12 times 1/1000. And that is the fraction we call 1/120 percent. So the our sample distance of 12 billion LY is growing 1/120 percent per million years.

And all the other large-scale distances (outside of gravitationally bound systems are growing at essentially that same percentage rate. (Ignoring minor local motions of stuff. this is the basic expansion rate as of year 10 billion.)

It is a simple rule of thumb and it applies to our own epoch as well. Today the Hubble radius is estimated to be 14.4 billion LY. So what is the percentage growth rate of cosmic distances?
Hint: it is somewhat LESS than 1/120 percent per million years. The percentage distance growth rate has been declining, as the Hubble radius (shown as the blue curve in the figure) has been increasing.

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Matterwave
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Mwave thanks for the response. I have been wanting someone to discuss this picture with to see what is or is not understood and what kind/amount of explanation is needed!

It's actually really simple: in a million years a distance that size, growing that rate, would increase by a million LY. So what percentage is that?
what is the fraction 1 million/12 billion ? Well it is 1/12 times 1/1000. And that is the fraction we call 1/120 percent. So the our sample distance of 12 billion LY is growing 1/120 percent per million years.

I'm a bit confused by this calculation. Where did you get 1 million LY from and the 1 million years? How do I read that off the graph? Wouldn't the growth rate actually be something like the derivative of the Blue curve?

EDIT: Nevermind, I got it. You are basically saying since that Hubble radius is the radius at which things are "moving away at the speed of light" from us, then the distance is growing at 1 million light years / million years. I think if you used 1 light year/year it would have been a bit more obvious haha.

marcus
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thanks for the comment! I may be able to work a reference to 1 LY per Y into a future explanation. You understand why one would also want to think of the percentage growth on a per million year basis. Because current percent rates are already pretty small even on that basis, like 1/144 % per million years . Don't want to boggle the reader (and have to use extra words) by changing to extreme smallness like "1/144 millionths of a percent" per year. Hard to imagine such a small fraction.
But one could make the offhand remark that the speed of light is 1 LY per Y or equivalently 106 LY per 106 Y, and then go from there.

BTW how easy is it to judge other Hubble radius sizes just looking at the blue curve? For example roughly what was R in year 1 billion? To me it looks like R was about 1.5 billion LY. At year 1 billion the blue curve is not quite halfway up from zero to 4, so it is less than 2. But more than 1. Judging by eye it looks like 1.5 to me.
If that's how you judge it too, then that tells you what the percentage rate of distance growth was, around year 1 billion.

(We could always refer to the table if we needed an exact number, but IMO it's nice to be able to read a simple curve plot figure visually like that.)

Matterwave
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Certainly this growth rate can be related to the Hubble constant right? After all, the Hubble constant can also be expressed as a frequency. 70km/s/Mpc converts to what in %/million years? (I'm lazy and Wolfram won't do this automatically).

Visually the curve is not too difficult to read. But at 1 Billion years, I can only tell it's between 1 and 2, not so much that it's 1.5 Gly.

It's been a while since I've done cosmology (several years), and so this stuff is slowly coming back to me lol.

Another thought: as the blue curve is always increasing, then this "%/million years" growth rate will always be decreasing right? How does this mesh with the fact that Hubble's constant is a constant? Am I confusing myself again? Was the Hubble constant much larger in the past?

Orodruin
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Certainly this growth rate can be related to the Hubble constant right? After all, the Hubble constant can also be expressed as a frequency. 70km/s/Mpc converts to what in %/million years? (I'm lazy and Wolfram won't do this automatically).

Just use that 1 %/Myr is the same as 0.01 Myr-1:
http://www.wolframalpha.com/input/?i=Hubble constant / (0.01 Myr^-1)
Insert whatever value you want if you want the Hubble constant at another time. Which brings us to ...
Another thought: as the blue curve is always increasing, then this "%/million years" growth rate will always be decreasing right? How does this mesh with the fact that Hubble's constant is a constant? Am I confusing myself again? Was the Hubble constant much larger in the past?
Yes. The "constant" in the "Hubble constant" is a misnomer based on that Hubble's original relationship was linear (he did not look far away enough). The Hubble constant at a given time is ##\dot a/a## and thus will change with time.

Matterwave
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Yes. The "constant" in the "Hubble constant" is a misnomer based on that Hubble's original relationship was linear (he did not look far away enough). The Hubble constant at a given time is ##\dot a/a## and thus will change with time.

But in order to coincide with this %/million year growth rate, it must be actually strictly decreasing whenever the blue curve is increasing. In the past, the Hubble constant must have been quite a bit larger than it is now. Is that true? I guess if the "hubble time is t=1/H" then yes...H must have been a lot larger back when t was a lot smaller...

Orodruin
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But in order to coincide with this %/million year growth rate, it must be actually strictly decreasing whenever the blue curve is increasing. In the past, the Hubble constant must have been quite a bit larger than it is now. Is that true?
Correct. If you look at inflation, the Hubble constant was was huge. The only way of having a constant Hubble constant (or Hubble parameter, which is less of a misnomer) is to have exponential expansion (##\dot a = Ha## with ##H## constant gives ##a \propto \exp(Ht)##).

Matterwave
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Correct. If you look at inflation, the Hubble constant was was huge. The only way of having a constant Hubble constant (or Hubble parameter, which is less of a misnomer) is to have exponential expansion (##\dot a = Ha## with ##H## constant gives ##a \propto \exp(Ht)##).
So, in fact, it sounds like the snail on a rubber band analogy is actually quite analogous to cosmology!

Hopefully at some point we can derive a rigorous proof that the coordinate distance as well as proper distance traveled by an object with "constant velocity" will or will not both diverge as time goes to infinity, and finally answer OP's question. Although, with the expansion being exponential, I doubt that the coordinate distance will diverge.

Orodruin
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So, in fact, it sounds like the snail on a rubber band analogy is actually quite analogous to cosmology!
Yes. With the caveat that if the snail is (locally) moving with a constant speed, then it is an analogy of a light signal and not a rocket traveling at subluminal speeds. For the rocket, the speed will also be decreasing as the universe expands due to its red-shifted momentum. Otherwise yes. If you have a rubber band that is being stretched a fixed rate such that ##a = kt + l_0##, then ##H = k/(kt+l_0) = 1/(t + t_0)##, where ##t_0 = l_0/k##, which is a decreasing Hubble parameter. In fact, the snail problem is much easier to solve if you borrow a trick from cosmology and work in comoving coordinates (see, e.g., https://www.physicsforums.com/threads/bug-moving-on-a-rubber-band.766909/page-2#post-4831642).

Edit: That thread seems to bring memories from a time long since past ... it was before I even got the HH medal. How young and innocent I was ...

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Matterwave
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Look at that...my intuition didn't guide me wrong (much)!

Orodruin
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