# What value of z (redshift) equals apparent superluminal recession?

• B
• Cerenkov
Cerenkov
Hello.

I have visited this site recently... https://lco.global/spacebook/light/redshift/ ...and was wondering at what value of z in the table below would galaxies appear to recede at superluminal velocities?

zTime the light has been travelingDistance to the object now
0.00007151 million years1 million light years
0.101.286 billion years1.349 billion light years
0.252.916 billion years3.260 billion light years
0.55.019 billion years5.936 billion light years
17.731 billion years10.147 billion light years
210.324 billion years15.424 billion light years
311.476 billion years18.594 billion light years
412.094 billion years20.745 billion light years
512.469 billion years22.322 billion light years
612.716 billion years23.542 billion light years
712.888 billion years24.521 billion light years
813.014 billion years25.329 billion light years
913.110 billion years26.011 billion light years
1013.184 billion years26.596 billion light years

When I read Ethan Siegel... This Is How Distant Galaxies Recede Away From Us At Faster-Than-Light Speeds (forbes.com)

Either way, there's a critical distance where the apparent recession speed of a galaxy will exceed the speed of light: around a distance of 13-to-15 billion light-years. Beyond that, galaxies appear to recede faster than light, but this isn't due to an actual superluminal motion, but rather to the fact that space itself is expanding, which causes the light from distant objects to redshift.

...on this subject and apply his statements to the above table, I conclude that galaxies with a z value of 8 and above appear to us to be receding at superluminal velocities.

Is this correct?

Thank you for any help given.

Cerenkov.

Moderator's note: Thread moved to Cosmology forum.

Cerenkov said:
apply his statements to the above table, I conclude that galaxies with a z value of 8 and above appear to us to be receding at superluminal velocities
I'm not sure how you get a value of 8, since Siegel's article gives a distance of 13 to 15 billion light-years, which from the table corresponds to a redshift of between 1 and 2. A redshift of 8 corresponds to a time the light has been traveling of just over 13 billion years, but Siegel's article talks about distance, not time.

Cerenkov said:
appear to us to be receding at superluminal velocities
This isn't quite correct. The "superluminal velocity" is the "recession velocity" the galaxy has now, which depends on its distance now. But we don't see the galaxy as it is now. We see it as it was when the light was emitted, i.e., at a time to our past given by the "time the light has been traveling" column in the table. And when the galaxy emitted that light, it wasn't ]Edit: might not have been, see posts #28 and #29] moving at superluminal speed relative to us. What Siegel is doing is taking that galaxy and predicting its distance from us now, based on the assumption that it is exactly comoving, and then calculating what its recession velocity would be at that distance now. But that doesn't describe anything we actually see.

Last edited:
Cerenkov said:
... appear to us to be receding at superluminal velocities
Just to be sure you are clear, the velocity (based on where it is now) is not apparent, it is real, BUT ... it is not proper motion, it is a recession velocity so no speeding tickets are issued. That is, things inside space cannot move faster than c, but space itself can expand (recession velocity) at greater than c.

FactChecker
PeterDonis said:
I'm not sure how you get a value of 8, since Siegel's article gives a distance of 13 to 15 billion light-years, which from the table corresponds to a redshift of between 1 and 2. A redshift of 8 corresponds to a time the light has been traveling of just over 13 billion years, but Siegel's article talks about distance, not time.

I think I can see what I did wrong Peter.

I conflated Siegel's values of distance with time, figuring (wrongly) that since light takes one year to travel one light year, time and distance were therefore equal and interchangeable.

So when he talked about that critical distance of 13 to 15 billion years I read down the column until I found a value matching that and wrongly concluded that the z redshift in question must therefore be 8 and above.

Thank you for putting me right.

Cerenkov.

PeterDonis said:
This isn't quite correct. The "superluminal velocity" is the "recession velocity" the galaxy has now, which depends on its distance now. But we don't see the galaxy as it is now. We see it as it was when the light was emitted, i.e., at a time to our past given by the "time the light has been traveling" column in the table. And when the galaxy emitted that light, it wasn't moving at superluminal speed relative to us. What Siegel is doing is taking that galaxy and predicting its distance from us now, based on the assumption that it is exactly comoving, and then calculating what its recession velocity would be at that distance now. But that doesn't describe anything we actually see.

That's another error on my part Peter.

Treating the cited values as if they existed in a fixed and static universe. Hence the mistake about treating time and distance as equivalent.

I must try to get my head around the fact that what we are discussing here is a dynamic system. That space is expanding and the speed of light is finite, which therefore means that the values under discussion will have changed over time.

Thank you,

Cerenkov.

pbuk
phinds said:
Just to be sure you are clear, the velocity (based on where it is now) is not apparent, it is real, BUT ... it is not proper motion, it is a recession velocity so no speeding tickets are issued. That is, things inside space cannot move faster than c, but space itself can expand (recession velocity) at greater than c.

Thanks phinds.

Yes, I'm aware that it is the expansion of space that is carrying the galaxies along for the ride and not the galaxies themselves moving through space. Hence there is no violation of relativity.

Nevertheless, it is sometimes difficult for a layman like myself to get the terminology right. That's because my use of the terms critically depends on my understanding of the underlying concepts. If my understanding is faulty then it naturally follows that the wording of my questions will be too.

I try my best and hope for the best when creating a thread like this. The correction I receive in this forum is very welcome and if I have to learn from my mistakes, then so be it.

Thank you,

Cerenkov.

phinds
I always go back to the excellent paper by Davis and Lineweaver for these kinds of questions. As you see in Figure 1, and as @PeterDonis said above, the answer to your question is a redshift between 1 and 2. With the formulae in that paper, you could calculate it more exactly if you want to.

diogenesNY, PeterDonis and phinds
Cerenkov said:
I must try to get my head around the fact that what we are discussing here is a dynamic system. That space is expanding and the speed of light is finite, which therefore means that the values under discussion will have changed over time.
Yes, I think this is a crucial thing to keep in mind.

phyzguy said:
I always go back to the excellent paper by Davis and Lineweaver for these kinds of questions. As you see in Figure 1, and as @PeterDonis said above, the answer to your question is a redshift between 1 and 2. With the formulae in that paper, you could calculate it more exactly if you want to.

I'm looking at figure 1 of the .pdf of the Davis Lineweaver paper right now phyzguy.

The catch is that I cannot look at it with the eyes of understanding that you, Peter Donis and phinds can.

The meaning of those curved lines, numbers and shaded areas are no doubt obvious to you guys.

Not so to me.

I'm going to need help in deciphering these hieroglyphs.

I'm not try to be difficult here - I'm just trying to tell it like it is for me.

What do you suggest?

Cerenkov said:
What do you suggest?
Try starting with the bottom diagram in Figure 1. That is the one that most clearly shows causal relationships, since the worldlines of (radial) light rays are 45 degree lines in that diagram, just as they are in a standard spacetime diagram in SR. And as you can see from that diagram, all of the lines except the "Hubble sphere" line are 45 degree lines, meaning that they describe radial light rays.

PeterDonis said:
Try starting with the bottom diagram in Figure 1. That is the one that most clearly shows causal relationships, since the worldlines of (radial) light rays are 45 degree lines in that diagram, just as they are in a standard spacetime diagram in SR. And as you can see from that diagram, all of the lines except the "Hubble sphere" line are 45 degree lines, meaning that they describe radial light rays.

I have that on my screen, Peter.

Yes, I think I see it now. Here's what is immediately apparent to me.

The central vertical axis represents 'here' and where the horizontal 'now' line crosses it is where we are, here and now on planet Earth. There's a light cone extending from us downwards to 0, the beginning of Conformal time. Our worldline extends downwards along the central axis 0 axis of Comoving Distance, expressed in Giga light years. As far as I understand only events within this light cone are causally connected to us and can affect us or be observed by us.

Is that sufficient?

Cerenkov said:
The central vertical axis represents 'here' and where the horizontal 'now' line crosses it is where we are, here and now on planet Earth.
Yes.

Cerenkov said:
There's a light cone extending from us downwards to 0, the beginning of Conformal time.
Yes.

Cerenkov said:
only events within this light cone are causally connected to us and can affect us or be observed by us
For "us here and now", yes, these are all true. But as we continue to move into the future, our "here and now" will move upward on the central worldline, which means our past light cone will also move upward. The "event horizon" is just the future limit of that light cone (i.e., it is the boundary of the region of spacetime containing all the events that will ever be able to causally affect us or be observed by us).

Ok, looking closer, here's what I see.

The dotted lines flanking our central worldline represent the worldlines at redshift values of 1, 3, 10 and 1,000. These worldlines are also physically removed from us through space by certain Comoving distances values, expressed in Giga light years.

The meaning of Conformal Time and Scale Factor will need explaining but the white-coloured central volume representing the Hubble sphere - that I do understand.

The legend under these diagrams tells me that all comoving objects beyond this Hubble sphere are receding from us faster than the speed of light. Looking where the boundary of the Hubble sphere intersects the horizontal 'Now' line I can see that the redshift value is between 1 and 3 - just what you and phyzguy said earlier in this thread.

Thanks,

Cerenkov.

PeterDonis said:
For "us here and now", yes, these are all true. But as we continue to move into the future, our "here and now" will move upward on the central worldline, which means our past light cone will also move upward. The "event horizon" is just the future limit of that light cone (i.e., it is the boundary of the region of spacetime containing all the events that will ever be able to causally affect us or be observed by us).

Ah, thank you. That would have been one of my questions. You pre-empted me. In a nice way.

Currently observable light that has been travelling towards us since the beginning of the universe, was emitted from comoving positions that are now 46 Glyr from us.

Yes.

Our light cone grows at the speed of light allowing us to see further and further into space and back in time and this would be shown on the diagram by its the outward growth to higher and higher Comoving distance values.

Cerenkov said:
The dotted lines flanking our central worldline represent the worldlines at redshift values of 1, 3, 10 and 1,000.
More precisely, they are worldlines for which light emitted from those comoving points at the instant they cross our past light cone "here and now" would have the labeled redshift when we receive the light "here and now".

Cerenkov said:
These worldlines are also physically removed from us through space by certain Comoving distances values, expressed in Giga light years.
Yes.

Cerenkov said:
The meaning of Conformal Time and Scale Factor will need explaining
Scale Factor is just the factor by which the universe has expanded at a given time, relative to "now" (the Scale Factor "now" is 1).

Conformal Time is just a rescaling of time to make the paths of radial light rays 45 degree lines. Basically it adjusts the scale of time according to the scale factor to achieve that.

Thank you so much for this help, Peter.

For these questions about horizons and related topics like the here discussed "Hubble Sphere", in the standard cosmological description of the expanding universe (FLRW metric), see

https://arxiv.org/abs/astro-ph/0310808

vanhees71 said:
For these questions about horizons and related topics like the here discussed "Hubble Sphere", in the standard cosmological description of the expanding universe (FLRW metric), see

https://arxiv.org/abs/astro-ph/0310808
Exactly. Which is why @phyzguy linked to it in post #9 and the OP started commenting about in post #11. In other words, you're late to the party

vanhees71
PeterDonis said:
More precisely, they are worldlines for which light emitted from those comoving points at the instant they cross our past light cone "here and now" would have the labeled redshift when we receive the light "here and now".

The light left these worldline locations, took time to reach the periphery of our light cone. And that elapsed time has to be factored in when calculating the redshift values.

The system is not at rest, fixed and static, as I wrongly supposed but all is movement and change.

PeterDonis said:
Scale Factor is just the factor by which the universe has expanded at a given time, relative to "now" (the Scale Factor "now" is 1).

Yes. I can see where our 'now' line cross the apex of our light cone. The scale factor value is 1.

I believe that I have a grasp of Comoving Distance because that is expressed in units that I understand. Light years. But could you please explain what the Scale Factor values, running from 0.001 to infinity actually mean? I can gain no traction on them unless I can somehow relate them to the universe I inhabit.

For example, the HR diagram deals with luminosity and temperature - two concepts that I can easily relate to. I experience both luminosity and temperature in my daily life. They are relatable concepts.

Even though I know that the universe has expanded since Conformal Time 0, the Scale Factor is still mysterious to me - and currently unrelatable to anything within my experience.

PeterDonis said:
Conformal Time is just a rescaling of time to make the paths of radial light rays 45 degree lines. Basically it adjusts the scale of time according to the scale factor to achieve that.

So this is just another way of expressing the information under discussion? A reorganization for the sake of clarity and not related to anything that actually occurs in reality? Much like a pie chart or a scatter plot?

Thank you,

Cerenkov.

Cerenkov said:
could you please explain what the Scale Factor values, running from 0.001 to infinity actually mean?
They are the factor by which you multiply Comoving Distance to get Proper Distance. If you compare the top diagram in Figure 1 with the others, you should be able to see the relationship (the comoving dotted lines work well for this).

Cerenkov said:
So this is just another way of expressing the information under discussion? A reorganization for the sake of clarity and not related to anything that actually occurs in reality?
Yes. The "real" time actually experienced by observers is Proper Time. Comoving Time is a convenient reorganization of the data but it does not correspond to anything any observer actually experiences.

vanhees71
PeterDonis said:
They are the factor by which you multiply Comoving Distance to get Proper Distance. If you compare the top diagram in Figure 1 with the others, you should be able to see the relationship (the comoving dotted lines work well for this).

Thank you, Peter.

I will do that, see what I can see and get back to you with my findings.

PeterDonis said:
Yes. The "real" time actually experienced by observers is Proper Time. Comoving Time is a convenient reorganization of the data but it does not correspond to anything any observer actually experiences.

Ok, so you've used the terms Comoving Distance and Comoving Time today. Also Proper Distance and Proper Time. Comoving Distance I now begin to understand.

Based upon your comments above, would Proper Distance and Proper Time be what any observer, located anywhere, would experience? Or do those terms only apply to us, on our central worldline?

And what of Comoving Time? Is that what is experienced by observers located on these other worldlines?

Thank you,

Cerenkov.

Thank you.

Cerenkov.

Cerenkov said:
Based upon your comments above, would Proper Distance and Proper Time be what any observer, located anywhere, would experience?
Proper Time is directly experienced; it is defined as the time an observer experiences along their worldline. So Proper Time is actually a property of a specific worldline, or a family of worldlines that all share some key property. In this case, the Proper Time shown in the diagram is Proper Time along comoving worldlines; the horizontal lines in the diagram correspond to surfaces on which all comoving observers have experienced the same Proper Time since the Big Bang. These are also surfaces on which the universe is spatially homogeneous and isotropic.

Proper Distance is not directly experienced since a single observer cannot directly measure it. They can only infer it from other measurements. But it is defined, at least for this case, as the distance that would be read off on an ideal ruler that lay between two comoving observers and moved along with them. "Expansion of the universe" is just another way of saying that this Proper Distance increases with Proper Time for comoving observers.

Cerenkov said:
Or do those terms only apply to us, on our central worldline?
No, they apply to any comoving worldline. See above.

Cerenkov said:
And what of Comoving Time? Is that what is experienced by observers located on these other worldlines?
That was a typo, I meant Conformal Time. Sorry for the mixup on my part.

vanhees71
PeterDonis said:
Proper Time is directly experienced; it is defined as the time an observer experiences along their worldline. So Proper Time is actually a property of a specific worldline, or a family of worldlines that all share some key property. In this case, the Proper Time shown in the diagram is Proper Time along comoving worldlines; the horizontal lines in the diagram correspond to surfaces on which all comoving observers have experienced the same Proper Time since the Big Bang. These are also surfaces on which the universe is spatially homogeneous and isotropic.

Ok, that seems reasonable. And it agrees with what I understand about Relativity. Specifically, that no observer occupies a special or privileged location or viewpoint. Therefore, observers located on their own particular worldlines experience their own Proper Times, each being as valid as any other.

Fortunately I'm now in a better position to understand what you mean about surfaces than I was a few weeks ago. Will Kinney's book, An Infinity of Worlds arrived on my doorstep and I've been reading it avidly.

A minor epiphany occurred when I saw his diagram of the Big Bang singularity displayed using Conformal Time. It showed a light cone resting on a flat surface, with a small arrow pointing to the cone's apex and the words 'You are here' written next to it.

His words... "On a diagram of space-time with two directions of space and one of time, infinite in extent: the big bang happens everywhere in an infinitely large space at once."

This instantly allowed me to reconcile the apparent mismatch between the pop-science diagrams showing the universe emerging from a point-like singularity and the statements that I often read in this forum, where the singularity is described not as a point but as a time.

On the back of that leap of understanding I can now take what you say about surfaces, apply the logic Kinney's diagram to it and square that with the light cone of Davis and Lineweaver's paper.

Therefore, these surfaces, where the universe is spatially homogenous and isotropic, correspond to horizontal lines on the diagram, with earlier surfaces being lower down, closer to the big bang, having lower Conformal Time values and lower Scale Factors. Different worldlines, physically separated from ours by different Comoving Distance values are at identical distances from the big bang in terms of Conformal Time and Scale Factor.

This neatly satisfies both Relativity and the Cosmological Principle. I can also see how, if the Friedmann equations hold good and the universe really is 'flat' then all such surfaces, representing different times from the big bang forwards, would indeed extend infinitely.

PeterDonis said:
Proper Distance is not directly experienced since a single observer cannot directly measure it. They can only infer it from other measurements. But it is defined, at least for this case, as the distance that would be read off on an ideal ruler that lay between two comoving observers and moved along with them. "Expansion of the universe" is just another way of saying that this Proper Distance increases with Proper Time for comoving observers.

Ok, I think I get that.

PeterDonis said:
No, they apply to any comoving worldline. See above.

Thank you.

PeterDonis said:
That was a typo, I meant Conformal Time. Sorry for the mixup on my part.

Not a problem, Peter.

And thank you once again. I really am enjoying this! Cerenkov.

Cerenkov said:
A minor epiphany occurred when I saw his diagram of the Big Bang singularity displayed using Conformal Time. It showed a light cone resting on a flat surface, with a small arrow pointing to the cone's apex and the words 'You are here' written next to it.
Yes, this would be similar to the bottom diagram in Figure 1 in the Davis & Lineweaver paper. The only clarification I would make is that the words next to the arrow should really be "You are here and now".

Cerenkov said:
Therefore, these surfaces, where the universe is spatially homogenous and isotropic, correspond to horizontal lines on the diagram, with earlier surfaces being lower down, closer to the big bang, having lower Conformal Time values and lower Scale Factors.
Yes.

Cerenkov said:
Different worldlines, physically separated from ours by different Comoving Distance values are at identical distances from the big bang
I would say at identical Proper Times from the Big Bang (as long as we are talking about comoving worldlines) on a given surface.

Cerenkov said:
if the Friedmann equations hold good and the universe really is 'flat' then all such surfaces, representing different times from the big bang forwards, would indeed extend infinitely.
Yes. This would also be true if the universe were open (i.e., spatially infinite but with negative spatial curvature instead of flag).

If the universe were closed (i.e., spatially finite with positive spatial curvature), the surfaces would have finite extent; the extent would be a fixed value of Comoving Distance and the extent in Proper Distance would vary with time.

PeterDonis said:
The "superluminal velocity" is the "recession velocity" the galaxy has now, which depends on its distance now. But we don't see the galaxy as it is now. We see it as it was when the light was emitted, i.e., at a time to our past given by the "time the light has been traveling" column in the table. And when the galaxy emitted that light, it wasn't moving at superluminal speed relative to us.
This isn't entirely correct. Galaxies with z>2 were receding at a speed greater than the speed of light at the time when they emitted the light we see today

Jaime Rudas said:
Galaxies with z>2 were receding at a speed greater than the speed of light at the time when they emitted the light we see today
Yes, you're right. For the benefit of @Cerenkov, the key thing is to look at where the Hubble sphere intersects our past light cone; in the bottom diagram in Figure 1 in the paper, that appears to occur around a redshift of 2.

• Cosmology
Replies
9
Views
4K
• Cosmology
Replies
39
Views
7K
• Cosmology
Replies
21
Views
3K