# Conundrum involving distance to object, universal expansion, age of universe, etc

I was recently reading about a galaxy or quasar (not sure which, so lets just say quasar) where the estimated distance to this thing was some very significant fraction of the age of the universe. What I mean is this: I can't recall the exact particulars, but assuming for argument's sake that the age of the universe is 13.8 billion years old, this thing was like 13 billion light years away. (Again, some detail here might be slightly off; the thing might have been estimated at 12 billion light years away - so focus on the general principle I am layout out here, not any particulars).

What I don't understand is this. If the thing is 13 billion light years away, then the light reaching my eyes as I look at it now took 13 billion years to get to me. Yet the age of the universe is 13.8 billion years old (again if I'm slightly wrong in some of these numbers, just accept them for now for argument's sake). It means the light must have left this object when the universe was much "smaller" or more compact, and things were much closer together. Which means the light should have passed my position a long time ago if I have been either "moving" or "expanding" (or some combination) away from the object at anything except some rather large percentage of the speed of light.

To try to boil this down, or restate it in simpler terms: 1) either it seems I am moving/expanding/red-shifting away from this object at some significant fraction of the speed of light (haven't done the numbers, but might be 80% or 90% or so) and have been for a while, or 2) something is "off" (the age of the universe, or the distance of this object).

or 3) there is something I'm not understanding. Which is why I came here.

I have never seen it stated that we are "expanding" or "redshifting" or [insert whatever term you like] at some significant fraction of the speed of light; I have only ever seen it stated that for some brief period of time after the big bang the universe could have (or did) expand faster than light, but again that was for like less than a second if memory serves.

At any rate, anyone care to explain?

Thanks.

• in some (strictly incorrect) sense you are indeed expanding at a velocity much larger than the speed of light. The catch is that in an expanding universe, velocity composition is not "Euclidian". – chris Apr 13 '14 at 16:37
• you might want to look at this? (e.g. their figure 2)mso.anu.edu.au/~charley/papers/DavisLineweaver04.pdf – chris Apr 13 '14 at 16:43
• This one might be easier to read? mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf quoting it: Baffled by the expansion of the universe? You’re not alone. Even astronomers frequently get it wrong ! – chris Apr 13 '14 at 16:59
• All of that made my eyes glaze over. I just need a plain-spoken, layman's description of why I'm wrong, and what is correct. If it is understandable, then it is describable in simple layman's language. If it isn't describable that way, it probably isn't understandable. – user3355020 Apr 13 '14 at 21:18
• In a simple language, 1) is correct in some sense. – chris Apr 13 '14 at 21:31

The 13 billion lightyears distance of the quasar mean 13 billion lightyears light-travel distance. In other words, the light took 13 billion years to reach us, independent of the distance the quasar is now away from us.

The current proper distance (a chain of rulers would measure) is called comoving distance; it is much larger than the light-travel distance for large redshifts.

The quasar is moving away from us in an accelerated fashion, and is now moving away faster than light. This means, light which is emitted now (in the sense of cosmic time) by the galaxy, the quasar may have evolved into in the meanwhile, will never reach us. The light, the quasar emitted 13 billion years ago, left the quasar just in time to leave the regions of the universe which later have been receding faster than light from us, to eventually reach us.

As a thought experiment, imagine walking with 5 km/h (our playground speed of light) on a rubber band which is expanding with a constant rate of 1 km/h per meter actual length of the band. (This leads to an exponential acceleration of the distance between the two marks on the band). If you start closer than 5 meters away from your goal, e.g. from a start mark (our quasar) 4.50 m away, you'll finally reach the goal mark. The start mark will soon be further away from the goal mark than 5 meters, therefore receding with more than 5 km/h from the goal shortly after you left the start mark. At the moment you arrive at the goal mark, the start mark will be much further away (comoving distance) than the distance you needed to walk (light travel distance). And you've been walking a longer distance than the (proper) distance between the marks was at the time you started walking.

Btw.: Acceleration is only felt as a force, when the velocity to the local rubber band (mataphoric space-time) is changed.

Example calculations with a protogalaxy of redshift $z=11.9$: Based on the Cosmology Calculator on this website, the cosmological parameters $H_0 = 67.11$ km/s/Mpc, $\Omega_{\Lambda} = 0.6825$ provided by the Planck project, and the scale factor $d(t) = d_0 / (1+z)$, setting $\Omega_M = 1- \Omega_{\Lambda} = 0.3175$, the age of the universe is $13.820$ Gyr, and the comoving distance of the protogalaxy is $d_0 = 32.644$ Gly.

The age of the universe, we see the protogalaxy (at redshift 11.9), was 0.370 Gyr, light-travel distance has been 13.450 Gly, proper distance was 2.531 Gly.

After the protogalaxy has been emitting light 0.370 Gyr after the big bang, the light travelled towards us through space of redshift beginning with 11.9 shrinking to 0; the light arrived at us 13.820 Gyr after the big bang. The comoving distance (to us) of the space traversed by the light started with 32.644 Gly shrinking to 0. The remaining distance, the light needed to travel, started with 13.450 Gly shrinking to 0. The proper distance between the protogalaxy and us started with 2.531 Gly increasing to 32.644 Gly due to the expansion of spacetime.

Here some intermediate states described by a couple of tuples, consisting of

• redshift $z$,
• according age $t$ of the universe (Gyr),
• comoving radial distance (at age $t$) of the emitted light, we can now detect from the protogalaxy (Gly),
• remaining light travel distance of that emitted light (Gly),
• proper distance of the protogalaxy at age $t$, according to $d(t) = d_0 / (1+z)$:

$$(11.9, 0.370, 32.644, 13.450, 2.531),$$ $$(11.0, 0.413, 32.115, 13.407, 2.720),$$ $$(10.0, 0.470, 31.453, 13.349, 2.968),$$ $$( 9.0, 0.543, 30.693, 13.277, 3.264),$$ $$( 8.0, 0.636, 29.811, 13.184, 3.627),$$ $$( 7.0, 0.759, 28.769, 13.061, 4.081),$$ $$( 6.0, 0.927, 27.511, 12.892, 4.663),$$ $$( 5.0, 1.168, 25.952, 12.651, 5.441),$$ $$( 4.0, 1.534, 23.952, 12.285, 6.529),$$ $$( 3.0, 2.139, 21.257, 11.680, 8.161),$$ $$( 2.0, 3.271, 17.362, 10.549, 10.881),$$ $$( 1.0, 5.845, 11.124, 7.974, 16.322),$$ $$( 0.0, 13.820, 0.0 , 0.0 , 32.644).$$

The Hubble parameter, meaning the expansion rate of space per fixed proper distance, is decreasing with time. This allowed the protogalaxy to recede almost with the speed of light, although it was just about 2.5 Gly away from us (proper distance) in the time, when it emitted the light we detect now. Nevertheless distant objects in this space accelerate away from us, since their increasing distance is multiplied with the expansion rate of space.

• Yeah, I "get" all of that, and thanks for going into it. But I'm not sure that answers my question, and I'm not sure that I've communicated it well enough. Here is a list of the most distant objects: en.wikipedia.org/wiki/…. One of them is 13.37 billion light years away. The point is, that's less than 0.5 years from the age of the universe. You might as well say we were on top of that object when the light left it. How could it possibly take that long to reach us when we were so close to it when the light left it? – user3355020 Apr 14 '14 at 19:08
• @user3355020 Because space is expanding. And btw, the Milky Way wasn't born at that time. – Yashbhatt Jul 26 '14 at 6:39

Couldn't comment as I do not have the proper rep yet, but regarding @user3355020 and the answer provided by @Gerald, (and me being far less educated than I ever could have imagined before reading this), I interpret it like this: (and I wouldn't be surprised if this is wrong) the speed of light is constant, but the expansion rate of the universe is not. The over-simplified observed effect is similar to the feeling of having a dream where you're in a hallway and you're running away from something as fast as you possibly can,focused on the end of the hallway, but the hallway seems to stretch more and more as time goes by. Much in the same way, light is the person running and 'space' is the hallway and earth is the opening at the end. Some light was able to get far enough in the beginning that the stretch did not impact it so much as to make it impossible to reach earth (it's just going to take it 13+ billion yrs), and some light (and from my understanding of the answer given by @Gerald all light after a certain 'time') will never reach earth.