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mgb_phys

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If it was stationary we could only see things appx 13.7bnlyr away, but things that were very close to us (or where we would eventually be) at the start of the universe have now moved to a point 46.5Bn lyr away while the light was on-route.

note - relativity doesn't put any limits on the speed the universe can expand, because it isn't transfering information

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the Hubble factor is about 70km/s per megaparsec

that's about 20km/s per Mlightyear distant.

All objects beyond the Hubble sphere are moving away faster than the speed of light.

mgb_phys is quite correct. because there is no frame of reference for the expansion of the universe it is not limited to lightspeed.

Normally we measure the redshift of a frequency emitted by a distant star to estimate its distance from us. You would expect redshift factor (Z) to be 1 for stars moving away at c. In fact it is about 1.5. This is down to the curvature of spacetime.

We currently can receive signals from stars etc moving away at about 4c which equates to about Z = 10.

For much more info look at http://arxiv.org/abs/astro-ph/0310808

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Chronos

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No it's not. It's our Hubble sphere.

read the paper.

read the paper.

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So, you will never be able to "see" beyond the point where the universe re-ionized (became transparent to photons).

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sylas

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The age of the universe is not the same as the distance to the observable horizon.

The universe is expanding. We know this as well as we know just about anything in science.

The "observable universe" means everything that we can see. However, of course, we see things very far away as they were very long ago. So "observable universe" usually means all the stuff we can see, whereever it is now.

The most distant light, which is about 13.7 billion years old, is the cosmic background radiation. That light was emitted about 0.00038 billion years after the Big Bang, or about 380,000 years after the Big Bang. What we see is actually hot glowing gas, mostly hydrogen, at an enormous redshift of about 1090, so that the light is no longer hot yellow light as emitted from a hot plasma, but is now microwave radiation.

Now... how far away is what we are looking at?

That's going to depend on how you define distance. There are lots of ways to define distance in cosmology, and they are different, because of the consequences of expansion.

One way to define distance is light travel time. The light took 13.7 billion years (the age of the universe less a tiny fraction) to get here.

Another way is "proper" distance -- this is the distance what you could measure if all of space was full of identical rulers, and you just add up all the rulers at any given instant in "proper" time between two locations.

SO... do we mean proper distance THEN when the light was emitted, from that gas to whatever stuff we were formed from as it was at that time?

Or do we mean proper distance NOW, from wherever that hot hydrogen is now: presumably all formed up into galaxies and so on like we are?

So far, I have given three different ways of defining distance; there are others!

Note that all these definitions all give values that are "model dependent"; in the sense that obtaining a distance using observable information (like redshift) requires calculations that use the model. The age of the universe is just as model dependent as the other definitions. Everything in science depends on some kind of model, or theory. The most useful distance definition of these three is probably the "proper distance now" definition, which is what is used in the Hubble relation, and which can be calculated using the ΛCMD model of the Big Bang.

You may find Professor Ned Wright's cosmology tutorial to be useful. Part 2 gets into the different ways distance is defined in cosmology. This is a widely recommended resource. He also gives a Cosmology Calculator, which can be used to find the distances to things in the universe, given the observed redshift, and given a particular model. The default parameters give the current best known model of the universe.

Type in 1090 for "z", and look at the result for "Flat" universe, or "General" (since the initial parameters are set up for the flat case anyway).

We have:

- The age of the universe is 13.666 billion years
- The light travel time is 13.665 billion years
- The comoving radial distance, which goes into Hubble's law, is 45.648 billion light years.

That last value is the radius of the observable universe. It is the "proper distance" to all the stuff we can see, but taken out to whereever it is NOW.

The proper radial distance at the time the light was emitted is not given with the calculator, but it is 45.648/(1+z) = 41.841 million light years; the universe has expanded by a factor of 1091 since then.

Some people are confused by this, as it looks like that stuff must have been moving faster than light. Quite so... it is indeed. Or, more correctly, the distance between us and that stuff is increasing at more than 3 times the speed of light, as the universe expands. (This rate of increase of the "proper distance" is not really the speed at which anything is moving through space. It is the rate of change of a separation, which turns out to be not quite the same thing.)

The reason we can see stuff that has a cosmological recession velocity greater than light speed is because that as light crosses space, it passes into regions that are receding more and more slowly. Eventually light crosses into regions receding at less than the speed of light. From that point, the light starts to get closer with time, in the proper distance co-ordinate system, and eventually it comes right into our local region of space, and we see with it. This is very non-technical, and like any non-technical answer it's wrong on some of the details; I'm just trying to give what may be a helpful intuition around a common stumbling block.

Cheers -- sylas

PS. After some discussion with other SAs, I have reworded some of this post to replace "measured" with "defined" in a number of places. We can't "measure" distances directly. The best we can do, at present, is measure things like the frequency of light, from which we can obtain a redshift, and then from that we calculate distances or ages, using some model.

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So my question is, at what distance do the various definitions start giving significantly different answers? Would that correspond to when the recession velocity becomes relativistic?

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sylas

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So my question is, at what distance do the various definitions start giving significantly different answers? Would that correspond to when the recession velocity becomes relativistic?

I'd say it is when the redshift becomes non-negligible, but perhaps that is a detail. If there is a difference in the scale of the universe between when the light was emitted and when it is observed, then distance definitions start to diverge.

For those interested in all the technical detail, I recommend Distance measures in cosmology by David Hogg (also at arXiv:astro-ph/9905116. It is intended to be purely pedagogical; to teach about the different definitions and how to calculate them.

Ned has described his calculator in some detail in A Cosmology Calculator for the World Wide Web, also at arXiv:astro-ph/0609593.

And then there are the non-technical explanations for various definitions. People who agree about how to use and calculate these distance values will still sometimes get into ding-dong fights over how best to explain them for others. My advice is to have a look at the various intuitive explanations, remember that every non-technical explanation omits something that someone considers important, and don't worry about it. Here's my own quick attempt, aiming to be really concise and not trying to cover everything. Because it is concise, each of these descriptions is incomplete and really needs a bit of technical detail to be defined unambiguously. Those who want that should read the references. I'm using Ned's variable name conventions.

If you look at Hogg's paper, you'll see a few additional definitions; and some of them are in Ned's calculator as well. I'm just going to give values for co-moving distance (D

- The co-moving distance is just the distance out to whatever we are looking at, but measured right now as if by a whole sequence of synchronized rulers at the same time. Same thing as "proper distance now". (D
_{now}) - Light travel time, or lookback time. How long the light has been traveling. (D
_{ltt}) - Transverse co-moving distance is the same as line of sight co-moving distance in a flat universe, which is what I am considering. In general, curvature can make a difference for large distances.
- Angular size distance. The distance you would expect normally given the angular size of an object in the sky by comparison with its real size. (D
_{A}) - Luminosity distance. The distance you would expect normally given the luminosity of the object in the sky by comparison with its real luminosity. (D
_{L})

Here is a table of values for different redshifts that are calculated from Ned's calculator using default parameters, which give a good match to our universe (Ω

[tex]\begin{array}{l|lll}

z & v & D_{ltt} & D_{now} \\

\hline

0 & 0 & 0 & 0 \\

0.01 & 0.010 & 0.137 & 0.137 \\

0.1 & 0.098 & 1.286 & 1.349 \\

0.5 & 0.445 & 5.019 & 6.137 \\

1.0 & 0.785 & 7.731 & 10.819 \\

1.408 & 1.000 & 9.082 & 13.779 \\

2 & 1.24 & 10.324 & 17.105 \\

3 & 1.53 & 11.476 & 21.071 \\

10 & 2.29 & 13.183 & 31.506 \\

100 & 3.06 & 13.649 & 42.157 \\

1000 & 3.31 & 13.665 & 45.587 \\

100000 & 3.37 & 13.665 & 46.507

\end{array}[/tex]

z & v & D_{ltt} & D_{now} \\

\hline

0 & 0 & 0 & 0 \\

0.01 & 0.010 & 0.137 & 0.137 \\

0.1 & 0.098 & 1.286 & 1.349 \\

0.5 & 0.445 & 5.019 & 6.137 \\

1.0 & 0.785 & 7.731 & 10.819 \\

1.408 & 1.000 & 9.082 & 13.779 \\

2 & 1.24 & 10.324 & 17.105 \\

3 & 1.53 & 11.476 & 21.071 \\

10 & 2.29 & 13.183 & 31.506 \\

100 & 3.06 & 13.649 & 42.157 \\

1000 & 3.31 & 13.665 & 45.587 \\

100000 & 3.37 & 13.665 & 46.507

\end{array}[/tex]

I'm not sure if that's going to help much, but there you go. I'm not even sure that the numbers are even used much for anything other than writing press releases to help people get a picture of what the universe is like. But I do think it can be worth nutting through what the numbers mean and how they are obtained to get a better appreciation of what an expanding universe is all about.

Cheers -- sylas

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Thanks Sylas. My take on this is, as long as *v/c* or *z* is a lot less than 1 (i.e. recession velocity is nonrelativistic), we have a good, consistent definition of distance. Otherwise ... http://www.urbandictionary.com/define.php?term=fuhgeddaboudit".

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Chronos

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sylas

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Sure. What makes it stand out is that it is the observable quantity, and not model dependent, and you can use it as a co-ordinate. It is equivalent to using "a" (scale factor) as a co-ordinate.

It's understandable, but it isn't a "distance", and using it to identify the edge of the observable universe tells you nothing at all. The edge of the observable universe is, by definition where the redshift diverges to infinite values. The redshift can be used as a kind of "time" like co-ordinate, however, since a given scale factor, in FRW models, identifies a hypersurface with a unique proper time... at least in models with no collapse.

It is not the same thing as light travel time, of course, which is one of the various model dependent quantities that can be calculated, given redshift and a specific model.

Cheers -- sylas

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thanks sylas that was a very good explanation

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Ich

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Some additional information on the distances sylas mentioned. I'll refer to http://www.astro.ucla.edu/~wright/cosmo_02.htm" [Broken]; there are redshift-distance diagrams for different universe models at the bottom of the page.

*at the time of emission*.

Therefore, in a flat FRW model like LCDM, it increases with redshift as we see further and further. When the source is (now) so far away that the emission time approaches the big bang, the object was actually very near then, so that the angular distance shrinks and approaches zero again for the (now) most distant objects. You can see this behaviour in the LCDM diagram to the right.

Also interesting: In an empty universe (middle diagram), FRW space is not flat, so the above reasoning is invalid. However, there is a flat space foliation of empty spacetime: the standard Minkowski metric, of course. Scroll up to the spacetime diagram in special relativistic (=Minkowski) coordinates; you see that "We also see that our past light cone crosses the worldline of the most distant galaxies at a special relativistic distance x = c*to/2.", so that the distance at the time of emission is to/2=1/(2H0). This means: H*D approaches 1/2 for the most remote objects.

So: For large z, luminosity distance does not correspond to any meaningful distance measure. The necessary correction for redshift is intentionally being omitted, as it would mix two observables. DL is a good observable, but a bad distance measure.

Now for the theoretical distances:

*not *a distance! It's rather a meaningful way of naming different members of the canonical family of observers, as it does not change with time.

[ personal opinion] Based on the unchanging comoving "distance", it is often claimed that such observers have no relative motion. In this paradigm, the increasing "actual" distance (see below) - a phenomenon normally known as "motion" - is dubbed "expansion of space". That's ok, but be warned: using every decent distance measurement you like, comoving observers actually have relative motion. Call it what you like, expansion*is *motion. [/ personal opinion]

If multiplied with an appropriately defined scale factor, it becomes the

"cosmological proper distance"

That's the distance you are most likely to see in a popular account (okay, maybe second rank after light travel time), on which all those statements like radius of the universe = 46 Gly or recession speed = 3.5 c are based.

It's called "proper" distance, because it can be traced back to what a suitable set of rulers would measure (see sylas' definition of "proper distance now"), and agrees to first order with any other sensible distance definition.

It is a common misconception that cosmological proper distance is as close to our common understanding of "distance" as it can get in cosmology. And that, consequently, every couterintuitive feature of those are a genuine physical conundrum, illustrating how cosmology defies common sense, rather than mere coordinate artifacts.

In reality, cosmological proper distance is measured with a set of rulers which are all in relative motion, while our intuitive (and SR-type) definition would be a set of rulers at rest wrt each other. The difference is very important when we deal with relativistic velocities, especially when the motion of the rulers changes with time (varying Hubble parameter).

That's an observable, and its definition is straightforward. If you choose your coordinates such that space is flat, it's clear that this distance is exactly the distanceAngular size distance. The distance you would expect normally given the angular size of an object in the sky by comparison with its real size. (DA)

Therefore, in a flat FRW model like LCDM, it increases with redshift as we see further and further. When the source is (now) so far away that the emission time approaches the big bang, the object was actually very near then, so that the angular distance shrinks and approaches zero again for the (now) most distant objects. You can see this behaviour in the LCDM diagram to the right.

Also interesting: In an empty universe (middle diagram), FRW space is not flat, so the above reasoning is invalid. However, there is a flat space foliation of empty spacetime: the standard Minkowski metric, of course. Scroll up to the spacetime diagram in special relativistic (=Minkowski) coordinates; you see that "We also see that our past light cone crosses the worldline of the most distant galaxies at a special relativistic distance x = c*to/2.", so that the distance at the time of emission is to/2=1/(2H0). This means: H*D approaches 1/2 for the most remote objects.

This is also an observabe, but it's not as straightforward as angular size distance: the luminosities are compared as if the emitting object were at rest wrt the observer, z=0. If the object is receding, as all distant sources are, their luminosity suffers from the http://demonstrations.wolfram.com/RelativisticHeadlightEffect/" [Broken]: they appear dimmer (=further away)than an object at rest at the same distance. Therefore, luminosity distance systematically overestimates the distance by a factor [itex]\sqrt{1+z}[/itex]. That's the main reason for the discrepancy between Dnow and DL in Ned's diagrams.Luminosity distance. The distance you would expect normally given the luminosity of the object in the sky by comparison with its real luminosity. (DL)

So: For large z, luminosity distance does not correspond to any meaningful distance measure. The necessary correction for redshift is intentionally being omitted, as it would mix two observables. DL is a good observable, but a bad distance measure.

Now for the theoretical distances:

One important thing here: comoving distance isThe co-moving distance is just the distance out to whatever we are looking at, but measured right now as if by a whole sequence of synchronized rulers at the same time. Same thing as "proper distance now". (Dnow)

[ personal opinion] Based on the unchanging comoving "distance", it is often claimed that such observers have no relative motion. In this paradigm, the increasing "actual" distance (see below) - a phenomenon normally known as "motion" - is dubbed "expansion of space". That's ok, but be warned: using every decent distance measurement you like, comoving observers actually have relative motion. Call it what you like, expansion

If multiplied with an appropriately defined scale factor, it becomes the

"cosmological proper distance"

That's the distance you are most likely to see in a popular account (okay, maybe second rank after light travel time), on which all those statements like radius of the universe = 46 Gly or recession speed = 3.5 c are based.

It's called "proper" distance, because it can be traced back to what a suitable set of rulers would measure (see sylas' definition of "proper distance now"), and agrees to first order with any other sensible distance definition.

It is a common misconception that cosmological proper distance is as close to our common understanding of "distance" as it can get in cosmology. And that, consequently, every couterintuitive feature of those are a genuine physical conundrum, illustrating how cosmology defies common sense, rather than mere coordinate artifacts.

In reality, cosmological proper distance is measured with a set of rulers which are all in relative motion, while our intuitive (and SR-type) definition would be a set of rulers at rest wrt each other. The difference is very important when we deal with relativistic velocities, especially when the motion of the rulers changes with time (varying Hubble parameter).

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- #17

sylas

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Some additional information on the distances sylas mentioned.

Thanks Ich, that was a very good explanation.

One important thing here: comoving distance isnota distance! It's rather a meaningful way of naming different members of the canonical family of observers, as it does not change with time.

[ personal opinion] Based on the unchanging comoving "distance", it is often claimed that such observers have no relative motion. In this paradigm, the increasing "actual" distance (see below) - a phenomen normally known as "motion" - is dubbed "expansion of space". That's ok, but be warned: using every decent distance measurement you like, comoving observers actually have relative motion. Call it what you like, expansionismotion. [/ personal opinion]

That also is precisely how I think of the matter.

If multiplied with an appropriately defined scale factor, it becomes the

"cosmological proper distance"

That's the distance you are most likely to see in a popular account (okay, maybe second rank after light travel time), on which all those statements like radius of the universe = 46 Gly or recession speed = 3.5 c are based.

It's called "proper" distance, because it can be traced back to what a suitable set of rulers would measure (see sylas' definition of "proper distance now"), and agrees to first order with any other sensible distance definition.

It is a common misconception that cosmological proper distance is as close to our comon understanding of "distance" as it can get in cosmology. And that, consequently, every couterintuitive feature of those are a genuine physical conundrum, illustrating how cosmology defies common sense, rather than mere coordinate artifacts.

In reality, cosmological proper distance is measured with a set of rulers which are all in relative motion, while our intuitive (and SR-type) definition would be a set of rulers at rest wrt each other. The difference is very important when we deal with relativistic velocities, especially when the motion of the rulers changes with time (varying Hubble parameter).

That is an excellent point. I have in the past been inclined to single out "proper distance" as the one which is closest to common understanding of distance, but as you note, the fact that the rulers are all moving (co-moving, in fact) means this not true. I take note and will revise my own attempts to describe the various distance definitions in use accordingly.

Thank you very much for this -- sylas

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DrChinese

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The age of the universe is not the same as the distance to the observable horizon...

Nice post, sylas! Includes all the key ideas together in a concise and explanatory fashion. Thanks for taking the time to write this.

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