# Size of the *observable* universe (noob questions about)

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## Main Question or Discussion Point

Size of the *observable* universe

Everybody should know that General Relativity allows distances to expand faster than the speed of light, and indeed it predicts that in many circumstances they will do so.

This is hardly news, Friedman figured out the standard expansion cosmology model in 1921 (long before Hubble found a pattern of expansion in the redshifts) simply on the basis of vintage 1915 General Relativity. Already in that 1921 model many of the distances increase at rates faster than light. It is so-to-speak typical. No object is going anywhere, i.e. approaching a destination. It's dynamic geometry, not ordinary motion.

Newcomers frequently show up puzzled by the usual estimate of the size of the observable (the stuff that we have already received light from).

It may help them to focus on an example which contains the essentials of what I think confuses them.

EXAMPLE: REDSHIFT Z = 1090. The Cosmic Microwave Background.

Microwave background radiation started life as orangish light in a hot fog, around 380,000 years after start of expansion. And its wavelengths are now longer by a factor of about 1090. The original mix of visible and infrared wavelengths have been extended by the same factor that largescale distances have increased during the intervening years while the light was in transit.

One way you can really focus in on this is to google "wright calculator" and put in 1090 for the redshift and see what you get.

It would be a big help if everybody had some experience with the standard cosmo model which is built into that calculator. This is at the heart of cosmology. It implements Friedman's 1921 model, with modern parameters determined from millions of datapoints of observation. The model has been amazingly successful. You should get some hands-on experience with it.

If you do that, put in z = 1090 and press "general" to get it to compute, it will tell you how far the crud was then, when it emitted the orangish light. And it will tell you how far the same stuff is now, today, as we are receiving the light in its present microwave form.

The distance then, it will tell you, was only 42 million lightyears (look where it says "angular size distance"). And the distance now, if it could be measured today by the concerted efforts of a chain of observers using standard radar ranging, would be 46 billion lightyears (that is where it says "comoving distance".).

In other words the distance to the crud has increased by a factor of around 1090. And neither that matter nor our matter (what became us) has moved significantly. The light has taken 13.7 billion years to get here. The matter that emitted it is 46 billion lightyears farther now than on the day the light was emitted.

I'm rounding off: 46 billion - 42 million = 46 billion. The little distance makes virtually no difference.

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## Answers and Replies

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Maybe I'm going about this the wrong way. I want this to be a completely non-technical thread that responds proactively to the kind of newcomer "Size of Universe" thread that we seem to be getting a couple of times a week.

If I'm off on the wrong foot, we'll find out and I'll just have to abandon this one and start over. What I'm responding to is when someone comes in and says

"The estimated age of expansion is 13.7 billion years. And I read that the current radius of the observable universe is 46 billion lightyears. All the stuff we can now see, all the stuff in the observable universe was concentrated in a small volume around the start of expansion. So how did some of it get out to 46 in a time of only 13.7?"

This is a good question and shows the person was thinking. How does one answer it? How do you? If someone has an idea they want to share of how to respond, please post it.

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I think these are some points to eventually make, maybe not in this order.

1. We're talking about the standard cosmology model. It fits the data well but it could still be modified especially as regards the very early universe.

2. The Background gives us a useful criterion of actual motion. If you are at rest relative Background then it is roughly the same temperature in all directions. If you are going somewhere you will see a Doppler hotspot ahead of you. Galaxy individual motions are small compared with typical expansion rates. Distance expansion is not motion.

3. Cosmology uses a concept of current distance where you imagine a chain of observers, each at rest relative to Background, co-operating so as to measure the distance at some given moment. It's called "comoving", to distinguish it from various alternative distance measures. The Hubble Law is in terms of this. It tells you that the presentday rate of increase of the current distance between two widely separated observers each of which is at rest is proportional to the current distance.

A more concise way to say the Hubble rate, in ordinary terms, is that at present largescale distances are increasing around 1/130 to 1/140 of a percent every million years. It doesn't sound like much but when you are talking about very large distances a small percentage like 1/130 of a percent can amount to something substantial.

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Now what I've just said is a bit dry and complicated. Maybe it is not right for the guy who shows up here and says "How did that stuff get out there 46 from here in a time of only 13.7???!!!"

4. Well, the stuff that is 46 billion lightyears from us, wasn't doing anything different from us. That matter wasn't moving, any more than we are!

5. It's reciprocal. Back in year 380,000 that matter over there was only 42 million lightyears from us and it sent us some of the CMB radiation we're now receiving.
And our matter, what eventually became us, sent CMB radiation off in their direction too. And they are receiving it now. It is not like they moved and we stood still, or vice versa.
And now their neighborhood probably looks pretty much the same as it does here. In the old days both places were hot partly ionized gas and now they've both thinned and curdled to stars and galaxies. Neither has moved but the distance has grown from 42 million to 46 billion.

6. The nub of it, the essential catch, is you have no right to expect geometry to always be Euclidean and static. In fact General Relativity is primarily a theory of dynamic geometry and it tells you precisely under what circumstances you can expect geometry to be Euclidean, and when you should expect it to be different.

In a sense it explains when and why you can expect pythagoras to be right, and why you can expect triangles to add to 180 degrees, approximately and most of the time, and it tells you under what circumstances you can't expect that to be a good approximation. It is the Mother of Geometry, in effect, and is what underlies both the geometry you learn in highschool and the geometry of Special Relativity. The upshot is, since GR is in charge of geometry, you can't expect it to follow Euclidean or Special Rel rules. If GR says for large distances between stationary observers to increase then they increase.

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Ich
Sorry to jump in again without am answer, but I believe it's appropriate to make some fine points - for the expert-level internal discussion, so to say.
So how did some of it get out to 46 in a time of only 13.7?
At v=.99 c, I can get to the Andromeda Galaxy (~2.5 million ly away) in ~350,000 years proper time. That makes v'~ 7 c. That's quite a lot, but does not violate the well known SR rule, because v' is not the standard way of measuring speeds.
How does this apply to your post?
And the distance now, if it could be measured today by the concerted efforts of a chain of observers using standard radar ranging, would be 46 billion lightyears
There's one point you didn't mention: you're not referring to standard radar ranging, but to a quite tricky setup: each radar station is meant to be comoving, i.e. at rest wrt the Hubble flow, but not to one another.
If you measured distance by standard radar ranging, with each station at rest wrt the origin, you'd either measure v<c or encounter a horizon and measure nothing at all.
I think it's important to emphasize that "relative velocity" as known in SR is not really well defined for distant objects in a curved spacetime, and that the "velocity" one is referring to is (conveniently, and with some reason) defined in a non-standard, but globally valid way. Similar to x/t' or x'/t in SR.

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The nub of it, the essential catch, is you have no right to expect geometry to always be Euclidean and static.
That's ok, but our definition of coordinates make for quite the same paradoxes in flat spacetime. Really, superluminal veloctities, in cosmology or not, do not need curved spacetime. Non-standard coordinates will do as well.

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Hello Ich,

The chain of observers, each of them at rest, is not intended to measure speed. They are intended to determine the distance (on that day when the measurement is planned for). This chain is a scheme that Ned Wright presented to give a pedagogical concrete idea of current distance.

You want enough observers so that each can be close enough to his neighbor so that he can measure the distance by an ordinary radar method in a short time. It is only a thought experiment of course and would involve a huge number of observers. They are close enough together (a light day? a light year? ) so they do not notice any expansion. Even if they are a million lightyears apart the expansion is only on the order of 1/140 of a percent, perhaps we will allow that much slop.

So, if you will allow me, I will continue to think of this as ordinary radar distance ranging (expansion effects too small to measure or to bother with). Add up the separation between neighbors to get the total distance.

You probably already understood this and I didn't need to say it. But it sounded from your post as if you thought I was talking about measuring a speed or increase rate, you used the symbol v.
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My aim here is to think about the newcomer with that recurrent question about the size of the observable universe and how best to respond. I think one should respond in a simple way that introduces common cosmology concepts like Hubble Law (which means co-moving distance and the concept of rest relative to CMB). This is partly for efficiency. Many come specifically with the figure of 46 billion lightyears for the radius of the observable. I think they could benefit from having that explained, if it could be done clearly in brief. Do you have some ideas of a helpful way to respond?

Really, superluminal veloctities, in cosmology or not, do not need curved spacetime. Non-standard coordinates will do as well.
My feeling about the non-standard coordinates you mention is that when a newcomer arrives with a really basic question about usual cosmology, the most common model and its numbers, then it is not the appropriate time to hit him with non-standard coordinates. There is too much risk of confusion. So in the context of this thread I am not interested in issues about non-standard coordinates. We need a consistent response that is as simple and coherent as possible.

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Ich
My aim here is to think about the newcomer with that recurrent question about the size of the observable universe and how best to respond.
That's my aim too, and I don't know yet what the best response would be. So please bear with me when I'm thinking out loud and take a "destructive" approach at first: defining what imho should not be said.
You know, I'm coming from the relativity subforum. There are severe penalties over there for talking about "actual motion". Never use this phrase, it does more harm than anything else.
2. The Background gives us a useful criterion of actual motion. If you are at rest relative Background then it is roughly the same temperature in all directions. If you are going somewhere you will see a Doppler hotspot ahead of you. Galaxy individual motions are small compared with typical expansion rates. Distance expansion is not motion.
We can't tell whether anything is actually moving. But we can tell when two things are in relative motioin wrt each other. For example, we are moving with ~600 km/s relative to the background. Or (not too) nearby galaxies are moving away from us. That's measurable. As these are themselves essentially at rest wrt their background, one must conlude that their background is moving away from ours.
A statement like "Expansion is not motion" is not generally justified by physics, it is how we choose to describe things when we use comoving coordinates. And it's dangerous, there are zillions of misconceptions that stem from this often used phrase: cosmological redshift cannot be explained by a doppler shift; meter sticks (and atoms) should expand also; if two particles start at rest wrt each other, their distance will increase according to the expansion; two particles at rest wrt each other should exhibit redshift; local celestial physics if fundamentally different from what one would expect using Newtonian or post-Newtonian reasoning...

Concerning "proper distance":
You want enough observers so that each can be close enough to his neighbor so that he can measure the distance by an ordinary radar method in a short time. It is only a thought experiment of course and would involve a huge number of observers. They are close enough together (a light day? a light year? ) so they do not notice any expansion. Even if they are a million lightyears apart the expansion is only on the order of 1/140 of a percent, perhaps we will allow that much slop.
If it were 1/140 of a percent, I would not argue.
I've played with Excel doing some numerical calculations, and came up with the following numbers: If we measured distance with the standard approach (meter sticks or radar chain, but every link at rest wrt its neighbours), it would not be 46 Gly, but 14.79 Gly. That difference is significant, and it explains most of what you wanted to explain - "How did that stuff get out there 46 from here in a time of only 13.7???!!!" - in a simple way: It didn't.
The difference is not because the measuring rods (or distances between radar observers) are getting larger. It is because cosmological distance is measured with moving rods, which are Lorentz-contracted. Or, in a more abstract way: one distance uses standard Einstein simultanteity, the other uses slices of constant proper time - similar to my Andromeda example.

I'll think more about your question, maybe I can come up with some more constructive input.

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...
You know, I'm coming from the relativity subforum...
Cosmology is not the same as abstract General Relativity. Cosmology deals with particular solutions of the Einstein equation, and has the extra feature of the Cosmic Microwave Background. This (among several other things) makes it special, as I'm sure you realize.

... If we measured distance with the standard approach (meter sticks or radar chain, but every link at rest wrt its neighbours), ...
Ich, you call that "standard" but I do not. I always specify that each observer should be at rest with respect to CMB.

In that case one gets a useful idea of distance which is commonly used in cosmology. It comes up in the calculators students use and in the Hubble Law. So there is no cognitive dissonance.

By contrast, with what you call "standard", if there is a long enough chain of observers all at rest wrt neighbor, then some of the observers are going to be burning up . Your observers will, some of them, be roasted by the Cosmic Microwave Background. It is their fitting punishment for trying to implement a perverse idea of distance .

And this proceedure you call "standard", besides burning up the observers, will result in an unfamiliar figure of distance which conflcts with what students typically see when they use the online cosmology calculators (which for practical purposes are the actual standard for us---they implement the standard cosmo model.) If your proposal were adopted it would cause confusion and waste everybody's time.

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... Even if they are a million lightyears apart the expansion is only on the order of 1/140 of a percent, perhaps we will allow that much slop...
...
If it were 1/140 of a percent, I would not argue.
...
Maybe there are other people besides Ich who don't get why million lightyear separation leads to slop on the order of 1/140 percent, so I should explain. I have been thru this kind of discussion several times already in other threads, but this is a good place to repeat.

In Cosmology we have a preferred global time which is observable*. Variously called universe time, Friedman time, cosmic time. Of course it can only be measured to some approximation. One way is by measuring the CMB temperature, which one can do only with finite accuracy.
(Whole sky average with dipole removed.)

There is also an absolute idea of rest, and simultaneity. Statements of the Hubble Law and the usual models in Cosmology employ these concepts.

So likewise does the routine idea of distance. That is the distance at a particular moment of universe time. As an instructive thought experiment, one can imagine measuring this---to give a kind of operational meaning.

Let's use some definite numbers so that we can see how "1/140 of a percent slop" comes into the picture if we put the observers about 1 million lightyears apart.

Imagine we have planned ahead and stationed around 1000 observers in a long line from A to B. They are all at rest relative to Background and they all have clocks telling Universe time. The plan is for them all simultaneously to measure the distance to their neighbor in the B-wards direction. Then we will add up the roughly 1000 increments to get the total distance.

But since they each use radar and they are approximately 1 million lightyears apart, it will take two million years for each to measure the distance to his neighbor!
This introduces imprecision on the order of 1/140 of one percent. Because distances out in open space are increasing roughly 1/140 percent per million years. One can allow for this, but it does cause some slop.
One could also reduce that slop by having more observers and stationing them closer together, so that the individual radar measurement takes a shorter time, say on the order of 1000 years instead of 1 million.

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*A good thing always to point out is that in contrast to Cosmology, in abstract General Relativity there is no global idea of time that is observable.
With a given system of coordinates one has the coordinate time, but this has no physical meaning and is not observable. And then for a particular observer there is that observer's time, but there is no unique natural choice. But here we are not working from the parochial standpoint of pure abstract General Relativity, so there are additional features to the picture.

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Ich
Hi marcus,

it seems that you completely misunderstood my post.

If your proposal were adopted it would cause confusion and waste everybody's time.
I did not make a proposal for a compulsory coordinate system. You, and the noob, may use wichever you want. And I would strongly advise that you teach the noob the basics of comoving coordinates. But don't tell him fairy tales.

A noob is wondering how something came so far in such a short time.
Why is he wodering? Obviously because he heard that nothing can go faster than light.
The basic notions of "distance" and "time" have a clear meaning in SR, where this speed limit comes from. In cosmology, however, we choose to use a different definition. That's how those numbers (13.7/46) arise.
If we adopted the SR definition - as close as possible in a generally curved spacetime - we'd have (13.7/14.7) instead. Maybe with some handwaving one could attribute the remaining difference to gravitational time dilatation in these coordinates. (I'm not sure how to do that right now.)
You don't have to adopt these coordinates, but you should recognize two things:
1. The noob's intuition is based on SR coordinates (if any). That's where he comes from.
2. If you use different coordinates, say so. And explain the difference, or at least say clearly that there are significant differences.

Ich, you call that "standard" but I do not.
You call it standard, too. But it's not standard cosmological coordinates.
In that case one gets a useful idea of distance which is commonly used in cosmology. It comes up in the calculators students use and in the Hubble Law. So there is no cognitive dissonance.
There is, as long as the students are led to believe that this idea of distance is as close as possible to what they're used to. It is not.
By contrast, with what you call "standard", if there is a long enough chain of observers all at rest wrt neighbor, then some of the observers are going to be burning up . Your observers will, some of them, be roasted by the Cosmic Microwave Background. It is their fitting punishment for trying to implement a perverse idea of distance .
They left their lives in the heroic effort to implement a simple radar chain. What's perverse about radar, radio detection and ranging?
And this proceedure you call "standard", besides burning up the observers, will result in an unfamiliar figure of distance
No. Everyone (except maybe some cosmologists) is familiar with this type of distance. It is the very definition of distance.
Students instead have to learn why cosmological distance is being used, how it is defined, and that it is fundamentally different.
Maybe there are other people besides Ich who don't get why million lightyear separation leads to slop on the order of 1/140 percent, so I should explain.
I did get it, because you used that 1/140% figure before. It's completely irrelevant to that discussion.

And then for a particular observer there is that observer's time, but there is no unique natural choice.
A unique natural choice does not forbid other choices, especially if they are as unique and natural, but for a different purpose. I ignore that "parochial" remark of yours, which is quite out of place.

Meanwhile, I have something constructive to say, too. But I'm not sure that this is the right place.