B How to do the calculations showing the Universe is flat?

[Joshua P]

Thanks, @Bandersnatch. Why is it that the comoving distance of the event horizon is getting smaller, but the proper distance of the event horizon is getting larger?

I'm going to list my other questions again, in case anyone has an answer. (Mostly regarding: https://www.physicsforums.com/insights/poor-mans-cmb-primer-part-4-cosmic-acoustics/ .)

Why is it that comoving distance can be solved for using the scale factor, with what's written in equations 18 and 20?

I'm still unable to get to an answer. If d_s = 3c_st_dec/a_dec and d_s/d_ls =
c_s(t_dec/t₀)^1/3, I can work backwards to see that d_ls must be 3t_dec^2/3t₀^1/3/a_dec.
I can't seem to get it by integrating.
If ∫ dt/a(t) = 3t/a(t) + c, then doing this integral from t_dec to t₀ would be 3t₀/a₀ - 3t_dec/a_dec, which is the same as 3(t₀ - t_dec/a_dec). Where have I gone wrong?

@Bandersnatch
Regarding the Insights article, is the distance traveled by light/sound between two points in the universe not affected by the expansion of the universe, during the time that it is travelling?

Thanks for everyone's help so far.

 

[bapowell]

Thanks, @Bandersnatch. Why is it that the comoving distance of the event horizon is getting smaller, but the proper distance of the event horizon is getting larger?

It's a matter of whether the horizon is growing more or less quickly than the expansion rate of the universe. When the universe accelerates (as it did during primordial inflation and now), the expansion rate is such that recession velocities at the horizon are greater than the growth rate of the horizon itself; hence, its comoving size is getting smaller in time. Meanwhile, the horizon is still growing physically and so the proper distance increases.

Regarding the Insights article question, sorry for the delay in responding. The comoving distance to the last scattering surface is approximately $3t_0/a_0$. Then $d_s/d_{ls} = c_s (a_0/a_{dec})(t_{dec}/t_0)$, with $(a_0/a_{dec}) = (t_0/t_{dec})^{2/3}$. This should give you the result.

Lastly, yes, the expansion affects the distance traveled between two comoving points.

 

[Joshua P]

@bapowell thank you for the new reply.

It's a matter of whether the horizon is growing more or less quickly than the expansion rate of the universe. When the universe accelerates (as it did during primordial inflation and now), the expansion rate is such that recession velocities at the horizon are greater than the growth rate of the horizon itself; hence, its comoving size is getting smaller in time. Meanwhile, the horizon is still growing physically and so the proper distance increases.

Does this mean that the actual amount of matter in the observable universe is decreasing as time goes on, even though the physical distance of the event horizon is increasing? Is what the comoving distance indicates?

Regarding the Insights article question, sorry for the delay in responding. The comoving distance to the last scattering surface is approximately $3t_0/a_0$. Then $d_s/d_{ls} = c_s (a_0/a_{dec})(t_{dec}/t_0)$, with $(a_0/a_{dec}) = (t_0/t_{dec})^{2/3}$. This should give you the result.

Thank you! I was really confused about this. However, is a₀ not equal to 1? The scale factor today is 1, right? But that would change the equation, and it wouldn't get the right answer. So... what am I misunderstanding?

Lastly, yes, the expansion affects the distance traveled between two comoving points.

So, if it does affect it, then why don't we use the proper distances in the Insights article, as it was originally?

 

[bapowell]

@bapowell
Does this mean that the actual amount of matter in the observable universe is decreasing as time goes on, even though the physical distance of the event horizon is increasing? Is what the comoving distance indicates?

Good insight. Indeed, objects do flow across the cosmological event horizon never to return...

However, is a₀ not equal to 1? The scale factor today is 1, right? But that would change the equation, and it wouldn't get the right answer. So... what am I misunderstanding?

You can make $a_0$ anything you like: it never shows up by itself, always in proportion to the scale factor at some other time. This ratio is known as redshift, and it is insensitive to the actual numerical value of the scale factor as a function of time. The actual numbers that go into the calculation are physical: the temperature today and the temperature at decoupling.

So, if it does affect it, then why don't we use the proper distances in the Insights article, as it was originally?

The angular scale of the sound horizon at decoupling is being computed today, using today's rulers. These rulers are bigger than they were at decoupling. So, the proper distance measured with these big rulers is actually quite a bit smaller than the actual size of the sound horizon at decoupling as measured by rulers back then. By using the comoving distances, we factor out this change in ruler size.

 

[Joshua P]

@bapowell, thanks again.

Good insight. Indeed, objects do flow across the cosmological event horizon never to return...

I didn't know about that before. That is incredibly cool. And I think I understand the comoving distance better.

You can make $a_0$ anything you like: it never shows up by itself, always in proportion to the scale factor at some other time. This ratio is known as redshift, and it is insensitive to the actual numerical value of the scale factor as a function of time. The actual numbers that go into the calculation are physical: the temperature today and the temperature at decoupling.

It's in proportion to the scale factor at some other time? Is a₀ not a(t₀)? (In your calculations, is t₀ not taken to be today, just as t_dec is the time of decoupling?)

Side question: why use temperature? I used the d_s/d_ls=c_s(t_dec/t₀)^1/3 equation you wrote just before equation 21, using the age of the universe today and age of the universe before decoupling. I got the right answer, so did you use temperature mainly because you could?

The angular scale of the sound horizon at decoupling is being computed today, using today's rulers. These rulers are bigger than they were at decoupling. So, the proper distance measured with these big rulers is actually quite a bit smaller than the actual size of the sound horizon at decoupling as measured by rulers back then. By using the comoving distances, we factor out this change in ruler size.

I'm completely lost by this. The speed of light has been constant since the beginning of our universe right? So why are our distance measurements subjective?

Let me create a hypothetical situation, in hope that it'll explain my confusion better. You have two galaxies in space, their comoving velocities are 0. They would still move away from one another due to expansion, right? Is the distance that light has to travel from one of those galaxies to the other not affected by those galaxies moving away?

EDIT: I just looked at the Insights article again, you did say that "one might expect the proper distance to the horizon to be c_st_dec... but this is wrong because it neglects the effect of expansion which aids in moving the wave along". What I was thinking of as the comoving distance was exactly that, distance = speed * time, not taking into account of the movements of the starting and end points. So, what exactly is comoving distance then? In what sense does it factor out the expansion of the universe? Why are they calculated the way they are (in equations 18 and 20)?

 

[George Jones]

Does this mean that the actual amount of matter in the observable universe is decreasing as time goes on

No, the actual amount of matter in observable universe in increasing. If something is outside the cosmological event horizon, then we have never seen it the past, we don't see it in now, and we will never see it in the future. It is possible, however, for something that we have not yet seen to be inside the event horizon. If this is the case, then at some time $t_1$ in the future we will first see the object. Once we see an object, we never lose sight (in principle) of the object, ever. This means that some objects move into the observable universe, and that no objects leave the observable universe.

 

[Joshua P]

@George Jones
Now I'm very confused. So what does it mean that the comoving distance of the event horizon is decreasing?

 

[George Jones]

@George Jones
Now I'm very confused. So what does it mean that the comoving distance of the event horizon is decreasing?

Are you familiar with spacetime diagrams in special relativity?

 

[bapowell]

It's true that there are points comoving with the expansion that begin inside the event horizon that eventually exit it. A galaxy at this point is leaving the cosmological horizon, though it will still be observable, hence George's clarification.

 

[Bandersnatch]

@George Jones
Now I'm very confused. So what does it mean that the comoving distance of the event horizon is decreasing?

Here, let me jump in. The two statements from posts #34 and #36 might seem contradictory at a first glance, but are in fact not.

If you think about galaxies NOW, some of them have just left the event horizon, so any light they emit NOW will never reach us. This is the sense @bapowell was talking about. They could communicate with us yesterday, but from now on they'll never be able to. So every day there's less galaxies available with which we could communicate by sending signals today (or vice versa).

But these same galaxies have been emitting light prior to getting past the horizon, and that light is still en route to be observed by us some time in the future. The closer to the horizon the light was emitted, the longer it'll take to see that light, with light emitted just at the horizon reaching us in infinite future = we'll always see these galaxies, but the image will be increasing more outdated.
Furthermore, there are galaxies which have long since passed the event horizon, but the light they emitted prior to doing that (long time ago) hasn't yet managed to get to us - but it will. So even though we will never be able to see them as they are NOW, we will eventually be able to observe them as they were in the past. In this sense, we get to see more galaxies every day.

All of this is actually very apparent on the graphs I posted earlier (#28). If you care, I once made a post elsewhere on how to read them, to be found here:
https://www.physicsforums.com/threa...increase-bc-of-expansion.912881/#post-5754083
Having said that

If something is outside the cosmological event horizon, then we have never seen it the past

I don't think that bit is right, unless you mean something else than I think you mean.
For example, a galaxy at z=10 is both outside the event horizon at the present epoch, and have been observable ever since something like 8 Gyr ago.

It is possible, however, for something that we have not yet seen to be inside the event horizon

Same thing. Such a something would have to be both outside our past light cone and be inside the event horizon at the present epoch - which is doable for events, but not for galaxies. If I'm reading the light cone graphs right, the last time that statement was true was when the z=10 galaxies were leaving the EH. All galaxies since then had already crossed the event horizon by the time they became observable.

 

[bapowell]

@bapowell, thanks again.
Side question: why use temperature? I used the d_s/d_ls=c_s(t_dec/t₀)^1/3 equation you wrote just before equation 21, using the age of the universe today and age of the universe before decoupling. I got the right answer, so did you use temperature mainly because you could?

Sure, you can use time if you want. I used temperature because I derived expressions for $T_0$ and $T_{dec}$ in an earlier article in the series.

Let me create a hypothetical situation, in hope that it'll explain my confusion better. You have two galaxies in space, their comoving velocities are 0. They would still move away from one another due to expansion, right? Is the distance that light has to travel from one of those galaxies to the other not affected by those galaxies moving away?

I hate to throw more references at you, but I wrote an Insights article a while back on expansion and horizons: https://www.physicsforums.com/insights/inflationary-misconceptions-basics-cosmological-horizons/. I think it will clear up much your concern regarding comoving distances.

 

[Joshua P]

Sorry for the delay in response. I was taking some time to process.
Thanks @Bandersnatch for helping clear things up.

All of this is actually very apparent on the graphs I posted earlier (#28). If you care, I once made a post elsewhere on how to read them, to be found here:

https://www.physicsforums.com/threa...increase-bc-of-expansion.912881/#post-5754083

So, after reading through this, I've realized I made a few false assumptions before. I believed that the event horizon was the edge of the observable universe. In this article, you explained that: "The presence of the event horizon indicates that no amount of waiting can make the observer at the 0 line see all the events."
This has made everyone's statements make a lot more sense.
What I don't understand is why this horizon exists. The horizon is a finite distance away, so why would it take an infinite amount of time for that light to reach us? I know it's due to the expansion, but I can't seem to intuitively grasp why.

I hate to throw more references at you, but I wrote an Insights article a while back on expansion and horizons: https://www.physicsforums.com/insights/inflationary-misconceptions-basics-cosmological-horizons/. I think it will clear up much your concern regarding comoving distances.

@bapowell, I love references. Thank you. I haven't had time to read this completely yet, and I'm sure it contains some answers to some of my questions above. I wanted to ask a couple questions about what I've read so far.

The angular scale of the sound horizon at decoupling is being computed today, using today's rulers. These rulers are bigger than they were at decoupling. So, the proper distance measured with these big rulers is actually quite a bit smaller than the actual size of the sound horizon at decoupling as measured by rulers back then. By using the comoving distances, we factor out this change in ruler size.

Fig. 2 shows that two objects with no peculiar velocity will move apart, but that the comoving distance between them stays the same (8 units), right? The units themselves increase. But isn't it the comoving units increase rather than the proper distance units? How does using comoving units facor out the change in ruler size?

Thanks, I'll keep reading the article.

 

[Bandersnatch]

What I don't understand is why this horizon exists. The horizon is a finite distance away, so why would it take an infinite amount of time for that light to reach us? I can't seem to intuitively grasp why the expansion would change this.

Imagine light emitted exactly at the event horizon. Let's say for simplicity it's in the far future, when the Hubble constant will have functionally stopped decreasing, and the event horizon coincides with the Hubble sphere (marking recession velocity = c) at some constant distance from the observer.
Then for every light-second this light signal makes traveling towards the observer, the expansion will carry it away one light-second away. It will never make any headway, and it will never be able to reach the observer. It would effectively 'hover' at that constant distance for eternity.
But if it was emitted just a tiny bit closer, even an infinitesimal distance closer, then for every light-second it travels, it'll be carried away by expansion just a little bit less than one light-second. And that tiny bit will mean that the signal will find itself even farther away from the event horizon, where expansion rate is even less, which will allow it to make even more headway, and eventually reach the observer.
The closer to the event horizon was the initial emission, the longer it takes to make those initial little advances. At the limit of the event horizon, it takes forever.

This should be actually explained somewhere in bapowell's second article, and much more clearly. And with visual aids to boot.

Also there's this often used analogy of an ant walking on a rubber band being stretched. If that band is 100 cm long, and it's being stretched by 1% every second, and there's an ant walking from one end to another at 1 cm/s, then it'll never get there. For every 1 cm it travels, the stretching of the band will take it back 1 cm.
But it the ant starts at 99 cm from the other end, then the stretching will take it back only 0.99 cm, so after 1 second it finds itself 98.99 away from its destination. The next second, the stretching carries it back 0.9899, i.e. even less than the last time.
In cosmological terms, the 1% is the Hubble parameter (in the far future when it's no longer decreasing); the 100 cm distance is the event horizon; the 1cm/s speed of ant is the speed of light; and the distance to which the initial starting point would have receded by the time the ant finally gets to the observer is the particle horizon, or what we'd call proper radius of the observable universe - remember that even as the ant was making these little, almost imperceptible headways every second, that initial distance was expanding exponentially (at $100*(1.01)^n$ for n seconds).

 

[Joshua P]

@Bandersnatch That actually makes a whole lot of sense. Bapowell's article does talk about how light is affected by the expansion, just as you described. Thanks.

The only thing I didn't quite get was the particle horizon / proper radius of the observable universe part. On the graphs you showed, the particle horizon is outside the event horizon. How can the edge of the observable universe be outside the event horizon?

 

[Bandersnatch]

How can the edge of the observable universe be outside the event horizon?

I'm not sure I can explain it better than with the ant analogy. The particle horizon is the distance to which the point on the expanding band where the ant was initially (at emission) would have receded by the time the ant (light) gets to the destination (is observed). That distance can be much larger than the event horizon, depending on where exactly the ant started from.

Event horizon tells you from where light emitted NOW can ever reach you. Particle horizon tells you something about the whereabouts of the emitters whose light was emitted 13.8 Gly ago.

Or maybe let's use the CMBR as an example. When the light we see today as CMBR was emitted, the emitter (hot plasma) was some 42-ish million light years away from another bit of hot plasma that eventually formed our galaxy. This light had to make slow headway against the expanding space, and after 13.8 billion years it finally reached us. That it did reach us tells you that initially the emitter must have been within the event horizon (however large it was back then).
During these 13.8 billion years, the original emission point was being carried away by expansion to its current (proper) distance of ~45 billion light years. This means that currently it is way beyond the event horizon, so whatever it emits now will never reach us.
Since it has left the event horizon long ago (I'm eyeballing the graphs here, but it looks like less than 1 Gyr after emission - its redshift is 1090, so it's around the dotted lines marked with 1000), we will never see the likely galaxies which will have evolved from that plasma whose glow we observe today as CMBR.

Also, and this is I believe shown in bapowell's second insight as well, the particle horizon has an equivalent meaning of: 'proper distance which the light emitted by us (meaning by the plasma at our location when CMBR was emitted) has managed to cover by now'. That's why the graphs show it as if it were a path of light starting at the bottom of the graph at the 'here' point, and going out.
The aliens in a galaxy at 45 Gly see our light as their CMBR. Tomorrow it'll be some aliens a bit further away.

 

[bapowell]

Fig. 2 shows that two objects with no peculiar velocity will move apart, but that the comoving distance between them stays the same (8 units), right? The units themselves increase. But isn't it the comoving units increase rather than the proper distance units? How does using comoving units facor out the change in ruler size?

There are no "comoving units" and "proper distance units"; there are only measuring rods that grow with the expansion. In Fig 2, the proper distance of the points grows on account of the expansion. But since neither point moves relative to the expanding space, the comoving distance, $r$,---the distance measured with the measuring rod that is itself growing with the expansion---does not change. It is in this sense that comoving distance "factors out" the expansion.

But notice that the proper distance, $x$, depends explicitly on the length of the measuring rod: for two points A and B with fixed separation (not growing with the expansion), as time goes on the ruler gets larger and the proper distance we'd infer from a measurement at later times would be smaller than at earlier times. To correct for this, we must rescale proper distance measurements by the scale factor; but this is just the comoving distance: $r = x/a$. This is why we use comoving distance in the computation of the angular scale at decoupling: because the sound horizon at decoupling is a fixed length.

 

[Joshua P]

@Bandersnatch Thanks, no questions. That cleared things up perfectly.
@bapowell, I'm still confused about this particular topic. Thanks for the response.

There are no "comoving units" and "proper distance units"; there are only measuring rods that grow with the expansion. In Fig 2, the proper distance of the points grows on account of the expansion. But since neither point moves relative to the expanding space, the comoving distance, $r$,---the distance measured with the measuring rod that is itself growing with the expansion---does not change. It is in this sense that comoving distance "factors out" the expansion.

The comoving distance doesn't change between the points in Fig. 2, and it is only in this sense that these points can be "8 units" apart both before and after a period of time in which space has expanded (right?). The proper distance between A and B would be seen to increase as space expanded, right?

But notice that the proper distance, $x$, depends explicitly on the length of the measuring rod: for two points A and B with fixed separation (not growing with the expansion), as time goes on the ruler gets larger and the proper distance we'd infer from a measurement at later times would be smaller than at earlier times. To correct for this, we must rescale proper distance measurements by the scale factor; but this is just the comoving distance: $r = x/a$. This is why we use comoving distance in the computation of the angular scale at decoupling: because the sound horizon at decoupling is a fixed length.

Why would the distance between two points with fixed separation (unaffected by expansion) be measured differently at two different times, if we're using proper distances?

 

[bapowell]

The comoving distance doesn't change between the points in Fig. 2, and it is only in this sense that these points can be "8 units" apart both before and after a period of time in which space has expanded (right?). The proper distance between A and B would be seen to increase as space expanded, right?

Right.

Why would the distance between two points with fixed separation (unaffected by expansion) be measured differently at two different times, if we're using proper distances?

Let's try this. The size of the sound horizon had a certain proper length at decoupling ($3c_s t_{dec}$), and this length has continued to grow along with the expansion up to the present time. It is this distance---that at the present time---we are working with when we compute the angular scale of the sound horizon at decoupling. Now, the comoving distance between two points A and B is always equal to the proper distance between them measured at the present time (since $a_0 = 1$). The comoving sound horizon at decoupling is $3 c_s t_{dec}/a_{dec}$, from the general definition $x_{prop} = a x_{com}$. Division by $a_{dec}$ has the effect of rescaling the proper distance at a particular time ($t_{dec}$) to a proper distance today.

BTW, I apologize for the lengthy interval between question and response: things have been quite busy around here lately.[/USER]

 

[Joshua P]

Let's try this. The size of the sound horizon had a certain proper length at decoupling ($3c_s t_{dec}$), and this length has continued to grow along with the expansion up to the present time. It is this distance---that at the present time---we are working with when we compute the angular scale of the sound horizon at decoupling. Now, the comoving distance between two points A and B is always equal to the proper distance between them measured at the present time (since $a_0 = 1$). The comoving sound horizon at decoupling is $3 c_s t_{dec}/a_{dec}$, from the general definition $x_{prop} = a x_{com}$. Division by $a_{dec}$ has the effect of rescaling the proper distance at a particular time ($t_{dec}$) to a proper distance today.

Okay, but why do we want the length of the sound horizon as it would be today? The CMB shows the sound horizon as it was at the actual time of decoupling, right? The sound horizon as it was at the time of decoupling is what we actually see now (on the CMB), so why aren't we using the proper distance of that?
Also, why didn't switching from proper distances to comoving distances change your result?

BTW, I apologize for the lengthy interval between question and response: things have been quite busy around here lately.

After all you've helped me, you've no reason to apologise. I'm also very busy at the moment. Thanks for making time.

Another question about the article (https://www.physicsforums.com/insights/inflationary-misconceptions-basics-cosmological-horizons/): On Fig 6, why is the yellow shading on the upper half? Any light we emit after t=0 will not reach them by time T. So it wouldn't classify as an "event that we have influenced by time T", right? Am I reading the graph wrong? (Maybe I'm just too tired.)
EDIT: Yes, I just completely misinterpreted the graph.

 

[bapowell]

The sound horizon as it was at the time of decoupling is what we actually see now (on the CMB), so why aren't we using the proper distance of that?

No, what we actually see now is the length of the sound horizon at decoupling after it has expanded along with the universe to the present time.

Also, why didn't switching from proper distances to comoving distances change your result?

.
Ha! Because what I calculated and what I typed were not consistent. I used the comoving distance to compute the answer, but when I typed up that part of the article I erroneously referenced the proper time

Another question about the article (https://www.physicsforums.com/insights/inflationary-misconceptions-basics-cosmological-horizons/): On Fig 6, why is the yellow shading on the upper half? Any light we emit after t=0 will not reach them by time T. So it wouldn't classify as an "event that we have influenced by time T", right? Am I reading the graph wrong? (Maybe I'm just too tired.)
EDIT: Yes, I just completely misinterpreted the graph.

Well, you are correct that light we emit later along our worldline won't reach "them" until after time T, but, yeah: everything within the yellow wedge can be causally connected to some position along our worldline.

 

[Joshua P]

No, what we actually see now is the length of the sound horizon at decoupling after it has expanded along with the universe to the present time.

Oh. That's because of the expansion of space? The photons spread out as they move towards us?

 

[bapowell]

Yes. More generally, all length scales grow with the expansion. I understand where you're getting caught up: the horizon size at decoupling is a physically significant length scale, and we should care about this distance at that time. Indeed, we do. But, that distance, say it was a million light years, has since increased along with the expansion and so the angle it subtends today on the last scattering sphere depends on its size today.

 

[Joshua P]

@bapowell thanks, I think I get it now.

This may be the first time I don't really have a question. I'm sure I'll have one soon.

Thanks to everyone for their help.