How to do the calculations showing the Universe is flat?

[Joshua P]

I've been trying to understand how we know that the observable universe is flat, and I'm having difficulty finding any sources that explain exactly how the calculations were done. On this WMAP website (https://map.gsfc.nasa.gov/mission/sgoals_parameters_geom.html), it says:
"A central feature of the microwave background fluctuations are randomly placed spots with an apparent size ~1 degree across. These are produced by sound waves that travel through the hot ionized gas in the universe at a known speed (the speed of light divided by the square root of 3) for a known length of time (375,000 years). By using the relation: distance = rate * time, we can infer the distance the sound travels, and thus the actual size of a typical hot (compressed) or cold (rarefacted) spot. By comparing the apparent size of the spots to their known actual size, we can measure a combination of the distance to the last scattering surface and the curvature of the light path between us and this surface, which depends on the geometry of the universe. Then If we independently know the Hubble constant, we can determine the distance to the last scattering surface and thus use the spot size to determine the geometry uniquely."
I was wondering how the "the actual size of a typical hot (compressed) or cold (rarefacted) spot" was calculated? How was the Hubble constant used? How did they ultimately show that the spots should be 1 degree across in a Euclidean universe? I understand that it should be basic Euclidean geometry, but I'm not quite understanding the problem. Please keep it simple for me to understand.

 

[Bandersnatch]

Hi Joshua P, welcome to PF!

Have a read through this Insights article:
https://www.physicsforums.com/insights/poor-mans-cmb-primer-part-4-cosmic-acoustics/
The oscillation size is discussed and calculated towards the end (eqs. 20 & 21), but reading it all might be beneficial to get the context.
See if that answers your questions, and get back to us if you need something clarified.

 

[Joshua P]

Thank you so much for the reply. I'm getting really excited.
I've been trying to get my head around the paper for a while now. I'm not a physicist (yet), so this is quite difficult for me.

It looks like the calculated angle of 1 degree is based on the actual size of the sound horizon. Is that right?
To prove the universe is flat, the actual size of the sound horizon has to match the apparent size of the sound horizon. I know that the CMB shows that it appears to be 1 degree apart. I was looking for a source that showed why the fluctuations should be 1 degree apart in a flat universe. I'm just confirming, but that is what it's doing?

I was wondering what the value "a" represented in all the equations, I know it has something to do with the Hubble constant? Called... the scale factor?
Also, C_s, the speed of sound in the primordial plasma, was said to equal the square root of 3 in this paper. The WMAP website said that the speed of sound through this hot ionized gas was the speed of light divided by the square root of 3. What am I missing?

There are other things I don't understand, but they depend on me understanding these things I've mentioned. Thanks.

 

[Bandersnatch]

I'll have to keep this short, since I've a deadline to meet that I just can't procrastinate (any more). So just a few points for now:

I was wondering what the value "a" represented in all the equations, I know it has something to do with the Hubble constant? Called... the scale factor?

The scale factor is defined as $a(t)=r(t)/x_0$, where r(t) is any arbitrarily chosen distance (e.g., between some two test galaxies), and $x_0$ is the value of this distance at the present time.
Intuitively, the scale factor tells you how much the universe (=all distances) at some time t will be/was larger/smaller than now. a=1 today; it was equal to 1/1000 when the universe was 1/1000 the current size; it'll be equal to 2 when the universe will have grown to twice the current size.

If the scale factor grows $\dot a >0$, the universe expands, and if $\dot a<0$ then it contracts. If $\ddot a>0$, the universe accelerates, and vice versa.

Hubble parameter is the rate of expansion. It is defined as $H^2=(\frac{\dot a}{a})^2$. Its intuitive meaning is the instantaneous percentage growth of the universe at a given time. E.g., the current value of the Hubble parameter, $H_0$ i.e. the 'Hubble constant', translates to something like 1/144 % growth per million years.

If you have the time to watch Leonard Susskind's cosmology lectures (available on youtube), he covers it in the first one. It's generally a great resource if you're new to the topic.

It looks like the calculated angle of 1 degree is based on the actual size of the sound horizon. Is that right?
To prove the universe is flat, the actual size of the sound horizon has to match the apparent size of the sound horizon. I know that the CMB shows that it appears to be 1 degree apart. I was looking for a source that showed why the fluctuations should be 1 degree apart in a flat universe. I'm just confirming, but that is what it's doing?

The one degree is calculated under the assumption that the universe is flat - the Friedmann equation given in (19) and used to get the scale factor, does not include the curvature parameter k, which was assumed to be equal to 0 (=flat). The full equation is $$H^2=(\frac{\dot a}{a})^2=\frac{8piG}{3}\rho - \frac{kc^2}{a^2}$$
With $k=0$ it reduces to eq. (19).

So, again, that's the size you'd expect if the universe were flat.

Also, Cs, the speed of sound in the primordial plasma, was said to equal the square root of 3 in this paper. The WMAP website said that the speed of sound through this hot ionized gas was the speed of light divided by the square root of 3. What am I missing?

You mean in that bit after the eq. 21? It's given as $c_s^{-1}=\sqrt{3}$, so it's actually $c_s=1/\sqrt{3}$, and it uses units where the speed of light is equal to 1 - so they're actually the same.

O.k., I really have to leave you here. Maybe somebody else will jump in while I'm gone, though. Heck, let me cast a bat-signal calling @bapowell (the author), so that you may get your info straight from the horse's mouth.

 

[kimbyd]

Thank you so much for the reply. I'm getting really excited.
I've been trying to get my head around the paper for a while now. I'm not a physicist (yet), so this is quite difficult for me.

It looks like the calculated angle of 1 degree is based on the actual size of the sound horizon. Is that right?
To prove the universe is flat, the actual size of the sound horizon has to match the apparent size of the sound horizon. I know that the CMB shows that it appears to be 1 degree apart. I was looking for a source that showed why the fluctuations should be 1 degree apart in a flat universe. I'm just confirming, but that is what it's doing?

I was wondering what the value "a" represented in all the equations, I know it has something to do with the Hubble constant? Called... the scale factor?
Also, C_s, the speed of sound in the primordial plasma, was said to equal the square root of 3 in this paper. The WMAP website said that the speed of sound through this hot ionized gas was the speed of light divided by the square root of 3. What am I missing?

There are other things I don't understand, but they depend on me understanding these things I've mentioned. Thanks.

Yes, that's a big part of it. The other way to look at it is to compare the average distances between relatively nearby galaxies to the distances expected from the CMB fluctuations. This large difference in distance between these galaxies and the surface of last scattering provides a long lever-arm with which to measure curvature.

 

[Joshua P]

Thank you so much, @Bandersnatch. You've been an amazing help.
@kimbyd, that's interesting. Distances of what objects are expected from the CMB fluctuations? The surface of last scattering? How does the difference in distance help measure curvature?

I have a bunch more questions about the Insights article (https://www.physicsforums.com/insights/poor-mans-cmb-primer-part-4-cosmic-acoustics/), if anyone else would be willing to answer them. (No need to answer all of them at once, of course.)

How does equation 19, the Friedman equation (where we've assumed k = 0) show that density is proportional to a^-3?
How did they then "find that in a matter dominated universe the scale factor grows as a power-law, a(t)∝t^2/3"?
I was able to work out how they got from there to calculate d_s, but I'm having difficulty doing the same thing to find x_ls from equation 20. Can anyone show me how its done?
And how do the units cancel out if c_s is measured in terms of the speed of light? Are the distances measured in lightyears?

 

[George Jones]

How does equation 19, the Friedman equation (where we've assumed k = 0) show that density is proportional to a^-3?

A heuristic way to see this: consider a box of volume $V$ that contains $N$ particles that each have mass $m$, so that the density of the stuff in the box in $\rho = Nm/V$. If the box follows the expansion of the universe, and if the particles don't have peculiar velocities, then the number of particles in the box stays constant as the box expands. Without loss of generality (WLOG), assume that the box is $a \times a \times a$ (with $a$ the scale factor of the universe, so that $V = a^3$. At time $t_1$, density is $\rho_1 = Nm/a_1^3$; at time $t_2$, density is $\rho_2 = Nm/a_2^3$. Hence, $\rho_2/\rho_1 = a_1^3/a_2^3$, i.e., the density of non-relativistic matter is inversely proportional to the cube of the scale factor.

A less heuristic way to to see this uses local conservation of energy and the specific form of the connection coefficients for Friedmann-Lemaitre-Robertson-Walker (FLRW) universes to give an equation that resembles the first law of thermodynamics,
$$\begin{align}
\frac{d}{dt} \left( \rho a^3 \right) &= -P \frac{d}{dt} \left( a^3 \right) \\
a^s d\rho + 3\rho a^2 da &= -3P a^2 da \\
a \frac{d\rho}{da} &= -3 \left( \rho + P \right)
\end{align}$$
In the above $c = 1$, and $P$ is pressure, which can be taken to be zero, since for non-relativistic matter, $P$ is much smaller than $\rho$. This gives
$$\begin{align}
\frac{d\rho}{\rho} &= -3 \frac{da}{a} \\
\int_{\rho_1}^{\rho_2} \frac{d\rho}{\rho} &= -3 \int_{a_1}^{a_2} \frac{da}{a} \\
\ln \frac{\rho_2}{\rho_1} &= \ln \frac{a_1^3}{a_2^3} \\
\frac{\rho_2}{\rho_1} &= \frac{a_1^3}{a_2^3}
\end{align}$$

How did they then "find that in a matter dominated universe the scale factor grows as a power-law, a(t)∝t^2/3"?

Substitute $H = \dot{a}/a$ and $\rho = K a^{-3}$ into (19). Here, $K$ is a constant of proportionality (and has nothing to do with spatial curvature).

 

[kimbyd]

Thank you so much, @Bandersnatch. You've been an amazing help.
@kimbyd, that's interesting. Distances of what objects are expected from the CMB fluctuations? The surface of last scattering? How does the difference in distance help measure curvature?

Think of it as the equivalent of summing the angles of a triangle.

In flat space, the angles of a triangle always add up to 180 degrees. In positively-curved space, the sum is greater. In negatively-curved space, the sum is smaller.

In principle, you could measure the curvature by just summing up the angles of a single triangle, and the larger the triangle the better.

The difficulty enters if there is some uncertainty as to precisely what the true length of the far-away side of the triangle is. We can't measure it directly: we have to infer from a model. We can resolve this degeneracy by comparing this triangle to another triangle measured in the more nearby universe. Relatively simple physics ties the triangle measured from the typical distances between galaxies and the triangle measured from distances between the temperature peaks on the CMB.

In detail, physicists don't actually set up triangles and measure them. This is just a heuristic device for explaining why it's helpful to look at both relatively near and relatively far objects for estimating curvature.

 

[kimbyd]

A heuristic way to see this: consider a box of volume $V$ that contains $N$ particles that each have mass $m$, so that the density of the stuff in the box in $\rho = Nm/V$. If the box follows the expansion of the universe, and if the particles don't have peculiar velocities, then the number of particles in the box stays constant as the box expands. Without loss of generality (WLOG), assume that the box is $a \times a \times a$ (with $a$ the scale factor of the universe, so that $V = a^3$. At time $t_1$, density is $\rho_1 = Nm/a_1^3$; at time $t_2$, density is $\rho_2 = Nm/a_2^3$. Hence, $\rho_2/\rho_1 = a_1^3/a_2^3$, i.e., the density of non-relativistic matter is inversely proportional to the cube of the scale factor.

A less heuristic way to to see this uses local conservation of energy and the specific form of the connection coefficients for Friedmann-Lemaitre-Robertson-Walker (FLRW) universes to give an equation that resembles the first law of thermodynamics,
$$\begin{align}
\frac{d}{dt} \left( \rho a^3 \right) &= -P \frac{d}{dt} \left( a^3 \right) \\
a^s d\rho + 3\rho a^2 da &= -3P a^2 da \\
a \frac{d\rho}{da} &= -3 \left( \rho + P \right)
\end{align}$$
In the above $c = 1$, and $P$ is pressure, which can be taken to be zero, since for non-relativistic matter, $P$ is much smaller than $\rho$. This gives
$$\begin{align}
\frac{d\rho}{\rho} &= -3 \frac{da}{a} \\
\int_{\rho_1}^{\rho_2} \frac{d\rho}{\rho} &= -3 \int_{a_1}^{a_2} \frac{da}{a} \\
\ln \frac{\rho_2}{\rho_1} &= \ln \frac{a_1^3}{a_2^3} \\
\frac{\rho_2}{\rho_1} &= \frac{a_1^3}{a_2^3}
\end{align}$$

Substitute $H = \dot{a}/a$ and $\rho = K a^{-3}$ into (19). Here, $K$ is a constant of proportionality (and has nothing to do with spatial curvature).

Or, to put it more simply:
That the density of matter scales as $1/a^{-3}$ is a result of the conservation of stress-energy. Since matter doesn't experience pressure, this reduces to just the conservation of energy, which for non-relativistic matter is just the mass, whose density just decreases as $1/a^{-3}$ as the universe expands.

To get the scaling of other forms of matter, you have to use an argument more similar to George Jones above. But matter is easy.

 

[Joshua P]

@George Jones thanks. How did you get your first equation? Does it come simply from equation 19?
Also, after you "Substitute H=(change in a)/a and p=Ka^-3 into (19)", how do you simplify? I don't know what to do with the (change in a) part.
@kimbyd, how is that "relatively simple physics ties the triangle measured from the typical distances between galaxies and the triangle measured from distances between the temperature peaks on the CMB"?

 

[kimbyd]

Here's a blog post that might help you to understand BAO measurements:
https://medium.com/starts-with-a-bang/what-the-hell-are-baryon-acoustic-oscillations-cfee6d726538

 

[George Jones]

@George Jones thanks. How did you get your first equation? Does it come simply from equation 19?

What is the first law of thermodynamics?

Also, after you "Substitute H=(change in a)/a and p=Ka^-3 into (19)", how do you simplify? I don't know what to do with the (change in a) part.

$$\begin{align}
H &= \sqrt{\frac{8 \pi G}{3} \rho} \\
\frac{1}{a} \frac{da}{dt} &= \sqrt{\frac{8 \pi G}{3} \frac{K}{a^3}} \\
a^{1/2} da &= \sqrt{\frac{8 \pi G}{3}} dt \\
\int_0^{a_1} a^{1/2} da &= \sqrt{\frac{8 \pi G}{3}} \int_0^{t_1} dt \\
\frac{2}{3} a_1^{3/2} &= \sqrt{\frac{8 \pi G}{3}} t_1
\end{align}$$

 

[Bandersnatch]

And how do the units cancel out if cs is measured in terms of the speed of light? Are the distances measured in lightyears?

That's one option. Other options include light-seconds (for c=1 light-second/second), light-fortnights (c=1 light-fortnight/fortnight), light-heartbeats (c=1 light-heartbeat/heartbeat), etc.
Doesn't matter which it is, as long as you're consistent with your units when you start evaluating numerical values.

 

[Joshua P]

@kimbyd thanks, that was an interesting article.
@Bandersnatch thanks again.
@George Jones thanks for the explanation (and your time), I understood that last one.
I took a look at the first law of thermodynamics, and a closer look at your answers.
I believe that the first law states: dU = dQ - PdV
So dQ is change in energy from the surroundings, but given that we are talking about the whole universe, there is none. Right?
PdV was clear enough for me. As you said a³ is volume. dU is said to be the change in internal energy. From what you wrote, I take it also to be change in mass in the universe, from the d(density*a³)/dt. Why?
In the insights article Bandersnatch originally cited, they used the first Friedman equation to show what you've shown. Is what you're doing related to that?

Seperate question: I was looking at the Friedman equations, the first one also includes the cosmological constant? We assume that to be zero here, right? Why?

 

[George Jones]

IPlease keep it simple for me to understand.

@Joshua P : I just realized that you tagged this thread as B. Given some of the previous posts in this thread, I had assumed it was I. What background level of physics and maths should we assume?

 

[Joshua P]

Well... I've made it most of the way through high school. I'm doing IB HL Maths and Physics, so I've done vectors and calculus (although only at a high school level). A lot of this physics is new to me. I've been doing research for about a month so far on this particular subject (curvature of the universe). That being said, I am looking to understand this subject in depth (and I want to know how the findings are made), though I know I have a ways to go. I apologise for asking such basic questions, and thanks to everyone for their patience. :)

 

[Bandersnatch]

No, if anyone, it's me who should apologise for not paying attention to the thread level and leading you into deeper waters than I probably should.

Would you like to take a step back, towards a more conversational manner? Maybe with suggestions for reading to get you up to speed?

 

[Joshua P]

@Bandersnatch You were simply answering my initial question; it was what I asked for, so thank you. The Insights article you gave me was exactly what I wanted.
It's quite difficult to explain what I'm looking for, but I'll try.
To give some context, I'm hoping to write a Maths paper on non-Euclidean geometry and what it has told us about the curvature of the universe. Since it's a Maths paper, I plan on simplifying the physics a lot. The entire essay aims to show that scientific theory tells us how far apart the fluctuations should be in a flat universe, and the apparent angle between the fluctuations on the CMB matches that, showing that the observable universe is more or less flat. Everything that everyone has given me so far is helping me write this essay.
I don't really need to fully understand any of the science (Eistein's field equations, the Friedman equations, or even the first law of thermodynamics) or justify it in my paper. I would just like to show that the science can be used to find the actual angle between fluctuations in a Euclidean universe. I'll more or less be quoting the middle of the Insights article, but first I want to understand how the Insights article reaches its conclusion. There are a bunch of places where I just didn't (or still don't) understand how it got from one point to the next. If I know how to explain the insights article (to a non-Physics-y person), I can do my essay. I would love some extra reading, I know I need it. But I don't think I need to understand all of the underlying physics in a whole lot of depth, just the main concepts?

 

[bapowell]

Sorry to take so long to respond to @Bandersnatch 's bat signal, but I'm available to help if there are still questions!

 

[Joshua P]

@bapowell Hi! Yes, I have more questions. There are still a couple questions that haven't been answered yet.

How does equation 19, the Friedman equation (where we've assumed k = 0) show that density is proportional to a^-3?

A less heuristic way to to see this uses local conservation of energy and the specific form of the connection coefficients for Friedmann-Lemaitre-Robertson-Walker (FLRW) universes to give an equation that resembles the first law of thermodynamics,
\begin{align} \frac{d}{dt} \left( \rho a^3 \right) &= -P \frac{d}{dt} \left( a^3 \right) \\ a^s d\rho + 3\rho a^2 da &= -3P a^2 da \\ a \frac{d\rho}{da} &= -3 \left( \rho + P \right) \end{align}
In the above c = 1, and P is pressure, which can be taken to be zero, since for non-relativistic matter, P is much smaller than ρ. This gives

\begin{align} \frac{d\rho}{\rho} &= -3 \frac{da}{a} \\ \int_{\rho_1}^{\rho_2} \frac{d\rho}{\rho} &= -3 \int_{a_1}^{a_2} \frac{da}{a} \\ \ln \frac{\rho_2}{\rho_1} &= \ln \frac{a_1^3}{a_2^3} \\ \frac{\rho_2}{\rho_1} &= \frac{a_1^3}{a_2^3} \end{align}

I took a look at the first law of thermodynamics, and a closer look at your answers.
I believe that the first law states: dU = dQ - PdV
So dQ is change in energy from the surroundings, but given that we are talking about the whole universe, there is none. Right?
PdV was clear enough for me. As you said a3 is volume. dU is said to be the change in internal energy. From what you wrote, I take it also to be change in mass in the universe, from the d(density*a3)/dt. Why?
In the insights article Bandersnatch originally cited, they used the first Friedman equation to show what you've shown. Is what you're doing related to that?

Seperate question: I was looking at the Friedman equations, the first one also includes the cosmological constant? We assume that to be zero here, right? Why?

 

[bapowell]

The Friedmann equation does not show that $\rho \sim a^{-3}$. This is seen using the continuity equation, which follows from energy conservation:
$$\dot{\rho} = -3\frac{\dot{a}}{a}(\rho + p)$$
For nonrelativistic matter, the pressure $p = 0$. Can you solve the equation that follows for $\rho$?

EDIT: I see George already discussed this approach above: is there something still unclear?

 

[bapowell]

Regarding the cosmological constant, even if nonzero it is irrelevant to the dynamics of the early universe because it is dwarfed by the energy density of radiation. Specifically during inflation, it is negligible compared to the energy density of the inflaton field. So, it's there, it's just too tiny to worry about and so we assume it's zero.

 

[Joshua P]

@bapowell Thanks for the reply.
I misunderstood before. In the Insights article (https://www.physicsforums.com/insights/poor-mans-cmb-primer-part-4-cosmic-acoustics/), I believe it says that solving the Friedman equation "relates the expansion rate to the homogeneous and isotropic density, with ρ∝a⁻³" (equation 18). What did I miss or misunderstand? And what is the Friedman equation actually showing here, as equation 18?

About the cosmological constant, that's interesting. In the Friedman equation, it's dwarfed because the universe was very dense and expanding very quickly? Another question: I thought inflation lasted less than a second; are we not talking about when the universe was 380,000 years old?

EDIT: I was reading Wikipedia (https://en.wikipedia.org/wiki/Friedmann_equations), the "continuity equation" you gave can be derived from the Friedman equations? How did they do it, exacly?

 

[bapowell]

@bapowell Thanks for the reply.
I misunderstood before. In the Insights article (https://www.physicsforums.com/insights/poor-mans-cmb-primer-part-4-cosmic-acoustics/), I believe it says that solving the Friedman equation "relates the expansion rate to the homogeneous and isotropic density, with ρ∝a⁻³" (equation 18). What did I miss or misunderstand? And what is the Friedman equation actually showing here, as equation 18?

The text following the Friedmann equation, Eq. (19), reads: "which relates the expansion rate to the homogeneous and isotropic density, with $\rho \propto a^{-3}$, since matter densities vary inversely with volume, $V \propto a^3$..." I apologize that this isn't more clear, but it's not saying that the FE provides this relation, but simply that it relates the expansion rate to the density. The fact that $\rho \propto a^{-3}$ is really a parenthetical in this statement, which I mention in order to specialize to the case of matter-dominated expansion.

About the cosmological constant, that's interesting. In the Friedman equation, it's dwarfed because the universe was very dense and expanding very quickly? Another question: I thought inflation lasted less than a second; are we not talking about when the universe was 380,000 years old?

It's dwarfed because it's small relative to the other energy components. At any time, the universe has three major energy components: radiation, cold matter, and cosmological constant (if it's nonzero). Today, the CC is the dominant energy component, but earlier in the history of the universe, radiation and matter were dominant (these eventually dilute away as the universe expands, leaving the CC in charge).

It's possible to suppose that inflation lasted for less than a second. The CMB was generated when the universe was around 380K years old, i.e. around 380K years after the big bang (which is taken to coincide with the end of the inflationary period). But the length of the inflationary period is a totally separate consideration from when the CMB was created.

EDIT: I was reading Wikipedia (https://en.wikipedia.org/wiki/Friedmann_equations), the "continuity equation" you gave can be derived from the Friedman equations? How did they do it, exacly?

Have you tried it? What happens if you take the time derivative of the first FE?

 

[Joshua P]

@bapowell Thanks for the explanations! About the deriving the continuity equation in Wikipedia article, I have tried, but I keep getting stuck. I tried to take the time derivative of the first FE, like you said, but I'm messing up somewhere.

Another question: In the Insights article, how do you get from equation 20 to find d_s/x_ls=c_s(t_dec/t₀)^1/3? I tried to use equation 20 to find x_ls in the same way equation 18 was used to find d_s, but I'm messing up somewhere. I understood how to get d_s.

 

[bapowell]

@bapowell Thanks for the explanations! About the deriving the continuity equation in Wikipedia article, I have tried, but I keep getting stuck. I tried to take the time derivative of the first FE, like you said, but I'm messing up somewhere.

What do you get for the time derivative of the first FE?

Another question: In the Insights article, how do you get from equation 20 to find d_s/x_ls=c_s(t_dec/t₀)^1/3? I tried to use equation 20 to find x_ls in the same way equation 18 was used to find d_s, but I'm messing up somewhere. I understood how to get d_s.

You caught a mistake, congratulations! The computation of the angular scale of the sound horizon involves the comoving quantities, but I've written it in terms of proper distances. The result is unchanged, but one needs to compute $d_s/d_{\rm ls}$ as comoving quantities (I've corrected the article, and swapped $x_{\rm ls}$ for $d_{\rm ls}$ to make the switch from proper to comoving units more apparent.) Thanks for catching this!

 

[Joshua P]

@bapowell
I didn't get the time derivative. I got stuck. I haven't graduated high school yet, I still have difficulty with this sort of thing. It's really not something I need to know, so nevermind. :)

I'm happy to have helped find a mistake, but I wasn't aware there was one. Why was it a mistake? Am I right in thinking that the comoving quantity factors out the expansion of the universe? Why does the expansion of the universe not matter here?
I don't quite understand comoving and proper distance. As Bandersnatch said, the scale factor relates the proper distance at any time to the proper distance today. x(t) = a(t)x₀. How is it also that the scale factor relates the proper distance to the comoving distance, by x(t) = a(t)r(t)? Does the comoving distance at epoch t equal the proper distance today?
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]

I don't quite understand comoving and proper distance. As Bandersnatch said, the scale factor relates the proper distance at any time to the proper distance today. x(t) = a(t)x0. How is it also that the scale factor relates the proper distance to the comoving distance, by x(t) = a(t)r(t)? Does the comoving distance at epoch t equal the proper distance today?

Yes. The numerical value of the comoving distance is by convention chosen to equal the present (proper) distance. It could have been the distance at any other time, in principle, but the current distance is simply a convenient choice. For all intents and purposes, $x_0$ is the comoving distance.

Comoving distance is a convenient measure in an expanding universe, since it exactly factors out the expansion, and tells you just about the relative positions of points in space.

For example, take a look at the two top graphs shown below, relating time and distance in our universe:

(graphs taken from: https://arxiv.org/abs/astro-ph/0310808)
Here, the dotted lines marked with redshifts (1, 3, 10, 1000...) can be visualised as some test galaxies. On the first graph, which uses only proper distances and proper time, the spatial relationship between those galaxies is hard to discern, especially towards the bottom. But if you factor out the expansion by adopting comoving distances, it becomes much more clearer.
Notice how at the horizontal line marked 'now' both the proper distance and the comoving distance to the same galaxies are equal.

The third graph does the same 'trick' with time, i.e. it scales the time with the scale factor, but it's less relevant here.

 

[Joshua P]

@Bandersnatch
Why is it that comoving distances, which factor out expansion, "scale with expansion"?
On those graphs, what is are the comoving and proper distances referring to? Distances of what exactly? Everything in the universe?
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 again for everyone's help.

 

[Bandersnatch]

Why is it that comoving distances, which factor out expansion, "scale with expansion"?

Oh, poo. That's what you get if you try to post while tipsy. What I should have said is that they're constant during expansion. It doesn't make sense as I wrote it. It shall be promptly corrected.

On those graphs, what is are the comoving and proper distances referring to? Distances of what exactly? Everything in the universe?

The graphs are meant to show relationships between various types of cosmological horizons. Such as the event horizon or the distance at which recession velocity reaches the speed of light. You can also see distances to the regions of space containing objects which emitted currently measured redshifts (dotted lines). In comoving coordinates those regions, just as any other given region in space, always have the same coordinates (same comoving distance from the observer).
Anyway, probably better not to focus on those too much, as they seem to be a distraction.

 

[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)?