Journey of an Observed Cosmic Microwave Background Photon

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• JimJCW
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TL;DR Summary
An observed cosmic microwave background photon spent 13.8 billion years to travel a distance of 42 million light years to reach our location. The question is: What happened?
Suppose we receive a cosmic microwave background (CMB) photon from space. According to the Big Bang model, this photon became free to travel when the universe was about 0.38 million years (Myr) old. At that time, it was about 42 million light years (Mly) away from our location. Because of the expansion of space, its wavelength was stretched by a factor of 1090, changed from infrared to microwave. Also, it takes the photon about 13.8 billion years (Gyr) to travel a distance of 42 Mly to reach our location. The question is: What happened? One may want to know what was the journey of the photon, i.e., its distance from our location as a function of the age of the universe. We can call it the journey of an observed CMB photon.

The following cosmological calculators can help us to answer the question:

Jorrie: LightCone7-2021-03-12 Cosmo-Calculator
http://jorrie.epizy.com/Lightcone7-2021-03-12/LightCone_Ho7.html?i=1

Nick Gnedin: Cosmological Calculator for the Flat Universe
https://home.fnal.gov/~gnedin/cc/

JimJCW

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The light did not travel 42 Mly to reach our location. While the distance at emission was 42 Mly, the distance traveled by the light was 13.8 billion ly as light travels 1 ly per year by definition. However, as the light traveled towards us, the space in front of it expanded, which increased the distance the light had to travel to finally result in that the light had to travel 13.8 billion ly to reach us. Similarly, the space behind the light also expanded, meaning that the distance to where the light was emitted from today is more than 13.8 billion ly away.

vanhees71, docnet, etotheipi and 1 other person
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The light did not travel 42 Mly to reach our location. While the distance at emission was 42 Mly, the distance traveled by the light was 13.8 billion ly as light travels 1 ly per year by definition. However, as the light traveled towards us, the space in front of it expanded, which increased the distance the light had to travel to finally result in that the light had to travel 13.8 billion ly to reach us. Similarly, the space behind the light also expanded, meaning that the distance to where the light was emitted from today is more than 13.8 billion ly away.

What we are saying is that the photon started 42 Mly away from our location at t=0.38 Myr, but when it reaches here it is already t=13.8 Gyr. Maybe we should not describe the journey as distance traveled.

I think all this makes it interesting to plot the proper distance of the photon from our location as a function of cosmological time, right?

JimJCW

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I think all this makes it interesting to plot the proper distance of the photon from our location as a function of cosmological time, right?
You can do that, but you need to be careful with how you present it and you will need to solve ##dx/dt = 1/a(t)## to find the comoving coordinate of the world-line first. You can then multiply by the scale factor to obtain the proper distance. Note that any time for which ##H d > 1##, where ##H## is the Hubble parameter and ##d## the comoving distance, the proper distance to the light signal will increase with time.

For example, in a matter dominated universe, where ##a \propto t^{2/3}##, the proper distance to the light signal will be given by
##
d(t) = (d_0 + 3t_0) \left(\frac t{t_0}\right)^{2/3} - 3 t
##
(unless I took a wrong turn somewhere in the computation), where ##t_0## is the time of emission and ##d_0## the proper distance at that time. Depending on the value of ##d_0##, the proper distance may start out growing, but will eventually decrease and reach zero.

docnet
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You can do that, but you need to be careful with how you present it and you will need to solve ##dx/dt = 1/a(t)## to find the comoving coordinate of the world-line first. You can then multiply by the scale factor to obtain the proper distance. Note that any time for which ##H d > 1##, where ##H## is the Hubble parameter and ##d## the comoving distance, the proper distance to the light signal will increase with time.

For example, in a matter dominated universe, where ##a \propto t^{2/3}##, the proper distance to the light signal will be given by
##
d(t) = (d_0 + 3t_0) \left(\frac t{t_0}\right)^{2/3} - 3 t
##
(unless I took a wrong turn somewhere in the computation), where ##t_0## is the time of emission and ##d_0## the proper distance at that time. Depending on the value of ##d_0##, the proper distance may start out growing, but will eventually decrease and reach zero.

Not everyone has the mathematical and theoretical training to do/think as you described. The two cosmological calculators mentioned can be used to produce the proper distance vs. time plot for the observed photon. I am thinking to outline the steps so that anyone can easily obtain the result. After that we can concentrate on the interpretation of the plot.

JimJCW

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The two cosmological calculators mentioned can be used to produce the proper distance vs. time plot for the observed photon. I am thinking to outline the steps so that anyone can easily obtain the result. After that we can concentrate on the interpretation of the plot.

Output of Jorrie’s calculator is in tabular form and has Chart function; we will use it here. To get the proper distance vs. time plot for the observed photon, we can use the following steps:
• Start the calculator: http://jorrie.epizy.com/Lightcone7-2021-03-12/LightCone_Ho7.html?i=1, it uses PLANCK Data (2015) as default input.
• Leave the Upper row redshift, zupper at 1090, z value of the observed CMB photon, and change the Lower row redshift, z(lower) to 0.
• Click Open Column Definition and Selection, keep only Cosmic Time and Dthen selected, and click Open Column Definition and Selection again to close it.
• Select Chart and click Calculate. The Dthen vs. Cosmic Time plot will appear.
See attachment.
Please help if you can. I tried to insert an image, but received the message,​
"Oops! We ran into some problems.
Security error occurred. Please press back, refresh the page, and try again."​
That's why I am using attachment here.​

I argue that this plot gives us the proper distance vs. time relation for the observed CMB photon. Here is my interpretation:

The photon, started 41.6 Mly from our location around t=0.38 Myr, was carried away by the rapid expansion of space. It moved as far away as 5.84 Gly. As the space expansion slowed down, it turned around near t=3.9 Gyr and traveled to us, spending total 13.8 Gyr for the journey. The result shows one of the complicate consequences of the expansion of space.​

@Jorrie @Jorrie @Jorrie, @Bandersnatch @Bandersnatch @Bandersnatch

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I argue that this plot gives us the proper distance vs. time relation for the observed CMB photon.
I mean, there's hardly a need to make an argument here, since that's pretty much literally what a light cone is, in those coordinates. Depending on the audience you'd want to reach, it might make sense explaining what a light cone is, and why a range of different emitters (we graph a range of z's after all) can be thought of as equivalent to the journey of a single photon.

As the space expansion slowed down, it turned around near t=3.9 Gyr and traveled to us
Myself, I'd probably avoid describing it as 'turning around'. In local space, the photon always travels towards the observer. While for some of its history it recedes, and then approaches, it never changes direction - which is what turning around implies.

As for embedding images, I usually just do a copy and paste, directly into the message box. Seems to work most of the time.
Oh, and I didn't get a notification for when you tagged me - maybe that's because of doing it three times, I wouldn't know. In any case, probably better to stick to typing @ and the name of whomever, just once.

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I mean, there's hardly a need to make an argument here, since that's pretty much literally what a light cone is, in those coordinates. Depending on the audience you'd want to reach, it might make sense explaining what a light cone is, and why a range of different emitters (we graph a range of z's after all) can be thought of as equivalent to the journey of a single photon.

Some people, like me, learn things by having physical pictures in their minds. I cannot do reasoning using light cone and world lines yet. Maybe you can start a new thread to explain those things or give references such as light cone 101, etc.

About treating D(then) vs. time plot as proper distance vs. time plot for the observed photon, in my mind, I am thinking about a relay race: a new photon taking over the ‘baton’ is like emitting a new photon at that location and continues the journey. I think Jorrie’s D(then) vs. time plot does that for us.

Myself, I'd probably avoid describing it as 'turning around'. In local space, the photon always travels towards the observer. While for some of its history it recedes, and then approaches, it never changes direction - which is what turning around implies.

You have a good point. Instead of saying 'turning around', maybe I should describe it in the following way:

The photon, started about 41.6 Mly from our location around t=0.38 Myr, was carried away by the rapid expansion of space [see the V(then)/c vs. Time plot shown below]. It moved as far away as 5.84 Gly until the recession velocity at the photon’s location reached V(then)/c=1 near t=3.9 Gyr. Then the proper distance of the photon started to decrease until it reached our location.​

As for embedding images, I usually just do a copy and paste, directly into the message box. Seems to work most of the time.

I did a Google search. It happens elsewhere too. It seems refreshing the page or logout and login again solves the problem.

Oh, and I didn't get a notification for when you tagged me - maybe that's because of doing it three times, I wouldn't know. In any case, probably better to stick to typing @ and the name of whomever, just once.

I learned from you in your thread,
A - Distances between observers using the Lightone7 calculator.
You mentioned,

One has to stand in front of a mirror, draw the ancient sigil of the electric monkey, and repeat his name three times. …​

@Bandersnatch (How is that?)

You mentioned,
That was meant as a joke. You know, as in that horror trope, where you call a name three times to summon the Slenderman, Beetlejuice, or whatever the monster of the hour is. I probably consume too much popculture. ;)

Klystron
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That was meant as a joke. You know, as in that horror trope, where you call a name three times to summon the Slenderman, Beetlejuice, or whatever the monster of the hour is. I probably consume too much popculture. ;)
That was on the first day I joined Physics Forums. I thought 'three times' is the trick. Now I know better.

Klystron
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I mean, there's hardly a need to make an argument here, since that's pretty much literally what a light cone is, in those coordinates.

Please provide some explanation about using light cone to justify my argument. I am reading about world lines, light cones, and space-time intervals, etc.

@Bandersnatch

@Jorrie

Sorry for the late response.
in my mind, I am thinking about a relay race: a new photon taking over the ‘baton’ is like emitting a new photon at that location and continues the journey.
I think that's an ok intuition. Maybe with making clear that the relay racers don't stop on meeting the next relay partner in line (i.e. passing by another emitter), but continue together. The light cone is, after all, the combined path of all the photons the observer receives at a given moment.

My objection earlier was just to what looks like a tautological reasoning: you use a calculator that is ostensibly for drawing light cones (it's even in the name) to draw a light cone, and then want to argue that it's a path of photons - i.e. a light cone. 'Well, duh' - is the kind of reaction you might get.
The point being, you don't have to 'argue' that this is what it is. You can just explain what it is. And your earlier efforts at doing that are just fine, I think.

Gold Member
My objection earlier was just to what looks like a tautological reasoning: you use a calculator that is ostensibly for drawing light cones (it's even in the name) to draw a light cone, and then want to argue that it's a path of photons - i.e. a light cone. 'Well, duh' - is the kind of reaction you might get.

I understand that in a static universe, an event on the past light cone would give the photon a straight-line journey. The journey of the photon in our case has a bow-shaped d vs. t plot due to the expansion of space. Please help me understand how to relate the space expansion to the light cone.

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I understand that in a static universe, an event on the past light cone would give the photon a straight-line journey. The journey of the photon in our case has a bow-shaped d vs. t plot due to the expansion of space. Please help me understand how to relate the space expansion to the light cone.
Bow-shaped according to whom? Light follows null geodesics. Which some people call "straight lines"; although, personally, I reserve the term straight line for Euclidean geometry.

andrew s 1905
Bow-shaped according to whom? Light follows null geodesics. Which some people call "straight lines"; although, personally, I reserve the term straight line for Euclidean geometry.
See top plot in figure 1 in this https://www.cambridge.org/core/jour...the-universe/EFEEEFD8D71E59F86DDA82FDF576EFD3 .

Due to the expansion of the Universe CMB radiation we see now was originally moving away from us as I understand it. Regards Andrew

andrew s 1905
The figure show the bow shaped light path.

Gold Member
Bow-shaped according to whom? Light follows null geodesics. Which some people call "straight lines"; although, personally, I reserve the term straight line for Euclidean geometry.
I think the D(then) vs. Time plot looks like a bow. In a non-expanding universe, it would be a straight line with very short length for the CMB photon we are considering, 42 Mly.

andrew s 1905
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I think the D(then) vs. Time plot looks like a bow. In a non-expanding universe, it would be a straight line with very short length for the CMB photon we are considering, 42 Mly.

View attachment 281046
If you change your coordinates, you'll get a linear graph if you want one. There is no physical significance in a graph like that. It certainly doesn't, in any meaningful way, represent light traveling on a "bow shaped" path.

Quite fundamentally an object accelerating in a straight line has a curved distance vs time graph - but it's still traveling in a straight line.

Orodruin
andrew s 1905
I think you are being rather pedantic. It is common to discuss gravitational lensing where light is "bent" by large concentrations of mass. The "bow" due to the expansion of the universe is just an extreem example.
Regards Andrew

Mentor
It certainly doesn't, in any meaningful way, represent light traveling on a "bow shaped" path.

I think this is a bit strong. The "proper distance vs. comoving time" graph is indeed bow-shaped. It is true that that graph is only one of several possible graphs that can be drawn, but it is one of them. (The Davis and Lineweaver paper that @andrew s 1905 linked to shows three different graphs; each one has its own advantages and disadvantages.)

It is also true that, since the path of the light ray in question is a geodesic, it is a "straight line in spacetime", not curved, in that sense. But that fact by itself doesn't tell you anything about the relationship between the path of the light ray through spacetime and the path through spacetime of anything else. It seems like that kind of relationship is what the OP is asking about.

an object accelerating in a straight line has a curved distance vs time graph - but it's still traveling in a straight line.

It's traveling in a straight line in space. But not in spacetime. Its path through spacetime is curved--that's what proper acceleration means, path curvature in spacetime.

But as above, knowing that a particular individual object's path through spacetime is curved or straight doesn't, in itself, tell you anything about its relationship with other paths through spacetime.

andrew s 1905
Mentor
In a non-expanding universe, it would be a straight line

More precisely, in flat Minkowski spacetime (which is a "universe" with nothing in it), in standard inertial coordinates, the path of a light ray is a straight line. But there is no meaningful way to compare paths in different spacetime geometries. So there is no meaningful way to compare the path of a light ray in our actual universe with any light ray's path in some other different "universe".

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It's traveling in a straight line in space. But not in spacetime.
I assumed we were talking about paths through space; not spacetime. As in:

It is common to discuss gravitational lensing where light is "bent" by large concentrations of mass. The "bow" due to the expansion of the universe is just an extreem example.
Regards Andrew

If everyone else has been talking about curved paths in spacetime, then I've misunderstood.

andrew s 1905
I understood the bending of light by mass to be a spacetime effect. Am I mistaken?

I understand that in a static universe, an event on the past light cone would give the photon a straight-line journey. The journey of the photon in our case has a bow-shaped d vs. t plot due to the expansion of space. Please help me understand how to relate the space expansion to the light cone.
The bow (or teardrop) shape arises naturally if the space expands while the photon travels - as you understand, it's that being carried away by expansion before approaching. But since it is still the path of a photon through space-time, it's still a light cone. Even if doesn't look terribly conical in those coordinates.
Have you read Ned Wright's cosmology tutorial? If not, see here:
http://www.astro.ucla.edu/~wright/cosmo_02.htm
In particular, it can be seen how in the immediate vicinity of each comoving observer the bow-like path (red) is approximately parallel to the local light cones (the drawings show future light cones instead of past ones, but the principle is the same).

Mentor
I understood the bending of light by mass to be a spacetime effect. Am I mistaken?

The path of a light ray through spacetime is a geodesic, i.e., straight. The term "bending" refers to what the light ray's path in space looks like if the spacetime is curved, for an appropriate choice of how spacetime is split into space and time.

andrew s 1905
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If you change your coordinates, you'll get a linear graph if you want one. There is no physical significance in a graph like that. It certainly doesn't, in any meaningful way, represent light traveling on a "bow shaped" path.

Quite fundamentally an object accelerating in a straight line has a curved distance vs time graph - but it's still traveling in a straight line.

The CMB photon we received had always been moving along the straight line pointing to the direction of its source. Its proper distance to our location had a small value at the start (42 Mly at t=0.38 Myr), increased to a large value (5.84 Gly at t=3.9 Gyr) due to space expansion, started to decrease when its recession speed became c, and finally reached our location (t=13.8 Gyr). The bow-shaped d vs. t plot describes this change of the proper distance with time, not a curved photon path in space.

PeroK
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I assumed we were talking about paths through space; not spacetime.

Not when you said this:

Light follows null geodesics.

This is a description of the light's path through spacetime, not space.

If we are talking about paths through space, then there are multiple possible "paths through space" depending on how "space" is defined and how "time" is defined (since the "path through space" is the object's position in "space" as a function of "time"). Each of the three graphs in the Davis and Lineweaver paper shows a different shape of the past light cone due to a different definition of "space" and a different definition of "time". One of those shapes is indeed "bow-shaped", so I don't think it's correct to say that such a description is not "meaningful". It's just as "meaningful" as the descriptions of the other two shapes. None of those shapes are "the" unique "path through space" of the light ray, but that's because there is no such thing as a unique "path through space" of anything in GR, since there is no unique split of any spacetime into "space" and "time"; there are always an infinite number of possible ways of doing that.

Mentor
The CMB photon we received had always been moving along the straight line pointing to the direction of its source.

It's always been moving along that direction in space, but "path in space" is not just a direction. "Path in space" means the set of "positions in space" that the object occupies. But in order to even make sense of that, you have to have a well-defined concept of "position in space". Proper distance in a particular direction from a fixed location, such as the Earth (or the place where the Earth would have been, for times before the Earth existed), is one way of defining such a concept. It is not the only one, as I have already pointed out; the Davis and Lineweaver paper gives three different ways of defining such a concept.

The bow-shaped d vs. t plot describes this change of the proper distance with time, not a curved photon path in space.

By this definition of "path in space", a photon in our universe has no well-defined "path in space" at all, since there is no "space" independent of time in our universe, so there is no way of defining a "position in space" for a photon. In other words, the concept of "position in space" that you are implicitly using in the above quote is not any of the valid concepts described in the Davis and Lineweaver paper; it's not a valid concept at all in an expanding universe.

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By this definition of "path in space", a photon in our universe has no well-defined "path in space" ...

What I am saying is that the bow-shaped d vs. time plot is different from a bow-shaped x-position vs. y-position plot.

Mentor
What I am saying is that the bow-shaped d vs. time plot is different from a bow-shaped x-position vs. y-position plot.

What bow-shaped x-position vs. y-position plot are you talking about?

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What bow-shaped x-position vs. y-position plot are you talking about?
Good luck!

Mentor
What bow-shaped x-position vs. y-position plot are you talking about?

Or, for that matter, what non bow-shaped x-position vs. y-position plot are you talking about? So far I have not seen any x-position vs. y-position plot described or referenced at all in this thread.

Gold Member
What bow-shaped x-position vs. y-position plot are you talking about?
I was referring to the comment made by PeroK (in #19):

If you change your coordinates, you'll get a linear graph if you want one. There is no physical significance in a graph like that. It certainly doesn't, in any meaningful way, represent light traveling on a "bow shaped" path.​

Mentor
I was referring to the comment made by PeroK (in #19):

Nothing in that comment is about a graph of x-position vs. y-position. So I still don't know what you are referring to.