Distance traveled by light accounting for expansion

In summary, the conversation discusses the concept of accounting for expansion when calculating the distance traveled by light in a given time interval. It is mentioned that the proper distance between comoving galaxies at a fixed time scales with the scale factor, and that by knowing the distance at emission and the relative increase of the scale factor until absorption, one can calculate the distance traveled by light. It is also noted that this method is not straightforward in a cosmos with radiation, matter, and a cosmological constant, and that the general method is to numerically integrate the 1st Friedman equation over the time or expansion factor range. The conversation also includes a recommendation to read BA Powell's work on inflationary misconceptions for further insights on cosmology.
  • #1
etotheipi
Suppose light travels during a time interval of t2 - t1, where the scale factors at t1 and t2 are a(t1) and a(t2) respectively.

If we consider an infinitesimally small interval of time dt during this interval, without accounting for expansion we would expect light to travel a distance cdt. How do we adjust this quantity to account for the factor by which this distance has increased during this time interval?

The goal is to determine the proper distance traveled by the light between these two times, and I think this just constitutes integrating the adjusted version of the quantity cdt over t1 to t2 but don't know how to find the expansion factor.

Please do let me know if I'm completely wrong, I'm still not at all comfortable with cosmological distances!
 
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  • #2
etotheipi said:
Suppose light travels during a time interval of t2 - t1, where the scale factors at t1 and t2 are a(t1) and a(t2) respectively.

If we consider an infinitesimally small interval of time dt during this interval, without accounting for expansion we would expect light to travel a distance cdt.
If you don't count for expansion then you obtain a distance at ##t=t_1## between the location at emission and our location then. The distance between the location of emission and the location of absorption by us now is not well defined in expanding spacetime.
 
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  • #3
timmdeeg said:
If you don't count for expansion then you obtain a distance at ##t=t_1## between the location at emission and our location then. The distance between the location of emission and the location of absorption by us now is not well defined in expanding space.

Thank you for clarifying this, it clears up some doubt.

On a side note, if one galaxy were to emit a photon of light which travels for say 10 billion years before reaching another galaxy (both of which are moving solely due to Hubble flow), is there a calculation one can perform (knowing the scale factors at the start and end times) that would give the initial or final distances between the galaxies?
 
  • #4
Well, the proper distance between comoving galaxies at a fixed time scales with the scale factor ##a##. So if you know the distance at ##t=t_1## (emission) and the relative increase of ##a## until ##t=t_2## then you know the distance at absorption. If ##a## grows by 50% then the distance grows to the same extent.
 
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  • #5
etotheipi said:
The goal is to determine the proper distance traveled by the light between these two times, and I think this just constitutes integrating the adjusted version of the quantity cdt over t1 to t2 but don't know how to find the expansion factor.

In a cosmos with radiation, matter and a cosmological constant (like ours), it is not straightforward to find the expansion factor change from a cosmological time interval, except approximately (if the latter is small and the expansion close to linear over time).

As you expected, the general method is to numerically integrate the 1st Friedman equation over the interval, either a time or an expansion factor range (to find the other one), knowing some constants derived from present observations - like the Hubble constant and the various energy densities.

The distance that light has traveled is then just cdt, as you said.

See Davies: http://arxiv.org/pdf/astro-ph/0402278v1.pdf Appendix A-1, specifically equation A.18 for the most useful form of the Friedman equation.



 
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  • #6
Jorrie said:
In a cosmos with radiation, matter and a cosmological constant (like ours), it is not straightforward to find the expansion factor change from a cosmological time interval, except approximately (if the latter is small and the expansion close to linear over time).

As you expected, the general method is to numerically integrate the 1st Friedman equation over the interval, either a time or an expansion factor range (to find the other one), knowing some constants derived from present observations - like the Hubble constant and the various energy densities.

The distance that light has traveled is then just cdt, as you said.

See Davies: http://arxiv.org/pdf/astro-ph/0402278v1.pdf Appendix A-1, specifically equation A.18 for the most useful form of the Friedman equation.

Thank you, this is really useful!
 
  • #7
etotheipi said:
On a side note, if one galaxy were to emit a photon of light which travels for say 10 billion years before reaching another galaxy (both of which are moving solely due to Hubble flow) ...
First off, you would use the reference frame of one or the other, so they are not both moving. Considering each to be moving and you have to say what they are moving relative to and then you've just unnecessarily complicated your calculations.

Take one as stationary with the other moving relative to it (and of course from the FOR of the other one, the "stationary" one is moving, but that's a different issue and irrelevant to your calculation).
 
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  • #8
etotheipi said:
Thank you, this is really useful!
On a general note, there is an excellent insight by BA Powell here on cosmology. Try searching PF for "inflationary misconceptions".
 
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  • #9
timmdeeg said:
Well, the proper distance between comoving galaxies at a fixed time scales with the scale factor ##a##. So if you know the distance at ##t=t_1## (emission) and the relative increase of ##a## until ##t=t_2## then you know the distance at absorption. If ##a## grows by 50% then the distance grows to the same extent.
Looking at the graph of ##a## vs ##t##, what you said is actually roughly true for a large chunk of cosmic time!

1566109248804.png

It is not true for the first 3 Giga years or so due to dominant radiation energy. Also not for the last 4 Gy due to the accelerated expansion, but still close.

It requires a bit of juggling of settings in the LightCone7 calculator to produce this particular chart.
 
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  • #11
Jorrie said:
Looking at the graph of ##a## vs ##t##, what you said is actually roughly true for a large chunk of cosmic time!

View attachment 248339
It is not true for the first 3 Giga years or so due to dominant radiation energy. Also not for the last 4 Gy due to the accelerated expansion, but still close.
But I was talking about proper distance between comoving objects vs. ##a##. Do you say scaling of ##a## with proper distance is epoch dependent?
 
  • #12
phinds said:
Take one as stationary with the other moving relative to it (and of course from the FOR of the other one, the "stationary" one is moving, but that's a different issue and irrelevant to your calculation).
Doesn't that fit better to flat spacetime?
 
  • #13
timmdeeg said:
But I was talking about proper distance between comoving objects vs. ##a##. Do you say scaling of ##a## with proper distance is epoch dependent?
In terms of the OP question, as I understood it, I think yes. Light does not travel the proper distance in expanding space, but simply ##c\Delta t##. By the time it arrives at the receiver, the proper distance between emitter and receiver is larger than the distance light actually traveled.
 
  • #14
Jorrie said:
In terms of the OP question, as I understood it, I think yes.
etotheipi said:
On a side note, if one galaxy were to emit a photon of light which travels for say 10 billion years before reaching another galaxy (both of which are moving solely due to Hubble flow), is there a calculation one can perform (knowing the scale factors at the start and end times) that would give the initial or final distances between the galaxies?
I've been answering this question.
 
  • #15
timmdeeg said:
etotheipi said:
On a side note, if one galaxy were to emit a photon of light which travels for say 10 billion years before reaching another galaxy (both of which are moving solely due to Hubble flow), is there a calculation one can perform (knowing the scale factors at the start and end times) that would give the initial or final distances between the galaxies?
I've been answering this question.
Hmmm, OK, but I thought that the OP's concern was that if you only have a light travel time and two scale factors, how do you find one of the mentioned proper distances.

The only practical way I can think of is to make Earth the receiving galaxy and then find out what the scale factor (or redshift) of a galaxy would be that emitted the observed light say 10 billion years ago, which means at cosmic time 3.8 Gyr. This can be found by integrating the Friedman equation backwards from today for 10 Gy and find the scale factor, which happens to be 0.368 (as can be seen approximately from the graph in #9 above).

The same integration can also yield the proper distance of that galaxy now, which happens to be 15.8 Gly. The distance from us at emission is than adjusted by the sale factor, as you said, giving 5.8 Gly.

This may perhaps be an over-elaboration, but it may help the OP.
 
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  • #16
Jorrie said:
This can be found by integrating the Friedman equation backwards from today for 10 Gy and find the scale factor, which happens to be 0.368 (as can be seen approximately from the graph in #9 above).
Yes this is what one want to know in practice and is really very helpful. It answers more than @etotheipi was asking as he assumes "knowing the scale factors at the start and end times" as given.
 
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  • #17
Jorrie said:
This can be found by integrating the Friedman equation backwards from today for 10 Gy and find the scale factor

But the OP's question is the other way around: you already know the scale factor at emission; what you want is the distances/times. (Note that in practice, we do know the scale factor at emission since we know the redshift, and the redshift is the ratio of the scale factors at emission and absorption.)
 
  • #18
PeterDonis said:
But the OP's question is the other way around: you already know the scale factor at emission; what you want is the distances/times. (Note that in practice, we do know the scale factor at emission since we know the redshift, and the redshift is the ratio of the scale factors at emission and absorption.)
It could be read both ways around, so I tried to answer this part of the OP question, where he asked about the scale factors from a given time interval.
etotheipi said:
The goal is to determine the proper distance traveled by the light between these two times, and I think this just constitutes integrating the adjusted version of the quantity cdt over t1 to t2 but don't know how to find the expansion factor.
 
  • #19
PeterDonis said:
Note that in practice, we do know the scale factor at emission since we know the redshift, and the redshift is the ratio of the scale factors at emission and absorption.
How do we know the scale factor at emission? Would't this require to know the scale factor at absorption?
 
  • #20
timmdeeg said:
How do we know the scale factor at emission?

The redshift ##1 + z# is the ratio of the scale factors.

timmdeeg said:
Would't this require to know the scale factor at absorption?

The scale factor at absorption for objects whose redshifts we observe is the scale factor now, which is ##1## by definition.

In the case given in the OP, we are told that we already know the scale factors at emission and absorption.
 
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  • #21
Jorrie said:
I tried to answer this part of the OP question, where he asked about the scale factors from a given time interval.

The OP also said we already know the scale factors.
 
  • #22
etotheipi said:
Thank you for clarifying this, it clears up some doubt.

On a side note, if one galaxy were to emit a photon of light which travels for say 10 billion years before reaching another galaxy (both of which are moving solely due to Hubble flow), is there a calculation one can perform (knowing the scale factors at the start and end times) that would give the initial or final distances between the galaxies?
This is my go-to reference resource for how distances are calculated in cosmology:
https://arxiv.org/abs/astro-ph/9905116
Hopefully it should answer all of your questions, if they haven't already been answered (personal note: I did not read this thread past this point, except to skim it and try to verify that this link hadn't yet been posted).
 
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  • #23
kimbyd said:
This is my go-to reference resource for how distances are calculated in cosmology:
https://arxiv.org/abs/astro-ph/9905116
Hopefully it should answer all of your questions, if they haven't already been answered (personal note: I did not read this thread past this point, except to skim it and try to verify that this link hadn't yet been posted).

Wow, that's amazing! Thanks!
 

1. How does the expansion of the universe affect the distance traveled by light?

The expansion of the universe causes the space between objects to increase, which means that the distance traveled by light also increases. This is because the light waves must travel through more space to reach their destination, resulting in a longer distance traveled.

2. Can light travel an infinite distance due to the expansion of the universe?

No, light cannot travel an infinite distance due to the expansion of the universe. While the expansion of the universe does increase the distance traveled by light, there is still a limit to how far light can travel before it becomes too weak to detect.

3. How does the speed of light play a role in accounting for expansion?

The speed of light is a constant in our universe, and it does not change due to the expansion of the universe. However, the distance traveled by light is affected by the expansion, which can give the illusion of the speed of light changing.

4. Does the distance traveled by light account for the age of the universe?

Yes, the distance traveled by light is used to calculate the age of the universe. This is because the light we see from distant objects has traveled through space for a certain amount of time before reaching our telescopes, giving us a glimpse into the past.

5. How does the distance traveled by light account for the redshift of galaxies?

The distance traveled by light is directly related to the redshift of galaxies. As light travels through expanding space, its wavelength is stretched, resulting in a longer wavelength and a shift towards the red end of the spectrum. This allows us to measure the distance of galaxies and their movement away from us.

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