Derivation of the formula for cosmological redshift

[andrewkirk]

I was hoping somebody could point me towards a derivation of the following formula for cosmological redshift:

z = R(t₀)/R(t_e)-1.

Wikipedia just presents the formula as a fait accompli and the only explanation is a vague reference to "stretched photons", which is not helpful.

I was hoping there is a fairly simple explanation of why the redshift is related to the ratio of the cosmological scale parameters, but so far I haven't found one.

Thanks very much.

 

[Chronos]

See Davis and Lineweaver - http://arxiv.org/abs/astro-ph/0310808

 

[andrewkirk]

Thank you for the reply Chronos. Are you sure it was the 2003 D&L paper you meant? Because that's the one I've been reading that led me to post the question (after looking unsuccessfully in Wikipedia for a derivation). The formula is presented without derivation at the top of p107. Is it derived somewhere else in the paper (I've only read part of it so far), or perhaps in their 2001 paper instead (which I don't have yet)?

 

[Chalnoth]

Well, photons are lengthened precisely by the amount of expansion. That is, the wavelength is simply multiplied by the expansion factor. The equation you are asking about is an explicit way of writing this, once you realize that (z+1) is the factor which multiplies the redshift.

If you want to know why the wavelength is expanded along with the expansion, well, I'm not sure of a direct way of deriving this result, but a roundabout way of doing it is to first demonstrate using the stress-energy tensor for a photon gas that the energy density drops as $1/a^4$. Since the number density of photons falls as $1/a^3$, it follows that the energy of each individual photon falls as $1/a$.

 

[marcus]

I was hoping somebody could point me towards a derivation of the following formula for cosmological redshift:

z = R(t₀)/R(t_e)-1.

Wikipedia just presents the formula as a fait accompli and the only explanation is a vague reference to "stretched photons", which is not helpful.

I was hoping there is a fairly simple explanation of why the redshift is related to the ratio of the cosmological scale parameters, but so far I haven't found one.

Thanks very much.

Smart question. You've surely done calculus so you're familiar with derivations which involve dividing an interval up finer and finer---taking the limit as epsilon goes to zero and soforth. Look up Bunn and Hogg's paper on arxiv.

 

[marcus]

I assume you know how to use arxiv. If unfamiliar or any difficulty, please say--glad to help!
Just put Bunn in one author field and Hogg in the other.

they show that the formula everbody uses, namely 1+z = R(now)/R(then),
is EQUIVALENT to what you get by dividing the path the light took into a lot of small segments each approximated with a local lorentz frame in which the increase of distance at the time the light passed thru that neighborhood could be treated as an ordinary motion with a infinitesimal DOPPLER shift.

So you add up the cumulative effect of that huge number of tiny Doppler shifts and, Lo and Behold, it adds up to everybody's favorite formula.

http://arxiv.org/multi?group=grp_physics&/find=Search
Turn the "title" field into "author" so you have two author fields.

 

[Chalnoth]

I assume you know how to use arxiv. If unfamiliar or any difficulty, please say--glad to help!
Just put Bunn in one author field and Hogg in the other.

It also happens to be the first thin that pops up in a Google search for "bunn hogg" (without quotes), at least for me :)

 

[Lino]

I have been reading the papers mentioned and some other sources, and I can see why the calculation and logic applies when there is consistent expansion. But if the rate of expansion changes (as per our current understanding), doesn't that invalidate the logic and hence the equation?

Regards,

Noel.

 

[andrewkirk]

a roundabout way of doing it is to first demonstrate using the stress-energy tensor for a photon gas that the energy density drops as $1/a^4$. Since the number density of photons falls as $1/a^3$, it follows that the energy of each individual photon falls as $1/a$.

That's a nice way of looking at it. I'm not capable of doing the first step as my thermodynamics is not up to scratch, but if I take the first claim as given, I can certainly see how the result follows based on what happens to the number density.

 

[andrewkirk]

Thank you Marcus for the paper. I now have a copy of Bunn and Hogg, which looks like it is exactly what I need. I shall start to plod my way though it, in a slow but determined fashion.

 

[George Jones]

Consider two observers who move with the Hubble flow, A at constant comoving $\chi =0$ and B at constant comoving $\chi =\chi_e$. At time $t_e$, B starts sending a light-signal to A, which A starts receiving at time $t_r$. The start ("initial photon") of the signal propagates along a lightlike worldline, so, assuming $\theta$ and $\phi$ are constant along this worldline

$$0 = ds^2 = dt^2 - a\left(t\right) d\chi^2 ,$$
giving $dt/a\left(t\right) = -d\chi$ and

$$\int_{t_e}^{t_r} \frac{dt}{a} = - \int_{\chi_e}^0 d\chi = \chi_e .$$
Assume that the signal lasts for exactly one period of the lightwave, so the "final photon" in the signal starts at B at time $t_e + T_e$ and is received by B at time $t_r + T_r$, where $T_e$ and $T_r$ are the periods of the light as measured by B and A respectively.

During the duration of the signal, the comoving coordinates of B and A don't change, and the worldline of the "final photon" gives

$$\int_{t_e + T_e}^{t_r + T_r} \frac{dt}{a} = - \int_{\chi_e}^0 d\chi = \chi_e .$$
Therefore,

$$\int_{t_e + T_e}^{t_r + T_r} \frac{dt}{a} = \int_{t_e}^{t_r} \frac{dt}{a}$$
and

$$\begin{align} 0 &= \int_{t_e}^{t_r} \frac{dt}{a} - \int_{t_e + T_e}^{t_r + T_r} \frac{dt}{a}\\ & = \int_{t_e}^{t_e + T_e} \frac{dt}{a} + \int_{t_e + T_e}^{t_r} \frac{dt}{a} - \int_{t_e + T_e}^{t_r + T_r} \frac{dt}{a}\\ &= \int_{t_e}^{t_e + T_e} \frac{dt}{a} - \left( \int_{t_r}^{t_e + T_e }\frac{dt}{a} + \int_{t_e + T_e}^{t_r + T_r} \frac{dt}{a} \right)\\ &= \int_{t_e}^{t_e + T_e} \frac{dt}{a} - \int_{t_r}^{t_r + T_r }\frac{dt}{a} \end{align}$$
During the time interval $T_e$ that B sends the signal, the scale factor does not change appreciably from $a\left(t_e\right)$; during the time interval $T_r$ that A receives the signal, the scale factor does not change appreciably from $a\left(t_r\right)$. Consequently, these values can be pulled outside the integrals giving

$$\begin{align} \frac{1}{a\left(t_e\right)} \int_{t_e}^{t_e + T_e} dt &= \frac{1}{a\left(t_r\right)} \int_{t_r}^{t_r + T_r }dt\\ \frac{T_e}{a\left(t_e\right)} = \frac{T_r}{a\left(t_r\right)} \end{align}$$
With $c = 1$, the period and wavelength of light are the same. This, together with

$$z = \frac{\lambda_r}{\lambda_e} - 1,$$
gives the result.

Sorry, after typing this in, I realized that I changed notation for the scale factor, and for your subscript "o".

 

[Lino]

... Assume that the signal lasts for exactly one period of the lightwave ... the comoving coordinates of B and A don't change ... During the time interval ... the scale factor does not change appreciably ...

George, Does this remain valid at very large cosmic scales? If the time interval is very long, then presumably the scale factor will change and thus the comoving nature of A and B will also change. I appreciate that (even if this correct) the time and distances would need to be very extreme to have any impact - (if it is correct) do you know, or could you point me in the direction of material that would help me understand how significant the time / distance measure would need to be?

Regards,

Noel.

 

[andrewkirk]

Lino the distance between the emitter and the receiver doesn't affect that, as the two time periods to which George is referring when he pulls the integrand out of the integrals are the time it takes for the emitter to emit the signal, which will be of the order of 10^-12 seconds if it is visible red light, and the time it takes for the receiver to receive the signal, which will be of the same order as the time taken to emit, only scaled up by the ratio of the scale factors at the time of emission and the time of reception: a(t_r)/a(t_ε).

It is not the time taken for the signal to travel from the emitter to the receiver, which of course is very long indeed.

Also the fact that A and B are comoving never changes as the definition of the comoving coordinate system requires that the coordinates of comoving objects remain fixed.

So the argument remains valid.

 

[andrewkirk]

George that is a really beautiful piece of reasoning you have written. It's a long time since I read something so technical that I understood first time and by which I was convinced. It should be pinned in a FAQ or something like that, so it doesn't get lost.

 

[Lino]

... So the argument remains valid.

Thanks Andrew. That makes sense. May I ask a follow-up question, which I hope is still on topic, but if not please feel free to ignore? If you consider an observation (of, say, a distant galaxy, rather than a signal between observers), I appreciate that individual observations will still be of reasonably short intervals, but what about the comparison between observations from one day to the next, or one week to the next, or one year to the next. Is the logic still sound?

Regards,

Noel.

 

[andrewkirk]

It depends on what aspect of the observations of the distant galaxy on one day and the next interests you. If the question is whether its redshift will have noticeably changed the answer would be no. The scale parameter changes far too slowly for those changes to be measurable over an interval of a day.

On the other hand, I think that the rate of time passing in the distant galaxy, as observed by us, will be the rate here divided by the redshift ratio. So if the redshift is such that the wavelength is doubled and the frequency halved then to us time in the distant galaxy will appear to be passing at half the rate that it is here. So if there is a pulsar in the galaxy that looks to us like it has a period of 2 seconds, it really has a period of only 1 second. Similarly, if we take observations of the galaxy a day apart according to our clocks then we are actually seeing what happened in the galaxy at an interval of only 12 hours apart.

 

[Lino]

Thanks Andrew. Much appreciated.

Regards,

Noel.

 

[George Jones]

George that is a really beautiful piece of reasoning you have written. It's a long time since I read something so technical that I understood first time and by which I was convinced. It should be pinned in a FAQ or something like that, so it doesn't get lost.

Thanks.

Thanks Andrew. That makes sense. May I ask a follow-up question, which I hope is still on topic, but if not please feel free to ignore? If you consider an observation (of, say, a distant galaxy, rather than a signal between observers), I appreciate that individual observations will still be of reasonably short intervals, but what about the comparison between observations from one day to the next, or one week to the next, or one year to the next. Is the logic still sound?

I don't know what you mean by "Is the logic still sound?"

If we watch a given galaxy over a long period, then, at any given time, redshift will be given by

$$z = \frac{R \left( t_o \right)}{R \left( t_e \right)}-1,$$
but $z$ will change over time because $t_o$ (for us) and $t_e$ (for the observed galaxy) both change over time. If we could directly observe this effect, it would be a fantastic way to test our models of the universe!

We are close to being able to do this, but, for economic and other reasons, such a project won't start for several decades. Once started, the project would take a couple of decades to start to get good results. From

http://arxiv.org/abs/0802.1532:

we find that a 42-m telescope is capable of unambiguously detecting the redshift drift over a period of ~20 yr using 4000 h of observing time. Such an experiment would provide independent evidence for the existence of dark energy without assuming spatial flatness, using any other cosmological constraints or making any other astrophysical assumption.

Also, redshifts of individual objects don't necessarily increase with time. Figure 1 from the above paper plots redshift versus time. The three red curves are for objects in our universe. As we watch (over many years) a distant, high redshift object, A, we will see the object's redshift decrease, reach a minimum, and then increase. If we watch a much closer, lower redshift object, B, we see the object's redshift only increase.

Roughly, when light left A, the universe was in a decelerating matter-dominated phase, and when light left B, the universe was in the accelerating dark energy-dominated phase.

 

[andrewkirk]

Looking again at George's derivation of the redshift formula, I see that it is just as valid for a local 'Doppler' effect as it is for distant galaxies where the redshift is typically described as being caused by the 'expansion of space'. All we have to do is define the comoving coordinate system as a time-dependent one in which the spatial coordinate distance between the observer and the emitter is constant. It even works for sound waves being emitted by ambulance sirens.

This rather nicely demonstrates that there is no intrinsic difference between redshifts from cosmological expansion and redshifts from local Doppler effects. They are just different ways of thinking about the same type of phenomenon.

I have now read the Bunn & Hogg paper to which Marcus referred in his post on page 1. That is a really excellent paper, and easy to understand. It generalises the redshift concept further and argues - convincingly to me - that cosmological redshifts, Doppler redshifts and gravitational redshifts are all the same phenomenon, viewed in different ways.

As Bunn & Hogg say, it is a pity that many physicists say there is some sort of fundamental difference between local Doppler redshifts and cosmological redshifts. This just sows unnecessary confusion because there isn't a fundamental difference, viewed from a GR perspective.

 

[andrewkirk]

I have a question about the Bunn & Hogg paper. I don't know if it'll get seen here in this old thread, but I'll try that first rather than just starting a new thread, since it's on the same topic.

In section III they suggest we parallel transport a distant galaxy's ancient four-velocity along the lightlike geodesic that reaches us now, and then measure the recessional velocity as
v_rel = sqrt(1-1/g(v_ob,v_em)²), where v_ob and v_em are the observer's current four-velocity and the emitter's parallel transported four-velocity respectively.

They then claim that this v_rel obeys the SR Doppler formula

sqrt((c+v_rel)/(c-v_rel)) = a(t₀)/a(t_em)
where a(t) is the cosmological scale factor at cosmic time t and t₀) and t_em are the cosmic time of observation and emission respectively.

They do not present the working as to how they arrive at this result, so my first question is whether anybody can point to a derivation of the result.

My second question is about the fact that the formula gives an imaginary redshift for an object receding faster than light. That seems to run into conflict with the statement in papers such as Davis & Lineweaver (2003) that we are able to see some galaxies outside the Hubble Sphere (both at the time they emitted the light we see now and ever since) that are receding from us faster than light, but with real redshifts in the range 1.46 - 6.6. How can this be reconciled? Is it because v_rel differs from the recessional velocities to which Davis & Lineweaver refer? If so, what is D&L's definition?

 

[George Jones]

They then claim that this v_rel obeys the SR Doppler formula

sqrt((c+v_rel)/(c-v_rel)) = a(t₀)/a(t_em)
where a(t) is the cosmological scale factor at cosmic time t and t₀) and t_em are the cosmic time of observation and emission respectively.

They do not present the working as to how they arrive at this result, so my first question is whether anybody can point to a derivation of the result.

This result is derived in section II of reference 28 for the Bunn and Hogg paper,

J.V. Narlikar, "Spectral shifts in general relativity," Am. J. Phys. 62, 903-907 (1994).

Have a look at this paper, and, in this thread, post any questions that you have about it.

My second question is about the fact that the formula gives an imaginary redshift for an object receding faster than light. That seems to run into conflict with the statement in papers such as Davis & Lineweaver (2003) that we are able to see some galaxies outside the Hubble Sphere (both at the time they emitted the light we see now and ever since) that are receding from us faster than light, but with real redshifts in the range 1.46 - 6.6. How can this be reconciled? Is it because v_rel differs from the recessional velocities to which Davis & Lineweaver refer?

Yes.

If so, what is D&L's definition?

Their (standard) definition is given is given in appendix A of their paper. Again, if you have any questions, just post.

 

[Lino]

... We are close to being able to do this, but, for economic and other reasons, such a project won't start for several decades. Once started, the project would take a couple of decades to start to get good results...

Thanks George, and apologies for the delayed appreciation. I'm still working my way through this, but it is making more sense.

Regards,

Noel.

 

[Lino]

... We are close to being able to do this, but, for economic and other reasons, such a project won't start for several decades. Once started, the project would take a couple of decades to start to get good results. ...

George, If this is off topic and should be a new thread, please feel free to let me know and action accordingly. (From searches that I have tried and what I could tell from the linked paper) I am surprised that something like this has not been tried todate. Given redshift measurements that have been taken of objects over the last (almost) century, is it not possible and appropriate to take a new measurement today and compare those to historic measurements of the same objects to achieve the same effect? I appreciate that this would not have the level of accuracy described in the paper that you linked to, but it seems (to me) that it should give a reasonable result (with the same level of accuracy that current redshift/distance relationships are known to).

(Again, if this has diverted from this thread, do please let me know and I will act accordingly.)

Regards,

Noel.

 

[andrewkirk]

George I'm working through the Narlikar paper. It's hard work as he seems to skip about 3-4 steps that I feel the need to write out between each of his steps. But it's worthwhile, and very good practice for me .

There are two steps in Part II. Cosmological and Doppler Shifts, that I cannot validate.

1. Deriving equation [20]. Starting with [18b] $U^i_O\bar{V}_{iS}=U^i_SV_{iS}$ I am able to get to:
$a(t_O)\bar{V}^{0}_{S}+a^2(t_O)\bar{V}^{1}_{S}=a(t_S)(V^{0}_{S}+V^{1}_{S}\frac{a(t_S)}{\sqrt{1-kr_S^2}})$, where the vector components on the RHS are in the FLRW basis, but Narlikar says that the right-hand side should just be $a(t_S)$, ie this implies that the parenthesis is equal to 1, but I cannot prove that.

If the vector components were in the inertial basis momentarily comoving with S we'd be OK, because we'd have $\vec{V}_S=[1,0,0,0]$in that basis. But they're not. They're in the FLRW basis.

I've attached a .doc file with my working for this.

2. In the paragraph arguing towards equation [21], Narlikar says "the radially outward direction from S is radially inwards at O". I don't follow this. There is no polar or spherical reference frame in use here that is centred at S, so presumably by radially outward from S he means radially outward in the FLRW frame, which is centred at O. But in that case radially outward at S is the same direction as radially outward at O, contrary to what Narlikar suggests.

Are you able to suggest anything about how I can fill in these steps?

Thanks very much.

 

[Chalnoth]

George, If this is off topic and should be a new thread, please feel free to let me know and action accordingly. (From searches that I have tried and what I could tell from the linked paper) I am surprised that something like this has not been tried todate. Given redshift measurements that have been taken of objects over the last (almost) century, is it not possible and appropriate to take a new measurement today and compare those to historic measurements of the same objects to achieve the same effect? I appreciate that this would not have the level of accuracy described in the paper that you linked to, but it seems (to me) that it should give a reasonable result (with the same level of accuracy that current redshift/distance relationships are known to).

(Again, if this has diverted from this thread, do please let me know and I will act accordingly.)

Regards,

Noel.

Our ability to measure redshift is only just now becoming accurate enough to measure these differences.

 

[George Jones]

If the vector components were in the inertial basis momentarily comoving with S we'd be OK, because we'd have $\vec{V}_S=[1,0,0,0]$in that basis. But they're not. They're in the FLRW basis.

It is also true in FLRW coordinates that $\vec{V}_S=\left[1,0,0,0\right]$. I haven't had a chance to look at 2. very much, but I think that it might just be case of poor wording.

Unfortunately, your document contained intermittent gibberish in my Word 2007 which made it unreadable.

 

[andrewkirk]

It is also true in FLRW coordinates that $\vec{V}_S=\left[1,0,0,0\right]$.

Thanks again George. I don't know why I didn't realize that before. I was wrongly assuming that S's velocity would have a spatial component in any coordinate system centred at O, but I forgot that, under the FLRW coordinates, it appears spatially stationary because it is comoving.
So item 1 now works fine!
I will turn my attention to item 2, which still puzzles me, with renewed confidence.

 

[George Jones]

2. In the paragraph arguing towards equation [21], Narlikar says "the radially outward direction from S is radially inwards at O". I don't follow this. There is no polar or spherical reference frame in use here that is centred at S, so presumably by radially outward from S he means radially outward in the FLRW frame, which is centred at O. But in that case radially outward at S is the same direction as radially outward at O, contrary to what Narlikar suggests.

I think I know what the words mean, but, to make sure, I decided to have a go at deriving (21) using math and ignoring the words. After some algebra, I was able to show $\left( \gamma V \right)^2 = \left( a\left( t_O \right) \tilde{V}^1_S \right)^2$. A bit more algebra gives (21).

I'll try and post the algebra tomorrow.

 

[andrewkirk]

One of the things that makes proving [21] particularly problematic for me is that it seems to require using the metric in the tangent space at O, but the metric is given in FLRW coordinates, which don't generate a useable basis for that tangent space, because the θ and $\phi$ (ie circumferential) basis vectors are non-existent and the r (radial) basis vector has an undefined direction (every direction is radial away from O). Hence the metric as given by [7] in FLRW coordinates appears to be inapplicable because it relates to these undefined basis vectors.

There must be some way around this, perhaps by expressing the metric in terms of the local Lorentz frame, but I haven't found it yet.

Edit: Thinking a bit more about this, I see that Narlikar's claim that, in the FLRW basis, $U^i(1)=A[a(a(t_O),-1,0,0]$ (line after [16]) is strictly meaningless, as it is written in terms of a basis that does not exist.

 

[andrewkirk]

Oh dear it's getting worse! I've now realized that equations [18], [19], [20] and [21] are all meaningless, because they use vector components $\bar{V}^0_S$ and $\bar{V}^1_S$ in a basis of $T_OM$ that does not exist: the basis derived from the hyperspherical comoving coordinates centred at O (given in equation [7]).

Just when I thought the whole thing almost made sense, it's starting to look like it's invalid.

 

[andrewkirk]

I think I have developed a solution to the problems with the Narlikar paper and will post it once I have written it up.

 

[andrewkirk]

Here's the solution as a pdf version of a word-processed file. I'll try to make a TeX version that is easier to read, and replace this one with that when I've done it. Any comments would be very welcome.

The file is essentially a proof that cosmological redshift due to expansion is equivalent to Doppler redshift.

 

[andrewkirk]

I have done a nice TeX version of the proof now, as well as fixing up the two hand-wavy bits to make them rigorous.
I posted it in the wrong thread though. The pdf is attached to the following post:https://www.physicsforums.com/showpost.php?p=4030816&postcount=12