## Cosmological redshift and doppler redshift

I am a bit confused here. Is cosmological redshift the same as doppler redshift?
This is from wikipedia:
"The redshift z often is described as a redshift velocity, which is the recessional velocity that would produce the same redshift if it were caused by a linear Doppler effect (which, however, is not the case, as the shift is caused in part by a cosmological expansion of space, and because the velocities involved are too large to use a non-relativistic formula for Doppler shift)"
If the cosmological redshift is based in the expansion of the metric, that means that the galaxies we observe at high z are not really receding at those superluminal speeds, right?
But the cosmological redshift is actually derived from the assumption that the redshift is doppler or recesional, at least in the fist years of modern cosmology, so it's a little confusing,can someone clear this up a little?
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 There is a difference. Doppler shift tells us what is happening to the object when the light was emitted (is it approaching, receding, at what speed), it does not take into account what happened to the light after it left its source. Cosmological redshift tells us what was happening to the light while it was on the way. Or think of this hypothetical scenario: Universe contracts, distant galaxy emits photon towards us. Now, some form of phantom energy kicks in, universe reverses to expansion, and for the remaining time of photons journey it expands (larger portion of journey). So although galaxy was initially approaching while emitting light, you end up with redshift.
 Ok, so a galaxy might have had any random movement (like towards us or away from us) ,what counts is the expansion of the metric on its light path from there to here,the stretching of the space, so to speak, at least that is how I understand it. And yet I think I read on some thread here that there is no such thing as expansion of the space, per se, but increase of distances between distant galaxies, but how would anybody tell one from the other?

## Cosmological redshift and doppler redshift

 Quote by TrickyDicky Ok, so a galaxy might have had any random movement (like towards us or away from us) ,what counts is the expansion of the metric on its light path from there to here,the stretching of the space, so to speak, at least that is how I understand it.
Yes, what counts is ratio of scale factor now and then. Cosmological redshift is the net result of the history of expansion during which light was on the journey. Doppler formula is used for local coordinates, but in cosmological terms it gives wrong answer.

 And yet I think I read on some thread here that there is no such thing as expansion of the space, per se, but increase of distances between distant galaxies, but how would anybody tell one from the other?
It is semantics question. Some people don't like to give to notion of space to much "substance". It is really not that important how do you call it, bottom line is the same - there is more space, and distances are increasing.

 Quote by Calimero bottom line is the same - there is more space, and distances are increasing.
How can it posibly be the same? Perhaps you could say that the stretching of space notion was historycally derived from the aparent general receding of galaxies based on the concept of Doppler effect. But that doesn't mean they are the same regardless semantics.If it was the same, of any two objects departing from each other we could say the space between them is stretching wich I don't think it's the case.

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Tricky,
First off (and this is really related to your question) you might be curious to know what is the absolute motion of the solar system, relative to the Background cosmologists use, and what is the overall motion of the Milkyway galaxy, relative to Background. We know the speed and direction in each case. It has been measured precisely by mapping the temperature of the microwave sky. This is slightly "hotter" in the direction that we are moving, and how much hotter tells us how fast. If you are, in fact, curious about that, you can start a thread or else look up some past threads where other PF people have asked about that.

 Quote by TrickyDicky If the cosmological redshift is based in the expansion of the metric, that means that the galaxies we observe at high z are not really receding at those superluminal speeds, right?
Not right. It's a semantic thing. In cosmology talk, "receding" can simply refer to the increase of distance, and not to motion relative to Background.
A "recession rate" can be superluminal. According to the Hubble law (v = Hd) if the distance d is large enough the recession rate v HAS to be superluminal. This is inherent in the standard form of the law which uses quantities defined by certain conventions.

So the galaxies at high z (any z > 1.4, which means most galaxies we can observe) really are receding at superluminal rates.

However they are approximately stationary relative to the CMB (cosmic microwave background).

 But the cosmological redshift is actually derived from the assumption that the redshift is doppler..., at least in the first years of modern cosmology, so it's a little confusing,can someone clear this up a little?
"In the first years of modern cosmology"---you mean Hubble's work in the 1930s? He was dealing with comparatively nearby, small z, galaxies. Like z < 0.1. At small distances the Doppler approximation works very well!

Over short timespans like 100 million years the recession rates do not change appreciably. In that simple case, you can think of the average rate a certain distance has been increasing as the equivalent to what it was doing when the light was emitted, and a simple Doppler picture works just fine.

So in 1930s cosmology, all the redshift data could be interpreted and discussed in a straightforward Doppler way. But that does not fit today's data or correspond to how the redshift arises in contemporary cosmologists' favorite model of the universe---the FRW (Friedman Robertson Walker)---sometimes with an L for Lamaitre.

In that standard math model of the universe, the wavelength expansion ratio 1+z equals the ratio by which the universe has expanded during the time the light has been traveling.
It depends on the whole expansion history while the light was in transit.

The standard model uses the idea of observers stationary relative to the Background (the ancient light--radiated by the most ancient visible matter, a roughly uniform hot gas).
All these observers experience the same time, called FRW time, or "universe time".
This is the time parameter used in the FRW model. The corresponding idea of distance is called "proper distance". It is what you would measure if you could freeze the expansion process at a given instant of universe time, and then use radar or light signals in the usual way.

The Hubble law v=Hd is expressed in terms of proper distance d and its rate of increase v at a certain moment of universe time. Everything in the law is time-dependent---the Hubble parameter changes too. So to be pedantic it should be written v(t) = H(t)d(t).

FRW model is based on general relativity, which allows distances to increase at rates exceeding the speed of light. Indeed as applied to cosmology via the Friedman model it effectively requires superluminal expansion rates.

The basic lesson, I guess, is that geometry is dynamic (that's what spacetime curvature is about.) You don't have the right to expect that the distance between two stationary observers will remain constant. General relativity is about dynamically changing geometry, and cosmology is based on that (not on special) so you have to retool your geometric intuition somewhat---change some expectations that were acquired in a static geometry situation.

 Quote by marcus In cosmology talk, "receding" can simply refer to the increase of distance, and not to motion relative to Background.
 Quote by marcus So the galaxies at high z (any z > 1.4, which means most galaxies we can observe) really are receding at superluminal rates. However they are approximately stationary relative to the CMB (cosmic microwave background).
Ok, so the Background is common for every object in the universe and works as a spatial absolute reference, maybe?

 Quote by marcus The standard model uses the idea of observers stationary relative to the Background (the ancient light--radiated by the most ancient visible matter, a roughly uniform hot gas). All these observers experience the same time, called FRW time, or "universe time". This is the time parameter used in the FRW model. The corresponding idea of distance is called "proper distance". It is what you would measure if you could freeze the expansion process at a given instant of universe time, and then use radar or light signals in the usual way.
And FRW time is kind like reference absolute time or "universe time",is that it?

 Quote by marcus So in 1930s cosmology, all the redshift data could be interpreted and discussed in a straightforward Doppler way. But that does not fit today's data or correspond to how the redshift arises in contemporary cosmologists' favorite model of the universe---the FRW (Friedman Robertson Walker)---sometimes with an L for Lamaitre.
But our contemporary cosmology model is based in those observations at low z by Hubble if I am not mistaken. BTW I think Friedmann , Walker and Robertson developed this metric in the 1920's and 30's.

 Quote by marcus You don't have the right to expect that the distance between two stationary observers will remain constant.
That's for sure, but this was the core of my question, if this two stationary -according to the Backgroun as you described- observers will not remain stationary according to each other (they observe each other receding) there is surely some geometrical trick going on here, since how can they be stationary in the reference spacetime you just described and receding at superluminal speeds at the same time?

 Quote by TrickyDicky How can it posibly be the same? Perhaps you could say that the stretching of space notion was historycally derived from the aparent general receding of galaxies based on the concept of Doppler effect. But that doesn't mean they are the same regardless semantics.If it was the same, of any two objects departing from each other we could say the space between them is stretching wich I don't think it's the case.
Historically, when people noticed that universe is expanding it was thought that it is due to some initial velocities that galaxies possessed, and that it must be slowing down due to the gravitation. Surprise came along with distant supernovae data which showed that universe is not slowing, but accelerating. So dark energy was hypothesized. As we still don't have a proper knowledge of what dark energy really is, other then simply plugging cosmological constant in Friedmann equations, we are describing accelerated expansion of universe as it looks like space itself is expanding. Of course, you can ask, how expanding space can push galaxies? Well it can't. And that is the trouble with that analogy. Other than that it provides good description of what is happening. So, Marcus gave you a really good explanation. It is expanding geometry.

 Quote by TrickyDicky Ok, so the Background is common for every object in the universe and works as a spatial absolute reference, maybe?
Yes, Background is common for every observer in the universe, and no, it is not spatial absolute reference (no such thing). As a spatial reference it can be only used to determine if observer is at fixed comoving spatial position (no hot spots and cold spots, he measures same temperature in every direction). It is also time reference.

What we are calling Cosmic Microwave Background Radiation are photons (radiation) that originated from single event around 300000+ years after the Big Bang. What is so special about them? Event that generated them happened everywhere.

So, universe is filled with highly isotropic radiation. What is happening to that radiation as time passes by? It is redshifted due to the space expansion, and its temperature drops, hence if observers measure the same temperature of CMB they are doing it in the same cosmological time.

 Quote by TrickyDicky That's for sure, but this was the core of my question, if this two stationary -according to the Backgroun as you described- observers will not remain stationary according to each other (they observe each other receding) there is surely some geometrical trick going on here, since how can they be stationary in the reference spacetime you just described and receding at superluminal speeds at the same time?
CMB radiation is undergoing exactly the same expansion as those observers, so they are stationary relative to the CMB.

 Quote by TrickyDicky And FRW time is kind like reference absolute time or "universe time",is that it?
It is simply the time that comoving observers measure since the big bang according to their own clocks. Or above definition via CMB temperature is also fine.
 Recognitions: Gold Member Science Advisor Tricky, thanks for replying to my post with a bunch of specific questions. I had to be away for a while and Calimero answered everything. I agree with his answers and they seem very clear, so I won't repeat. Looking forward to a new round, if you have further things to ask about.

 Quote by Calimero Yes, Background is common for every observer in the universe, and no, it is not spatial absolute reference (no such thing).
Right, not absolute space,since CMB is comoving (that would lead us back to a Newtonian world), but a spatial reference nevertheless, and as you said a cosmic time that if we believe in a Big Bang origin, (no other previous universes etc) is kind of absolute; isn't this a departure from Minkowskian spacetime? (in Specialr relativity you don't differentiate space from time and there is no such thing as cosmic time)

 Quote by Calimero CMB radiation is undergoing exactly the same expansion as those observers, so they are stationary relative to the CMB.
I think this is the key here. But if the CMB radiation is comoving with us, why do we see it as redshifted?

 Quote by TrickyDicky Right, not absolute space,since CMB is comoving (that would lead us back to a Newtonian world), but a spatial reference nevertheless, and as you said a cosmic time that if we believe in a Big Bang origin, (no other previous universes etc) is kind of absolute; isn't this a departure from Minkowskian spacetime? (in Specialr relativity you don't differentiate space from time and there is no such thing as cosmic time)
General relativity allows us to use any arbitrary coordinates, although some choices are more natural and easier to work with. That does not make them absolute in any way.

 Quote by TrickyDicky I think this is the key here. But if the CMB radiation is comoving with us, why do we see it as redshifted?
When we say 'stationary relative to the CMB' we think stationary relative to the CMB photon "soup". However photons are, of course, going at the speed of light.

Each successive photon of CMB that observer detects is coming from more distant place then previous one. That means that it spent more time traveling in expanding space, and thus higher redshift.

Thanks for the effort to explain
 Quote by Calimero General relativity allows us to use any arbitrary coordinates
Is this what they mean when it is said to be an invariant theory?
Does it apply to the metric too,so even if the FLWR metric sems the most convenient, others could be used in principle?

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I'll try to augment what Cali is saying.
 Quote by TrickyDicky I think this is the key here. But if the CMB radiation is comoving with us, why do we see it as redshifted?
Try not thinking Doppler, think expansion of distances. As distances expand, so do wavelengths, by the exact same proportion. This is the source of the redshift.

 Quote by TrickyDicky ,so even if the FLWR metric sems the most convenient, others could be used in principle?
the math expression of the metric depends on the coordinates. If you change coords (as you are free to do) you change the metric. So in a trivial sense sure. The standard picture can be morphed into many equivalent pictures.

BTW, I didn't read everything so I have to ask: do you know about the scale factor a(t)?
This is a factor which is defined within the context of the FLWR metric and it depends on universal time t. It tells the expansion history. It is arbitrarily normalized to be unity at present time----smaller in past, larger in future.
a(now) = 1.
Sometimes people call it "average distance between galaxies".

The actual definition of the cosmo redshift z (of light emitted back then and recieved now) is:

z+1 = a(now)/a(then)

z is the fractional gain in wavelength so that z+1 is the actual ratio of
wavelength(now)/wavelength(then)

So putting that together:
wavelength(now)/wavelength(then) = a(now)/a(then) = distance to it now/distance to it then

The z+1 ratio is the ratio that wavelengths have expanded AND the ratio that distances between galaxies have expanded while the light was in transit.

The basic cosmo model that everything is based on is the FRIEDMAN EQUATION model, which is a simple differential equation describing the evolution of a(t)

The math definition of the Hubble parameter H(t) that everybody uses is
H(t) = a'(t)/a(t).

So the FLWR metric and the Friedman equation model are a single package which if you buy it:
1. tells you the expansion history a(t) as a solution to a dif. eqn.
2. tells you the cosmo redshift z + 1 = a(day received)/a(day emitted)
3. tells you the Hubble parameter H(t) and how it evolves with time
4. slices the 4D into 3D space slices which all have the same Background temperature.
5. conforms with idea of observer at rest sees no microwave hotspot in sky. If he is at FLWR rest then he is at rest relative to Background.

A good thing to remember is that if you are at rest relative to Background, the light from the early hot gas before it began to condense and fall together into clumps, then you are in a sense at rest with respect to the ancient matter of the universe.

The FLWR metric is the natural one for an observer at rest to choose, since its space, its simultaneous moment everywhere, coincides with the observer's own idea of space.

The moral is, adjust your frame so that Doppler hotspots in the CMB sky go away and your microwave sky is as uniform temperature as possible. Factor the dipole out (because it is caused by individual random motion.)

 Quote by marcus adjust your frame so that Doppler hotspots in the CMB sky go away and your microwave sky is as uniform temperature as possible. Factor the dipole out (because it is caused by individual random motion.)
Just curious,the dipole according to http://antwrp.gsfc.nasa.gov/apod/ap090906.html indicates a speed for the local group of about 600 km/s, perhaps a bit high to be factored out as individual random motion?

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 Quote by TrickyDicky Just curious,the dipole according to http://antwrp.gsfc.nasa.gov/apod/ap090906.html indicates a speed for the local group of about 600 km/s, perhaps a bit high to be factored out as individual random motion?
Basically what we factor out is the solar system's own motion relative to Background. This is about 380 km/s in the direction of the constellation Leo. (you can get very precise coordinates if you want.)

This is what needs to be factored out of the data, like the CMB data itself, to make accurate maps, and the recession rates of galaxies in order to accurately estimate expansion.

You can think of that 380 as the resultant of adding up
1. the local group motion
2. plus the Milkyway motion within the local group
3. plus the solar system motion within the Milkyway
where vector addition causes some cancelation when there are opposing directions.

But thinking of it that way adds measurement uncertainty and unwanted extra baggage.

What matters is OUR motion relative background. which is around 380 km. And that is not "a bit too high to be factored out". And it HAS to be factored out to get good maps and good data.

And neither that 380 or the estimated 600 is especially high. Things in the world have random velocities of a few hundred km/s.
That is a small fraction of the speed of light. And it is just how it is. Structures fell together into wispy blobs and they picked up a few 100 clicks of speed and they are still falling.

It appears to be mostly in random directions but there are some vague largescale patterns of motion that some people think they have detected. The publicity has been overblown in some cases. In any case just marginal deviations from randomness, interesting if true but not IMO a revolutionary change in the basic picture.

You are on your own if you want to venture into that, maybe Cali or bapowell will help. I just wanted to comment on really basic stuff you were asking about, like FLWR and scalefactor a(t) and redshift. So I'll get out of the way now.
 An article by Tamara Davis in the current issue of Scientific American provides a 'popular science' level explanation how cosmological redshift can indeed be thought of as an accumulation of infinitesimal doppler shifts along a photon's path. This requires one to view the recession of distant galaxies as a kinematic event (actual motion through space) rather than an expansion of 'space itself'. The article cites to a paper by Bunn & Hogg which describes this interpretation in more technical terms. I think this view has become widely adopted in the last couple years, with the B&H paper considered to be very influential. Note that an 'accumulation' of infinitesimal doppler shifts yields a very different numerical answer than taking an end-to-end doppler calculation of the relative velocities of the emitter and receiver at the time of emission and reception respectively. The accumulation calculation yields a redshift figure which happens to be identical to the proportional amount by which the scale factor of the universe has expanded during the photon's journey. Thus the 'expanding space' and 'kinematic' interpretations of the cosmological redshift are absolutely indistinguishable from an observational and mathematical perspective.

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