# Density of the early Universe contributing to the red-shift?

• B
• D.S.Beyer

#### D.S.Beyer

Does the relative density of the early universe contribute to the red-shift of distant galaxies?
If so, by how much? How would this be calculated?

Assuming both the early universe and the current universe are flat, could the relative difference of their space time metric tensors contribute to the red-shift between them?

Was light from distant galaxies emitted at a relatively lower gravitational potential, due to higher density of the early universe?

Thank you all for your patience, I'm basically just struggling with time.

Nothing about the early universe could alter the speed of light and little beyond the rate of expansion would affect redshift. Little, if any, of the redshift of remote galaxies can be attributed to gravitational redshift. Even as far back as recombination the matter density of the universe was far too low to have any measurable effect on redshift.

Does the relative density of the early universe contribute to the red-shift of distant galaxies?
If so, by how much? How would this be calculated?
I'm not sure I understand this question. That the density of the universe decreases is pretty much another way of saying it's expanding. So one could say that the density change is responsible for the redshift. It feels a bit like attributing an effect to an effect, though, rather than attributing both effects to a cause. I might be wrong there.
Assuming both the early universe and the current universe are flat, could the relative difference of their space time metric tensors contribute to the red-shift between them?
This is definitely just a way of saying that the universe is expanding. The difference between the metric in the early universe and now is literally just the scale factor. The changing scale factor is responsible for the redshift.
Was light from distant galaxies emitted at a relatively lower gravitational potential, due to higher density of the early universe?
No. You can't define a gravitational potential in a non-stationary spacetime, and the FLRW spacetime used in cosmology is non-stationary. So this question can't be phrased in a way that makes sense in GR terms.

I think you are trying to ask if the cosmological redshift and gravitational redshift familiar from Pound-Rebka, the GPS system, or Interstellar have the same origin. The answer is yes and no.

Yes, because in both cases a light pulse takes longer to arrive (measured locally) than it took to emit (again, measured locally), and the root cause of that is that initially parallel light paths diverge due to spacetime curvature. No, because the geometries of spacetime in the two cases are radically different. That they're not flat (and hence locally parallel lines don't remain parallel everywhere) is pretty much all they have in common beyond basic properties.

PeterDonis
Does the relative density of the early universe contribute to the red-shift of distant galaxies?
If so, by how much? How would this be calculated?

Assuming both the early universe and the current universe are flat, could the relative difference of their space time metric tensors contribute to the red-shift between them?

Was light from distant galaxies emitted at a relatively lower gravitational potential, due to higher density of the early universe?

Thank you all for your patience, I'm basically just struggling with time.
These sorts of questions don't really help understand what's going on better.

The funny thing about General Relativity is that it doesn't actually say anything about the causes of any redshift. If you take a physical system, describe it in the form that General Relativity uses, you can always calculate what the redshift is. That's the easy part. But that calculation isn't a matter of adding up contributions like "doppler shift" or "gravitational redshift". You just get a single number from a complex calculation that tends not to resemble anything like those things.

Concepts like "doppler shift" and "gravitational redshift" stem from idealized scenarios designed to highlight their appearance, such as a photon traveling away from a black hole or two objects passing one another at high speed. But in general these idealized scenarios don't hold, and the reality is that the nature of reality is such that you can look at the exact same system in extremely different ways and it will look completely different!

For example, with the cosmological redshift, here are three completely equivalent ways of looking at the situation:
1. As the universe expands, the wavelength of photons also expands by the exact same factor.
2. Photons themselves have pressure, and if you draw a hypothetical, expanding box around a photon gas, that photon gas exerts pressure on the box's walls. That expansion causes the gas to do work on the walls of the hypothetical box, which causes the photon gas to lose energy. Thus, via energy conservation, the photons have to redshift. This description is an analogy, but it mirrors stress-energy conservation which General Relativity obeys.
3. You can look at redshift as a perspective effect, by tracing the path of the photon across the universe, and considering a set of observers that the photon crosses along its path, observers who are stationary with respect to the average expansion. As the photon goes from observer to observer, it picks up a little bit of a redshift because that observer is moving away from the location the photon is coming from. Add up all those little contributions from the perspectives of the various observers, and you get a net redshift.

I'm sure there are other ways to look at it as well, ways that are entirely mathematically-accurate.

Buzz Bloom, jeffinbath and Ibix
Photons themselves have pressure, and if you draw a hypothetical, expanding box around a photon gas, that photon gas exerts pressure on the box's walls. That expansion causes the gas to do work on the walls of the hypothetical box, which causes the photon gas to lose energy. Thus, via energy conservation, the photons have to redshift.

Wouldn't this heuristic argument also apply to a gas of ordinary particles? If so, it wouldn't explain why ordinary matter in the expanding universe doesn't redshift the way photons do. (More precisely, the energy density of ordinary matter decreases more slowly with expansion than the energy density of photons does; the extra decrease for photons is the photon redshift.)

Wouldn't this heuristic argument also apply to a gas of ordinary particles? If so, it wouldn't explain why ordinary matter in the expanding universe doesn't redshift the way photons do.
The momentum of ordinary particles also redshifts. The difference lies in the momentum of massless particles being equal to their energy.

Yes, because in both cases a light pulse takes longer to arrive (measured locally) than it took to emit (again, measured locally), and the root cause of that is that initially parallel light paths diverge due to spacetime curvature. No, because the geometries of spacetime in the two cases are radically different. That they're not flat (and hence locally parallel lines don't remain parallel everywhere) is pretty much all they have in common beyond basic properties.
This makes it sound a bit like the non-flat geometry is the source of the redshift, which is not the case (if it were you would not have Doppler shift in SR). The underlying cause for redshift is comparision of the observer/emitter 4-velocities with the 4-frequency of the light (which is parallel transported along the world line of the light signal).

The momentum of ordinary particles also redshifts. The difference lies in the momentum of massless particles being equal to their energy.

Ah, ok. So heuristic #2 in @kimbyd's post should read, not just "photons have pressure", but "photons have pressure of the same order of magnitude as their energy density". The latter condition is what makes their redshift significant.

Wouldn't this heuristic argument also apply to a gas of ordinary particles? If so, it wouldn't explain why ordinary matter in the expanding universe doesn't redshift the way photons do. (More precisely, the energy density of ordinary matter decreases more slowly with expansion than the energy density of photons does; the extra decrease for photons is the photon redshift.)
Yes, this describes the math involved in how a uniform perfect fluid changes over time in an expanding universe. And, as Orodruin mentioned, the fact that the pressure of non-relativistic matter is tiny in practice explains why non-relativistic matter is mostly constant in energy density.

But it isn't exactly constant! The expansion of the universe does actually cause normal matter to slow down too. The change in energy is so tiny compared to the mass of the matter that it's usually ignored. But when it comes to understanding the dynamics of galaxies on large scales, this effective friction is important.

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This makes it sound a bit like the non-flat geometry is the source of the redshift, which is not the case (if it were you would not have Doppler shift in SR). The underlying cause for redshift is comparision of the observer/emitter 4-velocities with the 4-frequency of the light (which is parallel transported along the world line of the light signal).
Isn't this a matter of how you choose to present it? In the general case there is a quadrilateral in spacetime whose corners are the following events: emit wavecrest 1, emit wavecrest 2, receive wavecrest 1 and receive wavecrest 2. Where there's redshift the interval between the "receive" events exceeds that between "emit" events. In flat spacetime that can only happen if you have relative velocity between the emitter and receiver. In curved spacetime "equivalent" observers such as pairs of hovering Schwarzschild observers or pairs of FLRW co-moving observers see redshift because of geodesic deviation between the null geodesics (or, at least, that's a possible interpretation.

I agree your explanation is better, and certainly more elegant. But I think mine is probably easier to get at B level. Without being wrong or too pop-science-y, I hope.

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You can calculate the matter density of the universe using the Friedmann equation:
. Just for the sake of argument, how big do you suppose 'p' [density of the universe] needs to get before relativistic corrections become important? [btw pc is the critical density of the universe] The answer, which should come as no surprise, is pretty darn big. It may be more helpful to use scale factor instead of density as an aid to visualizing how old the universe was before relativistic corrections started becoming a factor.

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You can calculate the matter density of the universe using the Friedmann equation:
. Just for the sake of argument, how big do you suppose 'p' [density of the universe] needs to get before relativistic corrections become important? [btw pc is the critical density of the universe] The answer, which should come as no surprise, is pretty darn big. It may be more helpful to use scale factor instead of density as an aid to visualizing how old the universe was before relativistic corrections started becoming a factor.
I'm not sure I understand your point. Don't the Friedmann equations follow from the FLRW solution to Einstein's field equations? So surely they're inherently relativistic?

I may well be missing something.

No, you're not missing anything. Of course, the expression given is based on the sum of the mass energy content of the universe, not just baryonic matter content, so indeed it is inherently relativistic. While, that does not alter the fact p must be pretty big relative to pc, before it has much effect on 'z', it also does not alter the fact my point was not entirely lucid.

I am surprised that 12 posts in nobody has mentioned the Sachs-Wolfe effect yet. It is certainly important for understanding the variations in the CMB, although minuscule compared to the total redshift.

I am surprised that 12 posts in nobody has mentioned the Sachs-Wolfe effect yet.
Probably because that's not the kind of density differences the OP was asking about?

Is it possible the higher density contributed to a higher refractive index and hence a net reduction in the effective speed of light?

@Ibix thanks for addressing each way I was trying to ask the question. I’d love to dig more into why we can’t define gravitational potentials of objects in the early universe.
@kimbyd awesome examples! The ‘pressure’ box is wild and makes the imagination race. Evidenced by multiple replies to that point.
@Chronos thanks for bringing in the math. Is there an analogous stellar object that is a comparable density to, say, the CMB?
@Orodruin I had never heard of the Sach-Wolfe effect. Thank you. I’ve got some reading to do.
@andrew s 1905 I enjoy this out of the box thinking.

Essentially with this question I am trying to understand how a densely packed universe, full of energy, does not warp spacetime any more than the one we see today. (@Chronos looking at you for the actually density numbers)

I assumed (likely incorrectly) that the light emitted at the CMB was at a lower gravitational potential than when it finally reaches us now.
Okay, now I’m going to embarrass myself with some diagrams drawn up in photoshop. (@Ibix I’m going to go for the GR counter point, don’t hold your breath)

Figure 1 shows the early universe. Matter is closely packed. The gravity wells of said matter are also packed together. Space-time is, on average, pretty flat within this soup of matter. If the universe was finite, and had an edge where this dense matter gave way to ‘empty’ space, the gravitational well would be pretty sever.

Figure 2 shows the current universe. Matter is spread out. Gravity wells plateau to increasing smaller curvature over the large distances, more closely approximating/reaching idealized ‘empty’ space. Space-time is, on average, pretty flat within this spread out matter.

Figure 3 shows the two states in relation. Making the 'empty' space-time curvature of a theoretically finite universe commonality. Though both states have an on average flatness, the early universe exists at a much lower gravitational potential than the current universe. The finite universe line is not really important, but does illustrate the idea better graphically.

I read about early galaxies being better at making stars, potentially because of the higher density of the early universe, which sent me down this line of questioning.
https://mcdonaldobservatory.org/news/releases/20151119

Ok, rip it up team.

I’m not out to prove anything. Just want to get my thoughts straight. Thanks for bearing with me, I really appreciate this community... even if I don't always understand it.

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@Ibix thanks for addressing each way I was trying to ask the question. I’d love to dig more into why we can’t define gravitational potentials of objects in the early universe.
Formally, because gravitational potential is the conserved quantity associated with the existence of a time-like Killing vector field, and there isn't one in the spacetime used in cosmology.

Very roughly, you can't define a potential unless you can go around in a circle and end up with everything being the same as it was. This is familiar from fluid dynamics where you have frictional losses - once you've gone round in a circle the fluid is hotter and you can't undo that. In cosmology, once you've gone around in a circle the universe is a bit larger and you can't undo that.

The existence of a time-like Killing vector field turns out to be a rigorous way of saying that nothing changed overall while you went round in a circle. Its non-existence means something did change irreversibly, so you can't define a potential.

Just to note - a Killing vector field isn't a real physical thing like an electric field. It's a solution to Killing's equation applied to the spacetime you infer from experiment - a formal description of the behaviour of spacetime.
@andrew s 1905 I enjoy this out of the box thinking.
The effect of changing refractive index is the Sachs-Wolfe effect @Orodruin mentioned. As I understand it, it's more a source of noise in measuring redshift than anything else.
Ok, rip it up team.
The problem with figure 1 is that your blue line "outside matter" doesn't exist even in principle. You are effectively trying to imagine what a universe full of uniform matter would look like if it weren't full of uniform matter. As I asked in another context recently, what colour is a sheep that is completely black and completely white at the same time? The only possible answer is "that makes no sense". A place outside the matter distribution in a universe completely filled with a uniform matter distribution is a concept that makes no sense.

Figure 2 has the same problem. If your figure 1 is a diagram of a sheep that is completely white and completely black, figure two is a diagram of one that is completely brown and completely white at the same time.

Hence your figure three doesn't work. The blue lines that you've joined together don't represent anything that you can define or measure. So, in the sheep analogy, you are sticking together a black and white sheep and a brown and white sheep and saying they're different because the white is the same but the other colour is different - ignoring the fact that neither sheep could actually exist.

Essentially all of this stems from you ignoring me when I said you couldn't define a potential. You are trying to equate potentials (that can't exist) at a point (that can't exist) outside a matter distribution in a universe completely filled with a uniform mass distribution, and hoping to get coherent sense out of the resulting mess.

If you want to reason about GR, rather than rote learn some facts, you have to learn the maths. Trying to guess how it works based on vague verbal descriptions won't produce a clear enough understanding - as you have just demonstrated.

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weirdoguy
The effect of changing refractive index is the Sachs-Wolfe effect @Orodruin mentioned. As I understand it, it's more a source of noise in measuring redshift than anything else.
This is not the Sachs-Wolfe effect. The Sachs-Wolfe effect deals with the CMB originating from regions with different density and therefore different gravitational redshift. Light from overdense regions will be more redshifted. This is visible in the CMB fluctuations.

Ibix
This is not the Sachs-Wolfe effect. The Sachs-Wolfe effect deals with the CMB originating from regions with different density and therefore different gravitational redshift. Light from overdense regions will be more redshifted. This is visible in the CMB fluctuations.
...and that's why I @'d Orodruin on that bit. Not sure what I read that I misunderstood so badly.

Figure 1 shows the early universe. Matter is closely packed.

That's not what you're showing. You're showing matter closely packed in a small region, surrounded by vacuum. That's not what the early universe was like. It had matter closely packed everywhere.

Space-time is, on average, pretty flat within this soup of matter.

No, it wasn't. Space was flat, but that's not the same as spacetime being flat.

If the universe was finite, and had an edge

It doesn't. See above.

Ok, rip it up team.

Your basic picture of the universe is wrong. See above.

Formally, because gravitational potential is the conserved quantity associated with the existence of a time-like Killing vector field, and there isn't one in the spacetime used in cosmology.

Very roughly, you can't define a potential unless you can go around in a circle and end up with everything being the same as it was. This is familiar from fluid dynamics where you have frictional losses - once you've gone round in a circle the fluid is hotter and you can't undo that. In cosmology, once you've gone around in a circle the universe is a bit larger and you can't undo that.

The existence of a time-like Killing vector field turns out to be a rigorous way of saying that nothing changed overall while you went round in a circle. Its non-existence means something did change irreversibly, so you can't define a potential.
To try to bring this down to how gravitational potentials are typically used in cosmology, they're often used when describing structure formation. In that situation, a non-zero gravitational potential is always associated with regions that have more or less density. The potential is identically zero if the universe is smooth, and there is no relationship between this potential and the average density at all: it's all about certain regions being more or less dense.

And, as it turns out, the only redshift (or blueshift) that is typically attributed to gravitational redshift stems from these potentials. The Sachs-Wolfe effect mentioned before, for example, stems from how photons fall into and climb out of these potential wells. In large scales in a matter-dominated universe, these potentials are constant with time (note: matter-dominated means that both normal matter and dark matter make up most of the density, as was the case early-on in our universe). This indicates that large overdense structures don't collapse so much as they were dense enough to begin with to stop expanding. And because these potentials are a constant, as a photon enters such a potential it picks up a gravitational blueshift. Then, as it leaves the potential it picks up a redshift which exactly cancels resulting in no net effect.

Things change when you add dark energy to the mix: with dark energy around, those gravitational potentials very slowly decay over time. So for very large potentials (think galaxy clusters and superclusters), in the millions of years it might take for a photon to cross the cluster, the cluster's potential has become more shallow, meaning that it picks up less of a redshift leaving than the blueshift it gained when falling into the potential. So overdense regions end up giving photons a little bit of an extra kick. For similar reasons, underdense regions result in a net redshift.

When you sum all these effects up, this is known as the Integrated Sachs-Wolfe Effect, and it is currently the strongest evidence for dark energy. It results in a relationship between CMB temperatures and nearby large-scale structure (for far-away stuff, the effect tends to average out to zero, and if they're really far away such that there was matter domination at the time, the effect would have been nearly zero anyway).

So to sum it all up, gravitational potentials do impact photons traveling through the universe, but those potentials are associated with deviations from the average density. The average density itself has zero impact.

Bernard OByrne and D.S.Beyer
@kimbyd and @Orodruin, after reading into the Sach-Wolfe effect, specifically the “Late-time Integrated Sach-Wolfe Effect”, this is answer to my initial questions. The “Late-time Integrated Sach-Wolfe Effect” deals with the large scale gravitational potentials of an expanding universe, from the CMB to the Earth, through time. The gravitational wells that a photon enters and escapes are the the ‘density’ of which I spoke.

Any further questions I have on this topic will be in a new thread, with the Late-time Integrated Sach-Wolfe Effect as a main focus.

Furthermore, I apologize for the vast confusion added to this thread by the diagrams. In retrospect they should have been half the size, with twice as much information. It was a risk, and it was a fail. @PeterDonis I will make a new thread with my basic picture of the universe in an attempt to explain myself better than this comedy of errors. It will most likely start off with this [http://1ucasvb.tumblr.com/post/142605511227/in-einsteins-general-theory-of-relativity-space]

@Ibix I read into Killing fields. I loved this PhysicsStackExchange description of them. [https://physics.stackexchange.com/questions/225436/why-are-killing-fields-relevant-in-physics] I would be interested in your opinion of this poetic take. I also read into the math, but my eyes began to fog over a little to be honest, and so…

Lastly, all cards on the table, I’ve definitely been watching way too much PBS Space Time lately. I have never taken an astro-physics course, and totally just think up questions based on pop-sci articles, wikipedia, and the odd paper. I absolutely love this stuff, but I am in no way classical learned.

That said, thank you all for your attention to these questions. I look forward to the next harsh burns and elixirs of truth and information.

Fascinating!
As a totally amateur cosmologist I have doubted the attempts to form a coherent theory of the evolution of the universe by introducing new concepts like dark matter and dark energy - which still fail to answer all the questions. When I studied physics, the 'big bang' was a revolutionary concept which was quickly adopted by (almost) all of the scientific community. After a while some discrepancies were identified concerning the the rate of expansion of the universe. The concept of 'dark matter' was introduced to explain these discrepancies (why not posit that black holes provided the extra mass required?). Anyway, this dark matter improved the situation, but did not answer all the questions. Another tweak was required. The idea of 'dark energy' was introduced. While this new concept helped towards a coherent theory, it still left some unanswered questions. It also caused me to question whether the repeated introduction of new concepts was a reasonable way to progress. It seemed to me it was like trying to force a round peg into a square hole.
What if there was a slight mistake in the original calculation of the rate of expansion? The effect of gravity on photons was not appreciated when the original calculations were done. It is now known that light is red shifted when it climbs out of a gravity well.
The early universe was far denser than it is today - an immense gravity well. While the red shift due to the photons climbing out of this well would be tiny compared with the red shift caused by the relative motion of the light source and ourselves I feel it could change the results of many calculations sufficiently to give a slightly changed picture of our universe - perhaps even giving a coherent theory of the evolution of the universe without the need for devices such as dark matter and dark energy.
Being new to this type of communication, I don't know how to register that I 'like' your post. I really do.

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Hi Bernard

(why not posit that black holes provided the extra mass required?)
It had been posited. These fall under the MACHO (massive compact halo objects) category of dark matter candidates, as opposed to WIMP (weakly interacting massive particles).
Both have been analysed to death, and the WIMPs were simply found to have less problems in the end.

The effect of gravity on photons was not appreciated when the original calculations were done. It is now known that light is red shifted when it climbs out of a gravity well.
The early universe was far denser than it is today - an immense gravity well. While the red shift due to the photons climbing out of this well would be tiny compared with the red shift caused by the relative motion of the light source and ourselves I feel it could change the results of many calculations sufficiently to give a slightly changed picture of our universe
That the universe was denser does not mean it was deeper in a gravity well. To talk about the depth of a gravity well you need to have a reference point that is somehow higher up. In the early universe, everything was denser everywhere, so there was no place far away from any mass that you could stand at and make a statement about how deep the entire rest of the universe was in a gravity well. For the same reason, there was no global well for the light to climb out of just because the universe underwent expansion.
There are local gravity wells associated with overdense regions in the universe, but they don't evolve without dark energy, and the influence of the latter is known and appreciated as the Sachs-Wolfe effect, already discussed in this this thread, e.g. in post #21 above.

Bernard OByrne and PeterDonis
Being new to this type of communication, I don't know how to register that I 'like' your post.

You should see a "Like" button near the bottom right of every post.

Bernard OByrne
Thanks so much for your response, Bandersnatch.
Not being a 'proper' cosmologist, I don't have the background to research my ideas, and perhaps I do not express myself very well.
I was aware that black holes were considered as a candidate for the dark energy, but were rejected. I was not aware of the difference between MACHO and WIMP objects, but find it interesting that that you say that 'WIMPs were simply found to have less problems in the end'. Forgive me, but 'less problems' seems to me to be an inadequate reason for incorporating them into the theory. There are still problems, and I was trying to resolve the dilemma to my own satisfaction.
Perhaps I am obsessed by the idea that a fundamental rethink is necessary to formulate a coherent theory.
As regards the gravity well, I realize I am not on a firm footing. I was considering that in our current position, billions of years after the light was emitted the ambient gravity is considerably less than it was then, which suggested to me that the light did have to climb out of a well. I acknowledge that I have nothing concrete to back up my feeling, but seeing the continued failure to come to a coherent theory - and the adding of new concepts to bring the approximations closer to observed facts - I was looking for a way to simplify the theory.
I freely admit that I do not (yet) 'get' your argument that there was no gravity well - it still seems to me that going from a very dense area to a much more sparse area would require energy - I see I have a long way to go.

Thanks again.

it still seems to me that going from a very dense area to a much more sparse area would require energy - I see I have a long way to go.
Not necessarily. The problem is that you haven't specified the movement of the components of the system. If you start with a dense region, but the particles in that region are all moving away from one another, then it will expand. It can even lose energy as it does so because those particles may slow down as they move away from one another.

But energy isn't conserved on cosmic scales in any event. So whether or not energy is required is moot: it can't be, because energy isn't conserved globally.