I Problem interpreting a Distance-Redshift Plot

[JohnnyGui]

I was looking at the following graph showing the relationship between redshift and distance for a constant, accelerating and decelerating expansion of the universe.

Source

Looking at the accelerating expansion line (red), I tried to reason why it would show a line that deviates upwards from the proportional one. I reasoned that it was so because, as a galaxy is further away, we would receive an older light, at the time when the galaxy was receding at a slower recession velocity than it was now. Thus, we receive the redshift based on an older (slower) velocity, meaning that redshift would not change too much as expected with a fixed increase in distance, making the line go upwards.

However, even though this reasoning gives me the expected line, when reviewing this reasoning I noticed that this is the ordinary Doppler shift which is not similar to the cosmological one. The problem is that when I reason according to the Cosmological Redshift, my conclusion tends to collide with the above graph.

My reasoning in the case of the cosmological redshift is as follows. In addition to redshift being based on the recession velocity, during the travel of emitted light towards us, its redshift would also adapt to changes in the expansion rate that occurs during that travel. Such that the redshift that we receive is representing the net result of recession velocity at the time the light was emitted + the changes in expansion rate until we received that light.

Thus, redshift of emitted light from a nearby galaxy would be based on a relatively small change in acceleration of the expansion until we receive it. In contrast, redshifted light of a very far galaxy, that is emitted a long time ago, is based on a large change in accelerated expansion rate since it was longer subject to it during its travel towards us.
Therefore, I’d conclude that light of a very far galaxy would be more redshifted than light of a nearby galaxy. And this would lead me to reason that the graph line for an accelerated expansion rate would have to deviate downwards from the proportional line since a fixed distance increase would give a larger redshift.

This reasoning does not match the accelerating graph line since it deviates upwards instead. The only explanation I could think of for this is because Cosmological Redshift is a combination of redshift based on the recession velocity + change in expansion rate, in such a way that the change in expansion rate was not sufficient to compensate for the relatively low recession velocity back at the time the light was emitted. However, this explanation would make a decelerating expansion rate show an even more upwards deviating line.

I would like to know where and why I reasoned wrong.

 

[PeterDonis]

I was looking at the following graph showing the relationship between redshift and distance

That's not what it shows. Look at the label of the y axis. It says "relative intensity of light". The relationship between that and distance is model dependent, so you can't just interpret the graph as comparing the redshift-distance relation for different models.

Also, you need to give a source for this graph. We can't discuss it if we don't know where it comes from and what the context is.

 

[JohnnyGui]

That's not what it shows. Look at the label of the y axis. It says "relative intensity of light". The relationship between that and distance is model dependent, so you can't just interpret the graph as comparing the redshift-distance relation for different models.

Also, you need to give a source for this graph. We can't discuss it if we don't know where it comes from and what the context is.

It also shows "Relative Distance" on the y-axis on the right side of the graph.
Source added to OP.

 

[PeterDonis]

It also shows "Relative Distance" on the y-axis on the right side of the graph.

Yes, but without units. I think that's because, as I said, the relationship is model-dependent, so they can't put units of distance in because there is no single relationship between light intensity and distance that applies to all three models shown in the graph.

It's hard to tell because the page you linked to is very sparse. This source appears to be focused on just summarizing facts, not teaching theory. You might want to try a cosmology textbook for a better understanding.

 

[JohnnyGui]

Yes, but without units. I think that's because, as I said, the relationship is model-dependent, so they can't put units of distance in because there is no single relationship between light intensity and distance that applies to all three models shown in the graph.

It's hard to tell because the page you linked to is very sparse. This source appears to be focused on just summarizing facts, not teaching theory. You might want to try a cosmology textbook for a better understanding.

Here's a plot with distance shown, from two research teams. Source.

The dotted line above the straight one implies an accelerating expansion. If this is a reliable graph, then my OP is directed at this one instead.

 

[kimbyd]

One way to tackle this kind of issue is to examine the extremes. Consider two cases: a flat universe with only a cosmological constant, and a flat universe with only matter. If you run through the calculations, the function for distance vs. redshift for the first case is:

D_M = {c z\over H_0}

Sweet, simple, to the point. This comes about because if you only have a cosmlogical constant, then the Hubble parameter is also a constant, so $v = Hd$ is a simple equation rather than a differential equation.

The matter-only case is a bit more complicated. Here $H$ changes over time (H decreases rather quickly as the universe expands), and that complicates the calculation. The answer ends up being, provided I did my math correctly:

D_M = {2 c \over H_0} \left(1 - {1 \over \sqrt{1+z}}\right)

You can use Wolfram to to graph these:
https://www.wolframalpha.com/input/?i=plot+d+=+z,+d+=+2(1-1/sqrt(z+1))

Note that for all values of $z > 0$, the matter-only case (the reddish line), the distances are smaller for the same redshift.

 

[PeterDonis]

Here's a plot with distance shown

They don't say which distance it is (there are different ways to measure the distance of observed objects like galaxies, light intensity is not the only one), but from the surrounding text I think it's luminosity distance (i.e., the intensity of the light we receive from the object). So it's basically the same as the graph you linked to in your OP. The problem is that converting this to an actual distance "now", i.e., how far away the object is from us at this instant of cosmological time, is model-dependent. The article does not appear to go into that. But as far as I can tell, it's still an issue, and it means you should not just interpret the "distance" in the graph as a spatial distance; it's more complicated than that.

The article also doesn't go into the dependence of the relationship between observed brightness and redshift, which depends on the spatial curvature in the model. The "constant expansion" empty universe has a different spatial curvature (open) from the "accelerating" and "decelerating" ones in the graphs in both of the articles you linked to (which are spatially flat). So that's another confounding factor.

Aside from the above, I think the key error you make in your OP is to try to separate out "Doppler" from "cosmological" redshift. The best way to interpret the redshift is the factor by which the universe expanded from emission to reception; more precisely, $1 + z$ is that factor, where $z$ is the redshift. So a redshift of $1$ means the universe doubled in size from emission of the light to reception (us seeing it). With that interpretation, it might be easier to see why the relationship between observed brightness and redshift is what it is for the different models.

 

[kimbyd]

They don't say which distance it is (there are different ways to measure the distance of observed objects like galaxies, light intensity is not the only one), but from the surrounding text I think it's luminosity distance (i.e., the intensity of the light we receive from the object). So it's basically the same as the graph you linked to in your OP. The problem is that converting this to an actual distance "now", i.e., how far away the object is from us at this instant of cosmological time, is model-dependent. The article does not appear to go into that. But as far as I can tell, it's still an issue, and it means you should not just interpret the "distance" in the graph as a spatial distance; it's more complicated than that.

For the purposes of this kind of plot, the specific distance used doesn't matter much. Typically they all differ from one another by some power of $(1+z)$. Multiplying or dividing by $(1+z)$ won't change the relative ordering of distance between objects.

 

[JohnnyGui]

Aside from the above, I think the key error you make in your OP is to try to separate out "Doppler" from "cosmological" redshift. The best way to interpret the redshift is the factor by which the universe expanded from emission to reception; more precisely, 1+z1+z1 + z is that factor, where zzz is the redshift. So a redshift of 111 means the universe doubled in size from emission of the light to reception (us seeing it). With that interpretation, it might be easier to see why the relationship between observed brightness and redshift is what it is for the different models.

That's the factor that I have been using. I'm trying to reason how redshift from a nearby galaxy would differ from a far one when I consider cosmological redshift being based both recession velocity plus changes in expansion rate.

One could describe this in a scenario of an observer receiving at t = 0 both old light from a far galaxy and recent light from a nearby one, at the same time. Let's say the old light has been emitted earlier at t = -100, at the time the expansion rate was relatively low, and ever since that time the expansion is accelerating. At t= -10, that old light passes a nearby galaxy and at that same moment the nearby galaxy emits its relatively recent light, such that both old and recent light arrive at the same time at the observer. At t=0 the observer receives both lights at the same time. Old light emitted at t=-100 came from a time when the expansion wasn't really fast but was longer subject to acceleration, recent light emitted at t=-10 came from a relatively faster expansion but was shorter subject to acceleration. Which of the 2 lights received is more redshifted, i.e. which of the 2 factors (time of emission vs duration of being subject to acceleration) "wins"?

EDIT: I also noticed that even in case of a constant expansion rate, if an observer receives both old and recent light from a far and nearby galaxy at the same time, he would calculate different H values. The old light is based on when the far galaxy had a high velocity at a smaller distance compared to the recent light that is based on a smaller velocity at a larger distance, since the universe expanded (at a constant rate) in the meantime. This makes me wonder how there's even a linear relationship between redshift and distance in case of a constant expansion while the observer should measure different H values depending on distance because there's a time factor in all this.

 

[Bandersnatch]

Looking at the accelerating expansion line (red), I tried to reason why it would show a line that deviates upwards from the proportional one. I reasoned that it was so because, as a galaxy is further away, we would receive an older light, at the time when the galaxy was receding at a slower recession velocity than it was now.

One could describe this in a scenario of an observer receiving at t = 0 both old light from a far galaxy and recent light from a nearby one, at the same time. Let's say the old light has been emitted earlier at t = -100, at the time the expansion rate was relatively low, and ever since that time the expansion is accelerating.

This is a misconception that might be throwing you off the track - whether our universe was in the accelerating stage or not, the rate of expansion (i.e. the Hubble parameter) always was and always will be going down. It would be going down even in an empty universe, and it would be constant only in a universe containing solely dark energy in the form of the cosmological constant. It would grow only if that dark energy wasn't constant, but also growing.
The accelerated expansion refers to the growth of the scale factor, not the expansion rate. It means that as the rate goes down, it approaches some positive, non-zero value, where reaching this rate in the far future is tantamount to achieving exponential growth of the scale factor.

Which of the 2 lights received is more redshifted, i.e. which of the 2 factors (time of emission vs duration of being subject to acceleration) "wins"?

This bit suggests another misconception.
Even if the expansion rate in the past weren't always decreasing, the redshift would still be more pronounced in light emitted from further away (= earlier), because redshift is the integrated result of expansion, and there was no time in the history of the universe when it was not expanding.
Using your example of older light joining more recent emissions: by the time the old light passes by the younger emitter, it has already accumulated some redshift due to the expansion the universe underwent during that time. Whatever additional redshift will be accumulated from that point onwards, equal in magnitude for both signals, will be added to the earlier effect. In case of the younger emission there's no pre-existing effect to add to, so older light will be always more redshifted.
The only possible case where the older light would arrive at the observer less redshifted than the younger one would be if by the time the two signals are joined together, the old light was blueshifted - i.e. if it had been traveling through a contracting universe that then started to expand.

Going back to the graph in the OP, since we know that the expansion rate is going down faster in universes with higher mass density, then light with some observed redshift would have to be emitted at different distances depending on that density - the denser the universe, the closer the emission (because in order to reach the current rate of expansion despite all that matter strongly decelerating it, the universe would have to have higher initial expansion rate, and consequently for a given emission distance, scale factor would have grown more and light would have accumulated more redshift).
Conversely, the lower the matter density (= higher proportion of dark energy density), the farther the emission must have been for the observed redshift. This is exactly what the graph shows.

 

[kimbyd]

That's the factor that I have been using. I'm trying to reason how redshift from a nearby galaxy would differ from a far one when I consider cosmological redshift being based both recession velocity plus changes in expansion rate.

Redshift is only impacted by the amount of expansion between us and the source. If distances in the universe have doubled, then the wavelength of light has doubled, meaning the redshift is $z=1$ (since wavelengths are multiplied by $z+1$).

 

[PeterDonis]

I'm trying to reason how redshift from a nearby galaxy would differ from a far one when I consider cosmological redshift being based both recession velocity plus changes in expansion rate.

And the point I was trying to make was that you can't split up the redshift into "recession velocity" and "changes in expansion rate". That doesn't work. You have to treat the redshift $1 + z$ as simply telling you by what factor the universe expanded from emission to reception of the light--you can't split it up into parts.

 

[JohnnyGui]

It seems I'm associating the H value too much with the plot. Assuming that measuring different H values at the same time should break the proportionality between z and distance.

I realized that I'd like to understand this further by knowing how the ratio of the observed wavelength/emitted wavelength is calculated in case of an accelerating expansion, expressed in terms of velocity change (acceleration), distance and time. I understand in case of a constant expansion $\frac{\lambda'}{\lambda} = 1 + \frac{H_0D}{c}$.
How do I calculate $\frac{\lambda'}{\lambda}$ in case of an accelerating expansion where the initial recession velocity $H_0 \cdot D$ changes over time? Do I have to integrate it over the time? Such that I'd have to integrate:
$$\frac{\lambda'}{\lambda} = \int \frac{H_0D +at}{c} dt$$

This bit suggests another misconception.
Even if the expansion rate in the past weren't always decreasing, the redshift would still be more pronounced in light emitted from further away (= earlier), because redshift is the integrated result of expansion, and there was no time in the history of the universe when it was not expanding.
Using your example of older light joining more recent emissions: by the time the old light passes by the younger emitter, it has already accumulated some redshift due to the expansion the universe underwent during that time. Whatever additional redshift will be accumulated from that point onwards, equal in magnitude for both signals, will be added to the earlier effect. In case of the younger emission there's no pre-existing effect to add to, so older light will be always more redshifted.
The only possible case where the older light would arrive at the observer less redshifted than the younger one would be if by the time the two signals are joined together, the old light was blueshifted - i.e. if it had been traveling through a contracting universe that then started to expand.

This cleared up that bit for me. Thanks.

The matter-only case is a bit more complicated. Here HHH changes over time (H decreases rather quickly as the universe expands), and that complicates the calculation. The answer ends up being, provided I did my math correctly:

DM=2cH0(1−1√1+z)DM=2cH0(1−11+z)​

Thanks. I'm curious how you deduced this formula.

 

[JohnnyGui]

I apologise for the double post (can't edit the post after leaving it for a while). Regarding the integration formula I deduced, what I meant is:

$$\frac{\lambda'}{\lambda} = \int 1+\frac{H_0D +at}{c} dt$$

Alternatively, if $z=\frac{Δ\lambda}{\lambda+Δ\lambda}$, then $z = \frac{Δ D}{D_0 + ΔD}$. If the $Δ D$ is achieved with constant acceleration, one could use the acceleration formula to calculate $ΔD$. Such that, $v\cdot t + \frac{1}{2}\cdot a \cdot t^2=ΔD$.

The acceleration formula is perhaps too simple to be used for an accelerated expansion, but if it suits sufficiently to understand the relationship between $z$ and distance, then if $v = H_0D_0$ and $t = \frac{D_0}{c}$, this shows that:
$$\frac{H_0D_0}{c} + a\cdot \frac{D_0}{c^2} = z$$
If this relationship between $z$ and $D_0$ is sufficient to understand the relationship between them, then plotting this relationship shows that $z$ is proportional to $D_0$ but with a larger slope than in the case of a constant expansion. But since $z$ here is on the y-axis and $D_0$ on the x-asis, that means that switching these axis, just like the graph in my OP, would make my acceleration graph line less steep than the constant expansion graph line. The graph in the OP is still showig the accelerated graph line being steeper than the constant one. How is this possible?

 

[JohnnyGui]

UPDATE:

I think I have found the culprit of my misunderstanding. I wasn't considering the plot in my OP to be the measured redshifts of different galaxies/stars at one moment in time.

When I noticed this, I used my earlier described scenario of receiving light from a star $A$ at a smaller distance and receiving light from star $B$ at a larger distance, both at the same moment during an an accelerating expansion. Using an initial Hubble value and distances $D_A$ and $D_B$ of the stars, all at $t=0$, I deduced a formula for the time $t$ at which the light of star $B$ would pass star $A$ while $A$ has been accelerating towards the light of star $B$, by solving the quardratic equation: $ΔD = H_0\cdot D_At + \frac{1}{2}at^2 + ct$. So that both lights would travel along each other towards earth.

Combining this with the increase in distance specific for star $A$ and $B$ until each of their light arrives at earth, I was able to conclude that for the redshift of both stars, $z_A$ and $z_B$, that if acceleration $a > 0$ then:
$$ z_A \cdot \frac{D_B}{D_A} > z_B $$
and if $a < 0$ (deceleration):
$$z_A \cdot \frac{D_B}{D_A} < z_B$$
Setting either the Hubble value or the distances of star $A$ and $B$ as variables always satisfies this.

I am aware that the simplistic constant acceleration formula is not applicable in the real cosmological expansion but it does show me in a simplistic way how redshift should behave during a changing expansion rate in general.

 

[JohnnyGui]

I've got a question on redshift in general.

I am aware that redshift caused by expansion is different from the regular Doppler shift. In the sense that the expansion redshift also adapts to changes in the expansion rate, even if the light already left the source. However, from what I understand, even if the expansion rate is constant the redshift caused by expansion is still different from regular Doppler shift because during its travel it passes regions of space that recess with different velocities w.r.t. us. If this is the case, how can the ratio of change in wavelength represent the change in distance ratio if it is also influenced by other space regions that it passes? Is this one of the reasons the equation doesn't apply for very large distances?

 

[PeterDonis]

I am aware that redshift caused by expansion is different from the regular Doppler shift.

This is not the right way to put it. Please re-read my post #12.

 

[timmdeeg]

I've got a question on redshift in general.

In the sense that the expansion redshift also adapts to changes in the expansion rate,

I'm not sure about your reasoning here. The expansion rate is given by $H=\dot a/a$, so as long as the universe expands $H$ decreases or is constant in the case of exponential expansion. There is no proportionality between $H$ and $a$, because $\dot a$ is independent of $a$.

even if the light already left the source. However, from what I understand, even if the expansion rate is constant the redshift caused by expansion is still different from regular Doppler shift because during its travel it passes regions of space that recess with different velocities w.r.t. us.

The redshift caused by expansion is independent from what happens in certain regions of space. It only depends on how the size of the universe was growing between the time of emission and absorption. If the size doubles (means $a$ doubles) the redshift doubles.

 

[JohnnyGui]

I'm not sure about your reasoning here. The expansion rate is given by H=˙a/aH=a˙/aH=\dot a/a, so as long as the universe expands HHH decreases or is constant in the case of exponential expansion. There is no proportionality between HHH and aaa, because ˙aa˙\dot a is independent of aaa.

What I meant is that redshift differs from the regular Dopplershift in the sense that it represents $ΔD/D$. If I understand correctly, this is because spacetime is curved in the case of an expanding universe so that the recession speed has a relationship with distance?

The redshift caused by expansion is independent from what happens in certain regions of space. It only depends on how the size of the universe was growing between the time of emission and absorption. If the size doubles (means aaa doubles) the redshift doubles.

My bad, I interpreted it the wrong way.

This is not the right way to put it. Please re-read my post #12.

So it differs from regular Doppler shift in the sense that it represents the increase in size w.r.t. the original size at emission of light?

 

[PeterDonis]

So it differs from regular Doppler shift in the sense that it represents the increase in size w.r.t. the original size at emission of light?

What you are calling "regular Doppler shift" is not something that can be split off in general from "Doppler shift due to expansion". There is only "observed shift". As I said in post #12. Read it again.

 

[timmdeeg]

What I meant is that redshift differs from the regular Dopplershift in the sense that it represents $ΔD/D$. If I understand correctly, this is because spacetime is curved in the case of an expanding universe so that the recession speed has a relationship with distance?

Originally the redshift was interpreted as Doppler shift in the context of special relativity. But that doesn't hold in an expanding universe. As far as I can tell the reason is that in curved spacetime a unambiguos definition of relative velocity isn't possible (with the exception of local measurements whereby the curvature is negligible) .

 

[PeterDonis]

As far as I can tell the reason is that in curved spacetime a unambiguos definition of relative velocity isn't possible

Yes.

 

[JohnnyGui]

Originally the redshift was interpreted as Doppler shift in the context of special relativity. But that doesn't hold in an expanding universe. As far as I can tell the reason is that in curved spacetime a unambiguos definition of relative velocity isn't possible (with the exception of local measurements whereby the curvature is negligible) .

Is curved spacetime different from the universie being curved spacially? If so, do both of these factors contribute to relative velocity not being unambiguous?

 

[PeterDonis]

Is curved spacetime different from the universie being curved spacially?

Not in the way you mean. Curved space is not something "added on" to curved spacetime. It's just that in any spacetime (flat or curved), you can choose coordinates so that "space" (a slice of constant coordinate time) is curved.

 

[JohnnyGui]

Not in the way you mean. Curved space is not something "added on" to curved spacetime. It's just that in any spacetime (flat or curved), you can choose coordinates so that "space" (a slice of constant coordinate time) is curved.

I didn't mean that it can be added to curved spacetime, I was asking if the definitions are different. Let's say there's a flat spacetime, but space A is curved. If a far away observer in another space doesn't know the amount of curvature of space A, can the observer therefore measure a different relative velocity of something that is moving in that space A compared to if the observer was standing in space A itself?

What you are calling "regular Doppler shift" is not something that can be split off in general from "Doppler shift due to expansion". There is only "observed shift". As I said in post #12. Read it again.

With mentioning the difference I didn't intend to say that one can distinguish it from the regular Doppler shift. I mentioned the difference that the redshift due to expansion has an extra property of that it represents the increase in size w.r.t. to the original size at emission of light, like you said in post #12, unlike regular Doppler shift.
Let's leave out the curvature of spacetime and space for argument's sake. If light is emitted from a source, and during its travel towards the observer, the source has accelerated due to an accelerating expansion, will that redshifted light still represent a correct $\frac{ΔD}{D}$ upon arrival to the observer, including the influence of acceleration on $ΔD$?

 

[PeterDonis]

Let's say there's a flat spacetime, but space A is curved.

"Space A is curved" isn't describing something physical. It's describing somebody's choice of coordinates. Observables don't depend on the choice of coordinates.

I mentioned the difference that the redshift due to expansion has an extra property of that it represents the increase in size w.r.t. to the original size at emission of light, like you said in post #12, unlike regular Doppler shift.

You're still not getting it. There is no "extra property". There is no "regular Doppler shift". There is only "redshift". What the redshift represents depends on the specifics of the spacetime geometry and the light and the object emitting the light and the observer receiving it. It doesn't depend on whether it is a "redshift due to expansion" or a "regular Doppler shift". When we say that the redshift we observe in light from distant galaxies tells us how much the universe has expanded while the light traveled, we mean that that's what that redshift means given that particular spacetime geometry (the FRW spacetime that describes our universe) and those objects within that geometry.

Let's leave out the curvature of spacetime and space for argument's sake

That changes the spacetime geometry, so it changes what the redshift means. See above.

If light is emitted from a source, and during its travel towards the observer, the source has accelerated due to an accelerating expansion

If spacetime is flat, there is no "accelerating expansion". That term describes a particular curved spacetime geometry. Change the geometry and you change what the redshift means. So your question doesn't make sense; you are basically asking "what does the redshift mean if spacetime is flat but isn't flat?" which is nonsense.

 

[JohnnyGui]

"Space A is curved" isn't describing something physical. It's describing somebody's choice of coordinates. Observables don't depend on the choice of coordinates.

Ok, got it.

You're still not getting it. There is no "extra property". There is no "regular Doppler shift". There is only "redshift". What the redshift represents depends on the specifics of the spacetime geometry and the light and the object emitting the light and the observer receiving it. It doesn't depend on whether it is a "redshift due to expansion" or a "regular Doppler shift". When we say that the redshift we observe in light from distant galaxies tells us how much the universe has expanded while the light traveled, we mean that that's what that redshift means given that particular spacetime geometry (the FRW spacetime that describes our universe) and those objects within that geometry.

Thanks for the explanation but I'm not sure if you understand what I'm saying. I googled and the last paragraph in this link is basically what I meant. I understand that regardless of the difference, in the case of cosmological redshift you can't split it into different forms of redshift.

If spacetime is flat, there is no "accelerating expansion". That term describes a particular curved spacetime geometry. Change the geometry and you change what the redshift means. So your question doesn't make sense; you are basically asking "what does the redshift mean if spacetime is flat but isn't flat?" which is nonsense.

You're right. Let there be a spacetime according to an accelerating expansion (curved). In that case, can the following question then be answered: If light is emitted from a source, and during its travel towards the observer, the source has accelerated due to an accelerating expansion, will that redshifted light still represent a correct $\frac{ΔD}{D}$ upon arrival to the observer, including the influence of acceleration on $ΔD$? According to the link I gave above, the answer seems to be yes?

 

[PeterDonis]

I googled and the last paragraph in this link is basically what I meant.

That paragraph is not a good source if you actually want to understand the physics involved. The redshift we observe in light from distant galaxies is not because the emitting galaxies moved away from us while the light was traveling, which is what that paragraph implies; the rule that what happens to the emitter doesn't affect the light once it's emitted holds just as well in cosmology as anywhere else.

Thinking of the light as being "stretched" by the expansion is better as a heuristic, since it at least makes it clear that it has nothing to do with the emitter having some effect on the light after it's emitted.

will that redshifted light still represent a correct $\frac{ΔD}{D}$ upon arrival to the observer

What does $\Delta D / D$ mean? How would you measure it?

 

[JohnnyGui]

What does ΔD/DΔD/D\Delta D / D mean? How would you measure it?

WIth $\frac{ΔD}{D}$ I mean the increase in distance $ΔD$ in $\frac{D}{c}$ time with respect to the original distance $D$ it moved from.
If one calculates $D$ using luminosity and knows the Hubble value $\frac{D}{c}$ time ago and how it changed over time until receiving the light, one could calculate $\frac{ΔD}{D}$ and check if it equals $1 + z$ and $\frac{Δ\lambda}{\lambda}$.

If this is all not possible to do, how can one be sure that redshift represents the factor by which the universe expanded in the first place?

 

[PeterDonis]

If one calculates $D$ using luminosity and knows the Hubble value $\frac{D}{c}$ time ago

The distance $D$ that you calculate using luminosity is not the distance "now". Nor is it the distance when the light was emitted. Nor is the time the light took to travel equal to $D / c$. The curvature of spacetime means you can't use this simple interpretation of any of these quantities.

Also, the "distance" itself is coordinate-dependent, as it is in any curved spacetime. When cosmologists talk about "distance", they mean distance in the standard FRW coordinates used in cosmology. But this "distance" can't be measured directly, because we don't have rulers billions of light years long, nor could we see where the distant galaxies fall on those rulers "now" even if we did.

What cosmologists actually do is to look at the relationships between the quantities that are observable: redshift, luminosity, and angular size are the main ones. We then compare those observed relationships with the relationships that are predicted by various possible spacetime geometries, in order to figure out which of those possible geometries represents the actual spacetime geometry of our universe. The distance $D$ is something we calculate after all of that has been done--i.e., it's just a number that cosmologists use as a convenient shorthand to describe the model the data says is the right one. It's not something we use to determine which model is right. We can't because, as above, we can't measure $D$.

If this is all not possible to do, how can one be sure that redshift represents the factor by which the universe expanded in the first place?

Because all of the possible spacetime geometries that could describe our universe have that property. So no matter which model our observations tell us is the right one, the redshift will represent the same thing. But note that, as above, this ratio of distances $D$ at reception and emission is not something we directly measure. So this property of the redshift is not something we can check by observation. Like the distances themselves, this property of the redshift is just a useful way of describing what it means in the model. It's not a test we make to see which model is right.

 

[JohnnyGui]

Because all of the possible spacetime geometries that could describe our universe have that property. So no matter which model our observations tell us is the right one, the redshift will represent the same thing. But note that, as above, this ratio of distances DDD at reception and emission is not something we directly measure. So this property of the redshift is not something we can check by observation. Like the distances themselves, this property of the redshift is just a useful way of describing what it means in the model. It's not a test we make to see which model is right.

I see. Never looked at it that way. So basically, the whole subject is actually about discussing the behaviour of redshift and $ΔD/D$ that our model(s) would predict.

the rule that what happens to the emitter doesn't affect the light once it's emitted holds just as well in cosmology as anywhere else

But according to our model, light does get affected by the change in expansion rate during its travel, even after it has been emitted by the source? Or is this also not true?

 

[PeterDonis]

So basically, the whole subject is actually about discussing the behaviour of redshift and $\Delta D / D$ that our model(s) would predict.

Yes.

according to our model, light does get affected by the change in expansion rate during its travel, even after it has been emitted by the source?

That's a matter of interpretation. The math can be interpreted as saying this, but it can also be interpreted as saying the light doesn't change at all with expansion, what changes is the "direction in spacetime" of the observer that will receive the light.

 

[Gerhard Mueller]

Why are we so sure, that there is no other effect than expansion of the universe and Doppler effect to explain the redshift?

If we live in a stationary universe of finite size, the theory of general relativity postulates a frequency shift in time!
If the universe is finite, then it must be curved. ( As an example the universe can be closed in the sense that if you travel endlessly in one direction you finally will (after a large time span) come to your starting point.)

According to the theory of general relativity, wave propagation across a curved region of space causes a frequency shift. This has been proved by the frequency shift of light from very massive stars.

 

[Drakkith]

Why are we so sure, that there is no other effect than expansion of the universe and Doppler effect to explain the redshift?

Because none of the the other possibilities investigated so far work. They either fail to accurately explain the redshift or require conditions that don't exist in the current universe.

If we live in a stationary universe of finite size, the theory of general relativity postulates a frequency shift in time!

It does not. Light is redshifted when coming out of a gravity well (as in your example of light from massive stars) but light moving through the universe is not moving out of a gravity well and would undergo no redshift in a static universe.

 

[timmdeeg]

A theoretical possibility to create redshift in a static universe are increasing masses.

http://www.nature.com/news/cosmologist-claims-universe-may-not-be-expanding-1.13379

 

[Drakkith]

A theoretical possibility to create redshift in a static universe are increasing masses.

http://www.nature.com/news/cosmologist-claims-universe-may-not-be-expanding-1.13379

You'll note that the paper in that article hasn't been peer-reviewed (at least not at the time the article was written). Let's stick to peer-reviewed sources please.

 

[timmdeeg]

You'll note that the paper in that article hasn't been peer-reviewed (at least not at the time the article was written). Let's stick to peer-reviewed sources please.

Sorry, I felt this comment in Nature "Although the paper has yet to be peer-reviewed, none of the experts contacted by Nature dismissed it as obviously wrong, and some of them found the idea worth pursuing. “I think it’s fascinating to explore this alternative representation,” says Hongsheng Zhao, a cosmologist at the University of St Andrews, UK. “His treatment seems rigorous enough to be entertained.” encouraging though and worthwhile to mention. My fault.

 

[JohnnyGui]

I have trouble understanding the relationship between a finite/infinite/accelerating/non-accelerating universe and the curvature of space-time or just the curvature of just space for that matter.

Is space-time curved in an accelerating universe merely because speed changes over time?
Why is space of the universe necessarily curved if it's finite and why not if it's infinite?

 

[PeterDonis]

Is space-time curved in an accelerating universe merely because speed changes over time?

It's even more basic than that. Spacetime in any expanding universe is curved. (Or in any contracting universe, but the expanding case is the one that's relevant to our actual universe.) The exact magnitude of the curvature depends on the details of how the rate of expansion changes with time; but the fact of curvature itself does not.

Why is space of the universe necessarily curved if it's finite and why not if it's infinite?

There is no necessary connection between these two things. There are models where the universe is spatially infinite but the spacelike slices of constant time are curved (such as an open universe, with density of mass/energy less than the critical density). There are also models where the universe is spatially finite but spacelike slices of constant time are flat--though AFAIK any such model has to have nontrivial spatial topology (for example, a flat 3-torus).

 

[George Jones]

Spacetime in any expanding universe is curved.

What about the Milne universe, which has scale factor given by$a(t)=t$?

 

[PeterDonis]

What about the Milne universe

Ah, yes, you're right, that's an edge case because the "expanding" coordinate chart is just an alternate chart on Minkowski spacetime.

 

[JohnnyGui]

It's even more basic than that. Spacetime in any expanding universe is curved. (Or in any contracting universe, but the expanding case is the one that's relevant to our actual universe.) The exact magnitude of the curvature depends on the details of how the rate of expansion changes with time; but the fact of curvature itself does not.

Ok. Is it then curved because in the case of expansion, the speed of expansion differs with distance from an observer?

There is no necessary connection between these two things. There are models where the universe is spatially infinite but the spacelike slices of constant time are curved (such as an open universe, with density of mass/energy less than the critical density). There are also models where the universe is spatially finite but spacelike slices of constant time are flat--though AFAIK any such model has to have nontrivial spatial topology (for example, a flat 3-torus).

Thanks. I'm curious how one would deduce the shape of the universe (like your mentioned flat 3-torus) by knowing if it's finite/infinite and has a flat/curved spacetime. Is there a way to explain this?

 

[PeterDonis]

Is it then curved because in the case of expansion, the speed of expansion differs with distance from an observer?

If by "the speed of expansion" you mean the observed redshift of light from distant objects, no. @George Jones pointed out the counterexample of the Milne universe (which is really just a non-standard coordinate chart on flat Minkowski spacetime). This universe is expanding, and obeys the Hubble redshift-distance relation, but the redshift observed by a given observer of light from a particular distant object does not change with time. Curvature means it does. Accelerating expansion is one particular case of this, but not the only one.

I'm curious how one would deduce the shape of the universe (like your mentioned flat 3-torus) by knowing if it's finite/infinite and has a flat/curved spacetime

Whether the universe is spatially flat or curved is a matter of your choice of coordinates, as I think I've said before. When people talk about the spatial geometry of the universe, they almost always are using the standard FRW (comoving) coordinates used in cosmology. As far as we can tell, the spatial geometry of our universe is flat in those coordinates.

In principle you could tell if the universe was spatially finite by looking for light that had circumnavigated it--gone all the way around and come back to its starting point. One way to spot this is by seeing duplicate images of the same object, in opposite directions on the sky. In practice, however, it can take a long, long time (much longer than the current age of the universe) for light to go all the way around--and in fact, in an accelerating universe, I don't think it's possible, because, heuristically, the spatial size of the universe increases faster than the light can cover the increased distance.

 

[JohnnyGui]

the Milne universe (which is really just a non-standard coordinate chart on flat Minkowski spacetime). This universe is expanding, and obeys the Hubble redshift-distance relation, but the redshift observed by a given observer of light from a particular distant object does not change with time

Is the Milne universe flat because the redshift not changing over time means that the expansion rate is constant over time (Hubble value decreases according to $1/t$)?

Also, I'm curious how redshift of an object can change over time apart from acceleration. If, for example, the redshift of a galaxy changes over time just because it has passed an area with a different gravity density (inhomogeneous universe) as it recesses, will one say that spacetime is indeed curved or that it's merely an illusion because of the different gravity densities in the universe?

Whether the universe is spatially flat or curved is a matter of your choice of coordinates, as I think I've said before. When people talk about the spatial geometry of the universe, they almost always are using the standard FRW (comoving) coordinates used in cosmology. As far as we can tell, the spatial geometry of our universe is flat in those coordinates.

I see. So for example, the 3-torus shape you mentioned is based on what the standard FRW coordinates predicts?

 

[PeterDonis]

Is the Milne universe flat because the redshift not changing over time means that the expansion rate is constant over time (Hubble value decreases according to $1/t$)?

You can deduce from that that the spacetime must be flat, yes. The usual method of deducing that is more direct, though--you just find a coordinate transformation that takes the metric to the Minkowski metric.

I'm curious how redshift of an object can change over time apart from acceleration

It can be either "acceleration" or "deceleration", if you think of it as the rate of change of the rate of change of comoving distance ("distance" in a spacelike slice of constant time according to comoving observers) with time (or, equivalently, as the second time derivative of the scale factor in standard FRW coordinates).

If, for example, the redshift of a galaxy changes over time just because it has passed an area with a different gravity density (inhomogeneous universe) as it recesses, will one say that spacetime is indeed curved

Yes, because different "gravity densities" in different parts of the universe means spacetime must be curved.

the 3-torus shape you mentioned is based on what the standard FRW coordinates predicts?

The coordinates alone can't tell you what the global topology of the universe is. A spatially flat universe of infinite size and a spatially flat 3-torus universe of finite size have exactly the same metric in terms of the coordinates.

 

[JohnnyGui]

It can be either "acceleration" or "deceleration", if you think of it as the rate of change of the rate of change of comoving distance ("distance" in a spacelike slice of constant time according to comoving observers) with time (or, equivalently, as the second time derivative of the scale factor in standard FRW coordinates).

Is there any model that predicts a change in redshift over time merely because of an "illusion"? So that there's a combination of spacetime being flat while redshift seems to change over time?

 

[PeterDonis]

Is there any model that predicts a change in redshift over time merely because of an "illusion"? So that there's a combination of spacetime being flat while redshift seems to change over time?

Not if the only objects whose redshifts we observe are moving inertially (i.e., in free fall). In flat spacetime, if I get in a rocket ship and start the engine and accelerate (in the sense of proper acceleration) away from you, the redshift you observe in light signals from me will increase with time. But distant galaxies don't have rocket engines.

 

[Adrian59]

That's not what it shows. Look at the label of the y axis. It says "relative intensity of light". The relationship between that and distance is model dependent, so you can't just interpret the graph as comparing the redshift-distance relation for different models.

Also, you need to give a source for this graph. We can't discuss it if we don't know where it comes from and what the context is.

M = m + 5 – 5 log10 (d); where M is the absolute magnitude, m is the observed magnitude and d is distance.

For a given M the relationship of m to d is logarithmic.

Looking at a single model, that is type 1a super novae are a standard candle, that assumes M is constant. This graph is similar if not identical to the one Saul Pelmutter used in his famous paper. Then the relationship between magnitude and red shift can in interpreted as a relationship between distance and red shift if the former is plotted on a log scale which it is.

The conclusion eventually was the universe contained something else and that was dark energy. If you are going to question this models assumption then the whole project is under question, including dark energy.

 

[Adrian59]

This is a misconception that might be throwing you off the track - whether our universe was in the accelerating stage or not, the rate of expansion (i.e. the Hubble parameter) always was and always will be going down. It would be going down even in an empty universe, and it would be constant only in a universe containing solely dark energy in the form of the cosmological constant. It would grow only if that dark energy wasn't constant, but also growing.
The accelerated expansion refers to the growth of the scale factor, not the expansion rate. It means that as the rate goes down, it approaches some positive, non-zero value, where reaching this rate in the far future is tantamount to achieving exponential growth of the scale factor.

I completely agree with your first point although they are related. In a non-matter universe v = H d; which can be rearranged as H = v/d.

However, H(t) = (dR/dt) / R, where R is the scale factor.

From Friedmann; [ (dR/dt) / R ]2 = 8 π G ρ / 3 - kc2 / R2 where there is now a density (ρ).

Or assuming a flat universe so k = 0;

[ (dR/dt) / R ]2 = 2 G M /R, with M now representing the total mass of the universe which can be assumed constant.

If you plot this you get a typical y=1/x type of plot which shows how the Hubble parameter is decreasing as suggested.

One could argue that the v = H d relationship is an observation at time t0 and the Freidmann derived one is showing the history of the universe. However, since we are in effect always looking back in time when observing distant objects I would say these expressions have a close relationship.

 

[Adrian59]

I was looking at the following graph showing the relationship between redshift and distance for a constant, accelerating and decelerating expansion of the universe.

'Looking at the accelerating expansion line (red), I tried to reason why it would show a line that deviates upwards from the proportional one. I reasoned that it was so because, as a galaxy is further away, we would receive an older light, at the time when the galaxy was receding at a slower recession velocity than it was now. Thus, we receive the redshift based on an older (slower) velocity, meaning that redshift would not change too much as expected with a fixed increase in distance, making the line go upwards.''This reasoning does not match the accelerating graph line since it deviates upwards instead. The only explanation I could think of for this is because Cosmological Redshift is a combination of redshift based on the recession velocity + change in expansion rate, in such a way that the change in expansion rate was not sufficient to compensate for the relatively low recession velocity back at the time the light was emitted. However, this explanation would make a decelerating expansion rate show an even more upwards deviating line'.

I would like to know where and why I reasoned wrong.

This has been an interesting thread for me and I have had to re-examine the assumptions that the graph of magnitude v red shift make which has been a large part of the discussion. I still think that your graph is very similar to the one Saul Pelmutter used in his famous paper and this was taken as a true reflection of distance v receding velocity. I did come to a similar conclusion to yourself wrt this graph and I still don't quite know what the answer is.