# I Problem interpreting a Distance-Redshift Plot

1. May 8, 2017

### 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.

Last edited: May 8, 2017
2. May 8, 2017

### Staff: Mentor

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.

3. May 8, 2017

### JohnnyGui

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

4. May 8, 2017

### Staff: Mentor

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.

5. May 8, 2017

### JohnnyGui

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.

Last edited: May 8, 2017
6. May 8, 2017

### 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.

7. May 8, 2017

### Staff: Mentor

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.

8. May 9, 2017

### kimbyd

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 $(1+z)$ wont change the relative ordering of distance between objects.

9. May 9, 2017

### JohnnyGui

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.

Last edited: May 9, 2017
10. May 9, 2017

### Bandersnatch

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.

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 travelling 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.

11. May 9, 2017

### kimbyd

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 $z+1$).

12. May 9, 2017

### Staff: Mentor

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.

13. May 10, 2017

### 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 realised 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 cleared up that bit for me. Thanks.

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

Last edited: May 10, 2017
14. May 11, 2017

### 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?

Last edited: May 11, 2017
15. May 15, 2017

### 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.

Last edited: May 15, 2017
16. May 18, 2017

### 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?

17. May 18, 2017

### Staff: Mentor

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

18. May 19, 2017

### timmdeeg

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 $a$.

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.

19. May 21, 2017

### JohnnyGui

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?

My bad, I interpreted it the wrong way.

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?

20. May 23, 2017

### Staff: Mentor

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.