Problem interpreting a Distance-Redshift Plot

[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.

 

[JohnnyGui]

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.

The way I understood it is by realising that the graph in my OP is a measure of redshifts at one moment in time (all redshifts measured at once).
Therefore, one would have to consider a scenario in which you measure the 2 redshifts of 2 lights both at once, one from a nearby star $A$ and another from a very far star $B$. To be able to measure them at once, you'd first have to calculate the new distance and velocity of star $A$ at the time at which the light of star $B$ passes star $A$ itself, during an acceleration. It's a quadratic formula in which you have to solve for the time duration until the light of star $B$ passes star $A$ (let's call that moment $T_A$). So star $A$ is approaching the light of star $B$ while accelerating while star $B$'s light is approaching star $A$ at $c$

Once you have calculated the new velocity and distance of star $A$ at $T_A$, you can calculate star $A$'s redshift by calculating how much distance star $A$ has traveled from $T_A$ (with acceleration) until its light reaches the Earth and divide that by the distance of star $A$ at $T_A$. Also, you can calculate the redshift of star $B$'s light that was emitted before $T_A$ (it's the light that passed star $A$ at $T_A$) at $T_B$ by calculating how much distance star $B$ has traveled from $T_B$ all up until it reached the earth, divided by the initial distance of star $B$.

You should get 2 redshifts that are not proportional with distance, such that the redshift of star $B$ is actually less than one would expect with the Hubble value deduced from star $A$'s redshift. I'm aware there are many factors into play but this is a basic concept that shows a simple case of the relation between redshift and acceleration. For more info on the calculation, see my post #15.

 

[Adrian59]

I would agree with this interpretation as I have now found the right expression for cosmological red shift as
1 + z = R(t ob ) with R being the scale factor,
R(t em)
which I think is the same as in your example.

 

[JohnnyGui]

I was wondering about the factors that cause redshift in our universe. So far I've concluded the following:

1. Redshift caused by leaving and/or approaching a gravity source
2. Redshift caused by expansion of the universe
3. Redshift caused by the own velocities of galaxies/stars/etc.

Am I missing something else? Is there a redshift that is caused by a different curvature of space far away from us or is this a physical meaningless conclusion since it's dependent on the choice of coordinates?

 

[PeterDonis]

I was wondering about the factors that cause redshift in our universe.

There are two: relative velocity (your #3--though it should be stated more carefully) and spacetime curvature (your #1 and #2).

Is there a redshift that is caused by a different curvature of space far away from us or is this a physical meaningless conclusion since it's dependent on the choice of coordinates?

"Curvature of space" itself depends on your choice of coordinates. But "different curvature of space far away" could also be a way (an imprecise way) of referring to being in a curved spacetime, which is one of the two factors above.

 

[JohnnyGui]

Thanks for the clarification.

"Curvature of space" itself depends on your choice of coordinates.

Correct me if I'm wrong, but if curvature of space itself is merely dependent of coordinates and have no physical meaning, then I have a hard time accepting that there are actual physical phenomena that result from this curvature of space, if there are any. I think I can accept that the angles of a triangle in a curved space would not equal 180 degrees if one uses a particular coordinate system. But are you saying that there isn't any redshift or time dilation caused by curvature of space itself if one uses a particular coordinate system that is not invariant to curvature?

 

[PeterDonis]

if curvature of space itself is merely dependent of coordinates and have no physical meaning, then I have a hard time accepting that there are actual physical phenomena that result from this curvature of space, if there are any.

The physical phenomena don't result from "curvature of space", they result from picking out particular physical events or sets of events. The events themselves, and the relationships between them, don't depend on your choice of coordinates.

I can accept that the angles of a triangle in a curved space would not equal 180 degrees if one uses a particular coordinate system.

The angles of a triangle are invariants; they don't depend on your choice of coordinates. But in order to define a triangle in the first place, you have to pick out three events (points in spacetime), and three sides (three curves in spacetime, each one connecting two of the points). Once you've picked out three points and three sides, the angles of the triangle, and the lengths of the sides, are invariants; they don't depend on your choice of coordinates. But whether all of the events and sides taken together, i.e., all of the points in a particular set of points that is the union of three points and three curves connecting them, are properly called a "triangle of points in space at a constant time" does depend on your choice of coordinates. They might not all have the same time coordinate in some choices of coordinates.

are you saying that there isn't any redshift or time dilation caused by curvature of space itself if one uses a particular coordinate system that is not invariant to curvature?

No. I'm saying that redshift and time dilation (assuming that by "time dilation" you mean some invariant such as the observed difference in elapsed proper time between two twins in a "twin paradox" scenario when they meet up again), since they are invariants, aren't caused by "curvature of space"; they are caused by something else. (The obvious cause in general is curvature of spacetime--not space--plus a choice of particular points and curves in spacetime, whose properties, like lengths and angles, are invariants.)