Inconsistencies between recession speeds and distance to far objects

[Halc]

I need help understanding the diagrams I see showing worldlines of distant objects on a cosmological scale.

From this paper http://people.virginia.edu/~dmw8f/astr5630/Topic16/t16_light_cones.html there appears this diagram, which appears in similar format on other sites.  It is called the concordance model here. Note that this is comoving coordinates, plotting proper distance, not a spacetime diagram of an inertial reference frame, which would not be able to foliate all of spacetime like this does.

It would seem that if space expansion was constant (no acceleration, no dark energy), then those worldlines (blue on left, dotted black on right) would be straight.  That they curve upward suggests that expansion is slowing, not accelerating. Accelerated expansion would curve them outward, no? It is here where I need an explanation.

Take for example GN-z11, the most distant galaxy visible.  It is listed on wiki as having a red shift of 11.09, and has a 'present proper distance' of 32 billion light years, which corresponds to a worldline labeled 'v = 2.3c'.  It would cross the red light cone at about t=400MY, proper distance of about 2.8 BLY.  See the redshift markings on the right side of the red light cone, and where 11 would be on that line.

Wiki agrees with this, saying what we see now was emitted when the universe was 400MY old.  It gives no proper distance at the time.

If it took .4BY to get 2.8BLY away, it was receding at an average speed of 7c up until then, but around 2c now.  That appears to be deceleration, not acceleration. Where am I going wrong?

 

[Halc]

Wiki does give emission proper distance if you look close.  I estimated it at 2.8 BLY based on inspection of the diagram above, but it says 2.66 BLY on the site.

So it went from 0 to 2.66 in 0.4 BY, or an average of 6.5c, and in the next 13.4 BY it went from there to 32 BLY distant, an average of 2.2c.

It is that falling off of recession speed that I'm trying to understand.  If expansion is supposedly accelerating due to dark energy or a positive cosmological constant, then why has the recession speed of GN-z11 fallen by at least a factor of 3 between the event that we see and its present speed?

[Mordred]

This graph doesn't look very accurate the redshift scale is nonlinear so that may be throwing me off. For example the CMB lies around z=1100. The Hubble horizon should be closer than the particle horizon. 

 At z=1100 you should have a recession speed of 3.2 c.

 Judging from the site descriptives and the graph this looks like a graph from an older dataset.

 Recession speed should not fall off. It's calculated by [math] v_{recessive}H_0 D[/math]

 

 

Edited June 9, 2020 by Mordred

[Halc]

1 hour ago, Mordred said:

This graph doesn't look very accurate the redshift scale is nonlinear so that may be throwing me off. For example the CMB lies around z=1100. The Hubble horizon should be closer than the particle horizon. 

 At z=1100 you should have a recession speed of 3.2 c.

 Judging from the site descriptives and the graph this looks like a graph from an older dataset.

The graph doesn't label redshifts beyond 10.  The hubble distance (v=c line on right) is closer than the particle horizon, which is the 'today's horizon' line on left. Not sure what 'Hubble horizon' is as distinct from Hubble distance. Are you aware of such a graph (especially one that shows redshifts and worldlines out to a good percentage of visible universe) with more realistic data?

Anyway, graph aside, the numbers quoted for GN-z11 are actual reported numbers, not something coming from a graph. My primary concern is those number and not a picture which may or may not accurately reflect reality.  It is unrealistic to draw a graph (comoving coordinates, proper distance) with GN-z11 crossing the events reported and not have that worldline curve upward (slowing).

Edited June 9, 2020 by Halc

[Mordred]

Hubble distance is the distance light would travel without expansion. So it should be a distance corresponding to universe age.

I will try and locate a more up to date graph for you when I get off work. There is a graph by Lineweaver and Davies with a good solid explanation. It is still an older dataset however it is far more accurate than this one. 

 Unfortunately the cosmocalc in my signature is still down it used to be able to produce these graphs.

Edited June 9, 2020 by Mordred

[Halc]

Here's another dated image, unsure of origin, but one I see used a lot:

GN-z11 worldline is very close to the 3rd dotted line.  The resolution is too low to see where (distance) it crosses the light cone in the upper image. 4th dotted line is close to today's CMB.

The lower image shows straight worldlines, and the upper can be supposedly generated from it by multiplying distances by that scalefactor on the right, but notice the scalefactor is either mislabeled or something, because there's no zero at the bottom, but rather something around 0.1, which would not produce a singularity as depicted in the upper diagram. It is that scalefactor that I suspect is the culprit. Notice the 0.2 is already closer to 0.4 than the spacing between the numbers above.  So I suspect it does go to 0, but very compressed near the bottom, which would be decelerating expansion (numbers getting further apart over time), not accelerating expansion.

The worldlines are curved just as they are in my prior diagram, and they would be straight if the scalefactor went evenly from 0 to 1, but that would put the emission-proper-distance of the 3rd dotted line at far less than 2.66 GLY.

No matter how accurate of a picture we get, either the number reported for distant things are wrong, or those worldlines really are curved.

The lower picture shows the difference between Hubble sphere (is Hubble Horizon?) and your description of Hubble distance. The latter would be a vertical worldline intersecting the event where blue 'now' and purple 'Hubble sphere' lines meet, correct?  But if there was no expansion, the lower picture would be meaningless as there would be no scalefactor. In fact, the Hubble distance as you define it would be the edge of the universe, which, without expansion, would effectlively be flat Minkowski spacetime.

I took the time to draw a picture of the universe using those coordinates rather than comoving coordinates.  I could not include dark energy, but by leaving that off, I could foliate all of spacetime with an inertial reference frame. The light cone becomes a straight line. Distant things like GN-z11 are not so distant since speeds add the relativistic way, not linear, so nothing recedes at superluminal speeds.  Alas, it fails empirical tests since really old things appear smaller (angular diameter) using the Minkowski spacetime, while they appear larger than younger 'closer' objects in reality.

Edited June 9, 2020 by Halc

[Mordred]

This will help you better understand those above spacetime diagrams they look to be very similar to the ones produced by Lineweaver and Davies.

The other factor you must also remember is the radius change of the observable universe. This gives the illusion of a curved worldine. However the worldline itself is close to flat. The volume changes are incorporated into these diagrams.

Hope this helps 

https://arxiv.org/abs/astro-ph/0310808

One of the main problems with these diagrams at higher z scales is that the time period between z=10 and Z=1100 is that the time period between that change is rather miniscule due to the non linearity of the Z scale. So it is incredibly difficult to show this on graph.

Edited June 9, 2020 by Mordred

[Mordred]

13 hours ago, Halc said:

Here's another dated image, unsure of origin, but one I see used a lot:

The origin is Tamara Davies doctorate dissertation paper.

https://arxiv.org/abs/astro-ph/0402278

Here is a link to a copy of it to fill in the details of those graphs.

 To me this actually explains a lot on the graph as were looking at much earlier datasets. At best we have the early WMAP dataset however more likely the COBE.

 A lot of development occurred since 2004 which the above graphs originated.

 For example equations. 2.3 to 2.9 are only good approximations in the small Z scales when the Hubble parameter is roughly linear. They do not work well beyond the Hubble horizon.

 Numerous corrections are needed and depend upon the evolution of the matter, radiation and Lambda evolution. This was a common  problem at that era of cosmology. A good paper covering the latter is

https://arxiv.org/abs/astro-ph/9905116

Distance measures in Cosmology. By Bunn and Hoggs 

Although it is slightly older paper it shows the knowledge of the era in how to apply cosmological redshift at greater distance.

 This is noted in the last link. You will see the equations used in the Tamara Davies paper discussed in the Bunn and Hogg paper.

In essence the wordlines did not compensate for the matter, radiation and Lambda evolution accurately at higher redshift values.

 

 

 

Edited June 10, 2020 by Mordred

[Halc]

On 6/9/2020 at 10:47 AM, Mordred said:

This will help you better understand those above spacetime diagrams they look to be very similar to the ones produced by Lineweaver and Davies.

OK, I've seen that paper before but didn't exactly see how it applied since I'm not committing any of the misconceptions mentioned.

That said, I read it more carefully and the misconception I'm making is assuming that scalefactor was a linear function, whereas it is in fact based on complicated solutions to Einstein's field equations in the FRW models.  They graph various models with this and that tuning, yielding this, which I've also seen before, but without making the connection:

The consensus model seems to be the purple one there, the only one giving an age-of-universe as about 13.8 BY. The slope of that line is the expansion rate at various times, and yes, expansion was much quicker in the first billion years than it is now, which accounts for the nonlinear scalefactor on the right side of the diagram in my prior post. It is even more evident in the similar diagram top of page 3 of the originally linked doc, which shows the future as well as the past, and shows the scalefactor once again compressing as the expansion accelerates from its current low level, which seems to have changed very little from its minimum about 5 BY ago. This nonlinear scalefactor accounts for the curvature of the worldlines in all these diagrams, including the one I first posted. You said that one was from old data, but I see nothing particularly wrong with it.

Problem is that popular articles talk about how the expansion is accelerating (dark energy and all), but not that it had been slowing in the past. So that prompted my initial post asking how GN-z11 could have got 2.66 BLY away in only 400 MY when its present recession velocity is only slightly over 2c.  That's a significant reduction in expansion rate that the texts seem to rarely talk about.

Anyway, Thanks Mordred for pointing me in the direction where I could find my answers.

[Mordred]

Excellent you got it in a nutshell and your welcome. The old data I mentioned involved various calculations such as the mentioned age of the universe but also the values in the matter and radiation equality. The latter affects the timeline from matter to Lambda eras. 

 Though in the latter the switch from matter dominant era which is a point of decelerating expansion. To the Lambda dominant era which would have accelerating expansion  would be roughly when the universe was 7.3 Gyrs.

 This varies according to datasets I've seen it as low as 6.7 Gyrs in some datasets. 

  Anyways glad to see you got the necessary answer one other side point is the Hubble parameter  is only constant at a particular time. For example at redshift  z=1100 It would be roughly 22,900 times greater than the roughly 70 km/sec/Mpc value today.

In point of detail even though the Hubble parameter is decreasing the universe expansion is accelerating when looked at the overall radius.

 This is a couple of main factors of the non linearity of the scale factor.

 

Using Stretch which is 1/a in the Lineweaver Davies notation the evolution of the Hubble parameter as a function of Z is.

[math]H = H_0 \sqrt{\Omega_\Lambda + (1-\Omega) S^2 + \Omega_m S^3 (1+S/S_{eq})}[/math]

However a more versatile form is

[math]H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}[/math]

Edited June 11, 2020 by Mordred