What is the distance at which gravity and the cosmological force are equal?

[PeterDonis]

I must point out that none of you have produced any observational evidence to back your claims.

This comment is out of line. The $\Lambda CDM$ model has a voluminous literature dealing with its predictions and comparison of those with observational evidence. Even in a "B" level thread, it is quite fair for other posters to expect you to have at least a basic knowledge of that. Asking for some references so you can learn more about the $\Lambda CDM$ model is fine (the Wikipedia article on the model, which @pervect linked to, has plenty of references to the literature, so that's a good place to start). Refusing to accept basic descriptions of what the $\Lambda CDM$ model says unless people produce observational evidence directly in this thread is not.

A graph of D against log(1+z) for very large z is consistent with the assumption that they are proportional

I have no idea what graph you are referring to, since the only graph that has been given so far in this thread directly contradicts your claim here, as I have already pointed out in post #26 of this thread.

 

[J O Linton]

I am well aware of the arguments in favour of the $\Lambda CDM$ model and am quite willing to accept that it is the best model we have at present. Please do not deliberately misunderstand me. I am not arguing that the expansion of the universe has been linear - only that in order to understand the basic properties of the universe we live in, a linear model is perfectly adequate, at least to a first approximation.
I attach a graph of log(Z) (Z = 1+z) against D. the data is taken from the NED-4D files available at https://ned.ipac.caltech.edu/level5/NED1D/intro.html.

In resepct of your other comment in post #26, by 'observable universe' I meant that part of the universe within the particle horizon. A linear universe has an infinite particle and an infinite event horizon. I believe I am right in saying that the $\Lambda CDM$ universe has an infinite particle horizon but a finite event horizon - i.e we can see all of the past universe but may not be able to communicate with all of the future universe. As I have already pointed out, the particle horizon of our universe extends at least as far as the CMBR.

 

[vanhees71]

The linear model is not adequate to describe the observations. As you say yourself to fit all (!) data, i.e., at least both the redshift-distance measurements (e.g., from Hubble space telescope) and the CMBR-fluctuation data (e.g., from the Planck satellite measurements) the cold-dark-matter model with a cosmological constant (including acceleration!) is the far best model, and the linear model fails. Whether or not this really is the final answer, nobody can say. There's at least the issue about the value of the Hubble constant when measured with different methods that has to be understood and be resolved.

 

[J O Linton]

I doubt if anyone has seriously tried to reconcile the observations with the linear model. The reason for this is, of course, that there are good theoretical reasons why the expansion cannot be linear because a linear universe is theoretically empty - which our universe patently is not! I am not trying to argue that the $\Lambda CDM$ model is wrong - only that, to a first approximation, the linear model makes a pretty good job of explaining the erssential features of our universe.

 

[Bandersnatch]

I doubt if anyone has seriously tried to reconcile the observations with the linear model. The reason for this is, of course, that there are good theoretical reasons why the expansion cannot be linear because a linear universe is theoretically empty - which our universe patently is not!

It's not just theoretical reasons. There are direct observables that can't be reconciled with the linear model. Most obviously the shape of the Hubble diagram (e.g. see Reiss' work, past and current). As one looks towards higher z's the shape of the curve will be different, depending on the value of the second derivative of $a$ (in particular, it will curve in opposite directions during acceleration/deceleration phases).
That's what first prompted the inclusion of Lambda into the model. And if you think about it, it couldn't have come from what you call theoretical reasons, as with matter - we can look around us and see that the universe is not empty, and decide that only models with matter in it should be considered; but the cosmological constant is not conspicuously advertising its existence, so there's no a priori reason to favour models with it over those without. Its inclusion in the model must come from observations of how the expansion unfolds over time.

Now, it's true that you can approximate a curve that slopes one way for about half it's length and the other way for the other half, as a line. This is contingent on not looking too closely and being fortunate enough to find oneself at the cosmic time when the two opposing contributions are about equal.
But you're doing more than that here - you're assuming the approximation will hold just as well in the future, which is what you're doing when you're making deductions about actual horizons.

If you forgive me a hyperbolic analogy, it's like acknowledging that the Earth being flat is a good local approximation, and - even though one is aware that the approximation diverges the farther (or closer) one looks, use the flat Earth model to make deductions about the entire planet.

A linear universe has an infinite particle and an infinite event horizon. I believe I am right in saying that the $\Lambda CDM$ universe has an infinite particle horizon but a finite event horizon - i.e we can see all of the past universe but may not be able to communicate with all of the future universe.

It is not correct. Yes, the particle horizon in the LCDM universe extends to infinity in terms of proper distance, but it's finite in terms of comoving distance. Which means there is a finite number of galaxies whose past states we can ever observe. Which is another way of saying that there exists an event horizon - the EH limits not only what galaxies you can send a signal to, but also what galaxies can send a signal to you.
This is different in a linear universe, where you can get to see all the galaxies, even at infinite distance, if you just wait long enough.

 

[PeterDonis]

I am not arguing that the expansion of the universe has been linear - only that in order to understand the basic properties of the universe we live in, a linear model is perfectly adequate, at least to a first approximation.

And that is the claim that, as you have already been told several times now, is wrong.

I attach a graph of log(Z) (Z = 1+z) against D. the data is taken from the NED-4D files available at https://ned.ipac.caltech.edu/level5/NED1D/intro.html.

You can always draw a linear fit through any set of points. That doesn't mean the linear fit is a good approximation.

A linear universe has an infinite particle and an infinite event horizon.

Wrong on the infinite particle horizon. The linear universe model does not occupy all of Minkowski spacetime. It only occupies the future light cone of the "Big Bang" event. The boundary of that future light cone is the particle horizon, and it is not infinite.

A linear universe model has no event horizon; calling it an "infinite event horizon" is not strictly correct, although I see what you mean by it.

I doubt if anyone has seriously tried to reconcile the observations with the linear model.

Sure they have. You already showed a graph of how that would work in post #23. All you would need to do is add the observations that support the $\Lambda CDM$ model and rule out the other models.

 

[PeterDonis]

the particle horizon in the LCDM universe extends to infinity in terms of proper distance

No, it doesn't. At any FRW coordinate time $t$, the proper distance to the particle horizon is finite (since it's just the scale factor times the comoving distance to the particle horizon, both of which are finite; the product of two finite numbers is finite).

 

[PeterDonis]

I believe I am right in saying that the universe has an infinite particle horizon

No, this is wrong. See my response to @Bandersnatch just now.

but a finite event horizon

Yes.

i.e we can see all of the past universe

No, we can't; our past light cone (which is sort of the "inverse" of the particle horizon) is finite. That will be true even into the infinite future, since that's what a finite event horizon means.

but may not be able to communicate with all of the future universe.

This is the finite event horizon considered in reverse--just as there are parts of the universe we will never see light signals from, there are parts of the universe we will never be able to send light signals to.

 

[PeterDonis]

This seems like a good time to post a link to Davis & Lineweaver 2003:

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

It corrects many common confusions about the various horizons and other features of the $\Lambda CDM$ model.

 

[Bandersnatch]

No, it doesn't. At any FRW coordinate time $t$, the proper distance to the particle horizon is finite (since it's just the scale factor times the comoving distance to the particle horizon, both of which are finite; the product of two finite numbers is finite).

I don't understand this objection. With time extending to infinity the scale factor grows to infinity. Doesn't it?

 

[PeterDonis]

With time extending to infinity the scale factor grows to infinity.

Time "infinity" is not part of the manifold. At every finite time, the scale factor is finite.

If you want to say "the proper distance to the particle horizon increases without bound", that would be fine. But just saying "infinite particle horizon" is not, because the particle horizon is not infinite at any time that anyone will ever experience.

 

[Bandersnatch]

Well, fair enough. Same as t=0 is not part of the manifold. Let's change 'to' to 'towards' and it should be fine, right?

 

[PeterDonis]

Let's change 'to' to 'towards' and it should be fine, right?

If you change both "to"s (the one after "time" and the one after "scale factor grows") to "towards", yes.

 

[PeterDonis]

the EH limits not only what galaxies you can send a signal to, but also what galaxies can send a signal to you.

Actually, the definition of the EH is the boundary of the region that contains galaxies that can send a signal to you. The fact that you can turn this around to obtain a limit on what galaxies you can send a signal to is not part of the definition of the EH, but a consequence of it in our universe.

 

[Bandersnatch]

'The existence of EH limits...' then. I would roll my eyes in the face of your ceasless nitpicky assaults. But I generally do agree one should be as precise as possible, wherever possible. So I appreciate the corrections.

 

[J O Linton]

I have studied Davis and Lineweavers excellent article referred to by PeterDonis in post #39 and it clarifies the issue as regards the particle horizon of the $\Lambda CDM$ model. In the caption to Fig 1 the authors state that the particle horizon is 46 Gly from us. The diagrams also indicate that the distance to the CMBR is only very slightly less than this. On page 20 they give a formal definition of the particle horizon $$ \chi_{ph}(t) = c \int_0^t \frac{dt'} {R(t')}$$.
If we assume a linear model for the expansion factor, this integral blows up to infinity because the rate of expansion is constant all the way down to t'=0. The $\Lambda CDM$ model, however assumes that in the early stage it was dominated by matter and that it started expanding according to the 2/3's power law. The gradient of this curve is infinite at t = 0. Under these circumstances, the integral is finite and the particle horizon works out to be 3cT₀ and in the absence of a cosmological constant the particle horizon would be at 41.4 Gly. Presumably Davis and Linewaever have integrated the whole expansion curve between 0 and T₀ numerically to include the contribution of the CC to get the stated figure of 46 Gly.
Now the whole discussion of horizons etc was sparked off by a rather off the cuff remark I made in post #13 that 'the universe has 'made a pretty good job of expanding linearly'. Clearly there is a considerable difference here. The $\Lambda CDM$ model predicts D_ph = 46 Gly, the linear model predicts infinity. But it should be realized that the former figure is extremely sensitive to the precise way in which the scale factor of the universe approaches zero as $t \to 0$. If the universe started to expand according to a different power law, the predicted particle horizon would be radically different. Since we have absolutely no experimental data on the laws of physics which pertain at temperatures above 500 million K (when the universe was about 70 years old), everything that we deduce on the basis of extrapolating into this region must be regarded as tentative speculation. The best we can say about the distance to the particle horizon is that, if it exists, it is at least as far away as the CMBR.
When it comes to the distance to the CMBR, Davis and Lineweaver are on firmer ground. They claim that the CMBR is at a distance of about 45 Gly. The linear model predicts a distance of 97 Gly (cT₀ log 1100). The agreement is not good - but then again, it is not bad either, considering the nature of the issue we are dealing with. I therefore continue to stand by my earlier remark in spite of the number of times I have been told that I am wrong.

 

[PeterDonis]

it should be realized that the former figure is extremely sensitive to the precise way in which the scale factor of the universe approaches zero

Which depends on which model you adopt. The Davis & Lineweaver paper is giving the result based on the $\Lambda CDM$ model, which is the best current model we have.

Also, the $t = 0$ point in the Davis & Lineweaver paper is not "the beginning of the universe", because the paper ignores inflation; its $t = 0$ is the hot, dense, rapidly expanding state that is the earliest state for which we have good evidence. Different models of what preceded that state will extend the Davis & Lineweaver diagrams further down below their $t = 0$ point in different ways; the article does not discuss that.

Since we have absolutely no experimental data on the laws of physics which pertain at temperatures above 500 million K (when the universe was about 70 years old)

I don't know where you are getting this from. The LHC reaches energies of roughly 10 TeV, which is roughly $10^{17}$ K.

I therefore continue to stand by my earlier remark in spite of the number of times I have been told that I am wrong.

Since you insist on taking this position despite repeated corrections, this thread is now closed.