Relativity and Cosmology

Hi Jorrie, interesting, but what's the point if it does not represent reality? The universe is in any case not homogeneous, so you can't treat it with Newton's shell theory.

SL

Hi SL, you asked about homogeneity issues of the universe.

If we take the big picture (large enough scale) the universe is remarkably homogeneous. At large scales, it also follows the simple Newtonian type Hubble law with intriguing closeness. As far as shell theory is concerned, it is still usable if you take a spherically symmetrical sample, e.g., a galaxy and some fairly empty space surrounding it. The problem is that one cannot use the simple gravitational collapse model that I used for the homogeneous case and reverse the infall to recession. A much more dynamic model needs to be constructed, which is quite difficult.

Jorrie

OK, I see. One more question.

You wrote that because the acceleration is directly proportional to r, the velocity at time t is also directly proportional to the r by the simple relationship v = at. However, r is changing during the time t, and so is the acceleration. Your calculation cannot be correct.

SL

Hi again SL. Good observation!

Yes, you are right, I should have said v = ∫a dt, from t = 0 to t1. The bottom line is however that if a is directly proportional to r for any given time, so would v be, hence the conclusions do not change.

I'll take the liberty to put such a note in my original post.

Jorrie

Hi Jorrie,

The result you calculate was quite unexpected, at least by me. Have you seen it written up in the textbooks or elsewhere?

I think it's very important to try to understand the relationship between Freidmann scale expansion/contraction at the level of a homogeneous observable universe, on the one hand, and normal Newtonian and GR gravitation metrics at the quasilocal nonhomogeneous level, on the other hand. It seems necessary that these different but related concepts be completely reconciled with each other. In particular, we need a mechanism to enable aggregating or integrating a collection of local gravitational forces as a means to calculate a normal Friedmann result.

Your example of a Newtonian Hubble-like contraction is a start in that direction. In this case, there is no need to talk about a "contraction of space." Masses simply move towards or away from each other through normal gravitational interaction, and the result looks precisely Freidmann-like. If it looks like Friedmann, then why not conclude that it is a quasilocal example of Friedmann?

As you suggest, we next need to find a quasilocal construct for the Einstein-de Sitter expansion which is analagous to the gravitational contraction construct. The expansion side is much more difficult to analogize however. Whereas gravity is present "now" and dynamically decelerates the overall expansion, the Einstein de-Sitter expansion itself is "pre-existing" and is not supposed to have any dynamic kinetic energy component at present. Other than dark energy of course, which (if it exists) currently accelerates the expansion. But the dynamic "negative pressure" mechanism of dark energy does not provide a useful construct for modelling the initial expansion, because dark energy increases along with the expansion, whereas the initial expansion itself does not gain kinetic energy over time; it has a fixed kinetic energy which is drained away (actually converted to potential energy) over time by gravity.

The idea of inflation is that the inflatons imparted a sort of powerful anti-gravity which accelerated the expansion to its peak rate and then was abruptly cut off, leaving the expansion to coast thereafter. One could imagine a reverse picture in an overdense universe, where gravity is applied for a period of time, causing the expansion to stop and contraction to begin. Then if all of the gravity could be abruptly "cut off", the contraction would continue coasting along anyway until all of the matter smashed together (or to some degree recoiled upon hitting each other).

In a situation where a region of space (such as a cluster) is gravitationally bound, I wonder if the underlying Friedmann expansion is "still there but supressed'; or if it is permanently eliminated. For example, if a cluster has peculiar motion, as it moves it continues to occupy new regions of space and vacate regions it occupied earlier. Presumably the newly-vacated spaces will resume expanding at the Friedmann rate. They will not have "forgotten" the original expansion (contrary to what Peacock said). It would make no sense that they are "re-taught" how to expand by neighboring regions of expanding space; they just "remember" all by themselves.

If the previous paragraph is correct, then my idea of "cutting off" gravity and allowing a contraction to continue coasting (to a singularity) may not work. Once gravity is "cut off", the underlying expansion will reassert itself, and the contraction will slow, or maybe even reverse itself and begin expanding again. (I realize that the idea of cutting off gravity abruptly is not physically realistic; but it is no more fantastic than the idea of an anti-gravity inflaton whose accelerative force abruptly cuts off.)

I'd like to say that gravity (negative acceleration) is equivalent to a negative-inflaton force (positive acceleration), but there's a big difference. Gravity is entirely localized around mass (inverse square law). But inflaton force is theorized to be entirely non-localized, or in any event there was no mass or other "stuff" around for inflaton force to be localized to while it was causing expansion to accelerate. That's another reason why I don't like inflation theory!

Sorry for the rambling, but I can't help feeling that someday somebody will figure out how expansion could be modeled as the exact flip side of the effect of gravity...

Jon

Hi Jon, you asked: "The result you calculate was quite unexpected, at least by me. Have you seen it written up in the textbooks or elsewhere?"

No, I haven't and I would like it if you will help check it's validity, especially my second figure. I think one can safely say that dots at the same distance will fall at the same rate. Given that k is roughly constant over the total sphere at any snapshot in time, are the radial and tangent rates of approaching each other approximately the same?

"... and the result looks precisely Freidmann-like. If it looks like Friedmann, then why not conclude that it is a quasilocal example of Friedmann?"

Yea, I suppose one can ask this, but we know that Friedmann handles expanding/contracting space (which can be superluminal) on top of peculiar movement. In the Newton case it is only peculiar movement, which can not be superluminal.

The other problem is that Friedmann fails if any significant non-homogeneity is introduced. Even with this Newton case, it becomes very, very difficult to calculate a non-homogeneous case. I'm busy with a homogeneous spherical collection with a finite, denser core region - rather simple sounding, but I'm struggling!

Jorrie

Hi again Jon.

I've got some preliminary results for the spherical distribution with a central clumping and it behaves more or less as expected.

Figure 1:

I've used a Hubble-like expansion of a homogeneous spherical distribution of matter (radius R) and then abruptly shifted 1% of the total mass into the 0.1R sub-sphere. This made the density of the sub-sphere 10 times the average density of the sphere. I then let some time t pass for the 'new' accelerations to work and finally approximated the velocities against r, simply by v = v₀+at, where v₀ is the Hubble-like recession rate and a was calculated using the mass inside each shell.

The straight blue line is the Hubble-like recession rate and the red curve is the recession rate resulting from the central inhomogeneity. The recession rate axis has an arbitrary scale so that the effects show up clearly. I had to fine-tune initial conditions a bit so that there is a collapsing portion.

Interestingly, an observer at the edge of the clump (10% of R) will view all the inner masses to fall towards her (she's falling in faster than all of them), while all outer masses will recede from her, at first non-Hubble-like, becoming more and more Hubble-like with range. Actually, if she observes far enough to left or right, things will be approximately Hubble-like recession.

If there was a CMB at the edge of the sphere for her to observe, she would have spotted a dipole in the CMB temperature, from which she could have deduced that her galaxy is moving towards the central mass concentration, her 'Great Attractor'.

Food for thought!

The graph of figure 1 signifies an apparent paradox: Edwin Hubble measured the original Hubble constant using the local galaxies, where the recession speeds should not have followed Hubble's law (inside the dip of the curve). This was later pointed out by Sandage and dubbed the 'Hubble-Sandage paradox'.

I've just found an interesting paper that 'solves the paradox': Chernin et al.: 'The very local Hubble flow'. It uses two effects: (i) initial collapse and chaotic interaction, throwing galaxies away from the barycenter, and (ii), you guess, the vacuum energy of Einstein's cosmological constant (dark energy).

The paper makes a lot of sense and is the best one that I've discovered on the issue of the 'paradox'. By its very definition, vacuum energy cannot 'clump together' like matter energy and hence smooths out any inhomogeneities. I'll try and model it into my very crude simulation and see what the curve will look like...

Jorrie

Hi Jorrie,

The Chernin et al article is excellent . Please let me know whether I have interpreted/interpolated some of its points correctly:

1. The "zero gravity surface" at 1.5-2 Mpc from the Milky Way-Andromeda barycenter defines the small cluster which is actually gravitationally bound. All of the (small) galaxies outside the ZGS are not actually gravitationally bound to our two major galaxies, even though they are referred to as part of the Local Group.

2. The paper says "low mass groups, like the Local Group, dominate, and the Hubble flow is not significantly disturbed around them." I read that to mean that many of the galaxies within most or all clusters do not comprise a single gravitationally bound object; a large cluster (or supercluster) may contain numerous relatively low mass groups (like the Local Group), the small "cores" of which now comprise the largest scale bound structures. That is, only a handful of large galaxies each bound core. The idea that only small core structures within clusters and superclusters are actually graviationally bound is a significant departure from my prior understanding. It should mean, in general, that the great majority of the volume within clusters and superclusters is expanding at about the average Hubble rate, rather than having essentially fixed radii. Which further means that the total percent of the volume of the observable universe which is bound (non-expanding) must be very tiny -- less than 1%. (Compare to Wiltshire's picture that 20-25% of total volume is bound).

3. The paper points out that within 4-8 Mpc of the Local Group barycenter, the difference in dispersion rates calculated with and without the cosmological constant is fairly small. I interpret that the reason why the difference is small is that even in the non-cosmological constant case, the small galaxies pick up the Hubble flow element resulting residually from the original expansion of space, which causes the same kind (but smaller magnitude) of smoothing/cooling effect that the cosmological constant does. Over larger distances, the effect of the cosmological constant begins to outweigh the residual effect of the original expansion.

4. The paper's model that most of the small galaxies originally formed within the Zero Gravity Surface and eventually escaped, due to the chaotic random peculiar motions of the churning set of objects within the ZGS, is an interesting and quite different picture of cluster formation. I had not heard the term "Little Bang" before, but I get the point. Clusters look kind of like fireworks; small galaxies "explode" rapidly outward at first from the ZGS core, then later their peculiar motions cool and become dominated by the cosmic Hubble flow, leaving their relative positions somewhat "frozen" over time (but expanding).

5. If the model described in the paper is basically correct, I think it goes a long way towards resolving the conceptual gulf between the homogeneous overall Hubble flow and the nonhomoegeneous local structure. It turns out that those gravitationally bound local structures are just a lot smaller than we had previously pictured them. Which means that the universe really is much more gravitationally homogeneous at quasilocal scales than it might otherwise have been.

6. I think your Newtonian shell model of quasilocal expansion and contraction fits quite neatly into the Chernin picture.

7. I would think that Wiltshire's idea can still make sense together with the Chernin picture. More rapid apparent expansion rates (as measured by wall observer clocks) within voids and other nonbound spaces could still substitute conceptually for the cosmological constant. However, it seems to me that the Chernin picture dramatically reduces the total volume of gravitationally bound regions (wall observers), and substitutes most of this volume with many nonbound mini-regions within clusters. Whether those mini-regions would expand at the same rate as the voids in the Wiltshire model isn't entirely clear to me.

Jon

Hi Jon, yes I agree.

I think your interpretations are good, except for perhaps this piece: "... I interpret that the reason why the difference is small is that even in the non-cosmological constant case, the small galaxies pick up the Hubble flow element resulting residually from the original expansion of space, ..."

I view the dispersion rates as small because of the kinematics of the "Little Bang", so it does not really need either of the other two effects to be 'smoothed', but I'm not sure...

Like you, I also think there may be elements of both Chernin and Wiltshire in the real universe, with neither one sufficient to stand alone.

You wrote: "Whether those mini-regions would expand at the same rate as the voids in the Wiltshire model isn't entirely clear to me."

If you consider expansion relative to the centers of voids and mini regions respectively, the larger voids must obviously expand faster. If you consider it from our vantage point, deep inside a mini region, it may make little difference. A lot of mini regions together should be equivalent to a large void with some scattered matter inside it, no so?

Jorrie

PS: It looks like Wallace & Co. on PF are mostly on vacation!

Hi Jorrie,

Does the modeling you did here bear any relationship to the LeMaitre-Tolman-Bondi (LTB) model for an inhomogeneous spherically symmetrical dust universe? The LTB model has a center of expansion, but seems to have been used quite a bit recently as a construct to assess inhomogeneous non FLRW alternatives to dark energy, etc. A couple of recent papers on the subject are http://arxiv.org/abs/astro-ph/0512006 and http://arxiv.org/PS_cache/astro-ph/pdf/0608/0608403v1.pdf

Jon

Hi Jon.

This inhomogeneous Newton model is similar, but quite far from the more formal and relativistic LTB model. As I presented it here, it has the "wrong" inhomogeneity in any case (an over-dense region centered on the observer). It will be interesting to see what happens if an under-dense region is centered on the observer.

In any case, being Newtonian, all it can give a s very rough "tool" for understanding some of the more formal models (almost like the Newton cannon ball analogy for the FRW model).

Jorrie

You said: "We can be on any galaxy and, provided that the universe is large enough so that we cannot observe an edge, we cannot tell whether we are in the center or not."

Um...an EDGE? I thought edges went out with the bathwater when we discarded the static universe model. I was under the impression that anything that could be considered an edge was either an energy density only, or so far back in space/time that we couldn't ever observe it, much less reach it. Or is this a semantics difficulty only?

Hi EnviroMan.

Remember, the quote you stated has to be read in the context of the original premise that I started with:

"Consider identical galaxies homogeneously and isotropically spread throughout a large piece of spherical space, in a hypothetical static scenario."

This hypothetical case could have an edge, the universe probably not. With this said, we cannot say for sure that the universe don't have an edge, simply because we cannot observe far enough...

Most commonly it is viewed as either infinite or closed back on itself (hyper-spherical) so that there is no edge, but who knows?

Jorrie

And again, there is the new Dominium explanation which aligns with Newton:

A graph for the observed expansion rate of the Universe has been generated that shows three distinct areas: initial rapid expansion, an inflection point, and the more modern slower and slowing rate acceleration of expansion.

In papers covering the initial period of rapid expansion, it has been put forward, by Guth and others, that in order to achieve a rate of expansion that matches the accepted graph there would be needed some form of "reverse-gravity." However, no conclusion is made to the dynamics or source of this reverse-gravity.

The new Dominium model does supply an answer that accounts for the reverse-gravity that has been postulated necessary in order to achieve the accepted rate of initial expansion: opposite-repel interactions between matter and antimatter proto-galaxies.

Assuming that the distribution of matter and antimatter was randomly and heterogeneously distributed, this reverse-gravity would be expected to be felt throughout the expanding Big Bang fireball. Therefore, one need not relativistic maths to supply an answer for this particular solution, only a rethinking of how/what forces were involved at this moment in time.