Are there any local effects of Einstein-de Sitter expansion?

[Bandersnatch]

The question is whether or not expansion in flat, matter-only universes (no lambda) has any effect whatsoever - no matter how negligible - on dynamics of small-scale systems.

Context:
It's a variation on the 'is Brooklyn expanding?' type of questions.

My understanding has always been that bound systems are not affected, period. If it's bound, then it's not expanding. Not that it's not measurable due to its negligible size (like with dark energy), but that there's no effect whatsoever.

I'm aware that this understanding comes mostly from the Newtonian description of expansion, which supplies this heuristic that it bears some similarities to inertial motion in local gravitational field. But I've never counted this bound=no expansion intuition among the many limitations of the description.

From what little I understand of GR, I gather that locally the FLRW metric is not applicable, due to the lack of homogeneity and isotropy, and instead one should use the Schwartzschild metric. Although perhaps it doesn't follow from this that there's necessarily no expansion on small scales?

On another forum, somebody brought my attention to this paper:
The influence of the cosmological expansion on local systems, F. I. Cooperstock, V. Faraoni, D. N. Vollick
...which seems to contradict my understanding.
I'm having a bit of a hard time following the reasoning. It appears to be assuming the FLRW metric to be valid on small scales, from which then follow some calculations of the effect the expansion has on dynamic systems (it is being compared to a force). Perhaps I've misunderstood the paper, but it seems pretty clear on that point.
From the wording in the article, it looks like it was, at least at the time (1998), not an unusual position to take that expansion should have a continuous effect on local systems.

I intend to read through at least some of the referenced work in hopes of gaining more insight, but maybe somebody else can educate me in the meantime.

To reiterate: is there a force-like effect from expansion acting on local systems in universes without dark energy?

I'd especially like to hear from @kimbyd @PeterDonis @Orodruin or @bapowell , if they can be bothered, but input of any knowledgeable person is welcome.

Cheers.

 

[kimbyd]

The question is whether or not expansion in flat, matter-only universes (no lambda) has any effect whatsoever - no matter how negligible - on dynamics of small-scale systems.

Context:
It's a variation on the 'is Brooklyn expanding?' type of questions.

My understanding has always been that bound systems are not affected, period. If it's bound, then it's not expanding. Not that it's not measurable due to its negligible size (like with dark energy), but that there's no effect whatsoever.

I'm aware that this understanding comes mostly from the Newtonian description of expansion, which supplies this heuristic that it bears some similarities to inertial motion in local gravitational field. But I never counted this bound=no expansion intuition among the many limitations of the description.

Yes, but it also applies to a full GR understanding of the situation.

This can be drawn from the perturbation theory approach to describing an expanding universe, where it is found that in the linear approximation (which is valid at very large scales, much larger than individual galaxies or galaxy clusters), overdense regions have constant gravitational potentials, provided there is no cosmological constant. This can be understood as stating that in the linear approximation, as the universe expands, large overdense regions don't so much collapse as they stop expanding. This should fit with our intuition based upon the behavior of bound systems like our solar system: once friction is no longer an issue, due to the gas and dust in the system either collapsing into dense objects or being blown away, the system settles into a steady state which remains mostly unchanged over very long periods of time.

If you read the above carefully, you may have noticed an important caveat: no cosmological constant. In the presence of a cosmological constant, these large gravitational potentials slowly decay over time (this is observed via the Integrated Sachs-Wolfe effect). This indicates that dark energy does have some local effects, which could in principle be measured. However, the value of the cosmological constant is just too small for its impact to be realistically measured with terrestrial experiments.

 

[PeterDonis]

is there a force-like effect from expansion acting on local systems in universes without dark energy?

There are several issues involved here.

First, as you note, the FRW metric is based on an assumption of homogeneity and isotropy, and obviously that assumption does not apply on small scales; our solar system is not homogeneous or isotropic, nor is our galaxy, nor our Local Group or nearby galaxy clusters. Only on very large scales (something like 100 megaparsecs or larger) do homogeneity and isotropy become reasonably close approximations. So to me, anyone who talks about the FRW metric applying on small scales like the solar system or our galaxy or a galaxy cluster is simply misstating things. (The paper you link to appears to me to fall into this category, since it's computing so-called "cosmological effects" in Fermi normal coordinates around a small scale bound system on the assumption that the FRW metric is valid there.)

Second, to know what metric to use on small scales, you have to know what the distribution of stress-energy is; that's how you solve the Einstein Field Equation and obtain a metric. The Schwarzschild metric, which you mention, describes the vacuum region around a single isolated spherically symmetric body; it is a reasonably good approximation, for example, for the near vicinity of the Earth (the Earth is not perfectly spherically symmetric, since it is rotating, but the effects of that are small enough that they can often be either ignored or finessed by adjusting the coefficients in the metric). For more complicated cases, like the solar system, perturbation methods are generally used: heuristically, you start with something like the Schwarzschild metric centered on the Sun, and then add in small adjustments due to the other planets. This still basically amounts to a metric that describes vacuum almost everywhere, with little "lumps" where the massive objects are; it certainly looks nothing like an FRW metric.

Third, as the paper you linked to notes, for the idealized case of a spherical region consisting of vacuum, surrounded by a spherically symmetric distribution of matter, the metric inside the spherical vacuum region is flat--i.e., the matter outside the region doesn't affect the metric inside at all (it's the same as if there were no matter anywhere). More generally, if you have a spherical region consisting of mostly vacuum with small lumps of matter here and there, surrounded by a spherically symmetric (on average) distribution of matter, the metric inside the spherical region will be determined entirely by the small lumps of matter inside it; the rest of the matter in the universe will have no effect. That is the basic argument for why the matter in the rest of the universe has no "expansion" effect on small scale systems like the solar system or our galaxy or a galaxy cluster: all of these cases meet the description I just gave.

(Note that the reason the above does not apply to dark energy is precisely that dark energy does not meet the description I just gave; its density is literally constant everywhere--at least that's the best current model we have--so there is no such thing in the presence of dark energy as a region of actual "vacuum"; there is always dark energy there, having its tiny little local effect.)

 

[Bandersnatch]

Thank you both. I think I'm slowly getting a better handle on it.
I have some more questions about the paper, specifically.

While I see @PeterDonis being rather dismissive of the article, I am somewhat reluctant to believe that the authors were unwittingly misstating the issue - especially seeing how well-cited the article is, and how it claims to be a continuation of a discussion going back many decades. For one, it would be odd if the authors didn't realize the FLRW metric is a solution for homogeneous and isotropic distributions only.

After giving it some more thought, this is what I think the paper does:
They acknowledge that there exist two domains described by different metrics - the large-scale FLRW one and the small-scale one where expansion does not happen. Given how the universe becomes gradually more and more homogeneous with scale, it is unreasonable to assume that there is a sharp cut-off separating the applicability of the two metrics. Not knowing at what specific scale the expansion should no longer have any effect, they take the approach of assuming that it happens across all scales in order to obtain the upper bound for the effects it could have on local systems. It's not attempting to show that there are such effects, but rather to estimate what is the lowest precision one would need if one wanted to test for any such effects.
Then they proceed to use motion with the Hubble flow as a perturbation in orbital dynamics in order to calulate that upper bound, and conclude that even given those unrealistic assumptions the effects are essentially unmeasurable.

Does the above seem like a good representation of the paper's intent?

 

[PeterDonis]

Given how the universe becomes gradually more and more homogeneous with scale, it is unreasonable to assume that there is a sharp cut-off separating the applicability of the two metrics.

It's not so much a question of a sharp cut-off vs. not, it's a question of whether you're looking at the overall dynamics of the universe as a whole, or the local dynamics of a particular system. If you're looking at the local dynamics of a particular system, like the solar system, then the argument I gave in my post about a spherically symmetric distribution of matter in the rest of the universe having zero effect applies. That's not a question of scale; it's a question of what you're looking at. A "local" system 100 megaparsecs across with a spherically symmetric distribution of matter outside it still satisfies the conditions of the argument. But the entire universe does not, precisely because it's the entire universe.

In other words, the whole point of the FRW metric is to investigate the dynamics of the universe as a whole--i.e., to investigate the case where you can't apply the argument "we have a local system with a spherically symmetric distribution of matter outside it". That's what it's for. Trying to use it to investigate cases where you can apply that argument is simply misrepresenting the purpose of the thing.

 

[Chronos]

It appears safe to say the minimum scale for cosmic expansion is a question that has evaded a definitive answer. These are the basic facts as presently known.. Galaxies at gigaparsec distances are receding faster than those at a megaparsec beyond any reasonable doubt. This has been measured with such painstaking care these number have been fine tuned down to within about 1.5 kn/sec. Equally not in doubt is the fact the expansion rate at gigaparsec distances is nonlinear to that at megaparsec distances. Despite the best efforts by our most talented mathematicians no formula has yet emerged that satisfactorily predicts the expansion rate across all distances. Such a splendid result would be entirely satisfied to any ordinary person, but, not for your typical O-C, twitchy and insecure scientist. Given thie nagging uncertainty over expansion that persists between billions and millions of light years, trying to explain how it works at puny comprehensible distances is a challenge no self respecting cosmic comic can resist. The popular answer is it doesn't. Gravitationally, not to mention even more tightly bound systems are as oblivious to expansion as a locomotive is to a collision with a gnat. Sounds pretty cowardly coming from a bunch of people claiming to have measured the distance to the moon to the nearest centimeter, but, that's where we are.

 

[Bandersnatch]

it's a question of whether you're looking at the overall dynamics of the universe as a whole, or the local dynamics of a particular system.

Right. But isn't the whole discussion in and around that paper about estimating the effect of global dynamics on local systems?
I (hope I) understand your objections to using FLRW across all scales - I'm just trying to pinpoint the utility of what Cooperstock et al. were doing.
Given how there's a clear distinction between the global and local behaviour of galaxies, and given how matter distribution has evolved from roughly homogeneous and isotropic at all scales (in the early universe) to clumpy, there should (?) be a scale at which the two behaviours transition into one another.
Doesn't assuming FLRW across all scales simply provide the upper bound on the effects of global expansion on local systems?

 

[PeterDonis]

isn't the whole discussion in and around that paper about estimating the effect of global dynamics on local systems?

It says it is, yes. But my point is that the paper is ignoring the very point it mentions near the start: that if the rest of the universe is spherically symmetric around a given region, there is no effect of global dynamics at all. The only such effects are if the spherical symmetry is not exact (which of course it isn't in our real universe); but the paper is not analyzing that case, it's specifically analyzing the idealized case where spherical symmetry is assumed to be exact, since that's what the global FRW metric describes.

See further comments below.

there's a clear distinction between the global and local behaviour of galaxies

And what is it?

matter distribution has evolved from roughly homogeneous and isotropic at all scales (in the early universe) to clumpy

Yes, that's true. But that evolution hasn't changed the (average) spherical symmetry of the rest of the universe relative to a particular local region.

Here's another way of looking at it: consider the universe as a 4-d spacetime, rather than 3-d "space" evolving in "time". In this 4-d spacetime, pick out the "world tube" that describes a particular spherical region surrounding a local system, like the solar system. Assume (as the paper does) that the rest of the universe is spherically symmetric around that region. Then the spacetime geometry inside that world tube is entirely determined by the stress-energy present within the "world tube"; the rest of the universe has no effect on it. That is a mathematical theorem.

But what about the expansion of the universe? you ask. The expansion of the universe has nothing to do with the geometry inside that world tube; it has to do with the global geometry of the entire 4-d spacetime, and how that "world tube" fits into it. For example, pick another localized system, say a solar system in a galaxy far, far away. This localized system also has a "world tube" around it, to which the same theorem described above applies: the geometry inside the "world tube" is unaffected by the rest of the universe outside it. But, for example, in an expanding universe, the proper distance between the world tubes, as measured in a spacelike hypersurface orthogonal to both (here we are assuming that both local systems are "comoving", which isn't really true of our solar system, but we can ignore that for this example), will not be constant; it will increase with time (here "time" means proper time for observers inside either world tube).

The expansion just described is a feature of the global spacetime geometry of the universe as a whole, described by the FRW metric. But asking how the global geometry affects the dynamics inside each world tube is missing the point. Each world tube "fits" into the global spacetime geometry in a particular way; that's all there is to it. There is no other global effect, since, by the theorem I mentioned, the geometry inside each world tube is unaffected by the rest of the universe outside it.

 

[PeterDonis]

the spacetime geometry inside that world tube is entirely determined by the stress-energy present within the "world tube"; the rest of the universe has no effect on it. That is a mathematical theorem.

Perhaps it's worth expanding on this, since it seems to contradict what the paper is saying. As I read the paper, here's what it says it is doing:

(1) Assume a global FRW metric.

(2) Pick a comoving observer's worldline within this metric, and construct Fermi normal coordinates describing a "world tube" around it.

(3) Compare the local dynamics described by those coordinates with the local dynamics in the "world tube" around a geodesic in flat spacetime.

(4) Notice that the two local dynamics are different, and call that a "global effect on local dynamics".

But here's what the paper is actually doing, IMO:

(1) Assume a global FRW metric.

(2) Pick a comoving observer's worldline within this metric, and construct Fermi normal coordinates describing a "world tube" around it.

(3) Compare the local dynamics described by those coordinates with the local dynamics in the "world tube" around a geodesic in flat spacetime--failing to notice that FRW spacetime is not flat.

(4) Notice that the two local dynamics are different, and call that a "global effect on local dynamics"--failing to notice that what it really is is a local description of the difference between flat spacetime and FRW spacetime.

To put it another way: FRW spacetime assumes exact homogeneity. That means that in the "world tube" surrounding a given geodesic worldline, there is a constant density of stress-energy (more precisely, constant at a given instant of proper time along the geodesic worldline--but decreasing with proper time along that worldline). And the local dynamics within the world tube is affected by that constant density of stress-energy within the world tube. It has nothing whatever to do with the stress-energy in the rest of the universe.

What the paper should be doing is this: start with a global FRW metric; pick out a spherically symmetric world tube along a comoving worldline; replace the stress-eenrgy inside that world tube with vacuum; then compute the local dynamics inside the world tube. By the theorem I mentioned in my previous post, these local dynamics must be those of flat spacetime. The extra terms that the paper has will not be present--because the stress-energy inside the world tube that produced them is no longer there.

For extra credit, one could then do things like: put a single isolate spherically symmetric body inside the world tube (with its center of mass on the geodesic worldline); then compute the local dynamics inside the world tube and verify that they are those of the Schwarzschild metric. Again, by the theorem I mentioned, that must be the case. But the paper, even after mentioning that theorem, does not seem to pay attention to its implications.

 

[PeterDonis]

A "local" system 100 megaparsecs across with a spherically symmetric distribution of matter outside it still satisfies the conditions of the argument.

I should expand on this too. There is a difference between this and the case of a "local" system the size of the solar system. Imagine doing what I described in my previous post just now, but with the "world tube" being 100 megaparsecs wide instead of, say, 1 parsec wide (a reasonable width for a "world tube" around a single solar system). Now replacing the stress-energy inside the "world tube" with vacuum might have a non-negligible effect on the global spacetime geometry (since that was obtained assuming homogeneous stress-energy density everywhere), which means the whole exercise becomes much more complicated, because you can't assume an exact FRW solution any more. A "world tube" 1 parsec wide doesn't raise that issue because it's so small compared to the size of our observable universe that the stress-energy removed will be negligible as regards the global dynamics.

So perhaps a better way of asking the "scale" question would be to ask, at what size of "world tube" does removing the stress-energy inside (or replacing a homogeneous stress-energy with actual lumps representing isolated stars, galaxies, etc.) start to have non-negligible effects on the global solution that describes the spacetime geometry of the entire universe?

 

[Bandersnatch]

I'll get back to you later, once I digest the responses so far. Thank you for your contributions. I'll also want to reference another paper, but that can wait too.

 

[timmdeeg]

But isn't the whole discussion in and around that paper about estimating the effect of global dynamics on local systems?

When I came across this paper some years ago my first impression was that Clopperstock et al intended to show that even if one assumes the ideal fluid locally the expansion on scales like the solar system, on the the galactic and even on the galactic cluster scale is "nevertheless essentially ignorable".
This impression was based on the statement: "In this paper, we assume that homogeneous isotropic expansion is actually universal and we analyze the consequences of this assumption".

 

[kimbyd]

It appears safe to say the minimum scale for cosmic expansion is a question that has evaded a definitive answer.

I disagree with this claim pretty strenuously. Linear perturbation theory describes the universe quite well on scales between the largest gravitationally-bound systems and the overall expansion.

The uncertainties arise mostly for more compact objects, such as galaxies and galaxy clusters. This is mostly due to the fact that the physics of normal matter is extraordinarily complicated, but there's also the fact that in the non-linear regime, the behavior of gravity becomes far harder to calculate (this is what N-body simulations are used for).

 

[Chronos]

Objection noted, Presumably the 57 citations to the OP paper are less dismissive.

 

[kimbyd]

Objection noted, Presumably the 57 citations to the OP paper are less dismissive.

I don't understand what your objection to my statement is. As near as I can tell, my statement is consistent with the paper quoted in the OP.

There was a flurry of work investigating, in great detail, the relationship between structure formation and expansion following the 1998 discovery of the accelerated expansion by two independent teams. The paper in the OP appears to be one early component in this effort. This is because one of the hypotheses put forward for explaining the accelerated expansion was the idea that it might be an illusion created by the fact that the equations to describe the expansion assume a homogeneous universe, where our universe is decidedly not homogeneous on smaller scales.

This effort broadly concluded that the non-linear effects of structure formation didn't change the apparent acceleration significantly, and the only way that inhomogeneities alone might potentially explain the accelerated expansion would be if we were very close to the center of a huge void. This possibility has since been ruled out by more detailed measurements of the universe.

To bring this back to the OP, I think it's fair to say that right now, the regime between where the overall expansion is the dominant factor and where it's essentially irrelevant is quite well-understood, and described by linear perturbation theory. Remaining uncertainties are mostly at smaller scales, with the details of the formation of galaxy clusters and galaxies.