The Universe is Flat (as in Not Flat)

I posed the following question on Reddit, based on the pondering expressed in My Universal (and Expanding) Struggle:

Recent observations tell us that the expansion of the universe is accelerating. Other observations tell us that the universe is flat. This seems to be in contradiction, if you follow this logic:

The critical density found via the first Friedmann equation is ρc=3H²/8πG. As Sean Carroll points out, if the universe is flat, then the density of the universe is equal that of the mass required to obtain a Schwarzschild radius of one Hubble length (the speed of light divided by the Hubble parameter) divided by the volume of a sphere with that radius. The implication is that that Hubble parameter is inversely proportional to the radius of the observable universe (note I said "proportional", which eliminates the question of whether that is the naive value [13.7 Mly] or the calculated value [46.6 Mly]) and consequently also inversely proportional to the age of the universe.

​How can this be squared with the observation that the rate of expansion of the universe is apparently increasing?

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I note that this issue is effectively mentioned at wikipedia where is it stated that:

The discovery in 1998 that q is apparently negative means that the universe could actually be older than 1/H. However, estimates of the age of the universe are very close to 1/H.

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but the issue is not taken up for discussion. The fact that the age of universe is strikingly close to 1/H seems like too much of a coincidence, particularly if the deceleration parameter, q, has varied during the life of the universe. It would put us in the middle of an era of the universe that would appear to contravene the Copernican principle. Or am I missing something?

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I got one response which was nice enough from u/nivlark:

The exact proportionality you describe only holds if q remains constant over the lifetime of the universe (and in particular, if it is equal to zero). In the absence of dark energy, we'd instead have a positive deceleration parameter and a universe younger than 1/H.

As to why the universe's age is very close to 1/H, we have the more complex situation of a time-varying deceleration parameter - dark energy only became dominant (i.e. expansion began to accelerate) relatively recently. Perhaps by coincidence, this means that 1/H has only recently 'caught up' to the age of the universe. The discrepancy between the two will widen in the future, eventually approaching some limiting value depending on the exact value of the deceleration parameter.

This led to me ask:

Does that mean that Sean Carroll is wrong when he writes "Note that a spatially flat universe remains spatially flat forever, so this isn’t telling us anything about the universe now; it always has been true, and will remain always true"?

Are you suggesting that the universe only appears to be flat (as per the WMAP and Planck surveys)?

Edit: I've read that if the universe is flat then q=1/2 (precisely, not more, not less), there may be caveats involved with that though.

u/nivlark responded with:

No, that is correct. A universe which is exactly flat will always be so, but it's an unstable equilibrium: deviations from flatness must grow such that non-flat universes become more open or closed with time.

Measurements of curvature from Planck &c. are consistent with flatness, but with some observational error (I have the number 0.4% in memory for the size of this error, but that may be out of date). So we can say that either the universe is exactly flat, or that it has a small non-zero amount of curvature consistent with these bounds. Neither is wholly uncontroversial: zero curvature suggests very finely-tuned initial conditions, while nonzero but small curvature requires a process like inflation to be invoked to produce the exceedingly small initial value of the curvature.

q=1/2 indicates a flat universe, but specifically one that is dominated by matter. A cosmological constant-dominated flat universe would instead have q=-1.

I just responded with “Thanks”, in part because I didn’t have more to ask at the time and in part because I’ve made enough of a fool of myself with mathematical questions, I don’t want to get into similar problems with physics.

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However … this answer by u/nivlark still bothered me.  He seemed to be saying, “yes, a universe which is exactly flat will always be so” and then immediately saying that there will inevitably be deviations from flatness (“it's an unstable equilibrium: deviations from flatness must grow such that non-flat universes become more open or closed with time”).  Let’s say that the universe started off not quite flat, but really close to flat.  The implication here is that the deviation can really only tend to one direction, because if it’s a tiny bit open and tended towards being a tiny bit closed, the universe would pass through exactly flat and he also said that exactly flat is flat forever.

Now, you could argue that the issue is tied in with variations not only over time, but also across space – local space to one observer might appear entirely flat, but another observer a cosmically significant distance away would see it as slightly open, or slightly closed.  That is, to be entirely flat, the universe would have to be eternally and universally flat, all the time, everywhere.

That would mean however that we just happen to be, just at the time that we first have the ability to measure the (local) density of the universe, just in the right place to measure that density to be completely consistent with a flat universe, neither a tiny bit open nor a tiny bit closed.

Which contravenes the Copernican principle, doesn’t it?

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The thinking above led to the following exchange:

neopolitan-

Can you confirm that you are happy with the fact that the Copernican principle is being contravened. In your argument you seem to be saying that the universe is not (entirely?) flat, not exactly flat, but it deviates from flat. However, our readings of the data, just now, just when we are just beginning (in cosmic timescales) to measure the curvature (or lack thereof) of the universe, we happen to be in an era and/or a sector in which our measurements tell us that the universe is flat.

This would, in a sense, make our era and/or sector special, would it not?

nivlark-

That isn't what I wrote. I said that the measurements we have are consistent with flatness, but that there is an observational error associated with those measurements which means that the best we can say is that the curvature is no greater than the magnitude of that error.

I then went on to say that there are some as-yet unsolved theoretical difficulties with both perfectly-flat and slightly-curved universes, so theory cannot help us by eliminating one of the possibilities.

The Copernican principle only applies to our spatial location: we do appear to occupy a privileged position in time. Whether by coincidence or by appeal to the anthropic principle, we appear to exist at an era when the densities of matter and dark energy are comparable, which will not be the case for the vast majority of the universe's lifetime.

neopolitan-

I know it wasn't what you wrote, that's why I asked for confirmation of what I interpreted from what you wrote (ie what you seemed to be saying, from my perspective). I'm sorry that I didn't make that more clear.

I agree that there is potential for observational error and there is also potential for what could be called "assumption error", since the measurements are based on certain assumptions, all of which might be perfectly correct but might also be slightly wrong (or more so).

And I might be extending the notion of the Copernican principle too far by considering a temporal aspect as well, but ... I suspect that we could be running into a simultaneity issue if we suggest that our position i(s) privileged only in time. There's an implication in your statement that the universe changed from matter dominated to dark energy dominated everywhere at the same time. Alternatively, we are in a part of the universe in which dark energy and matter are comparable (and/or in which the effects of there being a balance of matter and dark energy have manifested), which makes our location privileged as well.

Note, I have in mind the concept that I think of as "evenness" together with curvature, by which I mean that the extent to which the universe is flat or not, if it fluctuates as you suggest, won't be precisely the same everywhere - so it'd be "uneven". The flat universe that Sean Carroll referred to would also be even - flat everywhere, all the time. It seems to me that deviations from flat would also lead to deviations from even.

nivlark-

“There's an implication in your statement that the universe changed from matter dominated to dark energy dominated everywhere at the same time.”

This is the case...

“Alternatively, we are in a part of the universe in which dark energy and matter are comparable”

...as is this, and it is also true everywhere. By construction, we model a universe that is homogeneous on large scales, because that's what observations indicate to be the case.

“Note, I have in mind the concept that I think of as "evenness" together with curvature”

These are different quantities. The curvature referred to when talking about the flatness of the universe is a global quantity which is an intrinsic property of spacetime, and there's no theoretical basis to suspect it varies with position. However, 'local' curvature is produced by every massive object - this is what we perceive as gravitational fields. As a result of this the geometry of spacetime is lumpy/uneven on small scales (where 'small' here means galaxy-sized), and this can be the case irrespective of what the global curvature is. Cosmological models are applicable on much larger scales than this though, and so the real universe is very well-approximated by models of a perfectly homogeneous one.

neopolitan-

You haven't addressed the simultaneity issue associated with the entire universe fluctuating, or do you mean to do that by saying that the curvature is "an intrinsic property of spacetime"? If that is the case, would we not still expect to see the consequences of fluctuations in the intrinsic property of spacetime rippling through the universe due to simultaneity/relativism issues? Or do you suggest that we might if it weren't for the lumpiness of space at the galaxy level?

nivlark-

I don't know what the "simultaneity issue" you're referring to is. The global properties of a homogeneous universe are perceived to evolve simultaneously by any comoving observer (i.e. any observer who has no proper motion and is carried freely by expansion). This does not contradict relativity or the cosmological principles.

As I said in my previous comment, the flatness of the universe is such a global property. It does not depend on position. Superimposed on that global curvature is a time- and position-dependent local curvature, which occurs due to the presence and movement of mass. This has local effects, which we call 'gravity', but these are negligible on the scales relevant for cosmology because the magnitude of the local curvature falls with distance from the source (in classical language: the gravitational force weakens with distance).

neopolitan-

> I don't know what the "simultaneity issue" you're referring to is.

I'll try to explain, please forgive me if I don't use the precise terminology that you favour. There is a "slice" of the universe that constitute the comoving coordinates. It's this set that is normally referred to when considering the "shape of the universe" or the curvature. There's a reason for taking this particular slice such that the coordinates are comoving, namely that you can't really talk about the universe as a whole at a single point in time - the comoving coordinates set is as close as you can get (I'm assuming that it's basically the circular cow of a universe you need without anything in it to mess up the calculations). The comoving coordinates constitute a set of coordinates that are not collocated, and therefore a change of curvature that manifests across the entire set that is happening simultaneously is problematic.

Now I can accept that there is a process going on such that at the end of that process, no matter which localised subset of the comoving coordinates you consider, the curvature will fluctuate in the same way and therefore you'd see the entire manifold fluctuate at the same time (within the comoving frame). But if that were possible, it seems that that would be an alternative solution to the homogeneity of the CMB and inflation would not be necessary (and from what I read, something like inflation is necessary).

The conversion seemed to have died at that point, although it could be that it was the weekend and u/nivlark has a life.  Being the weekend, I did have some time to ponder though.

It’s possible that what u/nivlark is saying is that there’s a tendency to expansion (dark energy) and a tendency to contraction (gravity due to matter).  When the universe is small(ish) and there is a certain amount of matter in that small(ish) volume, then contraction has more sway than when the universe is larger.  When the density reduces to a certain point, we can say that dark energy now dominates; it would not be a punctuated transition but rather just a point of interest on a smooth curve.  If this were the case, then simultaneity would not be an issue.

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Please note the tag below "cynicism".  This is, in part, referring to the parenthetical "as in Not Flat" in the title.  My position is that the reason that all the measurements tell us that the universe is flat (at this time) is because the universe is actually flat.