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.

Inference? What Inference?

From the railway station in the distance came the sound of shunting trains, ringing and rumbling, softened almost into melody by the distance.

Chapter 1, Book 1, War of the Worlds, HG Wells

I was curious, after listening to his appearance on Unbelievable, as to whether Luke Barnes would advertise it.  I wrote in Come on out, Luke!  Just admit it! about how I think that he should just come clean about being a theist (a Christian theist at that).  Earlier, in Luke Barnes Exposed, I wrote about how Barnes had stumbled in his interview with Luke Muehlhauser and basically admitted to being a theist and how his recent book basically confirms it (so it's a bit strange how he is so coy with Justin Brierley, but I guess he's got into the habit of maintaining the illusion of plausible deniability).  What I further noted was that the strange fact that Barnes doesn't mention his appearance on the Pale Blue Dot podcast, a fact that I ascribed to his stumbling admission to theism during that discussion.

So, I wondered, would he advertise, on his blog Letters to Nature, his appearance on Unbelievable in which he as introduced as a christian (twice)?

The answer is no.  No mention of it at all.  No mention of it coming up (despite it being recorded a week before being published) and no mention of it having happened (at time of writing, in the middle of week following - but he does seem to have mentioned it three weeks after I initially posted this article, that is a month after the episode was published).

I did notice, however, that Luke has written a piece on Sean Carroll's new book, The Big Picture: On the Origins of Life, Meaning, and the Universe Itself.  Interesting, I thought, since Justin Brierley suggested that Carroll would be invited onto Unbelievable later in the year to go head to head with Barnes.  What a fortunate co-incidence!  (Note that while the episode was taped before Barnes' post went up, presumably the piece was written before, even though that too went up online the same day that Barnes posted about it, and Brierley may well have been oblivious.)

So, was this to be another arxiv paper, like his attack on Stenger?  No, this time it was going to be published by Inference: International Review of Science.

It was at this point that faint alarm bells started going off and I felt a jolt of adrenaline, that frisson that one feels at the beginning of a hunt.  Something wasn't quite right about this.  What was this journal that I had never heard of before?  And why does the word "inference" shoot up warnings for me?  And why does Barnes hasten to say that he did not choose the title?  Ah, it's called Good God!  Not a good start.

So I wandered over and looked at the About page (linked above).  Hm, interesting:

Although the editors have every intention of appealing to experts for advice, Inference is not a peer-reviewed journal.

We have no ideological, political, or religious agendas whatsoever.

So, it's a review, and/or a journal, but not a peer-reviewed journal.  And methinks they do protest too much (with questionable grammar) about their lack of agenda.  Why do they need to protest at all?  Taking a look at another open access on-line only journal (albeit peer reviewed), Science Advances (from AAAS), there's no mention of ideology, theology or agenda.  The word "ideology" does appear on their Statement on Scientific Transparency, Disclosure, and Responsibility, but in the context of unwelcome politicised and ideology-based intrusions into science.  There seems to be no need to declare that they are free of agenda.  Perhaps the intent, by the editors of Inference, is to imply that other journals are not free of agenda.

That word, "inference".  It still bothers me.

So, who are these people?  These editors who took Barnes' words and placed it below their own rubric, Good God!?  And who else has been published in this august journal?

Scanning down the list of editors, one name leaps out.  Contributing Editor, David Berlinski.  Early in the life of this journal, Jeffrey Shallit expressed misgivings about the Berlinski name appearing in the twitter feed associated with a new science journal, Mischa and Claire, and intuited that their father David was somehow involved (despite there not being any details of the editorial board available at that time).

So, Berlinski is a senior fellow at the Discovery Institute and a contributing editor to the journal that Barnes decides as the vehicle for his review of Sean Carroll's new book.  Interesting.  What about other authors?  There's Michael Denton, of course, another senior fellow from the Discovery Institute.  Then there's Arthur Cody, about whom I could find very little, but who wrote a piece against materialism.  And Tyler Hampton … tennis coach?  The apologist's go-to man for the BGV, Vilenkin, is there, writing about the beginning of the universe.  Interesting indeed.

It can't be said that every single article has intelligent design leanings, but … it's still an interesting choice by Barnes.  We have more information to work with though.  Barnes highlighted three articles that he noted as contributing to his decision.  There was Vilenkin's article as mentioned above and what appears to be a perfectly normal article by Jean-Pierre Luminet on the notion of a Holographic Universe.

There's also a reference to an article by Gregory Chaitin, although it's unclear which one since he's written for Inference three times - I could point at An Algorithmic God, but it's possible that Chaitin did not choose the name and it's not the most recent article by him published at Inference.  The most recent is Doing Mathematics Differently which starts with two quotes from Leibniz.  Both are from Philosophical Essays, but they are strangely out of order and one is a fragment:

And if someone traced a continuous line which is sometimes straight, sometimes circular, and sometimes of another nature, it is possible to find a notion, or rule, or equation common to all the points of this line, in virtue of which these very changes must occur. For example, there is no face whose contours are not part of a geometric line and cannot be traced in one stroke by a certain regular movement. But, when a rule is extremely complex, what is in conformity with it passes for irregular.

Thus, one can say, in whatever manner God might have created the world, it would always have been regular and in accordance with a certain general order. But God has chosen the most perfect world, that is, the one which is at the same time the simplest in hypotheses and the richest in phenomena, as might be a line in geometry whose construction is easy and whose properties and effects are extremely remarkable and widespread. I use these comparisons to sketch an imperfect likeness of divine Wisdom and to point out something that can at least elevate our minds to conceive in some way what cannot be sufficiently expressed. But I do not claim to explain in this way the great mystery upon which the entire universe depends.

Then there are the contents of the issue of Inference that sit alongside Barnes' piece.  Two are notable.

First there's a "Critical Essay" by David Berlinski & Juan Uriagereka, The Recovery of Case.

This initially appeals to the linguist and grammarian in me, especially as a person who continues to use the words "that" and "which", both of which are often dropped in modern English and are thus easily confused.  However, I noted an odd slide towards the end, when their examples stopped being so quirky and quaintly awkward and took on a different flavour.  Regarding the awkwardness, the authors provide an example of a complementiser in the sentence "Trottweiler prefers for Agnes to keep quiet" instead of the more natural sounding "Trottweiller prefers that Agnes keep quiet".  The thing here is (that) for isn't a complementiser in the example, it's a preposition and it seems to have been backwards engineered into a sentence which just seems wrong.  The preposition for is the only one associated with the noun "preference".  I think a far better example of for, as a complementizer, would be in a sentence where for is used instead of because, "we all sang him a nice song for he's a jolly good fellow" but even then, it would be less awkward to simply use because.

The last two examples in the article aren't as quirky, instead they dip into the bible, there's something there about Sampson and Delilah and then one about Cain and Able.   That's not the main issue though.  The main issue is hidden at the end of the third last paragraph where the authors ask "if human beings are notable in their possession of Universal Grammar, what explains the acquisition of Universal Grammar by the human species?"

Michael Denton has an answer - it's not evolution (posted at a Discovery Institute web-site, although that might not be immediately obvious, you have to go to the bottom of the right hand bar to see their logo).  So, despite a promising start, this is merely an attempt to promote intelligent design.  No surprises there given that it's co-written by a senior fellow of the Design Institute.

Then there is an "Experimental Review" by James Tour, Two Experiments in Abiogenesis.

James Tour is a synthetic organic chemist who is notable for being a "scientific dissenter from Darwinism" and who signed a petition to that effect for the Discovery Institute.  It's interesting that he should be deployed to comment on a couple of experiments with implications with regard to abiogenesis.

What I am not suggesting is that Inference is a journal that is literally awash in Intelligent Design propaganda (by which I mean metaphorically awash).  Not at all.  What I am suggesting is that Inference is a journal that acts as a platform on which high-brow Intelligent Design propaganda can sit side by side with mainstream science.  Another journal which is peer-reviewed, and does not have an editorial board that includes a senior fellow of the Discovery Institute, would be reluctant to publish the propaganda, but Inference seems to have no problems with publishing mainstream stuff that comes their way - because it aids their cause.

So, that word - inference - what was bothering me about it?  Two things really.  There's the common apologetic claim that they are using "inference to the best explanation", which sounds all scientific, except that it tends to stop at the first step (rather than going on to attempt falsification, prediction or hypothesis testing).  But inference in itself isn't a bad thing.  We all make inferences pretty much all the time.

I think what bothered me slightly and now bothers me somewhat more is the initially tenuous link to the Design Inference, a book by William Dembski - formally of, you guessed it, the Discovery Institute.  (His retirement from the institute and the ID community was only announced a couple of weeks ago, but his work on an ID book implies that he is not entirely retired.)

In earlier articles, I have written about Barnes' and his Templeton temptation.  That's bad enough, so I do hope that we are not about to see him fall into even worse company with the ID crowd.  There are plenty of other journals out there that Barnes could have sought out for publication.  Why did he choose this one?