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=3H*^{2}/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?

---

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.

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?

---

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:

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.

---

However … this still 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 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?

---

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.

---

Please note the tag below "cynicism". This is, in part, referring to the paranthetical "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.