On Friday, November 8, 2019 at 4:47:15 PM UTC-6, Alan Grayson wrote:
>
>
> https://gizmodo.com/a-round-universe-would-be-bad-news-for-physicists-1839647806
>
> Personally, FWIW, I don't believe the universe is open. When physicists
> claim it is "flat", they must mean "asymptotically flat", leaving the
> possibility of really being closed. AG
>
The Friedmann-LeMaitre-Robertson-Walker (FLRW) metric
ds^2 = dt^2 - (1 - kr^2)^{-1}[dr^2 - r^2(dθ^2 + sin^2θdφ^2)]
is flat for k = 0 and a sphere for k = 1. For k = -1 it is a hyperbolic
surface. I wrote on how the FLRW dynamics can be found with just Newtonian
mechanics and gravity on Stack exchange
<https://physics.stackexchange.com/questions/257476/how-did-the-universe-shift-from-dark-matter-dominated-to-dark-energy-dominate/257542#257542>.
The dynamical equation is found from the energy E_{tot} = 0 constraint
0 = (a'/a)^2 - 8πGρ/3c^2 + k/a^2,
where a is the scale factor for expansion or contraction of space and ρ the
density of the vacuum energy. For k not zero there is in addition to this
dynamical equation a term -k/a^2, where a is the scale parameter, or
related to the radius, of the spatial surface. For an expanding space this
term clearly becomes very small.
If the universe is a sphere and not flat it blows away a theory I worked
last decade on how a quantum critical point in inflationary physics.
However, it does gives support to a hypothesis I have that the pocket world
generated in the unstable vacuum of the inflating de Sitter manifold
"bubbles off" that manifold as a closed spatial surface. These pocket
worlds are regions where the high energy, so called false vacuum, of this
inflationary de Sitter manifold decays or transitions into a low energy
vacuum. See diagram below. Here the right side is stretched out to
illustrate there is a time delay for the transition as a "slow roll." If
this remains a pocket, similar to a bubble in Swiss cheese, there is then a
boundary. This boundary has some ugly features to it. However, this
boundary will have quantum field content to it, and this may then
transition into topological quantum numbers for a closed sphere. It is
again a type of quantum critical point.
I will say I am withholding judgment on this. The problem I see is that
with inflation and the general "flattening" that occurs with that this
deviation from flat to spheroidal geometry would be small. I estimate the
distance to the edge of the bubble or pocket world is around 2 trillion
light years, which would be observed here as highly red shifted quantum
gravitational data. If this region at the end of inflation by a phase
transition transformed the boundary information into topological numbers
for a sphere, or an spherical orbifold, this 3-dimensional sphere would now
be around 159 billion light years in radius. The volume of this 3-sphere
would be V = (8π/3)r^3 ~ 4x10^{24}ly^3 and the volume of the region we can
observe to the CMB is about 45billion light years out or 9x10^{22}ly^3 and
so angle deviations would be around .02rad (1.3 degrees). A telescope
looking at some region of the CMB would see a slight excess of
magnification. This is not impossible to detect, but given the state of the
art it is still tough. Also since Planck and the WMAP before it integrated
data this would tend to be somewhat integrated out.
LC
[image: inflaton potential.jpg]
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