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] -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/e72c2e32-bbed-45a6-a488-1d777208d347%40googlegroups.com.