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  Could torus spaces (defined by Planck length) exist?

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Let ϵc denote critical energy density, ϵt energy mass density at time t, c the speed of light, G the gravitational constant, H=˙a/a the Hubble parameter, Λ the cosmological constant, p Planck length, and Ep Planck energy. From the first order Friedmann equation,
H2=8πG3c2(ϵt+ϵΛ)κc2R20a,


and the quantum field theoretical assumption that, letting Vμ(p) denote the Planck volume,
ϵΛ=Λc28πG,EpVμ(p),

we may define ϵc=ϵt when curvature κ=0:
ϵc=3c28πGH2EpVμ(p).


There exists a closed manifold μ such that 
    Vμ(p)=8πGEp3c2H28πGϵc.

Let μ denote a torus (as a closed, compact 2–manifold). It should be noted that any topological space homeomorphic to a torus may be considered with the same treatment. The Planck volume is then,
    Vμ(p)=2π2R(p2)2,=GRπ22c3.


    The outer radius of μ is denoted by p2. If 0<p2<<1 denoted the inner radius, then μ degenerates into a double-covered sphere with radius R, which yields an undesirable Λ. Equating with (3) yields
    R=16Epc3π(3c2H28πGϵc).

    Given observational data, whereby ϵc7.81010Jm3,
    R1.92659953451048m.

The outer radius would then be approximately 21032 light years. A space of these 'tori' could technically exist since packing densities are higher than that of spheres (one could conjecture this as a minimum possible size of the universe – assuming total size is at least 31023 times larger than the observable). So, how could this be disproven? What implications would such a large outer radius have on other areas of physics and cosmology (seems absurd that the Planck length could describe such a large region of space – in terms of radii, not volume)? I would be grateful for any help on this.

asked Jul 21, 2021 in Theoretical Physics by jday [ no revision ]

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