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, ℓp 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(ℓp) denote the Planck volume,
ϵΛ=Λc28πG,≡EpVμ(ℓp),
we may define
ϵc=ϵt when curvature
κ=0:
ϵc=3c28πGH2−EpVμ(ℓp).
There exists a closed manifold μ such that
Vμ(ℓp)=8πGEp3c2H2−8πGϵc.
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(ℓp)=2π2R(ℓp2)2,=GRℏπ22c3.
The outer radius of
μ is denoted by
ℓp2. If
0<ℓp2<<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=16Epc3ℏπ(3c2H2−8πGϵc).
Given observational data, whereby
ϵc≈7.8⋅10−10Jm−3,
R≈1.9265995345⋅1048m.
The outer radius would then be approximately 2⋅1032 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 3⋅1023 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.