# Could torus spaces (defined by Planck length) exist?

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Let $\epsilon_c$ denote critical energy density, $\epsilon_t$ energy mass density at time $t$, $c$ the speed of light, $G$ the gravitational constant, $H=\dot a/ a$ the Hubble parameter, $\Lambda$ the cosmological constant, $\ell_p$ Planck length, and $E_p$ Planck energy. From the first order Friedmann equation,
\begin{equation}
H^2=\frac{8\pi G}{3 c^2}\left(\epsilon_t+\epsilon_\Lambda\right)-\frac{\kappa c^2}{R^2_0 a},
\end{equation}
and the quantum field theoretical assumption that, letting $V_\mu\left(\ell_p\right)$ denote the Planck volume,
\begin{equation*}
\begin{split}
\epsilon_\Lambda &= \frac{\Lambda c^2}{8\pi G},\\
&\equiv \frac{E_p}{V_\mu\left(\ell_p\right)},
\end{split}
\end{equation*}
we may define $\epsilon_c=\epsilon_t$ when curvature $\kappa=0$:
\begin{equation}
\epsilon_c=\frac{3c^2}{8\pi G}H^2-\frac{E_p}{V_\mu\left(\ell_p\right)}.
\end{equation}

There exists a closed manifold $\mu$ such that
\begin{equation}
V_\mu\left(\ell_p\right)=\frac{8\pi G E_p}{3c^2 H^2-8\pi G\epsilon_c}.
\end{equation}

Let $\mu$ 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,
\begin{equation*}
\begin{split}
V_\mu\left(\ell_p\right)&=2\pi^2 R\left(\frac{\ell_p}{2}\right)^2,\\
&=\frac{G R\hslash\pi^2}{2c^3}.
\end{split}
\end{equation*}
The outer radius of $\mu$ is denoted by $\frac{\ell_p}{2}$. If $0<\frac{\ell_p}{2}<<1$ denoted the inner radius, then $\mu$ degenerates into a double-covered sphere with radius $R$, which yields an undesirable $\Lambda$. Equating with (3) yields
\begin{equation}
R=\frac{16 E_p c^3}{\hslash\pi\left(3c^2 H^2-8\pi G\epsilon_c\right)}.
\end{equation}
Given observational data, whereby $\epsilon_c\approx7.8\cdot10^{-10}J m^{-3}$,
\begin{equation*}
R\approx 1.9265995345\cdot10^{48} m.
\end{equation*}

The outer radius would then be approximately $2\cdot10^{32}$ 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 $3\cdot10^{23}$ 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

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