The Friedmann integrals and physical vacuum in the framework of macroscopic extra dimensions

The structure and origin of the Friedmann integrals are analyzed within the framework of large extra dimensions proposed by Arkani-Hamed et all. (1998). It is demonstrated that the integrals might emerge from extra-dimension physics and reveal its signature. In the case of two extra dimensions, the integrals are expressed via the product $M_{\ast }R$ of the only fundamental energy $M_{\ast }$ and the size $R$ of the extra dimensions. The cosmic vacuum density turns out to be $\sim R^4$, in this case. The effective cut-off in the spectrum of the quantum zero-point oscillations at the frequency ${\omega }_V\sim 1\mathrm{/}R$ is assumed to be associated with the epoch when the Hubble radius becomes larger than $R$. This occurs at temperatures $\sim M_{\ast }$ in the first three picoseconds of the cosmic expansion.