A four-dimensional \Lambda CDM-type cosmological model induced from higher dimensions using a kinematical constraint

A class of cosmological solutions of higher dimensional Einstein field equations with the energy-momentum tensor of a homogeneous, isotropic fluid as the source are considered with an anisotropic metric that includes the direct sum of a 3-dimensional (physical, flat) external space metric and an $n$ -dimensional (compact, flat) internal space metric. A simple kinematical constraint is postulated that correlates the expansion rates of the external and internal spaces in terms of a real parameter $\lambda $ . A specific solution for which both the external and internal spaces expand at different rates is given analytically for $n=3$ . Assuming that the internal dimensions were at Planck length scales when the external space starts with a Big Bang ( $t=0$ ), they expand only 1.49 times and stay at Planck length scales even in the present age of the universe (13.7 Gyr). The effective four dimensional universe would exhibit a behavior consistent with our current understanding of the observed universe. It would start in a stiff fluid dominated phase and evolve through radiation dominated and pressureless matter dominated phases, eventually going into a de Sitter phase at late times.