Dynamical systems analysis of unimodular cosmology in $D=4+d$ dimensions

Abstract

We investigate the effective four-dimensional cosmology induced by unimodular gravity in $D=4+d$ dimensions, where the internal extra-dimensional volume is encoded in a scalar degree of freedom. After dimensional reduction, we show that the resulting FLRW equations admit a natural autonomous formulation whose phase-space structure differs qualitatively from that of general relativity. In the vacuum sector, the reduced system exhibits a continuous family of finite equilibrium points, $λ=dH$, together with well-defined asymptotic Poincaré directions. In the matter sector, we focus on the five-dimensional case $d=1$ and use the reduced Bianchi relation as the consistency condition that links the ordinary matter component to the internal-volume degree of freedom. The system is then closed by adopting the minimal higher-dimensional conservation prescription, according to which matter is diluted by both the external volume and the internal-volume modulus. This leads to a reduced matter--geometry dynamics with isolated critical points and a globally organized compactified flow. Numerical examples illustrate how the internal-volume degree of freedom affects the background evolution and the global phase-space structure. The comparison with $Λ$CDM is used only as a benchmark, while a full observational analysis and more general matter--geometry exchange prescriptions are left for future work.