On the emergence of the ${\bf\Lambda}$CDM model from self-interacting Brans-Dicke theory in ${\bf d= 5}$

We investigate whether a self-interacting Brans-Dicke theory in $d=5$ without matter and with a time-dependent metric can describe, after dimensional reduction to $d=4$, the FLRW model with accelerated expansion and non-relativistic matter. By rewriting the effective 4-dimensional theory as an autonomous three-dimensional dynamical system and studying its critical points, we show that the $\Lambda$CDM cosmology cannot emerge from such a model. This result suggests that a richer structure in $d=5$ may be needed to obtain the accelerated expansion as well as the matter content of the 4-dimensional universe.