Perturbative analysis of singularity-free cosmological solutions in unimodular Kaluza-Klein theory

TL;DR

This work derives an effective four-dimensional theory from unimodular five-dimensional Kaluza-Klein gravity, yielding a Brans-Dicke–like scalar coupled to gravity without an explicit Ricci scalar term. By imposing a flat FLRW background and a constant Ricci scalar, the authors obtain a non-singular bouncing cosmology and analyze perturbations in the presence of matter: tensor modes can remain finite and, in some cases, decouple from instabilities, while scalar perturbations are only sketched. The study shows that matter content allows scalar-field configurations that avoid sign changes of the extra scalar, which previously caused instabilities in the vacuum case, suggesting a route to stable, singularity-free cosmologies within this framework. A complete, quantitative perturbative analysis and potential observational signatures are identified as directions for future work.

Abstract

The unimodular version of the Kaluza-Klein theory is briefly recalled, and its projection on the $4$-dimensional spacetime is constructed. Imposing unimodularity condition on the $5$-dimensional Kaluza-Klein metric, det$g_{AB}=1$ is equivalent with introducing cosmological term in Einstein's equations in $4$ dimensions, and with scalar field of the Brans-Dicke type. Singularity-free cosmological solutions with scalar field and with matter sources are constructed, and their basic properties analyzed, along the results obtained in previous publications. In the present paper, attention is focussed on the perturbative analysis of cosmological solutions, providing a clue concerning their stability against small fluctuations.

Figures (4)

• Figure 1: Evolution of the tensorial mode for Eq. (\ref{['gw-p-p']}), with $A = 0$ and $B = 1$, with the initial conditions at $t = - 5$, $h(t_0) = h'(t_0) = 10^{-6}$. The parameters $k$ and $a_0$ were fixed equal to 1.
• Figure 2: Evolution of the tensorial mode for Eq. (\ref{['gw-p-p']}), with $A = 0.5$ and $B = 1$, with the initial conditions at $t = - 5$, $h(t_0) = h'(t_0) = 10^{-6}$. The parameters $k$ and $a_0$ were fixed equal to 1.
• Figure 3: Evolution of the tensorial mode for Eq. (\ref{['gw-p-p']}), with $A = 0$ and $B = 1$, with the initial conditions at $t = - 5$, $h(t_0) = h'(t_0) = 10^{-6}$. It has been choosen $k = 30$ and $a_0 = 1$. Oscillations appears near the bounce.
• Figure 4: Evolution of the tensorial mode for Eq. (\ref{['gw-dw-1']}), with $A = 0$ and $B = 1$, with the initial conditions at $t = - 5$, $h(t_0) = h'(t_0) = 10^{-6}$. The parameters $k$ and $a_0$ were fixed equal to 1.