Dynamical system analysis at background and perturbation levels: Quintessence in severe disadvantage comparing to $\mathrm{\Lambda }\mathrm{CDM}$

We perform for the first time a dynamical system analysis of both the background and perturbation equations, of $\mathrm{\Lambda }\mathrm{CDM}$ cosmology, and quintessence scenario with an exponential potential. In the former case the perturbations do not change the stability of the late-time attractor of the background equations, and the system still results in the dark-energy-dominated, de Sitter solution, having passed from the correct dark-matter era with $\gamma \approx 6/11$. However, in the case of quintessence the incorporation of perturbations changes the stability and properties of the background evolution, and the only conditionally stable points present an exponentially increasing matter clustering, not favored by observation; thus, this situation is not physically interesting. This result is a severe disadvantage of quintessence cosmology compared to the $\mathrm{\Lambda }\mathrm{CDM}$ paradigm.

Figure 1

The phase-space diagram for $\mathrm{\Lambda }\mathrm{CDM}$ cosmology, at both background and perturbation levels. At late times the system is attracted by the de Sitter point $P_3$. The thick line is the curve (18), which coincides with the unstable manifold of the matter-dominated solution $P_4$, and which for ${\mathrm{\Omega }}_m$ close to 1 gives $\gamma =\frac{6}{11}$ analytically as expected (see text).