New scaling solutions in cubic Horndeski theories

We propose a viable dark energy scenario in the presence of cubic Horndeski interactions and a standard scalar-field kinetic term with two exponential potentials. We show the existence of new scaling solutions along which the cubic coupling $G_3$ provides an important contribution to the field density that scales in the same way as the background fluid density. The solutions finally exit to the epoch of cosmic acceleration driven by a scalar-field dominated fixed point arising from the second exponential potential. We clarify the viable parameter space in which all the theoretically consistent conditions including those for the absence of ghost and Laplacian instabilities are satisfied on scaling and scalar-field dominated critical points. In comparison to quintessence with the same scalar potential, we find that the cubic coupling gives rise to some novel features: (i) the allowed model parameter space is wider in that a steeper potential can drive the cosmic acceleration; (ii) the dark energy equation of state $w_{\varphi }$ today can be closer to $-1$ relative to quintessence; (iii) even if the density associated with the cubic coupling dominates over the standard field density in the scaling era, the former contribution tends to be suppressed at low redshifts. We also compute quantities associated with the growth of matter perturbations and weak lensing potentials under the quasi-static approximation in the subhorizon limit and show that the cubic coupling leads to the modified evolution of perturbations which can be distinguished from quintessence.

Figure 1

Viable model parameter spaces (light blue) in the $(\lambda ,A)$ plane for the two cases: (i) ${\beta }_1=100$, ${\beta }_2=0.7$ (left) and (ii) ${\beta }_1=100$, ${\beta }_2=2.5$ (right). Each boundary is obtained by using the conditions (3.8, 3.9, 3.10) for points (a1) and (a2) as well as the conditions (3.16, 3.17, 3.18, 3.19, 3.20, 3.21) for point (b). The observational bounds (3.11) and (3.12) are also plotted, together with the region ${\mathrm{\Omega }}_{\varphi }^{(r)}<{10}^{-3}$. The labels M1, M2, M3, and M4 correspond to the G3 models presented in Table 2.

Figure 2

(Left) Evolution of the total fluid density ${\rho }_m+{\rho }_r$ (blue, solid line) and the scalar-field density ${\rho }_{\varphi }$ for M1 (orange, dashed line) and QE1 (green, dot-dashed line). The model parameters for M1 and QE1 are given in Table 2. The ICs of $x,y_1$ are chosen to be close to those of critical point (a), with $y_2$, ${\mathrm{\Omega }}_r$ realizing today’s density parameters ${\mathrm{\Omega }}_{\varphi }^{(0)}=0.68$ and ${\mathrm{\Omega }}_r^{(0)}={10}^{-4}$. (Right) Evolution of ${\rho }_m+{\rho }_r$ and ${\rho }_{\varphi }$ for the same model parameters as those in the left, but with different ICs: $x=0.015$, $y_1=0.04$ for M1 and $x=0.015$, $y_1=0.1$ for QE1.

Figure 3

Evolution of the propagation speed squared $c_s^2$ (solid line) and the kinetic term $Q_s$ (dotted line) versus $z+1$ for the model M1.

Figure 4

Evolution of ${\mathrm{\Omega }}_m$ (blue, dotted line), ${\mathrm{\Omega }}_r$ (orange, dot-dashed line), ${\mathrm{\Omega }}_{G_2}$ (red dashed line) and ${\mathrm{\Omega }}_{G_3}$ (green solid line) versus $z+1$ for the model M2. Note that ${\mathrm{\Omega }}_{G_2}$ and ${\mathrm{\Omega }}_{G_3}$ are the density parameters arising from the field Lagrangians $G_2$ and $-G_3\square \varphi$, respectively.

Figure 5

Evolution of ${\rho }_{\varphi }$ for the model M4 (orange, dashed line) and for QE2 (green, dot-dashed line) versus $z+1$, together with the background matter density ${\rho }_m+{\rho }_r$ (blue, solid line).

Figure 6

Variation of $w_{\varphi }$ versus $z+1$ for the models: QE1 (blue dashed line), M1 (dot-dashed orange line), M2 (green solid line), M3 (dotted black line), and M4 (long dashed red line).

Figure 7

Evolution of the quantity $\mu -1$ versus $z+1$ for the models M1 (orange, dot-dashed line), M2 (green, solid line), M3 (black, dotted line), and M4 (red, long dashed line).