Horndeski gravity in the swampland

We investigate the implications of string swampland criteria for alternative theories of gravity. Even though this has not rigorously been proven, there is some evidence that exact de Sitter solutions with a positive cosmological constant cannot be successfully embedded into string theory, and that the low energy effective field theories containing a scalar field $\pi$ with a potential $V$ in the habitable landscape should satisfy the swampland criteria $|V^{\prime }|/V\ge c\sim \mathcal{O}(1)$. As a paradigmatic class of modified gravity theories for inflation and dark energy, we consider the extensively studied family of Horndeski Lagrangians in view of cosmological observations. Apart from a possible potential term, they contain derivative self-interactions as the Galileon and nonminimal couplings to the gravity sector. Hence, our conclusions on the Galileon sector can be also applied to many other alternative theories with scalar helicity modes containing derivative interactions such as massive gravity and generalized Proca. In the presence of such derivative terms, the dynamics of the scalar mode is substantially modified, and imposing the cosmological evolution constrained by observations places tight constraints on $c$ within the swampland conjecture.

Figure 1

The figure shows the phase map of the autonomous system for the model with ${\alpha }_3=0$ and ${\alpha }_4=0$. The trajectories of $\{x_1,x_2\}$ on the $y=0$ plane are shown for $\alpha =-0.02$ and $\lambda =0.9$. The critical points $II$ (green), $III$ (brown), $IV$ (pink), $VI$ (red), and $VII$ (blue) are highlighted. The latter represents the scalar-field-dominated epoch, whereas $IV$ represents the matter-dominated phase. The critical point $I$ (corresponding to the radiation-dominated epoch) cannot be shown in this $y=0$ slice.

Figure 2

The figure shows the phase map of the autonomous system for the model with $x_1=0$ and $x_2=0$. The trajectories of $\{x_3,x_4\}$ on the $y=0$ plane are shown for $\alpha =-0.02$ and $\lambda =0.9$. The critical point (blue) represents the scalar-field-dominated epoch with $x_4\gg x_3\gg x_1^2$.

Figure 3

The figure shows the numerical solutions for (top) $x_i$ and (bottom) ${\mathrm{\Omega }}_i$ as a function of redshift $1+z$ for the model parameter $\alpha =-0.02$ and the initial conditions (at $z=1.02\times {10}^7$) $x_1=1.66\times {10}^{-13}$, $x_2=7.02\times {10}^{-25}$, $x_3=6.51\times {10}^{-22}$. $x_4=1.54\times {10}^{-18}$, $x_5=0.99985$, and $\lambda =0.9$. One clearly sees the transition between the radiation-, matter-, and scalar-field-dominated phases throughout the cosmic evolution.

Figure 4

The figure shows the allowed parameter space ($\lambda$, $\alpha$) after comparing the upper bound on the reconstructed equation of state $w(z)$ of the 1- and $2\text{−}\sigma$ contours of Fig. 21 of [19] with our numerical solutions of the Horndeski model. Luminal propagation speed for gravitational waves is assumed, i.e., we set ${\alpha }_4=0$.

Figure 5

The figure illustrates the allowed parameter space ($\lambda$, $\alpha$) of Euclid type stage-4 experiment.