Breaking of Vainshtein screening in scalar-tensor theories beyond Horndeski

The Horndeski theory of gravity is known as the most general scalar-tensor theory with second-order field equations. Recently, it was demonstrated by Gleyzes et al. that the Horndeski theory can further be generalized in such a way that although field equations are of third order, the number of propagating degrees of freedom remains the same. We study small-scale gravity in the generalized Horndeski theory, focusing in particular on an impact of the new derivative interaction beyond Horndeski on the Vainshtein screening mechanism. In the absence of the quintic Galileon term and its generalization, we show that the new interaction does not change the qualitative behavior of gravity outside and near the source: the two metric potentials coincide, $\mathrm{\Phi }=\mathrm{\Psi }(\sim r^{-1})$, while the gravitational coupling is given by the cosmological one and hence is time dependent in general. We find, however, that the gravitational field inside the source shows a novel behavior due to the interaction beyond Horndeski: the gravitational attraction is not determined solely from the enclosed mass and two potentials do not coincide, indicating breaking of the screening mechanism.

Figure 1

Exact numerical solutions for $x$, $y$, $z$. The profile $A(r)$ is such that $A\simeq \text{const}$ for $r\lesssim 0.01\times r_{\mathrm{V}}(≕r_{\mathrm{s}})$ (i.e., $r=0.01\times r_{\mathrm{V}}$ corresponds to the surface of a constant density star), as shown by the thin solid line. We assume that the system is static, and the coefficients are given by $\mathcal{F}=2$, $\mathcal{G}=1$, ${\xi }_1=1.5$, ${\xi }_2=1$, $\eta =0.5$, $\mu =1$, $\nu =0.5$, ${\alpha }_1=1.2$, ${\alpha }_2=1$, and ${\alpha }_{*}=-0.25$. The Vainshtein mechanism works well for $r_{\mathrm{s}}<r<r_{\mathrm{V}}$, giving $y/z\simeq 1.0$. However, for $r\gg r_{\mathrm{V}}$ we find $y/z\simeq 1.4$, while for $r\ll r_{\mathrm{s}}$ we have $y/z\simeq 1.6$, and hence standard gravity is not recovered there.