Vainshtein screening in a cosmological background in the most general second-order scalar-tensor theory

A generic second-order scalar-tensor theory contains a nonlinear derivative self-interaction of the scalar degree of freedom $\varphi$ à la Galileon models, which allows for the Vainshtein screening mechanism. We investigate this effect on subhorizon scales in a cosmological background, based on the most general second-order scalar-tensor theory. Our analysis takes into account all the relevant nonlinear terms and the effect of metric perturbations consistently. We derive an explicit form of Newton’s constant, which in general is time-dependent and hence is constrained from observations, as suggested earlier. It is argued that in the most general case the inverse-square law cannot be reproduced on the smallest scales. Some applications of our results are also presented.

Figure 1

Relation between the coefficients and the short-distance solution for Case I. In the shaded region one gets a real solution whose behavior at short distances is noted.

Figure 2

$r^2{\Phi }^{\prime }$, $r^2{\Psi }^{\prime }$, and $r^2{Q}^{\prime }$ as a function of $r$. $H^{-1}=1$, $c_h^2=1.2$, ${\alpha }_1={\alpha }_2=1$, ${\alpha }_0=2$, ${\beta }_2=2$, $\mu ={10}^{-10}$. Inside the Vainshtein radius $\gamma =1$ is reproduced, but the concrete value of the radius depends on the model under consideration [see Eqs. (47, 54)].

Figure 3

Coefficients for which a real solution exists (Case II).

Figure 4

Coefficients for which a real solution exists (Case II).

Figure 5

The ratio of the circular speed $V^2$ of a test particle in the generalized model to that in Newtonian gravity, as a function of radius $r(h^{-1}\mathrm{Mpc})$. We used the Hubble parameter at present $H=100h\mathrm{km}/\mathrm{s}/\mathrm{Mpc}$ where $h=0.7$, and the velocity dispersion $\sigma =100\mathrm{km}/\mathrm{s}$. Each line corresponds to the following cases: (a) $c_h^2=1$, ${\alpha }_0=-2.0$, ${\alpha }_2=1.0$, ${\beta }_0=1.0$, ${\alpha }_1={\beta }_1={\beta }_2={\gamma }_0=0.0$. (b) $c_h^2=0.9$, ${\alpha }_0=-2.0$, ${\alpha }_1=0.1$, ${\alpha }_2=0.9$, ${\beta }_0=0.9$, ${\beta }_1=-0.05$, ${\beta }_2=0.1$, ${\gamma }_0=-0.025$. (c) $c_h^2=0.9$, ${\alpha }_0=2.0$, ${\alpha }_1=1.0$, ${\alpha }_2=0.1$, ${\beta }_0=1.0$, ${\beta }_1=-0.05$, ${\beta }_2=0.1$, ${\gamma }_0=-0.025$. Dotted and dashed lines are given by Eqs. (73, 74), respectively.