Equivalence principle violation in Vainshtein screened two-body systems

In massive gravity, galileon, and braneworld explanations of cosmic acceleration, force modifications are screened by nonlinear derivative self-interactions of the scalar field mediating that force. Interactions between the field of a central body (“$A$”) and an orbiting body (“$B$”) imply that body $B$ does not move as a test body in the field of body $A$ if the orbit is smaller than the Vainshtein radius of body $B$. We find through numerical solutions of the joint field at the position of $B$ that the $A$-field Laplacian is nearly perfectly screened by the $B$ self-field, whereas first derivative or net forces are reduced in a manner that scales with the mass ratio of the bodies as $(M_B/M_A{)}^{3/5}$. The latter causes mass-dependent reductions in the universal perihelion precession rate due to the fifth force, with deviations for the Earth-Moon system at the $\sim 4\%$ level. In spite of universal coupling, which preserves the microscopic equivalence principle, the motion of macroscopic screened bodies depends on their mass providing in principle a means for testing the Vainshtein mechanism.

Figure 1

Schematic illustration of the two-body problem (not drawn to scale). Physical radii are shown by the solid lines whereas the Vainshtein radii, $r_{*A}$ and $r_{*B}$ are shown in dashed lines. Boundary conditions are set at $r=L$ and $z=\pm L$ with vanishing deviations from superposition of $A$ and $B$.

Figure 2

Two-body nonlinearity source function $F[r/d,z/d]\propto N[{\varphi }_A,{\varphi }_B]$ with $A$ at (0, 0) and $B$ at $(0,-1)$ with parameters of the fiducial model (see Table ). The region interior to the bodies, within the semicircles, is not shown.

Figure 3

Laplacian-screening statistic $Q_2={\nabla }^2{\varphi }_{\Delta }/{\nabla }^2{\varphi }_A$ (left: analytic; right: numerical). The red areas indicate a large relative deviation from the superposition solution in the Laplacian of the field. In the interior of body $B$ screening of the Laplacian of body $A$ is nearly complete whereas in the exterior there is a toroidal region of mutual screening. The analytic description is in good qualitative agreement near body $B$. Parameters are the fiducial choice of Table .

Figure 4

Force ($Q_1$, top) and Laplacian ($Q_2$, bottom) screening statistics along $z=-d$ as a function of the crossover scale $r_c$ (left) and absolute mass scale $r_{gA}=2G{M}_A$ (right). Other parameters including $M_B/M_A$ are set to the fiducial choices of Table  here and in the following figures. Arrows indicate the Vainshtein scale of body $B$, $r_{*B}$. Near the body the screening statistics are independent of $r_c$ and mass scale if $r_{*B}\gg d$.

Figure 5

Screening statistics as a function of the size of body $B$ $r_{sB}$ (left) and mass ratio $M_B/M_A$ (right) as in Fig. 4. $Q_1$ and $Q_2$ are independent of $r_{sB}$ in the exterior $r>r_{sB}$ and converge to constant values for $r<r_{sB}\ll d$. Likewise for $M_B/M_A$, $Q_2$ behaves according to analytic expectations for $r\gg d$ and both $Q_1$ and $Q_2$ scale with the mass ratio in the interior $r<r_{sB}=0.1$.

Figure 6

Test of Newton’s third law or momentum conservation. If the fields of $A$ and $B$ superimpose, then the ratio of forces would be $-\sqrt{M_B/M_A}$ (red dashed line) and violate the third law (solid blue line). Numerical results ($+$ points) show that the joint solution restores the third law through violation of the superposition principle ${\varphi }_{\Delta }\ne 0$.

Figure 7

Screening statistics at the center of body $B$, $Q_1(0)$, and $Q_2(0,-d)$ for various $r_{sB}$ and $M_B/M_A$ with other parameters held fixed to fiducial values. We show numerical results as points ($\times$) as well as the fitting functions in Eqs. (54, 55).

Figure 8

The fractional accuracy of the numerical results with the fiducial choice of parameters along $z=-d$.

Figure 9

Box size and resolution dependence of screening statistics. Left: changing $\alpha$ at fixed $N$, $h$ changes the box size $L$. Results are independent of $L$ near the bodies if $L\gg r_{*B}$ (arrows). Right: changing $N$ at fixed $\alpha$ changes the resolution $h$. Results are independent of $h$ if $h\ll r_{sB}$.