Testing long-distance modifications of gravity to 100 astronomical units

There are very few direct experimental tests of the inverse square law of gravity at distances comparable to the scale of the Solar System and beyond. Here we describe a possible space mission optimized to test the inverse square law at a scale of up to 100 AU. For example, sensitivity to a Yukawa correction with a strength of $10^{-7}$ times gravity and length scale of 100 AU is within reach, improving the current state of the art by over two orders of magnitude. This experiment would extend our understanding of gravity to the largest scale that can be reached with a direct probe using known technology. This would provide a powerful test of long-distance modifications of gravity including many theories motivated by dark matter or dark energy.

Figure 1

Existing $2\sigma$ experimental limits on new Yukawa forces with strength $|\alpha |$ relative to gravity as a function of the scale $\lambda$. The grey region, adapted from [1], is the current state of the art. The dashed line is the size of the “Pioneer anomaly,” which can be interpreted as a limit set by the trajectory of the Pioneer spacecraft. The dotted curve corresponds to the expected sensitivity of the experiment proposed here.

Figure 2

Conceptual view of the drag-free (DF) and relay spacecraft. The figure is not to scale, and the DF craft is shown in cross section. The DF craft uses microthrusters to remain centered on the free-falling “proof mass” (PM) while rotating in a plane perpendicular to the Sun to reduce thermal and gravitational backgrounds. The relay craft carries a high-gain antenna, always pointing towards Earth, for DSN ranging and communication. Communication and ranging between the two spacecraft are performed by small omnidirectional transponders.

Figure 3

Cross-sectional diagram of the rotating shell with a layer of vacuum insulation. The surfaces facing the vacuum are assumed to have emissivity $\epsilon$. While the material of the shell rotates at $\omega$, the direction of the inner and outer temperature differences $\mathrm{\Delta }T$ is fixed relative to the Sun.

Figure 4

Coordinate system used in Sec. 3d for the rotating shell. $\theta$ is the misalignment angle between the plane of rotation and the direction to the Sun.

Figure 5

Effect of fit parameters on experimental sensitivity to a new Yukawa force. The lowest curve results from fixing all parameters except $G{M}_{\text{Sun}}$ and $\alpha$; from bottom to top we add as free parameters the initial position and velocity and transverse acceleration, mass of the Kuiper belt, and shape of the Kuiper belt as described by its radius and offset from the ecliptic plane.

Figure 6

Expected $2\sigma$ exclusion limit on the strength $\alpha$ of a new Yukawa-type force with range $\lambda$. The blue solid curve is for conservative 1 m ranging precision, while the red dashed curve is an optimistic experiment with 10 cm ranging as the limiting uncertainty. The shaded region is excluded by planetary tests of Kepler’s third law.

Figure 7

Expected $2\sigma$ lower limit on the graviton Compton wavelength $m^{-1}$ of a new Vainshtein-type power-law contribution to the gravitational potential. For a particular theory with index $\nu$, the experiment will be sensitive to the region below the curve. The vertical lines at $\nu =\frac{1}{2}$ and $\nu =2$ correspond to DGP gravity and ghost-free ${\mathrm{\Lambda }}_3$ massive gravity respectively, and the horizontal line indicates the present-day Hubble scale which is the scale of interest for models that attempt to address the cosmological constant problem. The solid blue curve is for 1 m ranging precision, while the dashed red curve is an optimistic scenario with 10 cm ranging as the limiting uncertainty. The shaded area is excluded by lunar laser ranging.