The Torsion Pendulum's Contribution to Dark Energy

Bill Hamilton, Louisiana State University hamilton-at-phys.lsu.edu

This note is motivated by the recent publication[1] from the Eöt-Wash group with new experimental constraints on the nature of gravitational potential energy. Because of those constraints they infer limits to the size of proposed compact extra dimensions. Central to the Eöt-Wash experiment is a unique torsion balance. The torsion balance is frequently the instrument of choice when attempting to investigate weak forces because it can be constructed so as to have a very long period. This is another way to say that it can be modeled as a very weak spring. This in turn means that the instrument's deflection (angular deflection for the torsion balance) can be very large for a given applied force (torque). This is also the curse of using the instrument since deflections will be observed from a variety of undesired sources. The experimental challenge is to design the instrument to minimize the undesired deflections while maximizing the desired effect.

We usually think of a torsion balance as consisting of a long thin fiber supporting a dumbbell or a mass dipole. A gravitational gradient will exert a torque on this dipole and the torque can be determined either by measuring the angular deflection of the dipole or by measuring the frequency of oscillation of the pendulum and comparing it to its frequency of small oscillation about its equilibrium position. The modern torsion balance is mechanically more sophisticated and constructed to eliminate, as much as possible, undesired mass multipole moments[2,3] such as the dipole moment. The Eöt-Wash apparatus is designed to be, with the exception of small calibration masses and the mirrors used to reflect the measurement light beam, cylindrically symmetric. The detection element is a circular disk that forms the bottom of the cylinder. This detection disk is made sensitive to horizontal gravitational forces by drilling a pattern of holes in it (the Eöt-Wash group thinks of these holes as negative or missing mass). In the current experiment there are 42 holes in the detection disk, arranged in 21-fold azimuthal symmetry. Such an arrangement in isolation is insensitive to a gravitational gradient. This statement ignores the mass balancing that must be done to eliminate the mass multipole moments arising because of imperfections in construction and assembly.

A second circular plate is drilled with 42 holes in the same pattern and aligned with and placed below and close to the detector plate. The authors refer to this plate as the attractor plate. If its holes are directly aligned with the holes in the detector plate it will exert no torque on the torsion balance. However if the attractor plate is rotated about the common center axis so that the holes are displaced azimuthally its missing mass will interact with the missing mass of the detector plate and exert a torque on the balance, causing it to rotate. Rotating the attractor plate at an angular frequency $\omega$ will, because of the 21-fold symmetry, induce a primary torque at $21\omega$ on the torsion balance. There are also components of the torque at higher harmonics of $21\omega$.

From an experimental standpoint this is usually a very desirable setup. An input at a low frequency results in an output at a higher frequency. In the detection system the signal at $21\omega$ will not be contaminated by pickup from the source that drives the attractor plate at frequency $\omega$. Varying $\omega$ allows the experimenter to place the signal appropriately with respect to the free oscillation frequency of the torsion balance. For these experiments the free oscillation period was between 500 and 650 seconds.

The description to this point describes the principal attributes of most of the recent Eöt-Wash designs[4]. The new thing in this experiment is in the design of the lower attractor disk which is made of two circular disks, placed one on top of the other. The second lower disk is drilled with 21 holes with twice the diameter of each of the holes on the upper disk. The lower disk is rotated by 360/42 degrees with respect to the upper one so that the holes on the lower disk are directly below the mass between the holes on the upper disk. The thickness of the lower disk is chosen so that the missing mass of the lower disk combined with the missing mass of the upper disk would exert practically no torque on the detector plate if the inverse square law holds (emphasis by the Eöt-Wash group). Thus the experiment is designed to look for violations of the inverse square law at close distances.

I must, at this point, pay homage to those who built this and the previous experiments. A sensitive torsion balance is subject to being deflected by almost anything. Stray electric charges, magnetic impurities, unmodeled gravitational potentials; all of those are capable of causing extraneous excursions of the torsion balance. The spacing between the attractor disk and the detection disk is such that even the Casimir-Polder effect (which was only first measured in 1958 and precisely measured in 1997) has to be neutralized.

The authors parameterize[4,5] any violation of the inverse square law by adding a Yukawa term to the Newtonian potential:

$$V = - \frac{GM}{R} (1 + \alpha e^{-\frac{R}{\lambda}})$$

and assert that the torque would not cancel if $\lambda$ were less than the thickness of the upper disk which in this experiment is approximately 1 mm. They convert their measured angular responses into torques (the calibration is very elegant using a rotating external gravitational octopole) and plot the measured torques and calculated torques against the separation of the attractor and detection disks and display plots of the residuals. These residual plots must have gotten people in the lab very excited at one time. They eventually exclude any Yukawa type interaction with $\vert \alpha \vert =1$ and $\lambda > 56$ microns. These results are then used to make inferences about the allowed size of compact dimensions in some of the postulated corrections to Newtonian gravity.