Dark energy fifth forces in torsion pendulum experiments

The chameleon scalar field is a matter-coupled dark energy candidate whose nonlinear self-interaction partially screens its fifth force at laboratory scales. Nevertheless, small-scale experiments such as the torsion pendulum can provide powerful constraints on chameleon models. Here we develop a simple approximation for computing chameleon fifth forces in torsion pendulum experiments such as Eöt-Wash. We show that our approximation agrees well with published constraints on the quartic chameleon, and we use it to extend these constraints to a much wider range of models. Finally, we forecast the constraints which will result from the next-generation Eöt-Wash experiment, and show that this experiment will exclude a wide range of quantum-stable models.

Figure 1

Mass scalings of various chameleon models. Laboratory constraints are most suitable for probing models with $n\lesssim -1/2$ and $n>2$, while cosmological fifth force constraints probe the models $-1/2\lesssim n<1$ in which $m_{\mathrm{eff}}$ grows most rapidly with density.

Figure 2

Density (shaded yellow region) and field profile (solid blue line) for a thin planar slab inside a gap between two thick planar slabs. The horizontal axis shows the $z$ coordinate, the distance normal to the planes, while the vertical axis shows $\rho$ and $\varphi$ in arbitrary units. Distances, densities, and field values are labeled.

Figure 3

Factor $f_{\sup }$ by which the chameleon fifth force is suppressed by a shielding foil. For a large range of chameleon parameters $\gamma$, $n$, and $\beta$, the suppression factor, shown by points on the plot, is a function of the foil thickness $\Delta z_{\mathrm{foil}}$ in Compton wavelengths. $f_{\sup }$ is well approximated by the function $\mathrm{sech}(2m_{\mathrm{eff}}{z}_{\mathrm{foil}})$ (solid line). The thin-slab calculation of $f_{\sup }$, from Refs. 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, is shown by blue plus-shaped points; the scatter at large $m_{\mathrm{eff}}\Delta z_{\mathrm{foil}}$ is due to numerical error, since $\Delta \varphi (\mathrm{\text{foil}})$ is the difference of two nearly equal numbers. Solid black circles show $f_{\sup }$ computed directly from arbitrary-precision numerical integration of the equations of motion.

Figure 4

Chameleon field for a hole in the source mass moving past another one in the test mass. The chameleon potential is $V(\varphi )=\frac{\lambda }{4!}{\varphi }^4$ with $\lambda =1$, equivalent to (5) with $n=4$ and $\gamma =1/24$; the matter coupling is $\beta =1$.

Figure 5

One-dimensional plane-parallel approximation. Top panel: A feature, such as a groove or hole, on the source mass faces the test mass. Three points, $p$, $q$, and $r$, are labeled on the test mass, and $\Delta z(p)$, $\Delta z(q)$, and $\Delta z(r)$ are respectively, the distances between each of $p$, $q$, and $r$ and the nearest point on the source slab. Bottom panel: 1Dpp approximation at $r$.

Figure 6

Geometry of disks used in Eöt-Wash torsion pendulum experiment (from Ref. 47; not to scale). Left: Current experiment, with two rows of 21 holes in each disk. Right: Next-generation experiment, with 120 radial grooves in each disk.

Figure 7

Field on the surface of the source mass, with source and test holes offset, for $V(\varphi )=\frac{\lambda }{4!}{\varphi }^4$, $\lambda =1$, $\beta =1$, and $\Delta z_{\mathrm{S}\mathrm{\text{−}}\mathrm{T}}=0.2\mathrm{mm}$. The horizontal axis shows the tangential direction, and $x=0$ coincides with the center of the test mass hole. The 1Dpp approximation agrees quite well with the 3D numerical calculation of Ref. 47.

Figure 8

Similar to Fig. 7, but with (a) a greater source hole displacement, (b) $\Delta z_{\mathrm{S}\mathrm{\text{−}}\mathrm{T}}=0.1\mathrm{mm}$, (c) $\lambda =\beta =0.1$, and (d) $n=-1$, $\gamma =1$, $\beta =1$.

Figure 9

Comparison between the 1Dpp approximation and the 3D numerical computation of Ref. 47 for a ${\varphi }^4$ chameleon with $\lambda =\beta =1$ and disk separation $\Delta z_{\mathrm{S}\mathrm{\text{−}}\mathrm{T}}=0.1\mathrm{mm}$.

Figure 10

Comparison between the 1Dpp approximation and the 3D numerical computation of Ref. 47 for a ${\varphi }^4$ chameleon with $\lambda =1/10$ and $\beta =1$, as a function of $\Delta z_{\mathrm{S}\mathrm{\text{−}}\mathrm{T}}$.

Figure 11

1Dpp excluded regions (shaded light green). The black shaded regions identify models which are linear inside the source and test masses; these are excluded unless $\beta$ is very small. Top panel: Quartic chameleon, $V(\varphi )=\frac{\lambda }{4!}{\varphi }^4$. The long-dashed blue line shows the constraints of Refs. 38, 50. Models above the short-dashed purple line have large quantum corrections. Bottom panel: Inverse power law chameleon, $V(\varphi )=M_{\Lambda }^4(1+\gamma M_{\Lambda }/\varphi )$. All models shown have small quantum corrections.

Figure 12

Excluded region in the $\beta$, $n$ plane, for $\gamma =1$ (green shaded region), $\gamma =0.1$ (solid red line), and $\gamma =10$ (dashed blue line). Top panel: Positive $n$. For $\gamma =1$, models to the right of the purple dashed line have large quantum corrections; in the $\gamma =0.1$ case, all models shown pass the quantum stability test. For $\gamma =10$ and $n\ge 3$ no models are excluded. Bottom panel: Negative $n$. All models shown pass the quantum stability test.

Figure 13

Field on the surface of a wedge on the source disk, for $\lambda =100$, $\beta =10$, and $\Delta z_{\mathrm{S}\mathrm{\text{−}}\mathrm{T}}=0.1\mathrm{mm}$.

Figure 14

Torque as a function of rotation angle for the next-generation Eöt-Wash apparatus, assuming $\lambda =\beta =1$ and $\Delta z_{\mathrm{S}\mathrm{\text{−}}\mathrm{T}}=0.1\mathrm{mm}$. The 1Dpp approximation overestimates the numerical computation of Ref. 47 by $\sim 50\%$.

Figure 15

Forecast constraints from the next-generation Eöt-Wash apparatus. The light green shaded region is excluded; models in the black region are linear inside the disks. Top panel: $n=4$. Models inside the long-dashed blue curve are excluded by the current Eöt-Wash experiment, while models above the short-dashed purple line have large quantum corrections. Bottom panel: $n=-1$. Models below the short-dashed purple curve have large quantum corrections.

Figure 16

Forecast constraints for $\gamma =1$ from the next-generation Eöt-Wash apparatus. The light green shaded region is excluded; models in the black region are linear inside the disks. Top panel: Positive $n$. Models above the short-dashed curve have large quantum corrections. Bottom panel: Negative $n$. Models below the curve have large quantum corrections.

Figure 17

Dependence of 1Dpp forecasts on the torque bound (previously assumed to be ${\tau }_{\max }=0.003\mathrm{fN}·\mathrm{m}$) and the distance probed (previously assumed to be $\Delta z_{\mathrm{S}\mathrm{\text{−}}\mathrm{T}}=100\mu \mathrm{m}$).