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Experiments with levitated force sensor challenge theories of dark energy

Nature Physics volume 18, pages 1181–1185 (2022)Cite this article

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Abstract

Various theories have been proposed for the nature of dark energy. It is therefore essential to directly verify or rule out these theories by experiment. Despite substantial efforts in astrophysical observations and laboratory experiments, previous tests are not sufficiently accurate to provide decisive conclusions as to the validity of these theories. One particularly compelling idea—the so-called chameleon theory—describes an ultra-light scalar field that couples to normal-matter fields and leaves measurable effects that are not explained by the four fundamental interactions, a so-called fifth force. Here, using a diamagnetically levitated force sensor, we extend previous tests by nearly two orders of magnitude, covering the entire physically plausible parameter space of cosmologically viable chameleon models. We find no evidence for the fifth force predicted by chameleon models. Our results decisively rule out the basic chameleon model as a candidate for dark energy and demonstrate the robustness of laboratory experiments for unveiling the nature of dark energy in the future. The methodology developed here could be further applied to study a broad range of fundamental physics.

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Fig. 1: Schematic of experiment.
Fig. 2: Experimental data.
Fig. 3: Exclusion of the chameleon field.

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Data availability

Source data for all figures in the main text and extended data are provided with this paper. The raw simulation data used to produce Fig. 3 (in vtk format, about 1 TB in size) are available from the corresponding authors upon reasonable request. Source data are provided with this paper.

Code availability

The numerical code that supports the plots within this paper is available from the corresponding authors upon reasonable request.

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Acknowledgements

We thank D. Wu for the measurement of material magnetic susceptibility and X. Rong for helpful discussions. The silicon nitride window was fabricated in the micro-fabrication centre of the National Laboratory of Solid State Microstructures, and the numerical calculations were performed on the computing facilities at the High Performance Computing Center of Nanjing University. This work was supported by the Chinese Academy of Sciences (grants no. XDC07000000 and no. QYZDY-SSW-SLH004), the National Key R&D Program of China (grant no. 2018YFA0306600), the National Natural Science Foundation of China (grants no. 12075115, no. 12075116, no. 11890702 and no. 12150011), the Anhui Initiative in Quantum Information Technologies (grant no. AHY050000) and the Hefei Comprehensive National Science Center.

Author information

Author notes

  1. These authors contributed equally: Peiran Yin, Rui Li, Chengjiang Yin.

Authors and Affiliations

  1. National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, China

    Peiran Yin, Xiang Bian, Han Xie & Pu Huang

  2. CAS Key Laboratory of Microscale Magnetic Resonance and School of Physical Sciences, University of Science and Technology of China, Hefei, China

    Peiran Yin, Rui Li, Chang-Kui Duan & Jiangfeng Du

  3. CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, China

    Rui Li, Chang-Kui Duan & Jiangfeng Du

  4. School of Astronomy and Space Science, Nanjing University, Nanjing, China

    Chengjiang Yin, Xiangyu Xu & Jian-hua He

  5. Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing, China

    Chengjiang Yin, Xiangyu Xu & Jian-hua He

  6. Hefei National Laboratory, Hefei, China

    Jiangfeng Du

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Contributions

All authors contributed to the development and writing of this paper. P.H., J.H. and J.D. conceived the general idea. R.L., P.H., P.Y. and J.D. designed and built the experimental setup. P.Y., X.B. and H.X. performed the measurements. C.Y., X.X. and J.H. performed the theoretical calculations. J.H., P.H., J.D., P.Y., C.Y. and C.-K.D. wrote the manuscript. P.H., J.H. and J.D. supervised the research.

Corresponding authors

Correspondence to Pu Huang, Jian-hua He or Jiangfeng Du.

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Extended data

Extended Data Fig. 1 Numerical calculation of the chameleon field in our experiments.

a, Vertical cross section of the non-uniformly refined mesh used for FEM calculation. Each point represents a degree of freedom (DoF) with colors showing the matter density at the position of the point. The inset shows the grid around the electrical shield, where the finest resolution is about 0.25 μm. b, The distribution of the chameleon field ϕ around the test mass of the force sensor with (left half, phase angle θ = 0) and without (right half, θ = π/8) the source masses. Here θ = 0 is the phase illustrated by Fig. 1b in the main text. The horizontal black dashed line indicates the interface between air and vacuum. c, The distribution of ϕ along the central z-axis at different phases from θ = 0 (blue curves) to θ = π/8 (red curves).

Source data

Extended Data Fig. 2 Normalized phase functions of ‘fifth force’ felt by the force sensor.

The markers are the numerical calculations of the normalized variation of the chameleon force f(θ) under different phases (θ = 0 refers to the phase shown in Fig. 1b in the main text and rotates counter-clockwise). Curves are interpolated using quadratic splines. Different colors are for models with different Compton wavelengths in the air. These phase functions are close to each other and the difference between their power spectral density (PSD) magnitude at resonance frequency ≤0.5%.

Source data

Extended Data Fig. 3 Frequency stability of the force sensor and the line width of source masses.

a, The measured distribution of the resonant frequency drift δω0/2π. A single measurement of frequency is obtained from the fitted PSD of 4000 s. The measurement is fitted to a Gaussian distribution \(P\left( {\delta \omega _0} \right)\sim {{{\mathrm{e}}}}^{ - \frac{{\left( {\delta \omega _0} \right)^2}}{{2\sigma ^2}}}\), indicated by the red line. Here, we obtain the resonant frequency \(\omega _0/2\pi = 10.877\) Hz; the uncertainty of resonance frequency is given by \(\sigma /2\pi = 0.91\) mHz, with standard error of fitting 0.06 mHz. b, The measured i(ω) as a function of the frequency ω/2π. The blue dots indicate the measured laser intensity i(ω) for every 0.1 μHz. And the red line shows the calculated i(ω) of an ideal square wave signal with the same amplitude and period, they match with each other very well with errors smaller than the line width in the figure. The maximum of i(ω) is \(i(\omega )_{{{{\mathrm{max}}}}} = i\left( {\omega _{{{{\mathrm{dri}}}}}} \right) = 0.317\), corresponding to the measured drive frequency \(\omega _{{{{\mathrm{dri}}}}}/2\pi = 10.8759249\) Hz.

Source data

Extended Data Fig. 4 Thermal noise calibration.

a, The measured PSD of the voltage of photodiode at pressure 4×10−2 mbar. The dots represent the averaged PSD, the error bars show the standard deviation. And the red line is a Lorentz fitting to the dots, the corresponding line width is γ/2π = 0.079 Hz with standard error of fitting 0.004 Hz. b, The relative uncertainty of effective temperature as a function of the measurement time t. The dots with error bar represent the results from the measurement and the blue solid line is a fit with function \(f\left( x \right) = \sqrt {2/px} + q\). The grey dashed line shows the best fitted value of q, which is q = 0.075 with standard error of fitting 0.004.

Source data

Extended Data Table 1 Experiment parameters and deviation
Full size table

Supplementary information

Supplementary information

Supplementary Figs. 1–8, discussion and Tables 1 and 2.

Source data

Source Data Fig. 1

The field ϕ along the central z axis at two different rotation phases (Fig. 1c).

Source Data Fig. 2

Experiment data including the PSD (Fig. 2a), displacement response (Fig. 2b) and upper bound of the ‘fifth force’ (Fig. 2c).

Source Data Fig. 3

Acceleration of chameleon field for different parameter sets.

Source Data Extended Data Fig. 1

Chameleon field along the z axis for different phases.

Source Data Extended Data Fig. 2

Normalized phase functions for different parameter sets.

Source Data Extended Data Fig. 3

Measured distribution of the resonant frequency drift and the normalized laser intensity.

Source Data Extended Data Fig. 4

Thermal noise calibration of the force sensor at medium pressure.

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Yin, P., Li, R., Yin, C. et al. Experiments with levitated force sensor challenge theories of dark energy. Nat. Phys. 18, 1181–1185 (2022). https://doi.org/10.1038/s41567-022-01706-9

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