Background-Induced Forces from Quadratically Coupled Ultralight Dark Matter

Abstract: Quadratically coupled ultralight scalar dark matter behaves as a coherent classical field whose interactions with matter can induce a composition-dependent force through the dark matter background. We present a complete calculation of this background-induced force beyond the spherically symmetric approximation. Using a partial-wave treatment of dark-matter scattering, we determine its angular dependence and derive an analytic description valid even when the dark-matter wavelength is much smaller than the Earth's radius. We show for the first time that Earth screening generates a characteristic frequency-band structure, splitting the signal into multiple sidebands that provide a distinctive experimental signature. We further show that the relative amplitudes of these sidebands vary annually due to the Earth's motion through the dark-matter halo, enabling the construction of a complete signal template. As an application of these results, we re-evaluate constraints from the MICROSCOPE mission, which currently provides the strongest laboratory limits on equivalence-principle violations from ultralight dark matter. We further show that proposed space-based equivalence-principle experiments, such as Galileo Galilei and STE-QUEST, can significantly enhance their sensitivity to ultralight scalar dark matter by incorporating the full frequency-band information.