pith. sign in

arxiv: 2606.28491 · v1 · pith:OATRT4SAnew · submitted 2026-06-26 · ✦ hep-ph · gr-qc

A directional force template for quadratically coupled ultralight dark matter

Dawid Brzeminski , Aaron Pierce This is my paper

Pith reviewed 2026-06-30 01:13 UTC · model grok-4.3

classification ✦ hep-ph gr-qc
keywords ultralight dark matterquadratic couplingsequivalence principleMICROSCOPEmultipole expansionanisotropic forcescalar dark matterspace-based tests
0
0 comments X

The pith

Organizing the non-radial force from quadratically coupled ultralight scalar dark matter into multipoles yields a new template that could improve MICROSCOPE sensitivity by more than an order of magnitude above 10^{-9} eV.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Quadratic couplings between ultralight scalar dark matter and Standard Model fields distort the dark matter profile around Earth and induce a composition-dependent non-radial force. This force is suppressed near the surface but remains accessible to space-based tests. The authors organize the force into radial and polar multipole coefficients, project the result onto the MICROSCOPE measurement axis, and recast the existing radial-force bound using only the overlapping signal component. They then estimate the additional reach available from the non-overlapping component. A reader would care because the estimate shows a concrete path to stronger limits on this class of dark matter interactions without new hardware.

Core claim

The paper develops a directional force template by decomposing the anisotropic force into radial and polar multipole coefficients and projecting those coefficients onto the MICROSCOPE satellite's measurement axis. This template is applied to recast the published MICROSCOPE constraint on the overlapping signal and to quantify the sensitivity gain from an analysis that also uses the non-overlapping component, which exceeds an order of magnitude for dark matter masses greater than or equal to 10^{-9} eV.

What carries the argument

The directional force template formed by organizing the force into radial and polar multipole coefficients and projecting onto the MICROSCOPE measurement axis.

If this is right

  • The radial-force recast supplies a conservative, immediately usable bound on the quadratic couplings.
  • For masses above 10^{-9} eV the anisotropic regime opens an additional signal channel that is suppressed at the surface but visible from orbit.
  • An analysis that incorporates the full non-overlapping signal can improve sensitivity by more than an order of magnitude.
  • The template accounts for the composition dependence and the geometric suppression near Earth while remaining applicable to space-based equivalence-principle tests.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same multipole-projection method could be adapted to other orbiting equivalence-principle experiments that record directional data.
  • If the template holds, ground-based tests that average over many orientations would remain less sensitive than space-based ones in this mass range.
  • Numerical modeling of the field profile for specific quadratic operators would provide a direct check on the multipole decomposition used here.

Load-bearing premise

The non-radial force components can be cleanly separated into multipole coefficients whose projection onto the MICROSCOPE axis produces an independent signal not already captured by the radial template.

What would settle it

A detailed simulation or reanalysis of the MICROSCOPE data showing that the polar multipole projections are linearly dependent on the radial template would eliminate the estimated sensitivity gain.

read the original abstract

Quadratic couplings between ultralight scalar dark matter and Standard Model fields can produce a distorted dark-matter field profile around the Earth. Gradients in the field induce a non-radial, composition-dependent force that can be suppressed at the Earth's surface while remaining accessible to space-based experiments. The MICROSCOPE satellite, which searched for violations of the equivalence principle, can constrain this force, but existing results assume a radial force, and they cannot be directly translated into an optimal bound in the anisotropic regime. We develop a signal template for this regime by organizing the force into radial and polar multipole coefficients and projecting the force onto the MICROSCOPE measurement axis. We use this template to recast the published MICROSCOPE constraint using the component of the signal that overlaps with the radial-force template. We estimate the sensitivity gain that would be provided by an analysis utilizing the additional non-overlapping signal. Such an analysis could improve sensitivity to the couplings of quadratically coupled scalar dark matter by more than an order of magnitude relative to the radial-force recast for dark matter masses $\gtrsim 10^{-9}$ eV.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a directional force template for the non-radial, composition-dependent force induced by quadratically coupled ultralight scalar dark matter around Earth. The force is decomposed into radial and polar multipole coefficients, projected onto the MICROSCOPE satellite's measurement axis, and used to recast the published MICROSCOPE equivalence-principle constraint with the overlapping component of the radial template. The authors estimate that an analysis incorporating the additional non-overlapping signal component would improve sensitivity to the quadratic couplings by more than an order of magnitude for dark-matter masses ≳ 10^{-9} eV.

Significance. If the claimed statistical independence of the projected polar-multipole component holds after orbital averaging and axis projection, the work supplies a concrete method to extract additional constraining power from existing MICROSCOPE data in the anisotropic regime. The multipole-based template construction itself is a useful technical step for handling direction-dependent signals in satellite equivalence-principle tests.

major comments (2)
  1. [projection and template construction section] The load-bearing assumption that the polar-multipole projection is statistically independent of the radial template after projection onto the MICROSCOPE axis is not accompanied by an explicit overlap integral or covariance matrix. Without this calculation (presumably in the section deriving the projected templates), the claimed order-of-magnitude sensitivity gain cannot be verified and may be reduced if residual inner product remains after orbital averaging.
  2. [sensitivity estimate section] The sensitivity-gain estimate for m ≳ 10^{-9} eV is presented as a direct consequence of the non-overlapping fraction, yet no quantitative breakdown of that fraction (or its mass dependence) is supplied. The estimate therefore rests on an unshown numerical result rather than a demonstrated derivation.
minor comments (2)
  1. [abstract] The abstract refers to 'radial and polar multipole coefficients' without stating the maximum multipole order retained or the truncation criterion used.
  2. [introduction] Notation for the quadratic coupling constants and the dark-matter field amplitude should be defined once at first use and used consistently thereafter.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the need for explicit verification of the template projections and sensitivity estimates. We address each major comment below.

read point-by-point responses

  1. Referee: [projection and template construction section] The load-bearing assumption that the polar-multipole projection is statistically independent of the radial template after projection onto the MICROSCOPE axis is not accompanied by an explicit overlap integral or covariance matrix. Without this calculation (presumably in the section deriving the projected templates), the claimed order-of-magnitude sensitivity gain cannot be verified and may be reduced if residual inner product remains after orbital averaging.

    Authors: We agree that an explicit overlap integral and covariance matrix would make the independence claim fully verifiable. The multipole decomposition ensures orthogonality of radial and polar components in the unprojected basis; after orbital averaging and projection onto the MICROSCOPE axis, the inner product is suppressed. In the revised manuscript we will add the explicit numerical evaluation of the overlap integral (and associated covariance) in the projected-templates section, confirming that the residual is consistent with zero within integration tolerances. This will directly support the claimed sensitivity gain. revision: yes

  2. Referee: [sensitivity estimate section] The sensitivity-gain estimate for m ≳ 10^{-9} eV is presented as a direct consequence of the non-overlapping fraction, yet no quantitative breakdown of that fraction (or its mass dependence) is supplied. The estimate therefore rests on an unshown numerical result rather than a demonstrated derivation.

    Authors: The non-overlapping fraction is obtained by numerical integration of the projected signal power. We will include in the revised sensitivity section both a quantitative plot of the mass-dependent non-overlapping fraction and the explicit scaling relation (square-root of additional power) used to arrive at the order-of-magnitude improvement for m ≳ 10^{-9} eV, rendering the estimate fully traceable from the template construction. revision: yes

Circularity Check

0 steps flagged

No circularity: template construction and overlap estimate are methodological, not self-referential.

full rationale

The paper develops a directional force template by explicitly organizing the quadratic-coupling gradient into radial and polar multipole coefficients, then projects onto the MICROSCOPE axis. It recasts the published constraint using only the overlapping component and separately estimates the gain from the non-overlapping remainder. No equations, fitted parameters, or self-citations are supplied that reduce the claimed sensitivity improvement to a definition or prior fit by construction. The separation into multipoles is presented as an organizing step whose independence from the radial template is asserted on physical grounds (suppression at Earth's surface, satellite geometry), not derived from the result itself. This is a standard self-contained modeling exercise; the load-bearing assumption about non-overlap is an empirical modeling choice open to external verification rather than a circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; all ledger entries are therefore empty.

pith-pipeline@v0.9.1-grok · 5721 in / 1095 out tokens · 19834 ms · 2026-06-30T01:13:58.220917+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

32 extracted references · 29 canonical work pages · 8 internal anchors

  1. [1]

    K. A. Olive and M. Pospelov,Environmental dependence of masses and coupling constants,Phys. Rev. D77(2008) 043524, [arXiv:0709.3825]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  2. [2]

    Searching for dilaton dark matter with atomic clocks

    A. Arvanitaki, J. Huang, and K. Van Tilburg,Searching for dilaton dark matter with atomic clocks, Phys. Rev. D91(2015), no. 1 015015, [arXiv:1405.2925]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  3. [3]

    Brzeminski, Z

    D. Brzeminski, Z. Chacko, A. Dev, and A. Hook,Time-varying fine structure constant from naturally ultralight dark matter,Phys. Rev. D104(2021), no. 7 075019, [arXiv:2012.02787]

    work page arXiv 2021

  4. [4]

    Banerjee, G

    A. Banerjee, G. Perez, M. Safronova, I. Savoray, and A. Shalit,The phenomenology of quadratically coupled ultra light dark matter,JHEP10(2023) 042, [arXiv:2211.05174]

    work page arXiv 2023

  5. [5]

    Delaunay, M

    C. Delaunay, M. Geller, Z. Heller-Algazi, G. Perez, and K. Springmann,Natural ultralight dark matter: The quadratic twin,Phys. Rev. D113(2026), no. 3 035011, [arXiv:2507.12514]

    work page arXiv 2026

  6. [6]

    A. Hees, O. Minazzoli, E. Savalle, Y. V. Stadnik, and P. Wolf,Violation of the equivalence principle from light scalar dark matter,Phys. Rev. D98(2018), no. 6 064051, [arXiv:1807.04512]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  7. [7]

    Bouley, P

    T. Bouley, P. Sørensen, and T.-T. Yu,Constraints on ultralight scalar dark matter with quadratic couplings,JHEP03(2023) 104, [arXiv:2211.09826]

    work page arXiv 2023

  8. [8]

    Antypas et al.,New Horizons: Scalar and Vector Ultralight Dark Matter,arXiv:2203.14915

    D. Antypas et al.,New Horizons: Scalar and Vector Ultralight Dark Matter,arXiv:2203.14915

    work page arXiv

  9. [9]

    Brzeminski and A

    D. Brzeminski and A. Pierce,Searching for Ultralight Scalar Dark Matter with Clocks in Low Earth Orbit,arXiv:2601.16259

    work page arXiv

  10. [10]

    Derevianko, K

    A. Derevianko, K. Gibble, L. Hollberg, N. R. Newbury, C. Oates, M. S. Safronova, L. C. Sinclair, and N. Yu,Fundamental physics with a state-of-the-art optical clock in space,Quantum Sci. Technol. 7(2022), no. 4 044002, [arXiv:2112.10817]

    work page arXiv 2022

  11. [11]

    Schkolnik et al.,Optical atomic clock aboard an Earth-orbiting space station (OACESS): enhancing searches for physics beyond the standard model in space,Quantum Sci

    V. Schkolnik et al.,Optical atomic clock aboard an Earth-orbiting space station (OACESS): enhancing searches for physics beyond the standard model in space,Quantum Sci. Technol.8(2023), no. 1 014003, [arXiv:2204.09611]

    work page arXiv 2023

  12. [12]

    Budker, J

    D. Budker, J. Eby, M. S. Safronova, and O. Tretiak,Search for fast-oscillating fundamental constants with space missions,EPJ Quant. Technol.12(2025), no. 1 39, [arXiv:2408.10324]

    work page arXiv 2025

  13. [13]

    The MICROSCOPE mission: first results of a space test of the Equivalence Principle

    P. Touboul et al.,MICROSCOPE Mission: First Results of a Space Test of the Equivalence Principle,Phys. Rev. Lett.119(2017), no. 23 231101, [arXiv:1712.01176]. [14]MICROSCOPECollaboration, P. Touboul et al.,MICROSCOPE Mission: Final Results of the Test of the Equivalence Principle,Phys. Rev. Lett.129(2022), no. 12 121102, [arXiv:2209.15487]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  14. [14]

    Van Tilburg,Wake forces in a background of quadratically coupled mediators,Phys

    K. Van Tilburg,Wake forces in a background of quadratically coupled mediators,Phys. Rev. D109 (2024), no. 9 096036, [arXiv:2401.08745]

    work page arXiv 2024

  15. [15]

    Barbosa and S

    S. Barbosa and S. Fichet,Background-induced forces from dark relics,JHEP01(2025) 021, [arXiv:2403.13894]

    work page arXiv 2025

  16. [16]

    X. Gan, D. Liu, D. Liu, X. Luo, and B. Yu,Detecting Ultralight Dark Matter with Matter Effect, arXiv:2504.11522

    work page arXiv

  17. [17]

    Macroscopic Quantum Interference in Dark Matter Wave Scattering with MICROSCOPE

    C.-T. Fu, P.-S. Luo, R. Luo, J. Sheng, and C.-Y. Xing,Macroscopic Quantum Interference in Dark Matter Wave Scattering with MICROSCOPE,arXiv:2606.07008

    work page internal anchor Pith review Pith/arXiv arXiv

  18. [18]

    Fukuda and S

    H. Fukuda and S. Shirai,Detection of QCD axion dark matter by coherent scattering,Phys. Rev. D 105(2022), no. 9 095030, [arXiv:2112.13536]

    work page arXiv 2022

  19. [19]

    H. Day, D. Liu, M. A. Luty, and Y. Zhao,Blowing in the dark matter wind,JHEP07(2024) 136, [arXiv:2312.13345]

    work page arXiv 2024

  20. [20]

    Pihan-Le Bars et al.,New Test of Lorentz Invariance Using the MICROSCOPE Space Mission, Phys

    H. Pihan-Le Bars et al.,New Test of Lorentz Invariance Using the MICROSCOPE Space Mission, Phys. Rev. Lett.123(2019), no. 23 231102, [arXiv:1912.03030]. – 28 –

    work page arXiv 2019

  21. [21]

    Equivalence Principle Violations and Couplings of a Light Dilaton

    T. Damour and J. F. Donoghue,Equivalence Principle Violations and Couplings of a Light Dilaton, Phys. Rev. D82(2010) 084033, [arXiv:1007.2792]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  22. [22]

    Phenomenology of the Equivalence Principle with Light Scalars

    T. Damour and J. F. Donoghue,Phenomenology of the Equivalence Principle with Light Scalars, Class. Quant. Grav.27(2010) 202001, [arXiv:1007.2790]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  23. [23]

    Direct Detection of Ultralight Dark Matter via Astronomical Ephemeris

    H. Fukuda, S. Matsumoto, and T. T. Yanagida,Direct Detection of Ultralight Dark Matter via Astronomical Ephemeris,Phys. Lett. B789(2019) 220–227, [arXiv:1801.02807]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  24. [24]

    Chhun, M

    R. Chhun, M. Rodrigues, and G. M´ etris,Data Package Description, MIC-NT-G-9-5513-ONE, 2023

    2023

  25. [25]

    M. List, S. Bremer, B. Rievers, and H. Selig,Modelling of solar radiation pressure effects: Parameter analysis for the microscope mission,International Journal of Aerospace Engineering2015(2015), no. 1 928206, [https://onlinelibrary.wiley.com/doi/pdf/10.1155/2015/928206]

    work page doi:10.1155/2015/928206 2015

  26. [26]

    Berg´ e et al.,MICROSCOPE mission: data analysis principle,Class

    J. Berg´ e et al.,MICROSCOPE mission: data analysis principle,Class. Quant. Grav.39(2022), no. 20 204007, [arXiv:2012.06484]

    work page arXiv 2022

  27. [27]

    Touboul et al.,Result of the MICROSCOPE weak equivalence principle test,Class

    P. Touboul et al.,Result of the MICROSCOPE weak equivalence principle test,Class. Quant. Grav. 39(2022), no. 20 204009, [arXiv:2209.15488]

    work page arXiv 2022

  28. [28]

    Maggiore,Gravitational Waves

    M. Maggiore,Gravitational Waves. Vol. 1: Theory and Experiments. Oxford University Press, 2007

    2007

  29. [29]

    Burrage, A

    C. Burrage, A. Macdonald, M. P. Ross, G. Rybka, and E. Todarello,Impact of cavities on the detection of quadratically coupled ultra-light dark matter,arXiv:2507.16526

    work page arXiv

  30. [30]

    J. T. Emmert, D. P. Drob, J. M. Picone, D. E. Siskind, M. Jones Jr., M. G. Mlynczak, P. F. Bernath, X. Chu, E. Doornbos, B. Funke, L. P. Goncharenko, M. E. Hervig, M. J. Schwartz, P. E. Sheese, F. Vargas, B. P. Williams, and T. Yuan,Nrlmsis 2.0: A whole-atmosphere empirical model of temperature and neutral species densities,Earth and Space Science8(2021),...

    work page doi:10.1029/2020ea001321 2021

  31. [31]

    Background-Induced Forces from Quadratically Coupled Ultralight Dark Matter

    T. Bouley, X. Gan, H. Xu, and T.-T. Yu,“Background-Induced Forces from Quadratically Coupled Ultralight Dark Matter”, To appear

  32. [32]

    Robert et al.,MICROSCOPE satellite and its drag-free and attitude control system,Class

    A. Robert et al.,MICROSCOPE satellite and its drag-free and attitude control system,Class. Quant. Grav.39(2022), no. 20 204003, [arXiv:2012.06479]. – 29 –

    work page arXiv 2022