Completing the peculiar-acceleration power-spectrum test of the weak equivalence principle on cosmological scales: sensitivity to realistic violations, relativistic modeling, and observational feasibility

Lu, Zheng, Zhang, Li, Ding and Chan (Phys. Rev. D 112, 103542, 2025) proposed a direct test of the weak equivalence principle (EP) on cosmological scales: measure the line-of-sight peculiar-acceleration power spectrum of galaxies via the redshift-drift technique, form the estimator $E_{\rm ep}$ as the ratio of this power spectrum for two distinct tracer populations, and fit a single deviation parameter $\delta_{\rm ep}$; the ensemble expectation is unity if all tracers fall identically. The published paper validated the estimator on N-body mocks and demonstrated sensitivity to an explicitly phenomenological, mass-dependent EP-violation rescaling, but left open what stands between this proof of concept and a scientifically interpretable, observationally feasible test. The open problem consists of the following: (i) quantify the estimator's response to physically motivated EP-violating physics (e.g., dark-sector fifth forces) — detection thresholds, false-positive rates, and dependence on the form and scale-dependence of the violation; (ii) extend the estimator beyond its plane-parallel redshift-space, linear-scale treatment — to fully relativistic or angular statistics including the neglected observer-related, ISW-analog, and peculiar-velocity terms, and to quasi-linear and nonlinear scales; (iii) fold realistic observational systematics and noise into mock pipelines and marginalize over them; and (iv) establish quantitatively the spectroscopic precision, tracer density, and survey strategy under which the test becomes feasible. No later work resolving any of these axes is known in the literature to date; the problem remains open.

Background

The weak equivalence principle (EP) — the statement that the free fall of a body is independent of its composition, mass, and internal structure — is experimentally verified to exquisite precision in the laboratory and the Solar System, but only for ordinary baryonic matter over Solar-System distances. On cosmological scales, the dynamics of the Universe is dominated by dark matter and dark energy, and the validity of the EP for these components cannot be assumed: theories in which dark matter couples to a new light scalar field, or in which gravity is modified, generically predict that different mass tracers experience different gravitational accelerations. Because of this, several tests of the EP (or of the geodesic motion of dark matter, i.e., the cosmological Euler equation) have been proposed using the large-scale structure of the Universe: Kesden and Kamionkowski (2006) showed that a dark-matter-specific force would skew the leading-versus-trailing asymmetry of the tidal tails of disrupting satellite galaxies; Bonvin and Fleury (2018) showed that relativistic effects in galaxy number counts can test whether dark matter satisfies the Euler equation, with DESI/SKA-era forecasts at the 10% level; Umeh, Koyama and Crittenden (2021) used parity-odd multipoles of the galaxy cross-power spectrum and bispectrum, constraining the dark-matter–baryon relative velocity to below 28%; Castello, Zheng, Bonvin and Amendola (2025) constructed a model-independent null-test observable $E_P$ from the cross-correlation of two galaxy populations including relativistic corrections, forecastable at the 7–15% level with the Square Kilometre Array (SKA) at $z<0.6$; and Zheng, Schneider and Amendola (2026) generalized it to $\tilde E_P$, addressing degeneracies with dark-matter viscosity and modified gravity. All of these infer EP violation statistically from clustering observables.

Lu, Zheng, Zhang, Li, Ding and Chan (2025) proposed a more direct route based on real-time cosmology. The redshift drift — the slow secular change $\dot z$ of the redshift of a distant source, first discussed by Sandage (1962) and revived by Loeb (1998) — contains, beyond the cosmic signal, the line-of-sight peculiar acceleration of each source: to first order $\dot z \simeq a_{\parallel}/c$ for the acceleration-related part, with the full perturbative expression derived by Uzan, Clarkson and Ellis (2008). Since peculiar accelerations are of order cm/s per year per object, extracting them demands ultra-high spectroscopic stability over multi-year baselines; current facilities remain orders of magnitude away (Darling 2012 reached m/s-per-year limits; the dedicated ESPRESSO campaign of Trost et al. (2025, 2026) attains $\dot v\simeq -3.5\pm3.6\ {\rm m\,s^{-1}yr^{-1}}$ on the brightest known quasar over a two-year baseline).

The method of Lu et al. is as follows. Define the density-weighted line-of-sight peculiar-acceleration power spectrum $P_{a_{\parallel}a_{\parallel}}(k)=\langle|a_{\rm pec,\parallel}(\mathbf{k})|^2\rangle_{|\mathbf{k}|=k}$ for a tracer population, and the EP estimator

$E_{\rm ep}(k)=\frac{P^{(i)}_{a_{\parallel}a_{\parallel}}(k)}{P^{(j)}_{a_{\parallel}a_{\parallel}}(k)}$

as the ratio for two distinct tracer populations $i$ and $j$ (e.g., two halo mass bins, or two disjoint galaxy samples). Because the ratio is built from two highly correlated fields, cosmic variance — the sample-variance uncertainty from observing a single realization of the density field — largely cancels, and $\langle E_{\rm ep}\rangle=1$ is expected if the EP holds for these tracers. Deviations are quantified by fitting the constant model $E_{\rm ep}^{\rm model}=1+\delta_{\rm ep}$ across wavenumber bins, with the covariance estimated by jackknife resampling (systematically deleting sub-volumes and recomputing the statistic). Validation used a $\Lambda$CDM N-body simulation ($1024^3$ particles, $1\ {\rm Gpc}/h$ box, run with the Gadget-4 code) at four redshifts $z=0,0.5,1,1.5$ on linear scales $k\in[0.007,0.2]\ h/{\rm Mpc}$ (the plane-parallel approximation — treating the line of sight as a fixed direction, valid for small angular separations — is adopted, and the analysis is conducted in three-dimensional redshift space, with tracer positions mapped along the line of sight by their peculiar velocities, so that redshift-space distortions are included). Two tests were performed: dark-matter particle mocks with Gaussian redshift errors at forward-looking noise levels ($\sigma_{z_{\parallel}}=f\sigma_{\rm SKA}/\sqrt{t_{\rm obs}}$ with $\sigma_{\rm SKA}=10\ {\rm m/s}$, $t_{\rm obs}=12$ years, and $f=0.001$ or $0.002$, i.e., roughly cm/s per object), and halo mocks split into small-mass and large-mass bins. The halo test also injected artificial EP violations by rescaling each halo's acceleration by a mass-dependent factor $\lambda=1+\alpha(M/M_*)^{\beta}$ with $M_*=10^{12}M_\odot$: mild violations $(\alpha,\beta)=(0.1,0.1)$ were not detected, moderate violations $(0.1,0.3)$ were detected at $z=0$ ($\delta_{\rm ep}=0.20\pm0.06$), and strong violations $(0.1,0.5)$ were strongly rejected ($\delta_{\rm ep}=0.44\pm0.04$, $\chi^2=83.9$ at $z=0$). The assumed survey context is a hydrogen (HI) 21-cm galaxy survey with about $10^8$ galaxies at $z\le1.5$ and per-object redshift precision of order $1\ {\rm cm/s}$ — capabilities the authors state lie beyond the SKA.

What remains open between this proof of concept and a realizable observational test — beyond the authors' own future-work statement, which defers application to real, high-precision observational data and refinement under more complex, observationally realistic conditions with systematics marginalized — is the following: the violation model injected is purely phenomenological (an arbitrary mass-dependent rescaling, 'not derived from any specific underlying theory' in the authors' words) rather than a physical model such as a coupled dark-energy fifth force; the estimator, though built in redshift space under the plane-parallel approximation, is restricted to linear scales and neglects observer-related terms (argued by the authors to vanish in their estimator) and two source-related terms (one analogous to the integrated Sachs–Wolfe effect — the late-time evolution of gravitational potentials that also imprints on galaxy number counts — and one proportional to the peculiar velocity), approximations the authors defend as subdominant only within the validated regime; realistic instrumental systematics and non-Gaussian errors have not been folded into mock-observation pipelines; and no instrument or survey design has been shown to achieve the required precision and tracer density.

Problem Statement

Establish, before any application to real survey data, whether the peculiar-acceleration test of the weak equivalence principle proposed by Lu et al. (2025) is a scientifically viable and observationally feasible cosmological test. The common objective uniting the following deliverables is a validated, end-to-end characterization of the estimator $E_{\rm ep}$ (the ratio of density-weighted line-of-sight peculiar-acceleration power spectra of two tracer populations, summarized by the deviation parameter $\delta_{\rm ep}$) that a domain expert can use to decide whether the program may responsibly proceed to observational data. Specifically:

(i) Sensitivity to realistic violations: quantify the estimator's response to physically motivated EP-violating physics — for example dark-sector fifth forces or coupled dark-energy/dark-matter models implemented in cosmological simulations — determining detection thresholds, false-positive rates, and the dependence of detectability on the form, strength, and scale- and redshift-dependence of the violation, beyond the purely phenomenological mass-dependent rescaling already tested.

(ii) Modeling completeness: extend the estimator beyond the plane-parallel, linear-scale redshift-space treatment — to fully relativistic or angular statistics, including the neglected observer-related perturbative terms, the ISW-analog source term, and the term proportional to the peculiar velocity in the time derivative of the observed redshift, and to quasi-linear or nonlinear scales — and quantify the bias or additional variance each of these approximations introduces beyond the validated regime.

(iii) Realistic systematics: fold observationally realistic noise and systematics (instrument spectral calibration, time-dependent drifts, non-Gaussian measurement errors) into mock-observation pipelines and marginalize over them, so that inferred $\delta_{\rm ep}$ constraints are demonstrably robust rather than simulation-idealized.

(iv) Observational feasibility: determine quantitatively the conditions — per-object spectroscopic redshift precision, temporal baseline, tracer number density and selection, sky coverage, and instrument stability — under which the test reaches a stated detection significance on a specified class of EP violations, and assess whether any concrete instrument or survey design meets them.

Axes (iii) and (iv) correspond to the authors' own stated future work: applying the estimator to real, high-precision observational data, refining it under more complex, observationally realistic conditions, and marginalizing over systematic effects to ensure reliable inference. Axes (i) and (ii) address limitations the paper acknowledges: the injected violation model is 'not derived from any specific underlying theory', and the neglected relativistic terms together with the linear-scale restriction — defended by the authors as harmless within the validated regime — remain untested beyond it. These four deliverables are jointly required: none of them is independently sufficient, and the task is resolved only when all four are addressed. Each must be carried to the point of a quantitative, falsifiable statement — respectively, detection thresholds and false-positive rates; a quantified bias or added variance from each neglected term and from the scale restriction; demonstrated robustness of the inferred $\delta_{\rm ep}$ constraints under marginalized systematics; and explicit numerical conditions on precision, baseline, density and instrument stability together with whether any concrete design meets them. Characterizing these axes is what is needed to decide whether an observational cosmological EP test based on the peculiar-acceleration power spectrum can be realized and how its results would be interpreted.

References

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