Testing the Equivalence Principle on Cosmological Scales Using Peculiar Acceleration Power Spectrum

TL;DR

This study tackles the question of whether the weak Equivalence Principle (EP) holds on cosmological scales by proposing a direct test using the density-weighted peculiar acceleration power spectrum measured via redshift drift. It introduces the EP estimator $E_{ m ep}$, defined as the ratio of acceleration power spectra from two tracers, so that $\langle E_{ m ep} \rangle = 1$ if EP is valid and cosmic variance cancels in the ratio. The authors validate the approach with N-body simulations, employing both DM particle and DM halo mock catalogs, and analyze scales $k \in [0.007, 0.2]~h\mathrm{Mpc}^{-1}$ across redshifts $z \le 1.5$. In DM particle mocks, low observational noise yields $\delta_{\rm ep} \approx 0$ (i.e., $E_{ m ep} \approx 1$) with consistent chi-squared values, while high noise biases are evident at low $z$. In halo mocks, no-violation cases remain consistent with EP, whereas artificial violations parameterized by $(\alpha,\beta)$ produce detectable shifts in $\delta_{\rm ep}$ and rising $\chi^2$, especially at low redshift, demonstrating the estimator’s sensitivity. Overall, the results establish $E_{ m ep}$ as a robust tool for probing EP on cosmological scales and motivate its application to future high-precision datasets that extend beyond SKA.

Abstract

While the (weak) Equivalence Principle (EP) has been rigorously tested within the solar system, its validity on cosmological scales, particularly in the context of dark matter and dark energy, remains uncertain. In this study, we propose a novel method to test EP on cosmological scales by measuring the peculiar acceleration power spectrum of galaxies using the redshift drift technique. We develop an EP estimator, $E_{\rm ep}$, to evaluate the consistency of the peculiar acceleration power spectrum across different tracers. By calculating the ratio of the peculiar acceleration power spectra of tracers, the ensemble average of $E_{\rm ep}$ is expected to be unity if EP holds on cosmological scales for these tracers. We validate this estimator using N-body simulations, focusing on four redshift bins with $z\leq 1.5$ and scales of $k$ in the range of $0.007$ and $0.2$ $h/\rm Mpc$. By fitting a single parameter $δ_{\rm ep}$ across redshifts, we find that DM particle mocks without EP violation yield $δ_{\rm ep}$ consistent with zero under the small redshift measurement uncertainty case, while the large redshift uncertainty case slightly induces biases at low redshifts. In addition, when using DM halo mocks with controlled EP violations, no-violation and mild-violation cases show no significant detection, while moderate and strong violations produce statistically significant $δ_{\rm ep}$ values and high $χ^2$, especially at low redshifts, confirming the estimator's sensitivity. Taking advantage of advanced observing capabilities, such as next-generation facilities that extend beyond the Square Kilometer Array, the proposed method offers a promising approach for future cosmological tests of EP.

Figures (3)

• Figure 1: Measurement of $E_{\rm ep}$ for DM particles for testing EP. The ratio of acceleration power spectra for the two datasets--each containing $10^8$ particles--was computed after adding Gaussian redshift errors to each particle. These errors have zero mean and a standard deviation of $\sigma_{z_{\parallel}}$, corresponding to $f = 0.001$ (left) and $f = 0.002$ (right) for the low- and high-noise cases, respectively. Four redshift bins are considered, with $k$ values spanning the range $[0.007, 0.21]$ divided into 10 logarithmically spaced bins. The black solid line represents the expected value $E_{\rm ep} = 1$ under the exact validity of EP. The error bars correspond to the $1\sigma$ level, estimated from the jackknife scheme applied to the simulation box. The fitted values of $E_{\rm ep}$ are also shown for comparison. As shown, when $f=0.001$, the $E_{\rm ep}$ values remain consistent with unity within the $2\sigma$ confidence level across all $z$-bins. When $f=0.002$, the $E_{\rm ep}$ values exhibit slightly larger deviations from unity and increased uncertainties, particularly at low redshifts. Note that the bin centers have been slightly adjusted for clarity in the illustration.
• Figure 2: Same as Fig. \ref{['fig:ep-particle']}, but for the DM halo mock including cases with EP violation. The estimator $E_{\rm ep}$ is computed from the ratio of acceleration power spectra between the "large-mass" and "small-mass" halo catalogs. Shown are the estimated values of the EP violation parameter $\delta_{\rm ep}$ across different redshifts, for various levels of EP violation characterized by the parameters $(\alpha, \beta)$. The case with $(\alpha, \beta) = (0, 0)$ shows $\delta_{\rm ep} \approx 0$ across all redshifts, consistent with no violation. Mild violation scenarios such as $(0.1, 0.1)$ yield similar results, with $\delta_{\rm ep}$ remaining within $1\sigma$ uncertainties. In contrast, moderate $(0.1, 0.3)$ and strong $(0.1, 0.5)$ violations produce increasingly significant positive shifts in $\delta_{\rm ep}$, especially at lower redshifts, accompanied by rising $\chi^2$ values--indicating robust sensitivity of the estimator to EP-breaking signals.
• Figure 3: Comparison of halo acceleration power spectra from simulations and linear theory predictions for four redshift snapshots: small-mass halos (left) and large-mass halos (right).