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, $E_{\mathrm{ep}}$, 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 $E_{\mathrm{ep}}$ 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 $z\le 1.5$ and scales of $k$ in the range of 0.007 and $0.2h/\mathrm{Mpc}$. By fitting a single parameter ${\delta }_{\mathrm{ep}}$ across redshifts, we find that dark matter (DM) particle mocks without EP violation yield ${\delta }_{\mathrm{ep}}$ 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 ${\delta }_{\mathrm{ep}}$ values and high ${\chi }^2$, 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.

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• Astrophysical studies of gravity
• Large scale structure of the Universe