Testing the Equivalence Principle in Galaxy Clusters

TL;DR

This paper develops a model-independent test of the weak equivalence principle for dark matter on cluster scales by comparing the gravitational-potential depth, inferred from gravitational redshift, with the velocity dispersion of cluster galaxies via the Jeans equation, introducing a fifth-force parameter $\Gamma$. By measuring the shift and the width of the redshift-difference distribution $\Delta z = z_{\rm member}-z_{\rm BCG}$ across projected radii, the authors constrain $\Gamma$ through a Fisher-forecast framework that accounts for cluster density profiles (NFW), the cluster mass function slope $b$, and the BCG velocity dispersion, while treating non-gravitational DM interactions in a model-independent way. They show that the width (velocity dispersion) term provides the strongest constraint, but the shift is essential to break degeneracies with the mass distribution and density profile; in realistic scenarios, priors on population- and lensing-derived parameters improve the constraints on $\Gamma$. Current data can bound a possible dark-matter fifth force at roughly 7–14%, whereas next-generation surveys like DESI and Euclid are expected to reach the percent level, making galaxy clusters a powerful laboratory for testing fundamental properties of dark matter.

Abstract

Clusters of galaxies have been used to measure a subtle effect predicted by Einstein: gravitational redshift. This signal encodes pristine information about our Universe, since it is sensitive to the depth of the clusters' gravitational potential wells. In this work, we show how gravitational redshift can be used to test a fundamental physical principle: the weak equivalence principle. This principle stipulates that all matter falls in the same way in a gravitational potential, regardless of its nature. By comparing the amplitude of the gravitational redshift signal with the velocity dispersion in galaxy clusters, we build a novel test of this principle targeted to the unknown dark matter. Our test is sensitive to any additional interaction that would alter the way dark matter falls in gravitational potentials, hence leading to a violation of the equivalence principle. We show that currently available data can constrain the presence of a fifth force in clusters at the level of 7-14%, while the newest surveys will reach a precision of a few percents. This demonstrates the crucial role played by galaxy clusters in testing fundamental properties of dark matter.

Figures (7)

• Figure 1: We measure the distribution of the redshift difference between the cluster members and the BCG. The width of the distribution is directly sensitive to the linear Doppler effect, which can take any sign, leading to the variance in Eq. \ref{['eq:var']}. Gravitational redshift and second-order Doppler effects, on the other hand, always have the same sign and shift the mean of the distribution away from zero following Eq. \ref{['eq:mean']}. A breaking of the weak equivalence principle, encoded in the fifth force $\Gamma$, would modify the relation between gravitational redshift and velocities and can thus be tested by combining the shift with the width.
• Figure 2: 1$\sigma$ uncertainty on $\Gamma$ as a function of the number of galaxies $N$, with different assumptions about the other parameters. The black line shows the result where all parameters are fixed except $\Gamma$, while the green line corresponds to the case without any prior on any parameter. Intermediate lines all have a 20% Gaussian prior on $\mathcal{R}$ and $\sigma_{\rm BCG}^2$ and no prior on $c_v$, but they differ in terms of $b$, which is strongly degenerated with $\Gamma$: the blue line has no prior on $b$, while the light to dark red lines respectively have priors of (20%, 10%, 5%) on $b$. The vertical line corresponds to the number of galaxies considered in Wojtak:2011ia.
• Figure 3: Joint 1$\sigma$ constraints on $b$ and $\Gamma$ obtained from the variance only (red), the shift only (blue) and the combination of the two (black). In the left panel, we keep all the other parameters fixed at their fiducial values, while in the right panel, we also vary $c_v$ and marginalise over it.
• Figure 4: Joint 1$\sigma$ constraints on $c_v$ and $\Gamma$ for $N=125'000$. When the other parameters are kept fixed (black line), $c_v$ and $\Gamma$ are not degenerate. The other lines include a 20% Gaussian prior on $\mathcal{R}$ and $\sigma_{\rm BCG}^2$. We show the case with no prior on $b$ (blue line) and the cases with priors on $b$ of {5%, 10%, 20%} in shades from dark to light red. This illustrates how the constraints on $\Gamma$ degrade when $c_v$ and $b$ are both unknown.
• Figure 5: Comparison of the joint 1$\sigma$ constraints with the density-weighted binning (solid black line) and the optimal binning (red dashed line). All other parameters are marginalised over.