Gravitational Redshift from Galaxy Clusters -- a Relativistic Approach

TL;DR

This work derives a fully relativistic description of the gravitational redshift signal from galaxy clusters, including all relativistic corrections up to second order in the weak-field expansion, and treats density fluctuations nonperturbatively. By carefully averaging over cluster members with a galaxy-number counts weighting and accounting for velocity distributions and cluster symmetries, the authors show that the light-cone term cancels on constant-time hypersurfaces and provide a clean expression for the mean redshift difference driven by the gravitational potential, $-\\Delta\\Psi$, while second-order Doppler and evolution terms form the contaminant budget. They model stacked clusters with an NFW profile, connect the observable velocity dispersion to the potential via the Jeans equation, and quantify how BCG motion and off-centering modify both the width and the shift of the redshift distribution. The results clarify the relative size of gravitational redshift versus kinematic contaminants, emphasize the dependence on magnification bias $s_b$ and spectral index $\\alpha$, and establish a framework to test the weak equivalence principle (and, in a follow-up, the Euler equation) using cluster-scale observables.

Abstract

The light that we receive from clusters of galaxies is redshifted by the presence of the clusters' gravitational potential. This effect, known as gravitational redshift, was first detected from a sample of stacked clusters in 2011, by taking redshift differences between the centre of each cluster and the respective member galaxies. However, the interpretation of this result was later challenged by several studies, which emphasised the possible influence of additional kinematic effects on the observed signal, like the transverse Doppler effect. In this work, we present the first derivation of all such effects within a relativistic framework, accurate to third order in the weak-field approximation. This framework allows us to correctly capture the hierarchy of terms on the scale of clusters and at the same time account for all relativistic effects. We compare our result with previous literature and show that some terms of the same order of the transverse Doppler effect were not properly included, leading to an overestimation of the kinematic contamination. In particular, we do not find any contribution arising from the so-called light-cone effect and obtain a larger correction due the motion of the central galaxy. Our derivation is independent of the Euler equation, providing a straightforward framework to test the weak equivalence principle.

Figures (12)

• Figure 1: We can construct the distribution of the redshift difference for all members of the cluster situated at a fixed transverse separation from the BCG. The width of the distribution is governed by the linear Doppler term, while the shift (which is negative) is determined by gravitational redshift and by second-order Doppler contributions. Since the width is typically 100 times larger than the shift, it is necessary to have a very precise measurement of the shape (by stacking a larger number of clusters) to precisely determine the shift.
• Figure 2: Geometrical setup with the relevant quantities defined in a cluster at the emission time of the BCG (left panel) and a generic member galaxy (right panel). Note that we have defined $\boldsymbol{r} \equiv (\boldsymbol{R}\cdot \hat{\boldsymbol{\chi}}) \, \hat{\boldsymbol{\chi}}$. The motion of the member galaxy has been exaggerated for illustrative purposes. In both panels, $\boldsymbol{\chi}_{\rm BCG}$ denotes the comoving distance to the BCG at its emission time. For the quantities related to the member galaxy, we distinguish between their values at the emission time $\eta$ of the member galaxy (without any subscript) and at the emission time $\eta_e$ of the BCG (with the subscript $e$).
• Figure 3: Sketch of the gravitational redshift effect. The photon emitted by a galaxy is redshifted when escaping a gravitational potential to reach the observer. In order to extract this signal, we take the observed redshift difference between each member galaxy in the cluster and the BCG, which we consider as a proxy for the bottom of the cluster gravitational potential. On average, member galaxies are located higher in the potential, thus they are less gravitationally redshifted than the BCG. This yields a net measured blueshift with this method.
• Figure 4: The quantity $\Delta z$ is averaged over all galaxy members situated at the same $R_\perp$. While $R_\perp$ is uniquely defined in the flat sky, there are different possible definitions in the full sky. Here, we define $R_\perp = \sin(\Delta \theta) \chi_\mathrm{BCG}=\Delta\theta\,\chi_\mathrm{BCG}+\mathcal{O}(\epsilon^3_\mathcal{H})$, which means that we average over all member galaxies along the dashed blue segment.
• Figure 5: Illustration of the light-cone effect, showing the worldlines of the BCG (in red) and a member galaxy (in blue). On the left panel, the member galaxy is moving away from the BCG ($\dot{r}_e > 0$), leading to a change $\Delta r_e > \Delta \chi$, and hence $\frac{d \chi}{d r_e} > 1$. On the contrary, the member galaxy is moving towards the BCG on the right panel ($\dot{r}_e < 0$), leading to $\Delta r_e < \Delta \chi$, and hence $\frac{d \chi}{d r_e} < 1$.