Cluster gravitational redshifts: uncertainties and survey requirements

Abstract

We investigate the impact of observational and theoretical uncertainties in cluster gravitational redshifts as a probe of modified gravity using an end-to-end forecasting pipeline. We use a generative model to build a halo catalogue with $M_{500}\ge 3\times 10^{13}\,M_\odot$, populate haloes with member galaxies via a five-parameter halo occupation distribution (HOD), assign projected positions from radial density profiles, apply survey-like selections, and infer a linear rescaling of the gravitational potential, $α_\mathrm{MG}$, to parameterise modifications to general relativity (GR). We vary redshift uncertainties, radial and mass-redshift completeness, member abundance, minimum mass and maximum redshift, as well as mis-specify the clusters density and velocity profiles, centres, and mass function. We find that the intracluster velocity dispersion sets an effective floor: improving redshift precision beyond $σ_z\sim 10^{-4}(1+z)$ brings no improvement in the precision of $α_\mathrm{MG}$. Realistic redshift and mass cuts primarily remove low-mass haloes and have minimal impact on the $α_\mathrm{MG}$ precision. In this setting, we find that shallow, narrower spectroscopic surveys are preferable to deep, wide photometric ones for precise modified gravity constraints. We further find that mis-centring can mimic significant departures from GR. Baryonic deviations from a Navarro-Frenk-White profile and velocity anisotropies do not introduce appreciable biases. In the high-S/N regime of upcoming surveys, accurate determination of cluster centres will be essential to avoid interpreting systematic effects as new physics. The Spectroscopic Stage-5 Experiment and the Widefield Spectroscopic Telescope provide a clear route toward establishing gravitational redshifts as a competitive probe of modified gravity.

Figures (5)

• Figure 1: Model vs mock gravitational redshift in GR in the limit of no velocity dispersion (red) and the baseline case (black) with $1\sigma$ error bars. The sDGP and $f(R)$ predictions are shown in orange (dotted) and blue (dashed), respectively. We verify that the mock matches the model perfectly by artificially removing the velocity dispersions, and hence the transverse Doppler component, after which we recover perfect agreement.
• Figure 2: Predicted (red) vs mock (black) number of stacked galaxies per radial distance shell from the centre of the clusters as a function of halo mass.
• Figure 3: Summary of the inferred modified gravity parameter for the different variants considered in Sec. \ref{['sec:results']} compared to GR/$\Lambda$CDM (vertical black line). The yellow window highlights theoretical uncertainties (model mis-specifications). The baseline case refers to $\sigma_\mathrm{z}=10^{-4}(1+z)$, perfect member and halo completeness for $z_\mathrm{max}=2$ and $M_\mathrm{min}=3\times10^{13}\,M_\odot$.
• Figure 4: Forecasted uncertainty on the stacked gravitational redshift profile for a spectroscopic cluster catalogue obtained from a galaxy survey out to $z_\mathrm{max}=1$ assuming a member completeness $C_1=0.9$, $C_2=0.6$, $M_\mathrm{min}=10^{14}$ M$_\odot$ and $\log_{10}(M_{\rm cut,0}/M_\odot)=13.8$. This represents a relatively resource-efficient setting for Stage-V surveys which delivers $10\%$ precision on $\alpha_{\rm MG}$ in the absence of any mis-specifications. The mock data are generated assuming GR.
• Figure 5: (a) Gravitational redshift signal and (b) error on the signal as a function of radial distance from the clusters' centres' and (c) corresponding modified gravity parameter for different redshift uncertainties on the BCG, assuming spectroscopic uncertainty for the member galaxies.