Testing cosmic anisotropy with cluster scaling relations

We test claims of large-scale anisotropy in the local expansion rate using cluster scaling relations as distance indicators. Using a Bayesian forward model, we jointly fit the X-ray luminosity–temperature (LT) and thermal Sunyaev-Zel’dovich–temperature (YT) relations, marginalising over the latent cluster distances and modelling selection effects as well as peculiar velocities. The latter are modelled using reconstructions of the local peculiar velocity field where we self-consistently account for possible anisotropic redshift–distance relations via an approximate scheme. This treatment proves crucial to the inferred anisotropy and breaks the degeneracy between anisotropy in scaling relation normalisations and underlying cosmological anisotropy. We apply our method to 312 clusters at $z \lesssim 0.2$, testing dipolar, quadrupolar and general (pixelised) anisotropy models. Bayesian model selection finds no more than weak evidence for any anisotropic model. For dipole models, we obtain upper limits of $δH_0 / H_0 < 3.2%$ and bulk flow magnitude $< 1300,\mathrm{km,s^{-1}}$. Our results contrast with previous claims of statistically significant anisotropy from the same data, which we attribute to our principled forward modelling of both redshifts and scaling relation observables through latent distances and our treatment of the impact of anisotropic redshift–distance relations when modelling the local peculiar velocity field. Our work highlights the importance of accurately modelling peculiar velocities when testing isotropy with distance indicators, and motivates the further development of reconstructions that self-consistently treat large-scale deviations from the Hubble flow.

💡 Research Summary

This paper presents a rigorous Bayesian forward‑modelling analysis of galaxy‑cluster scaling relations to test claims of large‑scale anisotropy in the local expansion rate. The authors focus on two well‑studied cluster observables: the X‑ray luminosity–temperature (LT) relation and the thermal Sunyaev‑Zel’dovich–temperature (YT) relation. Both relations link a distance‑independent quantity (the intracluster temperature, T) to distance‑dependent observables (X‑ray luminosity L_X and integrated Compton‑y parameter Y_SZ). By predicting L_X and Y_SZ from the measured temperature and comparing to the observed fluxes, one can infer the cluster’s distance independently of its redshift.

The data set consists of 312 low‑redshift (z ≲ 0.2) clusters drawn from the extended HIghest X‑ray FLUx Galaxy Cluster Sample (eeHIFLUGCS). All clusters have measured X‑ray luminosities and temperatures; 273 also have Y_SZ measurements from Planck. The catalogue originally assumes a fiducial cosmology (H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_m = 0.3); the authors invert this to obtain raw fluxes before re‑introducing cosmology as a free parameter.

The core methodological innovation is to treat each cluster’s comoving distance r as a latent variable and to model the observed CMB‑frame redshift as the product of a cosmological redshift (derived from r via the standard ΛCDM distance–redshift relation) and a peculiar‑velocity redshift term, 1 + z_obs = (1 + z_cosmo)(1 + z_pec). Peculiar velocities V_pec are predicted from a reconstructed local density field (based on 2LPT or similar techniques). Crucially, the reconstruction itself depends on the assumed H₀ because the mapping from redshift to real‑space positions uses the distance–redshift conversion. This creates a feedback loop: an anisotropic H₀ changes the density field, which changes the predicted V_pec, which in turn alters the inferred redshifts. By modelling this loop explicitly, the authors break the perfect degeneracy between a dipole/quadrupole variation in H₀ and a corresponding dipole/quadrupole in the scaling‑relation normalisations.

The scaling relations are modelled hierarchically:
log L_X = α_LT log T + β_LT + ε_LT,
log Y_SZ = α_YT log T + β_YT + ε_YT,
with intrinsic scatters ε jointly Gaussian and correlated. Selection effects (flux limits, Y_SZ signal‑to‑noise cuts) and Malmquist bias are incorporated through explicit likelihood terms. The LT and YT relations are fitted jointly, allowing the data to constrain the intrinsic correlation between them.

Three families of anisotropy are tested: (i) a dipole or quadrupole modulation of H₀ across the sky, (ii) a dipole/quadrupole term added to the peculiar‑velocity field, and (iii) a fully pixelised (HEALPix) map of H₀ variations. For each model the Bayesian evidence (log Z) is computed, and model comparison is performed using Bayes factors.

The results are strikingly consistent across all tests: the evidence for any anisotropic model is at most weak, never reaching the conventional “strong” threshold. For the dipole H₀ model the 95 % upper limit on the fractional variation is δH₀/H₀ < 3.2 %, and the bulk‑flow amplitude is constrained to be < 1300 km s⁻¹. These limits are substantially tighter than those reported in earlier works that used the same cluster sample but did not marginalise over distances or include a self‑consistent treatment of peculiar velocities.

The authors attribute the discrepancy to the previous studies’ neglect of the coupling between H₀ anisotropy and the reconstructed velocity field. When this coupling is ignored, a dipole in H₀ can be mimicked by a corresponding shift in the scaling‑relation zero‑points, leading to a spurious detection of anisotropy. By explicitly modelling the redshift–distance conversion, the peculiar‑velocity field, and the selection function, the present analysis eliminates this degeneracy.

In conclusion, the paper demonstrates that, with a careful Bayesian forward model that jointly fits LT and YT relations, marginalises over latent distances, and self‑consistently incorporates peculiar velocities, the current low‑redshift cluster data show no statistically significant evidence for large‑scale anisotropy in the local expansion rate. The work underscores the importance of accurate velocity‑field reconstructions when using distance indicators to test isotropy and provides a framework that can be applied to upcoming, larger cluster samples (e.g., from eROSITA, SPT‑3G, or future CMB‑S4 surveys) to push these limits even further.