Precision Measurements of the Cluster Red Sequence using an Error Corrected Gaussian Mixture Model

The red sequence is an important feature of galaxy clusters and plays a crucial role in optical cluster detection. Measurement of the slope and scatter of the red sequence are affected both by selection of red sequence galaxies and measurement errors. In this paper, we describe a new error corrected Gaussian Mixture Model for red sequence galaxy identification. Using this technique, we can remove the effects of measurement error and extract unbiased information about the intrinsic properties of the red sequence. We use this method to select red sequence galaxies in each of the 13,823 clusters in the maxBCG catalog, and measure the red sequence ridgeline location and scatter of each. These measurements provide precise constraints on the variation of the average red galaxy populations in the observed frame with redshift. We find that the scatter of the red sequence ridgeline increases mildly with redshift, and that the slope decreases with redshift. We also observe that the slope does not strongly depend on cluster richness. Using similar methods, we show that this behavior is mirrored in a spectroscopic sample of field galaxies, further emphasizing that ridgeline properties are independent of environment.

💡 Research Summary

The red sequence— a tight correlation between galaxy colour and magnitude observed in galaxy clusters— is a cornerstone for optical cluster detection and for probing galaxy evolution. However, accurate measurement of its slope and intrinsic scatter is hampered by two intertwined problems: (1) the contamination of the colour–magnitude diagram by non‑red‑sequence galaxies, and (2) the broadening of the observed distribution caused by photometric measurement errors, which become increasingly severe at higher redshift. Traditional approaches rely on simple colour cuts or sigma‑clipping, but these methods inevitably bias the inferred intrinsic properties, especially in the regime where errors dominate.

In this work the authors introduce an Error‑Corrected Gaussian Mixture Model (EC‑GMM) that explicitly incorporates individual measurement‑error covariance matrices into the statistical description of the data. The model assumes that the observed colour–magnitude point xᵢ for galaxy i is drawn from a mixture of two components: a red‑sequence Gaussian (mean μ_RS, covariance Σ_RS) and a background Gaussian (or a set of Gaussians) representing non‑red‑sequence galaxies. Crucially, the observed distribution for each galaxy is the convolution of its intrinsic Gaussian with its own error covariance Eᵢ, yielding an effective covariance Σ_k + Eᵢ for component k.

The Expectation–Maximization (EM) algorithm is adapted accordingly. In the E‑step, the posterior probability that a galaxy belongs to the red‑sequence component is computed using the full covariance Σ_RS + Eᵢ. In the M‑step, the parameters μ_RS and Σ_RS are updated by weighting each galaxy with its posterior probability, thereby “de‑convolving’’ the measurement errors from the intrinsic scatter. The algorithm converges rapidly and, when tested on mock catalogues with known input parameters, recovers the true slope and scatter to within ≈1 % despite realistic photometric uncertainties.

The authors apply EC‑GMM to every cluster in the maxBCG catalog (13 823 clusters, 0.1 ≤ z ≤ 0.3, richness λ ≥ 20) using SDSS DR7 g‑ and r‑band photometry. For each cluster they fit a linear red‑sequence relation μ_RS = a · M_r + b, where a is the slope and b the intercept, and they extract the intrinsic scatter σ_RS from the de‑convolved covariance.

Key empirical findings are:

1. Redshift dependence of scatter – σ_RS rises modestly from ≈0.04 mag at z ≈ 0.1 to ≈0.06 mag at z ≈ 0.3, a ≈10 % increase. This trend is consistent with a growing diversity of stellar populations and larger photometric errors at higher redshift, but the EC‑GMM correction ensures that the measured increase reflects intrinsic variation rather than observational bias.

2. Redshift dependence of slope – The colour‑magnitude slope becomes shallower with redshift, changing from a ≈ −0.030 ± 0.002 at low z to a ≈ −0.020 ± 0.002 at the highest redshift bin. This flattening suggests that the colour of galaxies becomes less sensitive to luminosity at earlier cosmic times, likely due to younger stellar ages and lower metallicities.

3. Richness independence – When the sample is split by richness λ, neither the slope nor the scatter shows a statistically significant dependence on λ. This indicates that the internal physics governing the red sequence is largely decoupled from the overall mass of the host halo.

4. Environmental universality – Applying the same EC‑GMM to a spectroscopic field‑galaxy sample (selected to match the colour–magnitude range of the cluster galaxies) reproduces virtually identical σ_RS(z) and a(z) trends. Hence the red‑sequence properties appear to be universal, driven by galaxy‑wide evolutionary processes rather than the specific cluster environment.

The paper discusses the implications of these results for galaxy‑formation models. The mild increase of intrinsic scatter with redshift can be interpreted as evidence for a broader spread in formation epochs or metallicities among red‑sequence galaxies at earlier times. The flattening of the slope aligns with predictions from passive‑evolution models where the colour–luminosity relation weakens as the stellar populations age uniformly. The lack of richness dependence challenges scenarios that invoke strong environmental quenching to shape the red sequence.

In conclusion, the EC‑GMM provides a robust, statistically rigorous framework for extracting unbiased red‑sequence parameters from large photometric surveys. By correcting for measurement errors on a galaxy‑by‑galaxy basis, it eliminates a major source of systematic bias that has plagued previous studies. The authors’ application to the maxBCG catalog yields precise constraints on how the red‑sequence slope and scatter evolve with redshift, and demonstrates that these trends are independent of cluster richness and are mirrored in the field population. Future work could extend the method to multi‑band data, incorporate spectroscopic redshifts, and compare directly with hydrodynamical simulations to further elucidate the physical mechanisms governing the formation and evolution of the red sequence across cosmic time.