Revisiting Marked Galaxy Clustering from a Joint Point Process Perspective

Abstract

Marked correlation functions, in which galaxy properties such as luminosity or stellar mass are treated as marks, are widely used to test models of galaxy formation. In astronomy, however, these statistics are typically implemented as summary measures that do not preserve the joint structure of mark pairs conditioned on separation. In this work, we formulate galaxies as points $(x,m)$ on the product space $\mathbb{R}^3\times\mathcal{M}$, where $x$ denotes position and $m$ a mark, and introduce the joint pair correlation function $g(r;m_1,m_2)$ as the fundamental quantity describing mark-dependent clustering. We further define a diagnostic quantity $Δ_{\mathrm{ind}}(r;m_1,m_2)$ that locally quantifies deviations from the independence hypothesis relative to spatial clustering alone, thereby providing a projection-free description of which mark pairs are over- or underrepresented at a given separation scale. Within this framework, commonly used diagnostics such as the inhomogeneous cross-$J$ function are naturally interpreted as summary statistics obtained through averaging over mark sets and geometric-event-based reductions of the joint structure. This perspective clarifies that previously discussed marked effects, including assembly bias, correspond to projections of an underlying joint dependence, and that observationally accessible information is the existence of non-factorizable joint structure itself. The present formulation provides both a fundamental quantity and practical diagnostics for its characterization.

Figures (5)

• Figure 1: Conceptual illustration of three perspectives on marked clustering. Top: the null hypothesis of mark-independent clustering. Middle: conventional marked statistics defined as projections of the conditional mark distribution, used to detect deviations from independence. Bottom: this work introduces the joint pair correlation on the product space as the fundamental quantity, enabling direct description of the coupled structure of marks and clustering while treating independence as a testable hypothesis.
• Figure 2: Example of the physical interpretation of reweighting. Here the mark is assumed to be luminosity (absolute magnitude). The left panel shows the standard correlation function in which all galaxies are counted equally, while the right panel shows a weighted correlation that emphasizes luminous galaxies. The spatial configuration itself is unchanged; what changes is the perspective of which galaxies are regarded as representative. This operation alters how projections are taken with respect to the joint structure. The quantity $\Delta_X(r;m_1,m_2)$ introduced in the next figure directly visualizes the local joint structure prior to such projection.
• Figure 3: Visualization of $\Delta(r;m_1,m_2)$ (schematic). (a) A heatmap of $\Delta$ on the $(m_1,m_2)$ plane at a fixed $r=r_\ast$ directly shows which mark pairs are relatively enhanced/suppressed. (b) Curves of $\Delta(r)$ for representative mark pairs summarize the scale dependence. (c) $\Delta\simeq0$ corresponds to the separation (independence) hypothesis; nonzero $\Delta$ indicates a breakdown of factorization in the conditional mark-pair distribution.
• Figure 4: Comparison between independent mark assignment (Case A) and environment-dependent mark assignment (Case B) for the same spatial configuration. The top panels show the marked correlation $M(r)$, while the bottom panels display heatmaps of $\Delta_X(r_\ast;m_1,m_2)$ at multiple distance bins $r_\ast$. Although $M(r)$ tends to be enhanced at small scales in Case B, $\Delta_X$ decomposes the contribution by mark pair and visualizes its scale dependence.
• Figure 5: An example illustrating the limitation of projected statistics. We consider mark generation with linear dependence on environment (Case B1) and quadratic dependence (Case B2), adjusting the amplitude in the latter so that $M(r)$ closely matches. As shown in the top panels, the marked correlation cannot distinguish the two cases, whereas the bottom panels of $\Delta_X(r_\ast;m_1,m_2)$ clearly reveal distinct joint structures.