Generalized Fluctuation Theory of Galaxy Clustering With Continuous Mass Spectra

Abstract

We develop a continuum counts-in-cells fluctuation theory for gravitational clustering in galaxy populations described by a continuous comoving mass function \(\phi (m,z)\) within the quasi-equilibrium grand-canonical framework. Starting from the equal-mass and two-species fluctuation sectors, we construct a finite-resolution mass-bin representation, pass to the continuum limit and derive closed expressions for the internal energy, the pressure, and the first four count cumulants in the strict total-count sector. The latter is then rewritten in an intrinsic normalization based on the mean mass \(\bar{m} = M_1/M_0\), so that the observable hierarchy depends only on the mass function, the clustering parameter, and the thermodynamic control variables. The continuum construction recovers the equal-mass formulas exactly and reproduces the discrete two-species relations in the corresponding singular limit. In the weak-clustering regime, the leading spectral corrections are controlled by the moments \(M_3\), \(M_6\), and \(M_9\). Within the strict total-count criticality condition, the compressibility denominator is pinned to \(D_c=1/2\), so the Fano factor takes the fixed value \(F_c=2\) as an algebraic identity that follows directly from the criticality condition, whereas the higher reduced cumulants preserve explicit dependence on the mass spectrum through the response hierarchy \(A_r\). A truncated Schechter specialization for threshold-selected catalogs then makes the physical content explicit: raising the lower mass threshold lowers the critical clustering parameter while increasing the variance-, skewness-, and kurtosis-level responses of the counts-in-cells distribution. The resulting framework furnishes an analytical interface between quasi-equilibrium gravitational clustering theory and finite-resolution mass-selected one-point observables for realistic galaxy populations.