The Dynamical Mass Discrepancy as a Function of Baryonic Compactness

We present a strictly empirical analysis of the dynamical mass discrepancy as a function of baryonic compactness, treating the discrepancy as an effective, state-dependent response to baryonic boundary conditions. Our compactness proxy is closely related (up to geometric factors and $G$) to an aperture-averaged baryonic acceleration, and therefore connects naturally to the family of mass-discrepancy acceleration relations. We construct a unified and reproducible dataset spanning four galaxy regimes with strictly aperture-matched measurements: SPARC rotation-supported spirals (outer disk, $R=R_{\max}$;), LITTLE THINGS dwarf irregulars ($R=R_{\max}$;), SLACS strong-lens early-type galaxies (within the Einstein radius $R=R_{\rm Ein}$;), and MaNGA galaxies with Jeans Anisotropic Modeling (JAM) masses within one effective radius ($R=R_e$;). For each system, baryonic and dynamical masses are defined within the same physical aperture, and we introduce a minimal baryonic compactness proxy,
\begin{equation}
\Sigma \equiv \frac{M_b}{\pi R^2}.
\label{eq:Sigma}
\end{equation}

While the extensive excess mass $M_{\rm dyn}-M_b$ is physically transparent, comparisons across heterogeneous mass scales and apertures are more robust when expressed in terms of the intensive baryonic-efficiency variable
\begin{equation}
E \equiv \log_{10}\!\left(\frac{M_{\rm dyn}}{M_b}\right).
\label{eq:Edef}
\end{equation}
We therefore test the bivariate scaling relation
\begin{equation}
E = A' + \alpha' \log_{10} M_b + \delta' \log_{10} \Sigma,
\label{eq:EoS}
\end{equation}
with $A'$ absorbing the choice of units.

Across SPARC, LITTLE THINGS, and MaNGA, we measure a statistically strong and mutually consistent compactness response, $\delta' \simeq -0.4$ (SPARC: $-0.399\pm 0.038$; LITTLE THINGS: $-0.407\pm 0.131$; MaNGA: $-0.417\pm 0.006$), indicating that at fixed baryonic mass, more compact systems exhibit a smaller dynamical discrepancy. SLACS systems also show $\delta'<0$ but favor a steeper response in the high-density regime ($\delta'=-0.571\pm 0.069$); a formal SPARC-SLACS interaction test yields $\Delta\delta'$ with $p\simeq 0.078$, consistent with no difference at the $5\%$ level while indicating mild regime tension. In the MaNGA sample, expressing the discrepancy in terms of $E$ yields substantially higher coherence than the extensive residual $M_{\rm dyn}-M_b$ (typical $R^2\simeq 0.64$ with scatter $\sim 0.16$ dex), supporting baryonic compactness as a primary macroscopic organizing coordinate.

We further extend the analysis to the galaxy-cluster regime using the CCCP weak-lensing sample ($N=50$), with aperture-matched measurements at $R_{500}^{\rm WL}$ for both the total mass $M_{{\rm WL},500}$ and the gas mass $M_{{\rm gas},500}$}. Point estimates yield a steep compactness dependence ($\delta'=-1.326\pm 0.091$). The result is not driven by individual systems (leave-one-out and bootstrap resampling) and remains negative under Monte Carlo uncertainty propagation while explicitly scanning the mass--radius error covariance. For uncorrelated errors ($\rho=0$; 30,000 realizations), the median response is $\delta'_{50}=-0.834$ with a $95\%$ interval $[-1.340,-0.343]$, remaining fully negative for $\rho\ge -0.4$. These findings establish a cross-scale, compactness-driven phenomenology for the dynamical mass discrepancy, while leaving the underlying physical mechanism (dark matter, modified gravity, baryon-halo coupling, or alternative interpretations) explicitly open.