A Universal Relation Between Primordial Density-Potential Cross-correlation Coefficient and Spin Factor Distribution

Recent studies have revealed that the key properties of visible galaxies like their optical sizes, stellar ages, star formation rates and morphologies are closely linked with the angular momenta of their host dark matter halos. According to the linear tidal torque theory, the halo angular momentum, as a conserved quantity, is directly proportional to the primordial spin factor, $\tau$, defined as the degree of misalignment between the principal axes of the initial density and potential Hessian matrices, which were found by numerical experiments to follow a Gamma distribution, fully characterized by its mean and variance. In this study, we heuristically develop an analytic expression for the mean and variance of $\tau$ in terms of the initial density-potential cross-correlation coefficient, $q$. Analyzing a dataset from the Multiverse simulations performed for both of the flat $\Lambda$CDM and $w$CDM cosmologies, we prove that this analytic expression is universally valid in describing how the mean and variance of $\tau$ change with $q$, regardless of the smoothing scales for both of the cosmologies. Given the prior finding that the $\tau$-distribution can be reconstructed from the observable galaxy size distribution, this universal analytic expression may allow us to determine $q$ from the same observable via the mean and variance of $\tau$. We discuss a possibility of constraining the early universe physics from the reconstructed $q$ via our heuristic model, without suffering from cosmological degeneracies.