The matter power spectrum, $P(k)$, is one of the fundamental quantities in the study of large-scale structure in cosmology. Here, we study its small-scale asymptotic limit, and show that for cold dark matter in $d$ spatial dimensions, $P(k)$ has a universal $k^{-d}$ asymptotic scaling with the wave-number $k$, for $k \gg k_{\rm nl}$, where $k_{\rm nl}^{-1}$ denotes the length scale at which non-linearities in gravitational interactions become important. We propose a theoretical explanation for this scaling, based on a non-perturbative analysis of the system's phase-space structure. Gravitational collapse is shown to drive a turbulent phase-space flow of the quadratic Casimir invariant, where the linear and non-linear time scales are balanced, and this balance dictates the $k$ dependence of the power spectrum. A parallel is drawn to Batchelor turbulence in hydrodynamics, where large scales mix smaller ones via tidal interactions. The $k^{-d}$ scaling is also derived by expressing $P(k)$ as a phase-space integral in the framework of kinetic field theory, which is analysed by the saddle-point method; the dominant critical points of this integral are precisely those where the time scales are balanced. The coldness of the dark-matter distribution function -- its non-vanishing only on a $d$-dimensional sub-manifold of phase-space -- underpins both approaches. The theory is accompanied by $1\mathrm{D}$ Vlasov--Poisson simulations, which confirm it.
For the 696 trans-Neptunian objects (TNOs) with absolute magnitudes $5.5 < H_r < 8.2$ detected in the Dark Energy Survey (DES), we characterize the relationships between their dynamical state and physical properties -- namely $H_r$, indicating size; colors, indicating surface composition; and flux variation semi-amplitude $A$, indicating asphericity and surface inhomogeneity. We seek ``birth'' physical distributions that can recreate these parameters in every dynamical class. We show that the observed colors of these TNOs are consistent with 2 Gaussian distributions in $griz$ space, ``near-IR bright'' (NIRB) and ``near-IR faint'' (NIRF), presumably an inner and outer birth population, respectively. We find a model in which both the NIRB and NIRF $H_r$ and $A$ distributions are independent of current dynamical states, supporting their assignment as birth populations. All objects are consistent with a common rolling $p(H_r)$, but NIRF objects are significantly more variable. Cold classicals (CCs) are purely NIRF, while hot classical (HC), scattered, and detached TNOs are consistent with $\approx70\%$ NIRB, and resonances' NIRB fractions show significant variation. The NIRB component of the HCs and of some resonances have broader inclination distributions than the NIRFs, i.e. their current dynamics retains information about birth location. We find evidence for radial stratification within the birth NIRB population, in that HC NIRBs are on average redder than detached or scattered NIRBs; a similar effect distinguishes CCs from other NIRFs. We estimate total object counts and masses of each class within our $H_r$ range. These results will strongly constrain models of the outer solar system.