It is shown here that in a flat, cold dark matter (CDM) dominated Universe with positive cosmological constant (Λ), modelled in terms of a Newtonian and collisionless fluid, particle trajectories are analytical in time (representable by a convergent Taylor series) until at least a finite time after decoupling. The time variable used for this statement is the cosmic scale factor, i.e., the "a-time", and not the cosmic time. For this, a Lagrangian-coordinates formulation of the Euler-Poisson equations is employed, originally used by Cauchy for 3-D incompressible flow. Temporal analyticity for ΛCDM is found to be a consequence of novel explicit all-order recursion relations for the a-time Taylor coefficients of the Lagrangian displacement field, from which we derive the convergence of the a-time Taylor series. A lower bound for the a-time where analyticity is guaranteed and shell-crossing is ruled out is obtained, whose value depends only on Λ and on the initial spatial smoothness of the density field. The largest time interval is achieved when Λ vanishes, i.e., for an Einstein-de Sitter universe. Analyticity holds also if, instead of the a-time, one uses the linear structure growth D-time, but no simple recursion relations are then obtained. The analyticity result also holds when a curvature term is included in the Friedmann equation for the background, but inclusion of a radiation term arising from the primordial era spoils analyticity.
Physica D in pressWe are interested in the tail behavior of the pdf of mass density within the one and d-dimensional Burgers/adhesion model used, e.g., to model the formation of large-scale structures in the Universe after baryon-photon decoupling. We show that large densities are localized near "kurtoparabolic" singularities residing on space-time manifolds of codimension two (d ≤ 2) or higher (d ≥ 3). For smooth initial conditions, such singularities are obtained from the convex hull of the Lagrangian potential (the initial velocity potential minus a parabolic term). The singularities contribute universal power-law tails to the density pdf when the initial conditions are random. In one dimension the singularities are preshocks (nascent shocks), whereas in two and three dimensions they persist in time and correspond to boundaries of shocks; in all cases the corresponding density pdf has the exponent −7/2, originally proposed by E, Sinai (1997 Phys. Rev. Lett. 78, 1904) for the pdf of velocity gradients in one-dimensional forced Burgers turbulence. We also briefly consider models permitting particle crossings and thus multi-stream solutions, such as the Zel'dovich approximation and the (Jeans)-Vlasov-Poisson equation with single-stream initial data: they have singularities of codimension one, yielding power-law tails with exponent −3.su quell' immenso baratro di stelle sopra quei gruppi, sopra quegli ammassi, quel seminío, quel balenío di stelle Giovanni Pascoli, from La Vertigine [1]
Two prized papers, one by Augustin Cauchy in 1815, presented to the French Academy and the other by Hermann Hankel in 1861, presented to Göttingen University, contain major discoveries on vorticity dynamics whose impact is now quickly increasing. Cauchy found a Lagrangian formulation of 3D ideal incompressible flow in terms of three invariants that generalize to three dimensions the now well-known law of conservation of vorticity along fluid particle trajectories for two-dimensional flow. This has very recently been used to prove analyticity in time of fluid particle trajectories for 3D incompressible Euler flow and can be extended to compressible flow, in particular to cosmological dark matter. Hankel showed that Cauchy's formulation gives a very simple Lagrangian derivation of the Helmholtz vorticity-flux invariants and, in the middle of the proof, derived an intermediate result which is the conservation of the circulation of the velocity around a closed contour moving with the fluid. This circulation theorem was to be rediscovered independently by William Thomson (Kelvin) in 1869. Cauchy's invariants were only occasionally cited in the 19th century -besides Hankel, foremost by George Stokes and Maurice Lévy -and even less so in the 20th until they were rediscovered via Emmy Noether's theorem in the late 1960, but reattributed to Cauchy only at the end of the 20th century by Russian scientists.
It is shown by perturbation techniques and numerical simulations that the inverse cascade of kinkantikink annihilations, characteristic of the Kolmogorov flow in the slightly supercritical Reynolds number regime, is halted by the dispersive action of Rossby waves in the b-plane approximation. For b ! 0, the largest excited scale is~ln1͞b and differs strongly from what is predicted by standard dimensional phenomenology which ignores the depletion of nonlinearity. [S0031-9007 (99)09281-9] PACS numbers: 47.35. + i, 05.45.Yv, 47.27.Ak, 92.60.EkPlanetary-scale flow is subject to the competing effects of quasi-two-dimensional turbulence and Rossby waves (b effect). It is known from phenomenological arguments and numerical simulations that the inverse cascade which characterizes the large-scale dynamics of two-dimensional turbulence in planetary flow can be halted by Rossby wave dispersion and that the ensuing flow exhibits alternating jets [1][2][3][4][5]. The standard argument of Rhines [1] rests on a comparison between the local eddy turnover time and the period of Rossby waves. More generally,
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.