Back to the primordial Universe by a Monge–Ampère–Kantorovich optimization scheme

Mohayaee, Roya; Frisch, U.; Matarrese, S.; Sobolevski, A.

doi:10.1051/0004-6361:20030719

Cited by 26 publications

(29 citation statements)

References 35 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Furthermore such models can be employed to improve generic inhomogeneous collapse models (Buchert et al 2000;Kerscher et al 2001), and to enhance the power of reconstructions of initial data from present-day density and peculiar-velocity fields (e.g., Croft & Gaztañaga 1997;Brenier et al 2003;Mohayaee et al 2003 for recent efforts). In the latter context, the proposed non-perturbative approximation, based on the relation w ∝ P, Eq.…”

Section: Summary and Prospectsmentioning

confidence: 99%

Adhesive gravitational clustering

Buchert

Domínguez

2005

A&A

160

View full text Add to dashboard Cite

Abstract. The notion of adhesion has been advanced for the phenomenon of stabilization of large-scale structure emerging from gravitational instability of a cold medium. Recently, the physical origin of adhesion has been identified: a systematic derivation of the equations of motion for the density and the velocity fields leads naturally to the key equation of the "adhesion approximation" -however, under a set of strongly simplifying assumptions. In this work, we provide an evaluation of the current status of adhesive gravitational clustering and a clear explanation of the assumptions involved. Furthermore, we propose systematic generalizations with the aim to relax some of the simplifying assumptions. We start from the general Newtonian evolution equations for self-gravitating particles on an expanding Friedmann background and recover the popular "dust model" (pressureless fluid), which breaks down after the formation of density singularities; then we investigate, in a unified framework, two other models which, under the restrictions referred to above, lead to the "adhesion approximation". We apply the Eulerian and Lagrangian perturbative expansions to these new models and, finally, we discuss some non-perturbative results that may serve as starting points for workable approximations of non-linear structure formation in the multi-stream regime. In particular, we propose a new approximation that includes, in limiting cases, the standard "adhesion model" and the Eulerian as well as Lagrangian first-order approximations.Key words. gravitation -methods: analytical -cosmology: theory -cosmology: large-scale structure of Universe Gravitational clusteringThe present work aims to push analytical modeling of structure formation into a regime that may be placed between the formation epoch of large-scale structure and the onset of virialization of gravitationally bound objects. Phenomenologically, this regime is characterized by a stabilization of structures that formed out of gravitational instability of a cold medium. The physical origin of this adhesive clustering effect is the balance of gravitational forces and dynamical stresses in collisionless matter. The latter arise from a subsequently establishing multi-stream hierarchy within collapsing high-density regions. Although multi-stream forces tend to disperse structures, the resulting effect together with gravity tends to stabilize them. This regime is summarized by the term non-dissipative gravitational turbulence advanced by Gurevich & Zybin (1995). The models we investigate are a focus of current research, since Appendices A-C are only available in electronic form at http://www.edpsciences.org efforts to simulate Hubble volumes of the Universe and to understand galaxy halo formation are faced within a single approach. However, we take a more conservative point of view to understand the evolution of structure on galaxy cluster scales (for a recent theoretical attempt to address halo structure within kinetic theory see Ma & Bertschinger 2004).The current status of ...

show abstract

Section: Summary and Prospectsmentioning

confidence: 99%

Adhesive gravitational clustering

Buchert

Domínguez

2005

A&A

160

View full text Add to dashboard Cite

show abstract

“…Since (21) provides w as a local functional of the displacement field P, it may substantially enhance the power of reconstruction methods (Croft & Gaztañaga 1997;Susperregi & Buchert 1997;Courteau & Willick 2000;Courteau & Dekel 2001;Brenier et al 2003;Mohayaee et al 2003). As mentioned above, the integral (21) generalizes the assumption (10), which itself lies at the basis of most analytical models (as summarized in this paper) and which already furnishes a more general ansatz for reconstruction methods, compared to those that are currently implemented, e.g.…”

Section: Summary and Prospectsmentioning

confidence: 99%

The non-perturbative regime of cosmic structure formation

Buchert¹

2006

A&A

View full text Add to dashboard Cite

This paper focusses on the barely understood gap between the weakly nonlinear regime of structure formation and the onset of the virialized regime. While the former is accessed through perturbative calculations and the latter through virialization conditions incorporating dynamical stresses that arise in collisionless self-gravitating systems due to velocity dispersion forces, the addressed regime can only be understood through non-perturbative models. We here present an exact Lagrangian integral that provides a tool to access this regime. We derive a transport equation for the peculiar-gravitational field strength and integrate it along comoving trajectories of fluid elements. The so-obtained integral provides an exact expression that solves the longitudinal gravitational field equation in general. We argue that this integral provides a powerful approximation beyond the Lagrangian perturbative regime, and discuss its relation to known approximations, among them Lagrangian perturbation solutions including the Zel'dovich approximation and approximations for adhesive gravitational clustering, including the adhesion approximation. Furthermore, we propose an iteration scheme for a systematic analytical and numerical construction of trajectory fields. The integral may also be employed to improve inverse reconstruction techniques.

show abstract

“…We compared these results with those of the PIZA reconstruction method (see and Croft & Gaztañaga 1997), which gives a 2‐monotone but not necessarily optimal pairing between Lagrangian and Eulerian positions. We applied the PIZA method on the Δ x grid and obtained typically 30–40 per cent exactly reconstructed positions, but severe non‐uniqueness: for two different seeds of the random generator used to set up the initial tentative assignment, only about half of the exactly reconstructed positions were the same (see figs 3 and 7 of Mohayaee et al 2003 for an illustration). We also implemented a modification of the PIZA method establishing 3‐monotonicity (monotonicity with respect to interchanges of three points instead of pairs) and checked that it does not give a significant improvement over the original PIZA.…”

Section: Testing the Mak Reconstructionmentioning

confidence: 99%

Reconstruction of the early Universe as a convex optimization problem

Brenier

Frisch

Hénon

et al. 2003

Monthly Notices of the Royal Astronomical Society

Self Cite

156

212

View full text Add to dashboard Cite

We show that the deterministic past history of the Universe can be uniquely reconstructed from the knowledge of the present mass density field, the latter being inferred from the 3D distribution of luminous matter, assumed to be tracing the distribution of dark matter up to a known bias. Reconstruction ceases to be unique below those scales -- a few Mpc -- where multi-streaming becomes significant. Above 6 Mpc/h we propose and implement an effective Monge-Ampere-Kantorovich method of unique reconstruction. At such scales the Zel'dovich approximation is well satisfied and reconstruction becomes an instance of optimal mass transportation, a problem which goes back to Monge (1781). After discretization into N point masses one obtains an assignment problem that can be handled by effective algorithms with not more than cubic time complexity in N and reasonable CPU time requirements. Testing against N-body cosmological simulations gives over 60% of exactly reconstructed points. We apply several interrelated tools from optimization theory that were not used in cosmological reconstruction before, such as the Monge-Ampere equation, its relation to the mass transportation problem, the Kantorovich duality and the auction algorithm for optimal assignment. Self-contained discussion of relevant notions and techniques is provided.Comment: 26 pages, 14 figures; accepted to MNRAS. Version 2: numerous minour clarifications in the text, additional material on the history of the Monge-Ampere equation, improved description of the auction algorithm, updated bibliography. Version 3: several misprints correcte

show abstract

Back to the primordial Universe by a Monge–Ampère–Kantorovich optimization scheme

Cited by 26 publications

References 35 publications

Adhesive gravitational clustering

Adhesive gravitational clustering

The non-perturbative regime of cosmic structure formation

Reconstruction of the early Universe as a convex optimization problem

Contact Info

Product

Resources

About