Reconstruction of the early Universe as a convex optimization problem

Brenier, Yann; Frisch, U.; Hénon, M.; Loeper, Grégoire; Matarrese, S.; Mohayaee, Roya; Sobolevski, A.

doi:10.1046/j.1365-2966.2003.07106.x

Cited by 156 publications

(212 citation statements)

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“…Then, at shell crossing one eigenvalue goes through zero and becomes negative (generically the collapse proceeds at different rates along the three axes). Thus, as is known Brenier et al 2003;Bernardeau & Valageas 2010b), the onset of shell crossing is associated with the change in sign of the Hessian determinant of ϕ L and with the loss of convexity of the Lagrangian potential ϕ L (the Hessian matrix is no longer positive-definite).…”

Section: Nonperturbative Correctionmentioning

confidence: 99%

“…Thus, an efficient algorithm for building such a reconstruction (which can then be used to estimate the velocity field) is provided by the MongeAmpère-Kantorovich method, which neglects multistreaming (Brenier et al 2003;Mohayaee et al 2006). Similar methods are used to reconstruct the baryon acoustic oscillations (BAO) of the density power spectrum, in order to improve the accuracy of cosmological distance measurements and tighten the constraints on cosmological parameters such as the amount and evolution of dark energy (Eisenstein et al 2007;Seo et al 2010).…”

Section: Introductionmentioning

confidence: 99%

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Impact of shell crossing and scope of perturbative approaches, in real and redshift space

Valageas¹

2010

A&A

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Aims. We study the effect of nonperturbative corrections associated with the behavior of particles after shell crossing on the matter power spectrum. We compare their amplitude with the perturbative terms that can be obtained within the fluid description of the system, to estimate the range of scales where such perturbative approaches are relevant. Methods. We use the simple Zeldovich dynamics as a benchmark, as it allows the exact computation of the full nonlinear power spectrum and of perturbative terms at all orders. Then, we introduce a "sticky model" that coincides with the Zeldovich dynamics before shell crossing but shows a different behavior afterwards. Thus, their power spectra only differ in their nonperturbative terms. We consider both the real-space and redshift-space power spectra. Results. We find that the potential of perturbative schemes is greater at higher redshift for a ΛCDM cosmology. For the real-space power spectrum, one can go up to order 66 of perturbation theory at z = 3, and to order 9 at z = 0, before the nonperturbative correction surpasses the perturbative correction of that order. This allows us to increase the upper bound on k where systematic theoretical predictions may be obtained by perturbative schemes, beyond the linear regime, by a factor ∼26 at z = 3 and ∼6.5 at z = 0. This provides a strong motivation to study perturbative resummation schemes, especially at high redshifts z ≥ 1. In the context of cosmological reconstruction methods, the Monge-Ampère-Kantorovich scheme appears to be close to optimal at z = 0. There also seems to be little room for improvement over current reconstruction methods of the baryon acoustic oscillations at z = 0. This can be understood from the small number of perturbative terms that are relevant at z = 0, before nonperturbative corrections dominate. We also point out that the rise of the power spectrum on the transition scale to the nonlinear regime strongly depends on the behavior of the system after shell crossing. We find similar results for the redshift-space power spectrum, with characteristic wavenumbers that are shifted to lower values as redshift-space distortions amplify higher order terms of the perturbative expansions while decreasing the resummed nonlinear power at high k.

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Section: Nonperturbative Correctionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Impact of shell crossing and scope of perturbative approaches, in real and redshift space

Valageas¹

2010

A&A

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“…For the EdS model the Euler-Poisson equations take the following form (cf. Brenier et al 2003, and references therein):…”

Section: Basic Equations In Eulerian Coordinatesmentioning

confidence: 99%

Time-analyticity of Lagrangian particle trajectories in ideal fluid flow

2014

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It is known that the Eulerian and Lagrangian structures of fluid flow can be drastically different; for example, ideal fluid flow can have a trivial (static) Eulerian structure, while displaying chaotic streamlines. Here we show that ideal flow with limited spatial smoothness (an initial vorticity that is just a little better than continuous), nevertheless has time-analytic Lagrangian trajectories before the initial limited smoothness is lost. For proving such results we use a little-known Lagrangian formulation of ideal fluid flow derived by Cauchy in 1815 in a manuscript submitted for a prize of the French Academy. This formulation leads to simple recurrence relations among the time-Taylor coefficients of the Lagrangian map from initial to current fluid particle positions; the coefficients can then be bounded using elementary methods. We first consider various classes of incompressible fluid flow, governed by the Euler equations, and then turn to highly compressible flow governed by the Euler-Poisson equations, a case of cosmological relevance. The recurrence relation associated to the Lagrangian formulation of these incompressible and the compressible problems are so closely related that the proofs of time-analyticity are basically identical.

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“…A few years ago, a group of cosmologists and mathematicians came up with a novel idea: to use both the current state s(t 0 ) and the original state s(t r ) when predicting the past states s(t). This idea has led to successful predictions: specifically, at the large-scale level, they succeeded in using the 3K state and current locations of the galaxy clusters to "predict" (reconstruct) the velocities of these clusters (see, e.g., [1,4,8]; see also [5]). …”

Section: Explanation Of Main Cognitive Paradigms Of Physicsmentioning

confidence: 99%