THE RECONSTRUCTION OF THE EARLY UNIVERSE: GRAVITATION AND OPTIMAL TRANSPORTATION, A REVIEW Yann BRENIER CNRS-CMLS, Ecole Polytechnique, Palaiseau Optimal Transport in the Applied Sciences, RICAM-JKU, Linz, 8-12 Dec 2014 Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 1 / 25
RECONSTRUCTION OF THE EARLY UNIVERSE Following Peebles 1989, Frisch and coauthors (Nature 417) 2002, have managed to reconstruct the history of the universe from the observable distribution of galaxies. Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 2 / 25
THE SEMI-NEWTONIAN GRAVITATIONAL MODEL OF THE EARLY UNIVERSE (Zeldovich, Peebles...) The trajectory t → X ( t , a ) ∈ T 3 = R 3 / Z 3 of each "particle"(*) obeys d 2 X 2t dt 2 + dX � dt + ∇ ϕ ( t , X ) = 0 , 1 + t △ ϕ = ρ = T 3 δ ( x − X ( t , a )) da 3 where ρ ( t , x ) and ϕ ( t , x ) respectively denote the density field (supposed to be of unit average) and the gravitational potential. Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 3 / 25
THE SEMI-NEWTONIAN GRAVITATIONAL MODEL OF THE EARLY UNIVERSE (Zeldovich, Peebles...) The trajectory t → X ( t , a ) ∈ T 3 = R 3 / Z 3 of each "particle"(*) obeys d 2 X 2t dt 2 + dX � dt + ∇ ϕ ( t , X ) = 0 , 1 + t △ ϕ = ρ = T 3 δ ( x − X ( t , a )) da 3 where ρ ( t , x ) and ϕ ( t , x ) respectively denote the density field (supposed to be of unit average) and the gravitational potential. General relativity is taken into account only through the terms in red which take into account Big Bang effects. (*) for computations, a "particle" roughly corresponds to a ...cluster of galaxies! Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 3 / 25
THE INITIAL CONSTRAINTS Because of the Big-Bang terms, the equations d 2 X 2t dt 2 + dX � dt + ∇ ϕ ( t , X ( t , a )) = 0 , 1 + t △ ϕ = ρ = δ ( x − X ( t , a )) da 3 are degenerate at t = 0. At initial time, we get a continuum of particles. Their density is uniform and their initial velocity is "slaved" by the gravitational potential: Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 4 / 25
THE INITIAL CONSTRAINTS Because of the Big-Bang terms, the equations d 2 X 2t dt 2 + dX � dt + ∇ ϕ ( t , X ( t , a )) = 0 , 1 + t △ ϕ = ρ = δ ( x − X ( t , a )) da 3 are degenerate at t = 0. At initial time, we get a continuum of particles. Their density is uniform and their initial velocity is "slaved" by the gravitational potential: dX 0 ρ 0 ( x ) = 1 , X 0 ( a ) = a , dt ( a ) = −∇ ϕ 0 ( a ) Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 4 / 25
THE INITIAL CONSTRAINTS Because of the Big-Bang terms, the equations d 2 X 2t dt 2 + dX � dt + ∇ ϕ ( t , X ( t , a )) = 0 , 1 + t △ ϕ = ρ = δ ( x − X ( t , a )) da 3 are degenerate at t = 0. At initial time, we get a continuum of particles. Their density is uniform and their initial velocity is "slaved" by the gravitational potential: dX 0 ρ 0 ( x ) = 1 , X 0 ( a ) = a , dt ( a ) = −∇ ϕ 0 ( a ) which is totally different from usual N-body Newton’s gravitation. Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 4 / 25
ZELDOVICH APPROXIMATION A very simple approximate solution was proposed by Zeldovich ∼ 1970 for the semi-newtonian model d 2 X 2t dt 2 + dX � dt + ∇ ϕ ( t , X ) = 0 , ρ = δ ( x − X ( t , a )) da = 1 + t △ ϕ 3 Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 5 / 25
ZELDOVICH APPROXIMATION A very simple approximate solution was proposed by Zeldovich ∼ 1970 for the semi-newtonian model d 2 X 2t dt 2 + dX � dt + ∇ ϕ ( t , X ) = 0 , ρ = δ ( x − X ( t , a )) da = 1 + t △ ϕ 3 ρ ( t , x ) − 1 → : X ( t , a ) = a − t ∇ ϕ 0 ( a ) , △ ϕ 0 ( x ) = lim t t → 0 Each particle just travels with a constant velocity due to the initial density fluctuation, until a collision ocurs, which is somewhat reminiscent of Lucretius’ (99-55 BC) "DE RERUM NATURA". Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 5 / 25
1d Zeldovich solutions with sticky collisions horizontal : space / vertical : time 2.5 2 1.5 1 0.5 -1.5 -1 -0.5 0 0.5 1 1.5 Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 6 / 25
DE RERUM NATURA LIBER SECUNDUS 216 − 224 LUCRETIUS ( 99 − 55BC ) Quod nisi declinare solerent (corpora), omnia deorsum imbris uti guttae caderent per inane profundum ...Ita nihil umquam natura creasset. But if (corpora) were not in the habit of deviating, they would all fall straight down through the depths of the void, like drops of rain... In that case, nature would never have produced anything. Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 7 / 25
RECONSTRUCTING THE EARLY UNIVERSE? It is plausible (Peebles 1989) to reconstruct the early universe from the only knowledge of the observed density field ρ ( T , x ) . Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 8 / 25
RECONSTRUCTING THE EARLY UNIVERSE? It is plausible (Peebles 1989) to reconstruct the early universe from the only knowledge of the observed density field ρ ( T , x ) . As a matter of fact, the only initial condition we need to recover is ρ ( t , x ) − 1 ρ ′ 0 ( x ) = lim = △ ϕ 0 ( x ) t t ↓ 0 which is supposed to be a random field of very small amplitude according to the quantum theory of the VERY early universe. Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 8 / 25
MASS CONCENTRATIONS IN THE OBSERVED DISTRIBUTION OF MASS IN THE UNIVERSE Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 9 / 25
THE MONGE-AMPERE-KANTOROVICH PROBLEM Using the Zeldovich approximation, Uriel Frisch (and coauthors, Nature 417, 2002) observed that the reconstruction of the early universe is just the Monge problem with quadratic cost between the Lebesgue measure and the "observed" density field ρ ( T , x ) Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 10 / 25
THE MONGE-AMPERE-KANTOROVICH PROBLEM Using the Zeldovich approximation, Uriel Frisch (and coauthors, Nature 417, 2002) observed that the reconstruction of the early universe is just the Monge problem with quadratic cost between the Lebesgue measure and the "observed" density field ρ ( T , x ) if the Zeldovich map X ( T , a ) = a − T ∇ ϕ 0 ( a ) has convex potential. Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 10 / 25
THE MONGE-AMPERE-KANTOROVICH PROBLEM Using the Zeldovich approximation, Uriel Frisch (and coauthors, Nature 417, 2002) observed that the reconstruction of the early universe is just the Monge problem with quadratic cost between the Lebesgue measure and the "observed" density field ρ ( T , x ) if the Zeldovich map X ( T , a ) = a − T ∇ ϕ 0 ( a ) has convex potential. Frisch and collaborators designed an effective computational method, based on Bertsekas’ algorithm, for about 10 4 particles. Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 10 / 25
THE MONGE-AMPERE GRAVITATION MODEL The Monge approach to the problem has two drawbacks: Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 11 / 25
THE MONGE-AMPERE GRAVITATION MODEL The Monge approach to the problem has two drawbacks: i) it relies on the Zeldovich approximation which looks very crude; Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 11 / 25
THE MONGE-AMPERE GRAVITATION MODEL The Monge approach to the problem has two drawbacks: i) it relies on the Zeldovich approximation which looks very crude; ii) it rules out any collision effect. (As well known in optimal transport theory, collisions are possible only at the final time T .) Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 11 / 25
THE MONGE-AMPERE GRAVITATION MODEL The Monge approach to the problem has two drawbacks: i) it relies on the Zeldovich approximation which looks very crude; ii) it rules out any collision effect. (As well known in optimal transport theory, collisions are possible only at the final time T .) So, I have suggested a slightly different approach to the problem (Confluentes Math. 2011), still based on optimal transport theory, leaving several analytic and computational issues largely open. Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 11 / 25
MONGE-AMPERE GRAVITATION (MAG) In Confl. Math 2011, we substitute Monge-Ampère for Poisson: ρ ( t , x ) = det ( I + tD 2 ϕ ( t , x )) instead of ρ ( t , x ) = 1 + t △ ϕ ( t , x ) Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 12 / 25
MONGE-AMPERE GRAVITATION (MAG) In Confl. Math 2011, we substitute Monge-Ampère for Poisson: ρ ( t , x ) = det ( I + tD 2 ϕ ( t , x )) instead of ρ ( t , x ) = 1 + t △ ϕ ( t , x ) i) This is exact in 1d; Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 12 / 25
MONGE-AMPERE GRAVITATION (MAG) In Confl. Math 2011, we substitute Monge-Ampère for Poisson: ρ ( t , x ) = det ( I + tD 2 ϕ ( t , x )) instead of ρ ( t , x ) = 1 + t △ ϕ ( t , x ) i) This is exact in 1d; ii) asymptotically correct at early times and for weak fields; Yann Brenier (CNRS, Ecole Polytechnique) The EUR RICAM-JKU-LINZ 2014 12 / 25
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