École de Cosmologie VII
Reconstruction
de l'Univers primitif traité comme
un
problème d'optimisation convexe
Uriel
FRISCH
Cassini-OCA
Reconstruction
de l'Univers primitif traité comme
un
problème d'optimisation convexe
Uriel
FRISCH
Cassini-OCA
| We
show that the deterministic past history of the Universe can be
uniquely
reconstructed from the knowledge of the present mass 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 h-1Mpc we propose and implement
an efficient Monge-Ampère-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 efficient algorithms with
not more than N3 time complexity and reasonable CPU
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-Ampère 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. |
