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The numerical action method:prospects and problems
Steven Phelps, the Technion, HaifaWith: Adi Nusser, Vincent Desjacques, TechnionJanuary 2006
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Questions to be addressed by the numerical action program:
• What is the dynamical history of the nearby universe?
• What are the masses of nearby galaxies?
• What were the initial conditions?
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• Galaxies can be approximated, to high redshift, as non-merging point particles. Predicted paths are assumed to represent the center-of-mass motion of the galaxy+halo systems.
• The masses of galaxies are proportional to their luminosities. (can be relaxed)
• The cosmological parameters H0, Ω0, Λ are given (can also be relaxed)
Starting assumptions:
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The cosmological action:
In practice this is computed and solved per particle.
Numerical methods of finding stationary points in the action: Gradient method (Peebles 1989) Matrix inversion (Peebles 1994) Fast Action Method (Nusser and Branchini 2000)
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Mixed boundary conditions in the action
Initial time: early growth of velocities obeys linear perturbation theory (imposed by hand)
Final time: angular positions, and either redshifts or distances are fixed. The former is accomplished through a canonical transformation of the Hamiltonian:
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Predicted galaxy orbits for the Local Group
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Proper motion predictions for M31
Proper motion predictions for NGC 6822
Ensemble of 24 solutions (circles at center shows proposed SIM resolution)
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A high-resolution N-body simulation (initial timestep)
Stoehr F. et al, 2003, MNRAS 335, L84
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Radial distance errors in a numerical action solution
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Successful recovery of radial distances
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Numerical action distance predictions vs. Hubble-flow
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A view of the “sky” from the reference halo
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Predicted orbits vs. actual simulation halo orbits
Heaviest haloes All other haloes
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Errors in numerical action reconstruction
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Numerical action recovery of halo masses
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Conclusions
Prospects
• Spectacular predictions of proper motions that will be confirmed or disconfirmed experimentally (timescale of ~10 years).
• With proper motions, constraining individual masses of nearby galaxies may be possible.
Problems
• Some degeneracy in solutions.• High-density regions (poor distances and proper motions).• Degeneracies in orbits (good distances, poor proper motions).
The future
Constrained realizations of the Local Group.