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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.