1. 0 The numerical action method: prospects and problems Steven Phelps, the Technion, Haifa With: Adi Nusser, Vincent Desjacques, Technion January 2006

2. 1 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?

3. 2 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 H 0, Ω 0, Λ are given (can also be relaxed) Starting assumptions:

4. 3 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)

5. 4 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:

6. 5 Discretizing the action

7. 6 Predicted galaxy orbits for the Local Group

8. 7 Proper motion predictions for M31 Proper motion predictions for NGC 6822 Ensemble of 24 solutions (circles at center shows proposed SIM resolution)

9. 8 A high-resolution N-body simulation (initial timestep) Stoehr F. et al, 2003, MNRAS 335, L84

10. 9 Radial distance errors in a numerical action solution

11. 10 Successful recovery of radial distances

12. 11 Numerical action distance predictions vs. Hubble-flow

13. 12 A view of the “sky” from the reference halo

14. 13 Predicted orbits vs. actual simulation halo orbits Heaviest haloes All other haloes

15. 14 Errors in numerical action reconstruction

16. 15 Numerical action recovery of halo masses

17. 16 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.