1. Breaking of spherical symmetry in gravitational collapse

2. The problem: « Spherical Collapse » in Newtonian gravity N point particles randomly distributed in a sphere Cold i.e. velocities=0 “Turn on” Newtonian gravity... [Later: will add initial velocities and consider other “profiles”]

3. Questions: What is “final” state? (Is there one ?) What is evolution? How do these depend on initial conditions..? …..

4. Motivations Cosmological structure formation Galaxy formation Paradigmatic model to explore understanding of gravitational dynamics

5. The “spherical collapse model” in cosmology Single spherical overdensity embedded in expanding universe Analytical solution  singular collapse in a finite time With simple hypotheses: predict halo mass function from power spectrum of initial conditions From “turnaround” equivalent to an isolated cold sphere Internal structure of halos determined a priori by conditions at turnaround…

6. The “spherical collapse model” in galaxy formation Observed in 1980s that cold collapse can lead to virialized structures resembling elliptical galaxies Spherical symmetry breaking due to “radial orbit instability” Remains a toy model for galaxy formation: how much about galaxies can we understand using gravity alone ??

7. The Newtonian N -body problem: a general remark

8. Does evolution depend non-trivially on N (>>1)? Answer: At fixed time, NO, provided N is sufficiently large (“collisionless limit” ) On a sufficiently long time scale, YES, (“collisional relaxation”) Related to “discreteness issue” in cosmological simulations

9. 1: Cold quasi-uniform initial conditions

10. Evolution of a finite (initially) uniform system

13. Evolution of a self-gravitating system

14. Following scheme of dynamical evolution has emerged Initial Conditions  Quasi-Stationary State  Thermal Equilibrium (if defined) “violent relaxation” time-scale: Τ mf “thermal relaxation” time-scale: Τ coll ~ N α Τ mf (α > 0) Dynamics of “long-range interacting” systems

15. Questions… How is the singularity in collapse regulated at finite N? How do properties of virialized state depend on N?

16. Range of N explored: few x 10 2  few x 10 5 Simulate with GADGET Force is softened at a scale  Checks:  Energy conservation  Convergence tests for all of numerical parameters, and   Cross-check smaller N results with an N 2 code without smoothing Numerical simulations

17. Minimal size: theory At any finite N, the singularity is regulated by the perturbations. A minimal collapse radius R min can be estimated analytically, assuming Perturbations are small and evolve as in fluid limit Collapse stops when perturbations reach some given amplitude This gives

18. Minimal size: phenomenology In agreement with previous studies (Aarseth et al. 1988, Boily et al 2002) we find

19. Cold collapse The energy budget Total initial energy E 0 is asymptotically sum of three non-zero energies: W n : potential energy of bound (negative energy) particles K n : kinetic energy of bound (negative energy) particles K p : kinetic energy of ejected (positive energy) particles Further virialization implies Thus one unknown - choose K p. For mass likewise, we define f p = fraction of mass with positive energy Question: Do f p and K p depend on N?

20. Cold collapse Mass ejection Slow (approx logarithmic) increase in ejected mass as a function of N

21. Cold collapse Energy ejection

22. Cold collapse Mechanism of energy ejection Detailed study of collapse phase reveals mechanism of ejection: Outer particles lag (compared to SCM) on average, “arrive” late They then “scatter” off the inner re-expanding core With fixed “lagging mass”, the observed scaling can be recovered. [Could this be important for dark matter searches? ]

23. Question: Energy and mass are “not conserved” Can momentum and angular momentum be generated ? In principle YES if rotational symmetry is broken…

27. Symmetry breaking is due to amplification of initial finite N fluctuations.. [cf “Liu-Mestel-Shu mechanism”, “Zeldovich pancaking”..]

28. 2: Breaking of symmetry in cold collapse, power law density profiles

29. 2: Breaking of symmetry in cold collapse Final state as a function of α (b=0, N=10 4 )

30. Case 2: Breaking of symmetry in cold collapse N=10 5, α=1, b=0

31. This strong symmetry breaking is attributed to “radial orbit instability”: Virial equilibria with radial orbits are unstable to modes breaking spherical symmetry (Polyachenko 1981, Merritt & Aguilar 1985…)

32. 3: Mechanisms of symmetry breaking ?

33. “Plain” gravitational instability OR Radial orbit instability [Or both ? Or other? ]

34. Case 2: Breaking of symmetry in cold collapse N=10 5, α=1, b=0

35. 3: Symmetry breaking in “warm” collapse N=10 5, α=1, b=0.15

36. From cold to warm…

37. Final asymmetry as a function of N

38. Temporal evolution as a function of N

39. Conclusion of our study: Very cold/smaller N: GI playing central role “Cold-ish”/sufficiently large N: ROI dominates Warmer IC: GI only Angular momentum generation is most significant in first case.

40. Relevance to cosmology?? 1)Discreteness issue in cosmological simulations ROI apparently crucial in generation of NFW halos Is this instability being triggered by finite N fluctuations? “Convergence studies” are unlikely to have shown this.. Are we sure that other instabilities are not being seeded by finite N effects? 2) Cold vs Warm dark matter signatures?

41. Acknowledgement: My collaborators David BENHAIEM (LPNHE  La Sapienza, Rome) Tirawut WORRAKITPOONPON (Rajamangala University, Thailand) Francesco SYLOS LABINI (Rome, Italy) Bruno MARCOS (Nice, France)