1. Connecting the Galactic and Cosmological Length Scales: Dark Energy and The Cuspy-Core Problem By Achilles D. Speliotopoulos Talk Given at the Academia Sinica November 26, 2007

2. 100 kpc 10 Mpc 100 Mpc 4100 Mpc Galactic Supercluster Cluster Cosmological Range spans 5 orders of magnitude! Length scales of Phenomenon driven by Dark Matter and Dark Energy

3. The Cuspy-Core Problem

4. Dark Energy  DE =7.21x10 -29 g/cm 3

5. 100 kpc 10 Mpc 100 Mpc 4100 Mpc Galactic Supercluster Cluster Cosmological 14020 Mpc Links galactic and cosmological length scales Forming Galaxies = Lower Free Energy  8 = 0.68 ±0.11 (WMAP value = 0.761 -0.048 +0.049 ) Fractional density of matter that cannot be determined through gravity,  asymp = 0.196 ±0.017 Fractional density of matter that can be determined through gravity,  Dyn = 0.042 -0.026 +0.025 R   ±130 kpc

6. An Extended Lagrangian

7. Extended Geodesic Equations of Motion (GEOM) for massive test particles Only how fast D(x) changes matter! Curvature-dependent effective rest mass!

8. Extended GEOM for massless test particles (photons) Motion of massless test particles are not affected by the extension! Gravitational lensing and the deflection of light do not change!

9.  DE as the Cosmological Constant If T  =0, D(x) is a constant, the extended GEOM reduces to the GEOM!

10. T   Under Extended GEOM Spatial isotropy: Temporal variation v 2 = c 2 : Spatial variation: 1 st Law of Thermodynamics!

11. T  for Dust Under GEOM In the nonrelativistic limit!

12. A Choice for D(x) D’(x)<0 When  >>  DE /2 , D  0. Extended GEOM  GEOM. No observable 5 th force!

13. rHrH r II Region II v  v H Region III Region I v= v H r/r H A Model Galaxy

14. Idealized Velocity Curves

15. A Matter of Length Scales  -dependent length scale! Comparing length scales near the galactic core!

16. When  >>  DE /2  The Density Equation in Regions I and II Idealized density profiles! Our free energy!

17. The Solution in Region I Contributes positive term for  >0!

18. Free Energy Conjecture Like a Landau-Ginberg Theory, the system wants to be in a state the minimizes the Free Energy

19. The Solution in Region II Asymptotics Anzatz: f(r) <<  (r) for r large! Depends only on  DE,  , and symmetry! Contains no info on the structure of galaxy!

20. Perturbations The Decoupling of Length Scales Length scale set by DE No knowledge of galactic structure. Length scale set by r H Aside from BC, no knowledge of DE !.

21. The Free Energy in Region II Due only to  asymp. Independent of . ~ (  II 1 ) 2. Very small  = 2 gives state of lowest Free energy!

22. The Solution in Region III  <<  DE /2  f(u)  0 here! Density decreases exponentially fast here Fundamental scale is     DE /(1+4 1+a  ) 1/2  asymp (    DE /(1+4 1+a  ) 1/2 ) <<  DE /2  rasymp

23. Potentials What Can and Cannot be Seen Determined by  II.-  aymp Dominated by  aymp Dynamics driven by V eff not  ! Inferring mass from dynamics under gravity determines  II –  asymp. Mass of particles in  asymp cannot be “seen”!

24. The Link with Cosmology The theory naturally cuts off the density at ~ H /2 even though H was not put in at the beginning. What happens at the galactic scale is linked to the cosmological scale. Determined on galactic scale. WMAP Value:    = 1.51 ±0.011

25. Calculation of  8 Properties of the galaxy  8 dominated by . Result of rotation curves! Dominated by  asymp, H  1.

26.  8 from 1393 Galaxies Data Set De Blok et. al. (53)119.06.83.620.330.6130.0971.36 CF (348)179.12.97.430.350.840.180.43 Mathewson et. al. (935)169.51.915.190.420.6250.0891.34 Rubin et. Al. (57)223.37.61.240.142.790.822.46 Combined (1393)172.11.611.820.300.680.110.70 From WMAP : De Blok et. al. Data Set Rubin et. al. Data Set W. J. G. de Blok, S. S. McGaugh, A. Bosma, and V. C. Rubin, Astrophys. J. 552, L23 (2001). W. J. G. de Blok, and A. Bosma, Astro. Astrophys. 385, 816 (2002). S. S. McGaugh, V. C. Rubin, and W. J. G. de Blok, Astron. J. 122, 2381 (2001). V. C. Rubin, W. K. Ford, Jr., and N. Thonnard, Astrophys. J. 238, 471 (1980). V. C. Rubin, W. K. Ford, Jr., N. Thonnard, and D. Burstein, Astrophys. J. 261, 439 (1982). D. Burstein, V. C. Rubin, N. Thonnard, and W. K. Ford, Jr., Astrophys. J., part 1 253, 70 (1982). V. C. Rubin, D. Burstein, W. K. Ford, Jr., and N. Thonnard, Astrophys. J. 289, 81 (1985). Mathewson et. al. Data Set CF Data Set D. S Mathewson, V. L. Ford, and M. Buchhorn, Astrophys. J. Suppl. 82, 413 (1992).S. Courteau, Astron. J. 114, 2402 (1997).

27. Fractional Density of What Cannot be Seen Using Gravity Fractional density of non-baryonic (dark) matter from WMAP:

28. Fractional Density of What Can be Seen Using Gravity Fractional density of baryonic matter from WMAP:

29. Concluding Remarks Amazingly good agreement with WMAP Agreement supports Free Energy Conjecture R 200 =270 ±130 kpc, which agrees with observation   = 1.51 is small enough that it may be measurable in laboratory.  asymp =  m –  B and  Dyn   B. A numerical coincidence?