1. QUADRATIC POTENTIALS, AND ROTATION CURVES IN THE CONFORMAL THEORY James G. O’Brien APS April Meeting May 1 st, 2011

2. Galactic rotation rates  We would expect galactic rotation curves to look like curve A, but find they look like B.  This could be accounted for if there was a “halo” of unseen matter surrounding the galaxies.  These rotation rates were the original motivation for suggesting the existence of dark matter. Picture Source: http://en.wikipedia.org/wiki/Galaxy_rotation_problem

3. Conformal Theory where: The Conformal Theory was originally developed by Weyl, and later re- explored by Mannheim and Kazanas. It is a fourth order, scale invarient renormalizable gravitational theory:

4. Conformal Theory The Schwarzschild like solution in conformal theory can be solved via:

5. Conformal Theory

6. Yields after some work:

7. Conformal theory - Global Since the conformal theory uses a fourth Poisson equation, we are not free to use only the local considerations as in Newtonian gravity. We thus need to include a contribution from the cosmology, and inhomogeneities to the cosmology.

8. Cosmology term We can implement a Robertson Walker metric in static coordinates via the following transformation Brings the metric to the following form, which we can see can be written as conformal to flat, as

9. Cosmology Term cont’d. So in a topologically open RW cosmology, we introduce the universal linear potential, hence With three space Curvature K= Since the transformed metric is conformally equivalent to a co-moving Robertson Walker Metric, with spatial curvature written below, then when written as a static coordinate system, the comoving conformal cosmology behaves just like a static metric with universal linear and quadratic potentials. In Mannheim’s original work, the k (quadratic term) was left out, so that:

10. Original Fits

11. New Fits  We have Extended the Rotation Curve Sample for Conformal Gravity to 110 galaxies.  The Sample is comprised of the most recent data available (2005-2009)  Sample consists of galaxies of all morphologies including large HSB spirals, bulged spirals, small LSB spirals, and dwarfs.  Originally the idea was to use the same potential as in the 11 galaxy sample of Mannheim 1996.

13. Comparison to Dark Matter Fits  Universal, analytic solution, no matter what type of galaxy  No free parameters aside from the number of stars in a given galaxy (which has measurable bounds)  No Halo specification necessary.

14. Quadratic Term Vs. Linear Term  The addition of the quadratic term only becomes competitive at very large distances (about r>35kpc).  Thus we have isolated 17 of the largest galaxies which fit this criteria.  Addressed the issue of an infinite rise due to linear potentials.

18. Rest of the Sample  We now apply the full universal and local potentials of conformal gravity to the 110 galaxy sample.

39. Comparison to Dark Matter Fits  Universal, analytic solution, no matter what type of galaxy  No free parameters aside from the number of stars in a given galaxy (which has measurable bounds)  No Halo specification necessary.  Unlike MOND, Conformal Gravity is derived from a scalar action, and is not an ad-hoc modification.  Due to the addition of the quadratic potential, the theory presents the challenge to be falsifiable.

40. Conclusion  Conformal Theory clearly accommodates the latest rotation curve data in a parameter free way.  Accommodates HI dominant galaxies as well as spirals in a universal manner.  Future work is under way to test the conformal theory and the quadratic potentials via clusters.