Dark Matter and the Equivalence Principle Marc Kamionkowski Caltech (work done with Michael Kesden, astro- ph/ [PRL], ) 20 September 2006

Aristotle ( B.C.): Heavier things fall faster

Ioannes Phillipones (~600 AD): Observed objects fall ~same speed

Giambattista Benedetti (Venice, ): Proposed equality of free-fall rates (1586)

Simon Stevin (Flemish, ): Demonstrated equality of free fall experimentally (1586)

Galileo Galilei ( ): Leaning Tower story probably apocryphal, as arrived in Pisa ~1589, but did experiments with rolling balls Vincenzo Viviani, b. 1622

Isaac Newton ( ): Principia (1687): 5 of 70 people found the following review helpful: I can't believe people still believe this stuff, September 20, 2005 Reviewer: Jeff "Jeff" (Lakeland, FL, USA) - See all my reviews Jeff "Jeff" See all my reviews

Newton : pendulum composed of wood, gold, silver, lead, etc. Equivalence of inertial and gravitational mass ~ Later experiments, ~10 -5.

Roland von Eotvos ( ): used torsion balance (1889,1908) to demonstrate equivalence to ~ Used rotation of Earth to provide non-gravitational force (as opposed to string in pendulum).

Weak equivalence principle: All masses are accelerated the same way in a gravitational field. Einstein: motion of freely-falling particles are same in gravitational field and uniformly accelerated frame Einstein equivalence principle: laws of physics are same in any freely falling frame Central underpinning of general relativity

Further improvements (~ ) (Dicke et al., Braginsky et al…..) Replaced Earth’s g by Sun’s g and Earth’s rotation by its orbit around Sun. Achieved ~ Different elements have different (binding energy)/(mass), so have tested equivalence of free fall for electromagnetic energy and for strong interactions

Munich (1975): Free fall of free neutrons to ~10 -4.

What about gravitational binding energy? Strong equivalence principle: Gravitational binding energy falls the same way in a gravitational field……satisfied by GR, but not some alternatives (e.g., scalar-tensor theories)

Nordtvedt effect (1968): If SEP violated, Moon and Earth fall differently in Sun’s gravitational field, affecting Moon-Earth orbit. Tested by lunar laser ranging.

But what about dark matter? So far, all tests have been for g fields due to baryons and test masses made of baryons Stubbs (1991): Eotvos-like data correlated with Milky Way---different terrestrial materials fall similarly in g field due partly (~50%) to dark matter. I.e., baryon-DM force is still

But does dark matter fall same way in gravitational field? Does the force law, hold for dark matter as well? And if how would we know?

Usual DM tracers (e.g., rotation curves, lensing) probe DM mass distribution only. If G dm were different, could scale velocity distribution, in accordance with virial theorem, to self-consistently obtain same mass distribution.

Is G dm =G? Why bother asking? Curiosity….a fundamental prediction of GR Cosmic acceleration suggests gravity may be more complicated than we thought and/or that there may be new long-range interaction associated with new scalar fields E.g., new 1/r 2 force law for DM introduced in string theories (Gubser, Peebles, Farrar) Has been suggested to account for voids (Peebles, Gubser, Nusser), requiring new force law comparable in strength to gravity May occur in “chameleon” DM theories (Khoury, Mota, Shaw…)

E.g., if  is scalar field with coupling to (fermionic) DM particle  through Yukawa interaction,, leads to additional dm-dm static potential, Leading to an effective with For distances

How can we measure G dm ? Frieman-Gradwohl (1992): galactic halos in clusters would appear “heavier” in dynamical measurements, but effect degenerate with mass Mainini-Bonometto (2006): discussed baryon loss from clusters, but is nasty We considered: galaxies and their DM halos would be accelerated differently in cluster, giving rise to relative acceleration between galaxy and its halo. If strong enough, galaxy would get stripped from halo. But is nasty theoretically/observationally.

Instead, consider tidal streams of Sagittarius dwarf: Sgr is DM dominated so acts as DM tracer of Milky Way potential, while stripped stars act as baryonic tracers. Streams are long-lived and now well-observed with 2MASS and SDSS Detailed simulations compared with observations already provide remarkably precise constraints to Sgr mass, M/L, orbit, and Milky Way halo (e.g., Law, Johnston, Majewski 2005)

Majewski et al. 2003

Where do tidal streams come from?

What we anticipated: Orbits of streams with EP-violation would differ from those without…. What we found, is different, more striking, and in retrospect, easily understandable: If G dm > G, DM halo of Sgr accelerated toward MW more strongly than stellar Sgr. Stars in Sgr are thus displaced to larger MW radii, and thus leak out of Sgr at apocenter only from the far side, and not the near side, leading to a trailing tail, but no leading tail.

Simulations: Modified GADGET-2 to include different G dm Include active disk, bulge, halo Initial conditions from GALACTICS (Dubinski- Widrow) Use same mass distn for Sgr DM and stars 300,000 particles, 10K each for bulge and disk, 80K for halo, and 200K for satellite Runs for several orbits on CITA cluster

Stellar Streams of Sgr Dwarf

Leading-to-Trailing Stream Ratios Attractive force suppresses leading-to-trailing ratio CurveColor Standardblack Progradered Retrogradegreen Planar orbitblue Heavy diskcyan Massive Sgrmagenta

Conclusions Sgr tidal streams provide lab for testing 1/r 2 force law for dark matter Stronger force law for DM leads to depletion of leading tidal stream of Sgr dwarf Such an effect difficult to mimic by changing Sgr, MW masses, orbital parameters, etc. Conservative “by-eye” comparison with observation of roughly equal leading and trailing stream constrains DM force law to be within ~10% of that for baryons Estimate ~1% sensitivity with more detailed comparisons of data with model