Dark Matter and the Equivalence Principle

Marc Kamionkowski

Caltech

(work done with Michael Kesden, astro-ph/0606566 [PRL], 0608095)

20 September 2006

Aristotle (384-322 B.C.):Heavier things fall faster

Ioannes Phillipones (~600 AD):

Observed objects fall ~samespeed

Giambattista Benedetti(Venice, 1530-1590):

Proposed equality of free-fallrates (1586)

Simon Stevin (Flemish, 1548-1620):

Demonstrated equality of free fallexperimentally (1586)

Galileo Galilei (1564-1642):

Leaning Tower story probably apocryphal,as arrived in Pisa ~1589, but didexperiments with rolling balls

VincenzoViviani,b. 1622

Isaac Newton (1642-1727):

Principia (1687):

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Newton : pendulumcomposed of wood, gold, silver,lead, etc. Equivalence of inertial

and gravitational mass ~10-3. Later experiments, ~10-5.

Roland von Eotvos (1848-1919):used torsion balance (1889,1908) to demonstrateequivalence to ~10-9. Used rotation of Earthto provide non-gravitationalforce (as opposed to string inpendulum).

Weak equivalence principle:All masses are accelerated the same wayin a gravitational field.

Einstein: motion of freely-falling particles aresame in gravitational field and uniformlyaccelerated frame

Einstein equivalence principle: laws of physicsare same in any freely falling frame

Central underpinning of general relativity

Further improvements (~1960-1970)(Dicke et al., Braginsky et al…..)

Replaced Earth’s g by Sun’s g and Earth’s rotationby its orbit around Sun. Achieved ~10-12.

Different elements have different(binding energy)/(mass), sohave tested equivalence of freefall for electromagnetic energyand for strong interactions

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

What about gravitational bindingenergy?Strong equivalence principle: Gravitationalbinding energy falls the same way in agravitational field……satisfied by GR,but not some alternatives (e.g., scalar-tensortheories)

Nordtvedt effect (1968):

If SEP violated, Moon and Earthfall differently in Sun’s gravitationalfield, affecting Moon-Earth orbit.Tested by lunar laser ranging.

But what about dark matter?So far, all tests have been for g fields dueto baryons and test masses made of baryons

Stubbs (1991): Eotvos-like data correlated with Milky Way---different terrestrialmaterials fall similarly in g field due partly(~50%) to dark matter. I.e., baryon-DM forceis still

But does dark matter fall same wayin 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 Gdm were different, could scale velocitydistribution, in accordance with virial theorem,to self-consistently obtain same massdistribution.

Is Gdm=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 newscalar fields•E.g., new 1/r2 force law for DM introduced in stringtheories (Gubser, Peebles, Farrar)•Has been suggested to account for voids (Peebles,Gubser, Nusser), requiring new force law comparablein 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 Yukawainteraction, , leads to additionaldm-dm static potential,

Leading to an effectivewith

For distances

How can we measure Gdm?

Frieman-Gradwohl (1992): galactic halos in clusterswould appear “heavier” in dynamical measurements, but effect degenerate with massMainini-Bonometto (2006): discussed baryon lossfrom clusters, but is nasty

We considered: galaxies and their DM halos wouldbe accelerated differently in cluster, giving rise torelative acceleration between galaxy and its halo. Ifstrong 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 MilkyWay potential, while stripped stars act as baryonictracers.•Streams are long-lived and now well-observed with2MASS and SDSS•Detailed simulations compared with observationsalready provide remarkably precise constraints to Sgrmass, M/L, orbit, and Milky Way halo (e.g., Law,Johnston, Majewski 2005)

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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 inretrospect, easily understandable:

If Gdm > G, DM halo of Sgr accelerated toward MWmore strongly than stellar Sgr. Stars in Sgr are thusdisplaced to larger MW radii, and thus leak out of Sgrat 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 Gdm

•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

Curve Color

Standard black

Prograde red

Retrograde green

Planar orbit blue

Heavy disk cyan

Massive Sgr magenta

Conclusions• Sgr tidal streams provide lab for testing 1/r2 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