Modified Gravity vs. Dark Matter Successes of Dark Matter Why try anything else? Modified Gravity Scott Dodelson w/ Michele Liguori October 17, 2006

Four reasons to believe in dark matter Galactic Gravitational Potentials Cluster Gravitational Potentials Cosmology Theoretical Motivation

Potential Wells are much deeper than can be explained with visible matter We have measured this for many years on galactic scales Kepler: v=[GM/R]1/2

Fit Rotation Curves with Dark Matter

Bullet Cluster Gas clearly separated from potential peaks

Gravity is much stronger in clusters than it should be: This is seen in X-Ray studies as well as with gravitational lensing Tyson Sanders 1999

Cosmology Successes of Standard Model of Cosmology (Light Elements, CMB, Expansion) now supplemented by understanding of perturbations At z=1000, the photon/baryon distribution was smooth to one part in 10,000. Perturbations have grown since then by a factor of 1000 (if GR is correct)!

Simplest Explanation is Dark Matter Without dark matter, potential wells would be much shallower, and the universe would be much less clumpy Clumpiness Large Scales

Supersymmetry: Add partners to each particle in the Standard Model Beautiful theoretical idea invented long before it was realized that neutral, stable, massive, weakly interacting particles are needed: Neutralinos

Paves the way for a multi-prong Experimental Approach

Why consider Modified Gravity? Dark Matter has not been discovered yet. The game is not over! Recent Developments This is an age-old debate …

Remember how Neptune was discovered Formed a design in the beginning of this week, of investigating, as soon as possible after taking my degree, the irregularities of the motion of Uranus, which are yet unaccounted for; in order to find out whether they may be attributed to the action of an undiscovered planet beyond it; and if possible thence to determine the elements of its orbit, etc.. approximately, which would probably lead to its discovery. John Adams (not that one) Undergraduate Notebook, July 1841

Not everyone believed a new planet was responsible Adams informed Airy of his plans, but Airy did not grant observing time. Astronomer Royal, George Airy, believed deviation from 1/r2 force responsible for irregularities

By June 1846, both Adams and French astronomer LeVerrier had calculated positions Competition is a good thing: Airy instructed Cambridge Observatory to begin a search in July, 1846, and Neptune was discovered shortly thereafter.

Anomalous precession of Mercury’s perihelion went the other way LeVerrier assumed it was due to a small planet near the Sun and searched (in vain) for such a planet (Vulcan). We now know that this anomaly is due to a whole new theory of gravity.

How can gravity be modified to fit rotation curves? Change Newton’s Law far from a point mass Equate with centripetal acceleration, v2/r Expect to see largest deviation from Newton in largest galaxies

So Inferred Mass/Light ratio should be largest for large galaxies It isn’t!

But … the anomaly is most apparent at low accelerations Sanders & McGaugh 2002

So, modify Newton’s Law at low acceleration: For a point mass MOdified Newtonian Gravity (MOND, Milgrom 1983) Acceleration due to gravity New,fundamental scale

This leads to a simple prediction Expect stellar luminosity to be proportional to stellar mass

… which has been verified (Tully-Fisher Law) L~v4 Sanders & Veheijen 1998

You want pictures! Fit Rotation Curves of many galaxies w/ only one free parameter (recall 3 used in CDM).

You want pictures! Newtonian-inferred velocity from Stars Newtonian-inferred velocity from Gas

MOND does not do as well on galaxy clusters Sanders 1999

On cosmology, MOND is silent Not a comprehensive theory of gravity so cannot be applied to an almost homogeneous universe. We don’t even know if the true theory – which reduces to MOND in some limit – is consistent with an expanding universe. Need a relativistic theory which reduces to MOND

Scalar-Tensor Theory The metric appearing in the Einstein-Hilbert action is distinct from the metric coupling to matter (e.g. point particle) They are related by a conformal transformation

Equations of motion for a point particle in this theory In a weak gravitational field, the metric that appears in the Einstein-Hilbert action is where Φ is the standard Newtonian potential, obeying the Poisson equation. Then the eqn of motion for a point particle is Extra term, dominates when Standard term

MOND limit obtained by choosing Lφ Bekenstein & Milgrom 1984 There is a new fundamental mass scale in the Lagrangian

That may sound nutty, but remember …

We are in the market for new physics with a mass scale of order H0 Curvature of order a02 μ~a0 Quintessence Beyond Einstein-Hilbert

Scalar Tensor Theories face a huge hurdle All of these points are farther from Galactic centers than the visible matter. Light is deflected as it passes by distances far from visible matter in galaxies SDSS: Fischer et al. 2000

Theorem: Conformal Metrics have same null curves Bottom line: No extra lensing in scalar-tensor theories Bekenstein & Sanders 1994

Need to modify conformal relation between the 2 metrics with A,B functions of φ,μφ,μ also doesn’t work (Bekenstein & Sanders 1994). But, adding a new vector field Aμ so that does produce a theory with extra light deflection (Sanders 1997).

TeVeS (Bekenstein 2004) Two metrics related via (scalar,vector) as in Sanders theory; one has standard Einstein-Hilbert action, other couples to matter in standard fashion. Scalar action: Auxiliary scalar field added (χ) to make kinetic term standard; two parameters in potential V Vector action: F2 standard kinetic term for vector field; Lagrange multiplier, fixed by eqns of motion, enforces A2=-1; K is 3rd free parameter in model.

Scorecard Dark Matter Modified Gravity Rotation Curves Good Excellent Clusters Poor Cosmology ? Theoretical Motivation SUSY Hubble Scale

Zero Order Cosmology in TeVeS Metric coupling to matter is standard FRW: Scale factor a obeys a modified Friedmann equation Bekenstein 2004 Skordis, Mota, Ferreira, & Boehm 2006 Dodelson & Liguori 2006

Zero Order Cosmology in TeVeS with effective Newton constant and energy density of the scalar field

Zero Order Cosmology in TeVeS These corrections however are small so standard successes are retained Note the logarithmic growth of φ in the matter era 15/(4χ)

Inhomogeneities in TeVeS Skordis 2006 Skordis, Mota, Ferreira, & Boehm 2006 Dodelson & Liguori 2006 Perturb all fields: (metric, matter, radiation) + (scalar field, vector field) E.g., the perturbed metric is where a depends on time only and the two potentials depend on space and time.

Inhomogeneities in TeVeS Other fields are perturbed in the standard way; only the vector perturbation is subtle. Constraint leaves only 3 DOF’s. Two of these decouple from scalar perturbations, so we need track only the longitudinal component defined via:

Inhomogeneities in TeVeS Vector field satisfies second order differential eqn: The coefficients are complicated functions of the zero order time-dependent a and φ. In the matter era, Conformal time

Inhomogeneities in TeVeS Consider the homogeneous part of this equation: This has solutions: α~ηp with α decays until φ becomes large enough (recall log-growth). Then vector field starts growing.

Inhomogeneities in TeVeS Small K Large K Particular soln For large K, no growing mode: vector follows particular solution. For small K, growing mode comes to dominate.

Inhomogeneities in TeVeS This drives difference in the two gravitational potentials … Small K Large K

Inhomogeneities in TeVeS … which leads to enhanced growth in matter perturbations! Standard Growth Large K Small K

Scorecard Dark Matter Modified Gravity Rotation Curves Good Excellent Clusters Poor Cosmology ? + Theoretical Motivation SUSY Hubble Scale + Enhanced Growth

Conclusions Dark Matter explains a wide variety of phenomena, extremely well on largest scales and good enough on smallest scales. Modified Gravity is intriguing: it does well on small scales, poorly on intermediate scales, but there is no one theory that can be tested on cosmological scales. We are uncovering some hints: Theorists and Experimenters all have work to do!

In June 1845, the French also began the relevant calculations Urbain Le Verrier: I do not know whether M. Le Verrier is actually the most detestable man in France, but I am quite certain that he is the most detested.

This first search (by Challis) was unsuccessful Both Adams and LeVerrier refined their predictions… In September 1846, Dawes’ friend William Lassell, an amateur astronomer and a brewer by trade, had just completed building a large telescope that would be able to record the disk of the planet. He wrote to Lassell giving him Adams's predicted position. However Lassell had sprained his ankle and was confined to bed. He read the letter which he gave to his maid who then promptly lost it. His ankle was sufficiently recovered on the next night and he looked in vain for the letter with the predicted position.

LeVerrier wrote to German astronomer Galle on September 18, 1846 Galle discovered it in 30 minutes on September 23.