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

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

3. 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

4. Fit Rotation Curves with Dark Matter

5. Bullet Cluster
Gas clearly separated from potential peaks

6. 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

7. 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)!

8. 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

9. 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

10. Paves the way for a multi-prong Experimental Approach

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

12. 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

13. 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

14. 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.

15. 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.

16. 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

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

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

19. 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

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

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

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

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

24. MOND does not do as well on galaxy clusters
Sanders 1999

25. 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

26. 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

27. 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

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

29. That may sound nutty, but remember …

30. 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

31. 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

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

33. 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).

34. 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.

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

36. 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

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

38. 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)

39. 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.

40. 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:

41. 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

42. 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.

43. 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.

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

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

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

47. 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!

48. 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.

49. 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.

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