# Presentation: 518377

## Modified Gravity vs. Dark Matter

-  Successes of Dark Matter
-  Why try anything else?
-  Modified Gravity

- Scott Dodelson w/ Michele Liguori
- September 25, 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

- Advertisement: Special Colloquium Tomorrow by Douglas Clowe at 4PM, *A Direct Empirical Proof of the Existence of Dark Matter*

## Gravity is much stronger in clusters than it should be:

- This is seen in X-Ray studies as well as with gravitational lensing

*Sanders 1999*

*Tyson*

## 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

- Clumpiness

- Without dark matter, potential wells would be much shallower, and the universe would be much less clumpy

- 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

- Astronomer Royal, George Airy, believed deviation from 1/r2  force responsible for irregularities

- Adams informed Airy of his plans, but Airy did not grant observing time.

## 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.*

## By June 1846, both Adams and LeVerrier had calculated positions

- Competition is a good thing: Airy instructed Cambridge Observatory to begin a search in July, 1846.

## This first search (by Challis) was unsuccessful

- 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](http://www-history.mcs.st-and.ac.uk/%7Ehistory/Mathematicians/Adams.html)_'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.

- Both Adams and LeVerrier refined their predictions...

## LeVerrier wrote to German astronomer Galle on September 18, 1846

- Galle discovered it in 30 minutes on September 23.

## 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, *v**2**/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:

- Acceleration due to gravity

- New,fundamental scale

- For a point mass

- MOdified Newtonian Gravity (MOND, Milgrom 1983)

## 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!

- Galaxy w/ low acceleration: MOND regime

- Galaxy w/ high acceleration: Newtonian regime

- 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

- Quintessence

- Modified Gravity

*Curvature of order a**0**2*

*μ**~a**0*

## Scalar Tensor Theories face a huge hurdle

- Light is deflected as it passes by distances far from visible matter in galaxies

*SDSS: Fischer et al. 2000*

- All of these points are farther from Galactic centers than the visible matter.

## 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:

-  Vector action:

- Auxiliary scalar field added (χ) to make kinetic term standard; two parameters in potential V

- 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 | Excellent | Poor |
| Cosmology | Excellent | ? |
| 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

- 15/(4χ)

- Note the logarithmic growth of φ in the matter era

## 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**~**η**p* with

- α decays until φ becomes large enough (recall log-growth). Then vector field starts growing.

## Inhomogeneities in TeVeS

- For large K, no growing mode: vector follows 	particular solution.
- For small K, growing mode comes to dominate.

- Particular soln

- Large K

- Small K

## 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!

- Small K

- Large K

- Standard Growth

## Scorecard

|  | Dark Matter | Modified Gravity |
| --- | --- | --- |
| Rotation Curves | Good | Excellent |
| Clusters | Excellent | Poor |
| Cosmology | Excellent | ? + |
| 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!