Modified Gravity vs. Dark Matter

Successes of Dark Matter Why try anything else? Modified Gravity

Scott Dodelson w/ Michele LiguoriOctober 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

Chris Mihos Applet

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

Sanders 1999

Tyso

n

CosmologySuccesses of Standard Model of Cosmology (Light Elements, CMB, Expansion) now supplemented by understanding of perturbationsAt 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

ClumpinessWithout 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.

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)1(

02 r

rrMGag

constantrMG

rMGv r

0

2

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:

20 )/(rMGaaaa Ngg

Acceleration due to gravityNew,fundamental scale

For a point mass

1,1,1

)(

xxx

x

MOdified Newtonian Gravity (MOND, Milgrom 1983)

This leads to a simple prediction

MGavrMGa

rv

04

20

2

Expect stellar luminosity to be proportional to stellar mass

4vL

… 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

geg 2~

)~(~161 4 gRgxdG

SEH

dxdxgemdxdxgmSm

~

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

)(

dtvd

)21,21,21,21(~ diagg

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 whenStandard term

0a

MOND limit obtained by choosing Lφ

)/(8

20

,,

20 aFG

aL

Bekenstein & Milgrom 1984

eVHMpckm

Mpckmkm

crv

ca

gal

gal 3305

220 10

sec27

)005.0sec)(/103(sec)/200(

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 Beyond Einstein-Hilbert

Curvature of order a02

μ~a0

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

0~22

dxdxgedxdxgds

Bottom line: No extra lensing in scalar-tensor theories

Bekenstein & Sanders 1994

Need to modify conformal relation between the 2

metrics)~( ,

,2

BgAeg

with A,B functions of φ,μφ,μ also doesn’t work (Bekenstein & Sanders 1994).

)2sinh(2~2

AAgeg But, adding a new vector field Aμ so that

does produce a theory with extra light deflection (Sanders 1997).

TeVeS (Bekenstein 2004)

)()~(~161

,,4

VAAggxdG

S

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: )1(2~321 4

AAFKFgxdG

SA

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.

ScorecardDark Matter

Modified Gravity

Rotation Curves

GoodGood ExcellentExcellent

Clusters ExcellentExcellent PoorPoor

Cosmology ExcellentExcellent ??

Theoretical Motivation

SUSYSUSY Hubble ScaleHubble Scale

Zero Order Cosmology in TeVeS

),,,( 2222 aaaadiagg

38/ 2

effGadtda

Metric coupling to matter is standard FRW:

Scale factor a obeys a modified Friedmann equation

Bekenstein 2004Skordis, Mota, Ferreira, & Boehm 2006Dodelson & Liguori 2006

Zero Order Cosmology in TeVeS

VVG

e '

16

2

2

4

)]ln(/1[ addGeGeff

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 TeVeSSkordis 2006Skordis, Mota, Ferreira, & Boehm 2006Dodelson & Liguori 2006

Perturb all fields: (metric, matter, radiation) + (scalar field, vector field)

E.g., the perturbed metric is

)]21(),21(),21(),21([ 2222 aaaadiagg

where a depends on time only and the two potentials depend on space and time.

Inhomogeneities in TeVeS

,1 aeA

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

,21 Sbb

22/2412

Kb

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 time4

1 b

Inhomogeneities in TeVeSConsider the homogeneous part of this equation:

0)/241(242

K

This has solutions: α~ηp with

Kp /192121

23

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 solnLarge 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 KStandard Growth

ScorecardDark Matter

Modified Gravity

Rotation Curves

GoodGood ExcellentExcellent

Clusters ExcellentExcellent PoorPoorCosmology ExcellentExcellent ? ? ++Theoretical Motivation

SUSYSUSY Hubble ScaleHubble Scale++

Enhanced Enhanced GrowthGrowth

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

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.

Both Adams and LeVerrier refined their predictions…

LeVerrier wrote to German astronomer Galle on September 18,

1846Galle discovered it in 30 minutes on September 23.