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:
Tyson
This is seen in X-Ray
studies as well as
with gravitational
lensing
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.
Undergraduate Notebook, July 1841
John Adams
(not that one)
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
MG
r
a g  2 (1  )
r
r0
Equate with centripetal acceleration, v2/r
MG MG
v 

r


constant

r
r0
2
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:
MG
a g  ( a g / a0 )  a N  2
r
For a point mass
MOdified Newtonian Gravity
New,fundamental scale
(MOND, Milgrom 1983)
Acceleration due to gravity
1, x  1
 ( x) 
x, x  1
This leads to a simple prediction
2
a0 MG
v
4

 v  a0 MG
2
r
r
Expect stellar luminosity to be proportional to stellar mass
Lv
4
… which has been verified (TullyFisher 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
S EH
1
4
~ R( g~ )

d
x

g
16G 
is distinct from the metric coupling to matter (e.g. point
particle)
S m  m  g  dx  dx  m e  g~ dx  dx
They are related by a conformal transformation
2
~
g   e g 
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
g~  diag (1  2,1  2,1  2,1  2 )
where Φ is the standard Newtonian potential, obeying
the Poisson equation. Then the eqn of motion for a
point particle is


dv
 (   )
dt
Standard term Extra term, dominates
when   a0
MOND limit obtained by choosing Lφ
2
a0
2
,
L 
F (,  / a0 )
8G
Bekenstein & Milgrom 1984
There is a new fundamental mass scale in the Lagrangian
2
a0 vgal
(200km / sec) 2
km
33



27

H

10
eV
0
5
c crgal (3 10 km / sec)( 0.005Mpc )
sec Mpc
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


~
ds  g  dx dx  e g  dx dx  0
2


2
Bottom line: No extra lensing in scalar-tensor theories
Bekenstein & Sanders 1994
Need to modify conformal
relation between the 2 metrics
g   e
2
,
~
( Ag   B, )
with A,B functions of φ,μφ,μ also doesn’t work
(Bekenstein & Sanders 1994).
But, adding a new vector field Aμ so that
2
g   e
~
g   2 A A sinh( 2 )
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:
S  

1
4
~  ( g~   A A )   V (  )
d
x

g
,  ,
16G 

Auxiliary scalar field added (χ) to make kinetic term
standard; two parameters in potential V
 Vector action: S A  

1
4
~ KF  F  2 ( A A  1)
d
x

g


32G 
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
Clusters
Good
Excellent
Excellent
Poor
Cosmology
Excellent
?
Theoretical
Motivation
SUSY
Hubble Scale
Zero Order Cosmology in TeVeS
Metric coupling to matter is standard FRW:
g   diag (a , a , a , a )
2
2
2
2
Scale factor a obeys a modified Friedmann equation
 da / dt  8Geff
   

 
3
 a 
2
Bekenstein 2004
Skordis, Mota, Ferreira, & Boehm 2006
Dodelson & Liguori 2006
Zero Order Cosmology in TeVeS
with effective Newton constant
4
Geff
Ge

2
[1  d / d ln( a)]
and energy density of the scalar field
2
e
V 'V 
 
16G
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
g   diag[a 2 (1  2), a 2 (1  2), a 2 (1  2), a 2 (1  2)]
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.
A  ae


1    , 
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:
  b1  b2  S , 
The coefficients are complicated functions of the
zero order time-dependent a and φ.
In the matter era,
b1 
4

b2 
21  24 / K 
2
Conformal time
Inhomogeneities in TeVeS
Consider the homogeneous part of this equation:
4
   

2(1  24 / K )

2
 0
This has solutions: α~ηp with
3 1
p   
1  192 / K
2 2
α decays until φ becomes large enough (recall loggrowth). Then vector field starts growing.
Inhomogeneities in TeVeS
Small K
Particular soln
Large K
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
Clusters
Good
Excellent
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!
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.