Modified Gravity vs. Dark Matter

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Modified Gravity vs. Dark Matter

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

Scott Dodelson w/ Michele Liguori September 25,
2006
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Four reasons to believe in dark matter

• Galactic Gravitational Potentials
• Cluster Gravitational Potentials
• Cosmology
• Theoretical Motivation

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Potential Wells are much deeper than can be
explained with visible matter
We have measured this for many years on galactic
scales
Kepler vGM/R1/2
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Fit Rotation Curves with Dark Matter
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Bullet Cluster
Advertisement Special Colloquium Tomorrow by
Douglas Clowe at 4PM, A Direct Empirical Proof of
the Existence of Dark Matter
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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
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Cosmology
Successes of Standard Model of Cosmology (Light
Elements, CMB, Expansion) now supplemented by
understanding of perturbations
At z1000, 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)!
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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
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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
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Paves the way for a multi-prong Experimental
Approach
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Why consider Modified Gravity?

• Dark Matter has not been discovered yet. The
  game is not over!
• Recent Developments
• This is an age-old debate

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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
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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
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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.
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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.
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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.

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LeVerrier wrote to German astronomer Galle on
September 18, 1846
Galle discovered it in 30 minutes on September 23.
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Anomalous precession of Mercurys 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.
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How can gravity be modified to fit rotation
curves?

• Change Newtons Law far from a point mass

Equate with centripetal acceleration, v2/r
Expect to see largest deviation from Newton in
largest galaxies
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So Inferred Mass/Light ratio should be largest
for large galaxies
It isnt!
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But the anomaly is most apparent at low
accelerations
Sanders McGaugh 2002
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So, modify Newtons Law at low acceleration
MOdified Newtonian Gravity (MOND, Milgrom 1983)
For a point mass
Acceleration due to gravity
New,fundamental scale
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This leads to a simple prediction
Expect stellar luminosity to be proportional to
stellar mass
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which has been verified (Tully-Fisher Law)
Lv4
Sanders Veheijen 1998
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You want pictures!
Fit Rotation Curves of many galaxies w/ only one
free parameter (recall 3 used in CDM).
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You want pictures!
Galaxy w/ low acceleration MOND regime
Newtonian-inferred velocity from Stars
Galaxy w/ high acceleration Newtonian regime
Newtonian-inferred velocity from Gas
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MOND does not do as well on galaxy clusters
Sanders 1999
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On cosmology, MOND is silent
Not a comprehensive theory of gravity so cannot
be applied to an almost homogeneous universe. We
dont 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
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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
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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 F 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
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MOND limit obtained by choosing Lf
Bekenstein Milgrom 1984
There is a new fundamental mass scale in the
Lagrangian
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That may sound nutty, but remember
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We are in the market for new physics with a mass
scale of order H0
Curvature of order a02
µa0
Quintessence
Modified Gravity
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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
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Theorem Conformal Metrics have same null curves
Bottom line No extra lensing in scalar tensor
theories
Bekenstein Sanders 1994
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Need to modify conformal relation between the 2
metrics
with A,B functions of f,µf,µ also doesnt work
(Bekenstein Sanders 1994).
But, adding a new vector field Aµ so that
does produce a theory with extra light deflection
(Sanders 1997).
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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 (?) 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.
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Scorecard
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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
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Zero Order Cosmology in TeVeS
with effective Newton constant
and energy density of the scalar field
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Zero Order Cosmology in TeVeS
These corrections however are small so standard
successes are retained
Note the logarithmic growth of f in the matter era
15/(4?)
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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.
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Inhomogeneities in TeVeS
Other fields are perturbed in the standard way
only the vector perturbation is subtle.
Constraint leaves only 3 DOFs. Two of these
decouple from scalar perturbations, so we need
track only the longitudinal component defined via
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Inhomogeneities in TeVeS
Vector field satisfies second order differential
eqn
The coefficients are complicated functions of the
zero order time-dependent a and f.
In the matter era,
Conformal time
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Inhomogeneities in TeVeS
Consider the homogeneous part of this equation
This has solutions a?p with
a decays until f becomes large enough (recall
log-growth). Then vector field starts growing.
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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.
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Inhomogeneities in TeVeS
This drives difference in the two gravitational
potentials
Small K
Large K
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Inhomogeneities in TeVeS
which leads to enhanced growth in matter
perturbations!
Standard Growth
Large K
Small K
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Scorecard
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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!