Gravitational lensing with modified gravity

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Gravitational lensing with modified gravity

• Modified Newtonian Dynamics
• (MOND)

IAP, July 2007, R.H. Sanders
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-- true gravitational acceleration
-- Newtonian acceleration
-- fixed acceleration parameter
Flat rotation curves as
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Note that
Fundamental non-linearity necessary for TF
relation
(Sanders Verheijen 1998)
Ursa Major spirals
Not a prediction but fitting intercept
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LSB
HSB
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MOND in original form useful description of test
particle motion.
Problems for N-body system
The Pathology
An isolated system does not conserve linear or
angular momentum.
Center of mass of N-body system accelerates.
The Cure
Lagrangian-based theory (MOND as modified gravity)
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MOND as a modification of gravity
Bekenstein Milgrom (1984) aquadratic Lagrangian
Where
Modified Poisson equation--
Conservative!
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But what about gravitational lensing?
Is it so that
as in GR
where is determined from B-M equation?
Depends upon relativistic extension.
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Steps to a Relativistic Theory

• B-M clearly incomplete-- makes no prediction
  about cosmology or gravitational lensing.
• Need a relativistic theory!
• TeVeS Bekenstein 2004.
• Observed phenomenology of lensing has been a
  major input.
• Theory is also ad hoc and bottom up and
• probably not the last word.

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Covariant Extension of AQUAL
Two fields
AQUAL BM84
Scalar field Lagrangian
Interaction Lagrangian
But now is a scalar field. Complete
theory includes and Hilbert-Einstein action of
GR
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As before
so--
and..

This is a non-standard scalar-tensor theory in
the limit of large scalar field gradients
Brans-Dicke theory.
to be consistent with solar system experiments
Hint of a problem BD yields less deflection of
photons by sun than one would predict from
planetary motion.
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Fifth force from Scalar Field.
At large accelerations,
scalar force
But factor smaller than Newtonian
force.
Smaller accelerations
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The MOND phenomenology results from a 5th force,
in this case
mediated by a scalar field.
But what about equivalence principle?
Form of interaction Lagrangian means that there
exists a physical metric which is conformally
related to the Einstein metric.
Particles follow geodesics of physical metric,
not Einstein metric. In GR
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Gravitational Lensing the problem for
conformally coupled scalar fields
(Bekenstein Sanders 1994)
If Then
corresponds to
Scalar field does not influence the motion of
photons.
For clusters of galaxies
NOT TRUE!
(Fort Mellier 1994)
Non-conformal relation between physical and
Einstein metrics.
is normalized non-dynamical vector field
time-like in cosmological frame (preferred
frame).
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Disformal transformation -- stratified
theory (Ni 1972)
With aquadratic Lagrangian MOND
enhanced lensing.
as in GR (Sanders 1997)
But, non dynamic vector field violates General
Covariance.

No conserved
The cure-- dynamical vector field
(Bekenstein 2004)
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Tensor-Vector-Scalar theory (TeVeS)
The price of gravitational lensing a third field.
The reward
Form of disformal transform chosen such that
(not necessarily so)
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Predictions

• In general effective dark matter should coincide
  with
• detectable baryonic dark matter.
• No dark clusters no isolated mass concentrations
  where
• there are no baryons.
• 3. Galaxy-galaxy lensing round halos.
• 4. Necessity of 2 metrics different
  propagation
• speeds for gravitational and em waves.

But what about the bullet?
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A phenomenological problem for MOND
Mass of X-ray emitting clusters of galaxies
MOND
Newton
MOND helps, but still factor 2 or 3 discrepancy!
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With MOND clusters still require undetected
(dark) matter!
(The White 1984, Gerbal et al. 1992, Sanders
1999, 2003)
Bullet cluster
Clowe et al. 2006
No new problem for MOND but DM is
dissipationless!
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A falsification?
No, perhaps a prediction.
For example, non-baryonic dark matter exists!
Neutrinos
Only question is how much.
When meV neutrinos in thermal
equilibrium with photons.
Number density of neutrinos comparable to that of
photons.
per type, at present.
Three types of active neutrinos
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Neutrinos oscillate i.e., change type (flavor).
(e.g., Fukuda et al. 1998)
eV for most massive type.
Absolute masses not known, but experimentally
eV
(tritium beta decay)
If eV then
eV for all types
and
Possible that
and
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Non-interacting particles phase space density is
conserved!
(Tremaine Gunn 1979)
Even after gravitational collapse.
for one type
In final virialized object (cluster, galaxy)
An upper limit on density of neutrino fluid.
( is 1-d velocity dispersion)
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With and virial
relation we determine a scale
Can accumulate in clusters but..
neutrinos with mass of 1-2 eV do not cluster on
galaxy scale!
On scale of galaxies, mass of neutrino halo lt 1
baryonic mass.
We will know soon KATRIN tritium beta decay
experiment.
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Phantom density
(Milgrom 1986)
What is the implied density distribution in the
equivalent halo?
Assume
But really
then
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Equivalent Halo for a point mass
Equivalent dm halo has different distribution
than baryonic component.
But correlated--
kpc for
Peaks at
kpc for
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Recall
then
If and
(aspect of non-linear theory)
neg. phantom density
Dumbell configuration-- Milgrom 1986
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What you see (by way of convergence) is not what
you get (by way of surface density).
(Angus et al. 2006)
Example dumbell viewed edge-on
Dark matter ring!
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Strong lensing
Critical surface density
Now..
So--
Strong lensing always occurs in Newtonian
regime.
What you see is what you get.
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Conclusions

• Weak lensing For given mass distribution ray
  trace
• using BM field equation. Can be
  surprised.
• Strong lensing MOND doesnt help (much). What
  you
• see is what you get.
• Caution Relativistic extensions of MOND are
  still under
• development.