**Gravitational lensing with modified gravity** Modified Newtonian Dynamics (MOND) IAP, July 2007, R.H. Sanders

** Milgrom (1983)– modification of Newtonian dynamics:** One simple formula which subsumes wide range of phenomena-- or Asymptotically, and the observed form of the Tully-Fisher relation! -- true gravitational acceleration -- Newtonian acceleration -- fixed acceleration parameter Flat rotation curves as

** Note that** Fundamental non-linearity– necessary for TF relation: (Sanders & Verheijen1998) Ursa Major spirals Not a prediction– but fitting intercept

**LSB:** HSB:

**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)

**MOND as a modification of gravity:** Bekenstein & Milgrom (1984) aquadratic Lagrangian Where Modified Poisson equation-- Conservative!

**But what about gravitational lensing?** Is it so that as in GR where is determined from B-M equation? Depends upon relativistic extension.

**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.

**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:

**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.

**“Fifth force” from Scalar Field.** At large accelerations, scalar force But factor smaller than Newtonian force. Smaller accelerations:

** 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:

**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).

**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)

**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)

**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?

**A phenomenological problem for MOND:** Mass of X-ray emitting clusters of galaxies MOND: Newton: MOND helps, but still factor 2 or 3 discrepancy!

**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!

**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:

**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

**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)

**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 < 1% baryonic mass. We will know soon– KATRIN– tritium beta decay experiment.

**Phantom density** (Milgrom 1986) What is the implied density distribution in the equivalent halo? Assume But really then

**Equivalent Halo for a point mass ** Equivalent dm halo has different distribution than baryonic component. But correlated-- kpc for Peaks at: kpc for

**Recall– ** then If and (aspect of non-linear theory) neg. phantom density Dumbell configuration-- Milgrom 1986

**“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!

**Strong lensing** Critical surface density: Now….. So-- Strong lensing always occurs in “Newtonian” regime. What you see is what you get.

**Conclusions:** • Weak lensing: For given mass distribution– ray trace • using BM field equation. Can be surprised. • Strong lensing: MOND doesn’t help (much). What you • see is what you get. • Caution: Relativistic extensions of MOND are still under • development.