1. Similarity of DM/DE and TeVeS HongSheng Zhao @SUPA

2. ΛCDM  WDM  Tuned DM-DE (~ Tuned TeVeS μ )

3. I.Much ado about mu µ in disk galaxies with Famaey (Brussels), Angus (SUPA), Gentile (NMSU), Nipoti, Londrillo, Ciotti (Bologna) & high-z elliptical lensing galaxies with Bacon, Taylor, Horne (SUPA), Shan (Beijing) in cosmology with Skordis (Perimeter), Mota (Oslo)

4. III. TeVeS (Bekenstein 2004) Einstein-like equations for g , U , scalar  by varying the action w.r.t. each of these fields In a quasi-static system with a weak gravitational field,    N +  : dτ 2 = (1-2  ) dt 2 -(1+2  ) ( dx 2 + dy 2 + dz 2 ) where  obeys a B-M equation, and plays the role of the dark matter potential (dynamics and lensing are governed by the same physical metric g)

5. Define y = (3k’ 2 /a 0 2 ) Y, Y=(g  -U  U  ) ,  ,  where k’  k/ 4  is a parameter of the theory y  (  ) 2 >0 in quasi-static situation y  -(  /  t) 2 <0 in cosmology Equation for the scalar field in a quasi-static system . [  s (y)  ] = 4  G  (similar to B-M) In spherical symmetry gravity:  s =  / (1-  )

6. Analogy:  as dielectric and as inertia Conservative Field g or E = -▼Φ = d 2 x/dt 2 Poisson equation ρ = ▼·D or ▼·g N –Electric displacement D = μ E = E + PolarisationField –TeVeS-like theory g N = μ g = g - ScalarField Newtonian F = m inertia g, m inertia = m 

7. Scalar field in TeVeS resembles DM Scalar field (Polarisation Halo of luminous baryon  1-to-1 gravity) ~ tuned Dark Halo surrounding baryons NFW profile?  µ?

8. I. Constraining the law of  (Zhao & Famaey 2006, ApJ letters) Standard (I):  (x) = x / (1+x 2 ) 1/2 Exponential:  (x) = 1 - e -x Bekenstein toy model:  (x) = [(1+4x) 1/2 -1] / [(1+4x) 1/2 + 1] Simple (III):  (x) = x / (1+x)  s (g s ) =  /(1-  ) g s = g -  g  (g)

9. External Field Effect

10. Milgrom’s standard µ = x/(1+x 2 ) 1/2 Excellent description of data! –Deviation capped ~ a0. –Fast transition of law from Newtonian solar system into non- Newtonian outer galaxy. But … incompatible with TeVeS!

12. Fast transitions (implied by data & mu-standard) challenging for TeVeS Less freedom in mu than in 1-field MOND. –Bekenstein mu excluded by RC data. Scalar field must increase from galaxies (a0) to solar system –untailored mu will overrun Pioneer limit (<10a0).

13. Problems with earlier laws of  Milgrom’s standard  -law too sharp, – leads to multi-valued TeVeS L and problems with external field. Bekenstein’s  too gradual – unlike in real galaxy RC.

14. The standard and exponential functions are excluded (multi-valued), but good to fit RC’s Bekenstein’s toy is ok in TeVeS, but poor fit to the TVC of the Milky Way (Famaey & Binney 2005), same conclusion in spiral galaxy NGC 3198

15. A narrow range of mu is allowed alpha-model in Angus, Famaey, Zhao [Poster] & Gentile, Famaey, Zhao [Poster] Curl-field correction [Nipoti’s talk]

16. The Simple Function  = x / (1+x) =   / (a 0 +   ) gives a good fit to RC’s and corresponds to a plausible TeVeS  s (y) Simple ’s corresponding scalar field function is  s =   / (a 0 - α   ), α=1 (Zhao-Famaey)   =  s / (1+  s ) =    / a 0

17. Deeper Physics Beyond Simplicity? In spherical symmetry: Newtonian F N = m inertia g, m inertia = m  ExtraForce |F s |= m inertia a 0, with a 0 ~c 

18. Test mu where it is not made for: Elliptical Lenses, SNe, Cosmology!

19. Metric of Homogeneous Universe Hubble expansion in TeVeS

21. Scalar field density tracks matter density

22. As good as ΛCDM, but no Λ!

24. Updating Scores LCDM TeVeS Solar System ? ? Tides on Globulars & dSph Rot. curves HSB/LSB Lensing by Ellipticals/Clusters SNIa/CMB