1. A Generic Test of Modified Gravity Models which Emulate Dark Matter
arXiv: [astro-ph]
with Emre Kahya

2. Dark Matter vs Modified Gravity
Gμν =8πGΤμν works for solar system
But not for galaxies
Theory: v²=GM∕r
Obs: v²~√a0GM
Could be missing M
Or modified gravity

3. ds²=-B(r)dct²+A(r)dr²+r²dΩ²
A'/(r A) + (A - 1)/r² = (A/B) 8πGρ/c²
B'/(r B) - (A - 1)/r² = 0
Spherical Mass M
ε = 2GM/(c²r)
B = 1 - ε and A = 1/(1 - ε) ~ 1 + ε
ε ~ at rs ~ 8 kpc
With Isothermal Halo for r > rs
ε* = 2√a0GM/c²
B ~ 1 – ε + ε* ln(r/rs) and A ~ 1 + ε + ε*
ε* ~

4. No-Go Thm for metric models With Soussa, astro-ph/0307358
Єμν = 8πG Tμν ~ GM for gμν = ημν + hμν
hμν ~ v² ~√a0GM some Єμν ‘s ~ h²
If all Єμν ~ h² unstable!
Distinguished subsets
Divergence (0 to all orders)
Trace conf. invariant for Єμν ~ h¹
Extra force in conf. factor no lensing!

5. Five Assumptions Gravity carried by hμν with source Tμν
General coordinate invariance
Extra force in ultra-weak field regime
Stability (forbids all Єμν ~ h²)
Light couples conformally
Known models violate (1) & (5)
Violating (4) may also work

6. Known Models TeVeS Bekenstein (astro-ph/0403694)
OK Cosmology (astro-ph/ , , , )
SVTG
Moffat (gr-qc/ )
astro-ph/
Fields: gμν, Aμ & φ
Extra force from φ
Matter couples to
ĝμν =Exp(-2φ) gμν
+ 2 sinh(2φ)Aμ Aν
R term for solar system
Gravitons couple to gμν
ĝμν from GR with D.M.
gμν from GR w/o D.M.

7. Dark Matter Emulators ds²=-B(r)dt²+A(r)dr²+r²dΩ
Without Dark Matter, ε=2GM/(c²r)
B = 1-ε and A = 1/(1-ε) ~ 1+ε
ε ~ for rs ~ 8 kpc
Weak gravity waves see this geometry
With Isothermal Halo, ε*=2√a0GM/c²
B ~ 1-ε+ε*ln(r/rs) and A ~ 1+ε+ε*
ε* ~
Ordinary matter sees this geometry

8. Light-like Pulses from (0,xL)
Gravitons follow gμν to (t,xs)
B ~ 1 - ε and A ~ 1 + ε
ε = 2GM/(rc²) ~ at rs ~ 8 kpc
ν's and γ's follow ĝμν to (T,xs)
B ~ 1 – ε + ε* ln((r/rs) and A ~ 1 + ε + ε*
ε* ~ 2√a0GM/c² ~
Δt = T – t = # ε* Δx/c
SN 1987a: Δt ~ · 36.7 day ~ -5.3 day
~ hrs diff. between ν’s and γ’s irrelevant
Advanced LIGO will see to .8 Mpc!

9. Radial Prop: v/c = B(r)/A(r)
Gravitons: v/c ~ 1 – 2ε
ν’s & γ’s: V/c 1 - 2ε + ε* ln(r/rs) - ε*
Typically faster than gravitons
But depends on r and rs
Moore & Nelson, hep-ph/
(V-v)/c < 2 x E-15 for galaxy (maybe ok)
(V-v)/c < 2 x E-19 for extra-gal. (not ok)

10. Conclusions Mod. Gravity may explain rot. curves
But unstable if pure metric
Otherwise new fields and two metrics
Gravitons couple to gμν of GR w/o D.M.
Matter couples to ĝμν of GR with D.M.
Big time lag for gravitons vs ν’s & γ’s