A Generic Test of Modified
Gravity Models which Emulate
Dark Matter
arXiv:0705.0153 [astro-ph]
with Emre Kahya
Dark Matter vs Modified
Gravity
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Gμν =8πGΤμν works
for solar system
But not for galaxies
Theory: v²=GM∕r
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Obs: v²~√a0GM
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Could be missing M
Or modified gravity
ds²=-B(r)dct²+A(r)dr²+r²dΩ²
A'/(r A) + (A - 1)/r² = (A/B) 8πGρ/c²
2.
B'/(r B) - (A - 1)/r² = 0
Spherical Mass M
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ε = 2GM/(c²r)
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B = 1 - ε and A = 1/(1 - ε) ~ 1 + ε
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ε ~ .0000006 at rs ~ 8 kpc
With Isothermal Halo for r > rs
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ε* = 2√a0GM/c²
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B ~ 1 – ε + ε* ln(r/rs) and A ~ 1 + ε + ε*
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ε* ~ .0000006
1.
No-Go Thm for metric models
With Soussa, astro-ph/0307358
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Єμν = 8πG Tμν ~ GM for gμν = ημν + hμν
hμν ~ v² ~√a0GM
some Єμν ‘s ~ h²
If all Єμν ~ h²
unstable!
Distinguished subsets
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Divergence (0 to all orders)
Trace
conf. invariant for Єμν ~ h¹
Extra force in conf. factor
no lensing!
Five Assumptions
1.
2.
3.
4.
5.
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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
Known Models
TeVeS
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Bekenstein (astroph/0403694)
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OK Cosmology (astroph/0505519, 0606216,
0608602, 0611255)
SVTG
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Moffat (grqc/0506021)
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astro-ph/0506370
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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.
Dark Matter Emulators
ds²=-B(r)dt²+A(r)dr²+r²dΩ
Without Dark Matter, ε=2GM/(c²r)
1.
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B = 1-ε and A = 1/(1-ε) ~ 1+ε
ε ~ .0000006 for rs ~ 8 kpc
Weak gravity waves see this geometry
With Isothermal Halo, ε*=2√a0GM/c²
2.
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B ~ 1-ε+ε*ln(r/rs) and A ~ 1+ε+ε*
ε* ~ .0000006
Ordinary matter sees this geometry
Light-like Pulses from (0,xL)
Gravitons follow gμν to (t,xs)
1.
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B ~ 1 - ε and A ~ 1 + ε
ε = 2GM/(rc²) ~ .0000006 at rs ~ 8 kpc
ν's and γ's follow ĝμν to (T,xs)
2.
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B ~ 1 – ε + ε* ln((r/rs) and A ~ 1 + ε + ε*
ε* ~ 2√a0GM/c² ~.0000006
Δt = T – t = # ε* Δx/c
SN 1987a: Δt ~ -.144 · 36.7 day ~ -5.3 day
3.
4.
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~ hrs diff. between ν’s and γ’s irrelevant
Advanced LIGO will see to .8 Mpc!
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*
1.
2.
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Typically faster than gravitons
But depends on r and rs
Moore & Nelson, hep-ph/0106220
3.
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(V-v)/c < 2 x E-15 for galaxy (maybe ok)
(V-v)/c < 2 x E-19 for extra-gal. (not ok)
Conclusions
Mod. Gravity may explain rot. curves
But unstable if pure metric
Otherwise new fields and two metrics
1.
2.
3.
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4.
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