Conformal Gravity in the X-ray Cluster Abell 2029 Keith Horne SUPA St Andrews 10 8 K gas galaxies monster galaxy
Mar 23, 2016
Conformal Gravity in the X-ray
Cluster Abell 2029
Keith
Hor
ne S
UPA
St
Andr
ews
108K gas
galaxies
monster galaxy
Mannheim-Kazanas metric
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4α Wμν = Tμν = 0Source-free field equations:Static, spherical symmetry:
€
ds2 = −b(r) dt 2 + dr2
b(r)+ r2 dθ 2 + sin2 θ dφ2( )
€
b(r) =1− 3β γ −β 2 − 3β γ( )
r+ γ r −κ r2
Analogous results including rotation and charge.
Solar System Dynamics
€
βγ <<1, r2 << βγ
, κ = 0,
b(r)⇒ 1− 2βr
=1− 2G Mc 2 r
,
Schwarzschild metric:€
ds2 = −b(r) dt 2 + dr2
b(r)+ r2 dθ 2 + sin2 θ dφ2( )
€
b(r) =1− 3β γ −β 2 − 3β γ( )
r+ γ r −κ r2
€
β =G Mc 2
Fixes one parameter
Galaxy Rotation Curves
€
b(r) =1+ 2Φ(r)c 2 =1− 2β
r+ γ r −κ r2
Φ(r)c 2 = b(r) −1
2= − β
r+ γ
2r − κ
2r2
g(r)c 2 = 1
c 2
dΦdr
= βr2 + γ
2−κ r
V 2(r)c 2 = r g(r) = β
r+ γ
2r−κ r2
Linear potential gives a distance-independent inward acceleration.
Galaxy Rotation Curves
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g(r)c 2 = β
r2 + γ2
−κ r ⇒ g0mx 2 + m +1 ⎛ ⎝ ⎜
⎞ ⎠ ⎟
V 2(r)c 2 = β
r+ γ
2r−κ r2 ⇒ V0
2 mx
+ m x + x ⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
β ≡G Mc 2 γ = γ 0 m +1( ) κ = 0
V0 =100 km s−1 R0 = 24 kpc M0 = 5.6 ×1010 Msun
g0 = G M0
R02 = V0
2
R0
= γ 0 c 2
2=1.4 ×10−9 m s−1
Mannheim 1993, 1997.
Universal inward acceleration.
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m ≡ MM0
x ≡ rR0
( fits two more parameters)
Exterior mass matters !
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gg∞
= 2 x3
€
1− 13x 2
Mannheim€
g = 0
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g = GMr2
NewtonNo net force from
external shells
Hook’s law force toward
centre of external
shell
Rotation Curve Fits
starsgas
linear potenti
al
NGC 1560 poorest fit.
Data wiggles
follow the gas.
Fits adjust [M/L]* All mass in edge-on disk plane.
Abell 2029
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z = 0.0767
d ≈ c zH0
= 320 Mpc
Chandra X-ray Image of Abell 2029The galaxy cluster Abell 2029 is composed of thousands of galaxies enveloped in a gigantic cloud of hot gas, and an amount of dark matter equivalent to more than a hundred trillion Suns. At the center of this cluster is an enormous, elliptically shaped galaxy that is thought to have been formed from the mergers of many smaller galaxies.
200 Kpc
Probes gravity on 10x larger scales
X-ray Gas
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g(r) = 1ρ (r)
dP(r)dr
M(< r) = r2
Gρ (r)dP(r)
dr
€
dPdr
= −ρ (r) g(r) = − ρ (r) G M(< r)r2
observe : T(r), ρ (r), P(r) = ρ (r) kT(r)μ mH
Spherical symmetry + Hydrostatic Equilibrium:
Gravity and Total Mass profile:
X-ray Gas 3-300 kpc
T(r)
P(r)
(r)
cs(r)
v*
sin(i)
gals
Lewis, Stocke, Buote 2002.
Newtonian Analysis
90% Dark Matter
Required !€
g(r) ~ 3 ×108cm s−2 3− 300 kpc
M(< r) ~ 1014 Msunr
200 kpc
⎛ ⎝ ⎜
⎞ ⎠ ⎟2
Mgas + Mstars ~ 1013 Msunr
200 kpc
⎛ ⎝ ⎜
⎞ ⎠ ⎟
Gravity and Total Mass
profiles:g(r)
M(<r)
gasstar
s
gasstar
s
g(r)
M(<r)
gasstar
s
gasstar
s
g(r)
M(<r)
gasstar
s
gasstar
s
€
g(r) ~ 3 ×108cm s−2 3− 300 kpc
M(< r) ~ 1012 Msun r > 30 kpc
Mgas + Mstars ~ 1013 Msunr
200 kpc
⎛ ⎝ ⎜
⎞ ⎠ ⎟
Conformal Gravity
Dark Matter NOT
required !
Gravity and Total Mass
profiles:
Too Much Conformal Gravity!
Newton vs Conformal Gravity
g(r)
M(<r)
gasstar
s
gasstar
s
g(r)
M(<r)
gasstar
s
gasstar
s
Discussion Points
: ) Dark Matter is not neededto bind the X-ray Gas.: ( Too much Conformal Gravity !
• Conformal Gravity ruled out? (Not yet.)External material -- external Void ?Mannheim-Kazanas metric incomplete?Not in Higgs gauge --> vacuum polarisation.• Quadratic potential terms important?External shells of distant galaxies should generate a universal quadratic potential.• Gas not in hydrostatic equilibrium?Rotation/infall/outflow V > 1000 km/s?Doppler shift detectable in future X-ray spectroscopy.• Stars generate the gravity -- not the hot gas?Same problem as in the colliding clusters 1E0657-56 ?
Conformal Symmetry
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gμν ⇒ Ω2(x)gμν
Invariants: angles, velocities, light cones, causality.
Weyl action:
Clock ticks and rulers stretch by a factor that can vary in time and
space.
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IW = −α d4 x −g Cλμνκ C λμνκ∫= −2α d4 x −g Rμν R
μν − R2
3
⎛ ⎝ ⎜
⎞ ⎠ ⎟∫
€
g ≡ det gμν( ) Rμν ≡ Rαμνα R ≡ Rα
α
Conformal Matter Action
€
IM = − d4 x −g i ψ γ μ (x) ∂μ + Γμ (x)( ) + i h S[ ]ψ{∫
+ 12
Sα Sα − R12
S2 + λ S4 ⎫ ⎬ ⎭
€
δ IM
δψ= 0 ⇒ iγ μ (x) ∂μ + Γμ (x)[ ] − h S{ }ψ = 0
δ IM
δ S= 0 ⇒ Sα
;α + R6
S − 4 λ S3 = −hψψ
€
Sα ≡ ∂α S = ∂ S∂ xα
€
m = h S(x)Fermion mass:
€
mH2 = − R(x)
6
Higgs mass:
Equations of motion:
Ricci scalar: (negative spatial curvature).
(negative vacuum energy)
Dynamical Mass Generation
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δ IM
δψ= 0 ⇒ iγ μ (x) ∂μ + Γμ (x)[ ] − h S{ }ψ = 0
δ IM
δ S= 0 ⇒ Sα
;α + R6
S − 4 λ S3 = −hψψ
€
mH2 = − R(x)
6
Higgs mass:
€
m = h S(x)Fermion mass:
€
V (S) = R6
S2 − 4 λ S4
Symmetry Breaking Higgs potential:
€
R < 0
€
λ < 0 Vmin < 0
Higgs potential:
€
V (S)
€
R < 0
€
R > 0
Field Equations
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δ IW + IM( )δ gμν
= 0 ⇒ 4α Wμν = Tμν
€
Wμν = − 16
gμν R;α
;α + 23
R;μ ;ν + Rμν;α
;α − Rμα
;ν ;α − Rνα
;μ ;α
+ 23
RRμν − 2Rμα Rαν + 1
2gμν Rαβ R
αβ − R2
3
⎛ ⎝ ⎜
⎞ ⎠ ⎟
€
Tμν = iψ γ μ (x) ∂ν + Γν (x)[ ]ψ
+ 23
SμSν − 13S Sμ ;ν + 1
3gμν S Sα
;α − 12
Sα Sα
⎛ ⎝ ⎜
⎞ ⎠ ⎟
− S2
6Rμν − R
2gμν
⎛ ⎝ ⎜
⎞ ⎠ ⎟− λ S4gμν
Higgs Guage
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S(x) ⇒ Ω−1(x)S(x) = S0
€
Tμν ⇒ Tμνkin − S0
2
6Rμν − R
2gμν
⎛ ⎝ ⎜
⎞ ⎠ ⎟− λ S0
4gμν
€
Tμνkin = iψ γ μ (x) ∂ν + Γν (x)[ ]ψ
⇒ ρ + p( )UμUν + pgμν
= matter + geometry + vacuum
matter fields => perfect fluid
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gμν ⇒ Ω2(x)gμν
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m = h S(x) ⇒ h S0Fermion mass:
Conformal trajectories
Conformal trajectories are the geodessic trajectories in the Higgs gauge.
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xα (τ ) Uα ≡ dxα
dτdUα
dτ+ Γμν
α U μ Uν = −Sμ
Sgα μ + Uα U μ( )
Mannheim 1993.
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m = h S(x)Fermion mass:
€
IM = −m dτ∫ ⇒ −h S(x)dτ∫Trajectory for which action is
stationary.
Test particle action.
Trace Condition
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W αα = 0 ⇒ Tα
α = 0
0 = matter + geometry + vacuum
Ricci scalar in
the vacuum
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UαUα = −1
gαα = δα
α = D = 4
€
4α Wμν = Tμν
€
R = 24 λ S02
€
0 = Tαα = ρ + p( )UαUα + pgα
α + S02
6Rα
α − R2
gαα
⎛ ⎝ ⎜
⎞ ⎠ ⎟− λ S0
4
= −ρ + (D −1)p + 2 − D12
S02R − Dλ S0
4
0 = 3p − ρ − 16
S02R − 4 λ S0
4
MK metric -> Higgs guage
Mannheim-Kazanas metric is not in Higgs guage.Test particles will not follow MK geodessics.
Ricci scalar:
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R = 24 λ S02
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gμν ⇒ Ω2 gμν
S ⇒ Ω−1 S = S0
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Sα Sα − 16
S2R ⇒ Ω−2 Sα Sα − 16
S2R ⎛ ⎝ ⎜
⎞ ⎠ ⎟= −4 λ S0
4
Sα Sα − S2 γr
− β γr2 − 2κ
⎛ ⎝ ⎜
⎞ ⎠ ⎟= −4 λ S0
4Ω2
€
R = 6 γr
− β γr2 − 2κ
⎛ ⎝ ⎜
⎞ ⎠ ⎟
Thanks for Listening !