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Keith Horne SUPA St
Andrews
Conformal Gravity
in the X-ray Cluster
Abell 2029
galaxies
108K gas
monster
galaxy
Mannheim-Kazanas metric
4  W   T  0
Source-free field
equations:
Static, spherical
symmetry:
2
dr
2
2
2
2
2
2

ds  b(r) dt 
 r d  sin  d 
b(r)
b(r)  1 3   


 2  3   
r
  r   r2
Analogous results including rotation and
charge.


Solar System Dynamics
2
dr
2
2
2
2
ds  b(r) dt 
 r d  sin  d 
b(r)
2
2
b(r)  1 3   
 2  3   
r
Schwarzschild metric:

  1, r  ,   0,

2
2G M
2
b(r)  1
r
 1
2
c r
,
  r   r2
Fixes one
parameter
GM
 2
c

Galaxy Rotation Curves
b(r)
(r)
c2
g(r)
c2
V 2 (r)
2
c
2(r)
 1
2
c
b(r) 1

2
1 d

c 2 dr
2
2
 1
  r  r
r
 r g(r)


 


r



2
r

2
r2
  r
r
2
2

r


2
r r
2
Linear potential gives a distanceindependent inward acceleration.
Galaxy Rotation Curves
Mannheim 1993,
1997.
g(r)
 
 2   r
2
c
r
( fits two more
mparameters)

 g0  2  m  1
x


2 m
 V0   m x  x 
 x

2
V (r)
 
2

 r r
2
c
r 2
2
M
m
M0
r
x
R0
GM
 2
  0  m  1
 0
c
 10
1
V0  100 km s
R0  24 kpc M 0  5.6 10 M sun
G M 0 V0
0 c
9
1
g0 



1.4
10
m
s
2
R0
2
R0
2
2
Universal inward acceleration.
Exterior mass matters !
Newton
No net
force
from
external
shells
Hook’s 
law
force toward
centre of
external
shell

GM
g 2
r
g0

Mannheim
g 2x

g
3
1
1 2
3x
Rotation Curve Fits
Fits adjust [M/L]* All mass in edge-on disk plane.
linear
potentia
l
stars
gas
NGC 1560
poorest fit.
Data wiggles
follow the
gas.
Abell 2029
Probes gravity on 10x larger scales
200
Kpc
z  0.0767
cz
d
H0
 320 Mpc
Chandra X-ray Image of Abell 2029

The 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.
X-ray Gas
Spherical symmetry +
Hydrostatic Equilibrium:
dP
G M( r)
  (r) g(r)    (r)
dr
r2
(r) k T(r)
observe : T(r),  (r),
P(r) 
 mH
Gravity and Total Mass profile:
1 dP(r)
g(r) 
(r) dr
r
2
dP(r)
M( r) 
G (r) dr
X-ray Gas 3-300 kpc
Lewis, Stocke, Buote 2002.
(r)
T(r)
cs(r)
sgals
P(r)
v* sin(i)
Newtonian Analysis
Gravity and Total
Mass profiles:
g(r) ~ 3 10 8 cm s 2
g(r)
3  300 kpc
 r 
M( r) ~ 10 M sun 

200 kpc 
 r 
13
M gas  M stars ~ 10 M sun 

200 kpc 
gas
2
14
90% Dark
Matter
Required !
star
s
M(<r)
gas
star
s
Conformal Gravity
Gravity and Total
g(r)
Mass profiles:
g(r) ~ 3 10 8 cm s 2
g(r)
gas
star
s
3  300 kpc
gas
Dark Matter NOT
required !
star
s
r  30 kpc
gas
M(<r)

rstar 
13
M gas  M stars ~ 10 M sun 

s

200 kpc
12
M( r) ~ 10 M sun
Too Much Conformal
Gravity!
M(<r)
gas
star
s
Newton vs Conformal Gravity
g(r)
g(r)
gas
star
s
gas
star
s
M(<r)
gas
star
s
M(<r)
gas
star
s
Discussion Points
:)
Dark Matter is not needed
to 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
g   (x) g
2
Clock ticks and rulers stretch by a
factor that can vary in time and space.

Invariants: angles, velocities, light cones,
causality.
IW    d 4 x g C C 
2 
Weyl action:

R
4

 2   d x g R R  

g  det g  R  R


3 

RR

Conformal Matter Action
I    d x g i   (x)    (x) i h S 
4
M



1 
R 2
4 
 S S  S   S 

2
12
Equations of motion:
 IM

 0  i  (x)     (x) h S  0

 IM
R

 0  S ;  S  4  S 3  h 
S
6
Higgs
Fermion
S
S   S  
mass:
R(x)
mmass:
 h S(x)
2
x
mH  
6
Dynamical Mass Generation
 IM
 0  i   (x)     (x) h S  0

 IM
R

 0  S ;  S  4  S 3  h 
S
6
Higgs
Higgs
Fermion
mass:
potential:
R(x)
2
V
(S)
mmass:
 h S(x)
mH  
6
R0
Symmetry Breaking Higgs
potential:
R 2
4
R0
V (S)  S  4  S 
6


Ricci scalar:
(negative spatial
R0

  0 Vmin  0curvature).
Field Equations
 IW  IM 
 0  4  W   T
 g
1
2
;


;
W    g R ;  R; ;  R ;  R ; ;  R ; ;
6
3
2 

2
1
R
  R R  2 R  R  g R R  





3
2 
3 
T  i   (x)    (x)
 
2
1
1
1  
 S S  S S ;  g S S ;  S S 

3
3
3 
2

S 2 
R
 R  g   S 4 g

6 
2
Higgs Guage
g  2 (x) g
1
S(x)   (x) S(x)  S0
m  h S(x)  h S0
Fermion

mass:

T
 T
kin


S0 
R
4

R  g   S0 g

6 
2
2
= matter + geometry + vacuum
T  i   (x)    (x)
kin

  pUU  p g
matter fields => perfect fluid
Conformal trajectories
Mannheim 1993.
Fermion
mmass:
 h S(x)
Test particle action.
IM  m  d  h  S(x) d
Trajectory for which action is stationary.

dx


x ( )
U 

d

S  
dU



  U U   g  U  U  
d
S
Conformal trajectories are the geodessic
trajectories in the Higgs gauge.
Trace Condition
4  W   T
W   0  T   0
U U  1
g     D  4
2
S
R  
4

 0  T      pU U  p g   0 
R   g    S0


6 
2

2D 2
4
   (D 1) p 
S0 R  D  S0
12
1 2
4
0  3p    S0 R  4  S0
Ricci scalar
6
0 = matter + geometry + vacuum
in the
2
vacuum
R  24  S0
MK metric -> Higgs guage
Ricci scalar:
  

R  6  2  2 
r r

g  2 g
R  24  S0
2
1
S  
S S0
1 2
1 2 
4

2  
S S  S R   S S  S R 4  S0


6
6

 
4 2
 S  S  S 2 
  2  2  4  S0 

r r

Mannheim-Kazanas metric is not in Higgs guage.
Test particles will not follow MK geodessics.
Thanks for Listening !