Conformal Gravity in the X-ray

Cluster Abell 2029

Keit

h H

orn

e SU

PA

St

Andre

ws

108K gas

galaxies

monster galaxy

Mannheim-Kazanas metric

€

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 <<β

γ, κ = 0,

b(r)⇒ 1−2β

r=1−

2G M

c 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 M

c 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

dΦ

dr=

β

r2+

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

€

g(r)

c 2=

β

r2+

γ

2−κ r ⇒ g0

m

x 2+ m +1

⎛

⎝ ⎜

⎞

⎠ ⎟

V 2(r)

c 2=

β

r+

γ

2r−κ r2 ⇒ V0

2 m

x+ m x + x

⎛

⎝ ⎜

⎞

⎠ ⎟

€

β ≡G M

c 2γ = γ 0 m +1( ) κ = 0

V0 =100 km s−1 R0 = 24 kpc M0 = 5.6 ×1010 Msun

g0 =G M0

R02 =

V02

R0

=γ 0 c 2

2=1.4 ×10−9 m s−1

Mannheim 1993, 1997.

Universal inward acceleration.

€

m ≡M

M0

x ≡r

R0

( fits two more parameters)

Exterior mass matters !

€

g

g∞

=2 x

3

€

1−1

3x 2

Mannheim€

g = 0

€

g =GM

r2

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

€

z = 0.0767

d ≈c z

H0

= 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

€

g(r) =1

ρ (r)

dP(r)

dr

M(< r) =r2

Gρ (r)

dP(r)

dr

€

dP

dr= −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 Msun

r

200 kpc

⎛

⎝ ⎜

⎞

⎠ ⎟

2

Mgas + Mstars ~ 1013 Msun

r

200 kpc

⎛

⎝ ⎜

⎞

⎠ ⎟

Gravity and Total Mass

profiles:

g(r)

M(<r)

gas

stars

gas

stars

g(r)

M(<r)

gas

stars

gas

stars

g(r)

M(<r)

gasstar

s

gas

stars

€

g(r) ~ 3×108cm s−2 3 − 300 kpc

M(< r) ~ 1012 Msun r > 30 kpc

Mgas + Mstars ~ 1013 Msun

r

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)

gas

stars

gas

stars

g(r)

M(<r)

gasstar

s

gas

stars

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

€

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.

€

IW = −α d4 x −g Cλμνκ C λμνκ∫

= −2α d4 x −g Rμν Rμν −

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{∫

+1

2Sα Sα −

R

12S2 + λ S4 ⎫

⎬ ⎭

€

δ IM

δψ= 0 ⇒ iγ μ (x) ∂μ + Γμ (x)[ ] − h S{ }ψ = 0

δ IM

δ S= 0 ⇒ Sα

;α +R

6S − 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

€

δ IM

δψ= 0 ⇒ iγ μ (x) ∂μ + Γμ (x)[ ] − h S{ }ψ = 0

δ IM

δ S= 0 ⇒ Sα

;α +R

6S − 4 λ S3 = −hψψ

€

mH2 = −

R(x)

6

Higgs mass:

€

m = h S(x)Fermion mass:

€

V (S) =R

6S2 − 4 λ S4

Symmetry Breaking Higgs potential:

€

R < 0

€

λ < 0 Vmin < 0

Higgs potential:

€

V (S)

€

R < 0

€

R > 0

Field Equations

€

δ IW + IM( )δ gμν

= 0 ⇒ 4α Wμν = Tμν

€

Wμν = −1

6gμν R

;α;α +

2

3R;μ ;ν + Rμν

;α;α − Rμ

α;ν ;α − Rν

α;μ ;α

+2

3RRμν − 2Rμ

α Rαν +1

2gμν Rαβ R

αβ −R2

3

⎛

⎝ ⎜

⎞

⎠ ⎟

€

Tμν = iψ γ μ (x) ∂ν + Γν (x)[ ]ψ

+2

3SμSν −

1

3S Sμ ;ν +

1

3gμν S Sα

;α −1

2Sα Sα

⎛

⎝ ⎜

⎞

⎠ ⎟

−S2

6Rμν −

R

2gμν

⎛

⎝ ⎜

⎞

⎠ ⎟− λ S4gμν

Higgs Guage

€

S(x) ⇒ Ω−1(x)S(x) = S0

€

Tμν ⇒ Tμνkin −

S02

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

€

gμν ⇒ Ω2(x)gμν

€

m = h S(x) ⇒ h S0Fermion mass:

Conformal trajectories

Conformal trajectories are the geodessic trajectories in the Higgs gauge.

€

xα (τ ) Uα ≡dxα

dτ

dUα

dτ+ Γμν

α U μ Uν = −Sμ

Sgα μ +Uα U μ

( )

Mannheim 1993.

€

m = h S(x)Fermion mass:

€

IM = −m dτ∫ ⇒ −h S(x)dτ∫Trajectory for which action is

stationary.

Test particle action.

Trace Condition

€

W αα = 0 ⇒ Tα

α = 0

0 = matter + geometry + vacuum

Ricci scalar in

the vacuum

€

UαUα = −1

gαα = δα

α = D = 4

€

4α Wμν = Tμν

€

R = 24 λ S02

€

0 = Tαα = ρ + p( )UαUα + pgα

α +S0

2

6Rα

α −R

2gα

α ⎛

⎝ ⎜

⎞

⎠ ⎟− λ S0

4

= −ρ + (D −1)p +2 − D

12S0

2R − Dλ S04

0 = 3p − ρ −1

6S0

2R − 4 λ S04

MK metric -> Higgs guage

Mannheim-Kazanas metric is not in Higgs guage.

Test particles will not follow MK geodessics.

Ricci scalar:

€

R = 24 λ S02

€

gμν ⇒ Ω2 gμν

S ⇒ Ω−1 S = S0

€

Sα Sα −1

6S2R ⇒ Ω−2 Sα Sα −

1

6S2R

⎛

⎝ ⎜

⎞

⎠ ⎟= −4 λ S0

4

Sα Sα − S2 γ

r−

β γ

r2− 2κ

⎛

⎝ ⎜

⎞

⎠ ⎟= −4 λ S0

4Ω2

€

R = 6γ

r−

β γ

r2− 2κ

⎛

⎝ ⎜

⎞

⎠ ⎟

Thanks for Listening !