Connecting the Galactic and Cosmological Length Scales:

Dark Energy and The Cuspy-Core Problem

ByAchilles D. Speliotopoulos

Talk Given at the Academia SinicaNovember 26, 2007

100 kpc

10 Mpc

100 Mpc

4100 Mpc

Galactic

Supercluste

r

Cluster

Cosmologica

l

Range spans 5

orders

of magnitu

de!Len

gth scales

of Phen

omenon driv

en

by Dark M

atter a

nd Dark Energ

y

The Cuspy-Core Problem

0

20

40

60

80

100

120

140

160

180

0 2 4 6 8 10 12

Radius in kpc

Vel

ocit

y in

km

/s

Raw Data Psuedoisothermal Fit NFW Fit

/0

20

/1/

2,0,2/1

ss

Sim

S

iso

RrRr

Rr

Dark Energy

DE=7.21x10-29 g/cm3

100 kpc

10 Mpc

100 Mpc

4100 Mpc

Galactic

Supercluste

r

Cluster

Cosmologica

l

1402

0 Mpc

G

c

DE

DE • L

inks gala

ctic a

nd cosm

ological le

ngth scale

s

• Form

ing Gala

xies = Lower

Free Energ

y

• 8

= 0.68 ±0.11 (

WMAP valu

e = 0.761 -0.048

+0.049 )

• Frac

tional

densit

y of matt

er that

cannot b

e

dete

rmined

through grav

ity, asy

mp = 0.196 ±0.017

• Frac

tional

densit

y of matt

er that

can be

dete

rmined

through grav

ity, Dyn

= 0.042 -0.026+0.025

• R

±130

kpc

An Extended Lagrangian

dt

dx

dt

dxgGRcDmcL

v

DEExt

2

121

vv or cvv :Constraint 02

Extended Geodesic Equations of Motion (GEOM)for massive test particles

GRcDc

vvgc

dt

xDDE

/1log 2

22

2

2

GRcDcmp DE /1 2222

Only how fast D(x) changes matter!

Curvature-dependent effective rest mass!

Extended GEOMfor massless test particles (photons)

0/1 2

dt

dxGRcD

dt

DDE

0/12

22

dt

xD dt GRcDdt DE

Motion of massless test particles are not affected by the extension!

Gravitational lensing and the deflection of light do not change!

DE as the Cosmological Constant

Tc

G g

c

GRgR DE

42

8

2

1

42

84

c

TG

c

GR DE

22 /84/ GcTDGRcD DEDE

If T=0 , D(x) is a constant, the extended GEOM reduces to the GEOM!

TUnder Extended GEOM

pc

vvgvvT

2

dpVc d 2

0/841/8412 pGT DGT D pc DEDE

pc

vvgvv

c

pv

220

Spatial isotropy:

Temporal variationv2 = c2 :

Spatial variation:

1st Law of Thermodynamics!

Tfor Dust Under GEOM

0

22

)/84(1)/84(1

Gs D

ds GDccp

DE

DE

for c

p ile wh for c

p DEDE

DE

22~0~

222

In the nonrelativistic limit!

l! stilvv T dust-Ext

A Choice for D(x)

GRcD

GRcDc

dt

d

DE

DE

DE

2

/1

/42

2

2

2 x

0,1

11

t

dt

x

x t

dtxD 11

D’(x)<0

When >> DE/2, D 0. Extended GEOM GEOM.

No observable 5th force!

rH

rII

Region IIv vH

Region III

Region Iv= vHr/rH

A Model Galaxy

Idealized Velocity Curves

0

20

40

60

80

100

120

140

160

180

0 2 4 6 8 10 12

Radius in kpc

Vel

ocit

y in

km

/s

Raw Data Psuedoisothermal Fit NFW Fit

HHIdeal

HH

HIdeal r r for vrv ,rr for

r

rvrv

H

2H

HIdeal

H

2

HH

IdealCuspy r r for

r

rvrv ,rr for

r

rvrv

11

~

A Matter of Length Scales

22

2

8

/84

11)(

DEDEGrrf

11

2/1 841

1

DEDE

1

/8log

/log1 2

DE

DECoreCore

H

-dependentlength scale!

2/3Comparing length scalesnear the galactic core!

Grf

4)(

a

When >> DE/2The Density Equation in Regions I and II

8

1)( 2 DErrf

DE1/2

HHH

HHHru

r r for rr

rr for rrrf

,/

3

1/

)(

DE

DEDEDEDEDE fd

c F

8

88182

1

8

12

3

32/12

u

Idealized densityprofiles!

Our free energy!

The Solution in Region I

DE

DEDEDEDEDEI fd

c F

8

88182

1

8

12

3

32/12

u

132

861

1,

H

DEHDEIHI

rcF and F, minimizes 0 r

thermal)(pseudoiso 0 for ,r DEH

8

1 2

Contributes positive term for >0!

Free Energy Conjecture

Like a Landau-Ginberg Theory, the system wants to be in a state the minimizes the Free

Energy

The Solution in Region IIAsymptotics

asymp

DEasymp 8

10 2

-1 u r DE

asymp

1

11/1

212 ,

)1(

)31(2,

8

Anzatz: f(r) << (r) for r large!

integer even

integer oddDepends only on DE, , and symmetry!

Contains no info on the structure of galaxy!

PerturbationsThe Decoupling of Length Scales

HHHH

HasympII rrCCrrCr

r

r

rArr /logsin/logcos

3

10cossin0cos

2/5

121122

1

22ˆ,ˆ

ˆ

)1(

)31(2

3

1

)1(

)31(2IIIIII

IIHH u r

u

r

u

u

Length scale set by DE

No knowledge of galactic structure.

Length scale set by rH

Aside from BC, no knowledge of DE!.

The Free Energy in Region II

1II-asymp

IIasymp

IIII FFF F

asymp

DE2 - asymp

I uf dc~F8

)(3u

12/-5 ifr

12/-55/2 ifrr

5/2 firr

F

H

IIH

IIH

- asympII

)1/(25

2/5~

Due only to asymp.Independent of . ~ (II

1)2. Very small

= 2 gives state of lowest Free energy!

The Solution in Region III << DE/2

2

141

10

u

f(u)0 here!

Density decreases exponentially fast hereFundamental scale is DE/(1+41+a)1/2

asymp(DE/(1+41+a)1/2) << DE/2

DEIIr

141

rasymp

Potentials What Can and Cannot be Seen

EffDE VGRcDdt

d /1log 2

2

2 x

Vrf Eff2)(

G 24

)/(log31

1

412

22

HrrOuc

r

2/12 /log rOrrvV HHEff

Determined by II.-aymp

Dominated by aymp

Dynamics driven by Veff not !

Inferring mass from dynamics under gravity determines II – asymp.Mass of particles in asymp cannot be “seen”!

The Link with Cosmology

!52.0

413

82/31 HHIIr

The theory naturally cuts off the density at ~H /2even though H was not put in at the beginning.

What happens at the galactic scale is linked to the cosmological scale.

Determined on galactic scale.

WMAP Value: = 1.51±0.011

Calculation of 8

DE

8HHDE

Mpch

Mpchu vrMpchu

2/1

1**1

88

8,,,8

31

1

8

3 1/2

1

2208

2

2

8

2

8

828

)1()31(1

10

14

815

131

8

1

31

3

1

1313

4

11

1

y

MpchMpch

Mpch

MPCh

ry

c

vu H8

H12

21/28

8,

2

Properties of the galaxy

8 dominated by . Result of rotation curves!

Dominated by asymp, H 1.

8 from 1393 Galaxies

Data Set

De Blok et. al. (53) 119.0 6.8 3.62 0.33 0.613 0.097 1.36

CF (348) 179.1 2.9 7.43 0.35 0.84 0.18 0.43

Mathewson et. al. (935) 169.5 1.9 15.19 0.42 0.625 0.089 1.34

Rubin et. Al. (57) 223.3 7.6 1.24 0.14 2.79 0.82 2.46

Combined (1393) 172.1 1.6 11.82 0.30 0.68 0.11 0.70

testt *Hr

*Hv *

Hv 8*Hr 8

049.0048.08 761.0

From WMAP:

De Blok et. al. Data Set Rubin et. al. Data SetW. J. G. de Blok, S. S. McGaugh, A. Bosma, and V. C. Rubin, Astrophys. J. 552, L23 (2001).

W. J. G. de Blok, and A. Bosma, Astro. Astrophys. 385, 816 (2002).

S. S. McGaugh, V. C. Rubin, and W. J. G. de Blok, Astron. J. 122, 2381 (2001).

V. C. Rubin, W. K. Ford, Jr., and N. Thonnard, Astrophys. J. 238, 471 (1980).

V. C. Rubin, W. K. Ford, Jr., N. Thonnard, and D. Burstein, Astrophys. J. 261, 439 (1982).

D. Burstein, V. C. Rubin, N. Thonnard, and W. K. Ford, Jr., Astrophys. J., part 1 253, 70 (1982).

V. C. Rubin, D. Burstein, W. K. Ford, Jr., and N. Thonnard, Astrophys. J. 289, 81 (1985).

Mathewson et. al. Data Set CF Data SetD. S Mathewson, V. L. Ford, and M. Buchhorn, Astrophys. J. Suppl. 82, 413 (1992). S. Courteau, Astron. J. 114, 2402 (1997).

Fractional Density of What Cannot be SeenUsing Gravity

3)1/(1

12

2/ 411

312

31

1

8

3

H

H

casymp

rOH

025.0026.0196.0

Bm

11.0017.0 51.1,196.0 for asymp

Fractional density of non-baryonic (dark) matter from WMAP:

Fractional Density of What Can be SeenUsing Gravity

030.0031.0

2/ 042.0

asympm

c

asympII

DynH

0038.00039.00416.0

B

Fractional density of baryonic matter from WMAP:

Concluding Remarks

• Amazingly good agreement with WMAP

• Agreement supports Free Energy Conjecture

• R200=270±130 kpc, which agrees with observation

• = 1.51 is small enough that it may be measurable in laboratory.

• asymp = m – B and Dyn B . A numerical coincidence?