Post on 29-Dec-2015
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
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
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!
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: