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Fitting the luminosity data
from type Ia supernovae by
means of the cosmic defect
theory
Angelo Tartaglia
DIFIS – Politecnico and INFN
Torino, Italy
5th Italian-Sino Workshop on
Relativistic Astrophysics
Taipei - Hualien 29 May 2008
1
Plan of the talk





Starting point and motivation
Outline of the Cosmic Defect theory
Fit of the observational data
Defects and Vector Theories: general
Lagrangians
Open problems
5th Italian-Sino Workshop on
Relativistic Astrophysics
Taipei - Hualien 29 May 2008
2
The accelerated expansion
(luminosity data of SnIa’s)
5th Italian-Sino Workshop on
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Taipei - Hualien 29 May 2008
3
The power spectrum of CMB
k=0
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Presently agreed expansion
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5
Einstein equations
1
8 G
R  g  R  g   4 T
2
c
Spacetime geometry
“Matter”
Why is Λ on the left?
5th Italian-Sino Workshop on
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Taipei - Hualien 29 May 2008
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Something missing
•Inflation
•Gravity in
clusters and
galaxies
•Accelerated
expansion
Give up GR and
look for another
theory
5th Italian-Sino Workshop on
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There is something missing
Modify GR
Taipei - Hualien 29 May 2008
Introduce the
missing entities
7
1
8 G
R  g  R  (g  )  4 T 0   T 1  ...
2
c
Add “matter” components
Accept a four- (N-) dimentional spacetime manifold
Eos  p  wρc
Isotropy and homogeneity
5th Italian-Sino Workshop on
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Taipei - Hualien 29 May 2008
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Perfect fluid
8
Λ Cold Dark Matter
Simplest and most effective model for
the universe; however:
• “matter” must be 7 times what we “see”
(~30% of the source);
• Λ corresponds to 70% of the souce
but …
what is Λ?
5th Italian-Sino Workshop on
Relativistic Astrophysics
Taipei - Hualien 29 May 2008
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Trying a new approach:
The Cosmic Defect theory
Motivation, besides the simple fitting of the
data:
• describing the large scale behaviour of the
universe in terms of intrinsic properties of a
four-dimensional continuum;
• interpreting space-time as Einstein’s GR
ether
5th Italian-Sino Workshop on
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Taipei - Hualien 29 May 2008
10
Strain in a continuum
N-dimensional “sheet”
Strain induced by boundary conditions
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A defect
Internal “spontaneous” strain state
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Four-dimensional point defect
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Geometry, elasticity and defects
dx μ  φ μ a dy a
μ

x
dx μ  a dy a
y
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Taipei - Hualien 29 May 2008
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In a strained medium each point is in one to
one correspondence with points in the
unstrained state


x  y 
Intrinsic coordinates

Displacement
Extrinsic coord.
The new situation is diffeomorphic to the
old one
ξ is a function of x
5th Italian-Sino Workshop on
Relativistic Astrophysics
Taipei - Hualien 29 May 2008
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Induced metric
g     2 
α
β

ξ
1  μ ξ ν
ξ ξ 
ε μν   ν  μ  ηαβ μ ν 
2  x x
x x 
Strain tensor
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“Radial” displacement field (space
isotropy and homogeneity)
ξ  ut ,0,0,0
ξ  u
ξ 
u
ξ 
ξ 
 ;

;
 ur ;
 ur sin θ
2
t
t
r
θ
φ
1  kr
1  u  u 
 00  2   
2  t  t 
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2
2
2

u
u
u
2
2
2
 ;  rr 
;


r
;


r
sin



2

2 1  kr
2
2



Taipei - Hualien 29 May 2008
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The line element
2

dr
2
2
2
2
2
2
2
2
ds  dt  t 
 r dθ  r sin θdφ 
2
 1  kr

Unperturbed
2

 du 
ds 2  1   dt 2  t 2  u 2
 dt 
 dr 2
2
2
2
2
2



r
d


r
sin

d

2
 1  kr


Strained
k 0
 du 
dτ  1  dt  τ  t  u
 dt 
 dr 2
2
2
2
2
2
ds  dτ  τ  2τu 
 r dθ  r sin θdφ 
2
 1  kr

2
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2

2

Taipei - Hualien 29 May 2008
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A Robertson-Walker universe

ds  dτ  a dr  r dθ  r sin θdφ
2
2
2
2
2
2
2
2
2

a τ   τ  2τ uτ 
2
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How can we choose a Lagrangian
expressing the presence of the
defect?
Start from the phase space of a
Robertson-Walker universe and
look around for similar phase
spaces
5th Italian-Sino Workshop on
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Taipei - Hualien 29 May 2008
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Phase space analogy
Inertial expansion
FRW universe
Accelerated expansion
Decelerated expansion
Free motion
Point particle
Driving force
Braking force
5th Italian-Sino Workshop on
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Taipei - Hualien 29 May 2008
21
A simple classical problem
Motion of a point massive particle in a
viscous medium
b
S   Ldt
a
1 t  x 2
L e
mx
2
5th Italian-Sino Workshop on
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22
Invariant formulation of the
same problem
s2
    x
S  m  e
s1
ds
2
1 dx     x
L  m 1  2
e
c dt
5th Italian-Sino Workshop on
Relativistic Astrophysics
Taipei - Hualien 29 May 2008
23
Spacetime
“Dissipative” action integral
S  e
 
 g   
Rd
• Same structure as in the classical simple case
• The “viscous” properties of space-time are
contained in the vector field 
5th Italian-Sino Workshop on
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Robertson Walker symmetry
Isotropy and homogeneity in 3 dimensions
RW line element
2

dr
2
2
2
2
2
2
2 
ds  d  a 
 r d  sin d 
2
1  kr



k  0,1
The symmetry is induced by the presence of a “defect”
5th Italian-Sino Workshop on
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Taipei - Hualien 29 May 2008
25
Impose the 4-isotropy around the
origin and use cosmic time as the
“radial” coordinate
  (  ,0,0,0)
S  6 Vk  e
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
2
a(aa  a  k )d
Taipei - Hualien 29 May 2008
2
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Symmetry and application of the
minimal action principle do not
commute
Defect means
Symmetry first
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Lagrangian
L e
2
a(aa  a  k )
2
Non-trivial if
χ  constant
5th Italian-Sino Workshop on
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Divergence free vector


 g

,
0

 
a 
0

Q3
 3
a
3
5th Italian-Sino Workshop on
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The expansion rate
(a and time in units of Q)
a 
a
a
6
5/ 2
6


1/ 2
e
1
2a6
Choose the + sign
5th Italian-Sino Workshop on
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30
Expansion rate
Accelerated expansion
Asymptotic stop
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Expansion versus cosmic time
Acceleration
Inflation
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Fitting the data from SnIa
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The distance modulus
λm  m  M S  25  5 log c1  z   5 log 
z
0
a
H
a
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dζ
H ζ 
a0
a
1 z
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The energy function for the CD
theory
dL
 2
 χ2  6
4 3
H  a
 L  6κe  5  a a  κ 0 ρc a  W
da
a

W -ρc a
a  

1 / a 6  6
6κe
 5  a
a

4 3
2
5th Italian-Sino Workshop on
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Multicomponent cosmic fluid
pi  wi ρi c
Equation of state
2
31 wi 
Conservation law
a0
ρi  ρi 0 31 wi 
a
31 wi 
a0
W -c  ρi 0 3wi
a
2
i
a  

1 / a 6  6
6κe
 5  a
a

4
Expansion rate
5th Italian-Sino Workshop on
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Fitting the data (192 SnIa)
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Reduced 2 of the fits
ΛCDM
2 = 1.029
2 = 1.092
CD
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The Hubble parameter
H0 = (62.8 ± 1.7) km/sMpc
Most models
~64 km/sMpc
Observation
5th Italian-Sino Workshop on
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Taipei - Hualien 29 May 2008
~75 km/sMpc
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Weaknesses and open problems



Fitting the SnIa luminosity data with a
logarithmic function and two
parameters is “too easy”
The inflation is too strong and long
lasting (troubles with nucleosynthesis
and formation of structures)
The exponential in the action integral is
a poweful multiplier, but it should be
weakened
5th Italian-Sino Workshop on
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Taipei - Hualien 29 May 2008
40


The null divergence condition should be
a consequence of the singularity in
correspondence of the defect, rather
than a formal constraint imposed on the
vector.
Once it has been induced, the γ vector
has its own dynamics and energy content
which must be taken into account
5th Italian-Sino Workshop on
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41
General Lagrangian treatment
(non-exponential coupling)
e
δ αβ  γ α γ β
Rαβ  R  γ α γ β Rαβ  ...
γ
β
σ


Rαβ  ασ  σ α γ
Non-minimal
coupling
L
g


 R  λ α γ β  β γ α  μ  α γ α  ν α γ β  α γ β  γ 2 1  R 
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2
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The equations for a and 


Ca 2 χ  3Caaχ  3Baa  a 2  6B  3 A a 2 χ  0




3 2  Aχ 2 aa 2  6Bχχa 2 a  Cχ 2  χ 2 a 3  W
A  λ  ν  3μ  2
B  μ2
C  λ μν
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Special or trivial solutions
χ 0
aτ
2/ 3
Matter dominated FRW
χ  χ 0  constant  0
a  constant
2
B  A, any χ 0
3
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any aτ  W  0,
Taipei - Hualien 29 May 2008
χ0  
2
3B  2 A
44
Correspondences



Bimetric theories: “pre-shaped
container”
Vector-tensor theories
Curvature fluid
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Final remarks
The CD theory provides a consistent
physical interpretation of space-time
giving a heuristic tool to move across
the Lagrangian “engineering” mostly
driven by the formal search for the
desired result.
This conceptual framework looks
promising
5th Italian-Sino Workshop on
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A. Tartaglia, M. Capone, Int. Jour. Mod. Phys. D, 17, 275299 (2008)
A. Tartaglia, N. Radicella, Phys. Rev. D, 76, 083501 (2007)
A. Tartaglia, M. Capone, V. Cardone, N. Radicella,
arXiv:0801.1921, to appear on Int. Jour. Mod. Phys. D
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Thank you
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48
Why SnIa?
Accreting white dwarf
Supernova explosion
Tycho Brahé 1572 (Chandra’s image)
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SnIa is a good candle
Stable light curve
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Ether again
…. according to the general theory of relativity space
is endowed with physical qualities; in this sense,
therefore, there exists an ether. According to the
general theory of relativity space without ether is
unthinkable; for in such space there not only would be no
propagation of light, but also no possibility of existence for
standards of space and time (measuring-rods and clocks),
nor therefore any space-time intervals in the physical
sense. But this ether may not be thought of as
endowed with the quality characteristic of ponderable
media, as consisting of parts which may be tracked through
time. The idea of motion may not be applied to it.
Albert Einstein, Leiden, 1920
5th Italian-Sino Workshop on
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