5th italian-sino workshop on relativistic astrophysics taipei - hualien 29 may 2008 1 fitting the...

5th Italian-Sino Workshop on 5th Italian-Sino Workshop on Relativistic Astrophysics Relativistic Astrophysics

Taipei - Hualien 29 May 2008Taipei - Hualien 29 May 2008 11

Fitting the luminosity data from Fitting the luminosity data from type Ia supernovae by means type Ia supernovae by means of the cosmic defect theoryof the cosmic defect theory

Angelo TartagliaAngelo Tartaglia

DIFIS – Politecnico and INFN DIFIS – Politecnico and INFN Torino, ItalyTorino, Italy



Plan of the talkPlan of the talk Starting point and motivationStarting point and motivation Outline of the Cosmic Defect theoryOutline of the Cosmic Defect theory Fit of the observational dataFit of the observational data Defects and Vector Theories: general Defects and Vector Theories: general

LagrangiansLagrangians Open problemsOpen problems



The accelerated expansion The accelerated expansion (luminosity data of SnIa’s)(luminosity data of SnIa’s)



The power spectrum of CMBThe power spectrum of CMB

k=0



Presently agreed expansionPresently agreed expansion



Einstein equationsEinstein equations

Tc

GgRgR

4

8

2

1

Spacetime geometry “Matter”

Why is Λ on the left?



Something missingSomething missing

•Inflation•Gravity in clusters and galaxies•Accelerated expansion

There is something missing

Modify GRIntroduce the missing entities

Give up GR and look for another theory



...8

)(2

1104

TT

c

GgRgR

Accept a four- (N-) dimentional spacetime manifoldAdd “matter” components

Isotropy and homogeneityPerfect fluid

2cρwpEos



ΛΛ Cold Dark Matter 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?



Trying a new approach: Trying a new approach: The Cosmic Defect theoryThe 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



Strain in a continuumStrain in a continuumN-dimensional “sheet”N-dimensional “sheet”

Strain induced by boundary conditions



A defectA defect

Internal “spontaneous” strain state



Four-dimensional point defectFour-dimensional point defect



aa

μμ dyφdx

aa

μμ dy

y

xdx

Geometry, elasticity and defectsGeometry, elasticity and defects



In a strained medium each point is in one In a strained medium each point is in one to one correspondence with points in the to one correspondence with points in the

unstrained stateunstrained state

yx

The new situation is diffeomorphic to the old one

ξ is a function of x

Intrinsic coordinates Extrinsic coord.

Displacement



Induced metricInduced metric

2g

ν

β

μ

α

αβμν

ν

μμν x

ξ

x

ξη

x

ξ

x

ξε

2

1

Strain tensor



““Radial” displacement field Radial” displacement field (space (space isotropy and homogeneity)isotropy and homogeneity)

000 ,,,tuξ

θsinurφ

ξ;ur

θ

ξ;

kr

u

r

ξ;

t

u

t

ξ

21

22

22

2

2

22

00 sin2

;2

;12

;22

1r

ur

u

kr

u

t

u

t

urr



The line elementThe line element

22222

2

2222

1φdθsinrθdr

kr

drtdtds

22222

2

2222

12 φdθsinrθdr

kr

druτττdds

utτdtdt

duτd

1

22222

2

2222

22 sin

11 drdr

kr

drutdt

dt

duds

Unperturbed

Strained

0k



A Robertson-Walker universeA Robertson-Walker universe

τuτττa 222

222222222 φdθsinrθdrdraτdds



How can we choose a Lagrangian How can we choose a Lagrangian expressing the presence of the expressing the presence of the

defect?defect?

Start from the phase space of a Robertson-Walker universe and look around for similar phase

spaces



Phase space analogyPhase space analogy

FRW universeInertial expansion

Accelerated expansion

Decelerated expansion

Point particle

Free motion

Driving force

Braking force



A simple classical problemA simple classical problem

Motion of a point massive particle in a viscous medium

2

2

1xmeL

LdtS

xt

b

a



Invariant formulation of the Invariant formulation of the same problemsame problem

x

s

s

x

edt

xd

cmL

dsemS

2

2

11

2

1



SpacetimeSpacetime“Dissipative” action integral“Dissipative” action integral

RdeS g

• Same structure as in the classical simple case

• The “viscous” properties of space-time are contained in the vector field



Robertson Walker symmetryRobertson Walker symmetry

Isotropy and homogeneity in 3 dimensionsRW line element

1,0 sin1

22222

2222

kddr

kr

dradds

The symmetry is induced by the presence of a “defect”



Impose the 4-isotropy around the Impose the 4-isotropy around the origin and use cosmic time as the origin and use cosmic time as the

“radial” coordinate“radial” coordinate

dkaaaaeS )(6 22

kV

)0,0,0,(



Symmetry and application of the minimal action principle do

not commute

Defect means

Symmetry first



Non-trivial ifNon-trivial if

constantχ

)( 22

kaaaae L

Lagrangian



Divergence free Divergence free vectorvector

3

3

3

,

0

0

a

Q

a

g



The expansion rateThe expansion rate((a and time in units of Qa and time in units of Q))

62

1

2/16

2/5

6ae

a

aa

Choose the + sign



Expansion rateExpansion rate

Accelerated expansion

Asymptotic stop



Expansion versus cosmic timeExpansion versus cosmic time

Inflation

Acceleration



Fitting the data from Fitting the data from SnIaSnIa



z

Sm ζH

ζdlogzclogMmλ

051525

The distance modulusThe distance modulus

a

aH

z

aa

10



The energy function for the CD The energy function for the CD theorytheory

WLL

H

34

02

5

66

2

acρκaaa

eκad

da χ

a

aκe

acρ-a

a/5

1

342

66

6

W



Multicomponent cosmic fluidMulticomponent cosmic fluid

2cρwp iii

i

i

w

w

ii a

aρρ

13

130

0

aa

κe

a

aρ-c

aa/

w

w

ii i

i

51

3

130

04

2

66

6

W

Equation of state

Conservation law

Expansion rate



Fitting the data (192 SnIa)Fitting the data (192 SnIa)



ΛCDM 2 = 1.029

CD 2 = 1.092

Reduced Reduced 22 of the fits of the fits



The Hubble parameterThe Hubble parameter

H0 = (62.8 ± 1.7) km/sMpc

Most models ~64 km/sMpc

Observation ~75 km/sMpc



Weaknesses and open problemsWeaknesses and open problems

Fitting the SnIa luminosity data with Fitting the SnIa luminosity data with a logarithmic function and two a logarithmic function and two parameters is “too easy”parameters is “too easy”

The inflation is too strong and long The inflation is too strong and long lasting (troubles with nucleosynthesis lasting (troubles with nucleosynthesis and formation of structures)and formation of structures)

The exponential in the action integral The exponential in the action integral is a poweful multiplier, but it should is a poweful multiplier, but it should be weakenedbe weakened



The null divergence condition should The null divergence condition should be a consequence of the singularity in be a consequence of the singularity in correspondence of the defect, rather correspondence of the defect, rather than a formal constraint imposed on than a formal constraint imposed on the vector.the vector.

Once it has been induced, the Once it has been induced, the γγ vector vector has its own dynamics and energy has its own dynamics and energy content which must be taken into content which must be taken into accountaccount



General Lagrangian treatment General Lagrangian treatment (non-exponential coupling)(non-exponential coupling)

...RγγRRe αββα

αβγγδ βααβ

RγγγνγμγγλRg

βαβ

αα

αα

ββ

α 122L

σασσααβ

β γRγ

Non-minimalcoupling



The equations for The equations for aa and and

03633 222 χaABaaBaχaCaχCa

WaχχCaaχχBaaχA 322222 623

νμλC

μB

μνλA

2

23



Special or trivial solutionsSpecial or trivial solutions

320 /τaχ

00 constantχχ

03

2χ,ABa any constant

AB

χ,Wτa23

20 0

any

Matter dominated FRW



CorrespondencesCorrespondences

Bimetric theories: “pre-shaped Bimetric theories: “pre-shaped container”container”

Vector-tensor theoriesVector-tensor theories Curvature fluidCurvature fluid



Final remarksFinal remarks

The CD theory provides a consistent The CD theory provides a consistent physical interpretation of space-physical interpretation of space-time giving a heuristic tool to time giving a heuristic tool to move across the Lagrangian move across the Lagrangian “engineering” mostly driven by the “engineering” mostly driven by the formal search for the desired formal search for the desired result. result.

This conceptual framework looks This conceptual framework looks promisingpromising



A. Tartaglia, M. Capone, Int. Jour. Mod. Phys. D, 17, 275-299 (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



Thank you



Why SnIa?Why SnIa?

Accreting white dwarf

Supernova explosion

Tycho Brahé 1572 (Chandra’s image)



SnIa is a good candleSnIa is a good candle

Stable light curve



…. 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

Ether againEther again

5th italian-sino workshop on relativistic astrophysics taipei - hualien 29 may 2008 1 fitting the...

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