The Strained State Cosmology Angelo Tartaglia Politecnico di Torino - - PowerPoint PPT Presentation

The Strained State Cosmology

Angelo Tartaglia

Politecnico di Torino and INFN

Prologue

• It seems that something pushes space

to expand, but we do not know what it

• is. Apparently it does not produce other

effects

• It seems that, at a big enough scale,

localized gravitational effects exist whose source is not otherwise visible.

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Premises

• Apparently the gravitational interaction

is very well described as a geometric property of a four-dimensional Riemannian manifold

• Other fundamental interactions do not

share this geometric essence

 

T G 

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Axiomatic assumption

• The physical configuration of the world, in

all interactions, satisfies an universal principle of “economy”: the ‘least’ action principle

*   



S x d S

N

  L

Scalar N-form Lagrangian density

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What is L made of?

• Formal mathematical answer: any scalar

function of the state variables and their derivatives with respect to the (arbitrarily chosen) coordinates

• Intuitive physical answer: in analogy

with the Lagrangian densities, a posteriori built starting from the recognized physical laws validated by experiment

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Additional assumption

• The Lagrangian density must have the

highest possible ‘formal symmetry’ (the highest simplicity).

• Non-”simple” functional forms require

ad hoc motivations.

– Is reproducing a specific observed physical situation enough? – Is it possible to find universal ‘non-simple’ solutions?

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Tridimensional analogy

• Deformable continuous media
• Geometrizable interaction: elastic

interaction (with an external evolution parameter – time -)

• Macroscopic emergent representation,

from microscopic elementary interactions

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“Elastic” continua

 

 u

xμ Xa ξa

r r’

 

'  u

 

2 1



n N

X ,..., X , X f

 

X ,..., X , X h

n N 2 1





N+n N N

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Deformation

• Undifferenciated reference state  flat

manifold independent from the parameter.

• Geometry: Euclidean/Minkowskian
• Deformation due either to intrinsic

(defects) or ‘extrinsic’ (matter/energy) causes

• Riemannian geometry

j i ij

dx dx E dl 

2

ij ij j i ij

E g dx dx g dl  

2

globally

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The strain tensor

Lagrangian coordinates

Free energy

Second order scalars Lamé coefficients

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The stress (linear elasticity)

Hooke’s law

Elastic energy

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Generalization to 4 dimensions: Lagrangian density

 



           x d g R S

matter 4 2

2 2 2 1 L    

 

“Kinetic” term Potential term: “dark energy”

Geometry

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Lagrangian density and energy-momentum tensor

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 

g g g g g T e                     

             

           2 1 2 1 4 1

Robertson-Walker symmetry

 

2 2 2 2

dl a d dsnat    

 

  

 E g   2 1

O

Reference manifold Defect



Cosmic time Space Image space

O’ 

ρ

 =f(ρ)

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RW Lagrangian density

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Integration by parts

Euler Lagrange equations

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Solution

 

           3 8 3 2

2 2 3 2

  a a a B a a a   

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 

W a a B a a   

2 2 2

2 3 6  

Energy condition

 





    

  T B

e a W

The Hubble parameter with matter/radiation

       

2 / 1 2 2 2 3

1 1 4 1 1 3                           a z B z z c a a H

r m

   

G c2 8  

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Equation of state of the “strain fluid”

 

 

 

4 2 4 4 2 2 2

1 2 3 4 4 3 a a a B p a a B c

e e

        

 

2 2 2 4

3 1 2 3        a a a w

1 3 1   

   a a

w w

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Some cosmological tests

• SnIa luminosity
• Primordial nucleosynthesis (correct

proportion between He, D and hydrogen)

• CMB acoustic horizon
• BAO
• Structure formation after the

recombination era.

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Fitting the supernovae it works

λμ10-52m-2

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Bayesian posterior probability

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Optimal value of the parameters

     

2 52 1 a 3 27 m 2 52

m 10 06 012 B m kg 10 15 45 2 m 10 08 28 2 B          

   

. . / . . . .

4 r a

a 9 8 B  

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Matter or defects?

Flat reference manifold Curved natural manifold

2

  

 E g  

  

dx dx g dsnat 

2

  

dx dx E dsref 

2

Defect

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“Massive” gravity

SST Lagrangian density Fierz and Pauli Lagrangian density Linearized difference between the metric tensor and a Minkowski background

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Non-linear “massive” gravity

Introduce an auxiliary metric tensor f then from it and the background build a quantity H and write “Massive” gravity theories do not correspond to SST where:

• ε is “exact”
• the only metric tensor is g
• the reference manifold is Euclidean

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Questions

• Is the elasticity of space-time an

emerging property?

– May be. It involves the issue of the dualism space-time/matter-energy

• Is SST analogous to massive gravity?

– Not really

• Is SST a bimetric theory?

– No

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Questions

• Can defects get rid of matter?

– Troubles with quantum mechanics…

• Is this approach better than many
• thers?

– It depends on which criterion is used to judge what is best.

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Conclusion

• The idea is that

– Space-time is physical – there is a deformation energy density in space- time due to curvature.

• If we include a cosmic defect we obtain the

Robertson-Walker symmetry and the accelerated expansion.

• SST proposes an intuitive interpretation of Λ,
• r, more generally, of dark energy.

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References

• N. Radicella, M. Sereno, A. Tartaglia, MNRAS,

429, 1149-1155 (2013)

• Radicella N., Sereno M., Tartaglia A., CQG,

29, 115003 (2012)

• N. Radicella, M. Sereno, A. Tartaglia, IJMPD,

20, 1039 (2011)

• A. Tartaglia, ”The Strained State Cosmology”,

in Aspects of Today’s Cosmology, Ed. A. Antonio-Faus, InTech, Rijeka, p. 30-48 (2011).

• A. Tartaglia, N. Radicella, CQG, 27, 035001

(2010)

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