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. 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 2

Premises • Apparently the gravitational interaction is very well described as a geometric property of a four-dimensional Riemannian manifold   T G   • Other fundamental interactions do not share this geometric essence 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 3

Axiomatic assumption • The physical configuration of the world, in all interactions, satisfies an universal principle of “economy”: the ‘least’ action principle        N L S * d x S 0 Scalar Lagrangian density N-form 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 4

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 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 5

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? 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 6

Tridimensional analogy • Deformable continuous media • Geometrizable interaction: elastic interaction (with an external evolution parameter – time -) • Macroscopic emergent representation, from microscopic elementary interactions 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 7

“Elastic” continua N+n N   0 f X , X ,..., X  1 2 N  n ξ a    u '   h X , X ,..., X  0 1 2 N  n    u r x μ N r’ X a 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 8

Deformation • Undifferenciated reference state  flat manifold independent from the parameter. 2  i j dl E dx dx 0 ij • Geometry: Euclidean/Minkowskian • Deformation due either to intrinsic (defects) or ‘extrinsic’ (matter/energy) causes globally   2 i j dl g dx dx g E ij ij ij • Riemannian geometry 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 9

The strain tensor Lagrangian coordinates Free energy Lamé coefficients Second order scalars 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 10

The stress (linear elasticity) Hooke ’s law Elastic energy 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 11

Generalization to 4 dimensions: Lagrangian density     1            L 2 4 S R 2 2 g d x    matter   2 Potential term: “dark energy” “Kinetic” term Geometry 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 12

Lagrangian density and energy-momentum tensor   1 1          T e g g g         4 2   1                   g g          2 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 13

Robertson-Walker symmetry Image space Reference manifold  O’   2     2 2 2 ds nat d a dl O Defect   1 Cosmic time    g E    2   =f( ρ ) ρ Space 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 14

RW Lagrangian density Integration by parts 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 15

Euler Lagrange equations 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 16

Solution       2 a 3 B           2 2 a a a   3 2 a 8 a 3   B Energy 3 2      2 2 a a a W 6 condition a 2            B T W 0 a e 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 17

The Hubble parameter with matter/radiation 1 / 2     2       2        a B 1 z            3   H c 1 z 1 z 1   m 0 r 0 2   a 3 4   a   0 8    G c 2 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 18

Equation of state of the “strain fluid”   2   2 3 a    2 c B      e 4 4 2 4 a 3 a 2 a 1   w      4 2 2   B 3 a 2 a 1 2 3 a  p   e 4 a 4 1    w w 1    a a 0 3 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 19

Some cosmological tests • SnIa luminosity • Primordial nucleosynthesis (correct proportion between He, D and hydrogen) • CMB acoustic horizon • BAO • Structure formation after the recombination era. 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 20

Fitting the supernovae it works λ  μ  10 -52 m -2 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 21

Bayesian posterior probability 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 22

Optimal value of the parameters   B 2 28 0 08 10 52 m 2      . .   2 45 0 15 10 27 kg m 3      . . / m 0   1 B  0 012 0 06 10 m 52 2    . . a 0 8 4 B a   a r 0 0 9 0 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 23

Matter or defects? Flat reference manifold   ds ref  2 E dx dx  Defect  g E      2   ds nat  2 g dx dx Curved natural manifold  18/07/2014 FFP14 - Marseille- Angelo Tartaglia 24

“Massive” gravity SST Lagrangian density Fierz and Pauli Lagrangian density Linearized difference between the metric tensor and a Minkowski background 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 25

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 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 26

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 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 27

Questions • Can defects get rid of matter? – Troubles with quantum mechanics… • Is this approach better than many others? – It depends on which criterion is used to judge what is best. 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 28

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 , or, more generally, of dark energy. 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 29

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) 18/07/2014 FFP14 - Marseille- Angelo Tartaglia 30