May 28, 2010, Dezhou A Realistic Torsion Cosmological Model Li Xin-Zhou Shanghai United Center for Astrophysics, Shanghai Normal University.

May 28, 2010, Dezhou

A Realistic Torsion Cosmological Model

Li Xin-ZhouShanghai United Center for Astrophysics,

Shanghai Normal University


Two geometrical quantities

Torsion cosmology

Fit SNeIa

Analytical solutions of late-time in torsion cosmology

Summary

Contents

Li,Sun and Xi, PRD 79 (2009)Li,Sun and Xi, JCAP (2009)Ao,Li and Xi, preprint (2010)Li,Xi and Ao, Preprint (2010)


In various cosmological models, fundamental quantities are either physical (if they depend upon physical fields) or geometrical (if they are constructed from a spacetime geometry directly). Physical quantities are certainly model-dependent, while geometrical quantities are more universal.

replace physical fields by geometrical quantities in a cosmological theory.


Two basic geometrical subjects (tetrad and affine connection) have been discussed widely. Tetrad determines the (symmetric) metric and the local Lorentz frame, while affine connection defines the parallel transport and covariant derivative.

Einstein used (symmetric) metric to establish his General Relativity;

With these two geometrical subjects, we could find a realistic cosmology, in which we don’t have to introduce the mystical dark energy.

• Metric-compatible connection 1-form

• Orthonormal coframe 1-form

• Metric


[ ] aa dx

aae dx

0 0 i jijg


Pioneer Elie Joseph Cartan (1869-1951)

A geometry with an asymmetric Christoffel symbol is said to have torsion. Cartan has incorporated torsion into gravitational theory.

Cartan’s modification of Einstein’s theory attempts to take the spin density of elementary particles as the source of torsion.

A. S. Eddington, Proc.Roy.Soc.Lon.Ser.A99(1921)104

mentioned the notion of an asymmetry affinf connection in discussing possible extensions of GR


strong

weak electromagnetic

gravity can be described by local gauge theory.

Poincaré Gauge Theory gravity

Torsion Cosmology

Poincare gauge theory


PGT is based on Riemann-Cartan Geometry. It allows for dynamic torsion in addition to curvature.

To put gravitation into a gauge theory.

The connection dynamics (represented by torsion tensor) decomposes into 6 modes with certain spins and parity: 2±,1±,0±.


Two “scalar torsion” (0±) may well be the only physically accepted dynamic PGT torsion modes.

0+ or 0- has only a time component, then the homogeneous and isotropic cosmologies are naturally suitable for them.

Scalar modes


“pseudoscalar” 0- have small effects at late time of cosmology evolution, so we do not focus on this mode.

“scalar torsion” 0+ can be imagined as having

significant magnitude and being dramatically

noticed only through the non-linear equations.


Dynamical equationsThe torsion and curvature 2-forms are:

Which satisfy the Bianchi identities, respectively


Lagrangian density

where is the algebraically irreducible parts of the torsion, R is the scalar curvature and E is the pseudoscalar curvature. and are dimensionless parameters, have the same dimension with .


For a spatially flat Robertson-Walker cosmological model

where we have made the replacement

is Hubble parameter.

Dynamical equations

And


And the energy density of matter component is

The Newtonian limit requires .


Supernova 1998

Hi z Supernova TeamSupernova Cosmology Project

Two groups


The Discovery DataThe Discovery Data


In our model, the luminosity distance is

zdzH

Hzd

z

L

0

0

)()1(

Fit SNIa

For comparison with ΛCDM model: ΩM = 0.3, Ω Λ = 0.3 and χ2 = 177, χ2 /157 = 1.13.

The best fit for the torsion for the torsion parameters (a2, b) of the model are found by minimizing the quantity


Better model, better fit

We have obtained a better fit for our torsion

Cosmology!

Bao, CMB issues will be considered elsewhere.

　



Solutions with constant scalar curvatureWe consider the scalar curvature is constant as follows:


Solution IIWhen


Solution IIIWhen


Fate of universe

From the above formula, we get


Bifurcation


Solution of non-constant scalar curvature


We find an approximate formula up to order



Fit SNeIa We have obtained a better fit for our torsion Cosmology!We find three kinds of analytical solutions with a constant affine scalar curvature and a kind of expression with non-constant curvature. In the first case, it is not physical because the matter density will be negative. In the second case, it shows that the dark energy can be mimicked in the torsion cosmological model. In the third case, the charac-teristic of late-time evolution is similar to the universe of matter dominant. In the fourth case, we know the fate of universe that the universe would expand forever, slowly asymtotically to a halt.

Summary


Thanks!

May 28, 2010, Dezhou A Realistic Torsion Cosmological Model Li Xin-Zhou Shanghai United Center for Astrophysics, Shanghai Normal University.

Documents