Dark Universe or twisted Universe?

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Transcript Dark Universe or twisted Universe?

Dark Universe or twisted
Universe?
Einstein-Cartan theory.
Thomas Schucker : CPT Marseille France
Andre Tilquin :
CPPM Marseille France
THCA Tsinghua China
\
arXiv:1104.0160
arXiv:1109.4568
2
Einstein general relativity: quick reminder
Parallel transport and curvature
Limitation in GR
Einstein-Cartan general relativity
Torsion: what is that?
Parallel transport and torsion
Properties and advantages
Results on supernovae with a twisted Universe
Solving Einstein-Cartan equations
Effect of torsion on Hubble diagram
Summary and further work
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Related to curvature
ฮฑ
B
How to transport a vector or a frame in
A1
a curve space?
A2
A
-Using this procedure on a reference frame and in the limit of a null surface
defines the Einstein tensor: ๐บ๐œ‡๐œˆ = ๐‘…๐œ‡๐œˆ โˆ’ 1/2๐‘…๐‘”๐œ‡๐œˆ
which is symmetric
in ๐œ‡๐œˆ
4
-Geometry generates rotation
Einstein equation relates curvature with energy momentum
tensor
๐บ๐œ‡๐œˆ = 8๐œ‹๐บ๐‘‡๐œ‡๐œˆ
As a consequence of symmetric Riemann geometry, the energy momentum
tensor is symmetric:
๐‘‡๐œ‡๐œˆ = ๐‘‡๐œˆ๐œ‡
General relativity can accommodate particle with spin including spin-1/2
using vierbein formalism (all tensors are represented in terms of a chosen
basis of 4 independent orthogonal vectors field)
However it can not describes spin-orbite coupling because when spin and
orbital angular momentum are being exchanged, the momentum tensor is
known to be nonsymmetric. According to the general equation of
conservation of angular momentum:
๐œ‡๐œˆ๐›ผ
๐œ•๐›ผ ๐‘†
= ๐‘‡๐œ‡๐œˆ โˆ’ ๐‘‡ ๐œˆ๐œ‡ โ‰  0
Where ๐‘‡๐œ‡๐œˆ โˆ’ ๐‘‡ ๐œˆ๐œ‡ is the torque density = rate of conversion between orbital
momentum and spin.
Cartan => Torsion
5
*
curvature
torsion
๐‘Žโ€ฒ
โ„ต = ๐‘Ž. ๐‘
๐‘
t
t = โˆ’๐‘. ๐‘
๐‘โ€ฒ
๐‘
Cartan assumed that local torsion is related to spin ½ particles
โ„ต
6
translation
ฮฑ
B
A3
A1
A2
A
-In presence of torsion the infinitesimal parallelogram does not close
-Geometry generates translation
7
โ€ข Energy momentum is still the only source of space-time curvature
with the Newtonโ€™s constant being the coupling constant
โ€ข The source of torsion is half integer spin with the same coupling
constant
โ€ข Spin 1 particle is not source of torsion: photon not affected
โ€ข Photon and spin ½ particle (neutrino) geodesics are different
โ€ข Torsion doesnโ€™t propagate
โ€ข Itโ€™s non-vanishing only inside matter with half integer spin
โ€ข Theories of unification between gravity and standard model of particle
physics need a torsion field (loop quantum gravity)
โ€ข Supergravity is an Eistein-Cartan theory. Without torsion this theory loses its
supersymmetry.
โ€ข Torsion provides a consistency description of general relativity
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8๐œ‹๐บ๐‘‡๐œ‡๐œˆ
=
Energy-momentum
๐บ๐œ‡๐œˆ
curvature
Noether
theorem
geometry
Translations
rotations
Noether
theorem
geometry
Torsion
spin
Cartan
equation
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In space time with torsion there are 2 Einstein equations ( in vierbien
frame):
1. Equation for curvature:
๐บ๐‘Ž๐‘ = ๐‘…โˆ— ๐‘Ž๐‘ โˆ’ 1/2๐‘…โˆ— ๐‘”๐‘Ž๐‘
๐บ๐‘Ž๐‘ โˆ’โˆง ๐‘”๐‘Ž๐‘ = 8๐œ‹๐บ๐‘‡๐‘Ž๐‘
2. Equation for torsion (ฮฃ):
ฮฃ๐‘ ๐‘’ ๐‘‘ ๐œ–๐‘Ž๐‘๐‘๐‘‘ = โˆ’8๐œ‹๐บ๐‘†๐‘Ž๐‘
R* is the modify Ricci
tensor no more symmetric
๐‘†๐‘Ž๐‘ = spin tensor
In a maximally symmetric Universe the most general energy momentum
tensor has two functions of time:
The density : ๐œŒ ๐‘ก
With equation of state: ๐‘ ๐‘ก โ‰” ๐‘ค ๐œŒ(๐‘ก)
The pressure : ๐‘ ๐‘ก
The most general spin density
Even parity : ๐‘†0๐‘—๐‘˜ = โˆ’๐‘ (๐‘ก)๐›ฟ๐‘—๐‘˜
๐‘  ๐‘ก โ‰” ๐‘ค๐‘  ๐œŒ ๐‘ก
With โ€œequations of stateโ€:
๐‘  ๐‘ก โ‰” ๐‘ค๐‘  ๐œŒ(๐‘ก)
Odd parity : ๐‘†๐‘–๐‘—๐‘˜ = โˆ’๐‘ (๐‘ก)๐œ€๐‘–๐‘—๐‘˜
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In maximally symmetric and flat Universe Friedmann equations have 4
unknown functions of time: a,b,f and ฯ.
๐‘2 ๐‘ก โˆ’ ๐‘“ 2 (๐‘ก)
3
= ฮ› + 8๐œ‹๐บ๐œŒ ๐‘ก
๐‘Ž 2 (๐‘ก)
๐‘โ€ฒ (๐‘ก) ๐‘2 ๐‘ก โˆ’ ๐‘“ 2 (๐‘ก)
2
+
= ฮ› โˆ’ 8๐œ‹๐บ๐‘ ๐‘ก
๐‘Ž(๐‘ก)
๐‘Ž2 (๐‘ก)
๐‘Žโ€ฒ ๐‘ก โˆ’ ๐‘(๐‘ก)
3
= 8๐œ‹๐บ๐‘ค๐‘  ๐œŒ ๐‘ก
๐‘Ž(๐‘ก)
๐‘“(๐‘ก)
2
= 8๐œ‹๐บ๐‘ค๐‘  ๐œŒ(๐‘ก)
๐‘Ž(๐‘ก)
Using these expressions today and the dimensionless density ฮฉ = ๐œŒ(๐‘ก0)
ฮฉ๐‘š =
8๐œ‹๐บ๐œŒ0
3๐ป0 2
; ฮฉฮ› =
ฮ›
3๐ป0 2
; ฮฉ๐‘  = ๐‘ค๐‘  ๐ป0
8๐œ‹๐บ๐œŒ0
3๐ป0 2
; ฮฉ๐‘  = ๐‘ค๐‘  ๐ป0
๐œŒ๐‘ :
8๐œ‹๐บ๐œŒ0
3๐ป0 2
The Friedmann like closure relation reads:
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ฮฉ๐‘š + ฮฉฮ› + 2ฮฉ๐‘  โˆ’ ฮฉ๐‘  2 + ฮฉ๐‘  2 = 1
4
11
Supernovae of type Ia are almost standard candle:
There intrinsic luminosity (L) can be standardized at a level of
about 15%
Thus the apparent luminosity can be used as a distance indicator:
๐‘™ ๐‘ก =
๐ฟ
๐‘Ž(๐‘ก)2
4๐œ‹๐‘Ž0 2 ๐‘ฅ(๐‘ก)2 ๐‘Ž0
with ๐‘ฅ ๐‘ก =
๐‘ก0 ๐‘‘๐‘ก โ€ฒ
๐‘ก ๐‘Ž(๐‘ก โ€ฒ )
And the redshift as a scale factor measurement:
๐‘Ž0
๐œ†โˆ’๐œ†
๐‘ง = ๐‘Ž(๐‘ก)
โˆ’1 = ๐œ† 0
0
Because the geodesic equations for photons decouple to torsion, redshift and
luminosity have the same expression
->We just need to compute the scale factor
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We used the so called Union 2 sample containing 557 supernovae up
to a redshift of 1.5
๐œ‡ ๐‘ง = ๐‘š๐‘  + 2.5log ๐‘™ ๐‘ง
Standard cosmology fit gives (no flatness):
ms
โ„ฆm
โ„ฆฮ›
marginalized
0.35+0.10 โˆ’0.11
0.88+0.19 โˆ’0.11
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We use the full covariance matrix, taking into account systematic errors and
correlations to compute
๐œ’ 2 = ฮ”๐‘€๐‘‡ ๐‘‰ โˆ’1 ฮ”๐‘€
๐œŽ1 2
๐‘‰=
โ‹ฎ
๐œŒ๐‘›1 ๐œŽ๐‘› ๐œŽ1
โ‹ฏ ๐œŒ1๐‘› ๐œŽ1 ๐œŽ๐‘›
โ‹ฑ
โ‹ฎ
โ‹ฏ
๐œŽ๐‘› 2
๐œ‡๐‘กโ„Ž ๐‘ง1 , ฮฉ โˆ’ ๐œ‡1
โ‹ฎ
and ฮ”๐‘€ =
๐œ‡๐‘กโ„Ž ๐‘ง๐‘› , ฮฉ โˆ’ ๐œ‡๐‘›
The best cosmological parameters are computed by minimizing the
๐œ’2:
๐œ•๐œ’ 2
=0
๐œ•ฮฉ๐‘˜
Errors and contours are computed by using the frequentist prescription:
๐œ’ 2 ฮฉ๐‘š =
min ๐œ’ 2 (ฮฉ๐‘š , ฮฉ๐‘  , ๐‘š๐‘  โ‹ฏ ) + ๐‘  2
ฮฉ๐‘  ,๐‘š๐‘  โ‹ฏ
Free cosmological parameters are:
๐‘š๐‘  , ฮฉ๐‘š , ฮฉ๐‘  , ฮฉ๐‘  where ฮฉฮ› is deduced from Friedman like relation
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ฮฉ๐‘š + ฮฉฮ› + 2ฮฉ๐‘  โˆ’ ฮฉ๐‘  2 + ฮฉ๐‘  2 = 1
4
14
Even parity torsion: ฮฉ๐‘  = 0
โ„ฆm
โ„ฆฮ›
๐›€๐’”
0.09+0.30 โˆ’0.07 0.83+0.10 โˆ’0.16 0.04+0.01 โˆ’0.07
โ„ฆm
โ„ฆฮ›
Odd parity torsion: ฮฉ๐‘  = 0
โ„ฆm
โ„ฆฮ›
๐›€๐’”
0.27+0.03 โˆ’0.02 0.73+0.04 โˆ’0.11 0.0+0.22 โˆ’0.22
๐›€๐’”
0.08+0.27 โˆ’0.08 0.85+0.10 โˆ’0.15 0.04+0.02 โˆ’0.06
๐›€๐’”
0.0+0.1 โˆ’0.1
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Even parity torsion gives a prefer value for matter density equal to 0.09
ฮฉ๐‘š = 0.09+0.30 โˆ’0.07
The WMAP last results are:
ฮฉ๐‘ = 0.046 ± 0.003 and ฮฉ๐‘š = 0.27 ± 0.03
Supernovae results analyzed with torsion give a result statistically
compatible with both dark matter and baryonic matter.
However, torsion can contribute to a certain amount of dark matter.
Or better to say that torsion without dark matter is not incompatible
with Supernovae data.
More data or probes should be used to definitely conclude
16
We test the hypothesis of a null cosmological constant by using the log
likelihood ratio technic: Assume we want to test 2 different models, with one
include in the other:
๐‘š ๐‘  , ฮฉ๐‘  , ฮฉฮ› = 0 โ†’ ๐‘š ๐‘  , ฮฉ๐‘  , ฮฉฮ›
We can define the log likelihood ratio as:
๐‘€๐‘Ž๐‘ฅ โ„’ ๐‘š๐‘  , ฮฉ๐‘  , ฮฉฮ› = 0
๐‘… = โˆ’2๐ฟ๐‘›
๐‘คโ„Ž๐‘’๐‘Ÿ๐‘’ โ„’ =
2๐œ‹
๐‘€๐‘Ž๐‘ฅ โ„’ ๐‘š๐‘  , ฮฉ๐‘  , ฮฉฮ›
1
๐‘›/2
๐‘‰
๐‘’ โˆ’๐œ’
1/2
2 /2
๐‘… = ๐œ’ 2 ๐‘š๐‘–๐‘›,1 โˆ’ ๐œ’ 2 ๐‘š๐‘–๐‘›,2
The probability distribution of this variable is approximately a ๐œ’ 2 distribution
with a number of degree of freedom equal to the difference of ndofโ€™s = 1
Fore even parity: ฮ”๐œ’ 2 = 44.6 โ†’ ๐‘๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’ โ‰… 0.
โ†’ ๐‘Ÿ๐‘ข๐‘™๐‘™๐‘’๐‘‘ ๐‘œ๐‘ข๐‘ก
For odd parity : ฮ”๐œ’ 2 = 30.3 โ†’ ๐‘๐‘ฃ๐‘Ž๐‘™๐‘ข๐‘’ = 6 10โˆ’8 โ†’ ๐‘Ÿ๐‘ข๐‘™๐‘™๐‘’๐‘‘ ๐‘œ๐‘ข๐‘ก ๐‘Ž๐‘ก 5.4 ๐‘ ๐‘–๐‘”๐‘š๐‘Ž
This is not surprising because equation of state: ๐‘  ๐‘ก โ‰” ๐‘ค๐‘  ๐œŒ ๐‘ก
If acceleration today, then acceleration in the past: s ๐‘ก โ†— with ๐œŒ(๐‘ก)
In contradiction with previous publication (S. Capozziello et al. 2003)
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โ€ข Standard general relativity should be extended to account for spin-orbital
momentum coupling: Einstein-Cartan theory.
โ€ข If we apply torsion to cosmology we find:
โ€ข Torsion can contribute to dark matter at a certain amount
โ€ข Torsion as a source of dark energy is ruled out at more than 5 sigma
โ€ข However these results are encouraging enough to try to go further
โ€ข Look at galaxies rotation curves: Need to generalized the Schwarzschildโ€™s
equation. Work in progress.
โ€ข Use other probes:
โ€ข CMB/BAO/WL/Clusters: photons are not sensitive to torsion, but
dynamic is different, so everything should be recomputed. ๏Œ
โ€ข But we should not be too much excited by the Supernovae result on DM:
We found a spin energy density ฮฉ๐‘  of about 4% , corresponding to a
state parameter ๐‘ค๐‘  ~1/๐ป0 ~1017 ๐‘  which is 42 orders of magnitude away from the
โ„
โˆ’25 ๐‘  !
naïve value ๐‘ค๐‘  ~
2 ~10
๐‘š๐‘ ๐‘
Usual problem in cosmology i.e : ฮ› and vacuum energy!
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1) Torsion and curvature ?
2) Torsion and vacuum ?
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In the general case, assuming no special equation of state: ๐‘  ๐‘ก ๐‘Ž๐‘›๐‘‘ ๐‘  ๐‘ก are
free functions of time:
๐‘Ž
4๐œ‹๐บ
8๐œ‹๐บ
๐‘Ž ๐‘ก
=โˆ’
๐œŒ + 3๐‘ +
๐‘  ๐‘ก +
๐‘ (๐‘ก)
๐‘Ž
3
3
๐‘Ž ๐‘ก
Torsion is not source of gravity:
Odd parity torsion doesnโ€™t couple to dynamic (i.e curvature)
Even parity torsion couple to curvature through kinematic not dynamic
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(1)
*
1. Because geodesics are different for photons and spin ½ particles
(neutrino)
โ€ข Timing difference between photons and neutrinos in supernovae
explosion
1987A Supernovae
โ€ข Neutrino oscillation experiment OPERA. Time delay and supra
luminal neutrino
2. Rotation curve of galaxies or the modify Schwartsfield solution
โ€ข What is the effect of torsion on rotation curve of galaxy
3. Galaxies and cluster formation
4. The cosmological probes:
โ€ข Supernovae 1a
โ€ข CMB: effect of torsion in initial plasma (very high matter density)
โ€ข Weak lensing should not be affected
Lensing is gravitational coupling between curvature and photon
โ€ข Baryonic acoustic oscillation
Depends on the initial power spectrum
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Let consider the covariant derivative of a vector:๐ด๐œŽ
๐›ป๐œ‡ ๐ด๐œŽ = ๐ด๐œŽ ,๐œ‡ + ฮ“๐œ‡๐œˆ ๐œŽ ๐ด๐œˆ with ๐ด๐œŽ ,๐œ‡ = ๐œ•๐ด๐œŽ /๐œ•๐‘ฅ ๐œ‡ = ๐œ•๐œ‡ ๐ด๐œŽ
Where ฮ“๐œ‡๐œˆ ๐œŽ is the affine connection
This covariant derivative can be formally written as:
๐›ฟ๐œˆ ๐œŽ ๐›ป๐œ‡ = ๐›ฟ๐œˆ ๐œŽ ๐œ•๐œ‡ + ฮ“๐œ‡๐œˆ ๐œŽ
And compare to covariant derivative in QED: ๐ท๐œ‡ = ๐œ•๐œ‡ +๐‘–๐‘’๐ด๐œ‡
In geometric term, the affine connection is interpreted as the change of
vector during parallel transport along ๐‘‘๐‘ฅ๐œ‡ :
โˆ’ฮ“๐œ‡๐œˆ ๐œŽ ๐ด๐œˆ ๐‘‘๐‘ฅ ๐œ‡
And the curvature tensor is defined as the change of vector ๐ด๐œˆ
parallel transported around a closed path ๐œ‰๐œ‡ โˆถ
ฮ”๐ด๐œŽ =
1 ๐œˆ
๐ด ๐‘…๐›ฝ๐œ‡๐œˆ ๐œŽ
2
๐œ‰๐œ‡ ๐‘‘๐‘ฅ ๐›ฝ
22
โ„’ = ๐œ•๐œ‡ ๐œ™๐œ•๐œ‡ ๐œ™ โˆ— โˆ’๐‘š2 ๐œ™๐œ™ โˆ—
This Lagrangian is invariant under global rotation ๐œƒ in complex plane:
ฮธ
ฮธ
๐œ™ โ†’ ๐œ™๐‘’ โˆ’๐‘–๐œƒ
ฮธ
๐œ™ โˆ— โ†’ ๐œ™ โˆ— ๐‘’ ๐‘–๐œƒ
โ†’ ๐œ™๐œ™ โˆ— is invariant
๐œ•๐œ‡ ๐œ™ โ†’ ๐œ•๐œ‡ ๐œ™๐‘’ โˆ’๐‘–๐œƒ
ฮธ
๐œ•๐œ‡ ๐œ™ โˆ— โ†’ ๐œ•๐œ‡ ๐œ™ โˆ— ๐‘’ ๐‘–๐œƒ
โ†’ ๐œ•๐œ‡ ๐œ™๐œ•๐œ‡ ๐œ™ โˆ— is invariant
But is not invariant under local rotation ๐œƒ ๐‘ฅ๐œ‡ in complex plane:
ฮธ
๐œ•๐œ‡ ๐œ™ โ†’ ๐œ•๐œ‡ ๐œ™. ๐‘’ โˆ’๐‘–๐œƒ โˆ’ ๐‘–๐œ•๐œ‡ ๐œƒ ๐‘ฅ๐œ‡ . ๐œ™. ๐‘’ โˆ’๐‘–๐œƒ
=
๐œ•๐œ‡ ๐œ™ =
Variation of the field is assumed
to be linear in ๐œ™ ๐‘Ž๐‘›๐‘‘ ๐›ฟ๐‘ฅ๐œ‡ :
+
ฮ”๐œ™
๐›ฟ๐‘ฅ ๐œ‡
ฮ”๐œ™ = โˆ’๐‘–๐‘’๐ด๐œ‡ ๐›ฟ๐‘ฅ ๐œ‡ ๐œ™
+
๐ท๐œ‡ ๐œ™
๐ท๐œ‡ = ๐œ•๐œ‡ + ๐‘–๐‘’๐ด๐œ‡
23
๐œ ๐›ผ โˆ’ ฮ“๐œ‡๐œˆ ๐›ผ ๐œ’ ๐œ‡ ๐œ ๐œˆ
๐ต๐›ผ = ๐œ’ ๐›ผ + (๐œ ๐›ผ โˆ’ ฮ“๐œ‡๐œˆ ๐›ผ ๐œ’ ๐œ‡ ๐œ ๐œˆ )
๐œ’ ๐›ผ โˆ’ ฮ“๐œ‡๐œˆ ๐›ผ ๐œ๐œ‡ ๐œ’ ๐œˆ
๐œ’๐›ผ
๐ด๐›ผ = ๐œ ๐›ผ + (๐œ’ ๐›ผ โˆ’ ฮ“๐œ‡๐œˆ ๐›ผ ๐œ๐œ‡ ๐œ’ ๐œˆ )
๐œ๐›ผ
Then the difference ๐ถ ๐›ผ = ๐ด๐›ผ โˆ’ ๐ต๐›ผ is:
๐ถ ๐›ผ = 2๐‘†๐œ‡๐œˆ ๐›ผ ๐œ๐œ‡ ๐œ’ ๐œˆ
1
with ๐‘†๐œ‡๐œˆ ๐›ผ = ฮ“[๐œ‡๐œˆ] ๐›ผ = 2 ฮ“๐œ‡๐œˆ ๐›ผ โˆ’ ฮ“๐œˆ๐œ‡ ๐›ผ
Where ๐‘†๐œ‡๐œˆ ๐›ผ is defined as the torsion tensor
24
๐‘  ๐‘ก โ‰” ๐‘ค๐‘  ๐œŒ ๐‘ก
๐‘  ๐‘ก โ‰” ๐‘ค๐‘  ๐œŒ(๐‘ก)
โ€ข ๐œŒ ๐‘ก = any spin ½ matter density with null pressure
โ€ข ๐‘  ๐‘ก = spin density considered as a perfect fluid!
โ€ข ๐‘ค๐‘  has a dimension of time and is assumed to be constant
All physics are inside ws:
โ€ข Source of torsion is spin ½ particle
โ€ข Orbital momentum or spin 1 are not source of torsion.
โ€ข No spin orbital momentum coupling. Spin generates local torsion.
โ€ข It contains the Planck constant and GR and QM coupling. Expected
to be small.
โ€ข We assume it is not zero even though we donโ€™t know how spins
average?
โ€ข โ€ฆโ€ฆโ€ฆ
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๐‘ 2 ๐‘ก โˆ’ ๐‘“ 2 (๐‘ก)
3
= ฮ› + 8๐œ‹๐บ๐œŒ ๐‘ก
๐‘Ž2 (๐‘ก)
๐‘ โ€ฒ (๐‘ก) ๐‘ 2 ๐‘ก โˆ’ ๐‘“ 2 (๐‘ก)
2
+
= ฮ›
๐‘Ž(๐‘ก)
๐‘Ž2 (๐‘ก)
๐‘Žโ€ฒ ๐‘ก โˆ’ ๐‘(๐‘ก)
3
= 8๐œ‹๐บ๐‘ค๐‘  ๐œŒ ๐‘ก
๐‘Ž(๐‘ก)
๐‘“(๐‘ก)
2
= 8๐œ‹๐บ๐‘ค๐‘  ๐œŒ(๐‘ก)
๐‘Ž(๐‘ก)
We eliminate f(t) and ฯ(t)
We are left with 2 first order
differential equations and 2
unknown functions a(t) and
b(t)
We solve it numerically with the Runge-Kutta algorithm. a(t)
This is an iterative numerical algorithm
Example:
1. aโ€™(t) = 2 a(t)
a1
2. Start from an initial value a(t0)=a0
3. Compute the derivative aโ€™(t0)=2 a0
a0
4. Predict the new point at t0+ฮดt using Tailor
expansion : a(t1 = t0+ฮดt) = a0+2a0 ฮดt +โ€ฆ..= a1
5. Start from this new value a(t1)=a1 and iterate
t0
t1=t0+ฮดt t
We used a forth order Runge-Kutta algorithm with an adaptive step in time such
that the corresponding step in redshift is much smaller than the experimental
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redshift error (10-5)
In this paper they assume the same Friedmann equations for the torsion
fluid
๐‘Ž
4๐œ‹๐บ
=โˆ’
๐œŒ + 3๐‘
๐œŒ = ๐œŒ + ๐‘“ 2 ๐‘Ž๐‘›๐‘‘ ๐‘ = ๐‘ โˆ’ ๐‘“ 2
๐‘Ž
3
with
2
๐‘Ž
8๐œ‹๐บ
=
๐œŒ
๐‘Ž
3
3
The missing factor 3 implies torsion is
source of curvature and a constant โ€œfโ€
function with time which can be
interpreted as a cosmological constant.
At beginning I made the same kind of
mistake and I got ๏Š
Unfortunately itโ€™s wrong!
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In general case where s(t) and ๐‘  ๐‘ก are functions of time we have:
๐‘Ž 1
= ฮ› โˆ’ 4๐œ‹๐บ ๐œŒ + 3๐‘
๐‘Ž 3
8๐œ‹๐บ
๐‘Ž
+
๐‘  ๐‘ก + ๐‘ (๐‘ก)
3
๐‘Ž
โ€ข Odd parity torsion ๐‘ (๐‘ก) doesnโ€™t modify dynamic (Einstein curvature)
โ€ข Even parity torsion couple to gravit
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The Hilbert action yields the Einstein equation through the principle of
least action:
1
๐‘†=โˆ’
2๐œ…
4
๐‘… โˆ’๐‘”๐‘‘ ๐‘ฅ
R the Ricci scalar
๐‘” = ๐‘‘๐‘’๐‘ก ๐‘”๐œ‡๐œˆ
๐œ… = 8๐œ‹๐บ๐‘ โˆ’4
with
In presence of matter the action becomes:
๐‘†=
1
๐‘… + โ„’๐‘€
2๐œ…
โˆ’๐‘”๐‘‘ 4 ๐‘ฅ
The action principle ๐›ฟ๐‘† = 0 leads to:
๐›ฟ๐‘† =
๐›ฟ โˆ’๐‘”โ„’ ๐‘€
1 ๐›ฟ โˆ’๐‘”๐‘…
+
2๐œ… ๐›ฟ๐‘”๐œ‡๐œˆ
๐›ฟ๐‘”๐œ‡๐œˆ
๐›ฟ๐‘† =
1 ๐›ฟ๐‘…
๐‘… ๐›ฟ โˆ’๐‘”
+
2๐œ… ๐›ฟ๐‘”๐œ‡๐œˆ
โˆ’๐‘” ๐›ฟ๐‘”๐œ‡๐œˆ
๐›ฟ๐‘”๐œ‡๐œˆ ๐‘‘ 4 ๐‘ฅ
+
1 ๐›ฟ โˆ’๐‘”โ„’๐‘€
โˆ’๐‘”
๐›ฟ๐‘”๐œ‡๐œˆ
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๐›ฟ๐‘”๐œ‡๐œˆ โˆ’๐‘”๐‘‘4 ๐‘ฅ
Since the previous equation should hold for any ๐›ฟ๐‘”๐œ‡๐œˆ
๐›ฟ๐‘…
๐‘… ๐›ฟ โˆ’๐‘”
1 ๐›ฟ โˆ’๐‘”โ„’๐‘€
+
=
โˆ’2๐œ…
๐›ฟ๐‘”๐œ‡๐œˆ
โˆ’๐‘” ๐›ฟ๐‘”๐œ‡๐œˆ
โˆ’๐‘”
๐›ฟ๐‘”๐œ‡๐œˆ
๐›ฟ๐‘…
= ๐‘…๐œ‡๐œˆ
๐›ฟ๐‘”๐œ‡๐œˆ
1 ๐›ฟ โˆ’๐‘”
1
=
โˆ’
๐‘”
โˆ’๐‘” ๐›ฟ๐‘”๐œ‡๐œˆ
2 ๐œ‡๐œˆ
๐‘‡๐œ‡๐œˆ
โˆ’2 ๐›ฟ โˆ’๐‘”โ„’๐‘€
๐›ฟโ„’๐‘€
โ‰”
= โˆ’2 ๐œ‡๐œˆ + ๐‘”๐œ‡๐œˆ โ„’๐‘€
โˆ’๐‘”
๐›ฟ๐‘”๐œ‡๐œˆ
๐›ฟ๐‘”
1
8๐œ‹๐บ
๐‘…๐œ‡๐œˆ โˆ’ ๐‘”๐œ‡๐œˆ ๐‘… = 4 ๐‘‡๐œ‡๐œˆ
2
๐‘
The cosmological constant is introduced in the Lagrangian:
๐‘†=
1
๐‘… โˆ’ 2ฮ› + โ„’๐‘€
2๐œ…
1
8๐œ‹๐บ
๐‘…๐œ‡๐œˆ โˆ’ ๐‘”๐œ‡๐œˆ ๐‘… + ฮ›๐‘”๐œ‡๐œˆ = 4 ๐‘‡๐œ‡๐œˆ
2
๐‘
โˆ’๐‘”๐‘‘4 ๐‘ฅ
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