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
3
Related to curvature
α
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: 𝐺𝜇𝜈 = 𝑅𝜇𝜈 − 1/2𝑅𝑔𝜇𝜈
which is symmetric
in 𝜇𝜈
4
-Geometry generates rotation
Einstein equation relates curvature with energy momentum
tensor
𝐺𝜇𝜈 = 8𝜋𝐺𝑇𝜇𝜈
As a consequence of symmetric Riemann geometry, the energy momentum
tensor is symmetric:
𝑇𝜇𝜈 = 𝑇𝜈𝜇
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:
𝜇𝜈𝛼
𝜕𝛼 𝑆
= 𝑇𝜇𝜈 − 𝑇 𝜈𝜇 ≠ 0
Where 𝑇𝜇𝜈 − 𝑇 𝜈𝜇 is the torque density = rate of conversion between orbital
momentum and spin.
Cartan => Torsion
5
*
curvature
torsion
𝑎′
ℵ = 𝑎. 𝑏
𝑏
t
t = −𝑐. 𝑏
𝑐′
𝑐
Cartan assumed that local torsion is related to spin ½ particles
ℵ
6
translation
α
B
A3
A1
A2
A
-In presence of torsion the infinitesimal parallelogram does not close
-Geometry generates translation
7
• Energy momentum is still the only source of space-time curvature
with the Newton’s constant being the coupling constant
• The source of torsion is half integer spin with the same coupling
constant
• Spin 1 particle is not source of torsion: photon not affected
• Photon and spin ½ particle (neutrino) geodesics are different
• Torsion doesn’t propagate
• It’s non-vanishing only inside matter with half integer spin
• Theories of unification between gravity and standard model of particle
physics need a torsion field (loop quantum gravity)
• Supergravity is an Eistein-Cartan theory. Without torsion this theory loses its
supersymmetry.
• Torsion provides a consistency description of general relativity
8
8𝜋𝐺𝑇𝜇𝜈
=
Energy-momentum
𝐺𝜇𝜈
curvature
Noether
theorem
geometry
Translations
rotations
Noether
theorem
geometry
Torsion
spin
Cartan
equation
9
In space time with torsion there are 2 Einstein equations ( in vierbien
frame):
1. Equation for curvature:
𝐺𝑎𝑏 = 𝑅∗ 𝑎𝑏 − 1/2𝑅∗ 𝑔𝑎𝑏
𝐺𝑎𝑏 −∧ 𝑔𝑎𝑏 = 8𝜋𝐺𝑇𝑎𝑏
2. Equation for torsion (Σ):
Σ𝑐 𝑒 𝑑 𝜖𝑎𝑏𝑐𝑑 = −8𝜋𝐺𝑆𝑎𝑏
R* is the modify Ricci
tensor no more symmetric
𝑆𝑎𝑏 = spin tensor
In a maximally symmetric Universe the most general energy momentum
tensor has two functions of time:
The density : 𝜌 𝑡
With equation of state: 𝑝 𝑡 ≔ 𝑤 𝜌(𝑡)
The pressure : 𝑝 𝑡
The most general spin density
Even parity : 𝑆0𝑗𝑘 = −𝑠(𝑡)𝛿𝑗𝑘
𝑠 𝑡 ≔ 𝑤𝑠 𝜌 𝑡
With “equations of state”:
𝑠 𝑡 ≔ 𝑤𝑠 𝜌(𝑡)
Odd parity : 𝑆𝑖𝑗𝑘 = −𝑠(𝑡)𝜀𝑖𝑗𝑘
10
In maximally symmetric and flat Universe Friedmann equations have 4
unknown functions of time: a,b,f and ρ.
𝑏2 𝑡 − 𝑓 2 (𝑡)
3
= Λ + 8𝜋𝐺𝜌 𝑡
𝑎 2 (𝑡)
𝑏′ (𝑡) 𝑏2 𝑡 − 𝑓 2 (𝑡)
2
+
= Λ − 8𝜋𝐺𝑝 𝑡
𝑎(𝑡)
𝑎2 (𝑡)
𝑎′ 𝑡 − 𝑏(𝑡)
3
= 8𝜋𝐺𝑤𝑠 𝜌 𝑡
𝑎(𝑡)
𝑓(𝑡)
2
= 8𝜋𝐺𝑤𝑠 𝜌(𝑡)
𝑎(𝑡)
Using these expressions today and the dimensionless density Ω = 𝜌(𝑡0)
Ω𝑚 =
8𝜋𝐺𝜌0
3𝐻0 2
; ΩΛ =
Λ
3𝐻0 2
; Ω𝑠 = 𝑤𝑠 𝐻0
8𝜋𝐺𝜌0
3𝐻0 2
; Ω𝑠 = 𝑤𝑠 𝐻0
𝜌𝑐 :
8𝜋𝐺𝜌0
3𝐻0 2
The Friedmann like closure relation reads:
9
Ω𝑚 + ΩΛ + 2Ω𝑠 − Ω𝑠 2 + Ω𝑠 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:
𝑙 𝑡 =
𝐿
𝑎(𝑡)2
4𝜋𝑎0 2 𝑥(𝑡)2 𝑎0
with 𝑥 𝑡 =
𝑡0 𝑑𝑡 ′
𝑡 𝑎(𝑡 ′ )
And the redshift as a scale factor measurement:
𝑎0
𝜆−𝜆
𝑧 = 𝑎(𝑡)
−1 = 𝜆 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
12
We used the so called Union 2 sample containing 557 supernovae up
to a redshift of 1.5
𝜇 𝑧 = 𝑚𝑠 + 2.5log 𝑙 𝑧
Standard cosmology fit gives (no flatness):
ms
Ωm
ΩΛ
marginalized
0.35+0.10 −0.11
0.88+0.19 −0.11
13
We use the full covariance matrix, taking into account systematic errors and
correlations to compute
𝜒 2 = Δ𝑀𝑇 𝑉 −1 Δ𝑀
𝜎1 2
𝑉=
⋮
𝜌𝑛1 𝜎𝑛 𝜎1
⋯ 𝜌1𝑛 𝜎1 𝜎𝑛
⋱
⋮
⋯
𝜎𝑛 2
𝜇𝑡ℎ 𝑧1 , Ω − 𝜇1
⋮
and Δ𝑀 =
𝜇𝑡ℎ 𝑧𝑛 , Ω − 𝜇𝑛
The best cosmological parameters are computed by minimizing the
𝜒2:
𝜕𝜒 2
=0
𝜕Ω𝑘
Errors and contours are computed by using the frequentist prescription:
𝜒 2 Ω𝑚 =
min 𝜒 2 (Ω𝑚 , Ω𝑠 , 𝑚𝑠 ⋯ ) + 𝑠 2
Ω𝑠 ,𝑚𝑠 ⋯
Free cosmological parameters are:
𝑚𝑠 , Ω𝑚 , Ω𝑠 , Ω𝑠 where ΩΛ is deduced from Friedman like relation
9
Ω𝑚 + ΩΛ + 2Ω𝑠 − Ω𝑠 2 + Ω𝑠 2 = 1
4
14
Even parity torsion: Ω𝑠 = 0
Ωm
ΩΛ
𝛀𝒔
0.09+0.30 −0.07 0.83+0.10 −0.16 0.04+0.01 −0.07
Ωm
ΩΛ
Odd parity torsion: Ω𝑠 = 0
Ωm
ΩΛ
𝛀𝒔
0.27+0.03 −0.02 0.73+0.04 −0.11 0.0+0.22 −0.22
𝛀𝒔
0.08+0.27 −0.08 0.85+0.10 −0.15 0.04+0.02 −0.06
𝛀𝒔
0.0+0.1 −0.1
15
Even parity torsion gives a prefer value for matter density equal to 0.09
Ω𝑚 = 0.09+0.30 −0.07
The WMAP last results are:
Ω𝑏 = 0.046 ± 0.003 and Ω𝑚 = 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:
𝑚 𝑠 , Ω𝑠 , ΩΛ = 0 → 𝑚 𝑠 , Ω𝑠 , ΩΛ
We can define the log likelihood ratio as:
𝑀𝑎𝑥 ℒ 𝑚𝑠 , Ω𝑠 , ΩΛ = 0
𝑅 = −2𝐿𝑛
𝑤ℎ𝑒𝑟𝑒 ℒ =
2𝜋
𝑀𝑎𝑥 ℒ 𝑚𝑠 , Ω𝑠 , ΩΛ
1
𝑛/2
𝑉
𝑒 −𝜒
1/2
2 /2
𝑅 = 𝜒 2 𝑚𝑖𝑛,1 − 𝜒 2 𝑚𝑖𝑛,2
The probability distribution of this variable is approximately a 𝜒 2 distribution
with a number of degree of freedom equal to the difference of ndof’s = 1
Fore even parity: Δ𝜒 2 = 44.6 → 𝑝𝑣𝑎𝑙𝑢𝑒 ≅ 0.
→ 𝑟𝑢𝑙𝑙𝑒𝑑 𝑜𝑢𝑡
For odd parity : Δ𝜒 2 = 30.3 → 𝑝𝑣𝑎𝑙𝑢𝑒 = 6 10−8 → 𝑟𝑢𝑙𝑙𝑒𝑑 𝑜𝑢𝑡 𝑎𝑡 5.4 𝑠𝑖𝑔𝑚𝑎
This is not surprising because equation of state: 𝑠 𝑡 ≔ 𝑤𝑠 𝜌 𝑡
If acceleration today, then acceleration in the past: s 𝑡 ↗ with 𝜌(𝑡)
In contradiction with previous publication (S. Capozziello et al. 2003)
17
• Standard general relativity should be extended to account for spin-orbital
momentum coupling: Einstein-Cartan theory.
• If we apply torsion to cosmology we find:
• Torsion can contribute to dark matter at a certain amount
• Torsion as a source of dark energy is ruled out at more than 5 sigma
• However these results are encouraging enough to try to go further
• Look at galaxies rotation curves: Need to generalized the Schwarzschild’s
equation. Work in progress.
• Use other probes:
• CMB/BAO/WL/Clusters: photons are not sensitive to torsion, but
dynamic is different, so everything should be recomputed. 
• But we should not be too much excited by the Supernovae result on DM:
We found a spin energy density Ω𝑠 of about 4% , corresponding to a
state parameter 𝑤𝑠 ~1/𝐻0 ~1017 𝑠 which is 42 orders of magnitude away from the
ℏ
−25 𝑠 !
naïve value 𝑤𝑠 ~
2 ~10
𝑚𝑝 𝑐
Usual problem in cosmology i.e : Λ and vacuum energy!
18
1) Torsion and curvature ?
2) Torsion and vacuum ?
19
In the general case, assuming no special equation of state: 𝑠 𝑡 𝑎𝑛𝑑 𝑠 𝑡 are
free functions of time:
𝑎
4𝜋𝐺
8𝜋𝐺
𝑎 𝑡
=−
𝜌 + 3𝑝 +
𝑠 𝑡 +
𝑠(𝑡)
𝑎
3
3
𝑎 𝑡
Torsion is not source of gravity:
Odd parity torsion doesn’t couple to dynamic (i.e curvature)
Even parity torsion couple to curvature through kinematic not dynamic
20
(1)
*
1. Because geodesics are different for photons and spin ½ particles
(neutrino)
• Timing difference between photons and neutrinos in supernovae
explosion
1987A Supernovae
• Neutrino oscillation experiment OPERA. Time delay and supra
luminal neutrino
2. Rotation curve of galaxies or the modify Schwartsfield solution
• What is the effect of torsion on rotation curve of galaxy
3. Galaxies and cluster formation
4. The cosmological probes:
• Supernovae 1a
• CMB: effect of torsion in initial plasma (very high matter density)
• Weak lensing should not be affected
Lensing is gravitational coupling between curvature and photon
• Baryonic acoustic oscillation
Depends on the initial power spectrum
21
Let consider the covariant derivative of a vector:𝐴𝜎
𝛻𝜇 𝐴𝜎 = 𝐴𝜎 ,𝜇 + Γ𝜇𝜈 𝜎 𝐴𝜈 with 𝐴𝜎 ,𝜇 = 𝜕𝐴𝜎 /𝜕𝑥 𝜇 = 𝜕𝜇 𝐴𝜎
Where Γ𝜇𝜈 𝜎 is the affine connection
This covariant derivative can be formally written as:
𝛿𝜈 𝜎 𝛻𝜇 = 𝛿𝜈 𝜎 𝜕𝜇 + Γ𝜇𝜈 𝜎
And compare to covariant derivative in QED: 𝐷𝜇 = 𝜕𝜇 +𝑖𝑒𝐴𝜇
In geometric term, the affine connection is interpreted as the change of
vector during parallel transport along 𝑑𝑥𝜇 :
−Γ𝜇𝜈 𝜎 𝐴𝜈 𝑑𝑥 𝜇
And the curvature tensor is defined as the change of vector 𝐴𝜈
parallel transported around a closed path 𝜉𝜇 ∶
Δ𝐴𝜎 =
1 𝜈
𝐴 𝑅𝛽𝜇𝜈 𝜎
2
𝜉𝜇 𝑑𝑥 𝛽
22
ℒ = 𝜕𝜇 𝜙𝜕𝜇 𝜙 ∗ −𝑚2 𝜙𝜙 ∗
This Lagrangian is invariant under global rotation 𝜃 in complex plane:
θ
θ
𝜙 → 𝜙𝑒 −𝑖𝜃
θ
𝜙 ∗ → 𝜙 ∗ 𝑒 𝑖𝜃
→ 𝜙𝜙 ∗ is invariant
𝜕𝜇 𝜙 → 𝜕𝜇 𝜙𝑒 −𝑖𝜃
θ
𝜕𝜇 𝜙 ∗ → 𝜕𝜇 𝜙 ∗ 𝑒 𝑖𝜃
→ 𝜕𝜇 𝜙𝜕𝜇 𝜙 ∗ is invariant
But is not invariant under local rotation 𝜃 𝑥𝜇 in complex plane:
θ
𝜕𝜇 𝜙 → 𝜕𝜇 𝜙. 𝑒 −𝑖𝜃 − 𝑖𝜕𝜇 𝜃 𝑥𝜇 . 𝜙. 𝑒 −𝑖𝜃
=
𝜕𝜇 𝜙 =
Variation of the field is assumed
to be linear in 𝜙 𝑎𝑛𝑑 𝛿𝑥𝜇 :
+
Δ𝜙
𝛿𝑥 𝜇
Δ𝜙 = −𝑖𝑒𝐴𝜇 𝛿𝑥 𝜇 𝜙
+
𝐷𝜇 𝜙
𝐷𝜇 = 𝜕𝜇 + 𝑖𝑒𝐴𝜇
23
𝜁 𝛼 − Γ𝜇𝜈 𝛼 𝜒 𝜇 𝜁 𝜈
𝐵𝛼 = 𝜒 𝛼 + (𝜁 𝛼 − Γ𝜇𝜈 𝛼 𝜒 𝜇 𝜁 𝜈 )
𝜒 𝛼 − Γ𝜇𝜈 𝛼 𝜁𝜇 𝜒 𝜈
𝜒𝛼
𝐴𝛼 = 𝜁 𝛼 + (𝜒 𝛼 − Γ𝜇𝜈 𝛼 𝜁𝜇 𝜒 𝜈 )
𝜁𝛼
Then the difference 𝐶 𝛼 = 𝐴𝛼 − 𝐵𝛼 is:
𝐶 𝛼 = 2𝑆𝜇𝜈 𝛼 𝜁𝜇 𝜒 𝜈
1
with 𝑆𝜇𝜈 𝛼 = Γ[𝜇𝜈] 𝛼 = 2 Γ𝜇𝜈 𝛼 − Γ𝜈𝜇 𝛼
Where 𝑆𝜇𝜈 𝛼 is defined as the torsion tensor
24
𝑠 𝑡 ≔ 𝑤𝑠 𝜌 𝑡
𝑠 𝑡 ≔ 𝑤𝑠 𝜌(𝑡)
• 𝜌 𝑡 = any spin ½ matter density with null pressure
• 𝑠 𝑡 = spin density considered as a perfect fluid!
• 𝑤𝑠 has a dimension of time and is assumed to be constant
All physics are inside ws:
• Source of torsion is spin ½ particle
• Orbital momentum or spin 1 are not source of torsion.
• No spin orbital momentum coupling. Spin generates local torsion.
• It contains the Planck constant and GR and QM coupling. Expected
to be small.
• We assume it is not zero even though we don’t know how spins
average?
• ………
25
𝑏 2 𝑡 − 𝑓 2 (𝑡)
3
= Λ + 8𝜋𝐺𝜌 𝑡
𝑎2 (𝑡)
𝑏 ′ (𝑡) 𝑏 2 𝑡 − 𝑓 2 (𝑡)
2
+
= Λ
𝑎(𝑡)
𝑎2 (𝑡)
𝑎′ 𝑡 − 𝑏(𝑡)
3
= 8𝜋𝐺𝑤𝑠 𝜌 𝑡
𝑎(𝑡)
𝑓(𝑡)
2
= 8𝜋𝐺𝑤𝑠 𝜌(𝑡)
𝑎(𝑡)
We eliminate f(t) and ρ(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. a’(t) = 2 a(t)
a1
2. Start from an initial value a(t0)=a0
3. Compute the derivative a’(t0)=2 a0
a0
4. Predict the new point at t0+δt using Tailor
expansion : a(t1 = t0+δt) = a0+2a0 δt +…..= a1
5. Start from this new value a(t1)=a1 and iterate
t0
t1=t0+δt 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
26
redshift error (10-5)
In this paper they assume the same Friedmann equations for the torsion
fluid
𝑎
4𝜋𝐺
=−
𝜌 + 3𝑝
𝜌 = 𝜌 + 𝑓 2 𝑎𝑛𝑑 𝑝 = 𝑝 − 𝑓 2
𝑎
3
with
2
𝑎
8𝜋𝐺
=
𝜌
𝑎
3
3
The missing factor 3 implies torsion is
source of curvature and a constant “f”
function with time which can be
interpreted as a cosmological constant.
At beginning I made the same kind of
mistake and I got 
Unfortunately it’s wrong!
27
In general case where s(t) and 𝑠 𝑡 are functions of time we have:
𝑎 1
= Λ − 4𝜋𝐺 𝜌 + 3𝑝
𝑎 3
8𝜋𝐺
𝑎
+
𝑠 𝑡 + 𝑠(𝑡)
3
𝑎
• Odd parity torsion 𝑠(𝑡) doesn’t modify dynamic (Einstein curvature)
• Even parity torsion couple to gravit
28
The Hilbert action yields the Einstein equation through the principle of
least action:
1
𝑆=−
2𝜅
4
𝑅 −𝑔𝑑 𝑥
R the Ricci scalar
𝑔 = 𝑑𝑒𝑡 𝑔𝜇𝜈
𝜅 = 8𝜋𝐺𝑐 −4
with
In presence of matter the action becomes:
𝑆=
1
𝑅 + ℒ𝑀
2𝜅
−𝑔𝑑 4 𝑥
The action principle 𝛿𝑆 = 0 leads to:
𝛿𝑆 =
𝛿 −𝑔ℒ 𝑀
1 𝛿 −𝑔𝑅
+
2𝜅 𝛿𝑔𝜇𝜈
𝛿𝑔𝜇𝜈
𝛿𝑆 =
1 𝛿𝑅
𝑅 𝛿 −𝑔
+
2𝜅 𝛿𝑔𝜇𝜈
−𝑔 𝛿𝑔𝜇𝜈
𝛿𝑔𝜇𝜈 𝑑 4 𝑥
+
1 𝛿 −𝑔ℒ𝑀
−𝑔
𝛿𝑔𝜇𝜈
29
𝛿𝑔𝜇𝜈 −𝑔𝑑4 𝑥
Since the previous equation should hold for any 𝛿𝑔𝜇𝜈
𝛿𝑅
𝑅 𝛿 −𝑔
1 𝛿 −𝑔ℒ𝑀
+
=
−2𝜅
𝛿𝑔𝜇𝜈
−𝑔 𝛿𝑔𝜇𝜈
−𝑔
𝛿𝑔𝜇𝜈
𝛿𝑅
= 𝑅𝜇𝜈
𝛿𝑔𝜇𝜈
1 𝛿 −𝑔
1
=
−
𝑔
−𝑔 𝛿𝑔𝜇𝜈
2 𝜇𝜈
𝑇𝜇𝜈
−2 𝛿 −𝑔ℒ𝑀
𝛿ℒ𝑀
≔
= −2 𝜇𝜈 + 𝑔𝜇𝜈 ℒ𝑀
−𝑔
𝛿𝑔𝜇𝜈
𝛿𝑔
1
8𝜋𝐺
𝑅𝜇𝜈 − 𝑔𝜇𝜈 𝑅 = 4 𝑇𝜇𝜈
2
𝑐
The cosmological constant is introduced in the Lagrangian:
𝑆=
1
𝑅 − 2Λ + ℒ𝑀
2𝜅
1
8𝜋𝐺
𝑅𝜇𝜈 − 𝑔𝜇𝜈 𝑅 + Λ𝑔𝜇𝜈 = 4 𝑇𝜇𝜈
2
𝑐
−𝑔𝑑4 𝑥
30