E COLE N ATIONALE S UPÉRIEURE DE T ECHNIQUES A VANCÉES L ABORATOIRE DE M ATHÉMATIQUES A PPLIQUÉES The Universe as a dynamical system From Friedmann, to Bianchi passing by the Jungle Greco Seminar Monday, April 18 th , 2016 Jérôme Perez Ensta-ParisTech, Universite Paris Saclay
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E C O L E N A T I O N A L E S U P É R I E U R E D E T E C H N I Q U E S A V A N C É E S
L A B O R A T O I R E D E M A T H É M A T I Q U E S A P P L I Q U É E S
The Universe as a dynamical systemFrom Friedmann, to Bianchi passing by the Jungle
Greco SeminarMonday, April 18 th, 2016
Jérôme PerezEnsta-ParisTech, Universite Paris Saclay
Overview
A Dynamical Universe ?
Friedmann Universe
Bianchi Universe
Einstein Legacy
Einstein Legacy1905 - Special Relativity Principle −→
The equations of physicsare the same in all
galilean (inertial) frames
←− Minkowski : M4 = C± ∪ L ∪ A
Sm = −∫
mcds−∫
(
AαJα +
1
4µ0FαβF
αβ
)
dΩ
Einstein Legacy1905 - Special Relativity Principle −→
The equations of physicsare the same in all
galilean (inertial) frames
←− Minkowski : M4 = C± ∪ L ∪ A
Sm = −∫
mcds−∫
(
AαJα +
1
4µ0FαβF
αβ
)
dΩ
1907 - Equivalence principle ⇒ General relativityր
We [...] propose the complete equivalence between a gravitationnal field and theacceleration of the corresponding frame
The material content of the universe makes impossible the existence of aninertial frame at the universe scale !
The equations of physics are the same in all frames
We pass from M4 [ξα] to a Riemann manifold [xµ] in dimension 4
ds2 = ηαβdξαdξβ = ηαβ
∂ξα
∂xµ
∂ξβ
∂xνdxµdxν = gµνdx
µdxν
The universe becomes dynamical
R>0
Two parallels
can cross
The sum of triangle’s
angles is greater
than p
Geodesics are
arc of circlel
The universe becomes dynamical1915 - General Relativity χ = 8πGc−4
S = Sm −1
2χ
∫
gµνRµν
√−gdΩ with Rµν = Rµν(g) Ricci Tensor
variation of which gives : Gµν := Rµν −1
2gµνR = χTµν
R>0
Two parallels
can cross
The sum of triangle’s
angles is greater
than p
Geodesics are
arc of circlel
The universe becomes dynamical1915 - General Relativity χ = 8πGc−4
S = Sm −1
2χ
∫
gµνRµν
√−gdΩ with Rµν = Rµν(g) Ricci Tensor
variation of which gives : Gµν := Rµν −1
2gµνR = χTµν
1917 - Homogeneous, static and isotropic Universe (Einstein )
ds2 = −dt2 + a
(
dr2
1−Rr2+ r2dθ2 + r2 sin2 θdϕ2
)
R>0
Two parallels
can cross
The sum of triangle’s
angles is greater
than p
Geodesics are
arc of circlel
The universe becomes dynamical1915 - General Relativity χ = 8πGc−4
S = Sm −1
2χ
∫
gµνRµν
√−gdΩ with Rµν = Rµν(g) Ricci Tensor
variation of which gives : Gµν := Rµν −1
2gµνR = χTµν
1917 - Homogeneous, static and isotropic Universe (Einstein )
ds2 = −dt2 + a
(
dr2
1−Rr2+ r2dθ2 + r2 sin2 θdϕ2
)
R=0Two parallels
don’t cross
geodesics are straight
lines
The sum of any
triangle’s angles is p
The plane ...no static solutions
R>0
Two parallels
can cross
The sum of triangle’s
angles is greater
than p
Geodesics are
arc of circlel
The universe becomes dynamical1915 - General Relativity χ = 8πGc−4
S = Sm −1
2χ
∫
gµνRµν
√−gdΩ with Rµν = Rµν(g) Ricci Tensor
variation of which gives : Gµν := Rµν −1
2gµνR = χTµν
1917 - Homogeneous, static and isotropic Universe (Einstein )
ds2 = −dt2 + a
(
dr2
1−Rr2+ r2dθ2 + r2 sin2 θdϕ2
)
R<0
The sum of triangle’s
angles is less
than p
Two parallels can
diverge
Geodesics
are
hyperbolae
The hyperboloïd ...no static solutions
R>0
Two parallels
can cross
The sum of triangle’s
angles is greater
than p
Geodesics are
arc of circlel
The universe becomes dynamical1915 - General Relativity χ = 8πGc−4
S = Sm −1
2χ
∫
gµνRµν
√−gdΩ with Rµν = Rµν(g) Ricci Tensor
variation of which gives : Gµν := Rµν −1
2gµνR = χTµν
1917 - Homogeneous, static and isotropic Universe (Einstein )
ds2 = −dt2 + a
(
dr2
1−Rr2+ r2dθ2 + r2 sin2 θdϕ2
)
R>0
Two parallels
can cross
The sum of triangle’s
angles is greater
than p
Geodesics are
arc of circlel
The sphere ...allows a static solution
p = cste, ǫ = cste
a = cste, R = 6/a2
ifGµν + Λgµν = χTµν
Λ : Cosmological Const.
The universe becomes dynamical1915 - General Relativity χ = 8πGc−4
S = Sm −1
2χ
∫
gµνRµν
√−gdΩ with Rµν = Rµν(g) Ricci Tensor
variation of which gives : Gµν := Rµν −1
2gµνR = χTµν
1917 - Homogeneous, static and isotropic Universe (Einstein )
ds2 = −dt2 + a
(
dr2
1−Rr2+ r2dθ2 + r2 sin2 θdϕ2
)
R>0
Two parallels
can cross
The sum of triangle’s
angles is greater
than p
Geodesics are
arc of circlel
The sphere ...allows a static solution
p = cste, ǫ = cste
a = cste, R = 6/a2
ifGµν + Λgµν = χTµν
Λ : Cosmological Const.
Einstein Universe isunstable !
a(t) = a(1 + δa(t))
p(t) = p(1 + δp(t))
ǫ(t) = a(1 + δǫ(t))
δp(t) = ωδǫ(t)
a(t) diverges if ω > −1/3
Friedmann, Lemaitre and Hubble
1922 - 1924 : Alexandre Friedmann
Friedmann, Lemaitre and Hubble
1922 - 1924 : Alexandre Friedmann
(
1
a
da
dt
)2
+k
a2=
8πGǫ
3(F1)
1
a
d2a
dt2= −4πG
3(ǫ+ 3P ) (F2)
a3dP
dt=
d[
(ǫ+ P ) a3]
dt(F3)
Energy impulsion conservation
Friedmann, Lemaitre and Hubble
1922 - 1924 : Alexandre Friedmann
(
1
a
da
dt
)2
+k
a2=
8πGǫ
3(F1)
1
a
d2a
dt2= −4πG
3(ǫ+ 3P ) (F2)
a3dP
dt=
d[
(ǫ+ P ) a3]
dt(F3)
Energy impulsion conservation
(F2) : If ǫ+ 3P > 0 then(
a > 0,d2a
dt2< 0
)
⇒ a concave⇒ Big-Bang
Friedmann, Lemaitre and Hubble
1922 - 1924 : Alexandre Friedmann
(
1
a
da
dt
)2
+k
a2=
8πGǫ
3(F1)
1
a
d2a
dt2= −4πG
3(ǫ+ 3P ) (F2)
a3dP
dt=
d[
(ǫ+ P ) a3]
dt(F3)
Energy impulsion conservation
(F2) : If ǫ+ 3P > 0 then(
a > 0,d2a
dt2< 0
)
⇒ a concave⇒ Big-Bang
(F1) : Hubble’s Constant : H = a/a
Critical Density : ǫo =3H2
8πG= 1.87847(23)× 10−29 h2 · g · cm−3
We can mesure k = 83πGa2 (ǫ− ǫo)
Friedmann, Lemaitre and Hubble
1922 - 1924 : Alexandre Friedmann
(
1
a
da
dt
)2
+k
a2=
8πGǫ
3(F1)
1
a
d2a
dt2= −4πG
3(ǫ+ 3P ) (F2)
a3dP
dt=
d[
(ǫ+ P ) a3]
dt(F3)
Energy impulsion conservation
(F2) : If ǫ+ 3P > 0 then(
a > 0,d2a
dt2< 0
)
⇒ a concave⇒ Big-Bang
(F1) : Hubble’s Constant : H = a/a
Critical Density : ǫo =3H2
8πG= 1.87847(23)× 10−29 h2 · g · cm−3
We can mesure k = 83πGa2 (ǫ− ǫo)
Big controversy with Einstein but Friedmann dies in September ’25
Friedmann, Lemaitre and Hubble
Friedmann, Lemaitre and Hubble
1925 - Firsts observations Using cepheids stars, Hubble computes thedistance of "Islands Universes" closing the "Great Debate". Slipher measures asystematic red-shift in their spectra.
Friedmann, Lemaitre and Hubble
1925 - Firsts observations Using cepheids stars, Hubble computes thedistance of "Islands Universes" closing the "Great Debate". Slipher measures asystematic red-shift in their spectra.
1927 - Lemaıtre’s Idea Lemaître links observations and Friedmann’stheoretical results. He postulates "the birth of space".
Friedmann, Lemaitre and Hubble
1925 - Firsts observations Using cepheids stars, Hubble computes thedistance of "Islands Universes" closing the "Great Debate". Slipher measures asystematic red-shift in their spectra.
1927 - Lemaıtre’s Idea Lemaître links observations and Friedmann’stheoretical results. He postulates "the birth of space".
1929 - Hubble : The Universe is expanding !
The legend of Λ...
The legend of Λ...
1929 - Einstein’s Renunciation" The cosmological constant is my biggest mistake" =⇒ Λ = 0
The legend of Λ...
1929 - Einstein’s Renunciation" The cosmological constant is my biggest mistake" =⇒ Λ = 0
1990 - Cosmic candlesSystematic observation of White Dwarf SN’s shows a cosmic expansionacceleration (Nobel Prize 2011).
=⇒ Λ 6= 0
The legend of Λ...
1929 - Einstein’s Renunciation" The cosmological constant is my biggest mistake" =⇒ Λ = 0
1990 - Cosmic candlesSystematic observation of White Dwarf SN’s shows a cosmic expansionacceleration (Nobel Prize 2011).
=⇒ Λ 6= 0
A dynamical Universe
Very fun !
(
1
a
da
dt
)2
+k
a2=
8πGǫ
3+
Λ
3(F1)
1
a
d2a
dt2= −4πG
3(ǫ+ 3P ) +
Λ
3(F2)
a3dP
dt=
d[
(ǫ+ P ) a3]
dt(F3)
Impulsion-Energy conservation
Friedmann’s Universes Dynamics
Predator-Prey, competition
Building...
(
a
a
)2
=8πGǫ
3− k
a2+
Λ
3
a
a= −4πG
3(ǫ+ 3P ) +
Λ
3
ǫ = −3H (P + ǫ)
Building...
(
a
a
)2
=8πGǫ
3− k
a2+
Λ
3
a
a= −4πG
3(ǫ+ 3P ) +
Λ
3
ǫ = −3H (P + ǫ)
, setting
H (t) =a
a=
d (ln a)
dt
q (t) = − a
a
1
H2= − a a
a2
Ωm (t) =8πGǫ
3H2, Ωk (t) = −
k
a2H2
and ΩΛ (t) =Λ
3H2
Building...
(
a
a
)2
=8πGǫ
3− k
a2+
Λ
3
a
a= −4πG
3(ǫ+ 3P ) +
Λ
3
ǫ = −3H (P + ǫ)
, setting
H (t) =a
a=
d (ln a)
dt
q (t) = − a
a
1
H2= − a a
a2
Ωm (t) =8πGǫ
3H2, Ωk (t) = −
k
a2H2
and ΩΛ (t) =Λ
3H2
we obtain
Ωm +Ωk +ΩΛ = 1 (F1.1)
4πG
3H2(ǫ+ 3P ) = q +ΩΛ (F2.1)
ǫ = −3H (P + ǫ) (F3.1)
State equation
State equation
Barotropic : P = ωǫ = (Γ− 1) ǫ =(γ − 1)
3ǫ
State equation
Barotropic : P = ωǫ = (Γ− 1) ǫ =(γ − 1)
3ǫ
ω −1 0 1/3 2/3 1
Kind
of Matter
Quantum
Vacuum
Incoherent
Dust Gas
Photon
Ideal Gas
monoatomic
Ideal Gas
Stiff
matter
ω ∈ [−1, 1] , Γ ∈ [0, 2] , γ ∈ [−2, 4]
State equation
Barotropic : P = ωǫ = (Γ− 1) ǫ =(γ − 1)
3ǫ
ω −1 0 1/3 2/3 1
Kind
of Matter
Quantum
Vacuum
Incoherent
Dust Gas
Photon
Ideal Gas
monoatomic
Ideal Gas
Stiff
matter
ω ∈ [−1, 1] , Γ ∈ [0, 2] , γ ∈ [−2, 4]
Barotropic Friedmann’s Equations :
Ωk = 1 − Ωm − ΩΛ
q =Ωm (1 + 3ω)
2− ΩΛ
(ln ǫ)′= −3 (1 + ω) ′ =
d
d ln a
The dynamical system
The dynamical system
Ωk = 1 − Ωm − ΩΛ
Ω′m = Ωm [(1 + 3ω) (Ωm − 1)− 2ΩΛ]
Ω′Λ = ΩΛ [Ωm (1 + 3ω) + 2 (1− ΩΛ)]
The dynamical system
Ωk = 1 − Ωm − ΩΛ
Ω′m = Ωm [(1 + 3ω) (Ωm − 1)− 2ΩΛ]
Ω′Λ = ΩΛ [Ωm (1 + 3ω) + 2 (1− ΩΛ)]
setting γ = 1 + 3ω in the interval [−2, 4]
X ′ = Fγ (X) with X = [Ωm,ΩΛ]⊤ and Fγ :
∣
∣
∣
∣
∣
R2 → R2
(x, y) 7→ (f1 (x, y) , f2 (x, y))
where
f1 (x, y) = x (γx− 2y − γ)
f2 (x, y) = y (γx− 2y + 2)
Lotka-Volterra like equation
Equilibria
EquilibriaEquilibrium : X∗ = [x, y]
⊤= [Ωm,ΩΛ]
⊤ such that Fγ (X∗) = 0
x (γx− 2y − γ) = 0
y (γx− 2y + 2) = 0
EquilibriaEquilibrium : X∗ = [x, y]
⊤= [Ωm,ΩΛ]
⊤ such that Fγ (X∗) = 0
x (γx− 2y − γ) = 0
y (γx− 2y + 2) = 0There is 3 solutions :
EquilibriaEquilibrium : X∗ = [x, y]
⊤= [Ωm,ΩΛ]
⊤ such that Fγ (X∗) = 0
x (γx− 2y − γ) = 0
y (γx− 2y + 2) = 0There is 3 solutions :
de Sitter Universe : X∗1 = [0, 1]
⊤ and Ωk = 0
If a > 0 then a(t) ∝ e√
Λ3 t
Uncreated Universe in perpetual exponential expansion.
EquilibriaEquilibrium : X∗ = [x, y]
⊤= [Ωm,ΩΛ]
⊤ such that Fγ (X∗) = 0
x (γx− 2y − γ) = 0
y (γx− 2y + 2) = 0There is 3 solutions :
de Sitter Universe : X∗1 = [0, 1]
⊤ and Ωk = 0
If a > 0 then a(t) ∝ e√
Λ3 t
Uncreated Universe in perpetual exponential expansion.
Einstein-de Sitter Universe : X∗2 = [1, 0]
⊤ and Ωk = 0
If ω > −1 then a (t) ∝ t2
3(1+3ω)
Big-Bang followed by a decelerated expansion.
EquilibriaEquilibrium : X∗ = [x, y]
⊤= [Ωm,ΩΛ]
⊤ such that Fγ (X∗) = 0
x (γx− 2y − γ) = 0
y (γx− 2y + 2) = 0There is 3 solutions :
de Sitter Universe : X∗1 = [0, 1]
⊤ and Ωk = 0
If a > 0 then a(t) ∝ e√
Λ3 t
Uncreated Universe in perpetual exponential expansion.
Einstein-de Sitter Universe : X∗2 = [1, 0]
⊤ and Ωk = 0
If ω > −1 then a (t) ∝ t2
3(1+3ω)
Big-Bang followed by a decelerated expansion.
Milne Universe : X∗3 = [0, 0]
⊤ and Ωk = 1
k = −a2H2 : Hyperbolic universe with a(t) = a0t+ a0Linearly expanding Universe since Big-Bang : exotic cosmological models ?
Dynamic is a competition !
Ω′m = Ωm (γΩm − 2ΩΛ − γ)
Ω′Λ = ΩΛ (γΩm − 2ΩΛ + 2)
Competition between Ωm and ΩΛ "referred" by Ωk ;
3 equilibrium states :
• Matter (EdS) - γ−Hyperbolic ;
• Curvature (M) - γ−Hyperbolic ;
• Cosmological Constant (dS) - Stable.
The most competitive is always the Cosmological Constant : γ ∈ [−2, 4].
No Limit Cycle (Bendixon criteria, div(F ) has constant sign on [0, 1]2 ?
The fate of Friedmann’s Universes
If ω of Ωm is in ]− 1,−1/3[ :
The fate of Friedmann’s Universes
If ω of Ωm is in ]− 1/3, 1[ :
Coupled species : Jungle Universe
Coupled species : Jungle UniverseWithout any coupling between species (Ωi) the dynamic is fully degenerated :
x = (Ωb,Ωd,Ωr,Ωe)⊤
, x′ = diag(x) (r+ Ax)
with
A =
1 + 3ωb 1 + 3ωd 1 + 3ωr 1 + 3ωe
1 + 3ωb 1 + 3ωd 1 + 3ωr 1 + 3ωe
1 + 3ωb 1 + 3ωd 1 + 3ωr 1 + 3ωe
1 + 3ωb 1 + 3ωd 1 + 3ωr 1 + 3ωe
and r =
−1− 3ωb
−1− 3ωd
−1− 3ωr
−1− 3ωe
As rank(A) = 1, equilibria must lie on axes xi = 0, this is Friedmann’s dynamics.
Coupled species : Jungle UniverseWithout any coupling between species (Ωi) the dynamic is fully degenerated :
x = (Ωb,Ωd,Ωr,Ωe)⊤
, x′ = diag(x) (r+ Ax)
with
A =
1 + 3ωb 1 + 3ωd 1 + 3ωr 1 + 3ωe
1 + 3ωb 1 + 3ωd 1 + 3ωr 1 + 3ωe
1 + 3ωb 1 + 3ωd 1 + 3ωr 1 + 3ωe
1 + 3ωb 1 + 3ωd 1 + 3ωr 1 + 3ωe
and r =
−1− 3ωb
−1− 3ωd
−1− 3ωr
−1− 3ωe
As rank(A) = 1, equilibria must lie on axes xi = 0, this is Friedmann’s dynamics.
Introducing coupling between any barotropic components of the Universe, thedynamical systems becomes
xi = Ωi
ri = −(1 + 3ωi) (1)
Aij = 1 + 3ωj + εij with εij = −εji and εii = 0
The matrix A can have any rank, it can be invertible, equilibria can be everywhere,this is Jungle dynamics. [e.g. Perez et. al., 2014]
Dark coupling...
+r b
t=0
e
e
Coupling between dark energy and dark mater with ε = 4.
The radiative components (Ωr) and the baryonic matter (Ωb) dilutes and disappearswhile the dark component converges toward a limit cycle.
Other possibilities...
-0.1 0 0.1 0.2 0.3 0.4
0
0.2
0.4
0.15
0.25
0.35
0.45
I.C.
0
0.51 0
0.51
0.2
0.4
0.6
0.8
r
d I.C.
r
d
e=5/2
e=3/2
req
eeq
deq
eeq
req
Evolution of the three coupled density parameters, in the 3D phase space. Thebeginning of the orbit is overlined. Initial condition is indicated by a black dot. Relevant
equilibria are indicated by a star.
Other possibilities...
-0.1 0 0.1 0.2 0.3 0.4
0
0.2
0.4
0.15
0.25
0.35
0.45
I.C.
0
0.51 0
0.51
0.2
0.4
0.6
0.8
r
d I.C.
r
d
e=5/2
e=3/2
req
eeq
deq
eeq
req
Evolution of the three coupled density parameters, in the 3D phase space. Thebeginning of the orbit is overlined. Initial condition is indicated by a black dot. Relevant
equilibria are indicated by a star.
Camouflage in the jungle [Simon-Petit, J.P. & Yap, 2016]
10-1
100
101
½1
2
3
½
½
Density(arbitrary units)
Time (arbitrary units)
!1;e® (!1 = 0)
!2;e® (!2 = 0)
!3;e® (!3 = 0)
10-1
100
101
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Effectivebarotropic
index
Time (arbitrary units)
Camouflage in the jungle [Simon-Petit, J.P. & Yap, 2016]
Could dark energy emerge from the jungle coupling ?
10-1
100
101
½1
2
3
½
½
Density(arbitrary units)
Time (arbitrary units)
!1;e® (!1 = 0)
!2;e® (!2 = 0)
!3;e® (!3 = 0)
10-1
100
101
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Effectivebarotropic
index
Time (arbitrary units)
Camouflage in the jungle [Simon-Petit, J.P. & Yap, 2016]
Could dark energy emerge from the jungle coupling ?
The interaction term in the continuity equation of a fluid i reads
ρi = −3Hρi(1 + ωi) +n∑
j=1
ǫijHΩjρi
It actually modifies its equation of state which then describes a barotropic uid with aneffective time-dependent barotropic index ωeff
i = ωi −∑n
j=113ǫijΩj
10-1
100
101
½1
2
3
½
½
Density(arbitrary units)
Time (arbitrary units)
!1;e® (!1 = 0)
!2;e® (!2 = 0)
!3;e® (!3 = 0)
10-1
100
101
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Effectivebarotropic
index
Time (arbitrary units)
Camouflage in the jungle [Simon-Petit, J.P. & Yap, 2016]
Could dark energy emerge from the jungle coupling ?
The interaction term in the continuity equation of a fluid i reads
ρi = −3Hρi(1 + ωi) +n∑
j=1
ǫijHΩjρi
It actually modifies its equation of state which then describes a barotropic uid with aneffective time-dependent barotropic index ωeff
i = ωi −∑n
j=113ǫijΩj
Exemple :
10-1
100
101
½1
2
3
½
½
Density(arbitrary units)
Time (arbitrary units)
!1;e® (!1 = 0)
!2;e® (!2 = 0)
!3;e® (!3 = 0)
10-1
100
101
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Effectivebarotropic
index
Time (arbitrary units)
Jungle Interaction (ǫ12 = −2; ǫ23 = −3; ǫ13 = 0) between three dark matter fluids
Bianchi Universes
The Cosmological Billiard
Save General Relativity !
B
K L
Save General Relativity !
1915 A. Einstein : Gravitationnal Field Theory
B
K L
Save General Relativity !
1915 A. Einstein : Gravitationnal Field Theory
1922-27 A. Friedmann & G. Lemaître : Homogeneous and Isotropic solution(Big Bang ≈ 1960)
B
K L
Save General Relativity !
1915 A. Einstein : Gravitationnal Field Theory
1922-27 A. Friedmann & G. Lemaître : Homogeneous and Isotropic solution(Big Bang ≈ 1960)
1965-66 R. Penrose & S. Hawking : All solutions are singular !
B
K L
Save General Relativity !
1915 A. Einstein : Gravitationnal Field Theory
1922-27 A. Friedmann & G. Lemaître : Homogeneous and Isotropic solution(Big Bang ≈ 1960)
1965-66 R. Penrose & S. Hawking : All solutions are singular !
1969 V. Belinski, L. Khalatnikov & E. Lifchitz : Singularity may be chaotic ifUniverse is anisotropic !