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Exploring the Evolution of Dark Energy
and its Equation of State
Cristina Espana i Bonet
Universitat de Barcelona
12th February, 2008
Advisor: Dra. Pilar Ruiz-Lapuente
Overview
1 Motivacio
2 Introduction
3 Evolving Cosmological Constant
4 Non-Parametric Reconstructions
5 Future Perspectives
6 Summary and Conclusions
L’Univers
Si l’Univers fos estatic...
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
L’Univers
...pero esta en expansio (model de Big Bang)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
L’Univers
...i accelerada
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Com ho observem?
+ Necessitem un objecte delluminositat coneguda
+ La llum que rebem ensindica la distancia
Candela estandard
Candela estandard per exel·lencia:
Supernoves del tipus Ia (SNe Ia)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Com ho observem?
Necessitem un objecte delluminositat coneguda
La llum que rebem ensindica la distancia
+ Candela estandard
Candela estandard per exel·lencia:
Supernoves del tipus Ia (SNe Ia)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Com ho observem?
Necessitem un objecte delluminositat coneguda
La llum que rebem ensindica la distancia
Candela estandard
Candela estandard per exel·lencia:
Supernoves del tipus Ia (SNe Ia)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Com ho observem?
+ Sistema binari enacrecio
+ Explosio a MCh + Mateixalluminositat
Lluminositat inferior a Ld ⇒ mes lluny ⇒ expansio mes rapida
Lluminositat superior a Ld ⇒ mes a prop ⇒ expansio mes lenta
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Com ho observem?
Sistema binari enacrecio
Explosio a MCh Mateixalluminositat
Lluminositat inferior a Ld
⇒ mes lluny ⇒ expansio mes rapida
Lluminositat superior a Ld
⇒ mes a prop ⇒ expansio mes lenta
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Com ho observem?
Sistema binari enacrecio
Explosio a MCh Mateixalluminositat
Lluminositat inferior a Ld ⇒ mes lluny
⇒ expansio mes rapida
Lluminositat superior a Ld ⇒ mes a prop
⇒ expansio mes lenta
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Com ho observem?
Sistema binari enacrecio
Explosio a MCh Mateixalluminositat
Lluminositat inferior a Ld ⇒ mes lluny ⇒ expansio mes rapida
Lluminositat superior a Ld ⇒ mes a prop ⇒ expansio mes lenta
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Que observem?
+ Supernoves llunyanes: coherent amb BB estandard
Supernoves meitat de l’edat de l’Univers: sublluminoses
Supernoves properes: canvis imperceptibles
⇓Expansio recent accelerada
⇓Per que? i Perque?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Que observem?
Supernoves llunyanes: coherent amb BB estandard
+ Supernoves meitat de l’edat de l’Univers: sublluminoses
Supernoves properes: canvis imperceptibles
⇓Expansio recent accelerada
⇓Per que? i Perque?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Que observem?
Supernoves llunyanes: coherent amb BB estandard
Supernoves meitat de l’edat de l’Univers: sublluminoses
+ Supernoves properes: canvis imperceptibles
⇓Expansio recent accelerada
⇓Per que? i Perque?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Que observem?
Supernoves llunyanes: coherent amb BB estandard
Supernoves meitat de l’edat de l’Univers: sublluminoses
Supernoves properes: canvis imperceptibles
⇓Expansio recent accelerada
⇓Per que? i Perque?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Objectius
Estudi i caracteritzacio del component que accelera l’expansio,l’energia fosca, mitjancant:
1 Aproximacio directa:Construccio d’un model i contrast amb observacions
2 Aproximacio inversa:Reconstruccio del model subjacent a partir de lesobservacions
Eina principal:
1 Variacio de la magnitud de les SNe Ia amb la distancia
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Objectius
Estudi i caracteritzacio del component que accelera l’expansio,l’energia fosca, mitjancant:
1 Aproximacio directa:Construccio d’un model i contrast amb observacions
2 Aproximacio inversa:Reconstruccio del model subjacent a partir de lesobservacions
Eina principal:
1 Variacio de la magnitud de les SNe Ia amb la distancia
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Objectius
Estudi i caracteritzacio del component que accelera l’expansio,l’energia fosca, mitjancant:
1 Aproximacio directa:Construccio d’un model i contrast amb observacions
2 Aproximacio inversa:Reconstruccio del model subjacent a partir de lesobservacions
Eina principal:
1 Variacio de la magnitud de les SNe Ia amb la distancia
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Let’s start
1 Motivacio
2 IntroductionCosmological ConstantGeneral Dark Energy SourceObtaining Information with Observations
3 Evolving Cosmological Constant
4 Non-Parametric Reconstructions
5 Future Perspectives
6 Summary and Conclusions
Introduction
Why this thesis?
Clear evidence of theacceleration
Attribution to acosmological constant(CC)
I ...but other candidates
I ...possible evolution
What is the CC?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Introduction
Why this thesis?
Clear evidence of theacceleration
Attribution to acosmological constant(CC): Λ ≈ 10−47 GeV 4
I ...but other candidates
I ...possible evolution
What is the CC?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Introduction
Why this thesis?
Clear evidence of theacceleration
Attribution to acosmological constant(CC)...
I ...but other candidates
I ...possible evolution
What is the CC?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Introduction
Why this thesis?
Clear evidence of theacceleration
Attribution to acosmological constant(CC)...
I ...but other candidates
I ...possible evolution
What is the CC?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological constant (CC)The Einstein’s term
+ Interpretation:antigravitationalforce
Nothing forbids λ(t)
Gµν−λ(t)gµν = −8πGNTµν .
Inclusion of the CC inEinstein’s Field Equations
A. Einstein,Kosmologische betrachtungen zur allgemeinen Relativitatstheorie.S.-B. Preuss. Akad. Wiss. (1917), pp. 142-152.
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological constant (CC)The Einstein’s term
Interpretation:antigravitationalforce
+ Nothing forbids λ(t)
Gµν−λ(t)gµν = −8πGNTµν .
Inclusion of the CC inEinstein’s Field Equations
A. Einstein,Kosmologische betrachtungen zur allgemeinen Relativitatstheorie.S.-B. Preuss. Akad. Wiss. (1917), pp. 142-152.
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological constant (CC)Vacuum energy
+ Interpretation:vacuum energy
Inclusion of the CC as asource of Tµν :
Gµν = −8πGN(Tµν + Λgµν) .
In the vacuum: 〈Tµν〉 = Gµν〈Vφ〉
I Particles & fields 〈Vφ〉 = 0
I but Higgs 〈VH〉 ∝ m4H ≈ 108 GeV 4
=⇒ Λind ≈ 108 GeV 4
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological constant (CC)Vacuum energy
Interpretation:vacuum energy
Inclusion of the CC as asource of Tµν :
Gµν = −8πGN(Tµν + Λgµν) .
In the vacuum: 〈Tµν〉 = Gµν〈Vφ〉
I Particles & fields 〈Vφ〉 = 0
I but Higgs 〈VH〉 ∝ m4H ≈ 108 GeV 4
=⇒ Λind ≈ 108 GeV 4
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological constant (CC)Vacuum energy
Interpretation:vacuum energy
Inclusion of the CC as asource of Tµν :
Gµν = −8πGN(Tµν + Λgµν) .
In the vacuum: 〈Tµν〉 = Gµν〈Vφ〉
I Particles & fields 〈Vφ〉 = 0
I but Higgs 〈VH〉 ∝ m4H ≈ 108 GeV 4
=⇒ Λind ≈ 108 GeV 4
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological constant (CC)An only source?
Observations
Λobs ≈ 10−47 GeV 4
Theory
Λind ≈ 108 GeV 4
ΛEinstein
︸ ︷︷ ︸Λobs = ΛEinstein + Λind
|ΛEinstein| − |Λind | = 10−55 GeV 4
= 0.00000000000000000000000000000000000000000000000000000001!!
Cosmological Constant Problem
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological constant (CC)An only source?
Observations
Λobs ≈ 10−47 GeV 4
Theory
Λind ≈ 108 GeV 4
ΛEinstein︸ ︷︷ ︸Λobs = ΛEinstein + Λind
|ΛEinstein| − |Λind | = 10−55 GeV 4
= 0.00000000000000000000000000000000000000000000000000000001!!
Cosmological Constant Problem
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological constant (CC)An only source?
Observations
Λobs ≈ 10−47 GeV 4
Theory
Λind ≈ 108 GeV 4
ΛEinstein︸ ︷︷ ︸Λobs = ΛEinstein + Λind
|ΛEinstein| − |Λind | = 10−55 GeV 4
= 0.00000000000000000000000000000000000000000000000000000001!!
Cosmological Constant Problem
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
A general dark energy componentThe perfect fluid approach
+ Alternative: Inclusion of a perfect fluid extra componentcharacterized by its Equation of State (EoS)
p = wρ
Interpretation of the cosmological constant as an extracomponent of Tµν
=⇒ perfect fluid with ρ = Λ and p = −Λ=⇒ w = −1
In general, ρ(t) and p(t), so w(t)
But for Λ(t) still w = −1
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
A general dark energy componentThe perfect fluid approach
Alternative: Inclusion of a perfect fluid extra componentcharacterized by its Equation of State (EoS)
p = wρ
+ Interpretation of the cosmological constant as an extracomponent of Tµν
=⇒ perfect fluid with ρ = Λ and p = −Λ=⇒ w = −1
In general, ρ(t) and p(t), so w(t)
But for Λ(t) still w = −1
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
A general dark energy componentThe perfect fluid approach
Alternative: Inclusion of a perfect fluid extra componentcharacterized by its Equation of State (EoS)
p = wρ
Interpretation of the cosmological constant as an extracomponent of Tµν
=⇒ perfect fluid with ρ = Λ and p = −Λ=⇒ w = −1
+ In general, ρ(t) and p(t), so w(t)
But for Λ(t) still w = −1
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
A general dark energy componentThe perfect fluid approach
Alternative: Inclusion of a perfect fluid extra componentcharacterized by its Equation of State (EoS)
p = wρ
Interpretation of the cosmological constant as an extracomponent of Tµν
=⇒ perfect fluid with ρ = Λ and p = −Λ=⇒ w = −1
In general, ρ(t) and p(t), so w(t)
+ But for Λ(t) still w = −1
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
A general dark energy componentAlternatives
Every component is described by its equation of state:
wR = 1/3 radiationwM = 0 non-relativistic matter
wS = −1/3 cosmic stringswW = −2/3 domain wallswT = −1/3 textureswΛ = −1 (evolving) cosmological constant
wQ(t) > −1 (dwQ/dz > 0) quintessencewK (t) > −1 (dwK/dz < 0) k-essence
wPh(t) < −1 phantoms
Multitude of possibilities for dark energy...
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
A general dark energy componentAlternatives
Every component is described by its equation of state:
wR = 1/3 radiationwM = 0 non-relativistic matter
wS = −1/3 cosmic stringswW = −2/3 domain wallswT = −1/3 textureswΛ = −1 (evolving) cosmological constant
wQ(t) > −1 (dwQ/dz > 0) quintessencewK (t) > −1 (dwK/dz < 0) k-essence
wPh(t) < −1 phantoms
Multitude of possibilities for dark energy...
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
A general dark energy componentAlternatives
Every component is described by its equation of state:
wR = 1/3 radiationwM = 0 non-relativistic matter
wS = −1/3 cosmic stringswW = −2/3 domain wallswT = −1/3 textureswΛ = −1 (evolving) cosmological constant
wQ(t) > −1 (dwQ/dz > 0) quintessencewK (t) > −1 (dwK/dz < 0) k-essence
wPh(t) < −1 phantoms
Multitude of possibilities for dark energy...
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
A general dark energy componentAlternatives
Every component is described by its equation of state:
wR = 1/3 radiationwM = 0 non-relativistic matter
wS = −1/3 cosmic stringswW = −2/3 domain wallswT = −1/3 textureswΛ = −1 (evolving) cosmological constant
wQ(t) > −1 (dwQ/dz > 0) quintessencewK (t) > −1 (dwK/dz < 0) k-essence
wPh(t) < −1 phantoms
Multitude of possibilities for dark energy...
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
A general dark energy componentAlternatives
EoS index:
wR = 1/3wM = 0wS = −1/3wW = −2/3wT = −1/3wΛ = −1
wQ(t) > −1 (dwQ/dz > 0) =⇒wK (t) > −1 (dwK/dz < 0)wPh(t) < −1
Quintessence EoS
Weller & Albrecht (2002)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe IaMagnitude-redshift relation
The relation between models and observations is encoded in:
m(z, H0, Ω0M , Ω0
X ) = M + 5 log10
ˆH0 dL(z, H0, Ω
0M , Ω0
X )˜
dL(z, Ω0M , Ω0
X ) =c (1 + z)
H0
0B@Z z
0
dz ′qΩ0
M(1 + z)3 + ΩX (z)
1CAΩX (z) = Ω0
X exp
„3
Z z
0dz ′
1 + w(z ′)
1 + z ′
«
Double integral relating the observed m with w(z)I Smoothing of any possible evolutionI Increasing of degeneracy
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe IaMagnitude-redshift relation
The relation between models and observations is encoded in:
m(z, H0, Ω0M , Ω0
X ) = M + 5 log10
ˆH0 dL(z, H0, Ω
0M , Ω0
X )˜
dL(z, Ω0M , Ω0
X ) =c (1 + z)
H0
0B@Z z
0
dz ′qΩ0
M(1 + z)3 + ΩX (z)
1CAΩX (z) = Ω0
X exp
„3
Z z
0dz ′
1 + w(z ′)
1 + z ′
«
+ Double integral relating the observed m with w(z)
+ Smoothing of any possible evolution+ Increasing of degeneracy
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe IaMagnitude-redshift relation
The relation between models and observations is encoded in:
m(z, H0, Ω0M , Ω0
X ) = M + 5 log10
ˆH0 dL(z, H0, Ω
0M , Ω0
X )˜
dL(z, Ω0M , Ω0
X ) =c (1 + z)
H0
0B@Z z
0
dz ′qΩ0
M(1 + z)3 + ΩX (z)
1CAΩX (z) = Ω0
X exp
„3
Z z
0dz ′
1 + w(z ′)
1 + z ′
«
Double integral relating the observed m with w(z)
+ Smoothing of any possible evolution+ Increasing of degeneracy
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe Iam − z space
Well covered datarange: 0 < z < 1.5
Most models can befit to data
Small differencesamong models in therange
That’s a hard work!
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe Iam − z space
Well covered datarange: 0 < z < 1.5
Most models can befit to data
Small differencesamong models in therange
That’s a hard work!
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe IaData sets
Currently, 2 (+1) data sets available:
Riess et al.(2006)
182 SNe Ia
0.023 < z < 1.77
Wood-Vasey et al.(2007)
162 SNe Ia
0.015 < z < 0.96
Davis et al.(2007)
192 SNe Ia
0.015 < z < 1.77
Results do depend on the data set used. In the following,results for Riess et al. (2006) are shown.
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe IaData sets
Currently, 2 (+1) data sets available:
Riess et al.(2006)
182 SNe Ia
0.023 < z < 1.77
Wood-Vasey et al.(2007)
162 SNe Ia
0.015 < z < 0.96
Davis et al.(2007)
192 SNe Ia
0.015 < z < 1.77
Results do depend on the data set used. In the following,results for Riess et al. (2006) are shown.
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe IaComplementary probes and priors
+ Constraint on the curvature:
+ Flat Universe (WMAP3)
Constraint on ΩM :
I 0.27± 0.03 (clusters)
I BAO constraints
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe IaComplementary probes and priors
Constraint on the curvature:
+ Flat Universe (WMAP3)
Constraint on ΩM :
I 0.27± 0.03 (clusters)
I BAO constraints
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe IaComplementary probes and priors
Constraint on the curvature:
I Flat Universe (WMAP3)
+ Constraint on ΩM :
+ 0.27± 0.03 (clusters)
+ BAO constraints
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe IaComplementary probes and priors
Constraint on the curvature:
I Flat Universe (WMAP3)
Constraint on ΩM :
+ 0.27± 0.03 (clusters)
I BAO constraints
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Obtaining information with SNe IaComplementary probes and priors
Constraint on the curvature:
I Flat Universe (WMAP3)
Constraint on ΩM :
I 0.27± 0.03 (clusters)
+ BAO constraints
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The dark energy model
1 Motivacio
2 Introduction
3 Evolving Cosmological ConstantFundamentalsCosmological ScenariosObservational Constraints
4 Non-Parametric Reconstructions
5 Future Perspectives
6 Summary and Conclusions
Evolving Cosmological ConstantThe theory behind... in a few words
Quantum Field Theory(semiclassical approximation: fields in a curved space-time)
Vacuum action at low energies
SHE = −Z
d4x√−g
„1
16πGvacR + Λvac
«
Ultraviolet divergencies → regularization, renormalization
Scale invariance broken
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological ConstantThe theory behind... in a few words
Quantum Field Theory(semiclassical approximation: fields in a curved space-time)
⇓Vacuum action at low energies
SHE = −Z
d4x√−g
„1
16πGvacR + Λvac
«
Ultraviolet divergencies → regularization, renormalization
Scale invariance broken
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological ConstantThe theory behind... in a few words
Quantum Field Theory(semiclassical approximation: fields in a curved space-time)
⇓Vacuum action at low energies
SHE = −Z
d4x√−g
„1
16πGvacR + Λvac
«
⇓Ultraviolet divergencies → regularization, renormalization
Scale invariance broken
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological ConstantThe theory behind... in a few words
Quantum Field Theory(semiclassical approximation: fields in a curved space-time)
Vacuum action at low energies
SHE = −Z
d4x√−g
„1
16πGvacR + Λvac
«
⇓Ultraviolet divergencies → regularization, renormalization
⇓Scale invariance broken
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological ConstantThe theory behind... in a few words
Quantum Field Theory(semiclassical approximation: fields in a curved space-time)
Vacuum action at low energies
SHE = −Z
d4x√−g
„1
16πGvacR + Λvac
«
Ultraviolet divergencies → regularization, renormalization
⇓Scale invariance broken
⇓Renormalization Group Equations (RGE)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological Constantβ-function, RGE for the Cosmological Constant
+ For the CC, the dependence on the scale is encoded inthe β-function
µd
d ln µ
„Λ
8πG
«≡ βΛ =
1
(4π)2
0@Xi
Ai m4i + µ2
Xj
BjM2j + µ4
Xj
Cj + ...
1A
with:
Dependence on the renormalization scale µ
How is decoupling produced? Which are the active dofs?I Light particles?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological Constantβ-function, RGE for the Cosmological Constant
For the CC, the dependence on the scale is encoded inthe β-function
µd
d ln µ
„Λ
8πG
«≡ βΛ =
1
(4π)2
0@Xi
Ai m4i + µ2
Xj
BjM2j + µ4
Xj
Cj + ...
1A
with:
+ Dependence on the renormalization scale µ
How is decoupling produced? Which are the active dofs?I Light particles?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological Constantβ-function, RGE for the Cosmological Constant
For the CC, the dependence on the scale is encoded inthe β-function
µd
d ln µ
„Λ
8πG
«≡ βΛ =
1
(4π)2
0@Xi
Ai m4i + µ2
Xj
BjM2j + µ4
Xj
Cj + ...
1A
with:
Dependence on the renormalization scale µ
+ How is decoupling produced? Which are the active dofs?
+ Light particles?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological Constantβ-function, RGE for the Cosmological Constant
For the CC, the dependence on the scale is encoded inthe β-function
µd
d ln µ
„Λ
8πG
«≡ βΛ =
1
(4π)2
0@Xi
Ai m4i + µ2
Xj
BjM2j + µ4
Xj
Cj + ...
1A
with:
Dependence on the renormalization scale µ
How is decoupling produced? Which are the active dofs?
+ Light particles?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological Constantβ-function, RGE for the Cosmological Constant
For the CC, the dependence on the scale is encoded inthe β-function
µd
d ln µ
„Λ
8πG
«≡ βΛ =
1
(4π)2
0@Xi
Ai m4i + µ2
Xj
BjM2j + µ4
Xj
Cj + ...
1A
with:
Dependence on the renormalization scale µ
How is decoupling produced? Which are the active dofs?
+ Light particles? Heavy particles with soft decoupling?
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological ConstantCosmological scenarios
There are several choices in the literature. We contribute withone (Scenario 3) and test three of them:
Active dof Particles µ
Scenario 1 mi < µ neutrinos ρ1/4c (t)
Scenario 2 Mi > µ SM ρ1/4c (t)
Scenario 3 Mi > µ Plank H(t)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological ConstantCosmological scenarios
There are several choices in the literature. We contribute withone (Scenario 3) and test three of them:
Active dof Particles µ
Scenario 1 mi < µ neutrinos ρ1/4c (t)
Scenario 2 Mi > µ SM ρ1/4c (t)
Scenario 3 Mi > µ Plank H(t)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological ConstantCosmological scenarios
There are several choices in the literature. We contribute withone (Scenario 3) and test three of them:
Active dof Particles µ
Scenario 1 mi < µ neutrinos ρ1/4c (t)
Scenario 2 Mi > µ SM ρ1/4c (t)
Scenario 3 Mi > µ Plank H(t)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Evolving Cosmological ConstantCosmological scenarios
There are several choices in the literature. We contribute withone (Scenario 3) and test three of them:
Active dof Particles µ
Scenario 1 mi < µ neutrinos ρ1/4c (t)
Scenario 2 Mi > µ SM ρ1/4c (t)
Scenario 3 Mi > µ Plank H(t)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 3: Mi > µ, µ ∼ H(t)
Λ(H) = Λ0 +σ
2(4π)2M2(H2 − H2
0 ) RGE
Resolution of the system equation:
RGE + Friedmann Equation + Continuity Equation
Λ(z ; ν) = Λ0 + ρ0M
ν
1− ν
[(1 + z)3(1−ν) − 1
]− κ
1− 3ν
z (z + 2)
2+
ν
1− ν
[(1 + z)3(1−ν) − 1
]
κ ≡ −2 νΩ0K → proportional to curvature
ν ≡ σ12 π
M2
M2P→ cosmological index
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 3: Mi > µ, µ ∼ H(t)
Λ(H) = Λ0 +σ
2(4π)2M2(H2 − H2
0 ) RGE
Resolution of the system equation:
RGE + Friedmann Equation + Continuity Equation
Λ(z ; ν) = Λ0 + ρ0M
ν
1− ν
[(1 + z)3(1−ν) − 1
]− κ
1− 3ν
z (z + 2)
2+
ν
1− ν
[(1 + z)3(1−ν) − 1
]
κ ≡ −2 νΩ0K → proportional to curvature
ν ≡ σ12 π
M2
M2P→ cosmological index
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 3: Mi > µ, µ ∼ H(t)
Resolution of the system equation:
RGE + Friedmann Equation + Continuity Equation
Λ(z ; ν) = Λ0 + ρ0M
ν
1− ν
[(1 + z)3(1−ν) − 1
]− κ
1− 3ν
z (z + 2)
2+
ν
1− ν
[(1 + z)3(1−ν) − 1
]
κ ≡ −2 νΩ0K → proportional to curvature
ν ≡ σ12 π
M2
M2P→ cosmological index
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 3: Mi > µ, µ ∼ H(t)
Resolution of the system equation:
RGE + Friedmann Equation + Continuity Equation
Λ(z ; ν) = Λ0 + ρ0M
ν
1− ν
[(1 + z)3(1−ν) − 1
]− κ
1− 3ν
z (z + 2)
2+
ν
1− ν
[(1 + z)3(1−ν) − 1
]
+ κ ≡ −2 νΩ0K → proportional to curvature
ν ≡ σ12 π
M2
M2P→ cosmological index
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 3: Mi > µ, µ ∼ H(t)
Resolution of the system equation:
RGE + Friedmann Equation + Continuity Equation
Λ(z ; ν) = Λ0 + ρ0M
ν
1− ν
[(1 + z)3(1−ν) − 1
]− κ
1− 3ν
z (z + 2)
2+
ν
1− ν
[(1 + z)3(1−ν) − 1
]
κ ≡ −2 νΩ0K → proportional to curvature
+ ν ≡ σ12 π
M2
M2P→ cosmological index
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 3: Mi > µ, µ ∼ H(t)
But sometimes an image is better than words...
Where we assumed a flat universe with Ω0M = 0.3 and Ω0
Λ = 0.7.
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 3: Mi > µ, µ ∼ H(t)
But sometimes an image is better than words...
Where we assumed a flat universe with Ω0M = 0.3 and Ω0
Λ = 0.7.
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 3: Mi > µ, µ ∼ H(t)
All the functions describing the Universe in the standard CC
cosmology can be calculated here. For example:
The decelaration parameter, q
ν > 0 ⇒ acc. farther in time
ν < 0 ⇒ acc. closer in time
Again the plot corresponds to flat universe with Ω0M = 0.3 and Ω0
Λ = 0.7.
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 3: Mi > µ, µ ∼ H(t)
All the functions describing the Universe in the standard CC
cosmology can be calculated here. For example:
+ The decelaration parameter, q
ν > 0 ⇒ acc. farther in time
ν < 0 ⇒ acc. closer in time
Again the plot corresponds to flat universe with Ω0M = 0.3 and Ω0
Λ = 0.7.
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 3: Mi > µ, µ ∼ H(t)
All the functions describing the Universe in the standard CC
cosmology can be calculated here. For example:
The decelaration parameter, q
+ ν > 0 ⇒ acc. farther in time
+ ν < 0 ⇒ acc. closer in time
Again the plot corresponds to flat universe with Ω0M = 0.3 and Ω0
Λ = 0.7.
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Scenario 3Observational Constraints
1 free parameter, ν
Ω0M ν χ2
0.22+0.09−0.07 −0.5+0.3
−0.3 156.5
0.26+0.03−0.02 −0.3+0.2
−0.1 156.8
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 2: Mi > µ, µ ∼ ρ
1/4c (t)
A quick look into the other scenarios.Scenario 2
µ ∼ ρ1/4c (t)
Mi > µ, as in Scenario 3, SM particles active
Λ(µ) = Λ0 +1
(4π)2
"µ2 1
4
m2
H + 3m2Z + 6m2
W − 4X
i
Nim2i
!+ µ4
1
2
Xi
Ni −5
4
!#
µ2 term → huge evolution
I m2H = 4
∑i Nim
2i − 3m2
Z − 6m2W ≈ (550 GeV )2
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 2: Mi > µ, µ ∼ ρ
1/4c (t)
A quick look into the other scenarios.Scenario 2
+ µ ∼ ρ1/4c (t)
+ Mi > µ, as in Scenario 3, SM particles active
Λ(µ) = Λ0 +1
(4π)2
"µ2 1
4
m2
H + 3m2Z + 6m2
W − 4X
i
Nim2i
!+ µ4
1
2
Xi
Ni −5
4
!#
µ2 term → huge evolution
I m2H = 4
∑i Nim
2i − 3m2
Z − 6m2W ≈ (550 GeV )2
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 2: Mi > µ, µ ∼ ρ
1/4c (t)
A quick look into the other scenarios.Scenario 2
µ ∼ ρ1/4c (t)
Mi > µ, as in Scenario 3, SM particles active
Λ(µ) = Λ0 +1
(4π)2
"µ2 1
4
m2
H + 3m2Z + 6m2
W − 4X
i
Nim2i
!+ µ4
1
2
Xi
Ni −5
4
!#
µ2 term → huge evolution
I m2H = 4
∑i Nim
2i − 3m2
Z − 6m2W ≈ (550 GeV )2
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 2: Mi > µ, µ ∼ ρ
1/4c (t)
A quick look into the other scenarios.Scenario 2
µ ∼ ρ1/4c (t)
Mi > µ, as in Scenario 3, SM particles active
Λ(µ) = Λ0 +1
(4π)2
"µ2 1
4
m2
H + 3m2Z + 6m2
W − 4X
i
Nim2i
!+ µ4
1
2
Xi
Ni −5
4
!#
µ2 term → huge evolution
+ m2H = 4
∑i Nim
2i − 3m2
Z − 6m2W ≈ (550 GeV )2
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Scenario 2Observational Constraints
1 non-free parameter in SM
η ≡ 12
∑i Ni − 5
4= 10.75
Ω0M η χ2
0.35+0.04−0.04 +11+5
−55 158.6
0.35+0.04−0.04 −5+5
−5 158.6
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 1: mi < µ, µ ∼ ρ
1/4c (t)
Scenario 1
+ µ ∼ ρ1/4c (t), as in Scenario 2
+ mi < µ, only lightest neutrinos active
Λ(ρ) = Λ0 +1
(4π)2
(1
2m4
S − 4∑
ν
m4ν
)ln
ρ
ρ0RGE
Inclusion of the sterile neutrino to cover both signs ofevolution
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 1: mi < µ, µ ∼ ρ
1/4c (t)
Scenario 1
µ ∼ ρ1/4c (t), as in Scenario 2
mi < µ, only lightest neutrinos active
Λ(ρ) = Λ0 +1
(4π)2
(1
2m4
S − 4∑
ν
m4ν
)ln
ρ
ρ0RGE
Inclusion of the sterile neutrino to cover both signs ofevolution
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Cosmological scenariosScenario 1: mi < µ, µ ∼ ρ
1/4c (t)
Scenario 1
µ ∼ ρ1/4c (t), as in Scenario 2
mi < µ, only lightest neutrinos active
Λ(ρ) = Λ0 +1
(4π)2
(1
2m4
S − 4∑
ν
m4ν
)ln
ρ
ρ0RGE
+ Inclusion of the sterile neutrino to cover both signs ofevolution
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Scenario 1Observational Constraints
1 free (?) parameter
τ ≡ 12m4
S − 4∑
ν m4ν
Ω0M τ(10−9eV 4) χ2
0.20+0.10−0.08 −16+11
−12 156.5
0.27+0.02−0.04 −8+4
−5 157.1
mmaxν = 0.007± 0.006 eV
mmaxν = 0.006± 0.005 eV
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse approach
1 Motivacio
2 Introduction
3 Evolving Cosmological Constant
4 Non-Parametric ReconstructionsInverse ProblemInverse MethodResults
5 Future Perspectives
6 Summary and Conclusions
Inverse problemsThe concept
Theory Observations
Dark energy model
n χ2 fits
Inverse method
SNe Ia magnitudes
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Inverse problemsThe concept
TheoryForward problem
Observations
Dark energy model
n χ2 fits
Inverse method
SNe Ia magnitudes
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Inverse problemsThe concept
Theory
Inverse problem
Observations
Dark energy model
n χ2 fits
Inverse method
SNe Ia magnitudes
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Inverse problemsThe concept
Theory Observations
Dark energy model
n χ2 fits
Inverse method
SNe Ia magnitudes
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Inverse problemsInverse problem’s problems
Risks with inverse problems:
The solution does not necessary exist
The solution is not unique
The solution is not stable
One can minimize the difficulties by including a priori information
⇒ We use a probabilistic approach
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Inverse problemsInverse problem’s problems
Risks with inverse problems:
+ The solution does not necessary exist 7
The solution is not unique
The solution is not stable
One can minimize the difficulties by including a priori information
⇒ We use a probabilistic approach
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Inverse problemsInverse problem’s problems
Risks with inverse problems:
The solution does not necessary exist 7
+ The solution is not unique 4
The solution is not stable
One can minimize the difficulties by including a priori information
⇒ We use a probabilistic approach
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Inverse problemsInverse problem’s problems
Risks with inverse problems:
The solution does not necessary exist 7
The solution is not unique 4
+ The solution is not stable 4
One can minimize the difficulties by including a priori information
⇒ We use a probabilistic approach
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Inverse problemsInverse problem’s problems
Risks with inverse problems:
The solution does not necessary exist 7
The solution is not unique 4
The solution is not stable 4
One can minimize the difficulties by including a priori information
⇒ We use a probabilistic approach
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Inverse problemsThe method
Demands:
1 Inclusion of a priori information
2 Recover a function w(z) –or Λ(z)– instead ofparameterizations
Approach:
1 Through the Bayes theorem:
fpost(M|y) α L(y|M) fprior (M)
2 Working in functional spaces, were the functionals such asw(z) are defined in an infinite-dimensional space
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Inverse problemsThe method
Demands:
1 Inclusion of a priori information
2 Recover a function w(z) –or Λ(z)– instead ofparameterizations
Approach:
1 Through the Bayes theorem:
fpost(M|y) α L(y|M) fprior (M)
2 Working in functional spaces, were the functionals such asw(z) are defined in an infinite-dimensional space
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Inverse problemsThe method
Demands:
1 Inclusion of a priori information
2 Recover a function w(z) –or Λ(z)– instead ofparameterizations
Approach:
1 Through the Bayes theorem:
fpost(M|y) α L(y|M) fprior (M)
2 Working in functional spaces, were the functionals such asw(z) are defined in an infinite-dimensional space
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodMisfit function
Posterior distribution probability:
fpost(M|y) α exp [−S ]
S ≡ 12
(y− yth(M)
)∗C−1
y
(y− yth(M)
)
+ 12 (M−M0)
∗C−10 (M−M0)
Data y, covariance Cy , unknowns M
χ2 but... ∗ adjoint operator, scalar product in n-D space...
Information for the unknowns M: priors M0, C0
⇒ Data and unknowns treated at the same level
Goal: S minimization
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodMisfit function
Posterior distribution probability:
fpost(M|y) α exp [−S ]
S ≡ 12
(y− yth(M)
)∗C−1
y
(y− yth(M)
)
+ 12 (M−M0)
∗C−10 (M−M0)
+ Data y, covariance Cy , unknowns M
+ χ2 but... ∗ adjoint operator, scalar product in n-D space...
Information for the unknowns M: priors M0, C0
⇒ Data and unknowns treated at the same level
Goal: S minimization
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodMisfit function
Posterior distribution probability:
fpost(M|y) α exp [−S ]
S ≡ 12
(y− yth(M)
)∗C−1
y
(y− yth(M)
)+ 1
2 (M−M0)∗C−1
0 (M−M0)
Data y, covariance Cy , unknowns M
χ2 but... ∗ adjoint operator, scalar product in n-D space...
+ Information for the unknowns M: priors M0, C0
⇒ Data and unknowns treated at the same level
Goal: S minimization
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodMisfit function
Posterior distribution probability:
fpost(M|y) α exp [−S ]
S ≡ 12
(y− yth(M)
)∗C−1
y
(y− yth(M)
)+ 1
2 (M−M0)∗C−1
0 (M−M0)
Data y, covariance Cy , unknowns M
χ2 but... ∗ adjoint operator, scalar product in n-D space...
Information for the unknowns M: priors M0, C0
⇒ Data and unknowns treated at the same level
Goal: S minimization
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodMinimization
Minimization steps:
1 The minimization is done in the functional space using aNewton minimization method
2 A final functional equation for w(z) is obtained
3 At this point operators are discretized and concrete valuescalculated
Results:
1 Quite a long equation for w(z):
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodMinimization
Minimization steps:
1 The minimization is done in the functional space using aNewton minimization method
2 A final functional equation for w(z) is obtained
3 At this point operators are discretized and concrete valuescalculated
Results:
1 Quite a long equation for w(z):
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodMinimization
w[k+1](z) = w0(z) +NX
i=1
Wi [k]
Z zi
0Cw (z, z ′)gw [k](z
′)dz ′ ,
where Wi [k] =PN
j=1
“S−1
[k]
”i,j
Vj[k] ,
Vi = yi +∂y th
i
∂Ω0M
(Ω0M − Ω0
M0) +
∂y thi
∂w(z)· (w − w0)− y th
i (zi , Ω0M , w(z)) ,
Si,j = δi,jσiσj +∂y th
i
∂Ω0M
CΩ0M
∂y thj
∂Ω0M
+∂y th
i
∂w(z)·
Cw ·∂y th
j
∂w(z)
!.
And uncertainty:
σw(z)(z) =q
Cw(z)(z) =
vuutσ2w(z)
−Xi,j
Cw ·∂y th
i
∂w(z)(S−1)i,j
∂y thj
∂w(z)· Cw
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodMinimization
Results:
1 An equation for w(z) and the remaining unknowns
2 An estimation for the σ uncertainty
3 The resolving kernel K (z , zi ), a function at every zi indicatinghow well resolved it is
Comment:
Notice that both depend on the priorsI Monte Carlo exploration of the space of solutions
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodMinimization
Results:
1 An equation for w(z) and the remaining unknowns
2 An estimation for the σ uncertainty
3 The resolving kernel K (z , zi ), a function at every zi indicatinghow well resolved it is
Comment:
+ Notice that both depend on the priors
+ Monte Carlo exploration of the space of solutions
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodResults
For the dark energy equation of state w(z):
Riess et al. (2006) data
Prior: Ω0M = 0.27± 0.03
Prior: w(z)0 = −1± 0.5
Differs from Λ at morethan 1σ
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodResults
For the dark energy equation of state w(z):
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodResults
Obtaining confidence regions via a Monte Carlo exploration:
As before but:
1000 reconstructions
Explored range−3 < w(z) < 1
Still differs from Λ atmore than 1σ
Wider, more reliable,confidence regions
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
The inverse methodResults
One can do the same for the Cosmological Constant Λ(z)
1000 reconstructions
Explored range0.53 < ΩΛ(z) < 0.93
A constant CC valid atthe 1σ limit
Valid for a general ΩX (z)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
What the future holds in store
1 Motivacio
2 Introduction
3 Evolving Cosmological Constant
4 Non-Parametric Reconstructions
5 Future PerspectivesOncoming SurveysNon-Parametric ReconstructionsEvolving Cosmological Constant
6 Summary and Conclusions
Future PerspectivesOncoming surveys
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesOncoming surveys
Space observatory
2,000 SNe/year
0.1 < z < 1.7
Spectroscopic redshifts
Vs.
Ground telescope
250,000 SNe/year (wide survey)
10,000 SNe (deep survey)
0 < z < 0.9 (wide survey)
0 < z < 1.4 (deep survey)
Photometric redshifts
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesOncoming surveys
Space observatory
2,000 SNe/year
0.1 < z < 1.7
Spectroscopic redshifts
Vs.
Ground telescope
250,000 SNe/year (wide survey)
10,000 SNe (deep survey)
0 < z < 0.9 (wide survey)
0 < z < 1.4 (deep survey)
Photometric redshifts
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesOncoming surveys
Space observatory
2,000 SNe/year
0.1 < z < 1.7
Spectroscopic redshifts
Vs.
Ground telescope
250,000 SNe/year (wide survey)
10,000 SNe (deep survey)
0 < z < 0.9 (wide survey)
0 < z < 1.4 (deep survey)
Photometric redshifts
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesOncoming surveys
Space observatory
2,000 SNe/year
0.1 < z < 1.7
Spectroscopic redshifts
Vs.
Ground telescope
250,000 SNe/year (wide survey)
10,000 SNe (deep survey)
0 < z < 0.9 (wide survey)
0 < z < 1.4 (deep survey)
Photometric redshifts
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesOncoming surveys
Uncertaintiesin the data sets
LSST deep: σintr = 0.15, δz = 0.01 and σsys = 0.02
SNAP: σintr = 0.15, δz = 0.00 and σsys = 0.02 z/1.7
(+ SNFactory as low redshift anchor)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesOncoming surveys
Uncertaintiesin the data sets
LSST deep: σintr = 0.15, δz = 0.01 and σsys = 0.02
SNAP: σintr = 0.15, δz = 0.00 and σsys = 0.02 z/1.7
(+ SNFactory as low redshift anchor)
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesNon-parametric reconstructions
Flashback:Reconstruction for the dark energy equation of state w(z):
Riess et al. (2006) data
1000 reconstructions
Explored range−3 < w(z) < 1
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesNon-parametric reconstructions
For the dark energy equation of state w(z) with:
LSST SNAP
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesEvolving cosmological constant
Scenario 1 (mi < µ, µ ∼ ρ1/4c ) τ ≡ 1
2m4S − 4
∑ν m4
ν
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesEvolving cosmological constant
Scenario 1 (mi < µ, µ ∼ ρ1/4c ) τ ≡ 1
2m4S − 4
∑ν m4
ν
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesEvolving cosmological constant
Scenario 2 (Mi > µ, µ ∼ ρ1/4c ) η ≡ 1
2
∑i Ni − 5
4 = 10.75
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesEvolving cosmological constant
Scenario 2 (Mi > µ, µ ∼ ρ1/4c ) η ≡ 1
2
∑i Ni − 5
4 = 10.75
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesEvolving cosmological constant
Scenario 3 (Mi > µ, µ ∼ H) ν ≡ σ12 π
M2
M2P
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Future PerspectivesEvolving cosmological constant
Scenario 3 (Mi > µ, µ ∼ H) ν ≡ σ12 π
M2
M2P
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
In summary
1 Motivacio
2 Introduction
3 Evolving Cosmological Constant
4 Non-Parametric Reconstructions
5 Future Perspectives
6 Summary and Conclusions
Summary and conclusions
The problem
+ The Universe seems to be in accelerated expansion
Characterization of a dark energy source as the cause ofacceleration
Nowadays, it is a degenerated problem
Evolving Cosmological Constant
We motivate an evolving CC as a consequence of therenormalization effects in a Quantum Field Theory (QFT)
The running depends on the renormalization scale and theactive dofs
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
The problem
The Universe seems to be in accelerated expansion
+ Characterization of a dark energy source as the cause ofacceleration
Nowadays, it is a degenerated problem
Evolving Cosmological Constant
We motivate an evolving CC as a consequence of therenormalization effects in a Quantum Field Theory (QFT)
The running depends on the renormalization scale and theactive dofs
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
The problem
The Universe seems to be in accelerated expansion
Characterization of a dark energy source as the cause ofacceleration
+ Nowadays, it is a degenerated problem
Evolving Cosmological Constant
We motivate an evolving CC as a consequence of therenormalization effects in a Quantum Field Theory (QFT)
The running depends on the renormalization scale and theactive dofs
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
The problem
The Universe seems to be in accelerated expansion
Characterization of a dark energy source as the cause ofacceleration
Nowadays, it is a degenerated problem
Evolving Cosmological Constant
+ We motivate an evolving CC as a consequence of therenormalization effects in a Quantum Field Theory (QFT)
The running depends on the renormalization scale and theactive dofs
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
The problem
The Universe seems to be in accelerated expansion
Characterization of a dark energy source as the cause ofacceleration
Nowadays, it is a degenerated problem
Evolving Cosmological Constant
We motivate an evolving CC as a consequence of therenormalization effects in a Quantum Field Theory (QFT)
+ The running depends on the renormalization scale and theactive dofs
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
Evolving Cosmological Constant (continued)
+ In our approach (Scenario 3), particles with M ∼ MPl areresponsible for the running
This evolution affects the standard cosmological equations
Running must be small in order to be compatible withstructure formation and CMB
Such an small evolution is difficult to detect by observations,although it is mandatory within a QFT
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
Evolving Cosmological Constant (continued)
In our approach (Scenario 3), particles with M ∼ MPl areresponsible for the running
+ This evolution affects the standard cosmological equations
Running must be small in order to be compatible withstructure formation and CMB
Such an small evolution is difficult to detect by observations,although it is mandatory within a QFT
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
Evolving Cosmological Constant (continued)
In our approach (Scenario 3), particles with M ∼ MPl areresponsible for the running
This evolution affects the standard cosmological equations
+ Running must be small in order to be compatible withstructure formation and CMB
Such an small evolution is difficult to detect by observations,although it is mandatory within a QFT
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
Evolving Cosmological Constant (continued)
In our approach (Scenario 3), particles with M ∼ MPl areresponsible for the running
This evolution affects the standard cosmological equations
Running must be small in order to be compatible withstructure formation and CMB
+ Such an small evolution is difficult to detect by observations,although it is mandatory within a QFT
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
Non-parametric Reconstructions
+ We apply an inverse approach to estimate w(z) and Λ(z) in anon-parametric way
We introduce a priori information to regularize the inversion
Results depend on priors, but we include Monte Carloexplorations to overcome this limitation
Current data can already rule out a constant dark energysource at low redshift at 1σ level
Future surveys such as SNAP or LSST will confirm that pointup to redshift one
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
Non-parametric Reconstructions
We apply an inverse approach to estimate w(z) and Λ(z) in anon-parametric way
+ We introduce a priori information to regularize the inversion
Results depend on priors, but we include Monte Carloexplorations to overcome this limitation
Current data can already rule out a constant dark energysource at low redshift at 1σ level
Future surveys such as SNAP or LSST will confirm that pointup to redshift one
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
Non-parametric Reconstructions
We apply an inverse approach to estimate w(z) and Λ(z) in anon-parametric way
We introduce a priori information to regularize the inversion
+ Results depend on priors, but we include Monte Carloexplorations to overcome this limitation
Current data can already rule out a constant dark energysource at low redshift at 1σ level
Future surveys such as SNAP or LSST will confirm that pointup to redshift one
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
Non-parametric Reconstructions
We apply an inverse approach to estimate w(z) and Λ(z) in anon-parametric way
We introduce a priori information to regularize the inversion
Results depend on priors, but we include Monte Carloexplorations to overcome this limitation
+ Current data can already rule out a constant dark energysource at low redshift at 1σ level
Future surveys such as SNAP or LSST will confirm that pointup to redshift one
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Summary and conclusions
Non-parametric Reconstructions
We apply an inverse approach to estimate w(z) and Λ(z) in anon-parametric way
We introduce a priori information to regularize the inversion
Results depend on priors, but we include Monte Carloexplorations to overcome this limitation
Current data can already rule out a constant dark energysource at low redshift at 1σ level
+ Future surveys such as SNAP or LSST will confirm that pointup to redshift one
Cristina Espana i Bonet Exploring the Evolution of Dark Energy and its Equation of State
Exploring the Evolution of Dark Energy
and its Equation of State
Cristina Espana i Bonet
Universitat de Barcelona
12th February, 2008
Advisor: Dra. Pilar Ruiz-Lapuente