Exploring the Evolution of Dark Energy and its Equation of State

Transcription

1 Exploring the Evolution of Dark Energy and its Equation of State Advisor: Dra. Pilar Ruiz-Lapuente Universitat de Barcelona 12th February, 2008

2 Overview 1 Motivació 2 Introduction 3 Evolving Cosmological Constant 4 Non-Parametric Reconstructions 5 Future Perspectives 6 Summary and Conclusions

3 L Univers Si l Univers fos estàtic...

4 L Univers...però està en expansió (model de Big Bang)

5 L Univers...i accelerada

6 Com ho observem? Necessitem un objecte de lluminositat coneguda La llum que rebem ens indica la distància Candela estàndard

7 Com ho observem? Necessitem un objecte de lluminositat coneguda La llum que rebem ens indica la distància Candela estàndard

8 Com ho observem? Necessitem un objecte de lluminositat coneguda La llum que rebem ens indica la distància Candela estàndard Candela estàndard per exel lència: Supernoves del tipus Ia (SNe Ia)

9 Com ho observem? Sistema binari en acreció Explosió a M Ch Mateixa lluminositat

10 Com ho observem? Sistema binari en acreció Explosió a M Ch Mateixa lluminositat Lluminositat inferior a L d Lluminositat superior a L d

11 Com ho observem? Sistema binari en acreció Explosió a M Ch Mateixa lluminositat Lluminositat inferior a L d més lluny Lluminositat superior a L d més a prop

12 Com ho observem? Sistema binari en acreció Explosió a M Ch Mateixa lluminositat Lluminositat inferior a L d més lluny expansió més ràpida Lluminositat superior a L d més a prop expansió més lenta

13 Què observem? Supernoves llunyanes: coherent amb BB estàndard Supernoves meitat de l edat de l Univers: sublluminoses Supernoves properes: canvis imperceptibles

14 Què observem? Supernoves llunyanes: coherent amb BB estàndard Supernoves meitat de l edat de l Univers: sublluminoses Supernoves properes: canvis imperceptibles

15 Què observem? Supernoves llunyanes: coherent amb BB estàndard Supernoves meitat de l edat de l Univers: sublluminoses Supernoves properes: canvis imperceptibles

16 Què observem? Supernoves llunyanes: coherent amb BB estàndard Supernoves meitat de l edat de l Univers: sublluminoses Supernoves properes: canvis imperceptibles Expansió recent accelerada Per què? i Perquè?

17 Objectius Estudi i caracterització del component que accelera l expansió, l energia fosca, mitjançant: 1 Aproximació directa: Construcció d un model i contrast amb observacions 2 Aproximació inversa: Reconstrucció del model subjacent a partir de les observacions

18 Objectius Estudi i caracterització del component que accelera l expansió, l energia fosca, mitjançant: 1 Aproximació directa: Construcció d un model i contrast amb observacions 2 Aproximació inversa: Reconstrucció del model subjacent a partir de les observacions

19 Objectius Estudi i caracterització del component que accelera l expansió, l energia fosca, mitjançant: 1 Aproximació directa: Construcció d un model i contrast amb observacions 2 Aproximació inversa: Reconstrucció del model subjacent a partir de les observacions Eina principal: 1 Variació de la magnitud de les SNe Ia amb la distància

20 Let s start 1 Motivació 2 Introduction Cosmological Constant General Dark Energy Source Obtaining Information with Observations 3 Evolving Cosmological Constant 4 Non-Parametric Reconstructions 5 Future Perspectives 6 Summary and Conclusions

21 Introduction Why this thesis?

22 Introduction Why this thesis? Clear evidence of the acceleration Attribution to a cosmological constant (CC): Λ GeV 4

23 Introduction Why this thesis? Clear evidence of the acceleration Attribution to a cosmological constant (CC)......but other candidates...possible evolution

24 Introduction Why this thesis? Clear evidence of the acceleration Attribution to a cosmological constant (CC)......but other candidates...possible evolution What is the CC?

25 Cosmological constant (CC) The Einstein s term Interpretation: antigravitational force Inclusion of the CC in Einstein s Field Equations Nothing forbids λ(t) G µν λ(t)g µν = 8πG N T µν. A. Einstein, Kosmologische betrachtungen zur allgemeinen Relativitätstheorie. S.-B. Preuss. Akad. Wiss. (1917), pp

26 Cosmological constant (CC) The Einstein s term Interpretation: antigravitational force Inclusion of the CC in Einstein s Field Equations Nothing forbids λ(t) G µν λ(t)g µν = 8πG N T µν. A. Einstein, Kosmologische betrachtungen zur allgemeinen Relativitätstheorie. S.-B. Preuss. Akad. Wiss. (1917), pp

27 Cosmological constant (CC) Vacuum energy Interpretation: vacuum energy Inclusion of the CC as a source of T µν : G µν = 8πG N (T µν + Λg µν ). In the vacuum: T µν = G µν V φ Particles & fields Vφ = 0 but Higgs VH m 4 H 108 GeV 4 = Λ ind 10 8 GeV 4

28 Cosmological constant (CC) Vacuum energy Interpretation: vacuum energy Inclusion of the CC as a source of T µν : G µν = 8πG N (T µν + Λg µν ). In the vacuum: T µν = G µν V φ Particles & fields Vφ = 0 but Higgs VH m 4 H 108 GeV 4 = Λ ind 10 8 GeV 4

29 Cosmological constant (CC) Vacuum energy Interpretation: vacuum energy Inclusion of the CC as a source of T µν : G µν = 8πG N (T µν + Λg µν ). In the vacuum: T µν = G µν V φ Particles & fields Vφ = 0 but Higgs VH m 4 H 108 GeV 4 = Λ ind 10 8 GeV 4

30 Cosmological constant (CC) An only source? Observations Λ obs GeV 4 Theory Λ ind 10 8 GeV 4 Λ Einstein

31 Cosmological constant (CC) An only source? Observations Λ obs GeV 4 Theory Λ ind 10 8 GeV 4 }{{} Λ Einstein Λ obs = Λ Einstein + Λ ind

32 Cosmological constant (CC) An only source? Observations Λ obs GeV 4 Theory Λ ind 10 8 GeV 4 }{{} Λ Einstein Λ obs = Λ Einstein + Λ ind Λ Einstein Λ ind = GeV 4 = !! Cosmological Constant Problem

33 A general dark energy component The perfect fluid approach Alternative: Inclusion of a perfect fluid extra component characterized by its Equation of State (EoS) p = wρ Interpretation of the cosmological constant as an extra component of T µν = 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

34 A general dark energy component The perfect fluid approach Alternative: Inclusion of a perfect fluid extra component characterized by its Equation of State (EoS) p = wρ Interpretation of the cosmological constant as an extra component of T µν = 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

35 A general dark energy component The perfect fluid approach Alternative: Inclusion of a perfect fluid extra component characterized by its Equation of State (EoS) p = wρ Interpretation of the cosmological constant as an extra component of T µν = 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

36 A general dark energy component The perfect fluid approach Alternative: Inclusion of a perfect fluid extra component characterized by its Equation of State (EoS) p = wρ Interpretation of the cosmological constant as an extra component of T µν = 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

37 A general dark energy component Alternatives Every component is described by its equation of state: w R = 1/3 radiation w M = 0 non-relativistic matter w S = 1/3 cosmic strings w W = 2/3 domain walls w T = 1/3 textures w Λ = 1 (evolving) cosmological constant w Q (t) > 1 (dw Q /dz > 0) quintessence w K (t) > 1 (dw K /dz < 0) k-essence w Ph (t) < 1 phantoms

38 A general dark energy component Alternatives Every component is described by its equation of state: w R = 1/3 radiation w M = 0 non-relativistic matter w S = 1/3 cosmic strings w W = 2/3 domain walls w T = 1/3 textures w Λ = 1 (evolving) cosmological constant w Q (t) > 1 (dw Q /dz > 0) quintessence w K (t) > 1 (dw K /dz < 0) k-essence w Ph (t) < 1 phantoms

39 A general dark energy component Alternatives Every component is described by its equation of state: w R = 1/3 radiation w M = 0 non-relativistic matter w S = 1/3 cosmic strings w W = 2/3 domain walls w T = 1/3 textures w Λ = 1 (evolving) cosmological constant w Q (t) > 1 (dw Q /dz > 0) quintessence w K (t) > 1 (dw K /dz < 0) k-essence w Ph (t) < 1 phantoms

40 A general dark energy component Alternatives Every component is described by its equation of state: w R = 1/3 radiation w M = 0 non-relativistic matter w S = 1/3 cosmic strings w W = 2/3 domain walls w T = 1/3 textures w Λ = 1 (evolving) cosmological constant w Q (t) > 1 (dw Q /dz > 0) quintessence w K (t) > 1 (dw K /dz < 0) k-essence w Ph (t) < 1 phantoms Multitude of possibilities for dark energy...

41 A general dark energy component Alternatives EoS index: w R = 1/3 w M = 0 w S = 1/3 w W = 2/3 w T = 1/3 w Λ = 1 w Q (t) > 1 (dw Q /dz > 0) = w K (t) > 1 (dw K /dz < 0) w Ph (t) < 1 Quintessence EoS Weller & Albrecht (2002)

42 Obtaining information with SNe Ia Magnitude-redshift relation The relation between models and observations is encoded in: m(z, H 0, Ω 0 M, Ω0 X ) = M + 5 log 10 ˆH0 d L (z, H 0, Ω 0 M, ) Ω0 X 0 Z d L (z, Ω 0 M, c (1 + z) z B Ω0 X ) H 0 0 Z z Ω X (z) = Ω 0 X 3 exp 0 1 dz C q A Ω 0 M (1 + z)3 + Ω X (z) dz 1 + «w(z ) 1 + z Double integral relating the observed m with w(z) Smoothing of any possible evolution Increasing of degeneracy

43 Obtaining information with SNe Ia Magnitude-redshift relation The relation between models and observations is encoded in: m(z, H 0, Ω 0 M, Ω0 X ) = M + 5 log 10 ˆH0 d L (z, H 0, Ω 0 M, ) Ω0 X 0 Z d L (z, Ω 0 M, c (1 + z) z B Ω0 X ) H 0 0 Z z Ω X (z) = Ω 0 X 3 exp 0 1 dz C q A Ω 0 M (1 + z)3 + Ω X (z) dz 1 + «w(z ) 1 + z Double integral relating the observed m with w(z) Smoothing of any possible evolution Increasing of degeneracy

44 Obtaining information with SNe Ia Magnitude-redshift relation The relation between models and observations is encoded in: m(z, H 0, Ω 0 M, Ω0 X ) = M + 5 log 10 ˆH0 d L (z, H 0, Ω 0 M, ) Ω0 X 0 Z d L (z, Ω 0 M, c (1 + z) z B Ω0 X ) H 0 0 Z z Ω X (z) = Ω 0 X 3 exp 0 1 dz C q A Ω 0 M (1 + z)3 + Ω X (z) dz 1 + «w(z ) 1 + z Double integral relating the observed m with w(z) Smoothing of any possible evolution Increasing of degeneracy

45 Obtaining information with SNe Ia m z space Well covered data range: 0 < z < 1.5 Most models can be fit to data Small differences among models in the range

46 Obtaining information with SNe Ia m z space Well covered data range: 0 < z < 1.5 Most models can be fit to data Small differences among models in the range That s a hard work!

47 Obtaining information with SNe Ia Data sets Currently, 2 (+1) data sets available: Riess et al. (2006) 182 SNe Ia < z < 1.77 Wood-Vasey et al. (2007) 162 SNe Ia < z < 0.96 Davis et al. (2007) 192 SNe Ia < z < 1.77

48 Obtaining information with SNe Ia Data sets Currently, 2 (+1) data sets available: Riess et al. (2006) 182 SNe Ia < z < 1.77 Wood-Vasey et al. (2007) 162 SNe Ia < z < 0.96 Davis et al. (2007) 192 SNe Ia < z < 1.77 Results do depend on the data set used. In the following, results for Riess et al. (2006) are shown.

49 Obtaining information with SNe Ia Complementary probes and priors Constraint on the curvature: Flat Universe (WMAP3) Constraint on Ω M : 0.27 ± 0.03 (clusters) BAO constraints

50 Obtaining information with SNe Ia Complementary probes and priors Constraint on the curvature: Flat Universe (WMAP3) Constraint on Ω M : 0.27 ± 0.03 (clusters) BAO constraints

51 Obtaining information with SNe Ia Complementary probes and priors Constraint on the curvature: Flat Universe (WMAP3) Constraint on Ω M : 0.27 ± 0.03 (clusters) BAO constraints

52 Obtaining information with SNe Ia Complementary probes and priors Constraint on the curvature: Flat Universe (WMAP3) Constraint on Ω M : 0.27 ± 0.03 (clusters) BAO constraints

53 Obtaining information with SNe Ia Complementary probes and priors Constraint on the curvature: Flat Universe (WMAP3) Constraint on Ω M : 0.27 ± 0.03 (clusters) BAO constraints

54 The dark energy model 1 Motivació 2 Introduction 3 Evolving Cosmological Constant Fundamentals Cosmological Scenarios Observational Constraints 4 Non-Parametric Reconstructions 5 Future Perspectives 6 Summary and Conclusions

55 Evolving Cosmological Constant The theory behind... in a few words Quantum Field Theory (semiclassical approximation: fields in a curved space-time)

56 Evolving Cosmological Constant The theory behind... in a few words Quantum Field Theory (semiclassical approximation: fields in a curved space-time) Vacuum action at low energies Z S HE = d 4 x g «1 R + Λ vac 16πG vac

57 Evolving Cosmological Constant The theory behind... in a few words Quantum Field Theory (semiclassical approximation: fields in a curved space-time) Vacuum action at low energies Z S HE = Ultraviolet divergencies d 4 x g «1 R + Λ vac 16πG vac regularization, renormalization

58 Evolving Cosmological Constant The theory behind... in a few words Quantum Field Theory (semiclassical approximation: fields in a curved space-time) Vacuum action at low energies Z S HE = Ultraviolet divergencies d 4 x g «1 R + Λ vac 16πG vac regularization, renormalization Scale invariance broken

59 Evolving Cosmological Constant The theory behind... in a few words Quantum Field Theory (semiclassical approximation: fields in a curved space-time) Vacuum action at low energies Z S HE = Ultraviolet divergencies d 4 x g «1 R + Λ vac 16πG vac regularization, renormalization Scale invariance broken Renormalization Group Equations (RGE)

60 Evolving Cosmological Constant β-function, RGE for the Cosmological Constant For the CC, the dependence on the scale is encoded in the β-function 0 µ d «Λ β Λ = X d ln µ 8πG (4π) 2 i A i m 4 i + µ 2 X j B j M 2 j + µ 4 X j 1 C j +... A with: Dependence on the renormalization scale µ How is decoupling produced? Which are the active dofs? Light particles?

61 Evolving Cosmological Constant β-function, RGE for the Cosmological Constant For the CC, the dependence on the scale is encoded in the β-function 0 µ d «Λ β Λ = X d ln µ 8πG (4π) 2 i A i m 4 i + µ 2 X j B j M 2 j + µ 4 X j 1 C j +... A with: Dependence on the renormalization scale µ How is decoupling produced? Which are the active dofs? Light particles?

62 Evolving Cosmological Constant β-function, RGE for the Cosmological Constant For the CC, the dependence on the scale is encoded in the β-function 0 µ d «Λ β Λ = X d ln µ 8πG (4π) 2 i A i m 4 i + µ 2 X j B j M 2 j + µ 4 X j 1 C j +... A with: Dependence on the renormalization scale µ How is decoupling produced? Which are the active dofs? Light particles?

63 Evolving Cosmological Constant β-function, RGE for the Cosmological Constant For the CC, the dependence on the scale is encoded in the β-function 0 µ d «Λ β Λ = X d ln µ 8πG (4π) 2 i A i m 4 i + µ 2 X j B j M 2 j + µ 4 X j 1 C j +... A with: Dependence on the renormalization scale µ How is decoupling produced? Which are the active dofs? Light particles?

64 Evolving Cosmological Constant β-function, RGE for the Cosmological Constant For the CC, the dependence on the scale is encoded in the β-function 0 µ d «Λ β Λ = X d ln µ 8πG (4π) 2 i A i m 4 i + µ 2 X j B j M 2 j + µ 4 X j 1 C j +... A with: Dependence on the renormalization scale µ How is decoupling produced? Which are the active dofs? Light particles? Heavy particles with soft decoupling?

65 Evolving Cosmological Constant Cosmological scenarios There are several choices in the literature. We contribute with one (Scenario 3) and test three of them: Active dof Particles µ Scenario 1 m i < µ neutrinos ρ 1/4 c (t) Scenario 2 M i > µ SM ρ 1/4 c (t) Scenario 3 M i > µ Plank H(t)

66 Evolving Cosmological Constant Cosmological scenarios There are several choices in the literature. We contribute with one (Scenario 3) and test three of them: Active dof Particles µ Scenario 1 m i < µ neutrinos ρ 1/4 c (t) Scenario 2 M i > µ SM ρ 1/4 c (t) Scenario 3 M i > µ Plank H(t)

67 Evolving Cosmological Constant Cosmological scenarios There are several choices in the literature. We contribute with one (Scenario 3) and test three of them: Active dof Particles µ Scenario 1 m i < µ neutrinos ρ 1/4 c (t) Scenario 2 M i > µ SM ρ 1/4 c (t) Scenario 3 M i > µ Plank H(t)

68 Evolving Cosmological Constant Cosmological scenarios There are several choices in the literature. We contribute with one (Scenario 3) and test three of them: Active dof Particles µ Scenario 1 m i < µ neutrinos ρ 1/4 c (t) Scenario 2 M i > µ SM ρ 1/4 c (t) Scenario 3 M i > µ Plank H(t)

69 Cosmological scenarios Scenario 3: M i > µ, µ H(t) Λ(H) = Λ 0 + σ 2(4π) 2 M2 (H 2 H 2 0 ) RGE

70 Cosmological scenarios Scenario 3: M i > µ, µ H(t) Λ(H) = Λ 0 + σ 2(4π) 2 M2 (H 2 H 2 0 ) RGE Resolution of the system equation: { RGE + Friedmann Equation + Continuity Equation }

71 Cosmological scenarios Scenario 3: M i > µ, µ H(t) Resolution of the system equation: { RGE + Friedmann Equation + Continuity Equation } Λ(z; ν) = Λ 0 + ρ 0 ν M 1 ν κ { z (z + 2) 1 3ν 2 [ ] (1 + z) 3(1 ν) 1 + ν 1 ν [ ] } (1 + z) 3(1 ν) 1

72 Cosmological scenarios Scenario 3: M i > µ, µ H(t) Resolution of the system equation: { RGE + Friedmann Equation + Continuity Equation } Λ(z; ν) = Λ 0 + ρ 0 ν M 1 ν κ { z (z + 2) 1 3ν 2 [ ] (1 + z) 3(1 ν) 1 + ν 1 ν [ ] } (1 + z) 3(1 ν) 1 κ 2 νω 0 K ν σ 12 π M 2 M 2 P proportional to curvature cosmological index

73 Cosmological scenarios Scenario 3: M i > µ, µ H(t) Resolution of the system equation: { RGE + Friedmann Equation + Continuity Equation } Λ(z; ν) = Λ 0 + ρ 0 ν M 1 ν κ { z (z + 2) 1 3ν 2 [ ] (1 + z) 3(1 ν) 1 + ν 1 ν [ ] } (1 + z) 3(1 ν) 1 κ 2 νω 0 K ν σ 12 π M 2 M 2 P proportional to curvature cosmological index

74 Cosmological scenarios Scenario 3: M i > µ, µ H(t) But sometimes an image is better than words... Where we assumed a flat universe with Ω 0 M = 0.3 and Ω0 Λ = 0.7.

75 Cosmological scenarios Scenario 3: M i > µ, µ H(t) But sometimes an image is better than words... Where we assumed a flat universe with Ω 0 M = 0.3 and Ω0 Λ = 0.7.

76 Cosmological scenarios Scenario 3: M i > 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

77 Cosmological scenarios Scenario 3: M i > 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 Ω 0 M = 0.3 and Ω0 Λ = 0.7.

78 Cosmological scenarios Scenario 3: M i > 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 Ω 0 M = 0.3 and Ω0 Λ = 0.7.

79 Scenario 3 Observational Constraints 1 free parameter, ν Ω 0 M ν χ

80 Cosmological scenarios Scenario 2: M i > µ, µ ρ 1/4 c (t) µ ρ 1/4 c (t) A quick look into the other scenarios. Scenario 2 M i > µ, as in Scenario 3, SM particles active µ 2 term huge evolution m 2 H = 4 i N im 2 i 3m 2 Z 6m2 W (550 GeV )2

81 Cosmological scenarios Scenario 2: M i > µ, µ ρ 1/4 c (t) µ ρ 1/4 c (t) A quick look into the other scenarios. Scenario 2 M i > µ, as in Scenario 3, SM particles active µ 2 term huge evolution m 2 H = 4 i N im 2 i 3m 2 Z 6m2 W (550 GeV )2

82 Cosmological scenarios Scenario 2: M i > µ, µ ρ 1/4 c (t) µ ρ 1/4 c (t) A quick look into the other scenarios. Scenario 2 M i > µ, as in Scenario 3, SM particles active " Λ(µ) = Λ (4π) 2 µ m 2 H + 3m2 Z + 6m2 W 4 X i N i m 2 i! + µ 4 1 2!# X N i 5 4 i µ 2 term huge evolution m 2 H = 4 i N im 2 i 3m 2 Z 6m2 W (550 GeV )2

83 Cosmological scenarios Scenario 2: M i > µ, µ ρ 1/4 c (t) µ ρ 1/4 c (t) A quick look into the other scenarios. Scenario 2 M i > µ, as in Scenario 3, SM particles active " Λ(µ) = Λ (4π) 2 µ m 2 H + 3m2 Z + 6m2 W 4 X i N i m 2 i! + µ 4 1 2!# X N i 5 4 i µ 2 term huge evolution m 2 H = 4 i N im 2 i 3m 2 Z 6m2 W (550 GeV )2

84 Scenario 2 Observational Constraints 1 non-free parameter in SM η 1 2 i N i 5 = Ω 0 M η χ

85 Cosmological scenarios Scenario 1: m i < µ, µ ρ 1/4 c (t) Scenario 1 µ ρ 1/4 c (t), as in Scenario 2 m i < µ, only lightest neutrinos active Inclusion of the sterile neutrino to cover both signs of evolution

86 Cosmological scenarios Scenario 1: m i < µ, µ ρ 1/4 c (t) Scenario 1 µ ρ 1/4 c (t), as in Scenario 2 m i < µ, only lightest neutrinos active ( Λ(ρ) = Λ (4π) 2 2 m4 S 4 ν m 4 ν ) ln ρ ρ 0 RGE Inclusion of the sterile neutrino to cover both signs of evolution

87 Cosmological scenarios Scenario 1: m i < µ, µ ρ 1/4 c (t) Scenario 1 µ ρ 1/4 c (t), as in Scenario 2 m i < µ, only lightest neutrinos active ( Λ(ρ) = Λ (4π) 2 2 m4 S 4 ν m 4 ν ) ln ρ ρ 0 RGE Inclusion of the sterile neutrino to cover both signs of evolution

88 Scenario 1 Observational Constraints 1 free (?) parameter τ 1 2 m4 S 4 ν m4 ν Ω 0 M τ(10 9 ev 4 ) χ mν max mν max = ± ev = ± ev

89 The inverse approach 1 Motivació 2 Introduction 3 Evolving Cosmological Constant 4 Non-Parametric Reconstructions Inverse Problem Inverse Method Results 5 Future Perspectives 6 Summary and Conclusions

90 Inverse problems The concept Theory Observations

91 Inverse problems The concept Theory Forward problem Observations

92 Inverse problems The concept Theory Inverse problem Observations

93 Inverse problems The concept Theory Observations n χ 2 fits Dark energy model SNe Ia magnitudes Inverse method

94 Inverse problems Inverse problem s problems Risks with inverse problems: The solution does not necessary exist The solution is not unique The solution is not stable

95 Inverse problems Inverse problem s problems Risks with inverse problems: The solution does not necessary exist The solution is not unique The solution is not stable

96 Inverse problems Inverse problem s problems Risks with inverse problems: The solution does not necessary exist The solution is not unique The solution is not stable

97 Inverse problems Inverse problem s problems Risks with inverse problems: The solution does not necessary exist The solution is not unique The solution is not stable

98 Inverse problems Inverse 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

99 Inverse problems The method Demands: 1 Inclusion of a priori information 2 Recover a function w(z) or Λ(z) instead of parameterizations

100 Inverse problems The method Demands: 1 Inclusion of a priori information 2 Recover a function w(z) or Λ(z) instead of parameterizations Approach: 1 Through the Bayes theorem: f post (M y) α L(y M) f prior (M)

101 Inverse problems The method Demands: 1 Inclusion of a priori information 2 Recover a function w(z) or Λ(z) instead of parameterizations Approach: 1 Through the Bayes theorem: f post (M y) α L(y M) f prior (M) 2 Working in functional spaces, were the functionals such as w(z) are defined in an infinite-dimensional space

102 The inverse method Misfit function Posterior distribution probability: S 1 2 f post (M y) α exp [ S] ( y y th (M) ) ( C 1 y y y th (M) ) Data y, covariance C y, unknowns M χ 2 but... adjoint operator, scalar product in n-d space... Information for the unknowns M: priors M 0, C 0 Data and unknowns treated at the same level

103 The inverse method Misfit function Posterior distribution probability: S 1 2 f post (M y) α exp [ S] ( y y th (M) ) ( C 1 y y y th (M) ) Data y, covariance C y, unknowns M χ 2 but... adjoint operator, scalar product in n-d space... Information for the unknowns M: priors M 0, C 0 Data and unknowns treated at the same level

104 The inverse method Misfit function Posterior distribution probability: f post (M y) α exp [ S] S 1 2 ( y y th (M) ) ( C 1 y y y th (M) ) (M M 0) C 1 0 (M M 0) Data y, covariance C y, unknowns M χ 2 but... adjoint operator, scalar product in n-d space... Information for the unknowns M: priors M 0, C 0 Data and unknowns treated at the same level

105 The inverse method Misfit function Posterior distribution probability: f post (M y) α exp [ S] S 1 2 ( y y th (M) ) ( C 1 y y y th (M) ) (M M 0) C 1 0 (M M 0) Data y, covariance C y, unknowns M χ 2 but... adjoint operator, scalar product in n-d space... Information for the unknowns M: priors M 0, C 0 Data and unknowns treated at the same level Goal: S minimization

106 The inverse method Minimization Minimization steps: 1 The minimization is done in the functional space using a Newton minimization method 2 A final functional equation for w(z) is obtained 3 At this point operators are discretized and concrete values calculated Results: 1 Quite a long equation for w(z):

107 The inverse method Minimization Minimization steps: 1 The minimization is done in the functional space using a Newton minimization method 2 A final functional equation for w(z) is obtained 3 At this point operators are discretized and concrete values calculated Results: 1 Quite a long equation for w(z):

108 The inverse method Minimization where NX Z zi w [k+1] (z) = w 0 (z) + W i [k] C w (z, z )g w[k] (z )dz, i=1 0 W i[k] = P N j=1 S 1 [k] V j[k], i,j V i S i,j = y i + = δ i,j σ i σ j + y th i Ω 0 (Ω 0 M y th Ω0 i M 0 ) + M w(z) (w w 0) yi th (z i, Ω 0 M, w(z)), y th i Ω 0 M C y j th Ω 0 M Ω 0 + M y th i w(z) Cw y th j w(z)!. And uncertainty: σ w(z) (z) = v q u C w(z) (z) = tσw(z) 2 X i,j y th y th i j C w w(z) (S 1 ) i,j w(z) Cw

109 The inverse method Minimization Results: 1 An equation for w(z) and the remaining unknowns 2 An estimation for the σ uncertainty 3 The resolving kernel K(z, z i ), a function at every z i indicating how well resolved it is Comment: Notice that both depend on the priors Monte Carlo exploration of the space of solutions

110 The inverse method Minimization Results: 1 An equation for w(z) and the remaining unknowns 2 An estimation for the σ uncertainty 3 The resolving kernel K(z, z i ), a function at every z i indicating how well resolved it is Comment: Notice that both depend on the priors Monte Carlo exploration of the space of solutions

111 The inverse method Results For the dark energy equation of state w(z): Riess et al. (2006) data Prior: Ω 0 M = 0.27 ± 0.03 Prior: w(z) 0 = 1 ± 0.5 Differs from Λ at more than 1σ

112 The inverse method Results For the dark energy equation of state w(z):

113 The inverse method Results Obtaining confidence regions via a Monte Carlo exploration: As before but: 1000 reconstructions Explored range 3 < w(z) < 1 Still differs from Λ at more than 1σ Wider, more reliable, confidence regions

114 The inverse method Results One can do the same for the Cosmological Constant Λ(z) 1000 reconstructions Explored range 0.53 < Ω Λ (z) < 0.93 A constant CC valid at the 1σ limit Valid for a general Ω X (z)

115 What the future holds in store 1 Motivació 2 Introduction 3 Evolving Cosmological Constant 4 Non-Parametric Reconstructions 5 Future Perspectives Oncoming Surveys Non-Parametric Reconstructions Evolving Cosmological Constant 6 Summary and Conclusions

116 Future Perspectives Oncoming surveys

117 Future Perspectives Oncoming surveys Vs. Space observatory 2,000 SNe/year Ground telescope 250,000 SNe/year (wide survey) 10,000 SNe (deep survey) 0.1 < z < < z < 0.9 (wide survey) 0 < z < 1.4 (deep survey) Spectroscopic redshifts Photometric redshifts

118 Future Perspectives Oncoming surveys Vs. Space observatory 2,000 SNe/year Ground telescope 250,000 SNe/year (wide survey) 10,000 SNe (deep survey) 0.1 < z < < z < 0.9 (wide survey) 0 < z < 1.4 (deep survey) Spectroscopic redshifts Photometric redshifts

119 Future Perspectives Oncoming surveys Vs. Space observatory 2,000 SNe/year Ground telescope 250,000 SNe/year (wide survey) 10,000 SNe (deep survey) 0.1 < z < < z < 0.9 (wide survey) 0 < z < 1.4 (deep survey) Spectroscopic redshifts Photometric redshifts

120 Future Perspectives Oncoming surveys Vs. Space observatory 2,000 SNe/year Ground telescope 250,000 SNe/year (wide survey) 10,000 SNe (deep survey) 0.1 < z < < z < 0.9 (wide survey) 0 < z < 1.4 (deep survey) Spectroscopic redshifts Photometric redshifts

121 Future Perspectives Oncoming surveys Uncertainties in the data sets

122 Future Perspectives Oncoming surveys Uncertainties in the data sets LSST deep: σ intr = 0.15, δz = 0.01 and σ sys = 0.02 SNAP: σ intr = 0.15, δz = 0.00 and σ sys = 0.02 z/1.7 (+ SNFactory as low redshift anchor)

123 Future Perspectives Non-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

124 Future Perspectives Non-parametric reconstructions For the dark energy equation of state w(z) with: LSST SNAP

125 Future Perspectives Evolving cosmological constant Scenario 1 (m i < µ, µ ρ 1/4 c ) τ 1 2 m4 S 4 ν m4 ν

126 Future Perspectives Evolving cosmological constant Scenario 1 (m i < µ, µ ρ 1/4 c ) τ 1 2 m4 S 4 ν m4 ν

127 Future Perspectives Evolving cosmological constant Scenario 2 (M i > µ, µ ρ 1/4 c ) η 1 2 i N i 5 4 = 10.75

128 Future Perspectives Evolving cosmological constant Scenario 2 (M i > µ, µ ρ 1/4 c ) η 1 2 i N i 5 4 = 10.75

129 Future Perspectives Evolving cosmological constant Scenario 3 (M i > µ, µ H) ν σ 12 π M 2 M 2 P

130 Future Perspectives Evolving cosmological constant Scenario 3 (M i > µ, µ H) ν σ 12 π M 2 M 2 P

131 In summary 1 Motivació 2 Introduction 3 Evolving Cosmological Constant 4 Non-Parametric Reconstructions 5 Future Perspectives 6 Summary and Conclusions

132 Summary and conclusions The problem The Universe seems to be in accelerated expansion Characterization of a dark energy source as the cause of acceleration Nowadays, it is a degenerated problem

133 Summary and conclusions The problem The Universe seems to be in accelerated expansion Characterization of a dark energy source as the cause of acceleration Nowadays, it is a degenerated problem

134 Summary and conclusions The problem The Universe seems to be in accelerated expansion Characterization of a dark energy source as the cause of acceleration Nowadays, it is a degenerated problem

135 Summary and conclusions The problem The Universe seems to be in accelerated expansion Characterization of a dark energy source as the cause of acceleration Nowadays, it is a degenerated problem Evolving Cosmological Constant We motivate an evolving CC as a consequence of the renormalization effects in a Quantum Field Theory (QFT) The running depends on the renormalization scale and the active dofs

136 Summary and conclusions The problem The Universe seems to be in accelerated expansion Characterization of a dark energy source as the cause of acceleration Nowadays, it is a degenerated problem Evolving Cosmological Constant We motivate an evolving CC as a consequence of the renormalization effects in a Quantum Field Theory (QFT) The running depends on the renormalization scale and the active dofs

137 Summary and conclusions Evolving Cosmological Constant (continued) In our approach (Scenario 3), particles with M M Pl are responsible for the running This evolution affects the standard cosmological equations Running must be small in order to be compatible with structure formation and CMB Such an small evolution is difficult to detect by observations, although it is mandatory within a QFT

138 Summary and conclusions Evolving Cosmological Constant (continued) In our approach (Scenario 3), particles with M M Pl are responsible for the running This evolution affects the standard cosmological equations Running must be small in order to be compatible with structure formation and CMB Such an small evolution is difficult to detect by observations, although it is mandatory within a QFT

139 Summary and conclusions Evolving Cosmological Constant (continued) In our approach (Scenario 3), particles with M M Pl are responsible for the running This evolution affects the standard cosmological equations Running must be small in order to be compatible with structure formation and CMB Such an small evolution is difficult to detect by observations, although it is mandatory within a QFT

140 Summary and conclusions Evolving Cosmological Constant (continued) In our approach (Scenario 3), particles with M M Pl are responsible for the running This evolution affects the standard cosmological equations Running must be small in order to be compatible with structure formation and CMB Such an small evolution is difficult to detect by observations, although it is mandatory within a QFT

141 Summary and conclusions Non-parametric Reconstructions We apply an inverse approach to estimate w(z) and Λ(z) in a non-parametric way We introduce a priori information to regularize the inversion Results depend on priors, but we include Monte Carlo explorations to overcome this limitation Current data can already rule out a constant dark energy source at low redshift at 1σ level Future surveys such as SNAP or LSST will confirm that point up to redshift one

142 Summary and conclusions Non-parametric Reconstructions We apply an inverse approach to estimate w(z) and Λ(z) in a non-parametric way We introduce a priori information to regularize the inversion Results depend on priors, but we include Monte Carlo explorations to overcome this limitation Current data can already rule out a constant dark energy source at low redshift at 1σ level Future surveys such as SNAP or LSST will confirm that point up to redshift one

143 Summary and conclusions Non-parametric Reconstructions We apply an inverse approach to estimate w(z) and Λ(z) in a non-parametric way We introduce a priori information to regularize the inversion Results depend on priors, but we include Monte Carlo explorations to overcome this limitation Current data can already rule out a constant dark energy source at low redshift at 1σ level Future surveys such as SNAP or LSST will confirm that point up to redshift one

144 Summary and conclusions Non-parametric Reconstructions We apply an inverse approach to estimate w(z) and Λ(z) in a non-parametric way We introduce a priori information to regularize the inversion Results depend on priors, but we include Monte Carlo explorations to overcome this limitation Current data can already rule out a constant dark energy source at low redshift at 1σ level Future surveys such as SNAP or LSST will confirm that point up to redshift one

145 Summary and conclusions Non-parametric Reconstructions We apply an inverse approach to estimate w(z) and Λ(z) in a non-parametric way We introduce a priori information to regularize the inversion Results depend on priors, but we include Monte Carlo explorations to overcome this limitation Current data can already rule out a constant dark energy source at low redshift at 1σ level Future surveys such as SNAP or LSST will confirm that point up to redshift one

146 Exploring the Evolution of Dark Energy and its Equation of State Advisor: Dra. Pilar Ruiz-Lapuente Universitat de Barcelona 12th February, 2008