Fluctuations of cosmic parameters in the local universe

Transcription

1 Fluctuations of cosmic parameters in the local universe Alexander Wiegand Dominik Schwarz Fakultät für Physik Universität Bielefeld 6. Kosmologietag, Bielefeld 2011 A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

2 Outline 1 Measuring cosmic parameters 2 Intrinsic uctuations of cosmic parameters Local parameters in the average model Calculation of their uctuations 3 Results 4 Conclusion A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

3 Outline 1 Measuring cosmic parameters 2 Intrinsic uctuations of cosmic parameters Local parameters in the average model Calculation of their uctuations 3 Results 4 Conclusion A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

4 Cosmic energy budget 2011 dark energy 72 neutrinos 1 atoms 4 dark matter 23 Standard model of cosmology indicates that the universe consists of 95% dark physics What do these numbers mean? Where do we get them from? Is this a measurement of the energy content of the universe? A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

5 Parameters of the standard model Perhaps. A priori they are the values of the global standard model parameters Ω m := 8πG 3H 2 ϱ Ω k := k a 2 H 2 Ω Λ := Λ 3H 2 Sloan Digital Sky Survey that, by the Friedmann equation, add up to one Ω m + Ω k + Ω Λ = 1 Basic model assumption: homogeneity and isotropy. But locally the universe is inhomogeneous A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

6 Parameters of the standard model Perhaps. A priori they are the values of the global standard model parameters Ω m := 8πG 3H 2 ϱ Ω k := k a 2 H 2 Ω Λ := Λ 3H 2 Sloan Digital Sky Survey that, by the Friedmann equation, add up to one Ω m + Ω k + Ω Λ = 1 Basic model assumption: homogeneity and isotropy. But locally the universe is inhomogeneous A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

7 Measurement of cosmic parameters Parameters determined from the concordant results of dierent measurements Main experimental probes: CMB, SN IA, BAO/structure, H 0 Only the CMB is a global in that it probes the largest scales Particle Data Group Physics Letters B 667 (2008) A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

8 Importance of local parameter measurements WMAP 5 arxiv: HST Key Project arxiv:astro-ph/ WMAP has to be complemented with local measurements to give non degenerate results One possibility: H 0 Measurement in the local universe. A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

9 Why consider uctuations? Basic question: How typical is our local environment where we draw our data from? To address this question: Describe the local environment as average over inhomogeneous distribution A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

10 Outline 1 Measuring cosmic parameters 2 Intrinsic uctuations of cosmic parameters Local parameters in the average model Calculation of their uctuations 3 Results 4 Conclusion A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

11 General inhomogeneous model Locally introduce explicit averaging D f D (t) := f ( t,x 1,X 2,X 3) dµ g D dµ g Friedmann equations of the homogeneous model become domain dependent averaged parameters 3äD a D = 4πG ϱ D + Λ+Q D 3H 2 D = 8πG ϱ D 3 k a 2 + Λ 1 2 Q D 0 = t ϱ D + 3H D ϱ D, A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

12 General inhomogeneous model Locally introduce explicit averaging D f D (t) := f ( t,x 1,X 2,X 3) dµ g D dµ g Friedmann equations of the homogeneous model become domain dependent averaged parameters 3äD a D = 4πG ϱ D + Λ + Q D 3H 2 D = 8πG ϱ D 1 2 R D + Λ 1 2 Q D 0 = t ϱ D + 3H D ϱ D, A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

13 Localised cosmic parameters Global standard model parameters, become domain dependent averaged parameters Ω D m := 8πG 3HD 2 ϱ D Ω D Λ := Ω D k := k a 2 H 2 Λ 3H 2 D ΩD Q := Q D 6H 2 D Interesting quantity: Fluctuations from domain to domain of equal size, but at a dierent location in the universe σ (O D ) := ( ) O 2 D O D Ensemble average O D commuting with domain average A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

14 Localised cosmic parameters Global standard model parameters, become domain dependent averaged parameters Ω D m := 8πG 3HD 2 ϱ D Ω D Λ := Ω D R := R D 6H 2 D Λ 3H 2 D Ω D Q := Q D 6H 2 D Interesting quantity: Fluctuations from domain to domain of equal size, but at a dierent location in the universe σ (O D ) := ( ) O 2 D O D Ensemble average O D commuting with domain average A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

15 Localised cosmic parameters Global standard model parameters, become domain dependent averaged parameters Ω D m := 8πG 3HD 2 ϱ D Ω D Λ := Ω D R := R D 6H 2 D Λ 3H 2 D Ω D Q := Q D 6H 2 D Interesting quantity: Fluctuations from domain to domain of equal size, but at a dierent location in the universe σ (O D ) := ( ) O 2 D O D Ensemble average O D commuting with domain average A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

16 Localised cosmic parameters Global standard model parameters, become domain dependent averaged parameters Ω D m := 8πG 3HD 2 ϱ D Ω D Λ := Ω D R := R D 6H 2 D Λ 3H 2 D Ω D Q := Q D 6H 2 D Interesting quantity: Fluctuations from domain to domain of equal size, but at a dierent location in the universe σ (O D ) := ( ) O 2 D O D Ensemble average O D commuting with domain average A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

17 Calculation Method used: Standard rst order cosmological perturbation theory { ds 2 = a 2 (η) dη 2 + [(1 2ψ (1)) δ ij + D ij χ (1)] dx i dx j} The metric perturbation ψ (1) is directly linked to the density perturbation This may then be solved ψ (1) (a,x) = 1 3 δ (a,x) ζ (x) (0) 0 δ + a a δ = 4πGρ a δ (a) = D(a) δ 0, D ( 0 D(a) = a 2 F 1 1, 1 3 ; 11 ) 6 ; ca3 δ ; c = Ω Λ /Ω m A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

18 Calculation Method used: Standard rst order cosmological perturbation theory { ds 2 = a 2 (η) dη 2 + [(1 2ψ (1)) δ ij + D ij χ (1)] dx i dx j} The metric perturbation ψ (1) is directly linked to the density perturbation This may then be solved ψ (1) (a,x) = 1 3 δ (a,x) ζ (x) (0) 0 δ + a a δ = 4πGρ a δ (a) = D(a) δ 0, D ( 0 D(a) = a 2 F 1 1, 1 3 ; 11 ) 6 ; ca3 δ ; c = Ω Λ /Ω m A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

19 Calculation Method used: Standard rst order cosmological perturbation theory { ds 2 = a 2 (η) dη 2 + [(1 2ψ (1)) δ ij + D ij χ (1)] dx i dx j} The metric perturbation ψ (1) is directly linked to the density perturbation This may then be solved ψ (1) (a,x) = 1 3 δ (a,x) ζ (x) (0) 0 δ + a a δ = 4πGρ a δ (a) = D(a) δ 0, D ( 0 D(a) = a 2 F 1 1, 1 3 ; 11 ) 6 ; ca3 δ ; c = Ω Λ /Ω m A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

20 Hubble uctuations H D for example depends on background H and perturbation θ = 3 ( a ) a a ψ(1) or in terms of a D ) 1 5 a 2 D D 1 Ω D 3 a D0 0 m H D = H 0 ( Ωm a 3 D 0 Ω D m a 3 D 2 δ 0 D D 0 with the uctuation evolution function ( ) { f (a D ) := ΩD m (a D ) 5 a D 0.34 ΛCDM D 0 = D 0 3 a D EdS H D = H D (a D ) (1 12 ) f (a D) δ 0 D A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

21 Hubble uctuations H D for example depends on background H and perturbation θ = 3 ( a ) a a ψ(1) or in terms of a D ) 1 5 a 2 D D 1 Ω D 3 a D0 0 m H D = H 0 ( Ωm a 3 D 0 Ω D m a 3 D 2 δ 0 D D 0 with the uctuation evolution function ( ) { f (a D ) := ΩD m (a D ) 5 a D 0.34 ΛCDM D 0 = D 0 3 a D EdS H D = H D (a D ) (1 12 ) f (a D) δ 0 D A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

22 Hubble uctuations H D for example depends on background H and perturbation θ = 3 ( a ) a a ψ(1) or in terms of a D ) 1 5 a 2 D D 1 Ω D 3 a D0 0 m H D = H 0 ( Ωm a 3 D 0 Ω D m a 3 D 2 δ 0 D D 0 with the uctuation evolution function ( ) { f (a D ) := ΩD m (a D ) 5 a D 0.34 ΛCDM D 0 = D 0 3 a D EdS H D = H D (a D ) (1 12 ) f (a D) δ 0 D A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

23 Fluctuations of cosmic parameters Fluctuations of cosmic parameters δh D = 1 2 H D (a D ) f (a D ) σ D 0 = 0.17H 0 σ D 0 δω D m = Ω D m (a D )(1 + f (a D )) σ0 D = 0.40σ0 D ( ) δω D R = ΩD m (a D ) 1 + f (a D) σ D Ω D m (a D ) 0 = 0.64σ0 D δω D Λ = ΩD Λ (a D) f (a D ) σ D 0 = 0.24σ D 0 δω D Q = O ( (σ D 0 ) 2 ) with the denitions ( Ω D m = 1 + c(a D /a D0 ) 3) 1 ( ) σ D 2 0 := σ 2 ( δ 0 D ) = d 3 k P i (k) W D (k) W D ( k) R 3 A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

24 Fluctuations of cosmic parameters Fluctuations of cosmic parameters δh D = 1 2 H D (a D ) f (a D ) σ D 0 = 0.17H 0 σ D 0 δω D m = Ω D m (a D )(1 + f (a D )) σ0 D = 0.40σ0 D ( ) δω D R = ΩD m (a D ) 1 + f (a D) σ D Ω D m (a D ) 0 = 0.64σ0 D δω D Λ = ΩD Λ (a D) f (a D ) σ D 0 = 0.24σ D 0 δω D Q = O ( (σ D 0 ) 2 ) with the denitions ( Ω D m = 1 + c(a D /a D0 ) 3) 1 ( ) σ D 2 0 := σ 2 ( δ 0 D ) = d 3 k P i (k) W D (k) W D ( k) R 3 A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

25 Outline 1 Measuring cosmic parameters 2 Intrinsic uctuations of cosmic parameters Local parameters in the average model Calculation of their uctuations 3 Results 4 Conclusion A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

26 Cosmic variance on dierent survey geometries 50 2dF Σ z in SDSS Full sky redshift z Observational domains adapted to the real situation Cosmic variance non negligible for current surveys, but subdominant A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

27 Curvature uctuations on dierent domains in opening angle in z = 0.05 z = 0.1 z = 0.2 z = 0.35 in Α = 6 Α = 30 Α = 90 Α = survey depth in h 1 Mpc Strong angular dependence only for small angles Large uctuations of the curvature parameter even for deep windows A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

28 Cosmic variance and measurement errors in full sky 2dF SDSS m H H redshift z Curvature parameter shows the biggest uctuations Errors in large surveys not yet cosmic variance limited A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

29 Outlook 5.0 shot noise DV z in dF SDSS insf. volume Seikel and Schwarz Full sky redshift z Use uctuations to measure f (a D ) Fluctuations of H are promising, those of the acoustic distance scale not quite A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

30 Outline 1 Measuring cosmic parameters 2 Intrinsic uctuations of cosmic parameters Local parameters in the average model Calculation of their uctuations 3 Results 4 Conclusion A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

31 Conclusion Averaging formalism allows denition of local Ω parameters Perturbation of the curvature parameter strong even on comparably large scales Directions to explore Try to measure the Hubble uctuations in real SN data For larger redshifts: Use perturbed luminosity distance for inhomogeneous cosmologies A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

32 Conclusion Averaging formalism allows denition of local Ω parameters Perturbation of the curvature parameter strong even on comparably large scales Directions to explore Try to measure the Hubble uctuations in real SN data For larger redshifts: Use perturbed luminosity distance for inhomogeneous cosmologies A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21

33 Conclusion Averaging formalism allows denition of local Ω parameters Perturbation of the curvature parameter strong even on comparably large scales Directions to explore Try to measure the Hubble uctuations in real SN data For larger redshifts: Use perturbed luminosity distance for inhomogeneous cosmologies A. Wiegand (Universität Bielefeld) Fluctuations of cosmic parameters Kosmologietag / 21