Consistent modified gravity analysis of anisotropic galaxy clustering using BOSS DR11

Transcription

1 Consistent modified gravity analysis of anisotropic galaxy clustering using BOSS DR11 August APCTP-TUS Workshop arxiv: Yong-Seon Song with Atsushi Taruya, Kazuya Koyama, Eric Linder, Cris Sabiu, Takahiro Nishimichi, Gongbo Zhao, Teppei Okumura, Francis Bernadeau

2 Implication of cosmic acceleration Breaking down our knowledge of particle physics: we have limited knowledge of particle physics bounded by testable high energy, and our efforts to explain the cosmic acceleration turn out in vain. Alternative mechanism to generate fine tuned vacuum energy New unknown energy component Unification or coupling between dark sectors Breaking down our knowledge of gravitational physics: gravitational physics has been tested in solar system scales, and it is yet confirmed at horizon size, Presence of extra dimension Non-linear interaction to Einstein equation Failure of standard cosmology model: our understanding of the universe is still standing on assumptions: Inhomogeneous models: LTB, back reaction

3 Theoretical models to explain acceleration Breaking down our knowledge of particle physics: we have limited knowledge of particle physics bounded by testable high energy, and our efforts to explain the cosmic acceleration turn out in vain. Dynamical Dark Energy: modifying matter G μν = 4πG N T μν + ΔT μν Alternative mechanism to generate fine tuned vacuum energy New unknown energy component Unification or coupling between dark sectors Breaking down our knowledge of gravitational physics: gravitational physics has been tested in solar system scales, and it is yet confirmed at horizon size, Geometrical Dark Energy: modifying gravity G μν + ΔG μν = 4πG N T μν Presence of extra dimension Non-linear interaction to Einstein equation Failure of standard cosmology model: our understanding of the universe is still standing on assumptions: Inhomogeneous models: LTB, back reaction

4 Implication of cosmic acceleration Breaking down our knowledge of particle physics: we have limited knowledge of particle physics bounded by testable high energy, and our efforts to explain the cosmic acceleration turn out in vain. Alternative mechanism to generate fine tuned vacuum energy New unknown energy component Unification or coupling between dark sectors Breaking down our knowledge of gravitational physics: gravitational physics has been tested in solar system scales, and it is yet confirmed at horizon size, Presence of extra dimension Non-linear interaction to Einstein equation Failure of standard cosmology model: our understanding of the universe is still standing on assumptions: Inhomogeneous models: LTB, back reaction

5 Key observables in cosmological science Angular diameter distance D A : Exploiting BAO as standard rulers which measure the angular diameter distance and expansion rate as a function of redshift. Radial distance H -1 : Exploiting redshift distortions as intrinsic anisotropy to decompose the radial distance represented by the inverse of Hubble rate as a function of redshift. Coherent motion G θ : The coherent motion, or flow, of galaxies can be statistically estimated from their effect on the clustering measurements of large redshift surveys, or through the measurement of redshift space distortions.

6 Spectroscopy wide deep field survey BOSS DR11 catalogue

7 Structure formation Squeezing effect at large scales (Kaiser 1987) Finger of God effect at small scales (Jackson 1972) P s (k,μ) = P gg (k) + 2μ 2 P g θ(k) + μ 4 P θθ (k) P s (k,μ) = [P gg (k) + ΔP gg + 2μ 2 P gθ (k) + ΔP g θ + μ 4 P θθ (k) + ΔP θθ + μ 2 A(k) + μ 4 B(k) + μ 6 C(k) +... ] exp[-(kμσ p ) 2 ] Taruya, Nishimichi, Saito 2010; Taruya, Hiramatsu 2008; Taruya, Bernardeau, Nishimichi 2012

8 Theoretical model in configuration space P s (k,μ) = [Q 0 (k) + μ 2 Q 2 (k) + μ 4 Q 4 (k) + μ 6 Q 6 (k)] exp[-(kμσ p ) 2 ] ξ(σ,π) = d 3 k P(k,μ)e ikx = Σ ξ l (s) P l (ν) P 0 (k) = p 0 (k) ξ l (s) = i l k 2 dk P l (k) j l (ks) P 2 (k) = 5/2 [3p 1 (k) - p 0 (k)] P 4 (k) = 9/8 [35p 2 (k) - 30p 1 (k) + 3p 0 (k) ] P 6 (k) = 13/16 [231p 3 (k) - 315p 2 (k) - 105p 1 (k) + 5p 0 (k) ] : p n (k) = 1/2 [ γ(n+1/2,κ)/κ n+1/2 Q 0 (k) + γ(n+3/2,κ)/κ n+3/2 Q 2 (k) κ = k 2 σ 2 p + γ(n+5/2,κ)/κ n+5/2 Q 4 (k) + γ(n+7/2,κ)/κ n+7/2 Q 6 (k) YSS, Okumura, Taruya 2014 Taruya, Nichimishi, Saito 2010

9 Measured correlation functions using DR11 Parameter space is (D A, H -1, G δ, G ϴ, FoG) YSS, Sabiu, Okumura, Oh, Linder2014 Planck prediction Best fit model

10 YSS, Sabiu, Okumura, Oh, Linder2014 Measured coherent motion Results from BOSS maps

11 f(r) gravity Corrections are introduced in the Einstein-Hilbert Lagrangian to modify the general relativity, which gets influential only low curvature, e.g. late time & not dense region. The corrections can be adjusted to generate the cosmic acceleration, Carroll, Duvvuri, Trodden, Turner (2004:CDTT) cosmic acceleration was discovered with f(r) = -a/r. Ruled out Two distinct branches of f(r) gravity was found depending on the sign of second order derivative of f(r) in terms of R, f RR = d 2 f/dr 2 < 0 f RR = d 2 f/dr 2 > 0 Unstable Stable The original proposal of CDTT is ruled out due to instability. YSS, Hu, Sawicki (2007)

12 f(r) gravity Corrections are introduced in the Einstein-Hilbert Lagrangian to modify the general relativity, which gets influential only low curvature, e.g. late time & not dense region. The corrections can be adjusted to generate the cosmic acceleration, Carroll, Duvvuri, Trodden, Turner (2004:CDTT) cosmic acceleration was discovered with f(r) = -a/r. Ruled out The f(r) gravity model in this talk is given by, f(r) = -2 κ 2 ρ Λ + f R0 R 02 /R 2

13 Measured coherent motion Results from BOSS maps f R0 < 10-4 at 95% confidence limit

14 LSS of f(r) gravity Dynamic equations of perturbations dδ m /dt + θ m /a = 0 dθ m /dt + Hθ m = k 2 ψ/a k 2 φ = 3/2 H 02 Ω m δ m /a F(ε) k 2 ψ = -3/2 H 02 Ω m δ m /a G(ε) which are not closed without knowing ε evolution For the case of DGP, dynamics equations with extra variable are closed with a constraint equation, but for the case of f(r) gravity, it is closed with an extra dynamic equation of ε.

15 LSS of f(r) gravity Dynamic equations of perturbations dδ m /dt + θ m /a = 0 dθ m /dt + Hθ m = k 2 ψ/a k 2 φ = 3/2 H 02 Ω m δ m /a F(ε) k 2 ψ = -3/2 H 02 Ω m δ m /a G(ε) which are not closed without knowing ε evolution Mass screening effect: k 2 φ fr = φ GR F(ε) Geometrical anisotropy: k 2 φ fr + k 2 ψ fr = -3H 02 Ω m δ m /a [F(ε) - G(ε)] Change on photon trajectory: φ fr - ψ fr = (φ GR - ψ GR )

16 LSS of f(r) gravity Dynamic equations of perturbations dδ m /dt + θ m /a = 0 dθ m /dt + Hθ m = k 2 ψ/a k 2 φ = 3/2 H 02 Ω m δ m /a F(ε) k 2 ψ= -3/2 H 02 Ω m δ m /a G(ε) Introducing the Brans-Dicke parameter φ φ fr - ψ fr = φ k 2 ψ= -3/2 H 02 Ω m δ m /a - 1/2 k 2 φ (1+w BD ) k 2 /a 2 φ = 3H 02 Ω m δ m /a - I(φ) where I(φ) is given by I(φ) = M 1 (k)φ(k) + 1/2 d 3 k 1 d 3 k n M 1 (k) M n (k) φ(k 1 ) φ(k n )

17 LSS of f(r) gravity Dynamic equations of perturbations dδ m /dt + θ m /a = 0 dθ m /dt + Hθ m = k 2 ψ/a k 2 φ = 3/2 H 02 Ω m δ m /a F(ε) k 2 ψ= -3/2 H 02 Ω m δ m /a G(ε) Later time growth functions are given by, D δ (k,t) = G δ (t) F δ (k,t;m 1 ) D ϴ (k,t) = G ϴ (t) F ϴ (k,t;m 1 ) We are not able to constrain f(r) gravity models using measured growth functions with the assumption of coherent growing after last scattering surface.

18 Linear power spectra with running f(r) The growth function becomes late time scale dependent, and we are not able to use the previous constraints if true model is f(r) gravity

19 Parameterisation of f(r) gravity model f(r) = -2 κ 2 ρ Λ + f R0 R 02 /R 2

20 Parameterisation of f(r) gravity model f(r) = -2 κ 2 ρ Λ + f R0 R 02 /R 2

21 Parameterisation of f(r) gravity model f(r) = -2 κ 2 ρ Λ + f R0 R 02 /R 2 We find that both coherent growth factors and scale dependent growth factors are separable in the following sense, D δ (k,t) = G δ (t) F δ (k,t;m 1 ) D ϴ (k,t) = G ϴ (t) F ϴ (k,t;m 1 )

22 Parameterisation of f(r) gravity model f(r) = -2 κ 2 ρ Λ + f R0 R 02 /R 2 Parameter space is (D A, H -1, G δ, G ϴ, FoG, f R0 ) D δ (k,t) = G δ (t) F δ (k,t;m 1 ) D ϴ (k,t) = G ϴ (t) F ϴ (k,t;m 1 )

23 Structure formation of RSD Squeezing effect at large scales (Kaiser 1987) Finger of God effect at small scales (Jackson 1972) P s (k,μ) = P gg (k) + 2μ 2 P g θ(k) + μ 4 P θθ (k) P s (k,μ) = [P gg (k) + ΔP gg + 2μ 2 P gθ (k) + ΔP g θ + μ 4 P θθ (k) + ΔP θθ + μ 2 A(k) + μ 4 B(k) + μ 6 C(k) +... ] exp[-(kμσ p ) 2 ]

24 Structure formation of RSD The non-linear solution is derived from dδ m /dt + [(1+δ m )v m ]/a = 0 dv m /dt + Hv m + (v m )v m /a = - ψ/a φ fr - ψ fr = φ k 2 ψ= -3/2 H 02 Ω m δ m /a - 1/2 k 2 φ (1+w BD ) k 2 /a 2 φ = 3H 02 Ω m δ m /a - I(φ) P s (k,μ) = P gg (k) + 2μ 2 P g θ(k) + μ 4 P θθ (k) P s (k,μ) = [P gg (k) + ΔP gg + 2μ 2 P gθ (k) + ΔP g θ + μ 4 P θθ (k) + ΔP θθ + μ 2 A(k) + μ 4 B(k) + μ 6 C(k) +... ] exp[-(kμσ p ) 2 ]

25 Structure formation of RSD The higher order polynomials are given by, A(k,t) = b 3 Σ n Σ a,b μ 2n (G ϴ /b) 2a+b-1 d 3 k dr dx X [A n ab(r,x)b 2ab (p,k-p,-k) + A n ab(r,x)b 2ab (k-p,p,-k) B(k,t) = b 4 Σ n Σ a,b μ 2n (-G ϴ /b) 2a+b-1 d 3 k dr dx X B n ab(r,x)p a2 (k 1+r 2-2rx)P b2 (kr)/(1+r 2-2rx) a P s (k,μ) = P gg (k) + 2μ 2 P g θ(k) + μ 4 P θθ (k) P s (k,μ) = [P gg (k) + ΔP gg + 2μ 2 P gθ (k) + ΔP g θ + μ 4 P θθ (k) + ΔP θθ + μ 2 A(k) + μ 4 B(k) + μ 6 C(k) +... ] exp[-(kμσ p ) 2 ]

26 Correlation function of f(r) gravity model The variation of D δ The variation of D ϴ

27 The measurement and best fit models

28 Constraints on f(r) gravity model We find new constraints on f(r) gravity models using BOSS DR11 f R0 < at 95% confidence limit

29 Constraints on f(r) gravity model We find new constraints on f(r) gravity models using BOSS DR11 f R0 < at 95% confidence limit

30 Constraints on distance measures Measured distances are consistent with LCDM model

31 Constraints on growth functions D ϴ (k,t) = G ϴ (t) F ϴ (k,t;m 1 ) log f R0 G ϴ

32 Constraints on f(r) now and future Invisible difference from LCDM model using BOSS Need a factor of 10 improvement

33 Where we are, and where will we go?

34 The targeted galaxies in next generation

35 Degeneracy for coherent motions Squeezing effect at large scales (Kaiser 1987) Finger of God effect at small scales (Jackson 1972) P s (k,μ) = P gg (k) + 2μ 2 P g θ(k) + μ 4 P θθ (k) P s (k,μ) = [P gg (k) + ΔP gg + 2μ 2 P gθ (k) + ΔP g θ + μ 4 P θθ (k) + ΔP θθ + μ 2 A(k) + μ 4 B(k) + μ 6 C(k) +... ] exp[-(kμσ p ) 2 ] Taruya, Nishimichi, Saito 2010; Taruya, Hiramatsu 2008; Taruya, Bernardeau, Nishimichi 2012

36 Degeneracy for coherent motions Approach I: extending into non-linear scale Approach II: staying at linear scale but go to higher order points

37 Bispectrum Alcock-Paczynski effect Configuration in redshift space B(k 1,k 2,k 3,μ 1,μ 2 ) k 3 μ 2 μ k 1 1 k 2 YSS, Taruya, Oka 2015

38 Bispectrum Alcock-Paczynski effect Definition of FoG effect B(k 1,k 2,k 3,μ 1,μ 2 ) = D B FoG B PT (k 1,k 2,k 3,μ 1,μ 2 ) D B FoG = exp[-(k 12 μ 12 +k 22 μ 22 +k 32 μ 32 )σ p2 ] YSS, Taruya, Oka 2015

39 Bispectrum Alcock-Paczynski effect B PT in redshift space B(k 1,k 2,k 3,μ 1,μ 2 ) = D B FoG B PT (k 1,k 2,k 3,μ 1,μ 2 ) B PT (k 1,k 2,k 3,μ 1,μ 2 ) = 2[Z 2 (k 1,k 2 )Z 1 (k 1 )Z 2 (k 2 )P(k 1 )P(k 2 ) + cyclic ] Z 1 (k 1 ) = b+fμ 1 2 Z 2 (k 1,k 2 ) = b 2 /2 + bf 2 + fμ 12 G 2 + fk 12 μ 12 /2[μ 1 /k 1 Z 2 (k 2 )+μ 1 /k 1 Z 2 (k 2 )] YSS, Taruya, Oka 2015

40 Bispectrum Alcock-Paczynski effect AP projection B obs (k 1,k 2,k 3,μ 1,μ 2 ) = (ΔH -1 ) 2 (ΔD A ) 4 B(q 1,q 2,q 3,ν 1,ν 2 ) ΔH -1 =H -1 fid/h -1 true ΔD A =D A fid /D A true q i =α(μ i )k i ν i =μ i ΔH -1 /α(μ i ) α(μ i )={(ΔD A ) 2 + [(ΔH -1 ) 2 -(ΔD A ) 2 ]μ i2 } 1/2 ν ij =(ΔD A ) 2 η ij /α(μ i )α(μ j )+[(ΔH -1 ) 2 -(ΔD A ) 2 ]μ i μ j /α(μ i )α(μ j ) YSS, Taruya, Oka 2015

41 Error forecast using power and bi combination F αβ = Σ k Σ k1k2k3 ( S/ p α ) C -1 ( S/ p β ) S = ( P(k,μ) ) B(k 1,k 2,k 3,μ 1,μ 2 ) ( ) C -1 = M -MC PB C BB -1 -C BB -1 C BB -1 M C BB -1 +C BB -1 C Bp MC PB C BB -1 M = (C pp -C PB C BB -1 C BP ) -1 YSS, Taruya, Oka 2015

42 Degeneracy in coherent and random motions Power Spectrum Bispectrum

43 Measured coherent motion Results from BOSS maps

44 Future constraints Expectation from DESI Einstein s gravity will be tested at cosmological scale in near future

45 Future work Invisible difference from LCDM model using BOSS then can we tell the difference in future??

46 Conclusion We succeed in measuring both distances and growth function simultaneously using RSD, and ready to test Einstein s gravity at cosmological scales through duality between distances and growth functions. We understand all systematics due to non-linear physics, and the perturbative description works fine the resolution of current experiment, at least two point correlation level. Now we face new challenge to meet the precision level of the high resolution experiment like DESI. We work out the Alcock-Paczynski effect on bispectra, and find that the combined constraint of power spectrum and bispectrum improves the detectability of growth function. We initiate new roadmap to accomplish this combination for the future experiment.