Modified gravity as an alternative to dark energy. Lecture 3. Observational tests of MG models

Transcription

1 Modified gravity as an alternative to dark energy Lecture 3. Observational tests of MG models

2 Observational tests Assume that we manage to construct a model How well can we test the model and distinguish it from LCDM model? There have been a lot of activities Here I focus on DGP as this is one-parameter theory GR Scalartensor GR

3 Three regime of gravity -DGP r Einstein ( ) 1 r r 3 * g c Scalar tensor rc H 0 4D 5D Solar system CMB ISW CMB Weak lensing LSS SNe Cluster abundance Non-linear linear Growth factor ga ( ) Expansion history rz ()

4 Expansion history Friedmann equation H 1 1 8π G K = + + ρ + rc 4r 3 a m c cf. LCDM 1 4rH ( ) r r m K r 1 = Ω + Ω +Ω +Ω, Ω = c c c 1 =Ω m +Ω Λ +ΩK c 0

5 SNe + baryon oscillation SNLS + SDSS Gold set + SDSS (Fairbairn and Goobar astro-ph/051109) (Maartens and Majerotto astro-ph/ ) (cf. Alam and Sahni, astro-ph/ )

6 flat model conflicts with data H πg = ρm H r c ρ DE H r c w DE 1 = 1 +Ω ( a) m (Song, Hu and Sawicki) inclusion of curvature improves a fit

7 Linear perturbations (Cardoso et.al.) 1 Φ + = ( Φ+Ψ ) = 0 GR 1 Φ = ( Φ Ψ) CMB ISW

8 Enhancement of low multipoles QCDM has the same expansion history as DGP DGP is a poorer fit than LCDM at 5.3 level inclusion of large scale CMB has 30% contribution to this conclusion σ (Fang et.al )

9 Integrated Sachs-Wolfe effects ISW effects Sensitive to time variation of growth rate / MG Cross correlation to matter perturbations (Giannantonio et.al.)

10 Peculiar velocity Redshift distortion due to peculiar velocities of galaxies, red-shift space power spectrum of galaxies becomes anisotropic s ( 4 k μ σ ( ) ( ) ) v Pg = Pg k + μ Pg θ k + μ Pθθ F H ( z ) μ cos of angle between the line of sight and wave number θ = a & δ is divergence of peculiar velocities multi-pole moment expansion l l= 0,, Pk (, μ) = P( kl ) ( μ)(l + 1) l Non-linear effects

11 Quadrupole spectrum linear theory P k β β δ P k d a ( ) 1 4 /3+ 4 /7 d ln =, β = f / b, f = 0( ) 5(1+ β /3 + β /5) ln SDSS LRGs significantly high b: galaxy bias (Yamamoto et.al.) σ 8 is required Combining to the CMB will exclude the model significantly γ f =Ω, 0.68 m γ = (DGP)

12 Future forecast (Song and Percival) df SDSS VDSS f σ 8 LCDM Best fit DGP peculiar velocity can give bias-free measurements of σ = f σ θ 8 8 from P θθ

13 Weak lensing Sensitive to growth rate Linder s parametrisation ( a ( )) 0 m ga ( ) = exp dln aω ( a) γ 1 ga ( ) = δ / a dark energy γ = (1 + wz ( = 1)) DGP γ = 0.68 WFMOS Δ γ = 0.1 (Yamamoto et.al.)

14 Non-linear power spectrum So far, GR mapping formula is used this neglects the subtlety of non-linear recovery of GR on non-linear scales example from f(r) mapping formula fails (Oyaizu et.al ) Need to understand non-linear physics in MG N-body perturbation theory (Hiramatsu, Taruya, KK)

15 Summary We have enough observations! current data has an ability to exclude DGP at 5 sigma level against LCDM Structure formation test can give significant contributions large scale CMB anisotropies peculiar velocities weak lensing

16 Can we distinguish between MG and DE? Can we surely prove that the acceleration is driven by MG not by DE This is clearly impossible in the background structure formation test is essential (+ sensible assumption for dark energy perturbations)

17 Dark energy vs DGP Can we distinguish between dark energy in GR and DGP? w = 1 LCDM DGP w = 0.7 z rz ( ) = dzhz ( ) DGP model is fitted by 0 (Linder) 0 wa ( ) = w + w(1 a), w 0 = 0.78, w = 0.3 a a 1 (Dvali and Turner)

18 Expansion history vs growth rate (Lue.et.al, Koyama & Maartens, Koyama) Growth rate resolves the degeneracy ga ( ) δ = a LCDM dark energy DGP

19 Experiments ASSUME our universe is DGP braneworld but you do not want to believe this,so fit the data using dark energy model (Ishak,Upadhye and Spergel, astro-ph/ ) wz ( ) = w+ wz, 0 1 m(z): apparent magnitude R: CMB shift parameter G(a): Growth rate OR Inconsistent! SNe+CMB SNe+weak lensing

20 Observables Density perturbations galaxy clustering mass function of clusters δ = δρ ρ Peculiar velocity red-shift distortions θ = v i i internal dynamics of clusters/galaxies Lensing potential weak lensing ISW of CMB Φ Ψ

21 Equations under horizon Gravitational equations (GR) 8π G H = ρ, ρ = ρ 3 k a T T i i Φ= πga ρδ ρδ = ρδ 4 T T, T T i i i Equation of motion for matter (no interaction) k a & δ m Ψ= & θ + m θm = a Hθ m

22 Consistency condition Eliminate Newton constant from Friedmann eq. and Possion eq. α( kt, ) Φ Ψ +Ψ = da ( θm) k Ψ= dt k ( ) 3aH δt Peculiar velocity In GR, background α ( kt, ) = 1 Weak lensing This is written only by observables Galaxy distribution δ = b δ g T T

23 Different observables measure different physical quantities and they are all valuable if we just extend our theoretical prior to include MG δ Ψ Φ Ψ

24 How to combine various observations? Estimator E G ( Ψ Φ) = 3H a θ 1 0 (Zhang et.al ) galaxy-lensing and -velocity cross correlation E g Ωm dln δ ( a), f = f ( Ω ) dlna m

25 Current status SDSS LRGs (Reyes et.al) It is free from bias but also eliminates Information on modified Newton constant Φ= ( G ) eff b 4 π / δ g

26 Consistency test (Song and KK) Reconstruction of gravitational lensing from density distribution and peculiar velocity at each red-shift bin GR k Φ= 4π Ga ρt δt da ( θm) k Ψ= dt GR GR is assumed φ ( l / χ, z ) = Φ Ψ i i i Consistency test Reconstructed gravitational lensing Observed gravitational lensing

27 D power spectrum of lensing potential test of reconstruction in GR Ratio of reconstructed and true spectrum in MG recon C l α = log 10 Cl Information of bias is crucial

28 Summary Observational test on MG models geometrical test + structure formation theoretical models are required understanding of non-linear clustering is necessary Model independent test of GR peculiar velocity + WL + galaxy distribution need to find the best estimator understanding of systematic in observation is necessary

29 Objective Seek solutions to the question of dark energy By challenging conventional GR construct consistent theoretical models building on rapid progress in understanding the law of gravity beyond GR develop efficient ways to combine observational data sets to distinguish modified gravity models from dark energy models based on GR provide tests of GR on largest scales