Modified gravity. Kazuya Koyama ICG, University of Portsmouth

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1 Modified gravity Kazuya Koyama ICG, University of Portsmouth

2 Cosmic acceleration Cosmic acceleration Big surprise in cosmology Simplest best fit model LCDM 4D general relativity + cosmological const. H 2 8π G Λ = ρ + + m 3 3 K a 2 1 = Ω +Ω +Ω m Λ K

3 3 independent data sets intersect

4 Problem of LCDM Huge difference in scales (theory vs observation) ρ ρ Λ Λ obs theory Λ = 8π G M H M (10 ev) (10 GeV) 10 GeV fundamental pl > (10 GeV) 10 GeV No theory to explain this huge mismatch Important to reconsider all the assumptions that lead to this conclusion LCDM: 4D general relativity + cosmological const

5 Alternative models Alternative - modified gravity dark energy is important only at late times large scales / low energy modifications cf. precession of perihelion dark planet v GR

6 Is cosmology probing breakdown of GR on cosmological scales?

7 Many approaches Constrain specific MG models A few models are known f(r) ) gravity Higher-dimensional gravity (DGP model) Model independent tests of GR Song and Koyama arxiv: Zhang and Jain arxiv:

8 DGP model (Dvali, Gabadadze,Porrati) (5) (5) 4 S = d x g R d x g R Lm 32πGr + c 16πG + ( ) Crossover scale r c r c r < r c r > r c 4D Newtonian gravity 5D Newtonian gravity gravity leakage Infinite extra-dimension

9 Cosmology of DGP Friedmann equation H 2 8π G = H r 3 c early times 4D Friedmann late times Hr c ρ (Deffayet) 1 ρ 1 0 H rc H 0 r c 1 16π G + 4 d x Lm 4 d x 1 32πGrc 1 4rH ( ) 2 r r m K r = Ω + Ω +Ω +Ω, Ω = c c c c 0 g R 5 (5) (5) dx g R

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

11 Tension with data Tension with data open model improves the fits (Song, Sawicki, Hu astro-ph/ ) K=0 H πg = ρm H r c equivalent to dark energy model with 1 w = V w = 1 1 +Ω ( a) m (Lue.et.al) (LCDM)

12 Baryon acoustic oscillation WFMOS type survey (Yamamoto et.al) We can clearly distinguish between LCDM and DGP

13 Dark energy vs DGP Can we distinguish between dark energy in GR and DGP? w = 1 LCDM DGP w = 0.7 z rz ( ) = dzhz ( ) 0 DGP model is fitted by wa ( ) = w + w(1 a), w 0 0 = 0.78, w = 0.32 (Linder) a a 1 Dvali and Turner

14 Growth rate of structure formation Evolution of CDM over-density GR δ + 2Hδ δρ = 4πGρδ δ = If there is no dark energy δ a dark energy suppresses the gravitational collapse ρ DGP + 2H = 4 Geff δ δ π ρδ an additional modification due to modified gravity

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

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

17 Weak lensing WL is sensitive to the growth rate / MG GR DGP WFMOS { a ( )} ga ( ) = exp dln aω( a) γ 1 0 ( wz ) γ = ( = 1) γ = 0.68 γ (Yamamoto et.al) 0.1 (Linder)

18 Peculiar velocity Peculiar velocity θ = δ Red-shift distortion DE + MG models LCDM MG model (modification to GR) DE model (GR) f f = d d lnδ ln a (Wang)

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

20 Constraining MG models It is essential to combine many observations geometrical tests + structure formation Non-linear clustering is largely unknown theory must approach GR to satisfy solar system constraints N-body, renormalized perturbations Theorists should work hard no convincing models (yet!) f(r) ) models need contrive f(r) ) to avoid solar system constraints DGP model suffers from quantum (ghost) instability

21 Is it possible to test GR without assuming any theory? Song and Koyama arxiv: cf. Zhang and Jain arxiv:

22 Cosmological principle Friedman universe + perturbations Metric ds 2 = (1+ 2 Ψ ) dt 2 + a( t) 2 (1+ 2 Φ) dx Newtonian potential 3D curvature perturbation 3 Matter perturbations T T 0 ρ i = ( + δρ) = ( ρ + p ) v 0 0 i density perturbations peculiar velocity

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

24 Equations in Newton limit Equations of motion δ for matter 2 k m Ψ= θ 2 m + Hθ, m δm = a a Gravitational equations (GR) 2 8π G H = ρ, ρ = ρ 3 k a 2 2 T T i i Φ= 4π G ρδ ρδ = ρδ T T T T i i i Newton constant is universal in spacetime θ

25 Consistency equation Consistency equation Peculiar velocity α( kt, ) = 2 2 k ( ) 2 2 3aH δt Φ Ψ +Ψ k a 2 2 Ψ= da ( θ ) dt m bakcground Weak lensing Galaxy distribution In GR α ( kt, ) = 1 all quantities are observables no need to assume any theory!

26 Consistency condition If If α( kt, ) 1 Modified Gravity Friedman eq.. and/or Poisson eq.. are modified Consistency test all observables should be measured in the same time and at the same location Weak lensing measures an integrated effect of potentials along line of sight

27 Reconstruction of gravitational lensing from density distribution and peculiar velocity at each red-shift bin δ T θ m k a k a 2 Φ= πg ρδ 2 4 T T 2 2 Ψ= da ( θ ) dt m GR is assumed φ = Φ Ψ Reconstructed gravitational lensing Consistency test Observed gravitational lensing

28 2D power spectrum of lensing potential test of reconstruction in GR Ratio of reconstructed and true spectrum in MG reconst C l α = log 10 true Cl

29 Cosmological test of GR Systematic Systematic in every observation contaminates the test We can do the test on various scales also a subset of the test e.g. γ ( kt, ) = Ψ / Φ c.f Einstein ring + velocity dispersion of galaxies θ γ π σ 2 2 E = (1 + )2 ( v / c )( DLS / DS) γ = 0.98 ± 0.07 (Bolton et.al)

30 by Bhuvnesh Jain Ψ Φ Ψ

31 Conclusion 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 accurate observational data sets are required understanding of systematic in observation is necessary

32 Different observables measure different physical quantities and they are all valuable if we just extend our theoretical prior to include MG Φ wz ( Ψ) Φ Ψ