Structure formation in modified gravity models

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1 Structure formation in modified gravity models Kazuya Koyama Institute of Cosmology and Gravitation University of Portsmouth

2 Dark energy v modified gravity Is cosmology probing the breakdown of general relativity at large distance?

3 General relativity Why do we believe general relativity? Observational point of view GR is tested to very high accuracy by solar system experiments and pulsar timing measurements C. Will gr-qc/ Theoretical point of view GR is the unique metric theory in 4D that gives second order differential equations

4

5 Brans-Dicke theory Action ( ) 4 BD S d x R V V H M 0 pl f(r) gravity: BD 0 quasi-static approximations (neglecting time derivatives) ds (1 ) dt a( t) (1 ) dx 0 (3 BD) 8 G 1 4 G

6 Constraints on BD parameter Solutions (3 BD) 8 4 BD 4 BD 4 G, Geff G 3 BD 3 BD 1 BD 1 PPN parameter G BD 1 BD BD 5 1 (.1.3) 10 BD 40,000 This constraint excludes any detectable modifications in cosmology

7 General picture Largest scales gravity is modified so that the universe accelerates without dark energy Large scale structure scales gravity is still modified by a fifth force from scalar graviton 1 H 0 r * GR Modified gravity Scalar tensor Small scales (solar system) GR is recovered by screening mechanism

8 How to suppress the fifth force (1) 4 BD( ) S d x R V ( ) Lm[ g] GR is recovered if (i) the mass is large V '' (ii) the kinetic term is large BD These limits should be realised in environmentally (density) dependent way to avoid the recovery of GR on all scales

9 Chameleon/symmetron/dilaton Einstein frame 1 ( ) [ ( ) ] V ( ), V ( ) V ( ) ( A( ) 1) 4 SE d x g R V Lm A g eff eff A BD m 1 A 1

10 How we recover GR on small scales Chameleon mechanism (Khoury & Weltman)

11 How to suppress the fifth force () Vainshtein mechanism originally discussed in massive gravity rediscovered in DGP brane world model linear theory 3 8 G BD 1 4 G 0 even if gravity is weak, the scalar can be non-linear i j rc i j Ga 3 8 rc 1 H 0

12 Vainshtein mechanism Spherically symmetric solution for the scalar 3 3 rg V ( r), ( r) V d r r dr r r r 1 3 c g 8rr rv, rg GM 9 rg r V rc 4D Einstein rg r r, r g rc rg r r r g rc 4D BD rg, r 3 rg 4 r 3.95km 0.1 kpc 3000Mpc for the Sun

13 Solar system constraints The fractional change in the gravitational potential The anomalous perihelion precession r r r r r The vainshtein radius is shorter for a smaller object 5 Lunar laser ranging: the Erath-moon distance r km E M 1/ 3 3 M pl r EM , r EM 4 8 rm c r V rc 1 H 0

14

15 Generalisaitons Galileon models c Nicolis, Rattazzi, Trincherini Massive gravity models Galileon models are naturally realised in the decoupling limit De Rham, Gabadadze, Tolley

16 phenomenology

17 (1) Environment dependent screening In modified gravity models, dynamical mass inferred from velocity dispersions and lensing mass can be different k Ga k a ( ) / 4 (, ) k Ga k a 4 (, ) m m m m f(r) 1 [1: 4 / 3] The fifth force does not change geodesics of photon The fifth force enhances Newtonian gravity Difference between dynamical and lensing masses d( r) / dr M ( r) 1, ( ) / d ( r) / dr M [0:1/ 3] Schmidt

18 Environmental effects Environment D d / r, M M Halo mass NB NB Zhao, Li, Koyama r NB d D D1 1 D 1 D 11 Large bubbles =better screened (GR is recovered)

19 Environmental effects GR is recovered in large halos / dense environment D d / r, M M NB NB Zhao, Li, Koyama r NB d D10 D10 D 1 D 1 1 Large bubbles =better screened (GR is recovered)

20 Creating a screening map It is essential to find places where GR is not recovered Small galaxies in underdense regions SDSS galaxies within 00 Mpc Cabre, Vikram, Zhao, Jain, KK f R f R 0 6 GR is recovered

21 Astrophysical tests Apparent violation of equivalent principle HI gas: unscreened Hui, Nicolis, Stubbs Jain & VanderPlas Stellar disk: screened The rotation curve of HI gas is enhanced compare with stars The stellar disk is displaced from the HI gas disk This happens only in unscreened galaxies (i.e. dwarf galaxies in voids)

22 () Vainshtein mechanism Vainshtein mechanism dark matter halos Schmidt Screening is nrealy independent of environment and mass Modified force (differnece between lensing and dynamical mass mass

23 Observational implication Morphology dependence The non-linear term vanishes for 1D plane wave i j i j screening is weak in filaments Apparent equivalent principle violation Hui, Nicolis stars can feel an external field generated by large scale structure but a black hole does not due to no hair theorem BH central BH lag behind stars

24 Vainshtein screened two bodies Non-superposition Hiramatsu, Hu, KK, Schmidt Two body problem (cf. Earth-moon)

25 Second derivatives Near a small body B (moon) O( M / M ) A A B the interference term cancels the second derivative of the field from the large body (Earth) Earth Surface of body B Earth Moon

26 First derivatives Effects on the first derivative (non-radial force) is small (0) 0.56 M B Q1 M A precession anomaly per orbit 0.6 Q for earth-moon The motion of screened objects depend on their mass

27 Linerisation Hui, Nicolis If body B is outside of the Vainshtein radius of body A body B still feels the force from body A as we can add a constant gradient to the solution (Galileon symmetry!) B A A const. near body B Two screening mechanisms give very different pictures Vainshtein Chameleon long wavelength mode

28 Challenge for simulations Screening mechanism governed by a non-linear Poisson equation 4 GA( ) V '( ) N[, ] no superposition rule it is not possible to separate long and short range forces need to solve the non-linear Poisson equation on a mesh MLAPM Li, Zhao , Li, Barrow Zhao, Li, Koyama Oyaizu et.al, Schmidt et.al. ECOSMOG (based on RAMSES) Li, Zhao, Teyssier, Koyama Jennings et.al , Li et.al Brax et.al

29 Conclusion Modifications to GR generally introduce the fifth force, which should be screend 1) break equivalence principle and remove coupling to baryons Einstein frame - interacting dark energy models ) Environmentally (density) dependent screening Chameleon/Symmetron/dilaton models 3) Vainshtein mechanism massive gravity, Galileon models, braneworld models Non-linearity of the Poisson equation for the fifth force leads to rich phenomenology

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