Structure formation and observational tests. Kazuya Koyama University of Portsmouth

Transcription

1 Structure foration and observational tests Kazuya Koyaa University of Portsouth

2 How to test D/MG odels instein equations M G T, T G ( T ) 0 M D MG Background (hoogeneity & Isotropy) everything is deterined by the equation of state Sall Inhoogeneity i j ds a ( ) (1 ) d (1 ) ijdx dx Linear scalar perturbations with respect to 3-space (assued to be flat) diag (, P, P, P ) w P / H = a ' a

3 Cosological perturbation theory Fourie transforation and Decoposition, S k S S e ˆ, ˆ ki Si i ki S ki k ij ˆˆ i Sij kik j S, Aij AL ij AT Sij, Si 0 3 Gauge fixing i k x i i Kodaa & Sasaki Mukhanov, Feldan, Brandenberger Malik & Wands We assued the theory is invariant under diffeoorphis x x and used the Longitudinal gauge. The gauge invariance can be always restored by introducing additional fields (Stuckelberg fields)

4 Matter content nergy oentu tensor T ( I I ), ( I PI ) vi i ( P ) v, P P P I i i i I I I I I j I I j v i i j v S i, S i j : anisotropic stress Conservation of energy oentu tensor for now, we assue atter and dark coponent obeys the conservation independently di d 3 H ( P ) ( P )( kv 3 ) I I I I I d ( I PI ) vi 4 H PI PI I ( I PI ) d k 3 w P / 0, P 0, w P /, P,,

5 quations for linear perturbations instein equations k 4 Ga ( ), II I 3( I PI ) vi k ( ) 8 Ga P Conservation of energy oentu tensor for atter ' H, H' k ' H 1 H H volution of atter is deterined by the Newtonian potential Dark coponent affects the evolution through the Newtonian potential H k k / H v : velocity divergence k H / 1 '' H ' k

6 quations for dark coponent Conservation of dark coponent (assued to be held independently) d d Sound speed c s 3 H ( P ) ( P)( kv 3 ) d (1 w ) v 4 H P P ( P ) d k 3 P v0 v P ' w' P cs ( cs ca ) ', ca w : adiabatic sound speed k ' 3 H (1 w)

7 Classification (1) 1) LCDM k 4Ga 0 We define the growth function solution for in MD era, a as the growing ode at late ties, due to the cosological constant, gravity becoes weaker g '' H ' 4Ga 0 D Ga, H 8 / 3 D a H 8 Ga 3 D a ds LCD M LCDMdark energy a DG P

8 Classification () w ( z) w w D 0 a z 1 z ) sooth D using N ln a 0 '' H ' 4Ga D (1 3 DwD ) D D 0 3 w (1 ), 1 D D D a For a fixed present-day,if w 1,D density is D larger in the past suppressing the growth copared with LCDM D Linder astro-ph/030586

9 Growth rate Growth rate f d ln a dd d ln a D da f f w (1 ) f D f, (1 w ( z 1)) is insensitive to the equation of state (but the growth rate depends on through ) D w D wd Dossett & Ishak

10 Classification (3) Clustering D 0 ( 0) Let s consider a toy odel for dark coponent with non-zero sound speed P c s assuing that the dark coponent doinates the universe '' H ' csk 4Ga 0 4 Ga For k k pressure wins over gravity and does not grow J c s clustering D requires sall sound speed k a 1 J / : Jean's length

11 Quintessence Sound speed c s P 1, v 0 propagation speed v scalar field equation of otion k a 3 H cp S d x g V ( ) for standard kinetic ter, c thus the scalar field does no cluster p cs 1 below the horizon scale thus can be approxiated as sooth D 0

12 Quintessence Note that this does not ean we can ignore the perturbations of scalar field entirely Caldwell: An introduction to quintessence

13 K-essence/assive scalar field K-essence S d x gk X X K, X cs 1 (if K, X X K, XX ) K X K 1 4 ( ),, X, XX Massive scalar field scalar field perturbations oscillate with frequency averaging order any oscillations k c s ( k a ) 4a ( k) k / a

14 ffects on growth instein equations k 4a G 1 0 D '' H D ' 4Ga 1 D 0 clustering D acts like odifications of gravity for dark atter Anisotropic stress scalar field does not have anisotropic stress noral atter has sall anisotropic stress copared with density O( / G) O( a / k ) 0

15 Classification (4) Brasn-Dicke gravity BD 0 1 k 4 Ga k k 4 Ga ( ) k ( ) 8 Ga P 1 4Ga P 4Ga k f(r) gravity exaple S d x g R S [ g ], (1 ) 16 G 4 1 M 4 FR ( ) S d x g L 16G 3 3a 8 Ga, F 3F, R, RR

16 Growth in f(r) gravity Poisson equation 4 a / k k 4 G a 3 a / k large scales k k Sall scales 4 Ga 16 Ga 3 GR 1 BD Song et.al. astro-ph/ Sall scales D a Large scales Pogosian & Silvestri Gravity is odified on sall scales so we need screening echanis

17 Classification (5) So far, we assued that atter and dark coponent obey the conservation equation independently, but this is not necessarily the case. ( T ) 0 xaple: ' H Q T Although we do not odify gravity, this looks like odified gravity Q Q H' ' k ' H 1 ( ) H H H (we can evade local constraints by breaking strong equivalence principle) 1 ( ) [ ( ) ] A( ) exp M pl 4 S d x g R V SM A g

18 Zoology of D/MG odels LCDM ( ) Clustering D K-essence (, w, ) Sooth D (, w) Quintessence Modified gravity (, w,, ) (screening echaniss) Interacting D (, w,, Q ) (violation of equivalence principle)

19 Observations Background H( z) Supernovae: luinosity distance CMB/Baryon Acoustic Oscillation (BAO): angular diaeter distance SA uclid BOSS SA Planck

20 Observations Weak lensing i j ds a (1 ) d (1 ) ijdx dx Convergence (photons follow geodesic) ( ) 1 s ( n) d W( 0, n), W ( ) s geoetry Galaxy shape is deterined by shear which can be coputed fro convergence Bartelann & Schneider astro-ph/991508

21 Observations CMB Integrated Sachs-Wolfe (ISW) effect The tie variation of lensing potential causes a shift of photon teperature (, ) ( k) d k W j [ k( 0 0 )] lensing CMB is also lensed ( n) ( n d) lensed 1 W ( ) ( ) 1 d, ( n) d (, n), ( ) LSS W 0 W LSS

22 Observations Redshift distortions galaxies have peculiar velocities clustering of galaxies in redshift space is enhanced along the line of sight s ( k, ) ( k) ( k), k n k s r ( v n) n / H, n r / r Hailton astro-ph/ If the continuity equation holds, the velocity dispersion is related to the growth rate s ( k) ( k, ) ( k) 1 ( k) 1 f ' H ( k) d ln a dd f d ln a D da

23 Background expansion history Background expansion is deterined by the equation of state Paraetrisation w w w (1 a) w w D 0 a 0 a z 1 z Planck collaboration Linder astro-ph/

24 Model independent approach Principal coponent analysis Approxiate rrors on w ( ) D z 1 w ( z) w ( z) N D i i i1 w i with any stepwise constant values are highly correlated. We diagonalise the covariant atrix C ( w w )( w w ) ij i i j j 1 T C W W, W ( e1, e,...,), ij i ij 1 w ( z) e ( z) D i i i1 rrors on new paraeters N W ( w w ) i ij j j j are uncorrelated and given by i

25 Principal coponent Requires a prior to truncate poorly deterined eigen odes Zhao et.al

26 Model based paraetrisation Paraetrisation 1 w ( a) 3 1 V ', V ( ) s D ( a 0) Planck

27 xpansion history v structure growth LCDM/Sooth D There is a one-to-one correspondence between background expansion history and growth of structure rz () w 1 LCDM w 0.7 a LCDM M dark energy Sooth D Sooth D z a Linder astro-ph/050763

28 xpansion history v structure growth Clustering D/MG structure growth is controlled also by ven if it has the sae expansion history as sooth D, structure growth is different rz () w 1 LCDM w 0.7 Clustering D/MG Sooth D z a Clustering DG D/MG P a LCDM M dark energy Sooth D Linder astro-ph/050763

29 Consistency test Assue that the Universe is described by a clustering D/MG odel but we still try to fit the date using sooth D w( z) w w z, 0 1 SNe+CMB SNe+weak lensing Inconsistent! Ishak et.al. astro-ph/

30 Clustering D v Modified gravity Modified gravity odels have anisotropic stress 1 cf. BD gravity with BD 0 4Ga P 4Ga k this creates a difference between lensing potential and Newton potential 1 W lensing ass is not the sae as dynaical ass ex.) g paraeter G 3Ha 1 0 k ( ) 8 Ga P H' k ' H 1 H H

31 Reyes et.al