Constraining gravity on cosmic scales with large scale structure In collaboration with: Istvan Laszlo (Cornell) Rachel Bean Matipon Tangmatitham (Cornell) Cornell University Sarah Bridle (UCL) Scott Dodelson (Chicago) Donnacha Kirk (Imperial) Michele Liguori (Cambridge) Pengie Zhang (Shanghai) Rachel Bean: IPMU January 2012 Weak field tests of gravity • Terrestrial and Solar System – Lab tests on mm scales – Lunar and planetary ranging • Galactic – Galactic rotation curves and velocity dispersions – Satellite galaxy dynamics • Intergalactic and Cluster – Galaxy lensing and peculiar motions – Cluster dynamical, X-ray & lensing mass estimates • Cosmological – Late times: comparing lensing, peculiar velocity, galaxy position, ISW correlations – Early times: BBN, CMB peaks Rachel Bean: IPMU January 2012 k2φ = 4πGQa2ρ∆ (18) − 1 d(aθ) (1 + R) = k2 ψ (19) a dτ 2 k2ψ = k2Rφ shear stresses (20) − 2 R(a)= (21) 3β(a) 1 − 1 Q(a)=1 (22) − 3β(a) k2 [˙α + (1 + R) α Rη]= 12πGa2ρ(1 + w)σ (23) H − − ds2 = a(τ)2[ (24) Why modify gravity− on cosmic scales? • Cosmic acceleration = a modification of Einstein’s equations ds2 = a(τ)2[1 + 2ψ(x,t)]dτ 2 + a(τ)2[1 2φ(x,t)]dx2 (25) − − Modified Gµν =8πGTµν (26) gravity? New matter? interactions? Λ? Inhomogeneous mat f(gµν,R,Rµν) + Gµν universe?=8π GTµν (27) Can we distinguish whether accelerationmat DE evidence for modifiedGµν gravity,=8 πΛ Gor a Tnewµν type+ Tofµ matter?ν (28) Rachel Bean: IPMU January 2012 1 S = d4x√ g R + d4x√ g (29) − 16πG − Lmat 1 S = d4x√ g (R + f (R))+ d4x√ g (30) − 16πG 2 − Lmat 1 S = d4x√ g f (φ)R + d4x√ g (31) − 16πG 1 − Lmat 1 S = d4x√ g R + d4x√ g (32) − 16πG − Lmat,DE I-2 How might we modify gravity? • Active area of research, many different options • Scalar tensor gravity simple models for which we can model effects 1 4 11 4 S = d 4x√ g R + d 4x√ g mat (30) GR SS == dd xx√ gg RR++ dd xx√ gg matmat (30)(30) −− 1616ππGG −− LL (31)(31) 1 4 11 4 S = d 4x√ g (R + f2(R))+ d 4x√ g mat (32) f(R) gravity SS == dd xx√√ gg ((RR++ff22((RR)))+)+ dd xx√√ gg matmat (32)(32) −− 1616ππGG −− LL (33)(33) 1 4 11 4 Scalar tensor gravity SS == dd4xx√√ gg ff1((φφ))RR++ dd4xx√√ gg mat (34)(34) −− 1616ππGG 1 −− LLmat (35)(35) 5 (5) 1 (5) 4 S = d55x g(5)(5) 1 R(5)(5) + d44x√ g (36) Higher dimensional gravity e.g. DGP SS == dd xx gg (5) RR ++ dd xx√√ gg mat (36)(36) −− 1616ππGG(5)(5) −− LLmat (37)(37) 4 1 4 44 √ 1 44 √ SS == dd xx√√ gg RR++ dd xx√√ gg mat,DE (38)(38) −− 1616ππGG −− LLmat,DE Rachel Bean: IPMU January 2012 a¨a¨ 44ππGG == ((ρρm +3+3PPm)) (39)(39) aa −− 33 m m a¨a¨ 44ππGG == ((ρρ+3+3PP ++ρρDE +3+3PPDE)) (40)(40) aa −− 33 DE DE a¨a¨ 44ππGG stuffstuff ++ == ((ρρm +3+3PPm)) (41)(41) aa −− 33 m m I-3I-3 1 S = d4x√ g R + d4x√ g (30) − 16πG − Lmat (31) 1 S = d4x√ g (R + f (R))+ d4x√ g (32) − 16πG 2 − Lmat (33) 1 S = d4x√ g f (φ)R + d4x√ g (34) − 16πG 1 − Lmat (35) (5)3 mp S = d5x g(5) R(5) + d4x√ g (36) − 2 − Lmat (37) 1 S = d4x√ g R + d4x√ g (38) − 16πG − Lmat,DE a¨ 4πG = (ρ +3P ) (39) a − 3 m m Modifications to GR a¨ 4πG = (ρ +3P + ρDE +3PDE) (40) • Alter Friedmann and aacceleration− 3 equations at late times a¨ 4πG stuff + = (ρ +3P ) (41) a − 3 m m e.g. f(R) gravity a2 3 1 a¨ 4πG H2f + f + Hf˙ + f¨ + = (ρ +3P ) − R 6 2 R 2 R a − 3 e.g. DGP gravity H˙ a¨ 4πG + = (ρ +3P ) − rc a − 3 Rachel Bean: IPMU January 2012 I-3 Expansion history constraints:ANRV352-AA46-11 ARI 25 July 2008 2:1 Cosmic geometry Standard candles 46 a 44 (w0, wa) 68% confidence region (Ω , Ω ) = (0.3, 0.7) M Λ 42 (Ω , Ω ) = (0.3, 0.0) M Λ (Ω , Ω ) = (1.0, 0.0) M Λ 40 (mag) µ 38 Nearby ESSENCE 36 SNLS Riess et al. (2004) 34 1.5 b Standard rulers1.0 by University of Tokyo on 01/22/12. For personal use only. 0.5 0 (mag) µ ∆ Annu. Rev. Astro. Astrophys. 2008.46:385-432. Downloaded from www.annualreviews.org –0.5 –1.0 –1.5 10–2 10–1 100 Redshift (z) Figure 6 Type Ia supernovae (SNe Ia) results: ESSENCE (Equation of State: Supernovae Trace Cosmic Expansion) ( purple diamonds), SNLS (Supernova Legacy Survey) (green crosses), low-redshift SNe (orange starbursts), and the compilation of Riess et al. (2004), which includes many of the other published supernova distances plus those from the Hubble Space Telescope (blue squares). (a) Distance modulus versus redshift (z) measurements shown with four cosmological models: ! 0.3, !" 0.7(black short-dashed line); ! 0.3, !" 0(black M = = M = = long-dashed line); ! 1, !" 0(red dotted line); and the 68% CL allowed region in the w -w plane, assuming spatial flatness and a M = = 0 a prior of ! 0.27 0.03 ( yellow shading). (b) Binned distance modulus residuals from the ! 0.3, !" 0 model. Adapted from M = ± M = = Wood-Vasey et al. (2007). www.annualreviews.org Dark Energy 399 • Rachel Bean: IPMU January 2012 ANRV352-AA46-11 ARI 25 July 2008 2:1 2.0 a Complementary expansion history constraints No Big Bang 1.5 0 b BAO Ω Λ 1.0 –0.5 CMB SNe w 0.5 –1.0 CMB SNe BAO Flat 0 –1.5 0 0.5 1.0 0 0.1 0.2 0.3 0.4 0.5 Ω ΩM M Figure 8 Kowalski 2008 Rachel Bean: IPMU January 2012 (a) Constraints upon !M and !" in the consensus model (cosmological constant/cold dark matter model) using baryon acoustic oscillations (BAO), cosmic microwave background (CMB), and supernovae (SNe) measurements. (b) Constraints upon !M and constant w in the fiducial dark energy model using the same data sets. Reproduced from Kowalski et al. (2008). by University of Tokyo on 01/22/12. For personal use only. in these two models; although the mix of data used here differs from that in Table 1 (supernovae are included in Figure 8), the resulting constraints are consistent. Regarding Sandage’s two numbers, H0 and q0, Table 1 reflects both good agreement with and a smaller uncertainty than the direct H0 measurement based upon the extragalactic distance scale, H0 72 8 km/s/Mpc (Freedman et al. 2001). However, the parameter values in Table 1 are Annu. Rev. Astro. Astrophys. 2008.46:385-432. Downloaded from www.annualreviews.org = ± predicated on the correctness of the CDM paradigm for structure formation. The entries for q0 in Table 1 are derived from the other parameters using Equation 6. Direct determinations of q0 require either ultraprecise distances to objects at low redshift or precise distances to objects at moderate redshift. The former are still beyond reach, whereas for the latter the H0/q0 expansion is not valid. If we go beyond the restrictive assumptions of these two models, allowing both curvature and w to be free parameters, then the parameter values shift slightly and the errors increase, as expected. In this case, combining WMAP, SDSS, 2dFGRS (Two-Degree-Field Galaxy Redshift Survey), 0.016 and SNe Ia data, Spergel et al. (2007) yield w 1.08 0.12 and !0 1.026+0.015, whereas = − ± = − WMAP SDSS only bounds H to the range 61 84 km/s/Mpc at 95% confidence (Tegmark + 0 − et al. 2006), comparable to the accuracy of the HST Key Project measurement (Freedman et al. 2001). Once we abandon the assumption that w 1, there are no strong theoretical reasons for = − restricting our attention to constant w. A widely used and simple form that accommodates evolu- tion is w w (1 a)w (see Section 6). Future surveys with greater reach than that of present = 0 + − a experiments will aim to constrain models in which !M, !DE,w0, and wa are all free parame- ters (see Section 8). We note that the current observational constraints on such models are quite www.annualreviews.org Dark Energy 401 • mat f(gµν,R,Rµν) + Gµν =8πGTµν (30) mat f(gµν,R,Rµν) + Gµν =8πGTµν (30) mat DE Gµν =8πG Tµν + Tµν (31) mat DE Gµν =8πGTµν + Tµν (31) mat f(gµν,R,Rµν) + Gµν =8πGTµν (30) mat Gµν =8πGTµν (32) mat Gµν =8πGT (32) mat DE µν GµConformalν =8π NewtonianG Tµν + gaugeTµν (31) 4 1 4 S = d x√ g 2 R + d x√ g mat 2 (c) (33) 4 − 16k πφ1G+3 (φ˙ + 4 ψ)=− L 4πGa ρδ (1) S = d x√ g RH+ dHx√ g mat− (33) − 16πG mat2 − L 2 (c) Gµν =8πGTk (φ˙ + ψ)=4πGa ρ(1 + w(32))θ (34) (2) µν H (34) 4 1 2 4 2 S = d x√ g 1 (R k+ [fφ2(Rψ)])+ = 12d πxGa√ ρg(1 +matw)σ (35) (3) S 4 = d4x1√− g16πG 4(R + f (−R))+ d4x√ −g L (35) S = d x√ g −R +16πGd x√ g2 mat − Lmat(33) − 16πG − L (36) Synchronous gauge (36) 4 1 4 (34) S = d x4 √ g 1 1 f1(φ)R + d4 x√ g mat (37) S 4 = d x1√2− g16πG˙ f1(φ)R + 42d x(s√) −g Lmat (37) S = d x√ g k η−(R16+πhfG2(=R))+4πGad xρδ√ g− matL (35) (4) − 16πG − 2H − − L (38) 2 2 (s) (38) k η˙ 1=4πGa ρ(1 + w)θ (36) (5) 5 (5) 1 (5) 4 S = 2d x5 g (5) R(5) + 2 d4 x√ g mat (39) S 4 = k [˙dα +2x1 αg η]=(5) R412π+Ga ρd(1x +√w)gσ mat (39) S = d x√ g −−f (φ16)16Rππ+GG(5)d x√ g −−LL (37) (6) H1 − − mat − 16πG − L h˙ δP (s)(40)(40) δ˙(s) = (1 + w) θ(s) + 3 w δ(s) 1 (38) (s) (7) 4 4 1 − 44 2 − H δρ − S S == ddxx√√ 1gg RR++ d xx√√ gg mat,DEmat,DE (41)(41) S = d5x g(5) −− 1616ππRGG(5) + d4x√−−gLL (39) − 16πG(5) − Lmat 1 df − (40) Q = 1+ (8) 4 1a¨a¨ 44ππGG4 dR S = d x√ g =R= + d ((xρρ√m +3g PPmat,DEm)) (41) (42)(42) − 16πGa − 3 m− L m a − 3 a¨ 4πG Q 1=Q as (9) a¨=a¨ 44ππG(Gρm +3Pm) − 0 (42) a ==− 3 ((ρρ+3+3PP ++ ρDE +3+3PPDE)) (43)(43) aa −− 33 DE DE (10) s a¨ 4πG R 1=R0a (11) = (ρ +3P + ρDE +3PDE−) (43) a − 3 a¨ 4πG stuff +a¨ = 4πG (ρm +3Pm) (44) stuff + a= − 3 (ρm +3Pm) (44) a¨ 4aπG 3− 2 32 stuff + =TheQ 1+ inhomogeneous2(kρ2 κ a+3ρ(1P +)w) universe:(44) Matter k2η˙a= − 3 m m κ2a2ρ(1 + w)θ 2 3Q 2 2 1+∆T2 κ a ρ(1 + w) 2k φ˙ + ψ˙ (45) • Matter collects ∆T into˙ clumps,˙ and leaves voids: ∆˙ T QR ˙ 2T ˙Q∝˙ φ ++ Qψκ2a2ρ(1 + w) α (Q 1) α˙ + (45)Q˙ + Q(R 1) η density fluctuationsφT+ ψ 2 (45) + HT− ∝ H −∝H − − H − H (12) 1+ 3Q κ2a2ρ(1 + w) ρ(x,t) 2k2 δ(ρx(,tx,t) ) (46) δ(x,t) ≡ ρ(xρ¯,t) (46) ≡δ(x,tρ¯) (46) ≡ ρ¯ 2 2 • Motion over and abovek φ = the Hubble4πGa expansion:ρj∆j (13) − j peculiarθ velocityvθ fluctuationsv (47) (47) ≡∇· ≡∇ (14) θ v θ (47) ≡∇·∆ = δ +3 (1 + w) (15) d I-3 H k2 • Conservation(ax)= (a xof)+ energyav and momentum relate(48) dτ H evolution over timeI-3 to potentials 1 d(aθ) 2 δ˙ = θ +3φ˙ = k ψ (49) (16) − a dτ θ˙ = θ + k2ψ (50) −H k2ψ = k2φ shear stresses (17) Rachel Bean: IPMU JanuaryI-3 2012 − I-1 4 3 6 M6 5 M5 Scascade = d x√ g6 R6 + d x√ g5 R5 bulk − 2 4 brane − 2 − 2 4 M4 + d x√ g4 R4 + matter .