Constraining Gravity on Cosmic Scales with Large Scale Structure

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

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 evolution 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 parameters (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 . (1) 3 brane − 2 L − 4 M6 m6 = 3 . (2) M5

Conformal Newtonian gauge 3 M5 m5 = . (3) 2 2 2 (c) k φ +3 (φ˙ + ψ)= M4π4Ga ρδ (1) H H − 2 ˙ 2 (c) k (φ + ψ)=42πGa 1ρ/(13 + w)θ (2) HΛ5 =(m M4) . (4) k2 [φ ψ] = 125 πGa2ρ(1 + w)σ (3) − Synchronous gauge 4 3 1/7 Λ6 =(m6M5 ) (5) 1 k2η h˙ = 4πGa2ρδ(s) (4) − 2H − 3 2 2 (s) M5 k η˙5 =4πGa ρ3(1π/2 + w)θ 27 2 (5) S = d x√ g5 e− R5 (∂π) 5π 2 2 2 k [˙α +22 αbulk η]=− 12πGa ρ(1 +−w)16σ m6 (6) H − − ˙ 27 (s) 1 3 (s) 4 3π/2(s) h δP µ (s) 3 M5 δ˙ d=x√ (1q +ew−) θ K++ 3 ∂µπ∂ π nwπ +δ ((7)nπ) − − 32m2 (s) L 3 L brane − 2 − 6 H δρ − 2 4 M4 + d x√ q R4 + matter . (6) brane − 2 L 1 df − Q = 1+ (8) dR 2 2 (i) 3H M4 = ρm + ρπ , i Q 1=Qas (9) − 0 (10) 2 3 3 9 aHs π 3π/2 π 1 ρ M a 2 + a2H2 R 1=2 R0a 1 6e (11) , π 5 − 2 − ≡ 8m6 a − a − 2a − a The inhomogeneous universe: Metric and Matter 3 2 2 Q 1+ 2k2 κ a ρ(1 + w11) 3 12/5 k2η˙ = wπ = +κ2a2wρ(1 +(Hw)θ m6/a ) (7) 3Q 10 2 •2 Perturbed1+ κ2 ametric2ρ(1 +− w) 2k2 12/5 wπ 2 = 1/5(2 2 H m2 6/a ). (8) ˙ 2 ds˙ = Q(1−2 + 2ψ)dt + a (1 2φ)dx ˙ QR Q +−2 κ a ρ(1 + w) α− (Q 1) α˙ + Q + Q(R 1) η + H − H − H − − H − H (12) • Modified Einstein’s equations3Q relate2 2 matter and metric perturbations 1+ 2k2 κ a ρ(1 + w) 1 3 – Poissonw equation:= + Howw space(H respondsm /a to12 to/5 )local. density (9) π 2 2 6 2 2 12/5 Δ ∼ δ = density ﬂuctn in wπ k =1φ =/3(4πGa Hρj∆j m6/a ). (13) (10) − maer’s rest frame (θ=0) j (14) – Relate two potentials θ ∆ = δ +3 (1 + w) (15) k2ψ = k2φ shearH stressesk2 (11) − Typically shear negligible at late times φ≈ ψ I-1 gi = ikiψ (16)

Rachel Bean: IPMU January 2012

k2ψ = k2φ shear stresses (17) − I-1 Changing the relationship between φ and ψ

• Aim to describe phenomenological properties common to theories

• A modification to Poisson’s equation, Q k2φ = 4πGQa2ρ∆ − Q≠1: can be mimicked by – additional (dark energy?) perturbations, – modified dark matter evolution

• An inequality between Newton’s potentials, R ψ = Rφ

R≠1: not easily mimicked. – potential smoking gun for modified gravity? – Significant stresses exceptionally hard to create in non-relativistic fluids e.g. DM and dark energy.

Rachel Bean: IPMU January 2012 5

Here we have kept only the leading order term in kH . 1 Note that the exact choice of fζ is rarely important f(R) k/H =100 for observable quantities. Any choice will produce the PPF 0 correct behavior of the metric evolution since that de- 10 ! 2 pends only on enforcing ζ = O(kH ζ). Hence observ- 0.1 ables associated with gravitational redshifts and lensing are not sensitive to this choice. Only observables that A modified growth model – Theoretical examples -g depend on the comoving density on large scales beyond 1 7 the quasi-static regime are affected by this parameter. 0.1 Furthermore the super-horizon density perturbation in 0.01• 1.0 Newtonian gauge or any gauge where the density fluctu- DGP: Scale independent f(R) ation evolves as the metric fluctuation is also insensitive modifications cnl=0 100 to fζ .

On small scales, recovery of the modiﬁed Poisson equa- 2 2

i 0.9 0.1 1 # tion (10) from (16) implies 10 ! 100 /2

a /

) k − " Γ = fGΦ , (kH →∞) . (20) ( linear ! P

FIG. 3: Evolution and scale dependence of the metric ratio g 3 1 Q(1+R)/2 Finally to interpolate between these two limits we take in f(R) models compared with the PPF fit. Here B0 =0.4, k 0.8 the full equation of motion for Γ to be weff = −1andΩm =0.24. 10 2 2 ! 2 2 DGP 0.1 (1 + cΓkH ) Γ + Γ + cΓkH (Γ − fGΦ−) = S. (21) ! " PPF k/H0=1 For models where S → 0asa → 0 we take initial con- 100 ! ditions of Γ = Γ = 0 when the mode was above the 1.1 0.1 0.01 1 0.1 1 10 horizon. a k (Mpc/h) In summary, given an expansion history H(a), our 30 PPF parameterization is defined by 3 functions and 1 pa- 1.0 • f(R) gravity : scale dependent− i FIG. 6: Evolution and scale dependence of Φ in the DGP FIG. 7: PPF non-linear power spectrum ansatz for an f(R) rameter: the metric ratio g(ln a, kH ), the super-horizon ! models compared a PPF fit. Here Ωm =0.24 and the PPF / modifications model. The non-linear power spectrum is constrained to lie

relationship between the metric and density fζ (ln a), the " parameter cg =0.14. between two extremes: defined by halo-model mass functions quasi-static relationship or scaling of Newton constant ! k/H =0.1 with the quasi-static growth rate [cnl =0orP0(k)] and the 0.9 0 fG(ln a), and the relationship between the transition smooth dark energy growth rate with the same expansion Q(1+R)/2 scale and the Hubble scale cΓ. For models which modify history [cnl = ∞ or P∞(k)]. Here B0 =0.001, weff = −1and gravity only well after matter radiation equality, these f(R) IV. NON-LINEAR PARAMETERIZATION Ωm =0.24 with other parameters given in the text. relations for the metric, density and velocity evolution 0.8 PPF combined with the usual transfer functions completely As discussed in §II C, we expect that a successful modi- specify the linear observables of the model. In specific 0.1 1 fication of gravity will have a non-linear mechanism that 0.8 models, these functions can themselves be simply param- a suppresses modifications within dark matter halos. In f(R) eterized as we shall now show for the f(R) and DGP Rachel Bean: IPMU January 2012 this section, we construct a non-linearHu PPFand frameworkSawicki ‘07 models. FIG. 4: Evolution and scale dependence of Φ− in f(R) models based on the halo model of non-linear clustering. Al- 0.6 cnl=0 compared with the PPF fit. Here B0 =0.4, weff = −1and though a complete parameterized description of modified )-1 Ωm =0.24. k B. Models gravity in the non-linear regime is beyond the scope of ( f(R) 0.01 this work, the halo model framework allows us to incor- GR 0.4 /P

porate the main qualitative features expected in these )

In f(R) models, the Einstein-Hilbert action is supple- k models. Searching for these qualitative features can act (

mented by the addition of a free function of the Ricci by varying the action with respect to the metric. We P 0.1 scalar R. The critical property of these models is the ex- follow the parameterized approach of [24] where a choice as a first step for cosmological tests of gravity in the non- 0.2 1 istence of an extra scalar degree of freedom fR = df/dR of the expansion history through weff and the Compton linear regime. and the inverse-mass or Compton scale associated with scale today B0 ≡ B(ln a = 0) implicitly describes the Under the halo model, the non-linear matter power it. The square of this length in units of the Hubble length f(R) function and model. For illustrative purposes, we spectrum is composed of two pieces (see [32] for details is proportional to take Ωm =0.24 and weff = −1. and a review). One piece involves the correlations be- Given H(ln a) and B(ln a), the metric ratio at super- tween dark matter halos. As in general relativity, the 0.010.1 1 10 fRR ! H horizon scales comes from solving Eqn. (7) interactions between halos should be well described by k (Mpc/h) B = R ! , (22) 1+fR H linear theory. The other piece involves the correlations !! ! ! FIG. 8: Fractional difference in P (k) of the PPF non-linear 2 2 !! H B H ! within dark matter haloes. It is this term that we mainly where fRR = d f/dR . Below the Compton scale, the Φ + 1 − + + B Φ (23) f(R) ansatz from the smooth dark energy prediction with # H! 1 − B H $ seek to parameterize. metric ratio g →−1/3. the same expansion history. As cnl →∞deviations become ! !! ! Specifically given a linear power spectrum of density The evolution of B and the expansion history come H H B confined to the weakly non-linear to linear regime. The model + − + Φ =0, (kH → 0) . fluctuations P , the halo model defines the non-linear from solving the modified Friedmann equation obtained # H H! 1 − B $ L parameters are the same as in Fig. 7. spectrum as the sum of the one and two halo pieces

2 P (k)=I1(k)+I2 (k)PL(k) , (35) with function which describes the comoving number density 2 of haloes. y(M,k) is the Fourier transform of the halo dM M dn 2 density profile normalized to y(M,0) = 1 and b(M) is I1(k)= y (M,k) , ! M " ρ0 # $d ln M % the halo bias. Note that I2(k = 0) = 1 so that the linear dM M dn power spectrum is recovered on scales that are larger than I2(k)= b(M)y(M,k) , (36) the extent of the halos. ! M " ρ0 # d ln M A simple ansatz that restores general relativity in the where ρ0 = ρm(ln a = 0). Here the integrals are over the non-linear regime is that the mass function and halo pro- mass M of dark matter halos and dn/d ln M is the mass files remain unchanged from general relativity. Specifi- An common alternative parameterization

• One parameter measure of growth rate in CDM density perturbation

d ln δ f c Ω (a)γ ≡ d ln a ≡ m • γ∼0.55 in accelerated era, reasonably robust prediction of GR

• Can think of alternative measure to Q, no bearing on R 5

γ(a)=γGR(a)+∆γ

γGR

Rachel Bean: IPMU January 2012

−5 FIG. 1: The effect of changing Q [left panels] and R [right FIG. 2: The effect of changing ∆xH =0(GR), ±5 × 10 panels] in Model I, defined in (8). Shown are the CDM density in Model II defined in (11) for two different scale depen- fluctuation, ∆c,itsgrowthrate,f,andISWandweaklensing dences, nx =0.5ns [left panels] and nx =0.75ns [right pan- transfer functions, TI and Tκ,today,relativetotheirvalues els]. For this illustration we fix sx =1andincludeacut −1 for ΛCDM. Four values of Q0 and R0 are shown for a time- off, kx =0.01Mpc ,andallothercosmologicalparameters varying but scale-independent modification with s =3and fixed. Shown are the CDM density fluctuation, ∆c,itsgrowth kc = ∞, and all other cosmological parameters fixed. On rate, f,theISWtransferfunction,TI ,andequivalentvalue subhorizon scales, boosting the gravitational potential (Q> of Q,today,relativetotheirvaluesforΛCDM. A positive 1) boosts the growth of dark matter density perturbations ∆xH is equivalent to Q>1, suppressing cold dark matter and the lensing potential. If Q is increased, on subhorizon growth, that can give rise to f<0onlargescales,andboost- scales the boost in the gravitational potential opposes the ing the ISW component. For nx =0.5ns the modification late-time decay induced by accelerated expansion. The ISW has roughly the same spectral evolution as the CDM pertur- signal can be progressively reduced, canceled out (for Q a bations, leading to a scale independent effect on large scales. little larger than unity) and turn negative as Q increases. Increasing the spectral tilt, nx,exacerbatestheimpactofthe The modification has a larger and opposite effect on CDM modification for scales just below the horizon. growth on horizon scales, where the time evolution of the modification plays a key role. Increasing Q causes growth in ∆c to slow, or even decay at late times on the largest scales (with f = d ln ∆c/d ln a<0). The effect of increasing R IV. OBSERVATIONAL IMPACT OF A is quantitatively degenerate for subhorizon CDM growth and MODIFIED GROWTH HISTORY qualitatively similar otherwise, but with a slightly reduced amplitude. We can combine the CDM fluid equations, (2) and (3) 2 for w = cs = 0, to give a second order evolution equation for ∆c that makes no assumptions about the underlying gravitational theory,

f˙ ∆¨ + 1 1 ∆˙ + f ψ = c H − f c 1 ! H 1 " f˙ +3 φ¨ + ψ˙ + 1 1 (φ˙ + ψ)+ ˙ ψ (26) σ =0.05 and B 1; and a 10% intrusion of low-redshift H H − f1 H H B ! ! H " " galaxies into the≥ high redshift bins due to catastrophic photometric redshift errors is modeled by a third parame- with ter, C, using a a one-sided Gaussian prior with σC =0.10 2 2 f1 = k +12πGa (ρi + Pi). (27) and C 1. The correlation functions C1,2 are multiplied i by (A/B≥ )2, A2 and (AB)2/C respectively in the low, # medium and high redshift bins. On sub-horizon scales (26) reduces to the commonly Modeling and Figures of Merit (FoM)

• FoMs impose a simple, theory dependent prior – Assume scale independent, but allow simple redshift evolution

• γ FoM – 68% confidence limit in γ FoM =1/σ(γ)2 γGR+∆γ s γ(af)== γΩGRm((aa))+∆γ , ∆γ (z)=∆γ0a

• Q and R FoM Q(1+R)/2 – Photons trace φ+ψ ∼ Q(1+R) – Matter density related to φ ∼ Q – Area of 68% c.l. in Q and Q(1+R) plane FoM =1/A Q(a)=1+(Q 1)as,R(a)=1+(R 1)as 0 − 0 −

Rachel Bean: IPMU January 2012 Distinct behavior on large and small scales

Large scale suppression of growth Q,R>1 CDM Λ ratio to

CDM density perturbation densityCDM perturbation Small scale boosng of growth Q,R>1 -2 10

Rachel Bean: IPMU January 2012 Notation still converging…

– f (Stabenau & Jain 2006) – G /G (Song and Koyama 2008), Q eff – g (Guzik et al 2010) – µ (Zhao et al 2008) – V 2 /(G-V)-1 (Linder et al 2011)

– 1/(1+η) (Daniel et al 2008) R – 1/η (Song and Koyama 2008) – 1/γ (Zhao et al 2008) – 2G /V -1 (Linder et al 2011)

Rachel Bean: IPMU January 2012 ʹ ͹ Ǥ ʹ Ǧ ǡ ȋ !"#$%ʹͲͲͺȌǤ ǡ Ǥ Ǧ δͶΨȋ δͳͲΨȌ ǣ x ͵ &'()*Ǥ x Ͷ Ǧ+ Ǥ x ͷ+&'()*ǦǤ LSS observations: Galaxy positions • Photometric redshifts locate galaxies

in 3D (angular+redshift ) space z – Calibrate galaxy spectral energy densities (SED) – brightness vs frequency --against spectroscopic test set or templates

Spectroscopic • Tomography – split galaxies into redshift bins Photometric z – X-correlations between!"#$%&' ()*' ' +&,- z'./01.'-/&'2&%,0%3456&'0,'-/".'2/0-078'49#0%"-/3',0%'-/&' bins useful Credit: LSST Consortiumugrizy'.:.-&3'$."5#' &32"%"649'-&3294-&.'$5%&94-&;'-0'-/0.&'$.&;'"5'-/&'."3$94-"05<'1/&%&'1&'4229:'=0-/'34#5"-$;&'45;' for systematics .$%,46&' =%"#/-5&..' 2%"0%.*' >&?&6-"5#' #494@"&.' "5' -/&' 54%%01' =4;' 2/0-078' .-%"2' A9&,-B' 45;' 4;;"5#' ;&&2' C' D' 2/0-03&-%:' "32%0E&.' -/".' ,$%-/&%' 4.' ./015' 05' %"#/-*' F5' &46/' 64.&<' &32"%"649' .2&6-%49' -&3294-&.'4%&'$.&;<'50-'-/&'-&3294-&.'-/4-'#&5&%4-&;'-/&'."3$94-&;'64-490#.' Stage IV

dn/dz

z Rachel Bean: IPMU January 2012 LSS observations: Weak Lensing

Credit: NASA/ESA

Rachel Bean: IPMU January 2012 Disentangling dark energy and cosmic tests of gravity from weak lensing systematics 5

The noise for each survey is modeled as statistical errors Survey Parameters Stage III Stage IV given by

Area(sq. deg.) 5000 20000 2 "i"j γrms √2z0 0.8 0.9 N# = δij , (25) zmin 0.001 0.001 2nj zmax 33 ninj 1 N# = δij , (26) Ng 10 35 nj N 510 n " ph N i j =0, (27) σz0 0.07 0.05 # γrms 0.23 0.35 where γrms is the root mean square uncertainty in the shear measurement of the galaxies and nj is number of galaxies Table 1. Summary of the photometric large scale structure sur- th per steradian in j photometric redshift bin so i ni = Ng . vey specifications assumed for the Stage III and Stage IV survey: The survey specifications assumed in our analysis for survey area; median survey redshift, √2z ;minimumandmaxi- 0 the Stage III and IV surveys are given in TableP 1. mum redshifts observed, z and zmax;numberofgalaxies,per min We include complementary constraints from tempera- square arcminute, Ng;numberofphotometricredshiftbins,Nph; standard photometric redshift measurement error at z =0,σz0, ture (T) and E-mode polarisation (E) measurements from and the r.m.s. shear measurement error, γrms. aPlanck-likeCMBsurveyuptol =3000.Assummarised in Table 2, we model this by considering the three lowest frequency bands of the Planck HFI instrument, three chan- ν(GHz) 100 143 217 nels for temperature data and 2 for E mode polarisation,as described in the Planck Bluebook 7.Weassumeeachfre- f 0.8 0.8 0.8 sky quency channel has Gaussian beams of width θFWHM and θFWHM(arc min) 10.7 8.0 5.5 error in X = T,E of σX ,sothatthenoiseinchannelc is σ (µK) 5.4 6.0 13.1 T given by Disentangling dark energy and cosmic tests of gravity from weak lensing systematics 5 σE (µK) - 11.4 26.7 The noise for each survey is modeled as statistical errors Survey Parameters Stage III Stage IV 2 XX,c 2 #(#+1)θ given by /8ln(2) Area(sq. deg.) 5000 20000 FWHM,c 2 N =(σX,cθFWHM,c) e "i"j γrms , (28) # √2z0 0.8 0.9 N = δ , (25) # ij 2n Table 2. CMB survey specifications for a Planck-like survey. We zmin 0.001 0.001 j zmax 33 ninj 1 N 10 35 N# = δij , (26) model this on the temperature, T ,andE-mode polarisation spec- and over all channels,g nj N 510 n " ph N i j =0, (27) σz0 0.07 0.05 # ifications from three lowest frequency bands for the Planck HFI γrms 0.23 0.35 −1 − where γrms is the root mean square uncertainty in the shear 1 measurement of the galaxies and nj is number of galaxies instrument. XX Table 1. Summary of the photometricXX large scale structure sur- th per steradian in j photometric redshift bin so i ni = Ng . N# =vey specifications assumedN for#,c the Stage III and Stage IV. survey: The survey specifications assumed(29) in our analysis for survey area; median survey redshift, √2z ;minimumandmaxi- 0 the Stage III and IV surveys are given in TableP 1. mum redshifts" observed, zmin and zmax;numberofgalaxies,per# c We include complementary constraints from tempera- square arcminute,XNg“;numberofphotometricredshiftbins,” Nph; standard photometric redshift measurement error at z =0,σz0, ture (T) and E-mode polarisation (E) measurements from term and the mass source term must be different and the r.m.s. shear measurement error, γrms. aPlanck-likeCMBsurveyuptol =3000.Assummarised in Table 2, we model this by considering the three lowest frequency bands of the Planck HFI instrument, three chan- Q(χ)(R(χ)+1) 2.4 Intrinsicν(GHz) Alignments 100 143 217 nels for temperature data and 2 for E mode polarisation,as described in the Planck Bluebook 7.Weassumeeachfre- PδG(k, χ)= Pδδ(k, χ), (22) f 0.8 0.8 0.8 sky quency channel has Gaussian beams of width θFWHM and 2 θFWHM(arc min) 10.7 8.0 5.5 error in X = T,E of σX ,sothatthenoiseinchannelc is σ (µK) 5.4 6.0 13.1 » – Cosmic shearT describes the distortion ofgiven the by image of a dis- 2 σE (µK) - 11.4 26.7 2 Q(χ)(R(χ)+1) N XX,c =(σ θ )2 e#(#+1)θFWHM,c/8ln(2), (28) tant galaxy due to the bending of light# fromX,c FWHM,c that galaxy PGG(k, χ)= Pδδ(k, χ). (23) Table 2. CMB survey specifications for a Planck-like survey. We 2 model this on the temperature, T ,andE-mode polarisation spec- and over all channels, by gravityifications as from it three passes lowest frequency massive bands for the Planck large-scale HFI structure.−1 For a −1 » – instrument. XX XX th N# = N#,c . (29) galaxy in the i photo-z bin, the observed" ellipticity,c # &,of The growth of the dimensionless power spectrum Pδδ is it- X “ ” term andLSS the mass sourceobservations: term must be different Weak Lensing the galaxy can beQ(χ)(R written(χ)+1) as a sum2.4 of Intrinsic three Alignments independent self dependent on modified gravity parameters Q and R,as P (k, χ)= P (k, χ), (22) δG 2 δδ » – Cosmic shear describes the distortion of the image of a dis- contributions: the cosmic2 shear γG,theintrinsic,non-lensed Q(χ)(R(χ)+1) summarised by (7) and (8). P (k, χ)= P (k, χ). (23) tant galaxy due to the bending of light from that galaxy • 2D mapGG on the sky 2of galaxyδδ shape of the galaxy,» γI ,andapparentellipticityintroduced– by gravity as it passes massive large-scale structure. For a To obtain the lensing and galaxy position correlations in galaxy in the ith photo-z bin, the observed ellipticity, &,of ellipticitiesThe growth of the dimensionless power spectrum Pδδ is it- through instrumentalself dependent on modified gravity and parameters foregroundQ and R,as noise,the galaxy can&rnd be written, as a sum of three independent the modified gravity scenarios we integrate the full equations summarised by (7) and (8). contributions: the cosmic shear γG,theintrinsic,non-lensed To obtain the lensing and galaxy position correlations in shape of the galaxy, γI ,andapparentellipticityintroduced of motion using a modified version of CAMB (Lewis et al. i the modifiedi gravity scenariosi we integrate thei full equations through instrumental and foreground noise, &rnd, & (θ)=of motionγG using(θ)+ a modifiedγI version(θ)+ of CAMB&rnd (Lewis(θ et) al.. i i i i (30) & (θ)=γG(θ)+γI (θ)+&rnd(θ). (30) 2000). 2000). To support other researchers investigating the role of The cosmic shear signal γG is very small, and we cannot Themodified cosmic gravity shear models on large signal scale structureγG is observa- verymeasure small, directly and the intrinsic we shear cannot of any individual galaxy. To support other researchers investigating the role of tions, without having to integrate the full perturbation equa- To recover the cosmic shear, therefore, one averages over a • measureCoherent/correlatedtions, directly we provide the a fitting intrinsic function shape in the Appendixshear for of anynumber individual of galaxies on a small gala patch onxy. a sky. Assuming modified gravity models on large scale structure observa- the ratio, rfit(k, z), between a fiducial ΛCDM linear matter that their intrinsic ellipticities are distributed randomly, and that their light passes by similar large scale structure, the Todistortion recoverpower themeasured spectrum, cosmicPδδ,ΛCDM (k, zstatistically shear,) and the one for therefore, a modified one averages over a tions, without having to integrate the full perturbation equa- gravity model described in 2.1, parameterised by Q0,R0 and intrinsic ellipticities cancel in the two-point function, and s: we are left with the cosmic shear signal. number of galaxies on a small patch onIn reality, asky. the assumption Assuming that intrinsic ellipticities are tions, we provide a fitting function in the Appendix for Pδδ,fit(k, z; Q0,R0,s) rfit(k, z; Q0,R0,s) . (24) randomly distributed on the sky is inaccurate. There are ≡ P (k, z) • thatRandom their intrinsic ellipticities ellipticitiesδδ, ΛfromCDM are distributedtwo strains of intrinsic random alignmently, of galaxy and ellipticities,both the ratio, rfit(k, z), between a fiducial ΛCDM linear matter arising from the same physics of galaxy formation. thatinstrument, their2.3 light Survey noise, specifications passes randomly by similar large scaleThe measured structure, weak lensing signal reflects the a correlation power spectrum, Pδδ,ΛCDM (k, z) and the one for a modified in shapes arising from distant galaxies passing near the same orientedWe considergalaxies the impact not of including an IAs issue on cosmological foreground gravitational lens. However, if the background gravity model described in 2.1, parameterised by Q0,R0 and intrinsicconstraints ellipticities for a near-term Dark cancel Energy Task in Force the (DETF) two-point function, and Albrecht et al. (2006) Stage III survey, such as DES or SuMIRe, and a longer-term Stage IV survey, such as Eu- 7 www.rssd.esa.int/SA/PLANCK/docs/Bluebook ESA we are left with the cosmic shear signal. − − s: clid, LSST or WFIRST. SCI(2005)1 V 2.pdf • CorrelatedIn reality, intrinsic the assumption alignments that intrinsic ellipticities are c 2009 RAS, MNRAS 000,1–18 Pδδ,fit(k, z; Q0,R0,s) # rfit(k, z; Q0,R0,s) . (24) randomly(instrumental distributed & astrophysical) on the sky is inaccurate. There are ≡ Pδδ,ΛCDM (k, z) twoneed strains to be of disentangled intrinsic alignment from of galaxy ellipticitiesCredit: Williamson,,both Oluseyi, Roe 2007 arisingcosmological from the shear same physics of galaxy formation.

2.3 Survey speciﬁcations Rachel Bean:The IPMU measured January 2012 weak lensing signal reﬂects a correlation in shapes arising from distant galaxies passing near the same We consider the impact of including IAs on cosmological foreground gravitational lens. However, if the background constraints for a near-term Dark Energy Task Force (DETF) Albrecht et al. (2006) Stage III survey, such as DES or SuMIRe, and a longer-term Stage IV survey, such as Eu- 7 www.rssd.esa.int/SA/PLANCK/docs/Bluebook ESA − − clid, LSST or WFIRST. SCI(2005)1 V 2.pdf c 2009 RAS, MNRAS 000,1–18 # Astrophysical systematic: Intrinsic alignments

i j i j i j i j i j = γGγG + γGγI + γI γG + γI γI Lensed Intrinsic shear Correlation: Observed shear shear (GG) (II)

GI shear (anti) correlation

Credit: Benjamin Joachimi, iCosmo

Rachel Bean: IPMU January 2012 LSS observations: Redshift Space Distortions

• Spectroscopic survey (basis for BAO) contains rich clustering info “redshift space distortions”

• 3D map of galaxy positions in redshift space v z = z + pec obs cosmo c

• Small scales: Line of sight smearing from velocity dispersion in groups/clusters

• Large scales: Coherent infall The 2dF Galaxy Redshift Survey Team (2001) compresses halo image in redshift space

Rachel Bean: IPMU January 2012 LSS observations: Redshift Space Distortions

• Coherent peculiar motions of galaxies (Θ=θ/aH) can be statistically estimated from redshift space distortions

2 2 2 obs 2 4 k µ σv (z) Pg (k, z, µ)= Pgg(k, z)+2µ PgΘ(k, z)+µ PΘΘ(k, z) e− • Complementary φ and ψ dependence from lensing and ISW

d(aθ) = k2ψ dt

• 1D velocity dispersion, simple statistic (Velocity equiv to σ8) 1 σ2(z)= dkP (k, z) v 3(2π2) ΘΘ But aspiration to use all z, k information tomographically

Rachel Bean: IPMU January 2012 CMB temperature correlations

• CMB photons traveled through cosmic gravitational wells

• Sensitive to gravitational wells decay/growth – boost/decline in temperature – Integrated Sachs Wolfe “ISW” ∆T φ˙ + ψ˙ T ∼ • Important on large scales

• Integrated effect but useful none the less – Cross-correlations help extract z dependence

Rachel Bean: IPMU January 2012 Dissecting the CMB power spectrum

ISW Acousc peaks ) 2

K Light only covered these γ- e- scaering at me CMB µ distances (~Gpc) recently formed (z=1100)

Inial condions & late-me evoluon

Horizon at z=1100 Temperature correlaon (

Rachel Bean: IPMU January 2012 Adapted from WMAP collaboration Tying theory to observations

• Galaxy positions and motions – trace non-relativistic matter – Measure ψ – Biasing of tracer (galaxy) issue δg = bδm

• Weak lensing and CMB – trace relavitistic (photon path) – Sensitive to (φ+ψ) and time derivs – No bias (but plenty of systematics…)

• Complementarity of tracers key to testing gravity

Rachel Bean: IPMU January 2012 Correlating datasets

X • 2-point autocorrelation between χ ~1/k! observables X and Y = δg, θ, G,I, TCMB α Y χ

χmax dχ CXiYj = W i (χ)W j (χ)S (k , χ)S (k , χ) l χ2 X Y X l Y 0

Window function for Source function observable X in ith for observable X photoz bin for k=l/χ

Galaxy distribution Encodes and homogeneous perturbation background history growth history

Rachel Bean: IPMU January 2012 Cross correlations and tomography help mitigate astrophysical systematics

XiYi X5Yi Plots of Cl and Cl

gI disncve GG and GI ≈ cross z signature extended in z

II strongly peaked at same z

Photo z bin, i Photo z bin, i

Rachel Bean: IPMU January 2012 Laszlo, RB, Kirk, Bridle, MNRAS submitted 2011 Cross- correlations can break theory degeneracies

XiYi X5Yi Plots of Cl and Cl

≈

MG eﬀect evenly in cross-z spectra, IA’s z dependent

Photo z bin, i Photo z bin, i

Rachel Bean: IPMU January 2012 Laszlo, RB, Kirk, Bridle, MNRAS submitted 2011 Cross- correlations can break theory degeneracies

• Boosting φ,ψ (Q,R>1) boosts growth of LSS – CMB cools, ΔT reduced, and could be <0, opposite to acceleration • large scale CMB ~ ΔT2 falls then increases (not monotonic) • large scale CMB-gal monotic, as Q,R increase correlation falls, anti-correlation can occur

CMB TT correlation CMB galaxy correlation

≈

Rachel Bean: IPMU January 2012 RB & Tangmatitham PRD 2010 Three groups of extra galactic observations for testing gravity

II: Growth, up to some I: Background expansion III: Growth directly normalization

CMB angular diameter Galaxy autocorrelations CMB ISW autocorrelation distance Galaxy – ISW x-corrln Weak lensing Supernovae luminosity autocorrelation distance Xray and SZ galaxy cluster measurements Peculiar velocity distribution/ BAO angular/radial scale bulk flows Ly-alpha measurements

Rachel Bean: IPMU January 2012 Current constraints

• WMAP, SDSS LRG auto and ISW-x correlations, COSMOS lensing, Union SN1a

• ISW + ISW-galaxy correlations drive constraints ≈ • Principal degeneracy (φ+ψ) direction ~Q(1+R)/2

• MG FoM ~ 0.03

Rachel Bean: IPMU January 2012 RB & Tangmatitham PRD 2010 Putting it all in the mix

• A “smoking gun” for GR on cosmic scales (Zhang et al PRL 2007)

gG galaxy position-lensing correlation (Cl ) E ~ G gΘ redshift space – galaxy position correlation (Cl )

• Contrasts relativistic and non-relativistic tracers – Lensing: G ~ φ+ψ ~ Q(1+R), – Galaxy position and motion: g,Θ ~ ψ ∼ QR

• Independent of galaxy bias and initial conditions

gG 2 Cl b σ8 ~ gΘ 2 Cl b σ8

Rachel Bean: IPMU January 2012 Distinguishing between modified gravity and Λ

Cgκ E l g ∼ gΘ Cl GR DGP f(R)

TeVeS K=0.1 TeVeS K=0.09 TeVeS K=0.08

Zhang, Liguori, RB, Dodelson PRL 2007 Rachel Bean: IPMU January 2012 13 Vital proof of principle with SDSS LRG data

See Reina’s talk!

Reyes et al Nature 2010

Rachel Bean: IPMU January 2012 Figure 2 | Comparison of observational constraints with predictions from

GR and viable modified gravity theories. Estimates of EG(R) are shown with 1! error bars (s.d.) including the statistical error on the measurement19 of ! (filled circles). The grey shaded region indicates the 1! envelope of the mean

-1 EG over scales R = 10 – 50h Mpc, where the systematic effects are least important (see Supplementary Information). The horizontal line shows the mean prediction of the GR+"CDM model, EG = !m,0 / f , for the effective redshift of the measurement, z = 0.32. On the right side of the panel, labelled vertical bars show the predicted ranges from three different gravity theories: (i) GR+"CDM

( EG = 0.408 ± 0.029(1! )), (ii) a class of cosmologically-interesting models 27 in f (R) theory with Compton wavelength parameters B0 = 0.001! 0.1 9 ( EG = 0.328 ! 0.365 ), and (iii) a TeVeS model designed to match existing cosmological data and to produce a significant enhancement of the growth factor ( EG = 0.22 , shown with a nominal error bar of 10 per cent for clarity). What about future surveys?

• Fisher matrix analysis = Inverse covariance (error) matrix

1 ∂ta 1 ∂tb Covij− = Fij = Covab− ∂pi ∂pj

• Assumed cosmology and parameterization

2 2 2 p = Ωbh , Ωmh , Ωk, τ,w0,wa,Q0,Q0(1 + R0)/2,ns, ∆ (k0), { R +systematic nuisance parameters } • Datasets

κ κ , t = CTT,CTE,CEE,CTg1 ,...,CEg1 ,...Cg1g1 ,Cg1g2,,...,C Nph Nph { }

• Survey specifications – near future (stage III) and end of decade (stage IV) surveys – Stage III = Planck CMB + DES-like imaging + BOSS spectroscopic surveys – Stage IV = Planck CMB + EUCLID-like imaging and spectroscopy

Rachel Bean: IPMU January 2012 Fisher analysis caveats

• What you put in affects what you get out – Figures of merit insightful but also model dependent – Statistical uncertainties /survey assumptions – Systematic errors difficult but important! • Instrumental e.g. calibration uncertainties • Modeling: e.g. Photo z modeling errors, nonlinearity • Astrophysical: e.g. IAs , Hα z distribution, baryonic effects

• How sensitive are results to assumptions? – Intrinsic alignments: • Simple model (e.g. linear alignment model e.g. Hirata & Seljak 2004 ? Or allow for ignorance with gridded z and k bins? – Galaxy bias • Scale independent bias, or scale and z dependent? – Non-linearities: At what scale should one truncate the analysis?

• Buyer beware – risky to compare FoM unless apples-for-apples treatment

Rachel Bean: IPMU January 2012 Sensitivity to theory and systematics

95% confidence contours w0 GR no IA GR with IA systematic error

MG no IA wa MG with IA systematic error

Q0 (1+R0 )/2

Q Ω w0 wa 0 Rachel Bean: IPMU January 2012 Λ Laszlo, RB, Kirk, Bridle, MNRAS submitted 2011 Impact of cross-correlations

Rachel Bean: IPMU January 2012 Laszlo, RB, Kirk, Bridle, MNRAS submitted 2011 Assumptions about bias and IA model

Number of k and z bins for bias & IA nuisance parameters Rachel Bean: IPMU January 2012 Laszlo, RB, Kirk, Bridle, MNRAS submitted 2011 Assumptions about non-linear scales

Rachel Bean: IPMU January 2012 Laszlo, RB, Kirk, Bridle, MNRAS submitted 2011 Concluding thoughts

• Growth history key to testing large scale, weak field gravity – We can do more than measure w; important opportunity, not just a 2 for the price of 1. – Included in assessing upcoming missions. Not just piggy-back science.

• Complementary techniques and datasets important to break cosmological and systematic degeneracies – Multiple upcoming surveys DES, BOSS/BigBOSS, LSST, EUCLID, WFIRST

• Systematics modeling needs to be factored in FoM calculations – Characterization key to assessing relative strengths of different techniques/ survey specs. – Can significantly impact predictions (beware apples vs oranges)

• Figures of merit (DE & MG) useful but constraining – γ or f describe the LSS growth rate – R and Q contrast relativistic and NR tracers of gravity, closer tie to theory – isolating R≠1 could be a smoking gun for MG. – Phenomenological form, a choice, but may not be the answer. Mapping to theory is a further significant effort.

Rachel Bean: IPMU January 2012 WFIRST

year to a deep, weak lensing plus BAO survey, one matic errors consistently. It is clear from the figures year to a wide BAO survey, and six months to a super- that such comparisons can only be meaningful if the novae survey. The figure on the left was obtained us- systematic errors are treated uniformly. Even then, ing conservative Systematicsystematic error estimates; the oneassumptions on comparisons with different matter missions is difficult as it the right using the more optimistic estimates. doesn’t take into account the relative risk of different In Appendix C, we compare the results of this straw approaches or the likelihood that design goals will be man program with those for other proposed missions. met. In so doing we have made every effort to treat syste-

Figure 7: DETF FoM calculations for conservative and optimistic+RSD WFIRST assumptions. The stage II baseline (know- ledge upon completion of ongoing projects) is a DETF FoM ~50.

The expected sky coverage and projected number x NASA WFIRST SDT Interim report 2011 A photometric redshift uncertainty of V(z)= density of galaxies for the 3 different weak lensing sur- 0.04(1+z). veys we consider (WFIRST, LSST and Euclid) are giv- x A statistical shear measurement uncertainty,

Rachel Bean:en inIPMU Table January 3. 2012 VJ=0.35/2. WFIRST LSST Euclid x 50 logarithmically spaced bins in ell, Sky coverage (sq. 2640/yr 15,000 15,000 10

Section 3: Figures of Merit 25