Understanding Gravity on Cosmic Scales

Rachel Bean Cornell University

Work in collaboration with Istvan Laszlo (Cornell) Eva-Marie Mueller (Cornell) Scott Watson (Syracuse) Donnacha Kirk & Sarah Bridle (UCL)

Rachel Bean: Itzykson 2012 ANRV352-AA46-11 ARI 25 July 2008 2:1

2.0 a CosmicNo geometry: Expansion history constraints Standard candles 1.5 0 b BAO

Ω Λ 1.0 –0.5 CMB SNe

w Standard rulers

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 Kowalski 2008 Figure 8 Rachel Bean: Itzykson 2012 (a) Constraints upon !M and !" in the consensus model (cosmological constant/cold model) using baryon acoustic oscillations (BAO), cosmic microwave background (CMB), and supernovae (SNe) measurements. (b) Constraints upon !M and constant w in the fiducial 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 • 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) − Understanding cosmic acceleration

ds2 = a(τ)2[1 + 2ψ(x,t)]dτ 2 + a(τ)2[1 2φ(x,t)]dx2 (25) Cosmic− acceleration = a modification of Einstein’s− equations

Gµν =8πGTµν (26) Deviations New matter? from GR? interactions? Λ? mat f(gµν,R,Rµν) + Gµν =8πGTµν (27) Inhomogeneous universe?

Broad aim =Phenomenology Distinguish which sector: new gravity,mat newDE matter or Λ? Gµν =8πG Tµν + Tµν (28) Ambitious aim = Theoretical￿ model ￿ Learn something more about the underlying theory?

Rachel Bean: Itzykson 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 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 ￿ ￿

Distinguishinga¨ with4πG expansion history = (ρ +3P ) (39) a − 3 m m • Alter Friedmann and acceleration equations at late times a¨ 4πG a¨ = 4πG (ρ +3P + ρDE +3PDE) (40) a= − 3 (ρm +3Pm)+stuff a − 3 or 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: Itzykson 2012

I-3 There are benefits to asking more questions…

Rachel Bean: Itzykson 2012 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: Itzykson 2012 Phenomenological model of modified gravity

• Perturbed metric ds2 = (1 + 2ψ)dt2 + a2(1 2φ)dx2 − −

• 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, or 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: Itzykson 2012 Cosmological tests of gravity

• Non-relativistic tracers: Galaxy positions and motions

– Measure ψ ∼ Gmat =QRGN – Biasing of tracer (galaxy) issue

δg = bδm

• Relativistic tracers: Weak lensing and CMB

– Sensitive to (φ+ψ) ∼Glight =Q(1+R)GN and time derivs – In theory direct tracer of potential, but still uncertainties • stochasticity relating luminous and all

mass rg Dekel & Lahav ’99 • plenty of systematics (photoz, IAs…)

• Complementarity of tracers key to testing gravity Rachel Bean: Itzykson 2012  ‹‰—”‡ʹ†‡‘•–”ƒ–‡•Š‘™™‡‡š’‡ ––Š‡ƒ’’Ž‹ ƒ–‹‘‘ˆ’”‹‘”•–‘–Š‡’Š‘–‘‡–”‹  ”‡†•Š‹ˆ–ƒŽ‰‘”‹–Š™‹ŽŽ‹’”‘˜‡‘–Š‡—„‡”•ˆ‘”–Š‡„‹ƒ•ƒ†• ƒ––‡”–Šƒ–™‡’”‡•‡– ƒ„‘˜‡‹ƒ•™‡”–‘“—‡•–‹‘͹ Ǥ ‹‰—”‡ʹ•Š‘™•–Š‡’‡”ˆ‘”ƒ ‡‘ˆ–Š‡’Š‘–‘ǦœƒŽ‰‘”‹–Š „‡ˆ‘”‡ƒ†ƒˆ–‡”ƒ’’Ž›‹‰ƒƒ‰‹–—†‡’”‹‘”ǡƒ†ƒ•—”ˆƒ ‡„”‹‰Š–‡••’”‹‘”ȋˆ‘ŽŽ‘™‹‰ ƒ’’”‘š‹ƒ–‡”‡Žƒ–‹‘•ˆ”‘–ƒ„‡ƒ—!"#$%ʹͲͲͺȌǤ –Š‹•‡šƒ’Ž‡ǡ–Š‡‡’‹”‹ ƒŽ–‡’Žƒ–‡• ƒ”‡—”‡Žƒ–‡†–‘–Š‘•‡—•‡†‹–Š‡•‹—Žƒ–‹‘Ǥ—”ˆƒ ‡„”‹‰Š–‡••‹•ˆ‘—†–‘„‡˜‡”› ‡ˆˆ‡ –‹˜‡ƒ–”‡‘˜‹‰–Š‡›ƒǦ „”‡ƒ†‡‰‡‡”ƒ ‹‡•ƒ†”‡†— ‡•–Š‡ˆ”ƒ –‹‘‘ˆ ‘—–Ž‹‡”•ˆ‘”–Š‡ˆ—ŽŽ†‡’–Š”ƒ‰‡–‘δͶΨȋ ‘’ƒ”‡†–‘–Š‡• ‹‡ ‡”‡“—‹”‡‡–‘ˆδͳͲΨȌ ‡’”‡•‡–˜ƒ”‹ƒ–‹‘•ƒŽ‘‰–Š‡•‡•ƒ‡Ž‹‡•ǣ  x  ‹‰—”‡͵™‡•Š‘™–Š‡’Š‘–‘‡–”‹ ”‡†•Š‹ˆ–’‡”ˆ‘”ƒ ‡™‹–Š–Š‡”‡•–”‹ –‡† &'()*ˆ‹Ž–‡”•Ǥ x  ‹‰—”‡Ͷ™‡•Š‘™–Š‡’‡”ˆ‘”ƒ ‡™‹–Š–Š‡‡ƒ”Ǧ‹ˆ”ƒ”‡†’ƒ••„ƒ†•„—–‘+ „ƒ††ƒ–ƒǤ x  ‹‰—”‡ͷ™‡•Š‘™+&'()*’Ž—•–Š‡–™‘‡ƒ”Ǧ‹ˆ”ƒ”‡†’ƒ••„ƒ†•Ǥ  Complications : photometric redshifts    • Faster but less precise alternative to

spectroscopic z   z  • Essential for tomography  – Measuring evolution on dark energy – Cross-correlations between z bins  Spectroscopic useful for disentangling systematics and   Photometric z  !"#$%&' ()*' ' +&,-'./01.'-/&'2&%,0%3456&'0,'-/".'2/0-078'49#0%"-/3',0%'-/&'Credit: LSST Consortiumugrizy'.:.-&3'$."5#' • Sensitive to modeling&32"%"649'-&3294-&.'$5%&94-&;'-0'-/0.&'$.&;'"5'-/ &'."3$94-"05<'1/&%&'1&'4229:'=0-/'34#5"-$;&'45;' .$%,46&' =%"#/-5&..' 2%"0%.*' >&?&6-"5#' #494@"&.' "5' -/&' 54%%01' =4;' 2/0-078' .-%"2' A9&,-B' 45;' 4;;"5#' – galaxy distribution,;&&2' C' D' 2/0-03&-%:' "32%0E&.' -/".' ,$%-/&%' 4.' ./015' 05' %"#/-*' F5' &46/' 64.&<' &32"%"649' .2&6-%49' -&3294-&.'4%&'$.&;<'50-'-/&'-&3294-&.'-/4-'#&5&%4-&;'-/&'."3$94-&;'64-490#.' Stage IV – photo-z statistical accuracy, systematic offsets and catastrophic errors dn/dz

30

z Rachel Bean: Itzykson 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 σE (µK) - 11.4 26.7 Disentangling dark energy and cosmic tests of gravity from weak lensing systematics 5 2 The noise for each survey is modeled as statistical errors XX,c Survey Parameters Stage III Stage IV 2 #(#+1)θFWHM,c/8ln(2) N# =(σX,cθFWHM,c) e given by , (28) Area(sq. deg.) 5000 20000 2 "i"j γrms Table 2. CMB survey specifications for a Planck-like survey. We √2z0 0.8 0.9 N# = δij , (25) zmin 0.001 0.001 2nj model this on the temperature, T ,andE-mode polarisation spec- and over all channels,zmax 33 ninj 1 N# = δij , (26) Ng 10 35 nj N 510 n " ifications from three lowest frequency bands for the Planck HFI ph − N i j =0, (27) σz0 0.07 0.05 1 # γrms 0.23 0.35−1 instrument. XX XX where γrms is the root mean square uncertainty in the shear measurement of the galaxies and nj is number of galaxies N# =Table 1. Summary of theN photometric#,c large scale structure. sur- th (29) 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, √2z0;minimumandmaxi-# c the Stage III and IV surveys are given in TableP 1. mum redshifts observed, z and zmax;numberofgalaxies,per X “ min ” We include complementary constraints from tempera- square arcminute, Ng;numberofphotometricredshiftbins,Nph; ture (T) and E-mode polarisation (E) measurements from term and the mass source term must be different standard photometric redshift measurement error at z =0,σz0, and the r.m.s. shear measurement error, γrms. aPlanck-likeCMBsurveyuptol =3000.Assummarised in Table 2, we model this by considering the three lowest Q(χ)(R(χ)+1) 2.4 Intrinsic Alignments 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 PδG(k, χ)= Pδδ(k, χ), (22) described in the Planck Bluebook 7.Weassumeeachfre- f 0.8 0.8 0.8 2 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 Cosmic shearσ (µK) describes 5.4 6.0 the 13.1 distortion of the image of a dis- 2 T given by σE (µK) - 11.4 26.7 Q(χ)(R(χ)+1) 2 tant galaxy due to the bending of lightXX,c from that2 #(#+1) galaxyθFWHM,c/8ln(2) N# =(σX,cθFWHM,c) e , (28) PGG(k, χ)= Pδδ(k, χ). (23) Table 2. CMB survey specifications for a Planck-like survey. We 2 by gravitymodel as this on it the temperature, passesT ,and massiveE-mode polarisation large-scale spec- and over all channels, structure. For a ifications from three lowest frequency bands for the Planck HFI −1 » – −1 instrument. th XX XX galaxy in the i photo-z bin, the observedN# = ellipticity,N#,c . &,of (29) " c # The growth of the dimensionless power spectrum Pδδ is it- Complications : Weak lensingX “ ” distortions the galaxyterm and can the mass be source written term must be diff aserent a sum of three independent Q(χ)(R(χ)+1) 2.4 Intrinsic Alignments self dependent on modified gravity parameters Q and R,as P (k, χ)= P (k, χ), (22) δG 2 δδ contributions: the» cosmic– shear γG,theintrinsic,non-lensedCosmic shear describes the distortion of the image of a dis- summarised by (7) and (8). Q(χ)(R(χ)+1) 2 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 the noise, galaxy can& bernd 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 motionγ using(θ)+ a modifiedγ version(θ)+ of CAMB& (Lewis(θ et) al.. i i i i (30) G I rnd & (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 modified gravity models on large scale structure observa- measure directly the intrinsic shear of any individual galaxy. To support other researchers investigating the role of The cosmictions, without having shear to integrate signal the full perturbationγG is equa- veryTo small, recover the cosmic and shear, we therefore, cannot one averages over a • Correlationtions, we providein ellipticities a fitting function in the Appendix for number of galaxies on a small patch on a sky. Assuming that their intrinsic ellipticities are distributed randomly, and modified gravity models on large scale structure observa- measure directlythe ratio, rfit(k, z the), between intrinsic a fiducial ΛCDM linearshear matter of any individual galaxy. measuredpower spectrum, statisticallyPδδ,ΛCDM (k, z) and the one for a modified that their light passes by similar large scale structure, the intrinsic ellipticities cancel in the two-point function, and tions, without having to integrate the full perturbation equa- To recovergravity the model cosmicdescribed in 2.1, parameterised shear, by therefore,Q0,R0 and one averages over a – Randoms: ellipticities not an issue we are left with the cosmic shear signal. In reality, the assumption that intrinsic ellipticities are number of galaxiesPδδ,fit on(k, z; Q a0,R small0,s) patch on a sky. Assuming tions, we provide a fitting function in the Appendix for rfit(k, z; Q0,R0,s) . (24) randomly distributed on the sky is inaccurate. There are ≡ P (k, z) – Instrumental & astrophysicalδδ,ΛCDM two strains of intrinsic alignment of galaxy ellipticities,both the ratio, rfit(k, z), between a fiducial ΛCDM linear matter that their“contaminants” intrinsic ellipticities – shear are distributedarising from the same random physics of galaxyly, formation. and 2.3 Survey specifications The measured weak lensing signal reflects a correlation that theircalibration light uncertainties passes by similar largein shapes scale arising from structure, distant galaxies passing th neare the same power spectrum, Pδδ,ΛCDM (k, z) and the one for a modified 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) gravity model described in 2.1, parameterised by Q0,R0 and intrinsic– Correlated ellipticitiesAlbrecht et al. alignments (2006) Stage cancel III survey, need such in as theto DES or two-point function, and SuMIRe, and a longer-term Stage IV survey, such as Eu- 7 www.rssd.esa.int/SA/PLANCK/docs/Bluebook ESA be modeled and disentangled − − s: we are leftclid, with LSST or WFIRST. the cosmic shear signal.SCI(2005)1 V 2.pdf from cosmological shear c 2009 RAS, MNRAS 000,1–18 In reality,# the assumption that intrinsic ellipticities are 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) Credit: Williamson, Oluseyi, Roe 2007 δδ,ΛCDM two strains of intrinsic alignment of galaxy ellipticities,both arising from the same physics of galaxy formation. 2.3 Survey specifications Rachel Bean:The Itzykson measured 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 # Complications : Intrinsic alignments

• Significant astrophysical systematic

• Galaxies align in the potential gradient of their host halo ￿i￿j = γi γj + γi γj + γi γj + γi γj ￿ ￿ ￿ G G￿ ￿ G I ￿ ￿ I G￿ ￿ I I ￿ Correlation: Observed Cosmological Intrinsic (II) (GG)

GI shear (anti) correlation

Credit: Benjamin Joachimi, iCosmo Rachel Bean: Itzykson 2012 Cross- correlations and tomography: break degeneracy between systematics and theory

XiYi X5Yi Plots of Cl and Cl

≈

MG and IA have different z dependent signatures

Photo z bin, i Photo z bin, i

Rachel Bean: Itzykson 2012 Laszlo, Bean, Kirk, Bridle, 2012 Current constraints

• Multiple data – WMAP CMB, – SDSS LRG auto – SDSS-WMAP ISW cross correlation – COSMOS weak lensing, – Union SN1a

• ISW + ISW-galaxy correlations ≈ drive constraints

• Principal degeneracy – (φ+ψ)direction ~Q(1+R)/2

• “Figure of Merit” – 1/error ellipse area – MG FoM ~ 0.03

Rachel Bean: Itzykson 2012 Bean & Tangmatitham 2010 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: Itzykson 2012 Forecasting: what you put in=what you get out

• Figures of merit /Fisher insightful but

• Model dependent – e.g. w0/wa or functions of z?

• Systematic errors difficult but important! – Instrumental e.g. calibration uncertainties • Internal cross-checks: inter-filter, concurrent & repetition ≠ redundancy

– Modeling: e.g. Photo z modeling errors, nonlinearity • Access to ground based facilities, • Training sets, simulation suites

– Astrophysical: e.g. IAs , Hα z distribution, galaxy bias, baryonic effects • At what scale should one truncate the analysis? • Analytical modeling, gridded k& z bins, simulations?

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

Rachel Bean: Itzykson 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

Q0 (1+R0 )/2

Q Ω w0 wa 0 Rachel Bean: Itzykson 2012 Λ Laszlo, Bean, Kirk, Bridle, 2012 Impact of cross-correlations: reducing systematics, breaking theory degeneracies

Rachel Bean: Itzykson 2012 Laszlo, Bean, Kirk, Bridle, MNRAS 2012 Assumptions about bias and IA model

z

k

Number of k and z bins for bias & IA nuisance parameters Rachel Bean: Itzykson 2012 Laszlo, Bean, Kirk, Bridle, MNRAS 2012 *If* you understand non-linear scales they could make a big difference

On scales <~ a few Mpc • Baryonic effects? • Non-linear modeling? • Screening effects?

Include small scale modeling uncertainties in forecasts.

Rachel Bean: Itzykson 2012 Laszlo, Bean, Kirk, Bridle, MNRAS 2012 Ways to modify gravity?

• Scalar tensor gravity = simple models we can model effects for

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) • Active area of research, many different options, −−no16 16solutions,ππGG yet −− LLmat,DE ￿￿ ￿￿

• Common theme: A scalar degree of freedom a¨a¨ 44ππGG == ((ρρm +3+3PPm)) (39) a − 3 m m (39) Rachel Bean: Itzykson 2012 a − 3

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 higher derivatives, and so it suffices for the purpose of attempting to classify general theories of dark energy (see Appendix C).

5. Another issue that arises with respect to the higher derivative terms is the following. Is it really necessary to include such terms in an action when trying to write down the most general theory of gravity and a scalar field, in a derivative expansion? Weinberg [18] suggested that perhaps a more general class of theories is generated by including these terms and performing a reduction of order procedure on them, rather than by omitting them. However, since it is ultimately possible to obtain a theory that is perturbatively equivalent to the higher-derivative theory, and which has second order equations of motion, it should be possible just to write down the action for this reduced theory. In other words, an equivalent class of theories should be obtained simply by omitting all the higher-derivative terms from the start. We show explicitly in Sec. 4 that this is the case for the class of theoriesEffective considered here.field theory of acceleration 6. We fix the remaining field redefinition freedom by choosing a “gauge” in field space, thus fixing the action uniquely (see Sec. 4.2). • Can we tie phenomenology/data a step closer to theory? 1.3 Results and Implications • OurWrite final a action general is [Eq. action (4.5) below] as an expansion in derivative powers

m2 1 S = d4x√ g p R ( φ)2 U(φ) + S [eα(φ)g , ψ ] − 2 − 2 ∇ − m αβ m ￿ ￿ ￿ + ￿ d4x√ g a ( φ)4 + b T ( φ)2 + c Gµν φ φ − 1 ∇ 2 ∇ 1 ∇µ ∇ν ￿ ￿ + d R2 4RµνR + R Rµνσρ + d ￿µνλρC αβC + e T µνT + e T 2 . (1.2) 3 − µν µνσρ 4 µν λραβ 1 µν 2 ￿ ￿ ￿ Here the coefficients a1, b2 etc. are arbitrary functions of φ. The corresponding equations of – a1,a2 etc are all free functions of φ motion do not contain any higher derivative terms. This result generalizes that of Weinberg [18] – non-minimally coupled to metric in Einstein frame, “modified gravity” to include couplings to matter. We can summarize our key results as follows: • What observational properties might this type of action have? The most general action contains nine free functions of φ: U, α,a ,b ,c ,d ,d ,e ,e , as • 1 2 1 3 4 1 2 compared to the four functions that are needed when matter is not present [18].

There are a variety of different forms of the final theory that can be obtained using field • Park, Watson, Zurek 2011 Rachel Bean: Itzyksonredefinitions. 2012 In particular some of the matter-coupling termsBloomfield in the & action Flanagan can be2012 re- expressed as terms that involve only the quintessence field and metric. Specifically, the term T ( φ)2 term could be eliminated in favor of φ( φ)2,the( φ)4 could be eliminated ∇ ￿ ∇ ∇ in favor of a term T µν φ φ, or the Gµν φ φ term could be eliminated in favor of a ∇µ ∇ν ∇µ ∇ν term T µν φ φ (see Sec. 4.2). ∇µ ∇ν As mentioned above, one obtains the correct final action if one excludes throughout the • calculation all higher-derivative terms.

5 higher derivatives, and so it suffices for the purpose of attempting to classify general theories of dark energy (see Appendix C).

5. Another issue that arises with respect to the higher derivative terms is the following. Is it really necessary to include such terms in an action when trying to write down the most general theory of gravity and a scalar field, in a derivative expansion? Weinberg [18] suggested that perhaps a more general class of theories is generated by including these terms and performing a reduction of order procedure on them, rather than by omitting them. However, since it is ultimately possible to obtain a theory that is perturbatively equivalent to the higher-derivative theory, and which has second order equations of motion, it should be possible just to write down the action for this reduced theory. In other words, an equivalent class of theories should be obtained simply by omitting all the higher-derivative terms from the start. We show explicitly in Sec. 4 that this is the case for the class of theories considered here.

6. We fix the remaining field redefinition freedom by choosing a “gauge” in field space, thus fixingEffective the action uniquely field (see theory Sec. 4.2). of acceleration

1.3 Results and Implications • A subset of terms particularly relevant to late time evolution Our (Work final action with Eva-Marie is [Eq. (4.5) Mueller below] and Scott Watson)

m2 1 S = d4x√ g p R ( φ)2 U(φ) + S [eα(φ)g , ψ ] − 2 − 2 ∇ − m αβ m ￿ ￿ ￿ + ￿ d4x√ g a ( φ)4 + b T ( φ)2 + c Gµν φ φ − 1 ∇ 2 ∇ 1 ∇µ ∇ν ￿ ￿ + d R2 4RµνR + R Rµνσρ + d ￿µνλρC αβC + e T µνT + e T 2 . (1.2) 3 − µν µνσρ 4 µν λραβ 1 µν 2 ￿ ￿ ￿ Here the coefficients a1, b2 etc. are arbitrary functions of φ. The corresponding equations of motion do not contain any higher derivative terms. This result generalizes that of Weinberg [18] to include– Canonical couplings scalar to matter. field We– canNon-minimally summarize ourcoupled key results matter as follows: – A quartic term The most general action contains nine free functions of φ: U, α,a ,b ,c ,d ,d ,e ,e , as • – A Gauss-Bonnet (GB) term 1 2 1 3 4 1 2 –compared Other terms to the here four well functions constrained that areby early needed time when effects matter is not present [18]. There are a variety of different forms of the final theory that can be obtained using field Rachel Bean:• Itzykson 2012 redefinitions. In particular some of the matter-coupling terms in the action can be re- expressed as terms that involve only the quintessence field and metric. Specifically, the term T ( φ)2 term could be eliminated in favor of φ( φ)2,the( φ)4 could be eliminated ∇ ￿ ∇ ∇ in favor of a term T µν φ φ, or the Gµν φ φ term could be eliminated in favor of a ∇µ ∇ν ∇µ ∇ν term T µν φ φ (see Sec. 4.2). ∇µ ∇ν As mentioned above, one obtains the correct final action if one excludes throughout the • calculation all higher-derivative terms.

5 Effects of extensions to minimally coupled scalar

• Study attractor behavior of this action

M 2 1 S = d4x√ g p R ( φ)2 V (φ)+X(φ˙,H) f(φ)R2 + S [eF (φ)g ] E − 2 − 2 ∇ − − GB m µν ￿ ￿ ￿

2 φ F (φ) exp 2C ≡ ￿ ￿3 Mp ￿ ˙ β ˙4 X(φ,H) 2 2 φ ≡Mp H

• Effect on Friedmann and equations

2 2 1 2 3 3M H = ρ (φ)+ρ + φ˙ + V (φ) + 24φ˙f ￿(φ)H +3X p m γ 2 2 C H˙ X X˙ φ¨ +3Hφ˙ + V = ρ 24f H4 +1 + 3 12 ￿ m ￿ 2 ˙ ￿3 Mp − ￿H ￿ Hφ ￿− XH − ￿

Rachel Bean: Itzykson 2012 2

and complement them in a very different range in cur- vature. We then analyze local tests of gravity in III |fR0|=0.01 alone § and show that solar-system tests are fairly easy to 15 evade, provided gravity behaves similarly to general rela-

tivity in the galaxy. However, if cosmological deviations 2 m /

from general relativity are required to be large, the lat- |

) 10

ter condition is satisfied only with extreme and testable R ( f

changes to the galactic halo. We discuss these results in | n=1 IV. 5 § n=4

II. f(R) COSMOLOGY 0 0.0010.01 0.1 1 10 100 1000 In this section, we discuss the cosmological impact of R/m2 f(R)modelsoftheacceleration.Webeginin II A by in- § troducing a class of models that accelerate the expansion FIG. 1: Functional form of f(R)forn =1, 4, with normaliza- without a true cosmological constant but nonetheless in- tion parameters c1,c2 given by |fR0| =0.01 and a matching to cludes the phenomenology of ΛCDM as a limiting case. ΛCDM densities (see §II D). These functions transition from We then describe the background equations of motion zero to a constant as R exceeds m2.Thesharpnessofthetran- ( II B) and their representation as an equation for the sition increases with n and its position increases with |fR0|. scalar§ degree of freedom ( II C). Finally, we calculate During cosmological expansion, the background only reaches 2 § R/m ∼ 40 for |fR0| " 1andsothefunctionaldependence the expansion history ( II D) and linear power spectrum 2 ( II E) in our class of f§(R)models. for smaller R/m has no impact on the phenomenology. §

A. Model with n>0, and for convenience we take the mass scale κ2ρ¯ Ω h2 m2 0 =(8315Mpc)−2 m , (4) We consider a modification to the Einstein-Hilbert ac- ≡ 3 0.13 tion of the form [84] $ % whereρ ¯0 =¯ρ(ln a =0)istheaveragedensitytoday. c1 4 R + f(R) S = d x√ g + , (1) and c2 are dimensionless parameters. It is useful to note − 2κ2 Lm ! " # that where R is the Ricci scalar, which we will refer to as the 2 2 −1 2 κ ρ ρ Ωmh curvature, κ 8πG,and m is the matter Lagrangian. =1.228 1030 . (5) ≡ L m2 × 1g cm−3 0.13 Note that a constant f is simply a cosmological constant. $ %$ % We work in the Jordan frame throughout this paper. The sign of f(R)ischosensothatitssecondderivative We choose the functional form of f(R)tosatisfycer- tain observationally desirable properties. Firstly, the cos- d2f(R) f > 0(6) To mologyattract should or mimicnot?Λ ACDM matter in the high-redshift of effective regime potentials RR ≡ dR2 where it is well-tested by the CMB. Secondly, it should for R m2,toensurethat,athighdensity,thesolu- • acceleratePotentials thewith expansionexponential at or low power redshift law forms with anallow expan- attractor solutions tion is% stable at high-curvature [61]. This condition also sion– Independence history that is to close initial to conditionsΛCDM, but (for without better or a worse true e.g. f(R) cosmologicalAmendola constant. et al 2007) Thirdly, there should be sufficient implies that cosmological tests at high redshift remain degrees of freedom in the parametrization to encompass 6 the same as in general relativity (GR). For example, the – Transitions from matter to accelerative era 2 as broad a range of low-redshift phenomena as is cur- physical matter density Ωmh inferred from the CMB us- rently observationally acceptable. Finally, for the pur- ing GR remains valid for the f(R)models.Assuch,m is poses of constraining small deviations from general rela- abetterchoiceofscalethanH0 since it does not vary for tivity with cosmological and solar-system tests, it should f(R)modelsinthisclass.Afewexamplesofthef(R) include the phenomenology of ΛCDM as a limiting case. functions are shown in Fig. 1. These requirements suggest that we take There is no true cosmological constant introduced in RB, Flanagan,this class,Laszlo, unlike in the models of [85]. However, at cur- lim f(R)=const., Trodden 2008 2 R→∞ vatures high compared with m , f(R)maybeexpanded as • Can postulate potentialslim f(R that)=0 deny, attractors, and retrofit(2) to LCDM FIG. 1: Examples of evolution of the effective→ equation of state, weff ,incoupledscalarfielddarkmattermodelswithan ∝ − R 0 ∝ n exponentialbackground. potential V (φ) exp( φ/Mp)(leftpanel)andapowerlawpotentialV (φ) 1/φ (right panel). Cosmological 2 parameters are fixed to H0 =70,Ωc =0.25, Ωb =0.05, and C =0.1(black)andC =0.5(red).Bothmodelsfollowthe c1 2 c1 2 m couplingwhich– dependentEvades can attractor be disadvantageous satisfied in the matter dominated by a era general an dasymptotetocouplingindependentattractorsatlatetimeattractor class behavior of broken e.g. power f(R) Hus. and lim f(R) m + 2 m . (7) The timing of the transition between these two attractors is sensitive to both the potential and coupling parameters. For m2/R→0 c c R the exponential potential the dynamical attractor leads to anegligibledependenceoninitialconditions,shownherethrough ≈− 2 2 law modelsSawicki 2007 −8 −10 $ % comparing evolution with two different initial values of φi ≡ φ(a =10 ), φi =1Mp (full) and 10 Mp (dashed). For the power law potential, however, a sensitivity to initial conditions can exist in the2 transitionn era. This is accounted for in the 2 analysis by marginalizing over initial conditions. 2 c1(R/m ) Thus the limiting case of c1/c2 0atfixedc1/c2 is a cos- f(R)= m 2 n , (3) → − c2(R/m ) +1 mological constant in both cosmological and local tests Rachel Bean:As shownItzykson in Fig. 2012 1, for the exponential potential the and attractor behavior quickly takes over, and the initial con- ditions have no effect on the dynamical evolution. In the 6¨η + h¨ +2 (h˙ +6˙η) 2k2η =0. (3.22) case of the power law potential, however, we find there H − can still remain some sensitivity to the initial value of the j Here k is the comoving wavevector and θc = ikjvc is the scalar field during the transition between matter domi- gradient of the CDM peculiar velocity, vc.Alsowehave nated and accelerative attractors. As discussed in section specialized to units with Mp =1.Weincludejustthe III C, we account for this in the analysis by marginalizing effects of CDM and the scalar field, and neglect baryons over the initial value of φ. and radiation, for simplicity. The perturbed fluid equa- tions are

B. Evolution of linearized cosmological 1 δ˙ + h˙ + θ =0, (3.23) perturbations c 2 c d ! (aeαθ )=ak2α eαϕ, (3.24) As well as background evolution, we are also interested dτ c !! !! ! in the predicted evolution of density perturbations. We ϕ¨ +2 ϕ˙ + k2 + a2V + a2eαρ α +(α )2 ϕ write the inhomogeneous density and scalar field as H c 1 2 ! α = ! h˙ φ˙ a α e δcρc. " (3.25)#$ ρc(x, τ)=ρc(τ)(1 + δc(x, τ)), (3.20a) −2 − φ(x, τ)=φ(τ)+ϕ(x, τ). (3.20b) There exists an extra gauge degree of freedom that We use the notation of Ref. [81] to describe the perturbed preserves synchronous gauge, given by the the coordinate metric in synchronous gauge in terms of two functions transformations η(τ)andh(τ). The four independent components of the c k·x Einstein equation are then τ τ + 0 [ei ], (3.26) → a R ! ! 2 ˙ 2 α 2 k·x dτ 2k η h = a e ρc(δ + α ϕ) a V ϕ xj xj + kc [ikˆ ei ] , (3.27) − H − − → 0R j a φ˙ϕ˙, (3.21a) % − 2 2 α 2 2k η˙ = a e ρcθc + k φϕ˙ , (3.21b) where c0 is a constant and kˆk = kj/k.Underthistrans- ! h¨ +2 h˙ 2k2η = 3φ˙ϕ˙ +3a2V ϕ, (3.21c) formation the metric and matter perturbations transform H − − Effect on cosmological attractors

• Express dynamical equations in terms of dimensionless parameters

1 φ˙ 1 √V 1 √ργ x = ,y= ,z= , MpH √6 MpH √3 MpH √3 X f H2 4 µ ￿ Gauss-Bonnet attractor notesµ -X Eva-Maria Mueller2 2 = 12βx GB I-4 ≡ 3Mp H ≡ Mp

• Find if stationary solutions exist x’=y’=z’=µ’=0

F(R) GB LCDM

10-6 10-4 0.01 1.00 100

Rachel Bean: Itzykson 2012

Figure 1: Black solid line: λCDM, black dashed line: F(R), red dashed line: Gauss-Bonnet.

Lower panel: blue dashed line: ΩGB only, red dashed line: radiation - matter - scalar (kinetic + potential). This plot is for λ =4and α =20. Here H0 =71km/s/Mpc and Ωm(today)=0.27.

In Koivisto & Mota they dont have this bump, but actually I don’t understand their matter • dominated era: Ω =0.1875 How do they get this scaling solution?? φ ￿ I need y =0and x =0during the matter dominated era to get H =71km/s/Mpc. If not, • ￿ ￿ 0 I need to choose f0 so big that it messes up the radiation dominated era. As explained later f0V0 needs to be around a certain value to get the GB exit at a=1. This can be achieved by either increasing V or f . But if you increase V you end up with Ω =0during the matter 0 0 0 φ ￿ dominated era as in Fig.1 . Evolutionary attractor

• Matter dominated era: NMC and kinetic terms important – NMC cosmic grease w >0 – Quartic term adds to Hubble drag, acts to slow expansion 2 x = C (C, β) y =0,z =0,µGB =0 3 eff

• Accelerative era: – GB term gives an accelerative attractor (Koivisto and Mota 2006) λ x =0,y =1,µ = GB 8

– In absence of GB term quartic allows acceleration for broader ranges of potentials with λ>2

Rachel Bean: Itzykson 2012 Matter era

Rachel Bean: Itzykson 2012 Paper in prep with Eva-Marie Mueller and Scott Watson Accelerative era

Rachel Bean: Itzykson 2012 Paper in prep with Eva-Marie Mueller and Scott Watson Notes : EFT– Eva-Maria Mueller 22.2.2012 Constraints on beta and C

Impacts on geometric cosmological8 constraints

Geometric constraints CMB + BAO + SN

+ quartic

8

w/o quartic

FIG. 3: Joint 68% (dark shaded) and 95% (light shaded) constraints using combined WMAP CMB, SDSS matter power spectrum, SDSS and 2dFGRS baryon acoustic oscillation, ‘union’ type 1a supernovae datasets and HST H0 prior for the power law potential (blue) and exponential potential (red) for thethefractionalmatterdensity,Ωm and the effective scalar equation of state, wφ,(leftpanel),andthecoupling,C (right panel).

model with Ωm =0.25, ΩΛ =0.75 and with the same In Figure 3 we show 2D marginalized constraints for Hubble constant as the theory model. We also use con- the exponential and power law potential models from all straints on the expansion history from the Baryon Acous- the cosmological datasets combined. The 1D marginal- tic Oscillation data of the 2dFRGS and SDSS surveys ized constraint on the coupling in the exponential poten- [11], based on measurements of the ratio of the sound tial case is C < 0.037(0.067) at the 68% (95%) con- | | horizon at last scattering, rs(z∗), to the distance mea- fidence level. This represents a significant tightening of Ωm sure dV (z)atz =0.2andz =0.35. Since the dynamical constraints over previous analyses, for example [86] found attractor solutions, in the presence of a non-minimal cou- C < 0.1atthe98.6% level using CMB data from the Projecting| | into the β = 0 plane, C varies around 0.03. The red contours in the second diagram show a pling, can alter the background evolution in the matter Boomerang satellite. The potentialJordan frame (coupling to all maer) exponent, λ,iscon-± dominated era, one finds that the redshift of lastvariation scat- strained of 0.04, to be so aλ little< 0.95 bit at thebigger. 95% c.l.. It might be that I need to run the ”C only” mcmc a bit longer. At β =1 ± | | tering, z∗,cannolongerbeaccuratelyestimatedusingEinstein frame (just coupling to CDM) C varies aroundIn the power0.9 which law potential is pretty case consistent the 1D with marginal- the second plot. FIG. 3: Joint 68% (dark shaded) and 95% (light shaded) constraints using combined WMAP± CMB, SDSS matter power spectrum, SDSS and 2dFGRSthe fitting baryon formula acoustic of oscillation, Hu and Sugiyama ‘union’ type [85]. 1a Instead supernovaeized datasets constraint and HST onH the0 prior coupling for the is power comparable,Bean, with Flanagan, Laszlo,Trodden 2008 law potential (blue) andwe exponential calculate potential the redshiftRachel (red) of Bean: maximumfor thethefractionalmatterdensity, Itzykson visibility 2012 and use 0.026(Ωm 0and.055) the eCffective0.Paper034(0 scalar.066) equationin atprep the 68%with (95%) Eva-Marie Mueller and Scott Watson this as the appropriate measure for the redshift of last − − ≤ ≤ of state, wφ,(leftpanel),andthecoupling,C (right panel). c.l.. Again, the constraints on the coupling strength scattering. have improved with the increased precision and comple- In Figure 2 we show the complementary 2D marginal- mentary variety of the cosmological data, e.g. a pre- ized constraints for the exponential potential model in vious analysis with first year WMAP data alone found model with Ω =0.25, Ω =0.75 and with the same In Figure 3 we show 2D marginalized constraints for m lightΛ of the various cosmological datasets. The CMB C 0.085(0.159) at the 68% (95%) c.l. [87]. Within the Hubble constant as the theory model. We also use con- the exponential and≤ power law potential models from all 1 data (along with the HST prior on H0)providethebest range investigated, 6 n 6, the power law exponent, straints on the expansionindividual history fromconstraint the Baryon on the Acous-coupling strengththe cosmological with 1D n datasets,isnotconstrainedbythedata. combined.− ≤ The≤ 1D marginal- tic Oscillation data ofmarginalized the 2dFRGS constraints and SDSSC surveys0.13 at theized 95% constraint confi- on theIn both coupling the exponential in the exponential and power poten- law potential cases | | ≤ [11], based on measurementsdence level of (c.l.). the ratio TheType of the 1a sound supernovaetial alone case provide is C

• Upcoming datasets provide invaluable opportunity to test the origins of cosmic acceleration and weak field gravity on cosmic scales – Complementary techniques important to break cosmological and systematic degeneracies – Relativistic and non-relativistic LSS tracers of gravity equivalent to complementary geometric techniques to measure expansion history/ curvature

• Inclusion of systematics modeling essential to forecasting – Can significantly impact predictions (beware apples vs oranges) – Theory and systematics can be tightly coupled. – How important are specific systematics/ algorithms to model them to a technique’s ability to constrain DE?

• Phenomenological modeling/ FoMs useful but a high pass filter on full info. Mapping to theory is the ultimate goal. – Surveys will give us information about z and k dependence – General effective field theory for DE is a first step, with interesting implications for both expansion history and growth history

Rachel Bean: Itzykson 2012