Cosmological Tests of General Relativity
Rachel Bean (Cornell University)
Rachel Bean, Tes ng Gravity, Vancouver, January 2015 The challenge: to connect theory to observation
2 • While GR is a remarkably beautiful, and comparatively Category Theory References Scalar-Tensor theory [21, 22] minimal metric theory, it is not (incl. Brans-Dicke) the only one. We should test it! f(R)gravity [23, 24] f(G)theories [25–27] • Rich and complex theory Covariant Galileons [28–30] Horndeski Theories The Fab Four [31–34] space of cosmological K-inflation and K-essence [35, 36] relevant modifications to GR Generalized G-inflation [37, 38] Kinetic Gravity Braiding [39, 40] Quintessence (incl. • Motivated by cosmic [41–44] universally coupled models) acceleration, but implications Effective dark fluid [45] from horizon to solar system Einstein-Aether theory [46–49] Lorentz-Violating theories scales Ho˘rava-Lifschitz theory [50, 51] DGP (4D effective theory) [52, 53] > 2newdegreesoffreedom EBI gravity [54–58] Aim: look for and confidently TeVeS [59–61] extract out signatures from Baker et al 2011 TABLE I: A non-exhaustive list of theories that are suitable for PPF parameterization. We will not treat all of these explicitly cosmological observations? 2 µν µνρσ in the present paper. G = R − 4Rµν R + RµνρσR is the Gauss-Bonnet term.
Rachel Bean, Tes ng Gravity, Vancouver, January 2015 derlying our formalism are stated in Table II. PPN and formalism. PPF are highly complementary in their coverage of dif- ferent accessible gravitational regimes. PPN is restricted ii) A reader with a particular interest in one of the ex- to weak-field regimes on scales sufficiently small that lin- ample theories listed in Table I may wish to addition- ally read II B-II D to understand how the mapping ear perturbation theory about the Minkowski metric is § an accurate description of the spacetime. Unlike PPN, into the PPF format is performed, and the most rel- evant example(s) of III. PPF is valid for arbitrary background metrics (such as § the FRW metric) provided that perturbations to the cur- iii) A reader concerned with the concept of parameter- vature scalar remain small. PPF also assumes the valid- ized modified gravity in and of itself may also find ity of linear perturbation theory, so it is applicable to II F and IV useful for explaining how the approach large length-scales on which matter perturbations have presented§ § here can be concretely implemented (for not yet crossed the nonlinear threshold (indicated by example, in numerical codes). IV also discusses the δM (knl) 1); note that this boundary evolves with red- § ∼ connection of PPF to other parameterizations in the shift. present literature. Perturbative expansions like PPN and PPF cannot be used in the nonlinear, strong-field regime inhabited Our conclusions are summarized in V. §−2 by compact objects. However, this regime can still We will use the notation κ = MP =8πG and set be subjected to parameterized tests of gravity via elec- c = 1 unless stated otherwise. Our convention for the metric signature is ( , +, +, +). Dots will be used to in- tromagnetic observations [62, 63] and the Parameter- − ized Post-Einsteinian framework (PPE) for gravitational dicate differentiation with respect to conformal time and waveforms [64, 65]. Note that despite the similarity in hatted variables indicate gauge-invariant combinations, nomenclature, PPE is somewhat different to PPN and which are formed by adding appropriate metric fluctua- PPF, being a parameterization of observables rather than tions to a perturbed quantity (see II D). Note that this meansχ ˆ = χ. § theories themselves. ̸ The purpose of this paper is to present the formalism that will be used for our future results [66] and demon- strate its use through a number of worked examples. We would like to politely suggest three strategies for guiding II. THE PPF FORMALISM busy readers to the most relevant sections:
i) The casually-interested reader is recommended to as- A. Basic Principles similate the basic concepts and structure of the pa- rameterization from II A and II E, and glance at As stated in the introduction, the PPF framework sys- Table I to see some example§ theories§ covered by this tematically accounts for allowable extensions to the Ein- for both the LSST DESC and MS-DESI. Modifications to GR include the presence of extra degrees of freedom (e.g. Carroll et al. (2004)), massive gravitons (e.g. Hinterbichler (2012)), gravity pervading extra dimensions (e.g. Dvali et al. (2000)), and those which attempt to resolve the fine tuning cosmological constant problem through degravitation (Dvali et al. 2007; de Rham et al. 2008).
In stark contrast to ⇤ and quintessence, modifications to gravity can have marked effects on both the growth for both the LSST DESC and MS-DESI.of large scale Modifications structure and to the GR background include the expansion. presence It of is extra common degrees that ofmodels that modify GR are able to freedom (e.g. Carroll et al. (2004)),reproduce massive the gravitons distance measurements (e.g. Hinterbichler but alter (2012)), the growth gravity of pervading large scale extra structure, opening up the possibility dimensions (e.g. Dvali et al. (2000)),of testing and and those discriminating which attempt between to resolve the different the fine theories. tuning cosmological Generically, the Poisson equation relating constant problem through degravitationover-densities (Dvali to et al.gravitational 2007; de Rham potentials et al. is 2008). altered and the potential that determines geodesics of relativistic ( + )/2 In stark contrast to ⇤ and quintessence,particles, modifications in terms of the to gravity Newtonian can have gauge marked potentials effects on both, differsthe growth from that that determines the motion of non-relativistic particles, . Creating / =1during an accelerative era is extraordinarily difficult in of large scale structure and the background expansion. It is common that models6 that modify GR are able to reproduce the distance measurementsfluid models but alter of the dark growth energy of (Hularge 1998). scale structure, Measuring opening it, therefore, up the could possibility be a smoking gun of a deviation from GR. In Zhang et al. (2007b) we proposed a way to constrain / , by contrasting the motions of galaxies of testingTime-space and (Synchronous) discriminating between the different theories. Generically, the Poisson equation relating m2⌦ 2k2⌘˜˙ k (h˙ +6⌘˜˙) =(¯⇢ + P¯ )kv + m2⌦˙ k2⇡˙ +[⇢ + P ] k2⇡ (D.14) Time-space0 (Synchronous) 0 m andm the0 lensingQ Q distortions of light from distant objects that LSST and MS-DESI data will be idea for. over-densities toh gravitationali potentials is altered and the potential that determines geodesics of relativistic Space-space trace2 (Synchronous)2 ˙ ˙ ˙ ¯ 2 ˙ 2 2 m0⌦ 2k ⌘˜ k0(h +6⌘˜) =(¯⇢m + Pm)kv + m0⌦k ⇡˙ +[⇢Q + PQ] k ⇡ (D.14) ¨ 2 Time-space (Synchronous)particles,2 in termsh ˙ of2kh the2k0 Newtoniani ˙ gauge potentials ( + )/2, differs from that that determines the motion mSpace-space⌦ hH trace+ (Synchronous)⌘˜ ⌘˜ = P + PQ⇡Bean+(⇢Q + P andQ)˙⇡ her group have(D.15) developed software based on the publicly available CAMB and CosmoMC codes 0 3 3a2 a2 2 2 ˙ ˙ " ˙ ¯ #2 ˙ 2 2 m0⌦ 2k ⌘˜ k0(h +6⌘˜) =(¯¨ ⇢m + Pm)kv2 + m0⌦k ⇡˙ +[⇢Q + PQ] k ⇡ (D.14) 2 h 2k 2k0 /2 ˙=1 of non-relativisticm ⌦ hH˙ particles,+ ⌘˜ ⌘˜ = .P + Creating(Lewis⌦¨P˙ ⇡ +(⇢ + etP 2)˙ al.k⇡ 2000;h during Lewisk0 &(D.15) an Bridle accelerative 2002) era to perform is extraordinarily likelihood difficult analyses in and forecasting for generic pho- Space-space traceh (Synchronous)0 i 3 3a2 a2 + m2⌦˙ ⇡˙Q+¨⇡ +3QH⇡˙ +Q ⇡ + + ⇡ 3H2 + " # 0 ˙ 3 a2 36 a2 "⌦ ✓ ◆# ¨ fluid2 models of dark energy (Hu 1998).¨ Measuring2 ˙ it, therefore, could be a smoking gun of a deviation from 2 h ˙ 2k Space-space2k0 traceless˙ (Synchronous) tometric2 ⌦ and2 k spectroscopich 2 k0 surveys (Bean & Tangmatitham 2010; Laszlo et al. 2011; Kirk et al. 2011; m0⌦ hH + ⌘˜ ⌘˜ = P + PQ⇡ +(⇢Q + PQ)˙⇡ + m ⌦˙ ⇡˙ +¨⇡ +3H(D.15)⇡˙ + ⇡ + + ⇡ 3H + 3 3a2 a2 0 ⌦˙ 3 a2 3 a2 " Frameworks#m2⌦ 2k2 to tie" observationsk2 1 ✓ to theory◆# GR. In Zhang0 et⌘˜ al.3H (2007b)(h˙ +6⌘˜˙) (h¨ +6 we⌘˜¨) = proposedP¯ ⇧ + m2⌦˙ ⇡ a+ wayh˙ +6⌘˜˙ to constrain(D.16) / , by contrasting the motions of galaxies 2 ¨ 2 Mueller˙ m 0 &2 Bean 2013). This includes peculiar velocity, weak lensing and galaxy clustering correlations Space-space2 a 2 traceless˙ ⌦ (Synchronous) 2 k h 2 ak0 2 + m0⌦ ⇡˙ +¨⇡ +3H⇡˙ + ⇡ + + ⇡ 3H + ⇣ ⌘ 2 2⌦˙ 3 a2 3 a2 2 and the lensingWe can translatem0⌦ distortions2k" these into Newtonian˙ ˙ of gauge. light¨ and¨ The from required¯✓ cross-correlations distant gauge2 ˙ ◆#k transformations objects1 ˙ ˙ are detailed thatwith LSST the CMB. and MS-DESI The code data models will dark be idea energy for. and modified gravity using a variety ⌘˜ 3H(h +6⌘˜) (h +6⌘˜) = Pm⇧ + m0⌦ ⇡ + h +6⌘˜ (D.16) Space-space traceless (Synchronous)in Appendix• Modify B.2 a2 homogeneous and perturbeda2 2 Einstein’s equations ⇣ ⌘ 2 2 Time-time (Newtonian) 2 of phenomenological parameterizations, including the equation of state w(z), the Hubble expansion rate m0⌦ 2k We can translate these into2 Newtoniank 1 gauge. The required gauge transformations are detailed ⌘˜ 3H(h˙ +6⌘˜˙) (h¨ +6⌘˜¨) 2= P¯ ⇧ + m ⌦˙ ⇡ + h˙ +6⌘˜˙ (D.16) Bean2 andin Appendix her group B.2k 6km have0 developed2 software based on the publicly available CAMB and CosmoMC codes 2 a m2⌦(t) 0 N +6H( ˙aN + H 2 N ) 0 a2 ⇣H(z⌘) , the logarithmic growth factor, f (z), and its exponent, (z), and a parameterization directly related Time-time (Newtonian) g We can translate(Lewis these into et Newtonian al. 2000;N gauge.N The Lewis requiredN & gaugeN Bridle transformations 2002) are detailed to perform likelihood analyses and forecasting for generic pho- = ⇢ ⇢˙Q⇡ 2 2c(t)(⇡ ˙ ) in Appendix B. 2 2k 6k0 N ˙N N m0⌦(t) +6H( +toH modifications) 2 of the perturbed Einstein equations, Gmatter(z,k) and Glight(z,k), 2 ˙ N a22 ˙ k0 N N k N ˙N N Time-time (Newtonian)tometric and+ m0 spectroscopic⌦ 3⇡ H H + +3 surveysH ⇡˙ (Bean+ ⇡ 3 & + TangmatithamH (D.17) 2010; Laszlo et al. 2011; Kirk et al. 2011; N N a2 N N a2 2 = ⇢ ✓⇢˙Q⇡ 2c(t)(⇡◆ ˙ ) 2 2k 6k0 N ˙N N ⇣ ⌘ m0⌦(t) 2 Time-space +6H( (Newtonian)+ H ) 2 Muellera & Bean2 ˙ 2013).N 2 This˙ k0 includesN N peculiark N ˙N velocity,N 2 weak lensing and2 galaxy clustering2 correlations 2 + m0⌦ 3⇡ H H + +3H ⇡˙ + ⇡ 3 + H (D.17) N N 2N N2 ˙N N ¯ 2 N 2 N 2 2 ˙ 2 N N k = 4⇡Gmattera ⇢ ,k( + )= 8⇡Glighta ⇢ , (1) = ⇢ ⇢˙ ⇡ 2c(t)(m⇡ ˙0⌦(t)2 k ) + H =(¯ ⇢m + Pam)kv +[⇢Q + PQ] k ⇡ a+ m0⌦ k ⇡˙ (D.18) Q ✓ ◆ ⇣ ⌘ and cross-correlationsTime-spaceh (Newtonian)i with2 the CMB. The code models dark energy and modified gravity using a variety 2 ˙ N 2Space-space˙ k0 trace (Newtonian)N N k N ˙N N + m0⌦ 3⇡ H H + +3H ⇡˙ + ⇡ 3 + H (D.17) a22 2 ˙N N a2 ¯ N 2 2 N 2 ˙ 2 N N of phenomenological✓ 2 m0◆⌦¨(t)2k 2 + H˙ parameterizations,=(¯⇢m˙+ Pm˙ )kv2k+[⇢Q + PQ] includingk ⇡ + m0⌦k ⇡˙ the equation(D.18) of state w(z), the Hubble expansion rate m0⌦ 2 +2 3 H +2H +2H( +3⇣ )+where( ⌘ )⇢ is background density, k is comoving spatial, a, the scale factor and is the rest-frame, gauge Time-space (Newtonian) 3a2 Space-space traceh (Newtonian)i N ˙ N⇣ ⌘ N N 2 2H˙N(z),N the= P logarithmic+ P¯Q⇡ +(N ⇢Q + PQ growth)(˙⇡ 2 N ) factor,2 ˙ 2 N fN (z), and its exponent, (z), and a parameterization directly related m0⌦(t)2k + H =(¯⇢m + Pm)kv +[⇢Q + PQ] k ⇡ + m0⌦kinvariant,⇡˙ 2 g matter(D.18) perturbation. It includes general parameterizations for galaxy bias and intrinsic alignments 2 ¨ 2 ˙ ˙ ˙ 2k m0⌦ ⌦¨2 +2 3H +2H +2H( +3 )+ ( ) k 2 k2 Space-space traceh (Newtonian)i + m2⌦˙ (˙⇡N N )+¨⇡N ˙N 2 ˙N +3H ⇡˙ N 3a52H N + ⇡N 3H2 + 0 + ⇡N to modifications0 ˙ of the perturbed Einstein equations, 2 G2matter(z,k) and G (z,k), N"⌦ ˙ N⇣ ⌘ N N (Hirata & Seljaka 2004;3 a Laszlo# et al. 2011),light a simple, Gaussian model for photometric redshift errors and = P + PQ⇡ +(2k⇢2Q + PQ)(˙⇡ ) ✓ ◆ m2⌦ 2 ¨ +2 3H2 +2H˙ +2H( ˙ +3 ˙)+ ( ) (D.19) 0 ¨ 2 2 2 ⌦ N 3a N N N N N N N 2 k0 2 k N ⇤ ⇣ Space-space⌘ + m traceless⌦˙ (˙⇡ (Newtonian) )+¨⇡ ˙ 2 nonlinear˙ +3H⇡˙ 5H model+ ⇡ 3H based+ + on the⇡ CDM-modeled Halofit algorithm (Smith & Zaldarriaga 2011). = P N + P˙ ⇡N +(⇢ + P )(˙⇡N 0N ) ⌦˙ 2 2 a2 2 3 a2 2 Q Q Q " k 2 = 4⇡Gmatter2a ⇢ ✓,k◆ ( +# )= 8⇡G a ⇢ , (1) 2 k N N 2 k N light m ⌦(t) = P¯ ⇧ + m ⌦˙ 2 ⇡ (D.20)(D.19) ⌦¨ 0 2 mk 02 k 2 + m2⌦˙ (˙⇡N N )+¨⇡N ˙N 2 ˙N +3H⇡˙ N 5Ha N + ⇡ N 3H2 + 0 + a ⇡N 0 ˙ Space-space traceless (Newtonian) Proposeda2 3 a2 work: This existing software will be modularized and documented to integrate into the LSST "⌦ The matter terms also transform in going✓ to Newtonian◆ gauge. # 2 2 2 k N N 2 (D.19)2 k N where ⇢ is background density,N Bloomfield ket alis 2012a comoving¯ ˙ spatial, a, the scale factor and is the rest-frame, gauge ⇢m0⌦=(t) ⇢2+ ⇢¯˙ (h˙DESC+6 ⌘˜˙)= Pm⇧ and+ m0⌦ MS-DESI2 ⇡ analysis(D.21)(D.20) pipelines. Three specific projects to enhance the software are described in Space-space traceless (Newtonian) a m 2k2 a 2 invariant, matterThe2 matter perturbation. terms also transformN 2 in Itgoinga includes to Newtonian general gauge. parameterizations for galaxy bias and intrinsic alignments 2 k N N P 2= kP +NP˙ sections(h˙ +6⌘˜˙) C.1-C.3. The (D.22)improvements will ensure that the analysis pipelines are able to meet the required m ⌦(t) = P¯ ⇧ + m ⌦˙ ⇡ m 2 (D.20) 0 a2 m 0 a2 2k a2 See Alessandra’s Talk ⇢N = ⇢ + ⇢¯˙ (h˙ +6⌘˜˙) 2 (D.21) (Hirata & Seljak 2004;¯ LaszloN ¯ etm al.¯ 2011),2 a ˙ a simple,˙ Gaussian model for photometric redshift errors and The matter terms also transform in going to(¯⇢ Newtonianm + Pm)kv gauge.=(¯⇢m + Pmlevel)kv + Pm of+¯2⇢km both(h +6 theoretical⌘˜) (D.23) modeling and survey-specific systematic error characterization necessary to define Rachel Bean, Tes ng Gravity, Vancouver, January 2015 2 2 2 N a The matter anisotropic shear stressa isP invariant.= P + P˙m (h˙ +6⌘˜˙) (D.22) nonlinear modelN ˙ based˙ ˙ on the ⇤CDM-modeled2k2 Halofit algorithm (Smith & Zaldarriaga 2011). ⇢ = ⇢ + ⇢¯m(h +6⌘˜) 2 science requirements.(D.21) When Stage III data is made public, expected on this proposal’s timeframe, we 2k 35 2 2 N a a (¯⇢m + P¯m)kv =(¯⇢m + P¯m)kv + P¯m +¯⇢m (h˙ +6⌘˜˙) (D.23) P N = P + P˙ (h˙ +6⌘˜˙) (D.22)2 m 2k2 will analyze the data using this software pipeline, and integrate improvements in the intrinsic alignment, ProposedThe work: matter anisotropicThis shear existing stress is2 invariant. software will be modularized and documented to integrate into the LSST ¯ N ¯ ¯ a ˙ ˙ (¯⇢m + Pm)kv =(¯⇢m + Pm)kv + Pm +¯⇢m (h +6photometric⌘˜)35 (D.23) error and nonlinear modeling into the code. DESC and MS-DESI analysis2 pipelines. Three specific projects to enhance the software are described in The matter anisotropic shear stress is invariant. sections C.1-C.3.35 The improvements will ensure that the analysis pipelines are able to meet the required C.1. Detailed ties between dark energy theory and LSST and MS-DESI observations level of both theoretical modeling and survey-specific systematic error characterization necessary to define science requirements. When StageWhile III the data phenomenological is made public, parameterizations expected on this outlined proposal’s above timeframe, help translate we observations into broad dark will analyze the data using thisenergy software characteristics, pipeline, and more integrate needs improvements to be be done in the to connect intrinsic the alignment, data further to dark energy theory and photometric error and nonlinearastrophysically modeling into the relevant code. modifications to GR. Many classes of modified gravity theories are described by the general “Horndeski” action, the most general theory of a scalar field coupled to gravity for which the C.1. Detailed ties between dark energy theory and LSST and MS-DESI observations Narrative - 3 While the phenomenological parameterizations outlined above help translate observations into broad dark energy characteristics, more needs to be be done to connect the data further to dark energy theory and astrophysically relevant modifications to GR. Many classes of modified gravity theories are described by the general “Horndeski” action, the most general theory of a scalar field coupled to gravity for which the
Narrative - 3 forfor both both the the LSST LSST DESC DESC and and MS-DESI. MS-DESI. Modifications Modifications to GR to GR include include the the presence presence of extraof extra degrees degrees of of freedomfreedom (e.g. (e.g. Carroll Carroll et al. et al.(2004)), (2004)), massive massive gravitons gravitons (e.g. (e.g. Hinterbichler Hinterbichler (2012)), (2012)), gravity gravity pervading pervading extra extra dimensionsdimensions (e.g. (e.g. Dvali Dvali et al. et al. (2000)), (2000)), and and those those which which attempt attempt to resolve to resolve the the fine fine tuning tuning cosmological cosmological constantconstant problem problem through through degravitation degravitation (Dvali (Dvali et al. et 2007;al. 2007; de Rhamde Rham et al. et 2008).al. 2008).
In starkIn stark contrast contrast to ⇤ toand⇤ and quintessence, quintessence, modifications modifications to gravity to gravity can can have have marked marked effects effects on onboth both the the growth growth of largeof large scale scale structure structure and and the the background background expansion. expansion. It is It common is common that that models models that that modify modify GR GR are are able able to to reproducereproduce the the distance distance measurements measurements but but alter alter the the growth growth of large of large scale scale structure, structure, opening opening up upthe the possibility possibility ofof testing testing and and discriminating discriminating between between the the different different theories. theories. Generically, Generically, the the Poisson Poisson equation equation relating relating over-densitiesover-densities to gravitational to gravitational potentials potentials is altered is altered and and the the potential potential that that determines determines geodesics geodesics of relativistic of relativistic particles,particles, in terms in terms of the of the Newtonian Newtonian gauge gauge potentials potentials( +( +)/ 2),/ differs2, differs from from that that that that determines determines the the motion motion of non-relativisticof non-relativistic particles, particles, . Creating. Creating / /=1 =1duringduring an anaccelerative accelerative era era is extraordinarily is extraordinarily difficult difficult in in 6 6 fluidfluid models models of dark of dark energy energy (Hu (Hu 1998). 1998). Measuring Measuring it, therefore, it, therefore, could could be abe smoking a smoking gun gun of a of deviation a deviation from from GR.GR. In Zhang In Zhang et al. et al. (2007b) (2007b) we we proposed proposed a way a way to constrain to constrain / /, by, by contrasting contrasting the the motions motions of galaxies of galaxies andand the the lensing lensing distortions distortions of light of light from from distant distant objects objects that that LSST LSST and and MS-DESI MS-DESI data data will will be ideabe idea for. for.
BeanBean and and her her group group have have developed developed software software based based on on the the publicly publicly available available CAMB CAMB and and CosmoMC CosmoMC codes codes (Lewis(Lewis et al. et al. 2000; 2000; Lewis Lewis & Bridle & Bridle 2002) 2002) to perform to perform likelihood likelihood analyses analyses and and forecasting forecasting for for generic generic pho- pho- tometrictometric and and spectroscopic spectroscopic surveys surveys (Bean (Bean & Tangmatitham& Tangmatitham 2010; 2010; Laszlo Laszlo et al. et al. 2011; 2011; Kirk Kirk et al. et al. 2011; 2011; MuellerMueller & Bean & Bean 2013). 2013). This This includes includes peculiar peculiar velocity, velocity, weak weak lensing lensing and and galaxy galaxy clustering clustering correlations correlations andand cross-correlations cross-correlations with with the the CMB. CMB. The The code code models models dark dark energy energy and and modified modified gravity gravity using using a variety a variety Phenomenological modeling of gravity of phenomenologicalof phenomenological parameterizations, parameterizations, including including the the equation equation of stateof statew(zw)(,z the), the Hubble Hubble expansion expansion rate rate H(Hz)(,z the), the logarithmic logarithmic growth growth factor, factor,fg(fzg)(,z and), and its itsexponent, exponent, (z )(,z and), and a parameterization a parameterization directly directly related related • Modify perturbed Einstein’s equations to modificationsto modifications of the of the perturbed perturbed Einstein Einstein equations, equations,GmatterGmatter(z,k(z,k) and) andGlightGlight(z,k(z,k), ),
k2 k2= =4⇡G4⇡G a2⇢a 2⇢,k ,k2( 2(+ +)= )=8⇡G8⇡G a2⇢a 2⇢, , (1)(1) mattermatter lightlight wherewhere⇢ is⇢ backgroundis background density, density,k isk comovingis comoving spatial, spatial,a, thea, the scale scale factor factor and and is theis the rest-frame, rest-frame, gauge gauge • A modification to Poisson’s equation, G invariant,invariant, matter matter perturbation. perturbation. It includes It includes general general parameterizations parameterizationsmatter for for galaxy galaxy bias bias and and intrinsic intrinsic alignments alignments – Relevant at sub-horizon scales (Hirata(Hirata & Seljak & Seljak 2004; 2004; Laszlo Laszlo et al. et al. 2011), 2011), a simple, a simple, Gaussian Gaussian model model for for photometric photometric redshift redshift errors errors and and – Gmatter ≠1: can be mimicked by additional, dark sector clustering nonlinearnonlinear model model based based on onthe the⇤CDM-modeled⇤CDM-modeled Halofit Halofit algorithm algorithm (Smith (Smith & Zaldarriaga & Zaldarriaga 2011). 2011).
• An inequality between Newton’s potentials, Glight/Gmatter ProposedProposed work: work:ThisThis existing– existingRemains software relevant software will even will be at be modularized horizon modularized scales and and documented documented to integrate to integrate into into the the LSST LSST
DESCDESC and and MS-DESI MS-DESI analysis– G analysislight/G pipelines.matter pipelines.≠1: not Three easily Three mimicked. specific specific projects projects to enhance to enhance the the software software are are described described in in sectionssections C.1-C.3. C.1-C.3. The The improvements improvements• potential smoking will will ensure gun ensure for that modified that the the analysis gravity? analysis pipelines pipelines are are able able to meet to meet the the required required • Significant stresses exceptionally hard to create in non-relativistic levellevel of both of both theoretical theoretical modeling modelingfluids e.g. and andDM survey-specific and survey-specific dark energy. systematic systematic error error characterization characterization necessary necessary to define to define sciencescience requirements. requirements. When When Stage Stage III III data data is made is made public, public, expected expected on on this this proposal’s proposal’s timeframe, timeframe, we we willwill analyze analyze theRachel Bean, Tes ng Gravity, Vancouver, January 2015 the data data using using this this software software pipeline, pipeline, and and integrate integrate improvements improvements in the in the intrinsic intrinsic alignment, alignment, photometricphotometric error error and and nonlinear nonlinear modeling modeling into into the the code. code.
C.1.C.1. Detailed Detailed ties ties between between dark dark energy energy theory theory and and LSST LSST and and MS-DESI MS-DESI observations observations
WhileWhile the the phenomenological phenomenological parameterizations parameterizations outlined outlined above above help help translate translate observations observations into into broad broad dark dark energyenergy characteristics, characteristics, more more needs needs to be to be be be done done to connect to connect the the data data further further to dark to dark energy energy theory theory and and astrophysicallyastrophysically relevant relevant modifications modifications to GR. to GR. Many Many classes classes of modified of modified gravity gravity theories theories are are described described by by thethe general general “Horndeski” “Horndeski” action, action, the the most most general general theory theory of a of scalar a scalar field field coupled coupled to gravity to gravity for for which which the the
NarrativeNarrative - 3 - 3 3
Dots represent a derivative with respect to conformal leading to 2 time, ✓ is the peculiar velocity divergence, and cs and 1 d ln are the fluid’s sound speed and shear stress respectively. c 1 (1 ) ⌦ (a)(1 + 3w ).(18) CAMB evolves perturbations in the synchronous gauge f d ln a ⇡ i i 3 g i in which the metric perturbations are described by h and X ˙ 2 Dots represent a derivative with respect to conformal ⌘.leading A parameter to ↵ (h +6˙⌘)/2k relates metric and For a typical ⇤CDM+GR model, the conformal Newto- 2 ⌘ time, ✓ is the peculiar velocity divergence, and cs and matter perturbations in the synchronous and conformal nian growth rate on horizon scales is 11% lower, cor- Newtonian1 d ln gauges c [ADD CITES] [Ma and Bert] ⇠ are the fluid’s sound speed and shear stress respectively. 1 (1 ) ⌦ (a)(1 + 3w ).(18) responding to a larger, horizon scale 0.6. CAMB evolves perturbations in the synchronous gauge f d ln a ⇡ i i ⇠ g =˙↵ + ↵ i (11) in which the metric perturbations are described by h and H X ⌘. A parameter ↵ (h˙ +6˙⌘)/2k2 relates metric and For a typical ⇤CDM+GR= ⌘ ↵ model, the conformal Newto-(12) ⌘ H 2. Perturbed Einstein equations matter perturbations in the synchronous and conformal nian growth rate = on horizonS 3 (1 scales + w is)↵ 11% lower,(13) cor- i i i ⇠ Newtonian gauges [ADD CITES] [Ma and Bert] responding to a larger,S horizonH2 scale 0.6. ✓i = ✓i + ↵k . ⇠ (14) We define two variables, GM and GL, that can be scale =˙↵ + ↵ (11) and redshift dependent [? ][CHECK sign conventions] H where S denotes synchronous gauge matter perturba- = ⌘ ↵ (12) 2. Perturbed Einstein equations2 H tions. Note that ⌘kCAMB =(1 3K/k )k⌘, zCAMB = S 3K 2 2 i = i 3 (1 + wi)↵ (13) ˙ h/2k and CAMB = ↵k. 1 2 k = 4⇡Ga ⇢ GM (19) S H2 k ⇥ ✓i = ✓i + ↵k . (14) We define two variables, GM and GL, that can be scale ✓ ◆ To evolve the metric perturbations in time in GR, 2 2 and redshift dependent [? ][CHECK sign conventions] k ( + )= 8⇡Ga ⇢ GL. (20) where S denotes synchronous gauge matter perturba- CAMB calculates⌘ ˙ using the time-space perturbed Ein- ⇥ 2 stein equation, and then uses the time-time and time- tions. Note that ⌘kCAMB =(1 3K/k )k⌘, zCAMB = 3K where G is the gravitational. G = 1 and G =1 ˙ space equations1 to obtaink2 = an expression4⇡Ga2 ⇢ forG ↵.Anew(19) M L h/2k and CAMB = ↵k. k2 ⇥ M for GR. The four equations (8), (10), (19) and (20) To evolve the metric perturbations in time in GR, model for gravity✓ is◆ implemented in CAMB by altering 2 ˙ 2 form a complete set of equations to describe perturba- CAMB calculates⌘ ˙ using the time-space perturbed Ein- the methods to calculatek ( + )=↵ and8h⇡.Ga ⇢ GL. (20) ⇥ tion growth. stein equation, and then uses the time-time and time- The CAMB code defines perturbations relative to the where G is theS gravitational. GM = 1 and GL =1 [DISCUSS alternative models of gravity and the rel- space equations to obtain an expression for ↵.AnewCDM rest frame, ✓c = 0, so that CDM peculiar velocities for GR. The four equations (8),2 (10), (19) and (20) evance of the above GM and GL to them e.g. Linder model for gravity is implemented in CAMB by altering are purely dependent on ↵, ✓c = ↵k . the methods to calculate ↵ and h˙ . form a complete set of equations to describe perturba- papers (e.g. Daniel and Linder ones, or others?) will The CAMB code defines perturbations relative to the tion growth. likely discuss this, perhaps Amendola et al also]. S [DISCUSS alternative models of gravity and the rel- CDM rest frame, ✓c = 0, so that CDM peculiar velocities 1. Modified growth rate On subhorizon scales, (8) and (10) are combined to give 2 evance of the above GM and GL to them e.g. Linder are purely dependent on ↵, ✓c = ↵k . a second order equation for CDM perturbation growth papers (e.g. Daniel and Linder ones, or others?) will [CHECK THIS for X ˙ and k2 ] Alikely minimal discuss approach this, perhaps to modifying Amendola the growth et al also] history. is H ± to parameterize a deviation from GR by its impact on the 1. Modified growth rate On subhorizon scales, (8) and (10) are combined to give ¨ + ˙ + k2 0 (21) lineara second CDM growth order equation rate directly. for CDM In the perturbation epochs in growthwhich c H c ⇡ Alternativethe modification descriptor: is considered,˙ the the 2growth fiducial form rate, for thef & γ A minimal approach to modifying the growth history is [CHECK THIS for X and k ] g growth rate of the rest frameH CDM± density perturbations, GM therefore parameterizes the e↵ect of a modification to parameterize a deviation from GR by its impact on the S 2 to Einstein’s equations on the clustering properties of • whichAlternative are equal parameterization: to ¨ c+in ˙ CAMB,+ thek rate is of reasonably0 CDM growth fit (21) by linear CDM growth rate directly. In the epochs in which c H c ⇡ matter on subhorizon scales. Note on super horizon the modification is considered, the fiducial form for the modeling the logarithmic growth factor, G therefore parameterizes the e↵ect of a modification scales time derivatives of both and , and hence GL growth rate of the rest frame CDM density perturbations, M S S to Einstein’s equationsd ln onc the clustering properties of and GM ,a↵ect the growth of matter perturbations. which are equal to in CAMB, is reasonably fit by fg = ⌦m(a) , (15) c matter on subhorizon⌘ d ln scales.a Note on superFor horizonGR γ~0.55 on Equation (21) provides an explicit connection between modeling the logarithmic growth factor, subhorizon scales scales time derivatives of both and , and hence GL fg and GM on sub horizon scales S with =0.55 even to horizon scales [ADD CITES] . d ln c and GM ,a↵ect the growth of matter perturbations. fg = ⌦m(a) , (15) [DISCUSS alternative models of gravity and the values ⌘ d ln a • f relatedEquation to G (21) providesbut does an describe explicit G connectionG between 2fg d ln fg ˙ of gfg or theymatter produce, e.g. Amendolalight/ etmatter al p88-89 fg and GM on sub horizon scales GM 1+fg + + H2 . (22) with =0.55 even to horizon scales [ADD CITES] . mention some]. ⇡ 3⌦m d ln a " H # [DISCUSS alternative models of gravity and the values To calculate the transfer functions we assume that (8) 2fg d ln fg ˙ of fg or they produce, e.g. Amendola et al p88-89 and (13) hold and use Photon geodesics are determined by the sum of the GM 1+fg + + H2 . (22) mention some]. ⇡ 3⌦m " d ln a # potentials, + . Hence G describes the alteration ˙ H L To calculate the transfer functions we assume that (8) S S h to how an over or under dense region will modify the ˙ = fg = (16) and (13) hold and use Photon geodesicsc areH determinedc 2 by the sum of the propagation of light. potentials, + . Hence G describes the alteration ˙ ˙ L In comparison to the universal adoption of w to de- ˙S S h to defineto howh, an and over assume or under the method dense region to calculate will modify⌘ ˙ is the the c = fg c = (16) scribe the equations of state parameter for dark energy, H 2 samepropagation as in GR. of light. there have been a large number of di↵erent, but equiv- Rachel Bean, Tes ng Gravity, Vancouver, January 2015 NoteIn that comparison the fiducial to the fit universal in (15) should adoption not beof w appliedto de- to define h˙ , and assume the method to calculate⌘ ˙ is the alent notations to describe the modifications to the per- same as in GR. to conformalscribe the Newtonianequations of CDM state perturbationsparameter for dark on horizon energy, scales.there Using have (13) been and a large noting number that, of at di late↵erent, times, but⌘ ˙ equiv-h˙ turbed Einstein equations. In Table I, for convenience, Note that the fiducial fit in (15) should not be applied ⌧ we summarize a number of the notations in the literature. andalenth˙ 2 notationsk2↵, to describe the modifications to the per- to conformal Newtonian CDM perturbations on horizon ⇡ In parameterizing our alterations to GR in (19) and scales. Using (13) and noting that, at late times,⌘ ˙ h˙ turbed Einstein equations. In Table I, for convenience, 3 2f ˙ ˙ 2 ⌧ we summarize a numberc of the notations in the literature. (20) we need to modify the equations for⌘ ˙ and h during and h 2k ↵, S 1 H2 , (17) ⇡ In parameterizing c our⇡ alterationsk to GR in (19) and the epochs in which the modifications are present. We 2 c 3 f (20) we need to modify the equations for⌘ ˙ and h˙ during S 1 H2 , (17) c ⇡ k the epochs in which the modifications are present. We Four distinct cosmological tracers
1. Standard candles and rulers
2. Clustering of matter
3. Motion of non-relativistic tracers
4. Lensing distortion of light
Rachel Bean, Tes ng Gravity, Vancouver, January 2015 Different relations to gravitational map
II: Growth, up to some I: Background expansion III: Growth directly normalization
Probes of homogeneous Biased tracers of perturbed Direct tracers of perturbed geometry: gravitational field gravitational field
• CMB angular diameter • Galaxy distribution • CMB ISW correlation distance • Xray and SZ selected • Weak galaxy and cluster • Supernovae luminosity galaxy cluster distribution lensing correlation distance • Peculiar velocities/ bulk • BAO angular/radial scale flows of galaxies
Rachel Bean, Tes ng Gravity, Vancouver, January 2015 Complementary tracers for testing gravity
Masamune et al 2010
• Non-relativistic: Galaxy • Relativistic: Weak lensing & positions & motions CMB lensing & ISW
– Sensitive to ψ ∼ Gmat – Sensitive to (φ+ψ): Glight – Biased tracer – Direct tracer of potential, – Can be measured at specific z – but still plenty of systematics (more on this…) – Integrated line of sight info
• Contrasting both can get at Glight/Gmatter Zhang et al 2007
Rachel Bean, Tes ng Gravity, Vancouver, January 2015 Complementary tracers for testing gravity
Are these the unrealis c dreams of a theorist?
Masamune et al 2010
• Non-relativistic: Galaxy • Relativistic: Weak lensing & positions ‘g’& motions ‘θ’ CMB lensing & ISW
– Sensitive to ψ ∼ Gmat – Sensitive to (φ+ψ): Glight – Biased tracer – Direct tracer of potential, but – Can be measured at specific z still plenty of systematics (more on this…) – Integrated line of sight info
• Contrasting both can get at Glight/Gmatter Zhang et al 2007
Rachel Bean, Tes ng Gravity, Vancouver, January 2015 CANDELS photo-z investigation 9
Complications: Photometric redshifts
• Quicker (more galaxies) and concurrent with imaging, but less accurate than spectral z CANDELS photo-z investigation 23
• Challenge to refine photo-z estimates given disparate and incomplete spectroscopic samples
• Redshift probability distribution dispersed and biased relative to the true spectroscopic z.
Fig. 14.— An example of the photometric redshift probability distributions for one galaxy with spectroscopic redshift z=0.734. Blue lines show five individual codes (code 3B, 6E ,7C ,11H, and 13C) without correcting distributions so that they match the 68.3% confidence interval z criterion. Red lines show the distributions after corrections. Finally, the black line shows the sum of the individual distributions.
z = rms 10 Dahlen et al. In this example of the hierarchical) Bayesian method, combined with a magnitude limit appropriate for this par- 1+zspec we have used a simple assumption for U(z), i.e., that we ticular survey. In addition, it should be possible to let have no information if the measured Pm(z)i is wrong. the expected distribution be dependent on, e.g., apparent Furthermore, we have allowed fbad in the whole range magnitude or color. fbad=[0.0,1.0]. Alternatively, we can assume that there Instead of using a generic form for U(z), another pos- z =(zphot zspec) is at least some minimum probabilityspec that the actual mea- sibility is to dilate the given P(z) and use this for U(z). z surement are correct and let the badz fraction vary in the In this case we assume that the errors are underestimated range fbad=[0.0,x]. Repeating our analysis after varying if the measurement is bad, rather than having no infor- biasz = mean x does not change results significantly, however, there is mation. There are many possibilities when applying the a slight decrease in the outlier fraction and full rms when hierarchical Bayesian method as discussed in Lang & Hogg 1+zspec setting 0.3 Fig. 3.— The rms after excluding outliers (σO )andoutlierfractionsforthedifferent codes. The five codes with the lowest combination of scatter and outlier fractions are plotted in red. Black star symbols show the median of all codes, while the red stars show the median of the five codes with the smallest scatter. B. Joachimi and S. L. Bridle: Simultaneous measurement of cosmic shear and galaxy number density correlations 2000; Lee & Pen 2000; Catelan et al. 2001; Crittenden et al. Johnstonet al. 2007; Cacciato et al. 2009) and lensing mag- 2001; Jing 2002; Mackey et al. 2002; Hirata & Seljak 2004; nification (Broadhurst et al. 1995; Zhang & Pen 2005, 2006; Bridle & Abdalla 2007; Schneider & Bridle 2010), as well as van Waerbeke 2010). We follow the ansatz outlined in Bernstein the alignment of the spin or the shape of a galaxy with its own (2009) and extend the investigation by Bridle & King (2007) dark matter halo (e.g. Pen et al. 2000; van den Bosch et al. 2002; who considered the residual information content in galaxy shape Okumuraet al. 2009; Okumura& Jing 2009; Brainerdet al. correlations after marginalising over the parameters of twolog- 2009; see also Schäfer 2009). Intrinsic alignments have linear grid models representing the II and GI terms. We quantify also been investigated observationally, where non-vanishing the cross-calibration properties of the joint set of observables II and GI signals have been detected in several surveys and determine the requirements on cosmological surveys to effi- (Brown et al. 2002; Heymans et al. 2004; Mandelbaum et al. ciently apply this joint approach. 2006; Hirata et al. 2007; Brainerd et al. 2009). This paper is organised as follows: In Sect. 2 we give an The results of both theoretical studies and observations show overview on the two-point correlations that form part of the large variations, but most are consistent with a contamination of galaxy shape and number density observables, and we derive the order 10 % by both II and GI correlations for future surveys their explicit form. Two appendices provide further details. that further divide the galaxy sample into redshift slices (cosmic Section 3 demonstrates how we model the different signals and shear tomography). Hence, the control of intrinsic alignments in their dependence on cosmology. We introduce a general grid cosmic shear studies is crucial to obtain unbiased results oncos- parametrisation for the intrinsic alignments and the galaxybias. mological parameters. Accurate models would solve the prob- Furthermore we summarise our Fisher matrix formalism and the lem, but progress is hampered due to the dependence of intrin- figures of merit we employ. In Sect.4 we present our results on sic alignments on the intricacies of galaxy formation and evolu- the dependence of the parameter constraints on the freedom in tion within their dark matter environment. Currently, the level of the model of intrinsic alignments and galaxy bias, the character- models is crude, and partly only motivated phenomenologically istics of the redshift distributions, and the priors on the different (see e.g. Schneider & Bridle 2010 for recent progress). sets of nuisance parameters. Finally, in Sect.5 we summariseour The II contamination can be controlled relatively eas- findings and conclude. ily by excluding close pairs of galaxies from the analy- sis (King & Schneider 2002, 2003; Heymans & Heavens 2003; Takada & White 2004). Joachimi & Schneider (2008, 2009) in- 2. Two-point correlations from cosmological troduced a nulling technique which transforms the cosmic shear surveys data vector and discards all entries of the transformed data set that are potentially contaminated by the GI signal. While this Cosmological imaging surveys observe the angular positionsand approach only relies on the well-known redshift dependence the projected shapes of huge numbers of galaxies over increas- of gravitational lensing, King (2005) projects out the GI term ingly large areas on the sky. In addition, by means of multi- by making use of template functions. Furthermore the work colour photometry, it is possible to perform a tomographic anal- by Mandelbaum et al. (2006) and Hirata et al. (2007) suggests ysis, i.e. obtain coarse information about the line-of-sight dimen- that the intrinsic alignment may be dominated by luminous red sion in terms of photometricredshifts (photo-z).From the galaxy galaxies which could be eliminated from the cosmic shear cat- shapes in a given region of space, one can infer the ellipticity alogues. All these removal techniques require excellent redshift ϵ(i)(θ) = γ(i)(θ) + γ(i)(θ) + ϵ(i) (θ) , (1) information, and still they can cause a significant reductionin G I rnd the constraints on cosmology. where the superscript in parentheses assigns a photo-z bin i. Deep imaging surveys not only provide information about The observed ellipticity ϵ has contributions from the gravita- the shape of galaxies, but allow in addition for a measurementof tional shear γG and an intrinsic shear γI,whichiscausedby galaxy number densities, as well as cross-correlations between the alignment of a galaxy in its surrounding gravitational field. shape and number density information. This substantial exten- Moreover, ϵ is assumed to have an uncorrelated component ϵrnd, sion of the set of observables increases the cosmological infor- which accounts for the purely random part of the intrinsic orien- mation to be extracted and, more importantly, enables oneComplications: to in- tations and shapes ofLensed galaxies. Note galaxies that (1) is only valid if the ternally calibrate systematic effects (Hu & Jain 2004; Bernstein gravitational shear is weak, see e.g. Seitz & Schneider (1997); 2009). By adding galaxy number density information one adds Bartelmann & Schneider (2001) and for certain definitions of el- signals that are capable of pinning down the functional form• ofLensinglipticity. distorts observed volume and intrinsic alignments, but one also introduces as another system-magnifiesLikewise, faint the galaxies positions ( ofMoessner galaxies can & be used to construct atic, the galaxy bias, which quantifies the lack of knowledgeJainan 1997) estimate of the number density contrast about how galaxies, i.e. the visible baryonic matter, followthe underlying dark matter distribution. (i) θ (i) θ (i) θ (i) θ n ( ) = nm ( ) + ng ( ) + n ( ) , (2) It is the scope of this work to elucidate the perfor- rnd mance of a joint analysis of galaxy shape and number den- which is determined by the intrinsic number density contrast • Number depends on slope of sity information as regards the ability to constrain cosmo- of galaxies n and the alteration of galaxy counts due to lens- luminosity functiong logical parameter in presence of general and flexible mod- ing magnification nm.Anuncorrelatedshotnoisecontributionis (i) els of intrinsic alignments and galaxy bias. In doing so we added via nrnd.Incontrasttoϵ (θ)thenumberdensitycontrast (i) θ incorporate several cosmological signals which have• been Galaxyn ( )canobviouslynotbeestimatedfromindividualgalaxies.correlations must factor this in considered before as promising probes of cosmology them- One can understand n(i)(θ)astheensembleaverageoverahypo- selves, including galaxy clustering from photometric red- thetical, Poisson-distributed random field of which the observed shift surveys (Blake & Bridle 2005; Dolney et al. 2006; ZhanCninj galaxy= Cgig distributionj + Cgimj is+ oneCm particularigj + Cm representation.imj The formal 2006; Blake et al. 2007; Padmanabhan et al. 2007) galaxy- relation between the projected number density contrast as used Wolf et al 2003 galaxy lensing (e.g. Schneider & Rix 1997; Guzik & Seljak in (2) and the three-dimensional galaxy number density fluctua- 2001, 2002; Seljak 2002; Seljak et al. 2005; Yoo et al. 2006; tions will be provided below, see (12). 2 Rachel Bean, Tes ng Gravity, Vancouver, January 2015 B. Joachimi and S. L. Bridle: Simultaneous measurement of cosmic shear and galaxy number density correlations 2000; Lee & Pen 2000; Catelan et al. 2001; Crittenden et al. Johnstonet al. 2007; Cacciato et al. 2009) and lensing mag- 2001; Jing 2002; Mackey et al. 2002; Hirata & Seljak 2004; nification (Broadhurst et al. 1995; Zhang & Pen 2005, 2006; Bridle & Abdalla 2007; Schneider & Bridle 2010), as well as van Waerbeke 2010). We follow the ansatz outlined in Bernstein the alignment of the spin or the shape of a galaxy with its own (2009) and extend the investigation by Bridle & King (2007) dark matter halo (e.g. Pen et al. 2000; van den Bosch et al. 2002; who considered the residual information content in galaxy shape Okumuraet al. 2009; Okumura& Jing 2009; Brainerdet al. correlations after marginalising over the parameters of twolog- 2009; see also Schäfer 2009). Intrinsic alignments have linear grid models representing the II and GI terms. We quantify also been investigated observationally, where non-vanishing the cross-calibration properties of the joint set of observables II and GI signals have been detected in several surveys and determine the requirements on cosmological surveys to effi- (Brown et al. 2002; Heymans et al. 2004; Mandelbaum et al. ciently apply this joint approach. 2006; Hirata et al. 2007; Brainerd et al. 2009). This paper is organised as follows: In Sect. 2 we give an The results of both theoretical studies and observations show overview on the two-point correlations that form part of the large variations, but most are consistent with a contamination of galaxy shape and number density observables, and we derive the order 10 % by both II and GI correlations for future surveys their explicit form. Two appendices provide further details. that further divide the galaxy sample into redshift slices (cosmic Section 3 demonstrates how we model the different signals and shear tomography). Hence, the control of intrinsic alignments in their dependence on cosmology. We introduce a general grid cosmic shear studies is crucial to obtain unbiased results oncos- parametrisation for the intrinsic alignments and the galaxybias. mological parameters. Accurate models would solve the prob- Furthermore we summarise our Fisher matrix formalism and the lem, but progress is hampered due to the dependence of intrin- figures of merit we employ. In Sect.4 we present our results on sic alignments on the intricacies of galaxy formation and evolu- the dependence of the parameter constraints on the freedom in tion within their dark matter environment. Currently, the level of the model of intrinsic alignments and galaxy bias, the character- models is crude, and partly only motivated phenomenologically istics of the redshift distributions, and the priors on the different (see e.g. Schneider & Bridle 2010 for recent progress). sets of nuisance parameters. Finally, in Sect.5 we summariseour The II contamination can be controlled relatively eas- findings and conclude. ily by excluding close pairs of galaxies from the analy- sis (King & Schneider 2002, 2003; Heymans & Heavens 2003; Takada & White 2004). Joachimi & Schneider (2008, 2009) in- 2. Two-point correlations from cosmological troduced a nulling technique which transforms the cosmic shear surveys data vector and discards all entries of the transformed data set that are potentially contaminated by the GI signal. While this Cosmological imaging surveys observe the angular positionsand approach only relies on the well-known redshift dependence the projected shapes of huge numbers of galaxies over increas- of gravitational lensing, King (2005) projects out the GI term ingly large areas on the sky. In addition, by means of multi- by making use of template functions. FurthermoreComplications: the work colour Intrinsic photometry, alignments it is possible to perform a tomographic anal- by Mandelbaum et al. (2006) and Hirata et al. (2007) suggests ysis, i.e. obtain coarse information about the line-of-sight dimen- sion in terms of photometricredshifts (photo-z).From the galaxy that the intrinsic alignment may be dominated• byGalaxy luminous Intrinsic red shapes are aligned in their host halo galaxies which could be eliminated from the cosmic shear cat- shapes in a given region of space, one can infer the ellipticity alogues. All these removal techniques require excellent redshift ϵ(i)(θ) = γ(i)(θ) + γ(i)(θ) + ϵ(i) (θ) , (1) information, and still they can cause a significant reductionin G I rnd the constraints on cosmology. where the superscript in parentheses assigns a photo-z bin i. Deep imaging surveys not only provide• information Observed about shapeThe correlations observed ellipticityinclude bothϵ has lensed contributions and intrinsic from the terms gravita- the shape of galaxies, but allow in addition for a measurementof tional shear γG and an intrinsic shear γI,whichiscausedby galaxy number densities, as well as cross-correlations between the alignment of a galaxy in its surrounding gravitational field. shape and number density information. This substantial exten- Moreover, ϵ is assumed to have an uncorrelated component ϵrnd, sion of the set of observables increases the cosmological infor- which accounts for the purely random part of the intrinsic orien- mation to be extracted and, more importantly, enables one to in- tations and shapes of galaxies. Note that (1) is only valid if the ternally calibrate systematic effects (Hu & Jain 2004; Bernstein gravitational shear is weak, see e.g. Seitz & Schneider (1997); 2009). By adding galaxy number density information one adds C✏ ✏ Bartelmann= CG G &+ SchneiderCG I (2001)+ CI andG for+ certainCI I definitions of el- signals that are capable of pinning down the functional form ofi j lipticity.i j i j i j i j intrinsic alignments, but one also introduces as another system- CnLikewise,✏ = C theg positionsG + C ofg galaxiesI can be used to construct atic, the galaxy bias, which quantifies the lack of knowledge an estimatei j of thei numberj densityi j contrast about how galaxies, i.e. the visible baryonic• matter, The amplitude followthe of intrinsic alignments is a function galaxy type, underlying dark matter distribution. (i) θ (i) θ (i) θ (i) θ luminosity and redshiftn ( ) = nm ( ) + ng ( ) + n ( ) , (2) It is the scope of this work to elucidate the perfor- rnd mance of a joint analysis of galaxy shapeRachel Bean, Tes ng Gravity, Vancouver, January 2015 and number den- which is determined by the intrinsic number density contrast sity information as regards the ability to constrain cosmo- of galaxies ng and the alteration of galaxy counts due to lens- logical parameter in presence of general and flexible mod- ing magnification nm.Anuncorrelatedshotnoisecontributionis (i) els of intrinsic alignments and galaxy bias. In doing so we added via nrnd.Incontrasttoϵ (θ)thenumberdensitycontrast incorporate several cosmological signals which have been n(i)(θ)canobviouslynotbeestimatedfromindividualgalaxies. considered before as promising probes of cosmology them- One can understand n(i)(θ)astheensembleaverageoverahypo- selves, including galaxy clustering from photometric red- thetical, Poisson-distributed random field of which the observed shift surveys (Blake & Bridle 2005; Dolney et al. 2006; Zhan galaxy distribution is one particular representation. The formal 2006; Blake et al. 2007; Padmanabhan et al. 2007) galaxy- relation between the projected number density contrast as used galaxy lensing (e.g. Schneider & Rix 1997; Guzik & Seljak in (2) and the three-dimensional galaxy number density fluctua- 2001, 2002; Seljak 2002; Seljak et al. 2005; Yoo et al. 2006; tions will be provided below, see (12). 2 Complications: Shear contamination • Incomplete correction of the atmospheric and instrumental PSF can induce additive and multiplicative shear errors ✏i =(1+mi)✏i + a Spurious Shear in Weak Lensing with LSST 15 (a) Figure 9. Absolute spurious shear correlation function after com- LSST Project bining 10 years of r-andi-band LSSTChang data when et aal standard 2012 KSB algorithm is implemented and the PSF is modeled at dif- ferent levels: non-stochastic PSF knowledge only (dashed) and partial stochastic PSF knowledge from polynomial interpolation Rachel Bean, Tes ng Gravity, Vancouver, January 2015 of stars (solid). Red lines indicate the pessimistic case assuming Ne↵ =184while the green lines show the optimistic case when Ne↵ =368is assumed. All curves are the medians of 20 different realisations under the fiducial observing condition, with the error bars indicating the standard deviation in the 20 curves divided by p20. Negative values are plotted with open symbols. (b) analysis. From Appendix D, we estimate Ne↵ to be between 184 and 368, with Ne↵ =184being the most pessimistic scenario and Ne↵ =368being the most optimistic. Consider now the three scenarios described in Section 7, where KSB is used to correct for the PSF effects and the three levels of PSF modelling are assumed. For a hypothet- ical perfect PSF modelling technique, the shear errors in individual frames are already consistent with zero, so there is no need to discuss the combined results here. For the second case, we know only the non-stochastic component of the PSF. In this case, spurious shear correla- tions result from not modelling any of the stochastic compo- nent of the PSF shape. In the combined dataset, the latter contribution can be estimated by taking the solid spurious shear correlation function in Figure 8 (b) and multiply by 1/Ne↵ to account for the averaging of the stochastic spurious shear correlation. When both the non-stochastic and stochastic PSF com- (c) ponents are modeled using a 5th-order polynomial model fit- Figure 8. The absolute ellipticity correlation function for the ted to the stars, we assume that the smoothly varying non- fiducial galaxies before PSF correction (dotted black) and the stochastic PSF component is fully modeled and the spurious absolute shear correlation function after PSF correction (solid shear is mainly due to stochastic PSF modelling errors. The green) for (a) perfect PSF models, (b) perfect non-stochastic PSF combined shear correlation function then can be estimated models, and (c) PSF models constructed with a 5th-order polyno- by scaling the spurious shear correlation function in Figure 8 mial fit to bright stars. The absolute ellipticity correlation func- (c) by 1/Ne↵ . tion (dashed red) in each case is plotted for comparison. All curves The total expected spurious shear correlation functions are the medians of 20 different realisations under the fiducial ob- from combining Ne↵ exposures for the latter two cases are serving condition, with the error bars indicating the standard shown in Figure 9. deviation in the 20 curves divided by p20. Negative values are plotted with open symbols. c 2011 RAS, MNRAS 000,1–18 Coyote III 9 1.05 1.04 1.03 1.02 1.01 2 sim ∆ / 1 2 emu ∆ 0.99 σ 0.98 0.97 0.96 0.95 −2 −1 0 10 10 10 k [1/Mpc] 0.1 0.0 0.1 0.2 0.3 0.4 Complications:− Non linearFIG .10.—M004,M008,M013,M016,M020,andM026.Foreachsim-scales 0.005 0.010 0.020 0.050 0.100 0.200 0.500 1.000 2.000 ulation, we show six residuals corresponding to the different values of the k [1/Mpc] scale factor a.Theerrorsareontheorderof1%forthebulkofthedomain of interest. Considering that the tested emulators are builtonanincomplete design, this result is remarkably good. • FIGCosmological.8.—MedianMCMCdrawforthebandwidthfunction extractionσ in(black NL line). scalesof the codes,requires to structure modeling the directories whereto different runs Small values on this plot correspond to places where the spectra are compar- ativelycommensurate less smooth because of the baryon wiggles.accuracy In addition, all theof: power are performed in, and to submit the simulations to the comput- spectra are shown (37 models, 6 redshifts each) in light gray.Thepower ing queue system. For future efforts of this kind the adoption spectra– a scaled the and shiftedunderlying to fit the plot. σCDMis low on thenonlinear baryon acoustic powerand development spectrum of dedicated (for LCDM workflow capabilitiesand other for thes models)e oscillation scale to accommodate local wiggles, as expected. The vertical tasks must be considered. The number of tasks to be car- line shows the matching point between perturbation theory and the simula- tion outputs.– OnModeling the left of this, each baryonic replicate is identica effects,l(comparedtothe signal riedand out uncertainties, will become too large toon keep halo track ofconcentration without such realizations from the different simulation boxes) and smooth, therefore the tools. In addition, since large projects will require extensive σ behavior of has not much information. collaborations, software tools will make it easier to work ina14 team environment since each collaborator will have informa- 1.05 a=0.5 tion about previous tasks and results. An example of such a a=0.6 1.04 a=0.7 tool for cosmological simulation analysis and visualization is a=0.8 a=0.9 given in Anderson et al. (2008). 1.03 a=1.0 We carriedout the data analysisafter the runswere finished. 1.02 For very large simulations this is not very practical, and on- 1.01 the-fly analysis tools are required to minimize read and write 2 sim times and failures. This in turn requires that the code infras- ∆ / 1 tructure be tailored to the problem under consideration. 2 emu ∆ 0.99 (3) Serving the Data: Clearly, large simulation efforts can- not be carried out by a few individuals, and require possi- 0.98 bly community wide coordination. The simulation data will 0.97 be valuable for many different projects. It is therefore nec- essary to make the data from such simulation efforts pub- 0.96 licly available and serve them in a way that new science can 0.95 −2 −1 0 10 10 10 be extracted from different groups of simulations. Transfer- k [1/Mpc] ring large amounts of data is difficult because of limitations in communication bandwidth and also because of the large FIG.9.—Ratiooftheemulatorpredictiontothesmoothsimulatedpower storage requirements. It would therefore be desirable to have spectra for the M000 cosmology at six values of the scale factor a.Theerror computational resources dedicated to the database. In such exceeds 1% very slightly in only one part of the domain for the scale factors asituation,researcherswouldbeabletoruntheiranalysis a =0.7,0.8, and 0.9. Zentner et al 1212.1177 LawrenceFIG. 8: Biases et induced al 0912.4490 in the dark energy equationcodes of state on machinesFIG. 9: with Same direct as Fig. access 8, but for to the the bias database on wa from and a per- Stage in terms of storage. From theparameter Coyotew Universe0 from the runs analysis we of stored a STAGE IVform dark energy queriesIV on experiment. the data easily. Notice We that are the planning vertical axis to is make asymmetric the experiment as a function of the maximum multipoleCoyote used Universe to about database zero. available in the future and use it as 11 time snapshots plus theinfer initial cosmological conditions parameters. (particle For pos STAGEi- IV experiments, tions and velocities and halofour information) of the OWLS leading simulations to 250 lead GB to biasesamanageabletestbedforsuchservices. significantly larger than the others and it is useful to emphasize(4) this.Communication In with other Communities: The complex- Rachel Bean, Tes ng Gravity, Vancouver, January 2015 of data per run. For the 300 billion particle run this would tions in larger computational volumes. On another front, increase to 75 TB. Only verythe few main places panel, worldwide the biases that would result be from analyzingity of those the analysisit may task be possiblemakes it to necessary develop more to efficiently sophisticated col modelslab- four simulations without any model to account for baryonic for the influence of baryons on lensing power spectra that able to manage the resultingeff largeects. databases.These biases are clearly very large. Eachorate line andis la- communicate with other communities, for example can minimize biases in inferred cosmological parameters (2) Simulation Infrastructure:beled byRunning the name aof very the corresponding large num- OWLS simulationstatisticians, in computer scientists, and applied mathematicians. ber of simulations makes itthe necessary panel. The shadedto integrate (orange) the band, ma- shows theMany range of tools bi- thatwithout will a be significant essential cost for in statisticalprecision errors. cosmology As the cos- in ases that result after taking the best-fit concentrationthe future model havemological already community been invented learns – from the task Stage is III toexperiments find them jor parts of the analysis stepsto describe into the baryonic simulation effects to code analyze and these same four simu- such as DES and prepares for the Stage IV experiments to automate as much of thelations. mechanics The biases of here running are significantly the code reduced,and but useremain themof in the the coming best way decade, possible. such efforts should be a high prior- (submission, restarts) as possible.non-negligible For ( the∼ 1σ Coyote). The inset Univers panel showse results for the ity in order to maximize the scientific yields of the next project we developed severalremaining scripts fiveto generate OWLS simulations. the input In fi thisles case, the inner (or- generation6. ofCONCLUSIONS dark energy experiments. ange) and outer (blue) shaded bands are the same as in Fig. 5 for DES. In each of these cases, the mitigation procedure ren- ders biases in the dark energy equation of state parameter w0 smaller than the statistical error. Acknowledgments This work grew out of a working group meeting hosted Future work may be able to improve this situation. For by the Aspen Center for Physics. As such, this material one, simulations such as the OWLS simulations make pre- is based upon work supported in part by the National dictions for the properties of galaxies. It may be possible Science Foundation under Grant PHY-1066293 and the to compare the properties of galaxies in order to deter- hospitality of the Aspen Center for Physics. We are mine which simulations are more likely to represent the grateful to Marcel van Daalen, Joop Schaye, and the observed universe, and use this information to place pri- other members of the OWLS collaboration for making ors on additional parameters in mitigation schemes (the their simulation power spectra available. We thank Henk concentration parameters in our case, see Ref. [21]). The Hoekstra, Dragan Huterer, Jeffrey Newman, Bob Saka- OWLS collaboration has shown that the “AGN” simu- mano, Joop Schaye, and Risa Wechsler for helpful discus- lation describes the observed properties of galaxies most sions and comments on early drafts of this manuscript. successfully [27, 46], while our analysis of the “AGN” sim- ARZ and APH were funded by the Pittsburgh Parti- ulation for Stage IV experiments leaves a non-negligible cle physics, Astrophysics, and Cosmology Center (PITT residual bias. An important and necessary aspect of fu- PACC) at the University of Pittsburgh and by the Na- ture efforts to address these issues with simulations will tional Science Foundation through grant AST 0806367. be to develop lensing predictions from baryonic simula- APH is also supported by the U.S. Department of En- A single survey can’t give the full picture • Trade offs in – Techniques (SN1a, BAO,RSD, WL, Clusters + lensing, motions, positions) – Photometric speed vs. spectroscopic precision – Angular and spectral resolution – Astrophysical tracers used (LRGs, ELGs, Lya/QSOs, clusters) – Epochs, scales and environs being studied (cluster vs dwarf galaxies) • Much more than a DETF FoM: – Astrophysical & instrumental systematic mitigation as so easily summarized. – Readiness vs technological innovation – Survey area vs depth - repeat imaging, dithering, cadence and survey area overlap/config. • Future surveys distinct and complementary, both with each other and • ground based LSS surveys (LSST, DESI and others) • Planck and ground-based CMB gravitational lensing measurements 15 Rachel Bean, Tes ng Gravity, Vancouver, January 2015 Upcoming Surveys: Different strengths & systematics (based on publicly available data) Stage IV DESI LSST Euclid WFIRST-AFTA Starts, dura on ~2018, 5 yr 2020, 10 yr 2020 Q2, 7 yr ~2023, 5-6 yr Area (deg2) 14,000 (N) 20,000 (S) 15,000 (N + S) 2,400 (S) FoV (deg2) 7.9 10 0.54 0.281 Diameter (m) 4 (less 1.8+) 6.7 1.3 2.4 Spec. res. Δλ/λ 3-4000 (Nfib=5000) 250 (slitless) 550-800 (slitless) Spec. range 360-980 nm 1.1-2 µm 1.35-1.95 µm 20m Hα ELGs 20-30m LRGs/[OII] z = 1–2, BAO/RSD ELGs 0.6 < z < 1.7, ~20-50m Hα ELGs Z~0.7-2.1 2m [OIII] ELGS 1m QSOs/Lya 1.9 WFIRST photometric survey + CMB. GMatter,Glight constant 0 Rachel Bean, Tes ng Gravity, Vancouver, January 2015 Bean and Mueller, in prep Spec & Photo survey: constraints on γ0-γa Beware a strong theoretical prior significantly biases redshift range of interest for growth studies (as with equation of state) Rachel Bean, Tes ng Gravity, Vancouver, January 2015 Bean and Mueller, in prep Clusters as a probe of gravity • Cluster dynamics provide an Hot cluster alternative direct probe of the CMB γ gas gravitational potential. Energe c e- • Measurable via the kinetic Sunyaev Zeldovich (kSZ) effect Blue-shi ed γ Observatory Rachel Bean, Tes ng Gravity, Vancouver, January 2015 Clusters as a probe of gravity • Problem: kSZ has weak • Solution: isolate kSZ through frequency dependence and cross-correlation small amplitude (compared to thermal SZ) mean pairwise momentum Hand et. al 2012 ned.ipac.caltech.edu Rachel Bean, Tes ng Gravity, Vancouver, January 2015 kSZ pairwise dark energy constraints Current constraints: BOSS: Mueller, De Bernardis, Bean, Niemack 2014 Rachel Bean, Tes ng Gravity, Vancouver, January 2015 To conclude • Phenomenology, a powerful tool to connect theoretical modeling of gravity to observations on cosmological scales • Variety is critical to achieve required sensitivities over range of redshifts and environments: – Non-relativistic (peculiar velocities) and relativistic (lensing) – Different systematics (e.g. CMB and galaxy lensing) – Different epochs and scales – Different tracers: halo and field galaxies, clusters and CMB • Understanding and incorporating instrumental and astrophysical uncertainties key to realizing cosmological aims – Photometric redshift uncertainties – Effect of baryons processes on dark matter clustering – Disentangling instrumental (e.g. PSF) and astrophysical (e.g IAs) signals from cosmological Rachel Bean, Tes ng Gravity, Vancouver, January 2015