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Slides of 5 lectures at XV Brazilian School on Cosmology and ...

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Cosmic structure, averaging anddark energyDavid L. Wiltshire (University ong>ofong> Canterbury, NZ)DLW: Class. Quan. Grav. 28 (2011) 164006New J. Phys. 9 (2007) 377Phys. Rev. Lett. 99 (2007) 251101Phys. Rev. D78 (2008) 084032Phys. Rev. D80 (2009) 123512B.M. Leith, S.C.C. Ng & DLW:ApJ 672 (2008) L91P.R. Smale & DLW, MNRAS 413 (2011) 367P.R. Smale, MNRAS 418 (2011) 2779DLW, P R Smale, T Mong>atong>tsson and R Wong>atong>kinsarXiv:1201.5731, ApJ submitted15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.1/143


Plan ong>ofong> Lectures1. Whong>atong> is dust? - Fitting, coarse-graining and averaging2. Approaches to coarse-graining, averaging andbackreaction3. Timescape cosmology4. Observong>atong>ional tests ong>ofong> the timescape cosmology5. Variance ong>ofong> the Hubble flow15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.2/143


Lecture 1Whong>atong> is dust?15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.3/143


Whong>atong> is “dark energy”?Usual explanong>atong>ion: a homogeneous isotropic form ong>ofong>“stuff” which violong>atong>es the strong energy condition.(Locally pressure P = wρc 2 , w < − 1 3 .)Best-fit close to cosmological constant, Λ, w = −1.Cosmic coincidence: Why now? Why Ω Λ0∼ 2Ω M0, sothong>atong> a universe which has been decelerong>atong>ing for much ong>ofong>its history began accelerong>atong>ing only ong>atong> z ∼ 0.7?Onset ong>ofong> accelerong>atong>ion coincides also with the nonlineargrowth ong>ofong> large structuresAre we oversimplifying the geometry?Hypothesis: must understand nonlinear evolution withbackreaction, AND gravitong>atong>ional energy gradients withinthe inhomogeneous geometry15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.4/143


6df: voids & bubble walls (A. Fairall, UCT)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.5/143


From smooth to lumpyUniverse was very smooth ong>atong> time ong>ofong> last scong>atong>tering;fluctuong>atong>ions in the fluid were tiny (δρ/ρ ∼ 10 −5 in photonsand baryons; ∼ 10 −4 ,10 −3 in non–baryonic dark mong>atong>ter).FLRW approximong>atong>ion very good early on.Universe is very lumpy or inhomogeneous today.Recent surveys estimong>atong>e thong>atong> 40–50% ong>ofong> the volume ong>ofong>the universe is contained in voids ong>ofong> diameter 30h −1Mpc. [Hubble constant H 0= 100h km/s/Mpc] (Hoyle &Vogeley, ApJ 566 (2002) 641; 607 (2004) 751)Add some larger voids, and many smaller minivoids,and the universe is void–dominong>atong>ed ong>atong> present epoch.Clusters ong>ofong> galaxies are strung in filaments and bubblesaround these voids.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.6/143


Coarse-graining, averaging, backreactionCoarse-graining: Replace the the microphysics ong>ofong> agiven spacetime region by some collective degrees ong>ofong>freedom sufficient to describe physics on scales largerthan the coarse-graining scale. BOTTOM UP.Averaging: Consider overall macroscopic dynamics andevolution, without direct considerong>atong>ion ong>ofong> the details ong>ofong>the course-graining procedure. TOP DOWN.Often assumes the existence ong>ofong> a particular average,e.g., FLRW background, without showing thong>atong> suchan average exists.Backreaction: Consider the effects ong>ofong> departuresfrom the average, perturbong>atong>ive or nonperturbong>atong>ive, onthe average evolution.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.7/143


Fitting problem (Ellis 1984)On whong>atong> scale are Einstein’s field equong>atong>ions (EFEs)valid?G µν = 8πGc 4 T µνScale on which mong>atong>ter fields are coarse–grained toproduce the energy–momentum tensor on r.h.s. notprescribedgeneral relong>atong>ivity only well tested for isolong>atong>ed systems –e.g., solar system or binary pulsars – for which T µν = 0Usual approach: just pretendOther approaches: cut and paste exact solutions, e.g.,Einstein-Straus vacuole (1946) → Swiss cheesemodels; LTB vacuoles → meong>atong>ball models15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.8/143


Layers ong>ofong> coarse-graining in cosmology1. Atomic, molecular, ionic or nuclear particlescoarse-grained as fluid in early universe, voids, stars etc2. Collapsed objects – stars, black holes coarse-grainedas isolong>atong>ed objects;3. Stellar systems coarse-grained as dust particles withingalaxies;4. Galaxies coarse-grained as dust particles withinclusters;5. Clusters ong>ofong> galaxies as bound systems withinexpanding walls and filaments;6. Voids, walls and filaments combined as regions ong>ofong>different densities in a smoothed out expandingcosmological fluid.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.9/143


Coarse-graining: first steps(1) → (2) (fluid → universe, star etc): Realm ong>ofong> EFEs“mong>atong>ter tells space how to curve” – well-established forearly universe; vacuum geometries; starting point fordefining neutron stars etc(2) → (3) (isolong>atong>ed Schwarzschild, Kerr geometry →particle in fluid):Replace Weyl curvong>atong>ure → Ricci curvong>atong>ureNo formal coarse-graining solution; but reasonableto assume possible in terms ong>ofong> ADM-like mass (see,e.g., Korzyński 2010)Even for 2 particles, gravitong>atong>ional energy (bindingenergy etc) necessarily involvedNeglect inter-particle interactions → dustapproximong>atong>ion15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.10/143


Coarse-graining: further steps(2) → (3) (stars, gas → galaxies):Newtonian gravity usually assumedCooperstock and Tieu (2006,2007): claimednon-Newtonian properties possible for rigidly rotong>atong>ingdust (van Stockum metrics)Neill and DLW (in preparong>atong>ion): new van Stockummetrics for empirically observed density prong>ofong>iles donot solve galaxy rotong>atong>ion curve problem(3) → (4) (galaxies → galaxy clusters):Newtonian gravity, viral theorem usually assumedVirial theorem studied formally only in GRRealistic solutions not known; given Lemaître–Tolman–Bondi (LTB) solutions inapplicable15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.11/143


Coarse-graining: final steps(4) → (5) (bound galaxy clusters → expandingwalls/filaments)New ingredient: expanding space(1) + (5) → (6) (voids + walls/filaments → universe)New ingredient: “building blocks” themselves areexpandingEffective hierarchyg stellarµν → g galaxyµν → g clusterµν → g wallµν.g voidµν⎫⎪⎬⎪⎭ → guniverse µνHow does 〈T µ ν(g x µν)〉 → T µ ν(g y µν) ong>atong> each step?15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.12/143


Dilemma ong>ofong> gravitong>atong>ional energy. . .In GR spacetime carries energy & angular momentumG µν = 8πGc 4 T µνOn account ong>ofong> the strong equivalence principle, T µνcontains localizable energy–momentum onlyKinetic energy and energy associong>atong>ed with spong>atong>ialcurvong>atong>ure are in G µν : variong>atong>ions are “quasilocal”!Newtonian version, T − U = −V , ong>ofong> Friedmann equong>atong>ionȧ 2a 2 + kc2a 2 = 8πGρ3where T = 1 2 mȧ2 x 2 , U = − 1 2 kmc2 x 2 , V = − 4 3 πGρa2 x 2 m;r = a(t)x.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.13/143


Dilemma ong>ofong> gravitong>atong>ional energy. . .Each step ong>ofong> coarse-graining hierarchy involvescoarse-graining gravitong>atong>ional degrees ong>ofong> freedomHow do we define coarse-grained averages?〈G µ ν〉 = 〈g µλ R λν 〉 − 1 2 δµ ν〈g λρ R λρ 〉 = 8πGc 4 〈T µ ν〉How do we relong>atong>e coarse-grained “particle” mass etc tosub-system masses, angular momenta etc?How do we relong>atong>e metric invariants (rulers, clocks) ong>ofong>subsystem to those ong>ofong> coarse-grained system?FLRW model success with Ω C0= Ω M0− Ω B0, Ω Λ0,suggests simplifying physical principles to be foundSteps 2 to 5 in hierarchy may shed light on dark mong>atong>ter;steps 5 to 6 on dark energy (only consider long>atong>ter here)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.14/143


Whong>atong> is a cosmological particle (dust)?In FLRW one takes observers “comoving with the dust”Traditionally galaxies were regarded as dust. However,Neither galaxies nor galaxy clusters arehomogeneously distributed todayDust particles should have (on average) invariantmasses over the timescale ong>ofong> the problemMust coarse-grain over expanding fluid elements largerthan the largest typical structuresASIDE: Taking galaxies as dust leads to flawedargument against backreaction (Peebles 0910.5142)Φ Newton (galaxy) ∼v 2 gal/c 2 ∼ 10 −6ΛCDM self-consistent; but galaxies, clusters do notjustify FLRW background15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.15/143


Sandage-de Vaucouleurs paradoxMong>atong>ter homogeneity only observed ong>atong> > ∼ 100/h MpcscalesIf “the coins on the balloon” are galaxies, their peculiarvelocities should show greong>atong> stong>atong>istical scong>atong>ter on scalemuch smaller than ∼ 100/h MpcHowever, a nearly linear Hubble law flow begins ong>atong>scales above 1.5–2 Mpc from barycentre ong>ofong> local group.Moreover, the local flow is stong>atong>istically “quiet”; despite apossible 65/h Mpc Hubble bubble feong>atong>ure.Peculiar velocities are isotropized in FLRW universeswhich expand forever (regardless ong>ofong> dark energy); butong>atong>tempts to explain the paradox not a good fit to ΛCDMparameters (Axenides & Perivolaropoulos 2002).15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.16/143


Largest typical structuresSurvey Void diameter Density contrastPSCz (29.8 ± 3.5)h −1 Mpc δ ρ = −0.92 ± 0.03UZC (29.2 ± 2.7)h −1 Mpc δ ρ = −0.96 ± 0.012dF NGP (29.8 ± 5.3)h −1 Mpc δ ρ = −0.94 ± 0.022dF SGP (31.2 ± 5.3)h −1 Mpc δ ρ = −0.94 ± 0.02Dominant void stong>atong>istics in the Point Source Cong>atong>alogue Survey (PSCz), the Updong>atong>edZwicky Cong>atong>alogue (UZC), and the 2 degree Field Survey (2dF) North Galactic Pole(NGP) and South Galactic Pole (SGP), (Hoyle and Vogeley 2002,2004). Morerecent results ong>ofong> Pan et al. (2011) using SDSS Dong>atong>a Release 7 similar.Particle size should be a few times greong>atong>er than largesttypical structures (voids with δ ρ ≡ (ρ − ¯ρ)/¯ρ near -1)Coarse grain dust “particles” – fluid elements – ong>atong> Scaleong>ofong> Stong>atong>istical Homogeneity (SSH) ∼ 100/h Mpc15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.17/143


Scale ong>ofong> stong>atong>istical homogeneityCoincides roughly with Baryon Acoustic Oscillong>atong>ion(BAO) scaleUsing Friedmann equong>atong>ion for pressureless dusta 2 0 H2 0 (Ω M0 − 1) = a2 (t)H 2 (t)[Ω M(t) − 1]with δ t ≡ δρ/ρ = Ω M(t) − 1 ≃ A × 10 −4 ong>atong> z = 1090,t = 380 kyr when H ≃ 2/(3t), estimong>atong>e on scales > SSHδ 0≡( δρρ)0≃δ 0= 6% if A = 1, h = 0.65( ) 2 H δ tH 0(1 + z) 2 ≃ 0.025Ah 2Measurement 7% (Hogg et al, 2005), 8% (Sylos Labiniet al, SDSS-DR7 2009)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.18/143


Scale ong>ofong> stong>atong>istical homogeneityASIDE: Sylos-Labini et al define SSH as δρ/ρ → 0, anddisagree with Hogg et al who find SHS ong>atong> > ∼ 70/h Mpc(but fractal dimension D ≃ 2 for galaxy distribution up toong>atong> least 20/h Mpc)Inflong>atong>ion and cosmic variance imply some large scalevariong>atong>ion in average density, bounded on scales > SSHBAO feong>atong>ure observed in linear regime ong>ofong> FLRWperturbong>atong>ion theory implying SSH < ∼ BAO scaleOn scales below BAO scale density perturbong>atong>ionsδ t = A × 10 −4 amplified by acoustic waves, more so thesmaller the scale, eventually becoming nonlinearSpeculong>atong>ion: dominant void scale, diameter 30/h Mpc, isa rarefaction amplificong>atong>ion set by 2nd acoustic peak?15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.19/143


Lecture 2Approaches to coarse-graining,averaging and backreaction15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.20/143


I. Coarse-graining ong>atong> SSHIn timescape model we will coarse-grain “dust” ong>atong> SSHScale ong>atong> which fluid cell properties from cell to cellremain similar on average throughout evolution ong>ofong>universeNotion ong>ofong> “comoving with dust” will require clarificong>atong>ionVariance ong>ofong> expansion etc relong>atong>es more to internaldegrees ong>ofong> freedom ong>ofong> fluid particle than differencesbetween particlesCoarse-graining over internal gravitong>atong>ional degrees ong>ofong>freedom means thong>atong> we no longer deal with a singleglobal geometry: description ong>ofong> geometry is stong>atong>istical15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.21/143


Coarse-graining: other approachesLindquist-Wheeler (LW) (1959) long>atong>tice model: continuumFLRW model is only realised as an approximong>atong>ionLike Swiss cheese it assumes a simplified hierarchysph symmetricgµν → gµνuniverseClifton & Ferreira studied light propagong>atong>ion in thespong>atong>ially flong>atong> LW model, and initially concluded1 + z ≃ (1 + z FLRW) 7/10 .However, after correcting a numerical error, the resultswere no different to the standard Friedman case(Clifton, Ferreira & O’Donnell, arXiv:1110.3191)Symmetry implies Friedmann evolution still; like Swisscheese this limits the metric ong>ofong> the universe15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.22/143


Korzyński’s covariant coarse-grainingIsometrically embed the boundary ong>ofong> a comovingdust-filled domain (with S 2 topology, positive scalarcurvong>atong>ure) into a three-dimensional Euclidean spaceConstruct a “fictitious” 3-dimensional fluid velocity whichinduces the same infinitesimal metric deformong>atong>ion onthe embedded surface as “true” flow on domainboundary original spacetimeUse velocity field to uniquely assign coarse-grainedexpressions for the volume expansion and shear.Using pushforward ong>ofong> ADM shift vector similarly obtain acoarse-grained vorticity.Coarse-grained quantities are quasilocal functionalswhich depend only on the geometry ong>ofong> the domainboundary. Class. Quan. Grav. 27 (2010) 10501515th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.23/143


II. Averaging and backreactionIn general 〈G µ ν(g αβ )〉 ≠ G µ ν(〈g αβ 〉)〈G µ ν〉 need not be Einstein tensor for an exact geometry(1)〈G µ ν〉 = 〈g µλ R λν 〉 − 1 2 δµ ν〈g λρ R λρ 〉 = 8πGc 4 〈T µ ν〉E.g., Zalaletdinov (1992,1993) works with the averageinverse metric 〈g µν 〉 and the average Ricci tensor 〈R µν 〉,and writes(2)〈g µλ 〉〈R λν 〉 − 1 2 δµ ν〈g λρ 〉〈R λρ 〉 + C µ ν = 8πGc 4 〈T µ ν〉,Correlong>atong>ion functions C µ ν defined by difference ong>ofong> thel.h.s. ong>ofong> (1) and (2): these are backreaction terms15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.24/143


Averaging and backreactionAlternong>atong>ively define g µν = ḡ µν + δg µν , where ḡ µν ≡ 〈g µν 〉,with inverse ḡ λµ ≠ 〈g λµ 〉Determine a connection ¯Γ λ µν, curvong>atong>ure tensor ¯R µ νλρand Einstein tensor Ḡµ ν based on the averaged metric,ḡ µν , alone.Differences δΓ λ µν ≡ 〈Γλ µν 〉 − ¯Γ λ µν,δR µ νλρ ≡ 〈Rµ νλρ 〉 − ¯R µ νλρ , δR µν ≡ 〈R µν 〉 − ¯R µν etc, thenrepresent the backreactionAverage EFEs (1) may be writtenḠ µ ν + δG µ ν = 8πGc 4 〈T µ ν〉Processes ong>ofong> averaging and constructing Einsteintensor do not commute15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.25/143


Approaches to averagingThree main types1. Perturbong>atong>ive schemes about a given backgroundgeometry;2. Spacetime averages;3. Spong>atong>ial averages on hypersurfaces based on a 1 + 3foliong>atong>ion.Perturbong>atong>ive schemes deal with weak backreactionApproaches 2 and 3 can be fully nonlinear giving strongbackreactionNo obvious way to average tensors on a manifold, soextra assumptions or structure needed15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.26/143


Approach 1: Weak backreactionMuch argument about backreaction from perturbong>atong>iontheory near a FLRW background (e.g., Kolb et al. 2006)Deal with gradient expansions ong>ofong> potentials anddensitiesDebong>atong>e is largely about mong>atong>hemong>atong>ical consistency, andconclusions vary with assumptions madeDebong>atong>e shows there is a problem – perturbong>atong>ion theorydoes not converge – and if backreaction changes thebackground, then any single FLRW model may simplybe the wrong background ong>atong> present epochMany reviews; e.g., Clarkson, Ellis, Larena and Umeh,Rep. Prog. Phys. 74 (2011) 112901 [arXiv:1109.2484];Kolb, Class. Quan. Grav. 28 (2011) 16400915th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.27/143


Approach 2: Spacetime averagesAny process ong>ofong> taking an average will in general breakgeneral covarianceIf average cosmological geometry on scales no longersong>atong>isfies EFEs, need to revisit the role ong>ofong> generalcovariance plays in defining spacetime structure on thelargest scales from first principlesHow do coordinong>atong>es ong>ofong> a “fine–grained manifold” relong>atong>eto those ong>ofong> an average “coarse–grained manifold”?Zalaletdinov views general covariance as paramount;he introduces additional mong>atong>hemong>atong>ical structure toperform averaging ong>ofong> tensors covariantlyAim: consistently average the Cartan equong>atong>ions fromfirst principles, in analogy to averaging ong>ofong> microscopicMaxwell–Lorentz equong>atong>ions in electromagnetism15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.28/143


Zalaletdinov’s macroscopic gravityDefine bilocal averaging operong>atong>ors, A µ α(x,x ′ ), withsupport ong>atong> two points x ∈ M and x ′ ∈ MConstruct a bitensor extension, T µ ν(x,x ′ ), ong>ofong> tensorT µ ν(x) according toT µ ν(x,x ′ ) = A µ α ′ (x,x ′ )T α′ β ′(x′ )A β′ ν(x ′ ,x).Integrong>atong>es bitensor extension over a 4-dimensionalspacetime region, Σ ⊂ M, with volume V Σ , to obtainregional averageT µ ν (x) = 1V Σ∫Σd 4 x ′ √ −g(x ′ )T µ ν(x,x ′ ),Bitensor transforms as a tensor ong>atong> each point but is ascalar when integrong>atong>ed for regional average.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.29/143


Zalaletdinov’s macroscopic gravityMong>atong>hemong>atong>ical formalism which in the average bears aclose resemblance to general relong>atong>ivity itselfMacroscopic scale is assumed to be larger than themicroscopic scale, there is no scale in the final theoryIssues ong>ofong> coarse-graining ong>ofong> gravitong>atong>ional d.o.f. incosmological relong>atong>ivity may make the problem subtlydifferentCosmological applicong>atong>ions ong>ofong> Zalaletdinov’s formalismneed additional assumptionsAssume a spong>atong>ial averaging limit (Paranjape andSingh 2007)Assume the average is FLRW: correlong>atong>ion tensorsthen take form ong>ofong> a spong>atong>ial curvong>atong>ure (Coley, Pelavas& Zalaletdinov 2005)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.30/143


Approach 3: Spong>atong>ial averagesAverage scalar quantities only; e.g., given a congruenceong>ofong> observers, with tangent vector U µ , and projectionoperong>atong>or h µ ν = δ µ ν + U µ U ν then with B µν ≡ ∇ µ U ν ,accelerong>atong>ion a µ = U ν ∇ ν U µ , we haveB ⊥ µν ≡ h λ µh σ νB λσ = B µν + a µ U νθ µν = h λ µh σ νB ⊥ (λσ)expansionω µν = h λ µh σ νB ⊥ [λσ]vorticityθ = θ µ µ = ∇ µ U µσ µν = θ µν − 1 3 h µνθ shearexpansion scalarBuchert approach, assume ω µν = 0. Flow ishypersurface orthogonal, i.e., spacetime can foliong>atong>ed byspacelike hypersurfaces Σ t , in standard ADM formalism.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.31/143


The 3 + 1 decompositionFor a globally hyperbolic manifold, we can make a 3 + 1-splitwhereds 2 = −ω 0 ⊗ ω 0 + g ij (t,x) ω i ⊗ ω j ,ω 0ω i= N(t,x) dt= dx i + N i (t,x) dt.N(t,x k ) is the lapse function: measures differencebetween coordinong>atong>e time, t, and proper time, τ, oncurves normal to hypersurfaces Σ t , n α = (−N, 0, 0, 0)N i (t,x k ) is the shift vector: measures the differencebetween a spong>atong>ial point, p, and the point reached byfollowing the normal n.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.32/143


The 3 + 1 decompositionn µN i dtdx iΣ t+dtdτ = NdtΣ tx ix i + dx iWhen P = 0, i.e., for dust, in Buchert scheme mayconsistently chooseN i = 0: comoving coordinong>atong>esN = 1: normalised, n α n α = −1 (i.e., n µ ≡ U µ )15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.33/143


Buchert averagingAverage scalar quantities only on domain in spong>atong>ialhypersurface D ∈ Σ t ; e.g.,(∫〈R〉 ≡ d 3 x √3 )gR(t,x) /V(t)Dwhere V(t) = ∫ D d3 x √3 g, 3 g ≡ det( 3 g ij ) = − det( 4 g µν ).Now √3 g θ = √ − 4 g ∇ µ U µ = ∂ µ ( √ − 4 g U µ ) = ∂ t ( √3 g), so〈θ〉 = (∂ t V)/VGenerally for any scalar Ψ, get commutong>atong>ion rule∂ t 〈Ψ〉 − 〈∂ t Ψ〉 = 〈Ψθ〉 − 〈θ〉〈Ψ〉 = 〈Ψδθ〉 = 〈θ δΨ〉 = 〈δΨδθ〉where δΨ ≡ Ψ − 〈Ψ〉, δθ ≡ θ − 〈θ〉.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.34/143


Buchert-Ehlers-Carfora-Piotrkowska-Russ-Song>ofong>fel-Kasai-Börner equong>atong>ionsFor irrotong>atong>ional dust cosmologies, with energy density,ρ(t,x), expansion scalar, θ(t,x), and shear scalar, σ(t,x),where σ 2 = 1 2 σ µνσ µν , defining 3˙ā/ā ≡ 〈θ〉, we find averagecosmic evolution described by exact Buchert equong>atong>ions3 ˙ā 2(3) ā 23¨ā= 8πG〈ρ〉 − 1 2 〈R〉 − 1 2 Q(4)ā = −4πG〈ρ〉 + Q∂ t 〈ρ〉 + 3 ˙ā(5) ā 〈ρ〉 = 0∂ (ā6 t Q ) + ā 4 ∂ (ā2 t 〈R〉 ) (6)= 0Q ≡ 2 3(〈θ 2 〉 − 〈θ〉 2) − 2〈σ 2 〉15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.35/143


Backreaction in Buchert averagingKinemong>atong>ic backreaction term can also be writtenQ = 2 3 〈(δθ)2 〉 − 2〈σ 2 〉i.e., combines variance ong>ofong> expansion, and shear.Eq. (6) is required to ensure (3) is an integral ong>ofong> (4).Buchert equong>atong>ions look deceptively like Friedmannequong>atong>ions, but deal with stong>atong>istical quantitiesThe extent to which the back–reaction, Q, can lead toapparent cosmic accelerong>atong>ion or not has been thesubject ong>ofong> much debong>atong>e (e.g., Ishibashi & Wald 2006):How do stong>atong>istical quantities relong>atong>e to observables?Whong>atong> about the time slicing?How big is Q given reasonable initial conditions?15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.36/143


III. Average spong>atong>ial homogeneityMany approaches simply assume FLRW as average:all perturbong>atong>ive calculong>atong>ions about the FLRW universeany “asymptotically FLRW” LTB models with corespherical inhomogeneitythe Dyer-Roeder (1974) approachSwiss cheese and meong>atong>ball modelsspecific cosmological studies ong>ofong> spong>atong>ial averaging (Russet al 1997; Green & Wald 2011. . . )specific cosmological studies ong>ofong> Zalaletdinov’smacroscopic gravity (Coley et al 2005; Paranjape &Singh 2007,2008; van den Hoogen 2009)specific cosmological studies ong>ofong> general Constant Mean(extrinsic) Curvong>atong>ure (CMC) flows (Reiris 2009,2009)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.37/143


Notions ong>ofong> average spong>atong>ial homogeneityHow do we define average spong>atong>ial homogeneity?Assumption ong>ofong> FLRW average is restrictive, demanding3 separong>atong>e notions:1. Average spong>atong>ial homogeneity is described by class ong>ofong>ideal comoving observers with synchronized clocks.2. Average spong>atong>ial homogeneity is described byaverage surfaces ong>ofong> constant spong>atong>ial curvong>atong>ure.3. The expansion rong>atong>e ong>atong> which the ideal comovingobservers separong>atong>e within the hypersurfaces ong>ofong>average spong>atong>ial homogeneity is uniform.No need to demand all ong>ofong> these notions must holdtogether15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.38/143


Perturbong>atong>ive average homogeneityBardeen (1980) described gauge invariant FLRWperturbong>atong>ions in different foliong>atong>ions which take one orother property as more fundamentalcomoving hypersurfaces (and relong>atong>ed synchronousgauge) embody (1)minimal shear hypersurfaces (and relong>atong>ed Newtoniangauge) are one type ong>ofong> foliong>atong>ion relong>atong>ed to (2)uniform Hubble flow hypersurfaces embody (3)Bičak, Kong>atong>z & Lynden-Bell (2007) consider Machianfoliong>atong>ions (LIF coords uniquely determined by δT µ ν):uniform 3 R hypersurfaces – embody (2)minimal shear hypersurfacesuniform Hubble flow hypersurfaces plus minimal shiftdistortion gauge condition ong>ofong> Smarr and York (1978)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.39/143


Within a stong>atong>istically average cellNeed to consider relong>atong>ive position ong>ofong> observers overscales ong>ofong> tens ong>ofong> Mpc over which δρ/ρ ∼ −1.GR is a local theory: gradients in spong>atong>ial curvong>atong>ure andgravitong>atong>ional energy can lead to calibrong>atong>ion differencesbetween our rulers & clocks and volume average ones15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.40/143


The Copernican principleRetain Copernican Principle - we are ong>atong> an averageposition for observers in a galaxyObservers in bound systems are not ong>atong> a volumeaverage position in freely expanding spaceBy Copernican principle other average observersshould see an isotropic CMBBUT nothing in theory, principle nor observong>atong>iondemands thong>atong> such observers measure the same meanCMB temperong>atong>ure nor the same angular scales in theCMB anisotropiesAverage mass environment (galaxy) can differsignificantly from volume–average environment (void)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.41/143


Back to first principles. . .Standard approach assumes single global FLRW frameplus Newtonian perturbong>atong>ionsIn absence ong>ofong> exact background symmetries,Newtonian approximong>atong>ion requires a weak fieldapproximong>atong>ion about suitable stong>atong>ic Minkowski frameWhong>atong> is the largest scale on which the StrongEquivalence Principle can be applied?Need to address Mach’s principle: “Local inertial framesare determined through the distributions ong>ofong> energy andmomentum in the universe by some weighted averageong>ofong> the apparent motions”How does coarse-graining affect relong>atong>ive calibrong>atong>ion ong>ofong>clocks and rods, from local to global, to account foraverage effects ong>ofong> gravity?15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.42/143


Whong>atong> expands? Can’t tell!tHomogeneous isotropic volume expansion is locallyindistinguishable from equivalent motion in stong>atong>icMinkowski space; on local scalesz ≃ v c ≃ H 0l rc, H 0 = ȧ∣awhether z + 1 = a 0/a or z + 1 = √ (c + v)/(c − v).∣t015th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.43/143


Whong>atong> expands? Can’t tell!For homogeneous isotropic volume expansion wecannot tell whether particles are ong>atong> rest in an expandingspace, or moving equivalently in a stong>atong>ic Minkowskispace.In the actual universe volume expansion decelerong>atong>esbecause ong>ofong> the average regional density ong>ofong> mong>atong>terNeed to separong>atong>e non-propagong>atong>ing d.o.f., in particularregional density, from propagong>atong>ing modes: shape d.o.f.Is there a Minkowski space analogue, like Galileo’sship, or Einstein’s elevong>atong>or, even accounting for theaverage density ong>ofong> mong>atong>ter? Yes. . .15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.44/143


Semi-tethered long>atong>ticeExtend to decelerong>atong>ing motion over long time intervalsby Minkowski space analogue (semi-tethered long>atong>tice -indefinitely long tethers with one end fixed, one free endon spool, apply brakes syncronously ong>atong> each site)Brakes convert kinetic energy ong>ofong> expansion to heong>atong> andso to other formsBrake impulse can be arbitrary pre-determined functionong>ofong> local proper time; but provided it is synchronousdecelerong>atong>ion remains homogeneous and isotropic: nonet force on any long>atong>tice observer.Decelerong>atong>ion preserves inertia, by symmetry15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.45/143


Thought experimentstmore decelerong>atong>ionless decelerong>atong>iont 0PRD 78, 084032(2008)Thought experimentt iequivalent situong>atong>ions:SR: observers in disjoint regional semi-tethered long>atong>ticesvolume decelerong>atong>e ong>atong> different rong>atong>esThose who decelerong>atong>e more age less15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.46/143


Thought experimentsaverage t = consttmore denseless densegradient in PRD 78, 084032(2008)Thought experiment t last−scong>atong>teringequivalent situong>atong>ions:GR: regions ong>ofong> different density have different volumedecelerong>atong>ion (for same initial conditions)Those in denser region age less15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.47/143


Cosmological Equivalence PrincipleAt any event, always and everywhere, it is possible tochoose a suitably defined spacetime neighbourhood,the cosmological inertial region, in which averagemotions (timelike and null) can be described bygeodesics in a geometry which is Minkowski up tosome time-dependent conformal transformong>atong>ion,ds 2 CIF = a2 (η) [ −dη 2 + dr 2 + r 2 dΩ 2] ,Defines Cosmological Inertial Frame (CIF)Accounts for regional average effect ong>ofong> density in termsong>ofong> frames for which the stong>atong>e ong>ofong> rest in an expandingspace is indistinguishable from decelerong>atong>ing expansionong>ofong> particles moving in a stong>atong>ic space15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.48/143


Finite infinityVirialized=0 θ>0CollapsingFinite infinityθ0Define finite infinity, “fi ” as boundary to connectedregion within which average expansion vanishes 〈θ〉 = 0and expansion is positive outside.Shape ong>ofong> fi boundary irrelevant (minimal surfacegenerally): could typically contain a galaxy cluster.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.49/143


Cosmic “rest frame”Pong>atong>ch together CIFs for observers who see an isotropicCMB by taking surfaces ong>ofong> uniform volume expansion1〈l r (τ)dl r (τ)〉 = 1 dτ 3 〈θ〉 1 = 1 3 〈θ〉 2 = · · · = ¯H(τ)Average over regions in which (i) spong>atong>ial curvong>atong>ure iszero or negong>atong>ive; (ii) space is expanding ong>atong> theboundaries, ong>atong> least marginally.Solves the Sandage–de Vaucouleurs paradox implicitly.Voids appear to expand faster; but canonical rong>atong>e τ vfaster, locally measured expansion can still be uniform.Global average H av on large scales with respect to anyone set ong>ofong> clocks may differ from ¯H15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.50/143


Better formalism?CEP should be associong>atong>ed with a stong>atong>istical geometricalgauge principleEquivalent ong>ofong> general covariance for cosmologicalrelong>atong>ivity, determined by initial stong>atong>e ong>ofong> universeExpect equivalent descriptions ong>ofong> internal d.o.f. ong>ofong>coarse-grained cell: (i) Buchert description; (ii) minimalshear description; (iii) uniform Hubble flow (CMC)description. . .Since more than one geometrical description ispossible, pong>atong>ching goes beyond junction conditions forgeometries with prescribed T µνPrincipled “modificong>atong>ionong>ofong> general relong>atong>ivity15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.51/143


Lecture 3The timescape cosmology15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.52/143


Two/three scale modelSplit spong>atong>ial volume V = V i ā 3 as disjoint union ong>ofong>negong>atong>ively curved void fraction with scale factor a v andspong>atong>ially flong>atong> “wall” fraction with scale factor a w .ā 3 = f wi a w 3 + f vi a v 3 ≡ ā 3 (f w + f v )f w ≡ f wi a w 3 /ā 3 , f v ≡ f vi a v 3 /ā 3f vi = 1 − f wi is the fraction ong>ofong> present epoch horizonvolume which was in uncompensong>atong>ed underdenseperturbong>atong>ions ong>atong> last scong>atong>tering.¯H(t) = ˙āā = f wH w + f v H v ;H w ≡ 1 a wda wdt ,H v ≡ 1 a vda vdtHere t is the Buchert time parameter, considered as acollective coordinong>atong>e ong>ofong> dust cell coarse-grained ong>atong> SSH.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.53/143


Phenomenological lapse functionsAccording to Buchert average variance ong>ofong> θ will includeinternal variance ong>ofong> H w relong>atong>ive to H v .Note h r ≡ H w /H v < 1.Buchert time, t, is measured ong>atong> the volume averageposition: locong>atong>ions where the local Ricci curvong>atong>urescalar is the same as horizon volume averageIn timescape model, rong>atong>es ong>ofong> wall and void centreobservers who measure an isotropic CMB are fixed bythe uniform quasilocal Hubble flow condition, i.e.,1ādādt = 1 a wda wdτ w= 1 a vda vdτ v; or ¯H(t) = ¯γw H w = ¯γ vH vwhere ¯γ v= dtdτ v, ¯γ w= dtdτ w= 1 + (1 − h r )f v /h r , arephenomenological lapse functions (NOT ADM lapse).15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.54/143


Other ingredients〈R〉 = k v /a v 3 = k v f vi 2/3 f v 1/3 /ā 3 since k w = 0Assume thong>atong> average shear in SSH cell vanishes; moreprecisely neglect Q within voids and walls separong>atong>ely〈δθ 2 〉 w = 3 4 〈σ2 〉 w〈δθ 2 〉 v = 3 4 〈σ2 〉 vJustificong>atong>ion: for spherical voids expect 〈σ 2 〉 = 〈ω 2 〉 = 0;for walls expect 〈σ 2 〉 and 〈ω 2 〉 largely self-canceling.Only remaining backreaction is variance ong>ofong> relong>atong>ivevolume expansion ong>ofong> walls and voidsQ = 6f v (1 − f v ) (H v − H w ) 2 2f =˙ 2v3f v (1 − f v )Solutions known for: dust (DLW 2007);dust + Λ (Viaggiu, 2012), taking ¯γ w= ¯γ v= 1;dust + radiong>atong>ion (Duley, Nazer + DLW, in preparong>atong>ion)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.55/143


Bare cosmological parametersBuchert equong>atong>ions for volume averaged observer, withf v (t) = f vi a v 3 /ā 3 (void volume fraction) and k v < 0¯Ω M+ ¯Ω R+ ¯Ω k + ¯Ω Q= 1,ā −6 ∂ t(¯ΩQ ¯H2ā6 ) + ā −2 ∂ t(¯Ωk ¯H2ā2 ) = 0.where the bare parameters are¯Ω M= 8πG¯ρ M0ā3 03 ¯H 2 ā 3 , ¯ΩR = 8πG¯ρ R0ā4 03 ¯H 2 ā 4 ,¯Ω k = −k vf vi 2/3 f v1/3ā 2 ¯H 2 , ¯ΩQ =−f ˙ 2v9f v (1 − f v ) ¯H 2 .15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.56/143


Dust modelSpecialize to dust only; in fact ¯Ω Ris negligible by thetime ¯Ω Qis significant, so this is good enough. Solutionbehaves like Einstein-de Sitter ong>atong> early times, with f vtiny and clock differences negligible.Buchert equong>atong>ions, in terms ong>ofong> bare (volume average)quantities are then˙ā 2ā 2 +2f˙v9f v (1 − f v ) − α2 1/3f vā 2= 8πG3 ¯ρ ā 3 00ā 3 ,f¨f v + ˙ 2v (2fv − 1)2f v (1 − f v ) + 3 ˙ā fā ˙ v − 3α2 f 1/3 v (1 − f v )2ā 2 = 0,where α 2 = −k v f vi 2/3 .15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.57/143


General exact solution PRL 99, 251101Irrespective ong>ofong> physical interpretong>atong>ion the two scaleBuchert equong>atong>ion can be solved exactlya w = a w0 t 2/3a w0 ≡ ā 0[94f wi −1 (1 − ǫ i ) ¯Ω M0 ¯H02 ] 1/3; Cǫ ≡ ǫ i¯Ω M0f v01/3( ∣∣∣∣ ∣1∣√ u ∣∣∣ 2 ∣∣∣u(u + Cǫ ) − C ǫ ln + 1 + u ∣1)∣∣∣ 2C ǫ C ǫ¯Ω k0= ᾱ a 0(t + t ǫ )where u ≡ f v1/3ā/ā0= f vi 1/3 a v /ā 0, α = ā 0 ¯H0¯Ω1/2 k0 /f v0 1/6 ,and constraints relong>atong>e many constants.4 independent parameters: e.g., ¯H 0 , f v0 , ǫ i , f vi .15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.58/143


Tracker solution limitParameters ǫ i and f vi should be restricted by powerspectrum ong>atong> last scong>atong>tering, e.g.,( ) ( )δρ δρ= f vi ∼ −10 −6 to − 10 −5ρ ρHi viE.g., f vi ∼ 10 −3 , (δρ/ρ) vi ∼ −10 −3 ; or f vi ∼ −10 −2 ,(δρ/ρ) vi ∼ −10 −4 ; or f vi ∼ 3 × 10 −3 , (δρ/ρ) vi ∼ −3 × 10 −3 .The general solution possesses a tracking limit,equivalent to the exact solution with ǫ i = 0, t ǫ = 0, witha v = a v0 t (i.e., voids expand like Milne solution in t)Tracker reached within 1% by redshift z ∼ 37 forreasonable priorsEffectively, there are only two free parameters15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.59/143


PRL 99 (2007) 251101:Tracker solutionā = ā0 (3 ¯H 0 t) 2/3 [] 1/33f v0 ¯H0 t + (1 − f v0 )(2 + f v0 )2 + f v0f v =3f v0 ¯H0 t3f v0 ¯H0 t + (1 − f v0 )(2 + f v0 ) ,Other parameters (drop subscript w on ¯γ w ):¯γ = 1 + 1 2 f v = 3 2 ¯Ht¯Ω M= 4(1 − f v)(2 + f v ) 2 ; ¯Ωk =τ w = 2 3 t + 2(1 − f v0)(2 + f v0 )27f v0 ¯H09f v(2 + f v ) 2 ; ¯ΩQ = −f v (1 − f v )(2 + f v ) 2()9f v0 ¯H0 tln 1 +2(1 − f v0 )(2 + f v0 )15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.60/143


Physical interpretong>atong>ionAs pointed out by Ishibashi and Wald, need to relong>atong>eaverage quantities to physical observablesMy interpretong>atong>ion assumes coarse–graining ong>ofong> dust ong>atong>SSH; Buchert average domain D is horizon volume.Uniform quasi–local Hubble flow below this scale;volume average time, t, and average curvong>atong>ure, 〈R〉, notlocal observable for all isotropic observersObservers within galaxies assumed to be withinspong>atong>ially flong>atong> finite infinity regions with geometryds 2 fi = −dτ2 w + a w 2 (τ w ) [ dη 2 w + η2 w dΩ2]Determine dressed cosmological parameters, as if localclocks and rulers were extended to whole universe15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.61/143


Past light cone averageInterpret solution ong>ofong> Buchert equong>atong>ions by radial nullcone averageds 2 = −dt 2 + ā 2 (t) d¯η 2 + A(¯η,t) dΩ 2 ,where ∫ ¯η H0d¯η A(¯η,t) = ā 2 (t)V i (¯η H)/(4π).LTB metric but NOT an LTB solution15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.62/143


Physical interpretong>atong>ionConformally mong>atong>ch radial null geodesics ong>ofong> sphericalBuchert geometry to those ong>ofong> finite infinity geometrywith uniform local Hubble flow conditiondt = ā d¯η and dτ w = a w dη w . But dt = ¯γdτ w anda w = f wi −1/3 (1 − f v )ā. Hence on radial null geodesicsdη w =f wi 1/3 d¯η¯γ (1 − f v ) 1/3Define η w by integral ong>ofong> above on radial null-geodesics.Extend spong>atong>ially flong>atong> wall geometry to dressed geometryds 2 = −dτ 2 w+ a 2 (τ w ) [ d¯η 2 + r 2 w(¯η,τ w ) dΩ 2]where r w ≡ ¯γ (1 − f v ) 1/3 f wi −1/3 η w (¯η,τ w ), a = ā/¯γ.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.63/143


Dressed cosmological parametersN.B. The extension is NOT an isometryN.B.ds 2 = −dτ 2 fi w+ a 2 w (τ w ) [ dη 2 w+ ηwdΩ 2 2]→ ds 2 = −dτ 2 w+ a 2 [ d¯η 2 + rw(¯η,τ 2 w ) dΩ 2]Extended metric is an effective “spherical Buchertgeometry” adapted to wall rulers and clocks.Since d¯η = dt/ā = ¯γ dτ w /ā = dτ w /a, this leads to dressedparameters which do not sum to 1, e.g.,Ω M= ¯γ 3¯ΩM .Dressed average Hubble parameterH = 1 adadτ w= 1 ādādτ w− 1¯γd¯γdτ w15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.64/143


Dressed cosmological parametersH is greong>atong>er than wall Hubble rong>atong>e; smaller than voidHubble rong>atong>e measured by wall (or any one set ong>ofong>) clocks¯H(t) = 1 ādādt = 1 a vda vdτ v= 1 a wda wdτ w< H < 1 a vda vdτ wFor tracker solution H = (4f v 2 + f v + 4)/6tDressed average decelerong>atong>ion parameterq = −1H 2 a 2 d 2 adτ 2 wCan have q < 0 even though ¯q = −1¯H 2 ā 2 d 2 ādt 2 > 0; differenceong>ofong> clocks important.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.65/143


Apparent cosmic accelerong>atong>ionVolume average observer sees no apparent cosmicaccelerong>atong>ion¯q = 2 (1 − f v) 2(2 + f v ) 2 .As t → ∞, f v → 1 and ¯q → 0 + .A wall observer registers apparent cosmic accelerong>atong>ionq = − (1 − f v) (8f v 3 + 39f v 2 − 12f v − 8)(4 + fv + 4f v2 ) 2,Effective decelerong>atong>ion parameter starts ong>atong> q ∼ 1 2 , forsmall f v ; changes sign when f v = 0.58670773..., andapproaches q → 0 − ong>atong> long>atong>e times.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.66/143


Cosmic coincidence problem solvedSpong>atong>ial curvong>atong>ure gradients largely responsible forgravitong>atong>ional energy gradient giving clock rong>atong>e variance.Apparent accelerong>atong>ion starts when voids start to open.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.67/143


Redshift, luminosity distanceCosmological redshift (last term tracker solution)z + 1 = a a 0= ā0¯γā¯γ 0= (2 + f v)f v1/33f v01/3 ¯H0 t=2 4/3 t 1/3 (t + b)f v01/3 ¯H0 t(2t + 3b) 4/3 ,where b = 2(1 − f v0 )(2 + f v0 )/[9f v0 ¯H0 ]Dressed luminosity distance relong>atong>ion d L= (1 + z)Dwhere the effective comoving distance to a redshift z isD = a 0r w , withr w = ¯γ (1 − f v ) 1/3 ∫ t0tdt ′¯γ(t ′ )(1 − f v (t ′ )) 1/3 ā(t ′ ) .15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.68/143


Redshift, luminosity distancePerform integral for tracker solutionD A=D1 + z = d L(1 + z) 2 = (t) 2 3∫ t0t2dt ′(2 + f v (t ′ ))(t ′ ) 2/3= t 2/3 (F(t 0) − F(t))where()F(t) = 2t 1/3 + b1/36 ln (t 1/3 + b 1/3 ) 2t 2/3 − b 1/3 t 1/3 + b 2/3+ b1/3√3tan −1 (2t 1/3 − b 1/3√3 b 1/3).t given implicitly in terms ong>ofong> z by previous relong>atong>ion15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.69/143


Sample Hubble diagram with SneIaµ484644424038363432300 0.5 1 1.5 2zType Ia supernovae ong>ofong> Riess07 Gold dong>atong>a set fit with χ 2per degree ong>ofong> freedom = 0.9Stong>atong>istically indistinguishable from ΛCDM.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.70/143


Void fraction, lapse functionThe void fraction, f v , (solid line), and lapse ong>ofong> volume–averageobservers with respect to wall observers, ¯γ, (dotted line) as a functionong>ofong> redshift for the TS model with H 0= 62.0 km/s/Mpc, ¯γ 0= 1.38,f v0 = 0.759.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.71/143


CEP relong>atong>ive decelerong>atong>ion scale−1010 m/s 2 zα1.21.00.120.100.08α/(Hc)0.80.060.60.40.040.02α/(Hc)(i)0 0.05 0.1 0.15 0.2 0.25(ii)0 2 4 z 6 8 10By equivalence principle the instantaneous relong>atong>ive decelerong>atong>ion ong>ofong> backgrounds gives aninstantaneous 4-accelerong>atong>ion ong>ofong> magnitude α = H 0c¯γ ˙¯γ/( p¯γ 2 − 1) beyond which weakfield cosmological general relong>atong>ivity will be changed from Newtonian expectong>atong>ions: (i) asabsolute scale nearby; (ii) divided by Hubble parameter to large z.For z < ∼ 0.25, coincides with empirical MOND scaleα 0 = 1.2−0.2 +0.3 × 10−10 ms −2 h 2 75 = 8.1+2.5 −1.6 × 10−11 ms −2 forH 0= 61.7 km/s/Mpc.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.72/143


Age ong>ofong> universeThe expansion age τ w (z) for observers in galaxies (solid) and t(z)for volume-average observers (dashed) for previous TS model.Comparison ΛCDM model: H 0= 71.0 km/s/Mpc, Ω M0= 0.268,Ω Λ0= 0.732 (dot–dashed). Note: τ w0 = 14.6 Gyr, t 0= 18.6 Gyr.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.73/143


Alleviong>atong>ion ong>ofong> age problemOld structures seen ong>atong> large redshifts are a challengefor ΛCDM.Problem alleviong>atong>ed here; expansion age is increased, byan increasingly larger relong>atong>ive fraction ong>atong> largerredshifts, e.g., for best–fit valuesΛCDM τ = 0.85 Gyr ong>atong> z = 6.42, τ = 0.365 Gyr ong>atong> z = 11TS τ = 1.14 Gyr ong>atong> z = 6.42, τ = 0.563 Gyr ong>atong> z = 11Present age ong>ofong> universe for best-fit is τ 0≃ 14.7 Gyr forwall observer; t 0≃ 18.6 Gyr for volume–averageobserver.Would the under–emptiness ong>ofong> voids in NewtonianN-body simulong>atong>ions may be an issue in open universewith bare parameters ¯Ω M= 0.125, t 0≃ 18.6 Gyr?15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.74/143


Magnitude ong>ofong> backreaction–0.024–0.026–0.028–0.030–0.032–0.034–0.036–0.038–0.040–0.042Ω Q0 1 2 3 4 5 6zMagnitude ong>ofong> ¯Ω Qdetermines departure from FLRWevolution: it is 4.2% ong>atong> mostThere is a closest FLRW universe: open model with¯Ω M0= 0.125, t 0= 18.6 Gyr.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.75/143


Lecture 4Observong>atong>ional tests ong>ofong>the timescape cosmology15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.76/143


Test 1: SneIa luminosity distancesDifference ong>ofong> model apparent magnitude and thong>atong> ong>ofong>empty Milne universe ong>ofong> same H 0= 61.7 km/s/Mpc, forRiess 2007 “gold dong>atong>a”. Note: residual depends on theexpansion rong>atong>e ong>ofong> the Milne universe subtracted (2σlimits on H 0indicong>atong>ed by whiskers)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.77/143


Comparison ΛCDM modelsBest-fit spong>atong>ially flong>atong> ΛCDMH 0= 62.7 km/s/Mpc,Ω M0= 0.34, Ω Λ0= 0.66Riess astro-ph/0611572, p. 63H 0= 65 km/s/Mpc,Ω M0= 0.29, Ω Λ0= 0.7115th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.78/143


Test 1: SneIa luminosity distancesTwo free parameters H 0versus Ω M0(dressed shown here),or alternong>atong>ively “bare values”, constrained by Riess07 Golddong>atong>a fit. (Normalizong>atong>ion ong>ofong> H 0not constrained by SneIa.)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.79/143


Best fit parametersHubble constant H 0+ ∆H 0= 61.7 +1.2−1.1 km/s/Mpcpresent void volume fraction f v0 = 0.76 +0.12−0.09bare density parameter ¯Ω M0= 0.125 +0.060−0.069dressed density parameter Ω M0= 0.33 +0.11−0.16non–baryonic dark mong>atong>ter / baryonic mong>atong>ter mass rong>atong>io(¯Ω M0− ¯Ω B0)/¯Ω B0= 3.1 +2.5−2.4bare Hubble constant ¯H 0 = 48.2 +2.0−2.4 km/s/Mpcmean lapse function ¯γ 0= 1.381 +0.061−0.046decelerong>atong>ion parameter q 0= −0.0428 +0.0120−0.0002wall age universe τ 0= 14.7 +0.7−0.5 Gyr15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.80/143


Smale + DLW, MNRAS 413 (2011) 367SALT/SALTII fits (Constitution,SALT2,Union2) favourΛCDM over TS: ln B TS:ΛCDM = −1.06, −1.55, −3.46MLCS2k2 (fits MLCS17,MLCS31,SDSS-II) favour TSover ΛCDM: ln B TS:ΛCDM = 1.37, 1.55, 0.53Different MLCS fitters give different best-fit parameters;e.g. with cut ong>atong> stong>atong>istical homogeneity scale, forMLCS31 (Hicken et al 2009) Ω M0= 0.12 +0.12−0.11 ;MLCS17 (Hicken et al 2009) Ω M0= 0.19 +0.14−0.18 ;SDSS-II (Kessler et al 2009) Ω M0= 0.42 +0.10−0.10Supernovae systemong>atong>ics (reddening/extinction, intrinsiccolour variong>atong>ions) must be understood to distinguishmodelsForegrounds, and inclusion ong>ofong> SneIa below SSH animportant issue (more in next lecture)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.81/143


Supernovae systemong>atong>ics0.7Gold (167 SneIa)0.7SDSS−II 1st year (272 SneIa)0.60.60.50.5Ω m00.40.3Ω m00.40.30.20.20.10.1056 58 60 62 64 66 68H0056 58 60 62 64 66 68H0MLCS17 (219 SneIa)MLCS31 (219 SneIa)0.60.60.50.5Ω m00.40.3Ω m00.40.30.20.20.10.1056 58 60 62 64 66 68H0056 58 60 62 64 66 68H015th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.82/143


CMB anisotropiesPower in CMB temperong>atong>ure anisotropies versus angular size ong>ofong> fluctuong>atong>ion on sky15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.83/143


CMB temperong>atong>ure calibrong>atong>ionVolume average observers measure mean CMBtemperong>atong>ure¯T = ¯γ −1 Twhere T is CMB temperong>atong>ure measured by wallobservers; T 0= 2.725 K, ¯T = 1.975 K.Number density ong>ofong> photons ong>atong> the volume average is(¯n γ = 2ζ(3) k B ¯Tπ 2 c) 3= n γ¯γ 3 ,yielding ¯n γ0 = 4.105 ¯γ 0 −3 × 10 8 m −3 ong>atong> present.This is needed to calibrong>atong>e light element abundancesfrom primordial nucleosynthesis.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.84/143


Photon to baryon rong>atong>ioNumber density ong>ofong> baryons ong>atong> the volume average iswhere ¯γ ′ 0 ≡ 1¯H0 d¯γdtf B≡ ¯Ω B0/¯Ω M0.¯n B= 3 ¯H 2 0 ¯Ω B08πG m p= 11.2f B ¯Ω M0h 2(¯γ 0− ¯γ ′ 0 )2 m −3 ,∣t0, m p is the proton mass,Hence the average photon to baryon rong>atong>io isη Bγ= ¯n B¯n γ= 2.736 × 10−8 f B ¯ΩM0¯γ 3 0 h2(¯γ 0− ¯γ ′ 0 )2 ,as compared to η FLRW = 2.736 × 10 −8 f BΩ Mh 2 .15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.85/143


Li abundance anomaly(i)(ii)Expected abundances for different values ong>ofong> theparameter η 10≡ 10 10 η Bγ(left), and measurements(right) (Steigman 2006), with 1σ uncertainties.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.86/143


Li abundance anomalyBig-bangnucleosynthesis, lightelement abundancesand WMAP with ΛCDMcosmology.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.87/143


Resolution ong>ofong> Li abundance anomaly?Prior to WMAP in 2003 favoured rong>atong>io wasη Bγ= 4.6–5.6 × 10 −10 ; after WMAP η Bγ= 6.1 +0.3−0.2 × 10−10Conventional dressed parameter Ω M0= 0.33 for wallobserver means ¯Ω M0= 0.125 for the volume–average.Conventional theory predicts the volume–averagebaryon fraction – with old BBN favoured η Bγ:¯Ω B0≃ 0.027–0.033; but this translong>atong>es to a conventionaldressed baryon fraction parameter Ω B0≃ 0.072–0.088The rong>atong>io ong>ofong> baryonic mong>atong>ter to non–baryonic darkmong>atong>ter is increased to 1:3.Need fit to 2nd acoustic peak to tighten Ω B0further15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.88/143


CMB – calibrong>atong>ion ong>ofong> sound horizonPhysics ong>atong> last–scong>atong>tering same as perturbedEinstein–de Sitter model. Whong>atong> is changed is relong>atong>ivecalibrong>atong>ion between now and then.Estimong>atong>ed proper distance to comoving scale ong>ofong> thesound horizon ong>atong> any epoch for volume–averageobserver [¯x = ā/ā 0, so ¯x dec = ¯γ −10 (1 + z dec) −1 ]¯D s = ā(t)ā 0c√3 ¯H0∫ ¯xdec0d¯x√(1 + 0.75 ¯Ω B0¯x/¯Ω γ0)(¯Ω M0¯x + ¯Ω R0),For wall observer D s (τ) = ¯γ −1 D sVolume–average observer measures lower mean CMBtemperong>atong>ure ( ¯T 0= T 0/¯γ 0∼ 1.98 K, c.f. T 0∼ 2.73 K inwalls)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.89/143


Test 2: Angular diameter ong>ofong> sound horizonParameters within the (Ω M0,H 0) plane which fit the angularscale ong>ofong> the sound horizon δ = 0.01 rad deduced for WMAP,to within 2%, 4% and 6%, with η Bγ= 4.6–5.6 × 10 −10 .15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.90/143


Test 3: Baryon acoustic oscillong>atong>ion scaleParameters within the (Ω m ,H 0) plane which fit the effectivecomoving baryon acoustic oscillong>atong>ion scale ong>ofong> 104h −1 Mpc,as seen in 2dF, SDSS etc. Warning: crude estimong>atong>e.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.91/143


Agreement ong>ofong> independent testsBest–fit parameters: H 0= 61.7 −1.1 +1.2 km/s/Mpc, Ω m = 0.33 −0.16+0.11(1σ errors for SneIa only) [Leith, Ng & Wiltshire, ApJ 672(2008) L91]15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.92/143


Dressed “comoving distance” D(z)2H 0D(i)(ii)(iii)3.53H 0D(i)(ii)(iii)1.5112.521.50.501 2 3 4 5 6z0z110.50200 400 600 800 1000zBest-fit TS model (black line) compared to 3 spong>atong>ially flong>atong>ΛCDM models: (i) best–fit to WMAP5 only (Ω Λ = 0.75);(ii) joint WMAP5 + BAO + SneIa fit (Ω Λ = 0.72);(iii) best flong>atong> fit to (Riess07) SneIa only (Ω Λ = 0.66).15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.93/143


Dressed “comoving distance” D(z)TS model closest to best–fit ΛCDM to SneIa only result(Ω M0= 0.34) ong>atong> low redshift.TS model closest to best–fit WMAP5 only result,(Ω M0= 0.249) ong>atong> high redshiftOver the range 1 < z < 6.6 tested by gamma raybursters (GRBs) the TS model fits B. Schaefer’s sampleong>ofong> 69 GRBs very slightly better than ΛCDM, not enoughto be stong>atong>istically significant - PR Smale, MNRAS 418(2011) 2779Maximum difference in apparent magnitude betweenTS model and LCDM is only 0.18–0.34 mag ong>atong> z = 6.Will need much dong>atong>a to get stong>atong>istically significant test.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.94/143


Equivalent “equong>atong>ion ong>ofong> stong>atong>e”?110.50z1 2 3 4 5 60.50z1 2 3 4 5 6–0.5–0.5wL–1–1.5w L=“ ” −123 (1 + z) dD d2 DdzΩ M0(1 + z) 3 H 2 0“dDdzdz 2 + 1” 2− 1wL–1–1.5(i)–2(ii)–2A formal “dark energy equong>atong>ion ong>ofong> stong>atong>e” w L(z) for the best-fit TS model, f v0 = 0.76,calculong>atong>ed directly from r w (z): (i) Ω M0= 0.33; (ii) Ω M0= 0.279.Description by a “dark energy equong>atong>ion ong>ofong> stong>atong>e” makesno sense when there’s no physics behind it; but averagevalue w L ≃ −1 for z < 0.7 makes empirical sense.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.95/143


Tests ong>ofong> “equong>atong>ion ong>ofong> stong>atong>e”10-1w(z)-2-3-4UnionConstitution0 0.2 0.4 0.6 0.8 1zZhao and Zhang arXiv:0908.1568 find mild 95%evidence in favour ong>ofong> w(z) crossing the phantom dividefrom w > −1 to w < −1 in the range 0.25 < w < 0.75”Serra et al. arXiv:0908.3186 find “no evidence” ong>ofong>dynamical dark energy, but their analysis (above) alsoconsistent with TS15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.96/143


Sahni, Shafieloo and Starobinsky Om(z)Sahni, Shafieloo and Starobinsky propose a darkenergy diagnosticOm(z) =H 2 (z)H 2 0− 1(1 + z) 3 − 1 ,For FLRW models OM(z) = Ω M0for ΛCDM ∀z sinceOm(z) = Ω M0+ (1 − Ω M0) (1 + z)3(1+w) − 1(1 + z) 3 − 1.but there are differences for other dark energy models.Note: Om(z) → Ω M0ong>atong> large z for all w.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.97/143


Om(z) testOm0.60.50.40.30.20 1 2 3 4 5 6Om(z) for the tracker solution with best–fit value f v0 = 0.762 (solid line), and 1σ limitszOm(0) = 2 3 H′ | 0 = 2(8f v0 3 −3f v0 2 +4)(2+f v0 )(4f v0 2 +f v0 +4) 2 is larger than forDE modelsFor large z, does not asymptote to Ω M0but toOm(∞) = 2(1−f v0)(2+f v0 ) 3(4f v0 2 +f v0 +4) 2 ) < Ω M0if f v0 > 0.25.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.98/143


Sahni, Shafieloo and Starobinsky Om(z)(i)10.81 sigma CLbest fit(ii)10.80.60.6Om(z)0.40.40.20.200.2 0.4 0.6 0.8 1 1.2 1.4z00.2 0.4 0.6 0.8 1 1.2 1.4 1.6z(i) Om(z) fit by Shafieloo et al. to SN+BAO+CMB with w(z)=− 1 2 [1+tanh((z−z t)∆)];(ii) TS model prediction for Om(z) (NOT same w(z)) – best-fit and 1σ uncertaintiesShafieloo et al., arxiv:0903.5141, fit Om(z) with hint thong>atong>“dark energy is decaying”.Intercept Om(0) agrees well with TS model expectong>atong>ion15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.99/143


Alcock–Paczyński testf AP3(iii)(i)(ii)2.521.5f AP= 1 z˛ δθ ˛δz10 1 2 3 4 5 6z˛ for the tracker solution with f v0 = 0.762 (solid line) is compared to threespong>atong>ially flong>atong> ΛCDM models with the same values ong>ofong> (Ω M0,Ω Λ0) as in earlier figuresFor a comoving standard ruler subtending and angle θ,f AP= 1 δθz ∣δz∣ = HD = 3( 2t 2 + 3bt + 2b 2) (1 + z)D Az t (2t + 3b) 2 z15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.100/143


Baryon Acoustic Oscillong>atong>ion test function1.41.2H 0D V(i)(ii)(iii)10.80.60.40.201 2 3 4 5 6zH 0D V= H 0Df −1/3APfor the tracker solution with f v0 = 0.762 (solid line) is comparedto three spong>atong>ially flong>atong> ΛCDM models with the same values ong>ofong> (Ω M0,Ω Λ0) as in earlierfiguresBAO tests ong>ofong> galaxy clustering typically considerD V=[ zD2] 1/3= Df −1/3 .H(z)AP15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.101/143


Gaztañaga, Cabre and Hui 0807.3551z = 0.15-0.47 z = 0.15-0.30 z = 0.40-0.4715th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.102/143


Gaztañaga, Cabre and Hui 0807.3551redshift Ω M0h 2 Ω B0h 2 Ω C0/Ω B0range0.15-0.30 0.132 0.028 3.70.15-0.47 0.12 0.026 3.60.40-0.47 0.124 0.04 2.1WMAP5 fit to ΛCDM: Ω B0≃ 0.045, Ω C0/Ω B0≃ 6.1GCH bestfit: Ω B0= 0.079 ± 0.025, Ω C0/Ω B0≃ 3.6.TS prediction Ω B0= 0.080 +0.021−0.013 , Ω C0 /Ω B0 = 3.1+1.8 −1.3 withmong>atong>ch to WMAP5 sound horizon within 4%.Blake et al (2012) now claim Alcock–Paczyńskimeasurement in WiggleZ survey, fits ΛCDM well15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.103/143


H(z)/H 010H/H 0(iii)(ii)(i)86420 1 2 3 4 5 6H(z)/H 0for f v0 = 0.762 (solid line) is compared to three spong>atong>ially flong>atong> ΛCDM models: (i)(Ω M0,Ω Λ0) = (0.249,0.751); (ii) (Ω M0,Ω Λ0) = (0.279,0.721) (iii)(Ω M0,Ω Λ0) = (0.34,0.66);.zFunction H(z)/H 0displays quite different characteristicsFor 0 < z < ∼ 1.7, H(z)/H 0is larger for TS model, butvalue ong>ofong> H 0assumed also affects H(z) numerical value15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.104/143


Redshift time drift (Sandage–Loeb test)H(z)/H 0measurements are model dependent.However,1 dzH 0dτ = 1 + z − H H 0For ΛCDM1H 0dzdt = (1 + z) − √Ω M0(1 + z) 3 + Ω Λ0+ Ω k0(1 + z) 2 .For TS model1 dzH 0dτ = 1 + z − 3( 2t 2 + 3bt + 2b 2)H 0t (2t + 3b) 2 ,where t is given implicitly in terms ong>ofong> z.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.105/143


Redshift time drift (Sandage–Loeb test)0–0.51 2 3 4 5 6z–1–1.5–2–2.5–3(i)(ii)(iii)H −1 dz0 dτ for the TS model with f v0 = 0.762 (solid line) is compared to three spong>atong>ially flong>atong>ΛCDM models with the same values ong>ofong> (Ω M0,Ω Λ0) as in previous figures.Measurement is extremely challenging. May be feasibleover a 10–20 year period by precision measurements ong>ofong>the Lyman-α forest over redshift 2 < z < 5 with nextgenerong>atong>ion ong>ofong> Extremely Large Telescopes15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.106/143


Clarkson, Bassett and Lu homogeneity testFor FLRW equong>atong>ions, irrespective ong>ofong> dark energy modelΩ k0=B[H 0D(z)] 2 = constwhere B(z) ≡ [H(z)D ′ (z)] 2 − 1. ThusC(z) ≡ 1 + H 2 (DD ′′ − D ′2 ) + HH ′ DD ′ = 0for any homogeneous isotropic cosmology, irrespectiveong>ofong> DE.Clarkson, Bassett and Lu [PRL 101 (2008) 011301] callthis a “test ong>ofong> the Copernican principle”. However, it ismerely a test ong>ofong> (in)homogeneity.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.107/143


Clarkson, Bassett and Lu homogeneity test0.2B (z)0.150.10.05(a)0.20.10–0.1C (z)2 4 6 8 10 12 14 16 18 20z(a) –0.05 01 2 3 4 5 z 6(b)–0.2(b) –0.3(a) B ≡ [H(z)D ′ (z)] 2 − 1 for TS model with f v0 = 0.762 (solid line)and two ΛCDM models (dashed lines): (i) Ω M0= 0.28, Ω Λ0= 0.71,Ω k0= 0.01; (ii) Ω M0= 0.28, Ω Λ0= 0.73, Ω k0= −0.01; (b) C(z).Will give a powerful test ong>ofong> FLRW assumption in future,with quantitong>atong>ive different prediction for TS model.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.108/143


Lecture 5Variance ong>ofong> the Hubble flow15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.109/143


Apparent Hubble flow variance15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.110/143


Result: arXiv:1201.5371v2CMB dipole usually interpreted as result ong>ofong> a boost w.r.t.cosmic rest frame, composed ong>ofong> our motion w.r.t.barycentre ong>ofong> Local Group plus a motion ong>ofong> the LocalGroup ong>ofong> 635 km s −1 towards ? Greong>atong> Attractor?Shapley Concentrong>atong>ion ? ??But Shapley Supercluster, is ong>atong> > ∼ 138h −1 Mpc > Scale ong>ofong>Stong>atong>istical HomogeneityWe find Hubble flow is significantly more uniform in restframe ong>ofong> LG rong>atong>her than standard “rest frame ong>ofong> CMB”Suggests LG is not moving ong>atong> 635 km s −1 ; but ∃ 0.5%foreground anisotropy in distance-redshift relong>atong>ion fromforeground density gradient on< ∼ 65h −1 Mpc scales15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.111/143


Peculiar velocity formalismStandard framework, FLRW + Newtonian perturbong>atong>ions,assumes peculiar velocity fieldv pec = cz − H 0rgenerong>atong>ed byv(r) = H 0 Ω0.55 M04π∫d 3 r ′ δ m (r ′ ) (r′ − r)|r ′ − r| 3After 3 decades ong>ofong> work, despite contradictory claims,the v(r) does not to converge to LG velocity w.r.t. CMBAgreement on direction, not amplitude or scale (Lavauxet al 2010; Bilicki et al 2011; . . . )Suggestions ong>ofong> bulk flows inconsistent with ΛCDM(Wong>atong>kins, Feldman, Hudson 2009. . . )15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.112/143


Spherical averagesDetermine variong>atong>ion in Hubble flow by determiningbest-fit linear Hubble law in spherical shells15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.113/143


N. Li & D. Schwarz, PRD 78, 083531HST key dong>atong>a: 68 points, single shell (all points within r Mpcas r varied) – correlong>atong>ed result0.20.15(H D -H 0 )/H 00.10.050-0.0540 60 80 100 120 140 160 180r (Mpc) 15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.114/143


Analysis ong>ofong> COMPOSITE sampleUse COMPOSITE sample: Wong>atong>kins, Feldman &Hudson 2009, 2010, with 4,534 galaxy redshifts anddistances, includes most large surveys to 2009Distance methods: Tully Fisher, fundamental plane,surface brightness fluctuong>atong>ion; 103 supernovaedistances.average in independent spherical shellsCompute H s in 12.5h −1 Mpc shells; combine 3 shells> 112.5h −1 MpcUse dong>atong>a beyond 156.25h −1 Mpc as check on H 0normalisong>atong>ion – COMPOSITE sample is normalized to100hkm/s/Mpc15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.115/143


(a) 1: 0−12.5 h −1 Mpc N = 92. (b) 2: 12.5−25 h −1 Mpc N = 505.(c) 3: 25−37.5 h −1 Mpc N = 514. (d) 4: 37.5−50 h −1 Mpc N = 731.(e) 5: 50−62.5 h −1 Mpc N = 819. (f) 6: 62.5−75 h −1 Mpc N = 562.(g) 7: 75−87.5 h −1 Mpc N = 414. (h) 8: 87.5−100 h −1 Mpc N = 304.(i) 9: 100−112.5 h −1 Mpc N = 222. (j) 10: 112.5−156.25 h −1 Mpc N = 280.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.116/143(k) 11: 156.25−417.4 h −1 Mpc N = 91.


Radial variance δH s = (H s − H 0 )/H 0Two choices ong>ofong> shell boundaries (closed and opencircles); for each choice dong>atong>a points uncorrelong>atong>edAnalyse linear Hubble relong>atong>ion in rest frame ong>ofong> CMB;Local Group (LG); Local Sheet (LS). LS result veryclose to LG result.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.117/143


Bayesian comparison ong>ofong> uniformity100Very strong10ln BStrongPositive1Not worth morethan a bare mentionScale: Kass & Raftery (1995)2 20 40 60 80 100 120Inner distance cutong>ofong>f r s (h −1 Mpc)Hubble flow more uniform in LG frame than CMB framewith very strong evidence15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.118/143


But why try the LG frame?From viewpoint ong>ofong> the timescape model and in particularthe “Cosmological Equivalence Principle” in boundsystems the finite infinity region (or mong>atong>ter horizon) isthe standard ong>ofong> restVirialized=0 θ>0CollapsingFinite infinityθ015th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.119/143


Boosts and spurious monopole varianceH s determined by linear regression in each shellH s =(Ns∑i=1(cz i ) 2σ 2 i) (Ns∑i=1cz i r iσ 2 i) −1,Under boost cz i → cz i ′ = cz i + v cosφ i for uniformlydistributed dong>atong>a, linear terms cancel on opposite sidesong>ofong> skyH ′ s − H s∼(Ns∑i=1(v cosφ i ) 2σ 2 i= 〈(v cos φ i) 2 〉〈cz i r i 〉∼) (Ns∑v2i=12H 0〈r 2 i 〉cz i r iσ 2 i) −115th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.120/143


Angular varianceTwo approaches; fit1. McClure and Dyer (2007) method – can look ong>atong> highermultipoles∑ Ni=1H α =W i α cz i ri−1∑ Nj=1 W j αwhere with cos θ i = ⃗r grid · ⃗r i , σ θ= 25 ◦ (typically)W i α =1√2πσθexp( −θ2i2σ 2 θ)2. Simple dipoleczr = H 0 + b cos φ15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.121/143


McClure-Dyer Gaussian window averageActually for COMPOSITE sample better to fitH −1α =∑ Ni=1 W i α r i (cz i ) −1∑ Nj=1 W j α,W i α =σ 2 H −1i1( −θ2√ exp i2πσθ2σ 2 θ), σ H−1i= σ icz iCanonical value ong>ofong> σ θ= 25 ◦ used, but varied15 ◦ < σ θ< 40 ◦ with no significant change. However, 2σ θmust be greong>atong>er than Zone ong>ofong> Avoidance diameter.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.122/143


Hubble variance: CMB frame15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.123/143


Hubble variance: LG frame15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.124/143


Angular uncertainties LG frame15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.125/143


Hubble variance quadrupole/dipole rong>atong>ios100CMB frame insideCMB frame outsideLG frame insideLG frame outsideC 2 /C 1 (%)10110 20 30 40 50 60 70 80 90 100Distance split r 0 (/h Mpc)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.126/143


Value ong>ofong> β in cz r = H 0 + β cos φSlope β (h km/s/Mpc)252015105LG frameLG’ frameCMB frameCMB’ frame00 20 40 60 80 100 120 140〈r s 〉 (h −1 Mpc)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.127/143


Value ong>ofong> β in cz r = H 0 + β cos φSlope β (h km/s/Mpc)1412108642LG frameCMB frame00 20 40 60 80 100 120Inner shell-radius r o (h −1 Mpc)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.128/143


Dipole directionCMB frame: direction (l d ,b d ) remains within 1σ ong>ofong> thebulk flow direction (l,b) = (287 ◦ ± 9 ◦ , 8 ◦ ± 6 ◦ ) found byWong>atong>kins et al (2009) for 20h −1 ≤ r o ≤ 115h −1 Mpc; forlargest values remains consistent with bulk flowdirection (l,b) = (319 ◦ ± 18 ◦ , 7 ◦ ± 14 ◦ ) ong>ofong> Turnbull et al(2012)CMB dipole drops to minimum ong>atong> 40h −1 Mpc but thenincreases and remains 4.0–7.0σ from zero.LG frame: For 20h −1 < ∼ r o< ∼ 45h −1 Mpc while the dipoleis strong, direction is consistently in range(l d ,b d ) = (83 ◦ ± 6 ◦ , −39 ◦ ± 3 ◦ ) but angular position thenwanders once magnitude reduced to residual levels.For r o> ∼ 80h −1 Mpc the typical position ong>ofong> residual dipolediffers from thong>atong> inner dipole by 80 ◦ – 100 ◦ in angle l.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.129/143


15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.130/143


Correlong>atong>ion with residual CMB dipoleDigitize skymaps with HEALPIX, compute√ ∑Np αρ HT=¯σ−2 α (H α − ¯H)(T α −√ ¯T)[ ∑α ¯σ−2 α][∑α ¯σ−2 α (H α − ¯H) 2] [ ∑α (T α −¯T) 2]ρ HT= −0.92, (almost unchanged for 15 ◦ < σ θ< 40 ◦ )Alternong>atong>ively, t-test on raw dong>atong>a: null hypothesis thong>atong>maps uncorrelong>atong>ed is rejected ong>atong> 24.4σ.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.131/143


Correlong>atong>ion with CMB dipole as r o variedCorrelong>atong>ion coefficient0-0.2-0.4-0.6-0.8LG inner sphere, no I.V. wtgLS inner sphere, no I.V. wtgLG outer sphere, no I.V. wtgLS outer sphere, no I.V. wtgLG outer sphere, I.V wtgLS outer sphere, I.V. wtg-110 20 30 40 50 60 70 80 90Distance cut r 0 (h -1 Mpc)15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.132/143


Redshift-distance anisotropyAs long as T ∝ 1/a, where a 0/a = 1 + z for someappropriong>atong>e average, not necessarily FLRW, then smallchange, δz, in the redshift ong>ofong> the surface ong>ofong> photondecoupling – due to foreground structures – will inducea CMB temperong>atong>ure increment T = T 0+ δT , withδTT 0=−δz1 + z decWith z dec = 1089, δT = ±(5.77 ± 0.36) mK represents anincrement δz = ∓(2.31 ± 0.15) to last scong>atong>teringProposal: rong>atong>her than originong>atong>ing in a LG boost the±5.77 mK dipole is due to a small anisotropy in thedistance-redshift relong>atong>ion on scales < ∼ 65h −1 Mpc.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.133/143


Redshift-distance anisotropyFor spong>atong>ially flong>atong> ΛCDMD = cH 0∫ 1+zdec1dx√ΩΛ0 + Ω M0x 3 + Ω R0x 4For standard values Ω R0= 4.15h −2 × 10 −5 , h = 0.72Ω M0= 0.25, find δD = ∓(0.33 ± 0.02)h −1 Mpc;Ω M0= 0.30, find δD = ∓(0.32 ± 0.02)h −1 Mpc;timescape model similar.Assuming thong>atong> the redshift-distance relong>atong>ion anisotropyis due to foreground structures within 65h −1 Mpc then±0.35h −1 Mpc represents a ±0.5% effect15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.134/143


Why a strong CMB dipole?Ray tracing ong>ofong> CMB sky seen by ong>ofong>f-centre observer inLTB void gives |a 10| ≫ |a 20| ≫ |a 30| (Alnes andAmarzguioui 2006). E.g.,a 20a 10=√415(h in − h out )d ong>ofong>f2998 Mpcwhere H in 0 = 100h in km/s/Mpc,H out 0 = 100h out km/s/Mpc are Hubble constantsinside/outside void, d ong>ofong>f = distance ong>ofong> the observer fromcentre in Mpc.Even for relong>atong>ively large values d ong>ofong>f = 50h −1 Mpc andh in − h out = 0.2, we have a 20/a 10< ∼ 1%.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.135/143


Towards a new formalismFor z < z hom define D L= (1 + z)D = (1 + z) 2 D A, whereD(z) = c∫ z0dz sH s (z s ) .where by linear regression in shells, z s < z ≤ z s + σ zH s =(Ns∑i=1(cz i ) 2σ 2 i) (Ns∑i=1cz i r iσ 2 i) −1,Smoothing scale σ z greong>atong>er than largest typical boundstructures, e.g., σ z = 0.0042 for radial width 12.5h −1 Mpc.This gives the monopole, or spherical Hubble bubblevariance15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.136/143


Towards a new formalismFor each shell redshift H(z s ,θ,φ) − H(z s ) will giveangular corrections which should lead to an expansionong>ofong> D(z,θ,φ) in multipolesConvergence ong>ofong> Hubble flow variance to CMB dipole isthen obtained if(i) dipole anisotropy in D converges to fixed value forz > z conv ;(ii) residual anisotropy in D is ong>ofong> order order ong>ofong>±0.33h −1 Mpc, with exact value depending on thecosmological model.Standard peculiar velocity formalism and Hubblevariance formalism can then be directed compared andtested15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.137/143


Type Ia supernova systemong>atong>ics?SneIa are standardizable candles only; two popularmethods SALT/SALT–II, and MLCS2k2 yield differentresults when comparing cosmological resultsDegeneracy between intrinsic colour variong>atong>ions andreddening by dustHubble bubble is seen if R V= 3.1 (Milky way value); notif R V= 1.7 (ong>ofong>ten includes dong>atong>a 0.015 < ∼ z < ∼ z conv ≃ 0.022)Study independent ong>ofong> SneIa in 15 nearby galaxies givesR V= 2.77 ± 0.41 (Finkelman et al 2010, 2011)We find “Hubble bubble” independently ong>ofong> SneIaN.B. SneIa are standardized by minimizing H 0residualsin CMB frame. Union, Constitution compilong>atong>ions containmany SneIa in range 0.015 < ∼ z < ∼ 0.022 where CMBboost compensong>atong>es partly.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.138/143


Large angle CMB anomalies?Anomalies (varying significance) includepower asymmetry ong>ofong> northern/southern hemispheresalignment ong>ofong> the quadrupole and octupole etc;low quadrupole power;parity asymmetry; . . .Critical re-examinong>atong>ion required; e.g.light propagong>atong>ion through Hubble variance dipoleforegrounds may differ subtly from Lorentz boost dipoledipole subtraction is an integral part ong>ofong> the map-making;is galaxy correctly cleaned?Freeman et al (2006): 1–2% error in dipole subtractionmay resolve the power asymmetry anomaly.15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.139/143


Dark flow?Controversial claim ong>ofong> bulk flows by Kashlinsky et al(2009,2010) ong>ofong> 600 – 1, 000 km s −1 over large scales,coinciding with LG boost direction, using the kinemong>atong>icSunyaev-Zel’dovich effectClaim to have subtracted all possible primordial dipoles,quadrupoles, octupoles etc, so measurement is madein “cluster rest frames”Not seen by Osborne et al (2011), Hand et al (2012),Lavaux et al (2012) who use different techniquesHowever, we note Kashlinsky modelling ong>ofong> clustertemperong>atong>ure requires use ong>ofong> cluster redshift, and anisotropic distance-redshift relong>atong>ion is assumeda 1m = a KSZ1m+ a TSZ1m+ a CMB1m+ σ noise√Ncl15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.140/143


Comments on ISW amplitudeIntegrong>atong>ed Sachs–Wolfe (Rees-Sciama) effect needsrecomputong>atong>ion in timescape modelCorrelong>atong>ion ong>ofong> radio-galaxies, voids and superclustersetc with CMB positively detected and well established(Boughn and Crittenden 2004, . . . Granett, Neyrinck andSzapudi, 2008)Amplitude ong>ofong> effect consistently ong>ofong> order 2σ greong>atong>er thanLCDM prediction, or 3σ greong>atong>er according to recentdetailed calculong>atong>ion ong>ofong> Nong>atong>hadur, Hotchkiss and Sarkar(2012)Does departure ong>ofong> local Hubble flow variance fromΛCDM expectong>atong>ions may give insights about feong>atong>ures ong>ofong>inhomogeneities ong>atong> high redshift?15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.141/143


Conclusion/OutlookVariance ong>ofong> the Hubble flow over tens ong>ofong> megaparsecscannot be reduced to a boost; i.e. Eppur si espande!,(Abramowicz et al 2007) space really is expandingLarge CMB angle anomalies, and map-makingprocedures would need to be reconsidered ... are thecold spot etc foreground artifacts, or primordial“Dark flow” probably a systemong>atong>ic “error”Frame ong>ofong> minimum variance Hubble flow variance frameto be determinedImpact ong>ofong> rest frame choice, e.g., on nearbymeasurements in setting distance scale etc, needs tobe re-examinedOpportunity to develop new formalism and approachesto observong>atong>ional cosmology15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.142/143


ConclusionApparent cosmic accelerong>atong>ion can be understood purelywithin general relong>atong>ivity; by (i) treong>atong>ing geometry ong>ofong>universe more realistically; (ii) understandingfundamental aspects ong>ofong> general relong>atong>ivity which have notbeen fully explored – quasi–local gravitong>atong>ional energy,ong>ofong> gradients in spong>atong>ial curvong>atong>ure etc.“Timescape” model gives good fit to major independenttests ong>ofong> ΛCDM with new perspectives on many puzzles– e.g., primordial lithium abundance anomalyMany tests can be done to distinguish from ΛCDM.It is crucial thong>atong> ΛCDM assumptions such as Friedmannequong>atong>ion are not used in dong>atong>a reduction.Many details – averaging scheme etc – may change,but fundamental questions remain15th ong>Brazilianong> ong>Schoolong> on Cosmology and Gravitong>atong>ion, August 2012 – p.143/143

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