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

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

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

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Cosmic structure, averaging **and**dark energyDavid L. Wiltshire (University **on** **and** R W**on** **Cosmology** **and** Gravit**on**, August 2012 – p.1/143

Plan **and** averaging2. Approaches to coarse-graining, averaging **and**backreacti**on**3. Timescape cosmology4. Observ**on**al tests **on** **Cosmology** **and** Gravit**on**, August 2012 – p.2/143

Lecture 1Wh**on** **Cosmology** **and** Gravit**on**, August 2012 – p.3/143

Wh**on**: a homogeneous isotropic form **on**g energy c**on**diti**on**.(Locally pressure P = wρc 2 , w < − 1 3 .)Best-fit close to cosmological c**on**stant, Λ, w = −1.Cosmic coincidence: Why now? Why Ω Λ0∼ 2Ω M0, soth**on**ly **on** coincides also with the n**on**lineargrowth **and** n**on**linear evoluti**on** withbackreacti**on**, AND gravit**on**al energy gradients withinthe inhomogeneous geometry15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.4/143

6df: voids & bubble walls (A. Fairall, UCT)15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.5/143

From smooth to lumpyUniverse was very smooth **on**s in the fluid were tiny (δρ/ρ ∼ 10 −5 in phot**on**s**and** bary**on**s; ∼ 10 −4 ,10 −3 in n**on**–bary**on**ic dark m**on** very good early **on**.Universe is very lumpy or inhomogeneous today.Recent surveys estim**on**tained in voids **on**stant 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–domin**and** bubblesaround these voids.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.6/143

Coarse-graining, averaging, backreacti**on**Coarse-graining: Replace the the microphysics **on** by some collective degrees **on** scales largerthan the coarse-graining scale. BOTTOM UP.Averaging: C**on**sider overall macroscopic dynamics **and**evoluti**on**, without direct c**on**sider**on** **on**: C**on**sider the effects **on**perturb**on**the average evoluti**on**.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.7/143

Fitting problem (Ellis 1984)On wh**on**s (EFEs)valid?G µν = 8πGc 4 T µνScale **on** which m**on** r.h.s. notprescribedgeneral rel**on**ly well tested for isol**and** paste exact soluti**on**s, e.g.,Einstein-Straus vacuole (1946) → Swiss cheesemodels; LTB vacuoles → me**on** **Cosmology** **and** Gravit**on**, August 2012 – p.8/143

Layers **on**ic or nuclear particlescoarse-grained as fluid in early universe, voids, stars etc2. Collapsed objects – stars, black holes coarse-grainedas isol**and**ing walls **and** filaments;6. Voids, walls **and** filaments combined as regi**on**s **and**ingcosmological fluid.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.9/143

Coarse-graining: first steps(1) → (2) (fluid → universe, star etc): Realm **on** stars etc(2) → (3) (isol**on**; but reas**on**ableto assume possible in terms **on**al energy (bindingenergy etc) necessarily involvedNeglect inter-particle interacti**on**s → dustapproxim**on**15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.10/143

Coarse-graining: further steps(2) → (3) (stars, gas → galaxies):Newt**on**ian gravity usually assumedCooperstock **and** Tieu (2006,2007): claimedn**on**-Newt**on**ian properties possible for rigidly rot**and** DLW (in prepar**on**): new van Stockummetrics for empirically observed density pr**on**ot solve galaxy rot**on** curve problem(3) → (4) (galaxies → galaxy clusters):Newt**on**ian gravity, viral theorem usually assumedVirial theorem studied formally **on**ly in GRRealistic soluti**on**s not known; given Lemaître–Tolman–B**on**di (LTB) soluti**on**s inapplicable15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.11/143

Coarse-graining: final steps(4) → (5) (bound galaxy clusters → exp**and**ingwalls/filaments)New ingredient: exp**and**ing space(1) + (5) → (6) (voids + walls/filaments → universe)New ingredient: “building blocks” themselves areexp**and**ingEffective 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) **on** **Cosmology** **and** Gravit**on**, August 2012 – p.12/143

Dilemma **on**al energy. . .In GR spacetime carries energy & angular momentumG µν = 8πGc 4 T µνOn account **on**g equivalence principle, T µνc**on**tains localizable energy–momentum **on**lyKinetic energy **and** energy associ**on**s are “quasilocal”!Newt**on**ian versi**on**, T − U = −V , **on**ȧ 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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.13/143

Dilemma **on**al energy. . .Each step **on**al degrees **on** dark m**on** dark energy (**on**ly c**on**sider l**on** **Cosmology** **and** Gravit**on**, August 2012 – p.14/143

Wh**on**e takes observers “comoving with the dust”Traditi**on**ally galaxies were regarded as dust. However,Neither galaxies nor galaxy clusters arehomogeneously distributed todayDust particles should have (**on** average) invariantmasses over the timescale **and**ing fluid elements largerthan the largest typical structuresASIDE: Taking galaxies as dust leads to flawedargument against backreacti**on** (Peebles 0910.5142)Φ Newt**on** (galaxy) ∼v 2 gal/c 2 ∼ 10 −6ΛCDM self-c**on**sistent; but galaxies, clusters do notjustify FLRW background15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.15/143

S**and**age-de Vaucouleurs paradoxM**on**ly observed **on** the ballo**on**” are galaxies, their peculiarvelocities should show gre**on** scalemuch smaller than ∼ 100/h MpcHowever, a nearly linear Hubble law flow begins **and** forever (regardless **on** **Cosmology** **and** Gravit**on**, August 2012 – p.16/143

Largest typical structuresSurvey Void diameter Density c**on**trastPSCz (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 st**and** the 2 degree Field Survey (2dF) North Galactic Pole(NGP) **and** South Galactic Pole (SGP), (Hoyle **and** Vogeley 2002,2004). Morerecent results **on** **Cosmology** **and** Gravit**on**, August 2012 – p.17/143

Scale **on** Acoustic Oscill**on**(BAO) scaleUsing Friedmann equ**on** 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 **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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.18/143

Scale **and**disagree with Hogg et al who find SHS **on** D ≃ 2 for galaxy distributi**on** up to**on** **and** cosmic variance imply some large scalevari**on** in average density, bounded **on** scales > SSHBAO fe**on** theory implying SSH < ∼ BAO scaleOn scales below BAO scale density perturb**on**sδ t = A × 10 −4 amplified by acoustic waves, more so thesmaller the scale, eventually becoming n**on**linearSpecul**on**: dominant void scale, diameter 30/h Mpc, isa rarefacti**on** amplific**on** set by 2nd acoustic peak?15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.19/143

Lecture 2Approaches to coarse-graining,averaging **and** backreacti**on**15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.20/143

I. Coarse-graining **on** average throughout evoluti**on** **on** **on**Variance **on** etc rel**on**al degrees **on**ger deal with a singleglobal geometry: descripti**on** **on** **Cosmology** **and** Gravit**on**, August 2012 – p.21/143

Coarse-graining: other approachesLindquist-Wheeler (LW) (1959) l**on**tinuumFLRW model is **on**ly realised as an approxim**on**Like Swiss cheese it assumes a simplified hierarchysph symmetricgµν → gµνuniverseClift**on** & Ferreira studied light propag**on** in thesp**and** initially c**on**cluded1 + z ≃ (1 + z FLRW) 7/10 .However, after correcting a numerical error, the resultswere no different to the st**and**ard Friedman case(Clift**on**, Ferreira & O’D**on**nell, arXiv:1110.3191)Symmetry implies Friedmann evoluti**on** still; like Swisscheese this limits the metric **on** **Cosmology** **and** Gravit**on**, August 2012 – p.22/143

Korzyński’s covariant coarse-grainingIsometrically embed the boundary **on**al Euclidean spaceC**on**struct a “fictitious” 3-dimensi**on**al fluid velocity whichinduces the same infinitesimal metric deform**on** **on**the embedded surface as “true” flow **on** domainboundary original spacetimeUse velocity field to uniquely assign coarse-grainedexpressi**on**s for the volume expansi**on** **and** shear.Using pushforward **on**alswhich depend **on**ly **on** the geometry **on** **Cosmology** **and** Gravit**on**, August 2012 – p.23/143

II. Averaging **and** backreacti**on**In 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〉,Correl**on** functi**on**s C µ ν defined by difference **and** (2): these are backreacti**on** terms15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.24/143

Averaging **and** backreacti**on**Altern**on**necti**on** ¯Γ λ µν, curv**and** Einstein tensor Ḡµ ν based **on** the averaged metric,ḡ µν , al**on**e.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 backreacti**on**Average EFEs (1) may be writtenḠ µ ν + δG µ ν = 8πGc 4 〈T µ ν〉Processes **and** c**on**structing Einsteintensor do not commute15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.25/143

Approaches to averagingThree main types1. Perturb**on** hypersurfaces based **on** a 1 + 3foli**on**.Perturb**on**Approaches 2 **and** 3 can be fully n**on**linear giving str**on**gbackreacti**on**No obvious way to average tensors **on** a manifold, soextra assumpti**on**s or structure needed15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.26/143

Approach 1: Weak backreacti**on**Much argument about backreacti**on** from perturb**on**theory near a FLRW background (e.g., Kolb et al. 2006)Deal with gradient expansi**on**s **and**densitiesDeb**on**sistency, **and**c**on**clusi**on**s vary with assumpti**on**s madeDeb**on** theorydoes not c**on**verge – **and** if backreacti**on** changes thebackground, then any single FLRW model may simplybe the wr**on**g background **on**, Ellis, Larena **and** Umeh,Rep. Prog. Phys. 74 (2011) 112901 [arXiv:1109.2484];Kolb, Class. Quan. Grav. 28 (2011) 16400915th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.27/143

Approach 2: Spacetime averagesAny process **on** scales no l**on**gers**on** thelargest scales from first principlesHow do coordin**on**al m**on**sistently average the Cartan equ**on**s fromfirst principles, in analogy to averaging **on**s in electromagnetism15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.28/143

Zalaletdinov’s macroscopic gravityDefine bilocal averaging oper**and** x ′ ∈ MC**on**struct a bitensor extensi**on**, T µ ν(x,x ′ ), **on** over a 4-dimensi**on**alspacetime regi**on**, Σ ⊂ M, with volume V Σ , to obtainregi**on**al averageT µ ν (x) = 1V Σ∫Σd 4 x ′ √ −g(x ′ )T µ ν(x,x ′ ),Bitensor transforms as a tensor **on**al average.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.29/143

Zalaletdinov’s macroscopic gravityM**on**al d.o.f. incosmological rel**on**s **on**al assumpti**on**sAssume a sp**and**Singh 2007)Assume the average is FLRW: correl**on** tensorsthen take form **on** **Cosmology** **and** Gravit**on**, August 2012 – p.30/143

Approach 3: Sp**on**ly; e.g., given a c**on**gruence**and** projecti**on**oper**on** 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)expansi**on**ω µν = h λ µh σ νB ⊥ [λσ]vorticityθ = θ µ µ = ∇ µ U µσ µν = θ µν − 1 3 h µνθ shearexpansi**on** scalarBuchert approach, assume ω µν = 0. Flow ishypersurface orthog**on**al, i.e., spacetime can foli**and**ard ADM formalism.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.31/143

The 3 + 1 decompositi**on**For 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 functi**on**: measures differencebetween coordin**and** proper time, τ, **on**curves normal to hypersurfaces Σ t , n α = (−N, 0, 0, 0)N i (t,x k ) is the shift vector: measures the differencebetween a sp**and** the point reached byfollowing the normal n.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.32/143

The 3 + 1 decompositi**on**n µN i dtdx iΣ t+dtdτ = NdtΣ tx ix i + dx iWhen P = 0, i.e., for dust, in Buchert scheme mayc**on**sistently chooseN i = 0: comoving coordin**on** **Cosmology** **and** Gravit**on**, August 2012 – p.33/143

Buchert averagingAverage scalar quantities **on**ly **on** domain in sp**on** rule∂ t 〈Ψ〉 − 〈∂ t Ψ〉 = 〈Ψθ〉 − 〈θ〉〈Ψ〉 = 〈Ψδθ〉 = 〈θ δΨ〉 = 〈δΨδθ〉where δΨ ≡ Ψ − 〈Ψ〉, δθ ≡ θ − 〈θ〉.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.34/143

Buchert-Ehlers-Carfora-Piotrkowska-Russ-S**on**sFor irrot**on**al dust cosmologies, with energy density,ρ(t,x), expansi**on** 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 evoluti**on** described by exact Buchert equ**on**s3 ˙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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.35/143

Backreacti**on** in Buchert averagingKinem**on** term can also be writtenQ = 2 3 〈(δθ)2 〉 − 2〈σ 2 〉i.e., combines variance **on**, **and** shear.Eq. (6) is required to ensure (3) is an integral **on**s look deceptively like Friedmannequ**on**s, but deal with st**on**, Q, can lead toapparent cosmic acceler**on** or not has been thesubject **on**able initial c**on**diti**on**s?15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.36/143

III. Average sp**on**s about the FLRW universeany “asymptotically FLRW” LTB models with corespherical inhomogeneitythe Dyer-Roeder (1974) approachSwiss cheese **and** me**on**stant Mean(extrinsic) Curv**on** **Cosmology** **and** Gravit**on**, August 2012 – p.37/143

Noti**on**s **on** **and**ing3 separ**on**s:1. Average sp**on**ized clocks.2. Average sp**on**stant sp**on** r**and** all **on**s must holdtogether15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.38/143

Perturb**on**s in different foli**on**s which take **on**e orother property as more fundamentalcomoving hypersurfaces (**and** rel**on**ousgauge) embody (1)minimal shear hypersurfaces (**and** rel**on**iangauge) are **on**e type **on** rel**on**sider Machianfoli**on**s (LIF coords uniquely determined by δT µ ν):uniform 3 R hypersurfaces – embody (2)minimal shear hypersurfacesuniform Hubble flow hypersurfaces plus minimal shiftdistorti**on** gauge c**on**diti**on** **and** York (1978)15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.39/143

Within a st**on**sider rel**on** **and**gravit**on**al energy can lead to calibr**on** differencesbetween our rulers & clocks **and** volume average **on**es15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.40/143

The Copernican principleRetain Copernican Principle - we are **on** for observers in a galaxyObservers in bound systems are not **on** in freely exp**and**ing spaceBy Copernican principle other average observersshould see an isotropic CMBBUT nothing in theory, principle nor observ**on**dem**and**s th**on**ment (galaxy) can differsignificantly from volume–average envir**on**ment (void)15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.41/143

Back to first principles. . .St**and**ard approach assumes single global FLRW frameplus Newt**on**ian perturb**on**sIn absence **on**ian approxim**on** requires a weak fieldapproxim**on** about suitable st**on** which the Str**on**gEquivalence Principle can be applied?Need to address Mach’s principle: “Local inertial framesare determined through the distributi**on**s **and**momentum in the universe by some weighted average**on**s”How does coarse-graining affect rel**on** **and** rods, from local to global, to account foraverage effects **on** **Cosmology** **and** Gravit**on**, August 2012 – p.42/143

Wh**and**s? Can’t tell!tHomogeneous isotropic volume expansi**on** is locallyindistinguishable from equivalent moti**on** in st**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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.43/143

Wh**and**s? Can’t tell!For homogeneous isotropic volume expansi**on** wecannot tell whether particles are **and**ingspace, or moving equivalently in a st**on** deceler**on**al density **on**-propag**on**al density, from propag**on** **Cosmology** **and** Gravit**on**, August 2012 – p.44/143

Semi-tethered l**on** over l**on**g time intervalsby Minkowski space analogue (semi-tethered l**on**g tethers with **on**e end fixed, **on**e free end**on** spool, apply brakes syncr**on**ously **on**vert kinetic energy **on** to he**and**so to other formsBrake impulse can be arbitrary pre-determined functi**on****on**ousdeceler**on** remains homogeneous **and** isotropic: n**on**et force **on** any l**on** preserves inertia, by symmetry15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.45/143

Thought experimentstmore deceler**on**less deceler**on**t 0PRD 78, 084032(2008)Thought experimentt iequivalent situ**on**s:SR: observers in disjoint regi**on**al semi-tethered l**on** **Cosmology** **and** Gravit**on**, August 2012 – p.46/143

Thought experimentsaverage t = c**on**sttmore denseless densegradient in PRD 78, 084032(2008)Thought experiment t last−sc**on**s:GR: regi**on**s **on** (for same initial c**on**diti**on**s)Those in denser regi**on** age less15th **on** **Cosmology** **and** Gravit**on**, 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 regi**on**, in which averagemoti**on**s (timelike **and** null) can be described bygeodesics in a geometry which is Minkowski up tosome time-dependent c**on**formal transform**on**,ds 2 CIF = a2 (η) [ −dη 2 + dr 2 + r 2 dΩ 2] ,Defines Cosmological Inertial Frame (CIF)Accounts for regi**on**al average effect **and**ingspace is indistinguishable from deceler**on****on** **Cosmology** **and** Gravit**on**, August 2012 – p.48/143

Finite infinityVirialized=0 θ>0CollapsingFinite infinityθ0Define finite infinity, “fi ” as boundary to c**on**nectedregi**on** within which average expansi**on** vanishes 〈θ〉 = 0**and** expansi**on** is positive outside.Shape **on**tain a galaxy cluster.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.49/143

Cosmic “rest frame”P**on**1〈l r (τ)dl r (τ)〉 = 1 dτ 3 〈θ〉 1 = 1 3 〈θ〉 2 = · · · = ¯H(τ)Average over regi**on**s in which (i) sp**and**ing **and**age–de Vaucouleurs paradox implicitly.Voids appear to exp**and** faster; but can**on**ical r**on** can still be uniform.Global average H av **on** large scales with respect to any**on**e set **on** **Cosmology** **and** Gravit**on**, August 2012 – p.50/143

Better formalism?CEP should be associ**on**s **on**; (ii) minimalshear descripti**on**; (iii) uniform Hubble flow (CMC)descripti**on**. . .Since more than **on**e geometrical descripti**on** ispossible, p**on**d juncti**on** c**on**diti**on**s forgeometries with prescribed T µνPrincipled “modific**on**” **on** **Cosmology** **and** Gravit**on**, August 2012 – p.51/143

Lecture 3The timescape cosmology15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.52/143

Two/three scale modelSplit sp**on** **on** with scale factor a v **and**sp**on** 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 fracti**on** **on**volume which was in uncompens**on**s **on**sidered as acollective coordin**on** **Cosmology** **and** Gravit**on**, August 2012 – p.53/143

Phenomenological lapse functi**on**sAccording to Buchert average variance **on**: loc**on**s where the local Ricci curv**on** volume averageIn timescape model, r**and** void centreobservers who measure an isotropic CMB are fixed bythe uniform quasilocal Hubble flow c**on**diti**on**, 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 functi**on**s (NOT ADM lapse).15th **on** **Cosmology** **and** Gravit**on**, 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 th**and** walls separ**on**: for spherical voids expect 〈σ 2 〉 = 〈ω 2 〉 = 0;for walls expect 〈σ 2 〉 **and** 〈ω 2 〉 largely self-canceling.Only remaining backreacti**on** is variance **on** **and** voidsQ = 6f v (1 − f v ) (H v − H w ) 2 2f =˙ 2v3f v (1 − f v )Soluti**on**s known for: dust (DLW 2007);dust + Λ (Viaggiu, 2012), taking ¯γ w= ¯γ v= 1;dust + radi**on** (Duley, Nazer + DLW, in prepar**on**)15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.55/143

Bare cosmological parametersBuchert equ**on**s for volume averaged observer, withf v (t) = f vi a v 3 /ā 3 (void volume fracti**on**) **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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.56/143

Dust modelSpecialize to dust **on**ly; in fact ¯Ω Ris negligible by thetime ¯Ω Qis significant, so this is good enough. Soluti**on**behaves like Einstein-de Sitter **and** clock differences negligible.Buchert equ**on**s, in terms **on** **Cosmology** **and** Gravit**on**, August 2012 – p.57/143

General exact soluti**on** PRL 99, 251101Irrespective **on** the two scaleBuchert equ**on** 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** c**on**straints rel**on**stants.4 independent parameters: e.g., ¯H 0 , f v0 , ǫ i , f vi .15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.58/143

Tracker soluti**on** limitParameters ǫ i **and** f vi should be restricted by powerspectrum **on** possesses a tracking limit,equivalent to the exact soluti**on** with ǫ i = 0, t ǫ = 0, witha v = a v0 t (i.e., voids exp**and** like Milne soluti**on** in t)Tracker reached within 1% by redshift z ∼ 37 forreas**on**able priorsEffectively, there are **on**ly two free parameters15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.59/143

PRL 99 (2007) 251101:Tracker soluti**on**ā = 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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.60/143

Physical interpret**on**As pointed out by Ishibashi **and** Wald, need to rel**on** assumes coarse–graining **on** volume.Uniform quasi–local Hubble flow below this scale;volume average time, t, **and** average curv**on**s 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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.61/143

Past light c**on**e averageInterpret soluti**on** **on**s by radial nullc**on**e 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 soluti**on**15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.62/143

Physical interpret**on**C**on**formally m**on**diti**on**dt = ā d¯η **and** dτ w = a w dη w . But dt = ¯γdτ w **and**a 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 **on** radial null-geodesics.Extend sp**on** **Cosmology** **and** Gravit**on**, August 2012 – p.63/143

Dressed cosmological parametersN.B. The extensi**on** 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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.64/143

Dressed cosmological parametersH is gre**on**e set **on** H = (4f v 2 + f v + 4)/6tDressed average deceler**on** 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; difference**on** **Cosmology** **and** Gravit**on**, August 2012 – p.65/143

Apparent cosmic acceler**on**Volume average observer sees no apparent cosmicacceler**on**¯q = 2 (1 − f v) 2(2 + f v ) 2 .As t → ∞, f v → 1 **and** ¯q → 0 + .A wall observer registers apparent cosmic acceler**on**q = − (1 − f v) (8f v 3 + 39f v 2 − 12f v − 8)(4 + fv + 4f v2 ) 2,Effective deceler**on** parameter starts **and**approaches q → 0 − **on** **Cosmology** **and** Gravit**on**, August 2012 – p.66/143

Cosmic coincidence problem solvedSp**on**sible forgravit**on**al energy gradient giving clock r**on** starts when voids start to open.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.67/143

Redshift, luminosity distanceCosmological redshift (last term tracker soluti**on**)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 rel**on** 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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.68/143

Redshift, luminosity distancePerform integral for tracker soluti**on**D 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 **on**15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.69/143

Sample Hubble diagram with SneIaµ484644424038363432300 0.5 1 1.5 2zType Ia supernovae **on** **Cosmology** **and** Gravit**on**, August 2012 – p.70/143

Void fracti**on**, lapse functi**on**The void fracti**on**, f v , (solid line), **and** lapse **on****on** **Cosmology** **and** Gravit**on**, August 2012 – p.71/143

CEP rel**on** 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 rel**on** **on** **on**d which weakfield cosmological general rel**on**ian expect**on**s: (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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.72/143

Age **on** age τ w (z) for observers in galaxies (solid) **and** t(z)for volume-average observers (dashed) for previous TS model.Comparis**on** 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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.73/143

Allevi**on** **on** age is increased, byan increasingly larger rel**on** **on**ianN-body simul**on**s may be an issue in open universewith bare parameters ¯Ω M= 0.125, t 0≃ 18.6 Gyr?15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.74/143

Magnitude **on**–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 **on**: it is 4.2% **on** **Cosmology** **and** Gravit**on**, August 2012 – p.75/143

Lecture 4Observ**on**al tests **on** **Cosmology** **and** Gravit**on**, August 2012 – p.76/143

Test 1: SneIa luminosity distancesDifference **and** th**on** theexpansi**on** r**on** H 0indic**on** **Cosmology** **and** Gravit**on**, August 2012 – p.77/143

Comparis**on** ΛCDM modelsBest-fit sp**on** **Cosmology** **and** Gravit**on**, August 2012 – p.78/143

Test 1: SneIa luminosity distancesTwo free parameters H 0versus Ω M0(dressed shown here),or altern**on**strained by Riess07 Goldd**on** **on**strained by SneIa.)15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.79/143

Best fit parametersHubble c**on**stant H 0+ ∆H 0= 61.7 +1.2−1.1 km/s/Mpcpresent void volume fracti**on** f v0 = 0.76 +0.12−0.09bare density parameter ¯Ω M0= 0.125 +0.060−0.069dressed density parameter Ω M0= 0.33 +0.11−0.16n**on**–bary**on**ic dark m**on**ic m**on**stant ¯H 0 = 48.2 +2.0−2.4 km/s/Mpcmean lapse functi**on** ¯γ 0= 1.381 +0.061−0.046deceler**on** parameter q 0= −0.0428 +0.0120−0.0002wall age universe τ 0= 14.7 +0.7−0.5 Gyr15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.80/143

Smale + DLW, MNRAS 413 (2011) 367SALT/SALTII fits (C**on**stituti**on**,SALT2,Uni**on**2) 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 **on**, intrinsiccolour vari**on**s) must be understood to distinguishmodelsForegrounds, **and** inclusi**on** **on** **Cosmology** **and** Gravit**on**, August 2012 – p.81/143

Supernovae system**on** **Cosmology** **and** Gravit**on**, August 2012 – p.82/143

CMB anisotropiesPower in CMB temper**on** **on** sky15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.83/143

CMB temper**on**Volume average observers measure mean CMBtemper**on**s **on** **Cosmology** **and** Gravit**on**, August 2012 – p.84/143

Phot**on** to bary**on** r**on**s **on** mass,Hence the average phot**on** to bary**on** r**on** **Cosmology** **and** Gravit**on**, August 2012 – p.85/143

Li abundance anomaly(i)(ii)Expected abundances for different values **and** measurements(right) (Steigman 2006), with 1σ uncertainties.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.86/143

Li abundance anomalyBig-bangnucleosynthesis, lightelement abundances**and** WMAP with ΛCDMcosmology.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.87/143

Resoluti**on** **on**venti**on**al dressed parameter Ω M0= 0.33 for wallobserver means ¯Ω M0= 0.125 for the volume–average.C**on**venti**on**al theory predicts the volume–averagebary**on** fracti**on** – with old BBN favoured η Bγ:¯Ω B0≃ 0.027–0.033; but this transl**on**venti**on**aldressed bary**on** fracti**on** parameter Ω B0≃ 0.072–0.088The r**on**ic m**on**–bary**on**ic darkm**on** **Cosmology** **and** Gravit**on**, August 2012 – p.88/143

CMB – calibr**on** **on**Physics **on** between now **and** then.Estim**on** **on** **Cosmology** **and** Gravit**on**, August 2012 – p.89/143

Test 2: Angular diameter **on**Parameters within the (Ω M0,H 0) plane which fit the angularscale **on** δ = 0.01 rad deduced for WMAP,to within 2%, 4% **and** 6%, with η Bγ= 4.6–5.6 × 10 −10 .15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.90/143

Test 3: Bary**on** acoustic oscill**on** scaleParameters within the (Ω m ,H 0) plane which fit the effectivecomoving bary**on** acoustic oscill**on** scale **on** **Cosmology** **and** Gravit**on**, August 2012 – p.91/143

Agreement **on**ly) [Leith, Ng & Wiltshire, ApJ 672(2008) L91]15th **on** **Cosmology** **and** Gravit**on**, 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 sp**on**ly (Ω Λ = 0.75);(ii) joint WMAP5 + BAO + SneIa fit (Ω Λ = 0.72);(iii) best fl**on**ly (Ω Λ = 0.66).15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.93/143

Dressed “comoving distance” D(z)TS model closest to best–fit ΛCDM to SneIa **on**ly result(Ω M0= 0.34) **on**ly result,(Ω M0= 0.249) **and** LCDM is **on**ly 0.18–0.34 mag **on** **Cosmology** **and** Gravit**on**, August 2012 – p.94/143

Equivalent “equ**on** **on** **on** by a “dark energy equ**on** **on** **Cosmology** **and** Gravit**on**, August 2012 – p.95/143

Tests **on** **on**C**on**stituti**on**0 0.2 0.4 0.6 0.8 1zZhao **and** Zhang arXiv:0908.1568 find mild 95%evidence in favour **on**sistent with TS15th **on** **Cosmology** **and** Gravit**on**, 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 M0**on** **Cosmology** **and** Gravit**on**, August 2012 – p.97/143

Om(z) testOm0.60.50.40.30.20 1 2 3 4 5 6Om(z) for the tracker soluti**on** 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 **on** **Cosmology** **and** Gravit**on**, 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 predicti**on** for Om(z) (NOT same w(z)) – best-fit **and** 1σ uncertaintiesShafieloo et al., arxiv:0903.5141, fit Om(z) with hint th**on**15th **on** **Cosmology** **and** Gravit**on**, 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 soluti**on** with f v0 = 0.762 (solid line) is compared to threesp**and**ard 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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.100/143

Bary**on** Acoustic Oscill**on** test functi**on**1.41.2H 0D V(i)(ii)(iii)10.80.60.40.201 2 3 4 5 6zH 0D V= H 0Df −1/3APfor the tracker soluti**on** with f v0 = 0.762 (solid line) is comparedto three sp**on**siderD V=[ zD2] 1/3= Df −1/3 .H(z)AP15th **on** **Cosmology** **and** Gravit**on**, 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 **on** **Cosmology** **and** Gravit**on**, 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 predicti**on** Ω B0= 0.080 +0.021−0.013 , Ω C0 /Ω B0 = 3.1+1.8 −1.3 withm**on** within 4%.Blake et al (2012) now claim Alcock–Paczyńskimeasurement in WiggleZ survey, fits ΛCDM well15th **on** **Cosmology** **and** Gravit**on**, 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 sp**on** H(z)/H 0displays quite different characteristicsFor 0 < z < ∼ 1.7, H(z)/H 0is larger for TS model, butvalue **on** **Cosmology** **and** Gravit**on**, August 2012 – p.104/143

Redshift time drift (S**and**age–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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.105/143

Redshift time drift (S**and**age–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 sp**on** measurements **on** **on** **Cosmology** **and** Gravit**on**, August 2012 – p.106/143

Clarks**on**, Bassett **and** Lu homogeneity testFor FLRW equ**on**s, irrespective **on**stwhere B(z) ≡ [H(z)D ′ (z)] 2 − 1. ThusC(z) ≡ 1 + H 2 (DD ′′ − D ′2 ) + HH ′ DD ′ = 0for any homogeneous isotropic cosmology, irrespective**on**, Bassett **and** Lu [PRL 101 (2008) 011301] callthis a “test **on** **Cosmology** **and** Gravit**on**, August 2012 – p.107/143

Clarks**on**, 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 **on** in future,with quantit**on** for TS model.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.108/143

Lecture 5Variance **on** **Cosmology** **and** Gravit**on**, August 2012 – p.109/143

Apparent Hubble flow variance15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.110/143

Result: arXiv:1201.5371v2CMB dipole usually interpreted as result **on** w.r.t.barycentre **on** **on**centr**on** ? ??But Shapley Supercluster, is **and**ard “rest frame **on** fromforeground density gradient **on**< ∼ 65h −1 Mpc scales15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.111/143

Peculiar velocity formalismSt**and**ard framework, FLRW + Newt**on**ian perturb**on**s,assumes peculiar velocity fieldv pec = cz − H 0rgener**on**tradictory claims,the v(r) does not to c**on**verge to LG velocity w.r.t. CMBAgreement **on** directi**on**, not amplitude or scale (Lavauxet al 2010; Bilicki et al 2011; . . . )Suggesti**on**s **on**sistent with ΛCDM(W**on** 2009. . . )15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.112/143

Spherical averagesDetermine vari**on** in Hubble flow by determiningbest-fit linear Hubble law in spherical shells15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.113/143

N. Li & D. Schwarz, PRD 78, 083531HST key d**on** **Cosmology** **and** Gravit**on**, August 2012 – p.114/143

Analysis **on** 2009, 2010, with 4,534 galaxy redshifts **and**distances, includes most large surveys to 2009Distance methods: Tully Fisher, fundamental plane,surface brightness fluctu**on**; 103 supernovaedistances.average in independent spherical shellsCompute H s in 12.5h −1 Mpc shells; combine 3 shells> 112.5h −1 MpcUse d**on**d 156.25h −1 Mpc as check **on** H 0normalis**on** – COMPOSITE sample is normalized to100hkm/s/Mpc15th **on** **Cosmology** **and** Gravit**on**, 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 **on** **Cosmology** **and** Gravit**on**, 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 **and** opencircles); for each choice d**on** in rest frame **on** **Cosmology** **and** Gravit**on**, August 2012 – p.117/143

Bayesian comparis**on** **on**g10ln BStr**on**gPositive1Not worth morethan a bare menti**on**Scale: Kass & Raftery (1995)2 20 40 60 80 100 120Inner distance cut**on**g evidence15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.118/143

But why try the LG frame?From viewpoint **and** in particularthe “Cosmological Equivalence Principle” in boundsystems the finite infinity regi**on** (or m**on**) isthe st**and**ard **on** **Cosmology** **and** Gravit**on**, August 2012 – p.119/143

Boosts **and** spurious m**on**opole varianceH s determined by linear regressi**on** 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 d**on** opposite sides**on** **Cosmology** **and** Gravit**on**, August 2012 – p.120/143

Angular varianceTwo approaches; fit1. McClure **and** Dyer (2007) method – can look **on** **Cosmology** **and** Gravit**on**, 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 iCan**on**ical value **on**e **on** **Cosmology** **and** Gravit**on**, August 2012 – p.122/143

Hubble variance: CMB frame15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.123/143

Hubble variance: LG frame15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.124/143

Angular uncertainties LG frame15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.125/143

Hubble variance quadrupole/dipole r**on** **Cosmology** **and** Gravit**on**, August 2012 – p.126/143

Value **on** **Cosmology** **and** Gravit**on**, August 2012 – p.127/143

Value **on** **Cosmology** **and** Gravit**on**, August 2012 – p.128/143

Dipole directi**on**CMB frame: directi**on** (l d ,b d ) remains within 1σ **on** (l,b) = (287 ◦ ± 9 ◦ , 8 ◦ ± 6 ◦ ) found byW**on**sistent with bulk flowdirecti**on** (l,b) = (319 ◦ ± 18 ◦ , 7 ◦ ± 14 ◦ ) **and** remains 4.0–7.0σ from zero.LG frame: For 20h −1 < ∼ r o< ∼ 45h −1 Mpc while the dipoleis str**on**g, directi**on** is c**on**sistently in range(l d ,b d ) = (83 ◦ ± 6 ◦ , −39 ◦ ± 3 ◦ ) but angular positi**on** thenw**and**ers **on**ce magnitude reduced to residual levels.For r o> ∼ 80h −1 Mpc the typical positi**on** **on** **Cosmology** **and** Gravit**on**, August 2012 – p.129/143

15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.130/143

Correl**on** 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 ◦ )Altern**on** raw d**on** **Cosmology** **and** Gravit**on**, August 2012 – p.131/143

Correl**on** with CMB dipole as r o variedCorrel**on** 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 **on** **Cosmology** **and** Gravit**on**, August 2012 – p.132/143

Redshift-distance anisotropyAs l**on**g as T ∝ 1/a, where a 0/a = 1 + z for someappropri**on**decoupling – due to foreground structures – will inducea CMB temper**on** **on** scales < ∼ 65h −1 Mpc.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.133/143

Redshift-distance anisotropyFor sp**and**ard 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 th**on** anisotropyis due to foreground structures within 65h −1 Mpc then±0.35h −1 Mpc represents a ±0.5% effect15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.134/143

Why a str**on**g CMB dipole?Ray tracing **and**Amarzguioui 2006). E.g.,a 20a 10=√415(h in − h out )d **on**stantsinside/outside void, d **and**h in − h out = 0.2, we have a 20/a 10< ∼ 1%.15th **on** **Cosmology** **and** Gravit**on**, 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 regressi**on** 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 gre**on**opole, or spherical Hubble bubblevariance15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.136/143

Towards a new formalismFor each shell redshift H(z s ,θ,φ) − H(z s ) will giveangular correcti**on**s which should lead to an expansi**on****on**vergence **on**verges to fixed value forz > z c**on**v ;(ii) residual anisotropy in D is **on** thecosmological model.St**and**ard peculiar velocity formalism **and** Hubblevariance formalism can then be directed compared **and**tested15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.137/143

Type Ia supernova system**and**ardizable c**and**les **on**ly; two popularmethods SALT/SALT–II, **and** MLCS2k2 yield differentresults when comparing cosmological resultsDegeneracy between intrinsic colour vari**on**s **and**reddening by dustHubble bubble is seen if R V= 3.1 (Milky way value); notif R V= 1.7 (**on**v ≃ 0.022)Study independent **and**ardized by minimizing H 0residualsin CMB frame. Uni**on**, C**on**stituti**on** compil**on**s c**on**tainmany SneIa in range 0.015 < ∼ z < ∼ 0.022 where CMBboost compens**on** **Cosmology** **and** Gravit**on**, August 2012 – p.138/143

Large angle CMB anomalies?Anomalies (varying significance) includepower asymmetry **and** octupole etc;low quadrupole power;parity asymmetry; . . .Critical re-examin**on** required; e.g.light propag**on** through Hubble variance dipoleforegrounds may differ subtly from Lorentz boost dipoledipole subtracti**on** is an integral part **on**may resolve the power asymmetry anomaly.15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.139/143

Dark flow?C**on**troversial claim **on**, using the kinem**and** et al (2012),Lavaux et al (2012) who use different techniquesHowever, we note Kashlinsky modelling **and** anisotropic distance-redshift rel**on** is assumeda 1m = a KSZ1m+ a TSZ1m+ a CMB1m+ σ noise√Ncl15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.140/143

Comments **on** ISW amplitudeIntegr**on** in timescape modelCorrel**on** **and** superclustersetc with CMB positively detected **and** well established(Boughn **and** Crittenden 2004, . . . Granett, Neyrinck **and**Szapudi, 2008)Amplitude **on**sistently **on**, or 3σ gre**on** **and** Sarkar(2012)Does departure **on**s may give insights about fe**on** **Cosmology** **and** Gravit**on**, August 2012 – p.141/143

C**on**clusi**on**/OutlookVariance **and**e!,(Abramowicz et al 2007) space really is exp**and**ingLarge CMB angle anomalies, **and** map-makingprocedures would need to be rec**on**sidered ... are thecold spot etc foreground artifacts, or primordial“Dark flow” probably a system**on** nearbymeasurements in setting distance scale etc, needs tobe re-examinedOpportunity to develop new formalism **and** approachesto observ**on**al cosmology15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.142/143

C**on**clusi**on**Apparent cosmic acceler**on** can be understood purelywithin general rel**and**ingfundamental aspects **on**al energy,**on** many puzzles– e.g., primordial lithium abundance anomalyMany tests can be d**on**e to distinguish from ΛCDM.It is crucial th**on**s such as Friedmannequ**on** are not used in d**on**.Many details – averaging scheme etc – may change,but fundamental questi**on**s remain15th **on** **Cosmology** **and** Gravit**on**, August 2012 – p.143/143