Astro 4PT Lecture Notes Set 1. Wayne Hu

Astro 4PT Lecture Notes Set 1 Wyne Hu

References Reltivistic Cosmologicl Perturbtion Theory Infltion Drk Energy Modified Grvity Cosmic Microwve Bckground Lrge Scle Structure Brdeen (1980), PRD 22 1882 Kodm & Sski (1984), Prog. Th. Phys. Supp., 78, 1 Mukhnov, Feldmn, Brndenberger (1992), Phys. Reports, 215, 203 Mlik & Wnds (2009), Phys. Reports, 475, 1

Covrint Perturbtion Theory Covrint = tkes sme form in ll coordinte systems Invrint = tkes the sme vlue in ll coordinte systems Fundmentl equtions: Einstein equtions, covrint conservtion of stress-energy tensor: G µν = 8πGT µν µ T µν = 0 Preserve generl covrince by keeping ll degrees of freedom: 10 for ech symmetric 4 4 tensor 1 2 3 4 5 6 7 8 9 10

Metric Tensor Useful to think in 3 + 1 lnguge since there re preferred sptil surfces where the stress tensor is nerly homogeneous In generl this is n Arnowitt-Deser-Misner (ADM) split Specilize to the cse of nerly FRW metric g 00 = 2, g ij = 2 γ ij. where the 0 component is conforml time η = dt/ nd γ ij is sptil metric of constnt curvture K = H 2 0(Ω tot 1). (3) R = 6K 2

Metric Tensor First define the slicing (lpse function A, shift function B i ) g 00 = 2 (1 2A), g 0i = 2 B i, A defines the lpse of proper time between 3-surfces wheres B i defines the threding or reltionship between the 3-coordintes of the surfces This bsorbs 1+3=4 degrees of freedom in the metric, remining 6 is in the sptil surfces which we prmeterize s g ij = 2 (γ ij 2H L γ ij 2H ij T ). here (1) H L perturbtion to the sptil curvture; (5) H ij T trce-free distortion to sptil metric (which lso cn perturb the curvture)

Curvture Perturbtion Curvture perturbtion on the 3D slice δ[ (3) R] = 4 2 ( 2 + 3K ) H L + 2 2 i j H ij T Note tht we will often loosely refer to H L s the curvture perturbtion We will see tht mny representtions hve H T = 0 It is esier to work with dimensionless quntity First exmple of 3-sclr - SVT decomposition

Mtter Tensor Likewise expnd the mtter stress energy tensor round homogeneous density ρ nd pressure p: T 0 0 = ρ δρ, T 0 i = (ρ + p)(v i B i ), T i 0 = (ρ + p)v i, T i j = (p + δp)δ i j + pπ i j, (1) δρ density perturbtion; (3) v i vector velocity, (1) δp pressure perturbtion; (5) Π ij n nisotropic stress perturbtion So fr this is fully generl nd pplies to ny type of mtter or coordinte choice including non-linerities in the mtter, e.g. sclr fields, cosmologicl defects, exotic drk energy.

Counting DOF s 20 Vribles (10 metric; 10 mtter) 10 4 Einstein equtions Conservtion equtions +4 Binchi identities 4 Guge (coordinte choice 1 time, 3 spce) 6 Degrees of freedom Without loss of generlity these cn be tken to be the 6 components of the mtter stress tensor For the bckground, specify p() or equivlently w() p()/ρ() the eqution of stte prmeter.

Homogeneous Einstein Equtions Einstein (Friedmnn) equtions: ( 1 ) 2 d = K dt + 8πG 2 3 ρ [= 1 ( 1 ) 2 ȧ ] d 2 dt 2 = 4πG 3 (ρ + 3p) [= 1 2 d dη so tht w p/ρ < 1/3 for ccelertion Conservtion eqution µ T µν = 0 implies ρ ρ = 3(1 + w)ȧ overdots re conforml time but eqully true with coordinte time ȧ ]

Homogeneous Einstein Equtions Counting exercise: 20 Vribles (10 metric; 10 mtter) 17 2 1 Homogeneity nd Isotropy Einstein equtions Conservtion equtions +1 Binchi identities 1 Degree of freedom without loss of generlity choose rtio of homogeneous & isotropic component of the stress tensor to the density w() = p()/ρ().

Accelertion Implies Negtive Pressure Role of stresses in the bckground cosmology Homogeneous Einstein equtions G µν = 8πGT µν imply the two Friedmnn equtions (flt universe, or ssociting curvture ρ K = 3K/8πG 2 ) ( 1 d dt ) 2 = 8πG 3 ρ 1 d 2 = 4πG (ρ + 3p) dt 2 3 so tht the totl eqution of stte w p/ρ < 1/3 for ccelertion Conservtion eqution µ T µν = 0 implies ρ ρ = 3(1 + w)ȧ so tht ρ must scle more slowly thn 2

Questions regrding Drk Energy Coincidence: given the very different sclings of mtter nd drk energy with, why re they comprble now? Stbility: why doesn t negtive pressure imply ccelerted collpse? or why doesn t the vcuum suck? Answer: stbility is ssocited with stress (pressure) grdients not stress (pressure) itself. Exmple: the cosmologicl constnt w Λ = 1, constnt in time nd spce no grdients. Exmple: sclr field where w = p/ρ δp/δρ = sound speed non-dibtic stress To quntify the perturbtion response let s exmine the norml mode decomposition

Sclr, Vector, Tensor In liner perturbtion theory, perturbtions my be seprted by their trnsformtion properties under 3D rottion nd trnsltion. The eigenfunctions of the Lplcin opertor form complete set 2 Q (0) = k 2 Q (0) S, 2 Q (±1) i = k 2 Q (±1) i V, 2 Q (±2) ij = k 2 Q (±2) ij T, Vector nd tensor modes stisfy divergence-free nd trnsverse-trceless conditions i Q (±1) i = 0 i Q (±2) ij = 0 γ ij Q (±2) ij = 0

Vector nd Tensor Quntities A sclr mode crries with it ssocited vector (curl-free) nd tensor (longitudinl) quntities A vector mode crries nd ssocited tensor (trce nd divergence free) quntities A tensor mode hs only tensor (trce nd divergence free) These re built from the mode bsis out of covrint derivtives nd the metric Q (0) i = k 1 i Q (0), Q (0) ij = (k 2 i j + 1 3 γ ij)q (0), Q (±1) ij = 1 2k [ iq (±1) j + j Q (±1) i ],

Sptilly Flt Cse For sptilly flt bckground metric, hrmonics re relted to plne wves: Q (0) = exp(ik x) Q (±1) i = i (ê 1 ± iê 2 ) i exp(ik x) 2 Q (±2) 3 ij = 8 (ê 1 ± iê 2 ) i (ê 1 ± iê 2 ) j exp(ik x) where ê 3 k. Chosen s spin sttes, c.f. polriztion. For vectors, the hrmonic points in direction orthogonl to k suitble for the vorticl component of vector

Sptilly Flt Cse Tensor hrmonics re the trnsverse trceless guge representtion Tensor mplitude relted to the more trditionl h + [(e 1 ) i (e 1 ) j (e 2 ) i (e 2 ) j ], h [(e 1 ) i (e 2 ) j + (e 2 ) i (e 1 ) j ] s h + ± ih = 6H ( 2) T H (±2) T proportionl to the right nd left circulrly polrized mplitudes of grvittionl wves with normliztion tht is convenient to mtch the sclr nd vector definitions

Perturbtion k-modes For the kth eigenmode, the sclr components become A(x) = A(k) Q (0), H L (x) = H L (k) Q (0), δρ(x) = δρ(k) Q (0), δp(x) = δp(k) Q (0), the vectors components become 1 B i (x) = B (m) (k) Q (m) i, v i (x) = m= 1 nd the tensors components H T ij (x) = 2 m= 2 H (m) T (k) Q (m) ij, Π ij (x) = 2 m= 2 1 m= 1 v (m) (k) Q (m) Π (m) (k) Q (m) ij, Note tht the curvture perturbtion only involves sclrs i, δ[ (3) R] = 4 2 (k2 3K)(H (0) L + 1 3 H(0) T )Q(0)

Covrint Sclr Equtions DOF counting exercise 4 2 8 Vribles (4 metric; 4 mtter) Einstein equtions Conservtion equtions +2 Binchi identities 2 Guge (coordinte choice 1 time, 1 spce) 2 Degrees of freedom without loss of generlity choose sclr components of the stress tensor δp, Π.

Covrint Sclr Equtions Einstein equtions (suppressing 0) superscripts (k 2 3K)[H L + 1 3 H T ] 3(ȧ )2 A + 3ȧ + ȧ ḢL kb = = 4πG 2 δρ, 00 Poisson Eqution k 2 (A + H L + 1 ( ) d 3 H T ) + dη + 2ȧ (kb ḢT ) = 8πG 2 pπ, ij Anisotropy Eqution ȧ A ḢL 1 K 3ḢT k 2 (kb ḢT ) = 4πG 2 (ρ + p)(v B)/k, 0i Momentum Eqution [ (ȧ ) ] 2 2ä 2 + ȧ [ d dη k2 d A 3 dη + ȧ ] (ḢL + 1 3 kb) = 4πG 2 (δp + 1 δρ), ii Accelertion Eqution 3

Covrint Sclr Equtions Conservtion equtions: continuity nd Nvier Stokes [ d dη + 4ȧ [ ] d dη + 3ȧ δρ + 3ȧ δp = (ρ + p)(kv + 3ḢL), ] [ ] (ρ + p) (v B) k = δp 2 3 (1 3 K )pπ + (ρ + p)a, k2 Equtions re not independent since µ G µν = 0 vi the Binchi identities. Relted to the bility to choose coordinte system or guge to represent the perturbtions.

Seprte Universes For perturbtions lrger thn the horizon, locl observer should just see different (seprte) FRW universe Sclr equtions should be equivlent to n ppropritely rempped Friedmnn eqution Unit norml vector N µ to constnt time hypersurfces N 0 = (1 + AQ), N i = 0 N 0 = 1 (1 AQ), N i = BQ i Expnsion of sptil volume per proper time is given by 4-divergence µ N µ θ = 3H(1 AQ) + k 3 BQ + ḢLQ

Sher nd Accelertion Other pieces of ν N µ give the vorticity, sher nd ccelertion with ν N µ ω µν + σ µν + 1 3 θp µν µ V ν P µν = g µν + N µ N ν ω µν = P α µ P β ν ( β N α α N β ) σ µν = 1 2 P α µ P β ν ( β N α + α N β ) 1 3 θp µν µ = ( ν V µ )V ν projection, trce free ntisymmetric, symmetric nd ccelertion

Sher nd Accelertion Vorticity ω µν = 0, σ 00 = σ 0i = 0 = 0 Remining perturbed quntities re the sptil sher nd ccelertion σ ij = (ḢT kb)q ij i = kaq i A convenient choice of coordintes might be sher free Ḣ T kb = 0 A lone is relted to the perturbed ccelertion

Seprte Universes So the e-foldings of the expnsion re given by dτ = (1 + AQ)dη N = = dτ 1 3 θ (ȧ dη + ḢLQ + 1 ) 3 kbq Thus if kb cn be ignored s k 0 then H L plys the role of locl chnge in the scle fctor, more generlly B plys the role of Eulerin Lgrngin coordintes. Chnge in H L between seprte universes relted to chnge in number of e-folds: so clled δn pproch, simplifying equtions by using N s time vrible to bsorb locl scle fctor effects We shll see tht for dibtic perturbtions p(ρ) tht Ḣ L = 0 for n pproprite choice of slicing. This conservtion lw plys n importnt role in simplifying clcultions

Seprte Universes Dropping the ḢL nd B terms, we get tht the 00 Einstein equtions t k 0 re (ȧ ) 2 A = 4πG 3 2 δρ which is to be compred to the Friedmnn eqution H 2 + K 2 = 8πG 3 ρ Noting tht H = H(1 AQ) we get 2δH H = 8πG 3 δρq = 2AQ H 2 = 2AQ 2 (ȧ ) 2

And the spce-spce piece [ 2ä 2 Seprte Universes (ȧ ) 2 + ȧ ] d A = 4πG dη 3 2 (δp + 3δρ) which is to be compred with the ccelertion eqution d (H) = 4πG dη 3 2 (p + 3ρ) gin expnding H = H(1 AQ) nd lso dη = (1 + AQ)d η d d (H) = (1 AQ) dη d η d d (H) 2AQ d η d η (H)[1 AQ] ȧ + ȧ d d η AQ

Finlly the continuity eqution is to be compred to Seprte Universes δρ + 3ȧ (δρ + δp) = 0 d ρ = 3(H)(ρ + p) dη which gin with the substitutions becomes (1 AQ) d ( ρ + δρq) = 3(H)(1 AQ)[ ρ + p] 3(H)[δρ + δp]q d η d d η δρ = 3ȧ(δρ + δp)

Covrint Vector Equtions Einstein equtions (1 2K/k 2 )(kb (±1) Ḣ(±1) T ) [ d dη + 2ȧ Conservtion Equtions = 16πG 2 (ρ + p)(v (±1) B (±1) )/k, ] (kb (±1) Ḣ(±1) T ) = 8πG 2 pπ (±1). [ ] d dη + 4ȧ [(ρ + p)(v (±1) B (±1) )/k] = 1 2 (1 2K/k2 )pπ (±1), Grvity provides no source to vorticity decy

Covrint Vector Equtions DOF counting exercise 4 2 8 Vribles (4 metric; 4 mtter) Einstein equtions Conservtion equtions +2 Binchi identities 2 Guge (coordinte choice 1 time, 1 spce) 2 Degrees of freedom without loss of generlity choose vector components of the stress tensor Π (±1).

Einstein eqution [ d 2 dη 2 + 2ȧ Covrint Tensor Eqution DOF counting exercise ] d dη + (k2 + 2K) H (±2) T = 8πG 2 pπ (±2). 4 Vribles (2 metric; 2 mtter) 2 0 Einstein equtions Conservtion equtions +0 Binchi identities 0 Guge (coordinte choice 1 time, 1 spce) 2 Degrees of freedom wlog choose tensor components of the stress tensor Π (±2).

Arbitrry Drk Components Totl stress energy tensor cn be broken up into individul pieces Drk components interct only through grvity nd so stisfy seprte conservtion equtions Einstein eqution source remins the sum of components. To specify n rbitrry drk component, give the behvior of the stress tensor: 6 components: δp, Π (i), where i = 2,..., 2. Mny types of drk components (drk mtter, sclr fields, mssive neutrinos,..) hve simple forms for their stress tensor in terms of the energy density, i.e. described by equtions of stte. An eqution of stte for the bckground w = p/ρ is not sufficient to determine the behvior of the perturbtions.

Guge Metric nd mtter fluctutions tke on different vlues in different coordinte system No such thing s guge invrint density perturbtion! Generl coordinte trnsformtion: η = η + T x i = x i + L i free to choose (T, L i ) to simplify equtions or physics - corresponds to choice of slicing nd threding in ADM. Decompose these into sclr T, L (0) nd vector hrmonics L (±1).

Guge g µν nd T µν trnsform s tensors, so components in different frmes cn be relted g µν ( η, x i ) = xα x µ x β x ν g µν(η, x i ) = xα x µ x β x ν g µν( η T Q, x i LQ i ) Fluctutions re compred t the sme coordinte positions (not sme spce time positions) between the two guges For exmple with T Q perturbtion, n event lbeled with η =const. nd x =const. represents different time with respect to the underlying homogeneous nd isotropic bckground

Sclr Metric: Guge Trnsformtion à = A T ȧ T, B = B + L + kt, H L = H L k 3 L ȧ T, H T = H T + kl, HL + 1 3 H T = H L + 1 3 H T ȧ T curvture perturbtion depends on slicing not threding Sclr Mtter (Jth component): δ ρ J = δρ J ρ J T, δ p J = δp J ṗ J T, ṽ J = v J + L, density nd pressure likewise depend on slicing only

Guge Trnsformtion Vector: B (±1) = B (±1) + L (±1), H (±1) T = H (±1) T + kl (±1), ṽ (±1) J = v (±1) J + L (±1), Sptil vector hs no bckground component hence no dependence on slicing t first order Tensor: no dependence on slicing or threding t first order Guge trnsformtions nd covrint representtion cn be extended to higher orders A coordinte system is fully specified if there is n explicit prescription for (T, L i ) or for sclrs (T, L)

Slicing Common choices for slicing T : set something geometric to zero Proper time slicing A = 0: proper time between slices corresponds to coordinte time T llows c/ freedom Comoving (velocity orthogonl) slicing: v B = 0, mtter 4 velocity is relted to N ν nd orthogonl to slicing - T fixed Newtonin (sher free) slicing: Ḣ T kb = 0, expnsion rte is isotropic, sher free, T fixed Uniform expnsion slicing: (ȧ/)a + ḢL + kb/3 = 0, perturbtion to the volume expnsion rte θ vnishes, T fixed Flt (constnt curvture) slicing, δ (3) R = 0, (H L + H T /3 = 0), T fixed Constnt density slicing, δρ I = 0, T fixed

Threding Threding specifies the reltionship between constnt sptil coordintes between slices nd is determined by L Typiclly involves condition on v, B, H T Orthogonl threding B = 0, constnt sptil coordintes orthogonl to slicing (zero shift), llows L = c trnsltionl freedom Comoving threding v = 0, llows L = c trnsltionl freedom. Isotropic threding H T = 0, fully fixes L

Newtonin (Longitudinl) Guge Newtonin (sher free slicing, isotropic threding): B = H T = 0 Ψ Ã (Newtonin potentil) Φ H L (Newtonin curvture) L = H T /k T = B/k + ḢT /k 2 Good: intuitive Newtonin like grvity; mtter nd metric lgebriclly relted; commonly chosen for nlytic CMB nd lensing work Bd: numericlly unstble

Newtonin (Longitudinl) Guge Newtonin (sher free) slicing, isotropic threding B = H T = 0 : [ (k 2 3K)Φ = 4πG 2 δρ + 3ȧ ] (ρ + p)v/k k 2 (Ψ + Φ) = 8πG 2 pπ Anisotropy so Ψ = Φ if nisotropic stress Π = 0 nd Poisson + Momentum [ ] d dη + 3ȧ δρ + 3ȧ δp = (ρ + p)(kv + 3 Φ), [ ] d dη + 4ȧ (ρ + p)v = kδp 2 3 (1 3 K )p kπ + (ρ + p) kψ, k2 Newtonin competition between stress (pressure nd viscosity) nd potentil grdients Note: Poisson source is the density perturbtion on comoving slicing

Totl Mtter Guge Totl mtter: (comoving threding, isotropic slicing) B = ṽ (T 0 i = 0) H T = 0 ξ = Ã R = H L (comoving curvture) = δ (totl density pert) T = (v B)/k L = H T /k Good: Algebric reltions between mtter nd metric; comoving curvture perturbtion obeys conservtion lw Bd: Non-intuitive threding involving v

Totl Mtter Guge Euler eqution becomes n lgebric reltion between stress nd potentil (ρ + p)ξ = δp + 2 3 ( 1 3K ) pπ k 2 Einstein eqution lcks momentum density source ȧ ξ Ṙ K k 2 kv = 0 Combine: R is conserved if stress fluctutions negligible, e.g. bove the horizon if K H 2 Ṙ + Kv/k = ȧ [ δp ρ + p + 2 ( 1 3K ) ] p 3 k 2 ρ + p Π 0

Guge Invrint Approch Guge trnsformtion rules llow vribles which tke on geometric mening in one choice of slicing nd threding to be ccessed from vribles on nother choice Functionl form of the reltionship between the vribles is guge invrint (not the vrible vlues themselves! i.e. eqution is covrint) E.g. comoving curvture nd density perturbtions R = H L + 1 3 H T ȧ (v B)/k ρ = δρ + 3(ρ + p)ȧ (v B)/k

Newtonin-Totl Mtter Hybrid With the guge in(or co)vrint pproch, express vribles of one guge in terms of those in nother llows mixture in the equtions of motion Exmple: Newtonin curvture nd comoving density (k 2 3K)Φ = 4πG 2 ρ ordinry Poisson eqution then implies Φ pproximtely constnt if stresses negligible. Exmple: Exct Newtonin curvture bove the horizon derived through comoving curvture conservtion Guge trnsformtion Φ = R + ȧ v k

Hybrid Guge Invrint Approch Einstein eqution to eliminte velocity ȧ Ψ Φ = 4πG 2 (ρ + p)v/k Friedmnn eqution with no sptil curvture (ȧ With Φ = 0 nd Ψ Φ ȧ ) 2 = 8πG 3 2 ρ v k = 2 3(1 + w) Φ

Newtonin-Totl Mtter Hybrid Combining guge trnsformtion with velocity reltion Φ = 3 + 3w 5 + 3w R Usge: clculte R from infltion determines Φ for ny choice of mtter content or cusl evolution. Exmple: Sclr field ( quintessence drk energy) equtions in totl mtter guge imply sound speed δp/δρ = 1 independent of potentil V (φ). Solve in synchronous guge.

Synchronous Guge Synchronous: (proper time slicing, orthogonl threding ) Ã = B = 0 η T H L 1 3 H T h L 6H L T = 1 L = dηa + c 1 1 dη(b + kt ) + c 2 Good: stble, the choice of numericl codes Bd: residul guge freedom in constnts c 1, c 2 must be specified s n initil condition, intrinsiclly reltivistic, threding conditions breks down beyond liner regime if c 1 is fixed to CDM comoving.

The Einstein equtions give Synchronous Guge η T K 2k 2 (ḣl + 6 η T ) = 4πG 2 (ρ + p) v k, ḧ L + ȧ while the conservtion equtions give ḣl = 8πG 2 (δρ + 3δp), [ ] d dη + 3ȧ δρ J + 3ȧ δp J = (ρ J + p J )(kv J + 1 2ḣL), [ ] d dη + 4ȧ (ρ J + p J ) v J k = δp J 2 3 (1 3 K k 2 )p JΠ J. Lck of lpse A implies no grvittionl forces in Nvier-Stokes eqution. Hence for stress free mtter like cold drk mtter, zero velocity initilly implies zero velocity lwys.

The Einstein equtions give Synchronous Guge η T K 2k 2 (ḣl + 6 η T ) = 4πG 2 (ρ + p) v k, ḧ L + ȧ = 8πG 2 (δρ + 3δp), ḣl (k 2 3K)η T + 1 ȧ = 4πG 2 δρ 2 ḣl [choose (1 & 2) or (1 & 3)] while the conservtion equtions give [ ] d dη + 3ȧ δρ J + 3ȧ δp J = (ρ J + p J )(kv J + 1 2ḣL), [ ] d dη + 4ȧ (ρ J + p J ) v J k = δp J 2 3 (1 3 K k 2 )p JΠ J.

Synchronous Guge Lck of lpse A implies no grvittionl forces in Nvier-Stokes eqution. Hence for stress free mtter like cold drk mtter, zero velocity initilly implies zero velocity lwys. Choosing the momentum nd ccelertion Einstein equtions is good since for CDM domintion, curvture η T is conserved nd ḣl is simple to solve for. Choosing the momentum nd Poisson equtions is good when the eqution of stte of the mtter is complicted since δp is not involved. This is the choice of CAMB. Cution: since the curvture η T ppers nd it hs zero CDM source, subtle effects like drk energy perturbtions re importnt everywhere

Sptilly Flt Guge Sptilly Flt (flt slicing, isotropic threding): H L = H T = 0 L = H T /k Ã, B = metric perturbtions (ȧ ) 1 ( T = H L + 1 ) 3 H T Good: elimintes sptil metric in evolution equtions; useful in infltionry clcultions (Mukhnov et l) Bd: non-intuitive slicing (no curvture!) nd threding Cution: perturbtion evolution is governed by the behvior of stress fluctutions nd n isotropic stress fluctution δp is guge dependent.

Uniform Density Guge Uniform density: (constnt density slicing, isotropic threding) H T = 0, ζ I H L B I B A I A T = δρ I ρ I L = H T /k Good: Curvture conserved involves only stress energy conservtion; simplifies isocurvture tretment Bd: non intuitive slicing (no density pert! problems beyond liner regime) nd threding

Uniform Density Guge Einstein equtions with I s the totl or dominnt species (ȧ (k 2 3K)ζ I 3 ) 2 A I + 3ȧ ζ I + ȧ kb I = 0, ȧ A I ζ I K k B I = 4πG 2 (ρ + p) v B I k The conservtion equtions (if J = I then δρ J = 0) [ d dη + 4ȧ [ ] d dη + 3ȧ ] δρ J + 3ȧ δp J = (ρ J + p J )(kv J + 3 ζ I ), (ρ J + p J ) v J B I k = δp J 2 3 (1 3 K k 2 )p JΠ J + (ρ J + p J )A I.,

Uniform Density Guge Conservtion of curvture - single component I ζ I = ȧ δp I ρ I + p I 1 3 kv I. Since δρ I = 0, δp I is the non-dibtic stress nd curvture is constnt s k 0 for dibtic fluctutions p I (ρ I ). Note tht this conservtion lw does not involve the Einstein equtions t ll: just locl energy momentum conservtion so it is vlid for lternte theories of grvity Curvture on comoving slices R nd ζ I relted by ζ I = R + 1 3 ρ I. (ρ I + p I ) comoving nd coincide bove the horizon for dibtic fluctutions

Uniform Density Guge Simple reltionship to density fluctutions in the sptilly flt guge ζ I = 1 3 δ ρ I. (ρ I + p I ) flt For ech prticle species δρ/(ρ + p) = δn/n, the number density fluctution Multiple ζ J crry informtion bout number density fluctutions between species ζ J constnt component by component outside horizon if ech component is dibtic p J (ρ J ).

Vector Guges Vector guge depends only on threding L Poisson guge: orthogonl threding B (±1) = 0, leves constnt L trnsltionl freedom Isotropic guge: isotropic threding H (±1) T = 0, fixes L To first order sclr nd vector guge conditions cn be chosen seprtely More cre required for second nd higher order where sclrs nd vectors mix