Cosmological Perturbations Power Spectrum Vector Perturbations Tensor Perturbations

Theoretical cosmology

David Wands

Homogeneous cosmology

Perturbation Theoretical cosmology theory Scalar perturbations Fourier transforms Cosmological perturbations Power spectrum Vector perturbations Tensor perturbations

Metric David Wands perturbations Geometrical interpretation Gauge dependence Institute of Cosmology and Gravitation Particular gauges Conformal University of Portsmouth Newtonian/Longitudinal gauge Uniform-density gauge Equating gauge-invariant 5th Tah Poe School on Cosmology variables Einstein equations Einstein equations in an arbitrary gauge Recovering Newtonian ﬂuid equations Redshift-space distortions Recovering Newtonian ﬂuid equations Standard model of structure formation primordial perturbations in cosmic microwave background

gravitational instability

large-scale structure of our Universe

new observational data offers precision tests of • cosmological parameters • the nature of the primordial perturbations Inflation: initial false vacuum state drives accelerated expansion zero-point fluctuations yield spectrum of perturbations Theoretical References cosmology David Wands

Homogeneous cosmology

Perturbation theory Scalar perturbations I Malik and Wands, Phys Rep 475, 1 (2009), Fourier transforms Power spectrum arXiv:0809.4944 Vector perturbations Tensor perturbations

I Bardeen, Phys Rev D22, 1882 (1980) Metric perturbations I Kodama and Sasaki, Prog Theor Phys Supp 78, 1 Geometrical interpretation Gauge dependence (1984) Particular gauges Conformal Bassett, Tsujikawa and Wands, Rev Mod Phys (2005), Newtonian/Longitudinal I gauge astro-ph/0507632 Uniform-density gauge Equating gauge-invariant variables

Homogeneous cosmology

Perturbation theory Homogeneous cosmology Scalar perturbations Fourier transforms Power spectrum Vector perturbations Perturbation theory Tensor perturbations Metric perturbations Geometrical interpretation Gauge dependence Particular gauges Metric perturbations Conformal Newtonian/Longitudinal gauge Uniform-density gauge Equating gauge-invariant variables Einstein equations Einstein equations Einstein equations in an arbitrary gauge Recovering Newtonian ﬂuid equations Redshift-space distortions Recovering Newtonian ﬂuid equations I alternative (conformal) time coordinate, d⌧ = cdt/a:

ds2 = a2(⌧) d⌧ 2 + dX 2 ⇥ ⇤ I Henceforth assume K = 0, ﬂat space

Einstein equations Einstein equations in an arbitrary gauge Recovering Newtonian ﬂuid equations Redshift-space distortions Recovering Newtonian ﬂuid equations u n t FRW cosmology preferred coordinates for homogeneous and isotropic space x

preferred space+time split in FRW cosmology breaks symmetry of Einstein’s theory I Henceforth assume K = 0, ﬂat space

Theoretical FLRW metric cosmology David Wands I 4D spacetime split into 1+3 Homogeneous I Friedmann-Lemaitre-Robertson-Walker (FLRW) line cosmology element: Perturbation 2 2 2 2 2 theory ds = c dt + a (t)dX Scalar perturbations Fourier transforms Power spectrum Vector perturbations I time + homogeneous and isotropic space Tensor perturbations dynamical scale factor, a(t), where a =1today Metric I 0 perturbations I maximally-symmetric 3-space, curvature K Geometrical interpretation Gauge dependence 2 Particular gauges 2 dr 2 2 2 2 Conformal dX = + r d✓ +sin ✓d Newtonian/Longitudinal 1 Kr 2 gauge Uniform-density gauge Equating gauge-invariant I alternative (conformal) time coordinate, d⌧ = cdt/a: variables Einstein equations 2 2 2 2 Einstein equations in an ds = a (⌧) d⌧ + dX arbitrary gauge Recovering Newtonian fluid equations ⇥ ⇤ Redshift-space distortions Recovering Newtonian fluid equations Theoretical FLRW metric cosmology David Wands I 4D spacetime split into 1+3 Homogeneous I Friedmann-Lemaitre-Robertson-Walker (FLRW) line cosmology element: Perturbation 2 2 2 2 2 theory ds = c dt + a (t)dX Scalar perturbations Fourier transforms Power spectrum Vector perturbations I time + homogeneous and isotropic space Tensor perturbations dynamical scale factor, a(t), where a =1today Metric I 0 perturbations I maximally-symmetric 3-space, curvature K Geometrical interpretation Gauge dependence 2 Particular gauges 2 dr 2 2 2 2 Conformal dX = + r d✓ +sin ✓d Newtonian/Longitudinal 1 Kr 2 gauge Uniform-density gauge Equating gauge-invariant I alternative (conformal) time coordinate, d⌧ = cdt/a: variables Einstein equations 2 2 2 2 Einstein equations in an ds = a (⌧) d⌧ + dX arbitrary gauge Recovering Newtonian fluid equations ⇥ ⇤ Redshift-space distortions Recovering Newtonian fluid equations I Henceforth assume K = 0, flat space Theoretical Scalar perturbations cosmology David Wands

I Scalar quantity, e.g., density at fixed point P is Homogeneous cosmology invariant under change of coordinates Perturbation Split into background (homogeneous) part and a theory I Scalar perturbations perturbation (inhomogeneous): Fourier transforms Power spectrum Vector perturbations ⇢(t, ~x)=¯⇢(t)+⇢(t, ~x) Tensor perturbations Metric perturbations Geometrical interpretation Gauge dependence I expand perturbation order-by-order in a small Particular gauges Conformal parameter, ": Newtonian/Longitudinal gauge Uniform-density gauge 1 2 Equating gauge-invariant ⇢(t, ~x)="1⇢(t, ~x)+ " 2⇢(t, ~x)+... variables 2 Einstein equations Einstein equations in an I keep only terms at first order in " linear peturbations arbitrary gauge ) Recovering Newtonian fluid equations ⇢(t, ~x)=" ⇢(t, ~x) Redshift-space distortions 1 Recovering Newtonian fluid equations Theoretical Expanding equations order-by-order cosmology David Wands I e.g., non-relativistic continuity equation for density ⇢(t, ~x) Homogeneous @⇢ cosmology + ~ (⇢~v) = 0 (1) Perturbation @t r · theory Scalar perturbations expand density and velocity order-by-order Fourier transforms Power spectrum 1 2 Vector perturbations ⇢(t, ~x)=¯⇢(t)+"1⇢(t, ~x)+ " 2⇢(t, ~x)+... Tensor perturbations 2 Metric 1 2 perturbations ~v(t, ~x)= "1~v(t, ~x)+ " 2~v(t, ~x)+... Geometrical interpretation 2 Gauge dependence Particular gauges substitute into Eq. (1) Conformal Newtonian/Longitudinal gauge Uniform-density gauge @ 1 2 ⇢¯ + " ⇢ + " ⇢ + ... Equating gauge-invariant @t 1 2 2 variables ✓ ◆ Einstein equations 1 2 Einstein equations in an +~ ⇢¯ + " ⇢ + " ⇢ + ... arbitrary gauge 1 2 Recovering Newtonian fluid r · 2 equations ✓ ◆ Redshift-space distortions 1 2 Recovering Newtonian " ~v + " ~v + ... =0 fluid equations ⇥ 1 2 2 ✓ ◆ Theoretical Perturbation equations order-by-order cosmology David Wands I collect terms order-by-order in " Homogeneous cosmology

@ Perturbation ⇢¯ theory @t Scalar perturbations @ Fourier transforms +" 1⇢ + ~ (¯⇢1~v) Power spectrum @t r · Vector perturbations ⇢ Tensor perturbations 1 2 @ ~ Metric + " 2⇢ + (¯⇢2~v +21⇢1~v) + ...=0 perturbations 2 @t r · Geometrical interpretation ⇢ Gauge dependence Particular gauges I solve order-by-order in " Conformal Newtonian/Longitudinal gauge @ Uniform-density gauge ⇢¯ =0 ⇢¯ = C Equating gauge-invariant @t ) variables @ Einstein equations ~ Einstein equations in an 1⇢ + C 1~v =0 arbitrary gauge @t r · Recovering Newtonian fluid equations @ ~ ~ Redshift-space distortions 2⇢ + C 2~v = 2C (1⇢1~v) Recovering Newtonian @t r · r · fluid equations Theoretical Fourier transform cosmology David Wands I Field in real space is an integral over Fourier modes: Homogeneous cosmology 3 d k i~k.~x Perturbation ⇢(t, ~x)= 3 ⇢~k (t) e theory (2⇡) Scalar perturbations Z Fourier transforms Power spectrum I Fourier modes are eigenfunctions of the spatial Vector perturbations Laplacian: Tensor perturbations 2 i~k.~x 2 i~k.~x Metric e = k e perturbations r Geometrical interpretation ⇣ ⌘ Gauge dependence which provide a complete orthonormal basis: Particular gauges Conformal Newtonian/Longitudinal gauge ~ ~ 3 ik1.~x ik2.~x 3 (3) ~ ~ Uniform-density gauge d xe e =(2⇡) k1 k2 Equating gauge-invariant variables Z ⇣ ⌘ Einstein equations Coecient in Fourier space is integral over real space: Einstein equations in an I arbitrary gauge Recovering Newtonian fluid equations 3 i~k.~x Redshift-space distortions ⇢~ (t)= d x ⇢(t, ~x) e Recovering Newtonian k fluid equations Z Theoretical Statistical distribution cosmology David Wands

Homogeneous cosmology

Perturbation theory Scalar perturbations Fourier transforms I theory Power spectrum Vector perturbations describes properties of distribution = ensemble, Tensor perturbations assumed isotropic Metric perturbations ( ... = average over all possible realisations) Geometrical interpretation h i Gauge dependence I observations Particular gauges Conformal Newtonian/Longitudinal describe one realisation from the distribution gauge Uniform-density gauge Equating gauge-invariant variables

Einstein equations Einstein equations in an arbitrary gauge Recovering Newtonian fluid equations Redshift-space distortions Recovering Newtonian fluid equations Theoretical Power spectrum cosmology I defined by the correlation of two modes in Fourier space: David Wands 3 3 Homogeneous ⇢ ⇢ =(2⇡) P⇢(k1) ~k1 + ~k2 cosmology ~k1 ~k2 h i Perturbation ⇣ ⌘ theory note: P⇢(k) only a function of wavenumber k, not Scalar perturbations wavevector ~k, for an isotropic distribution Fourier transforms Power spectrum I Variance in real space: (exercise for reader!) Vector perturbations Tensor perturbations 3 3 d ~k1d ~k2 ~ ~ Metric 2 i(k1+k2) ~x perturbations ⇢ (~x) = 6 ⇢~k ⇢~k e · h i h (2⇡) 1 2 i Geometrical interpretation Z Gauge dependence 3~ 3~ Particular gauges d k d k ~ ~ Conformal 1 2 i(k1+k2) ~x = ⇢~ ⇢~ e · Newtonian/Longitudinal (2⇡)6 h k1 k2 i gauge Z Uniform-density gauge 3 Equating gauge-invariant d ~k1 variables = 3 P⇢(k1)= d ln k1 ⇢(k1) Einstein equations (2⇡) P Einstein equations in an Z Z arbitrary gauge Recovering Newtonian fluid I dimensionless power spectrum per log k: equations 3 Redshift-space distortions 4⇡k Recovering Newtonian (k)= P (k) fluid equations P⇢ (2⇡)3 ⇢ Theoretical Higher-order statistics cosmology David Wands

Homogeneous I Bispectrum cosmology Perturbation 3 3 ~ ~ ~ theory ⇢ ⇢ ⇢ =(2⇡) B⇢(k1, k2,3 ) k1 + k2 + k3 Scalar perturbations ~k1 ~k2 ~k3 h i Fourier transforms ⇣ ⌘ Power spectrum Vector perturbations Bispectrum is zero for Gaussian perturbations (and for Tensor perturbations

all odd moments) Metric perturbations I We will take ﬁrst-order perturbations to be Gaussian: Geometrical interpretation Gauge dependence Particular gauges 1⇢ 1⇢ 1⇢ =0 Conformal ~k1 ~k2 ~k3 Newtonian/Longitudinal h i gauge Uniform-density gauge Equating gauge-invariant Second- and higher-order perturbations are variables

non-Gaussian. Einstein equations Einstein equations in an arbitrary gauge Recovering Newtonian fluid 2⇢ 1⇢ 1⇢ =0 equations ~k1 ~k2 ~k3 h i6 Redshift-space distortions Recovering Newtonian fluid equations Theoretical Vector perturbations cosmology I decompose any 3-vector: V~ = ~ V (s) + V~ (v) David Wands r (s) I scalar (longitudinal/potential) flow: ~ ~ V =0 Homogeneous r⇥r cosmology I vector (transverse/divergence-free) flow: ~ V~ (v) =0 r · Perturbation theory Scalar perturbations I Fourier transform Fourier transforms Power spectrum I scalar Vector perturbations 3 d k ~ Tensor perturbations V (s)(t, ~x)= V (s)(t) eik.~x (2⇡)3 ~k Metric Z perturbations I vector Geometrical interpretation Gauge dependence 3 Particular gauges d k (v) (v) ~ ~ (v) ˜ ~ ik.~x Conformal V (t, ~x)= 3 V~ (t)e~~k + V~ (t)e˜~k e Newtonian/Longitudinal (2⇡) k k gauge Z n o Uniform-density gauge Equating gauge-invariant ~ variables where e~~k and e˜~k are orthonormal polarisation vectors: Einstein equations ~ ~ ~ Einstein equations in an e~~k e~~k = e˜~k e˜~k =1, e~~k e˜~k =0 arbitrary gauge · · · Recovering Newtonian fluid ~ equations transverse to wavevector k: Redshift-space distortions Recovering Newtonian fluid equations ~k e~ = ~k ~e˜ =0 · ~k · ~k Theoretical Vector perturbations cosmology David Wands

Homogeneous cosmology

Perturbation theory Scalar perturbations Fourier transforms Power spectrum putting it all together Vector perturbations Tensor perturbations

3 Metric d k (s) (v) (v) ~ ~ ~ ~ ˜ ik.~x perturbations V (t, ~x)= 3 ikV~ (t)+e~~k V~ (t)+e˜~k V~ (t) e (2⇡) k k k Geometrical interpretation Z n o Gauge dependence Particular gauges Conformal Newtonian/Longitudinal gauge Uniform-density gauge Equating gauge-invariant variables

Homogeneous cosmology

Perturbation theory Scalar perturbations I decompose any 3-tensor: Fourier transforms Tij = ij C + i j S +(1/2) ( i Vj + j Vi )+hij Power spectrum r r r r Vector perturbations I scalars C and S are longitudinal/potential Tensor perturbations i I vector Vi is transverse: Vi =0 Metric r perturbations I tensor hij is transverse and trace-free: Geometrical interpretation Gauge dependence i j i Particular gauges hij = hij =0, hi =0 Conformal r r Newtonian/Longitudinal gauge Uniform-density gauge Equating gauge-invariant variables

Einstein equations Einstein equations in an arbitrary gauge Recovering Newtonian fluid equations Redshift-space distortions Recovering Newtonian fluid equations Theoretical Tensor perturbations cosmology David Wands I Fourier transform: 3 Homogeneous d k (+) (+) ( ) ( ) i~k.~x cosmology ⇥ ⇥ hij (t, ~x)= 3 h~ (t)q~ + h~ (t)q~ e (2⇡) k kij k kij Perturbation Z n o theory where polarisation tensors Scalar perturbations Fourier transforms Power spectrum (+) 1 Vector perturbations q = e~ e~ e˜~ e˜~ Tensor perturbations ~kij p2 ki kj ki kj ⇣ ⌘ Metric ( ) 1 perturbations q ⇥ = e e˜ +˜e e Geometrical interpretation ~kij ~ki ~kj ~ki ~kj p2 Gauge dependence ⇣ ⌘ Particular gauges Conformal and e~ ande ˜~ are orthonormal, transverse vectors, Newtonian/Longitudinal ki ki gauge such that (exercise for reader!) Uniform-density gauge Equating gauge-invariant (+) ij (+) ( ) ij ( ) (+) ij ( ) variables q q = q ⇥ q ⇥ =1, q q ⇥ =0 ~k ~kij ~k ~kij ~k ~kij Einstein equations Einstein equations in an (+)i ( )i arbitrary gauge ~ Recovering Newtonian fluid tracefree q = q ⇥ =0andtransversetok: equations ~ki ~ki Redshift-space distortions Recovering Newtonian i (+) i ( ) fluid equations k q = k q ⇥ =0 ~kij ~kij Theoretical Metric perturbations cosmology David Wands I Split metric into spatially-flat FLRW background and Homogeneous inhomogeneous perturbation: cosmology

Perturbation gµ⌫ =¯gµ⌫ + gµ⌫ . theory Scalar perturbations Fourier transforms I Background: Power spectrum Vector perturbations 2 2 Tensor perturbations g¯00 = a , g¯0i =0, g¯ij = a ij (2) Metric perturbations Geometrical interpretation I Perturbation: Gauge dependence Particular gauges 2 Conformal g00 =2a A Newtonian/Longitudinal gauge 2 Uniform-density gauge g0i = a ( i B Si ) Equating gauge-invariant r variables g = a2 (2C +2 E + F + F + h ) ij ij ri rj ri j rj i ij Einstein equations Einstein equations in an arbitrary gauge Recovering Newtonian ﬂuid equations I 4 scalars: A, B, C, E Redshift-space distortions Recovering Newtonian I 2 vectors: Si , Fi ﬂuid equations I 1 tensor: hij Theoretical Metric perturbations cosmology David Wands

Homogeneous cosmology I Perturbed line-element including only scalar Perturbation theory perturbations: Scalar perturbations Fourier transforms 2 µ ⌫ Power spectrum ds = gµ⌫dx dx Vector perturbations Tensor perturbations 2 2 i = a (⌧) (1 + 2A)d⌧ + 2(@i B)dx d⌧ Metric perturbations i j +[(1+2 C)ij + 2(@ij E)] dx dx Geometrical interpretation } Gauge dependence Particular gauges Conformal where four scalar perturbations are Newtonian/Longitudinal gauge I A = lapse perturbation Uniform-density gauge i Equating gauge-invariant I @i B = @B/@x = shift perturbation variables I C = spatial curvature perturbation Einstein equations 2 i j Einstein equations in an I @ij E = @ E/@x @x =o↵-diagonal spatial perturbation arbitrary gauge Recovering Newtonian fluid equations Redshift-space distortions Recovering Newtonian fluid equations Theoretical Geometrical interpretation cosmology David Wands I Temporal gauge (time-slicing) in 4D spacetime defines a hypersurface orthogonal 4-vector field: Homogeneous cosmology @⌧ Perturbation theory Nµ µ / @x Scalar perturbations µ Fourier transforms normalise such that NµN = 1. Power spectrum Vector perturbations I intrinsic curvature of constant ⌧ hypersurfaces: Tensor perturbations 4 Metric (3) 2 perturbations R = 2 C a r Geometrical interpretation Gauge dependence I expansion of constant ⌧ hypersurfaces: Particular gauges Conformal Newtonian/Longitudinal 3 a0 1 2 gauge ✓ = (1 A)+C 0 + Uniform-density gauge a a 3r Equating gauge-invariant ✓ ◆ variables I shear: Einstein equations Einstein equations in an arbitrary gauge 1 2 ij = i j , = E 0 B Recovering Newtonian fluid r r 3r equations ✓ ◆ Redshift-space distortions Recovering Newtonian I acceleration: fluid equations a = A i ri u n t FRW cosmology preferred coordinates for homogeneous and isotropic space x

preferred space+time split in FRW cosmology breaks symmetry of Einstein’s theory u n t FRW cosmology

x no unique choice of time (slicing) and space coordinates (threading) in an inhomogeneous universe u t n FRW cosmology + perturbations

arbitrary gauge (t,x) x

gauge problem: find different perturbations in different gauges Theoretical Gauge dependence cosmology David Wands

Homogeneous I Scalar quantity, e.g., density, ⇢ P , at given point P is cosmology | invariant Perturbation ˜ theory ⇢(⌧, ~x) P =˜⇢(˜⌧,~x) P Scalar perturbations | | Fourier transforms Power spectrum under ﬁrst-order (scalar) change of coordinates: Vector perturbations Tensor perturbations

Metric ⌧˜ = ⌧ + ⌧(⌧, ~x) perturbations ˜ ~ Geometrical interpretation ~x = ~x + x(⌧, ~x) Gauge dependence r Particular gauges Conformal Newtonian/Longitudinal I but background-perturbation split is gauge-dependent gauge Uniform-density gauge Equating gauge-invariant ⇢0(⌧)+⇢ P = ⇢0(˜⌧)+⇢ P variables | | Einstein equations ⇢ P = ⇢ P + ⇢0(⌧) ⇢0(˜⌧) Einstein equations in an ) | | arbitrary gauge e Recovering Newtonian fluid = ⇢ P ⇢0⌧ (3) equations e | Redshift-space distortions Recovering Newtonian fluid equations Theoretical Linear gauge transformation rules cosmology David Wands I Coordinate change: Homogeneous time-slicing:⌧ ˜ ⌧ + ⌧(⌧, ~x) cosmology ! ˜ Perturbation spatial-threading: ~x ~x + ~ x(⌧, ~x) theory ! r Scalar perturbations Fourier transforms I Gauge transformations: Power spectrum Vector perturbations density: ⇢ = ⇢ ⇢0⌧ Tensor perturbations Metric pressure: P = P P0⌧ perturbations ei i i Geometrical interpretation velocity:v ˜ = v + @ x (4) Gauge dependence Particular gauges f Conformal Newtonian/Longitudinal including three metric transformations independent of gauge spatial-threading: Uniform-density gauge Equating gauge-invariant variables a ˜ 0 Einstein equations lapse: A = A ⌧ ⌧0 Einstein equations in an a arbitrary gauge Recovering Newtonian fluid ˜ a0 equations curvature: C = C ⌧ Redshift-space distortions a Recovering Newtonian fluid equations shear: ˜ = E˜0 B˜ = ⌧ (5) u n t FRW cosmology

x synchronous+comoving with pressureless cold dark matter time-slicing orthogonal to comoving worldlines u t n FRW cosmology + perturbations

comoving-Lagrangian q coordinates (t,q) Theoretical Conformal Newtonian/Longitudinal gauge cosmology David Wands

I pick a gauge to completely ﬁx the coordinates Homogeneous cosmology

I for example: longitudinal gauge (zero-shear time-slices): Perturbation theory Scalar perturbations I set ˜ =0whichrequiresatransform⌧ = Fourier transforms ! Power spectrum I we then have Vector perturbations Tensor perturbations ⇢ ⇢ density: = 0 Metric ⌘ ⇢ ! ⇢ perturbations Geometrical interpretation (6) Gauge dependence e Particular gauges Conformal Newtonian/Longitudinal including two gauge-invariant metric perturbations: gauge Uniform-density gauge Equating gauge-invariant a0 variables lapse: A A 0 ! ⌘ a Einstein equations Einstein equations in an a0 arbitrary gauge curvature: C C (7) Recovering Newtonian ﬂuid ! ⌘ a equations Redshift-space distortions Recovering Newtonian ﬂuid equations u n t FRW cosmology

Poisson = conformal Newtonian = longitudinal gauge hypersurface-orthogonal 4-vector field n is shear-free u t’ n FRW cosmology + perturbations Poisson gauge x coordinates (t’,x) u n t FRW cosmology

time-slicing orthogonal to comoving worldlines spatial threading is same as Poisson gauge (Eulerian, not Lagrangian) u t n FRW cosmology + perturbations total-matter x coordinates (t,x) Standard Newtonian+Gaussian initial fields Gaussian primordial metric fluctuations ζ(x) from inflation + linear Einstein-Boltzmann code (e.g., CMBfast, CAMB, CLASS) Gaussian initial Newtonian potential =(3/5)⇣

Gaussian initial matter density using Poisson equation 2 =4⇡Ga2⇢¯ r Gaussian initial displacement ~ . ~ = r Newtonian N-body simulations 2 =4⇡Ga2⇢¯ r ˙ + ~ .((1 + ))~v =0 r ~v˙ + ~v +(~v.~ )~v = ~ H r r Theoretical Uniform-density gauge cosmology David Wands I pick a gauge to completely fix the coordinates I for example: uniform-density time-slices: Homogeneous cosmology I set ⇢ ⇢ =0whichrequiresatransform⌧ = ⇢/⇢0 ! Perturbation I we then have theory e Scalar perturbations density: ⇢ ⇢ =0 Fourier transforms ! Power spectrum 2 Vector perturbations pressure: P P = Pnad P cs ⇢ (8) Tensor perturbations ! e ⌘ where c2 = P /⇢ = adiabatic sound speed. Metric s 0 0 f perturbations I gauge-invariant metric perturbation: Geometrical interpretation Gauge dependence a ⇢ Particular gauges 0 Conformal curvature: C ⇣ C Newtonian/Longitudinal ! ⌘ a ⇢0 gauge Uniform-density gauge Equating gauge-invariant variables I more generally, for any fluid with density ⇢↵(⌧, ~x)we can identify the curvature perturbation on Einstein equations Einstein equations in an uniform-↵-density time-slices: arbitrary gauge Recovering Newtonian fluid equations a0 ⇢↵ Redshift-space distortions ⇣↵ C Recovering Newtonian fluid equations ⌘ a ⇢↵0 Theoretical Equating gauge-invariant variables cosmology David Wands

Homogeneous I Gauge-invariant variables are not unique, and they are cosmology not independent Perturbation theory I for example, the curvature on uniform-density Scalar perturbations Fourier transforms time-slices can be written in terms of the longitudinal Power spectrum Vector perturbations gauge metric potential and density contrast: Tensor perturbations Metric 1 perturbations ⇣↵ + ↵ Geometrical interpretation ⌘ 3(1 + w↵) Gauge dependence Particular gauges Conformal Newtonian/Longitudinal for example, for radiation and non-relativistic matter: gauge Uniform-density gauge Equating gauge-invariant 1 variables ⇣ + ⌘ 4 Einstein equations Einstein equations in an 1 arbitrary gauge Recovering Newtonian ﬂuid ⇣m + m (9) equations ⌘ 3 Redshift-space distortions Recovering Newtonian ﬂuid equations Theoretical multiple component cosmology cosmology David Wands

Homogeneous cosmology Local energy conservation Perturbation I theory Scalar perturbations Fourier transforms ⇢↵0 = 3(1 + w↵)(a0/a)⇢↵ Power spectrum Vector perturbations Tensor perturbations where w = P /⇢ leads to ↵ ↵ ↵ Metric perturbations 1 ⇢↵ Geometrical interpretation ⇣↵ = C + Gauge dependence 3(1 + w ) ⇢ Particular gauges ↵ ↵ Conformal Newtonian/Longitudinal gauge Uniform-density gauge I curvature perturbation, C, on ⇢↵ =0time-slices Equating gauge-invariant variables density perturbation, ⇢ /⇢ , on C =0time-slices I ↵ ↵ Einstein equations I conserved for barotropic fluids, P↵(⇢↵), on large scales. Einstein equations in an arbitrary gauge Recovering Newtonian fluid equations Redshift-space distortions Recovering Newtonian fluid equations Theoretical multiple component cosmology cosmology David Wands I Primordial plasma (e.g., at epoch of primordial nucleosynthesis when T 1 MeV) Homogeneous ⇡ cosmology I photons, baryons, neutrinos, cold dark matter + dark Perturbation energy? theory Scalar perturbations I total curvature/dimensionless density perturbation: Fourier transforms Power spectrum Vector perturbations 1+w↵ Tensor perturbations ⇣ = ⇣↵ 1+w Metric ↵ X perturbations Geometrical interpretation conserved on large scales for adiabatic perturbations Gauge dependence Particular gauges I isocurvature/relative entropy perturbation: Conformal Newtonian/Longitudinal gauge Uniform-density gauge S↵ =3(⇣↵ ⇣ ) Equating gauge-invariant variables for example, matter-isocurvature perturbation: Einstein equations I Einstein equations in an arbitrary gauge ⇢m 3 ⇢ Recovering Newtonian fluid Sm =3(⇣m ⇣ )= equations ⇢m 4 ⇢ Redshift-space distortions Recovering Newtonian fluid equations conserved on large scales Cosmological perturbations on large scales æ n ö dn • adiabatic perturbations e.g., g g dnB d ç ÷ µ - = 0 è nB ø ng nB – perturb along the background trajectory y dx dy = = dT x! y! – e.g, single-field perturbations along slow-roll attractor x – adiabatic perturbations stay adiabatic • entropy perturbations

• perturb off the background trajectory y dx dy ¹ x! y! • e.g., baryon-photon isocurvature perturbation: x Conserved cosmological perturbations Lyth & Wands 2003

time t2

t1 A B

space For every quantity, x, that obeys a local conservation equation dx = y(x) , e.g. r! = -3Hr dN m m where dN = Hdt is the locally-defined expansion along comoving worldlines there is a conserved perturbation d x z º dN = x y(x) where perturbation d x = xA - xB is a evaluated on hypersurfaces separated by uniform expansion DN=Dlna examples: (i) total energy conservation: dr = H -1r! = -3(r + P) dN for perfect fluid / adiabatic perturbations, P=P(r) dr Þ z = conserved r 3(r + P)

(ii) energy conservation for non-interacting perfect fluids:

-1 dri H r!i = -3(ri + Pi ) where Pi = Pi (ri ) Þ z i = 3(ri + Pi )

(iii) conserved particle/quantum numbers (e.g., B, B-L,…)

-1 dni H n!i = -3ni Þ z i = 3ni microwave background signatures:

2 2 Cl = A ´ + B ´ + 2 A B cosD ´

adiabatic CDM isocurvature correlation

Bucher, Moodley & Turok ’00 n = 1 } S Trotta, Riazuelo & Durrer ’01 Amendola, Gordon, Wands & Sasaki ’01 best-fit to Boomerang, Maxima & DASI

B/A = 0.3, cosD = +1, nS = 0.8

wb = 0.02, wcdm = 0.1, WL = 0.7 Portsmouth o island city on the south coast of England o historic home of the Royal Navy o University of Portsmouth founded 1992 o Institute of Cosmology and Gravitation established in 2002 o 60+ researchers (academic staff, postdoctoral researchers, phd students and visiting researchers) Portsmouth o island city on the south coast of England o historic home of the Royal Navy o University of Portsmouth founded 1992 o Institute of Cosmology and Gravitation established in 2002 o 60+ researchers (academic staff, postdoctoral researchers, phd students and visiting researchers) Portsmouth o island city on the south coast of England o historic home of the Royal Navy o University of Portsmouth founded 1992 o Institute of Cosmology and Gravitation established in 2002 o 60+ researchers (academic staff, postdoctoral researchers, phd students and visiting researchers) Theoretical Einstein equations in an arbitrary gauge cosmology David Wands Evolution equations Homogeneous trace and trace-free spatial part of Einstein equations: cosmology

Perturbation 2 2 2 2 C 00 +2 C 0 A0 (2 0 + )A = 4⇡Ga P + ⇧ , theory H H H H 3r Scalar perturbations ✓ ◆ Fourier transforms 2 0 +2 A C =8⇡Ga ⇧ , Power spectrum H Vector perturbations Tensor perturbations

Energy+momentum constraints Metric time-time and time-space components: perturbations Geometrical interpretation 2 2 Gauge dependence 3 (C 0 A) (C )=4⇡Ga ⇢ , H H r H Particular gauges 2 Conformal C 0 A =4⇡Ga (⇢ + P)(v + B) . Newtonian/Longitudinal H gauge Uniform-density gauge Equating gauge-invariant Energy+momentum conservation variables

Fluid continuity and Euler equations: Einstein equations 2 Einstein equations in an ⇢0 +3 (⇢ + P) + 3(⇢ + P)C 0 +(⇢ + P) (v + E 0)=0, arbitrary gauge Recovering Newtonian ﬂuid H r equations 2 1 2 2 Redshift-space distortions (v + B)0 +(1 3cs ) (v + B)+A + P + ⇧ =0. H ⇢ + P 3r Recovering Newtonian ✓ ◆ ﬂuid equations Theoretical Constraint equations in conformal Newtonian cosmology gauge David Wands Homogeneous cosmology

A = , B =0 , C = , E =0 Perturbation theory Scalar perturbations Energy and momentum constraint equations Fourier transforms Power spectrum 2 2 Vector perturbations 3 (0 ) =4⇡Ga ⇢ , Tensor perturbations H H r 2 Metric 0 = 4⇡Ga (⇢ + P)V . perturbations H Geometrical interpretation Gauge dependence eliminate 0 gives Poisson equation: Particular gauges H Conformal Newtonian/Longitudinal 2 gauge = 4⇡G⇢ , Uniform-density gauge r2 c Equating gauge-invariant a variables Einstein equations where gauge-invariant comoving energy density (i.e., ⇢ in Einstein equations in an arbitrary gauge Recovering Newtonian ﬂuid v + B = 0 gauge) equations Redshift-space distortions Recovering Newtonian ⇢ ⇢ +3 (⇢ + P)V ﬂuid equations c ⌘ H Theoretical Fluid equations in arbitrary gauge cosmology Fluid continuity and Euler equations: David Wands 2 ⇢0 +3 (⇢ + P) + 3(⇢ + P)C 0 +(⇢ + P) (v + E 0)=0, Homogeneous H r cosmology 2 1 2 2 (v + B)0 +(1 3cs ) (v + B)+A + P + ⇧ =0. Perturbation H ⇢ + P 3r theory ✓ ◆ Scalar perturbations Fourier transforms For zero pressure perturbations (e.g., ⇤CDM) Power spectrum Vector perturbations 2 Tensor perturbations ⇢0 +3 ⇢ +3⇢C 0 + ⇢ (v + E 0)=0, H r Metric (v + B)0 + (v + B)+A =0. perturbations H Geometrical interpretation Gauge dependence In comoving gauge (v + B = 0) Particular gauges Conformal Newtonian/Longitudinal 2 gauge ⇢c0 +3 ⇢c +3⇢Cc0 + ⇢ V =0, H r Uniform-density gauge Equating gauge-invariant Ac =0. variables

Einstein equations and momentum constraint then reduces to Cc0 = 0. Einstein equations in an arbitrary gauge In conformal Newtonian gauge (E = B = 0) Euler equation Recovering Newtonian fluid equations becomes Redshift-space distortions Recovering Newtonian fluid equations V 0 + V + =0. H Theoretical Recovering Newtonian fluid equations cosmology Changing to density contrast David Wands

⇢ Homogeneous cosmology ⌘ ⇢ Perturbation theory we then have for pressureless matter Scalar perturbations Fourier transforms I Poisson equation for conformal Newtonian potential: Power spectrum Vector perturbations Tensor perturbations 2 3 2 = c Metric r 2H perturbations Geometrical interpretation Gauge dependence I Continuity equation for comoving density: Particular gauges Conformal 2 Newtonian/Longitudinal c0 + V =0 gauge r Uniform-density gauge Equating gauge-invariant I Euler equation for conformal Newtonian velocity (V~ = ~ V ): variables r Einstein equations Einstein equations in an V~ 0 + V~ = ~ arbitrary gauge Recovering Newtonian fluid H r equations Redshift-space distortions Coincide exactly to first-order Newtonian perturbation equations Recovering Newtonian in an expanding cosmology, using = for zero anisotropic fluid equations stress. Theoretical Redshift-space distortions cosmology David Wands

Homogeneous cosmology

Perturbation Continuity equation for comoving density contrast: theory Scalar perturbations Fourier transforms 2 Power spectrum c0 + V =0 r Vector perturbations Tensor perturbations 2 divergence of peculiar velocities, ✓ V , seen in galaxy Metric ⌘r perturbations redshift surveys Geometrical interpretation 2 2 2 Gauge dependence ✓ = f c Particular gauges h i h i Conformal Newtonian/Longitudinal Important probe of growth of structure given by f 8,where gauge 2 Uniform-density gauge is the variance of the matter power spectrum on 8 Mpc Equating gauge-invariant 8 variables scales ( 1). ⇡ Einstein equations Einstein equations in an arbitrary gauge Recovering Newtonian ﬂuid equations Redshift-space distortions Recovering Newtonian ﬂuid equations Theoretical Second-order equation for cosmology Continuity equation for comoving density contrast: David Wands

2 Homogeneous 0 + V =0 c r cosmology Perturbation Taking time derivative theory 2 Scalar perturbations c00 + V 0 =0 Fourier transforms r Power spectrum Vector perturbations plus Euler equation Tensor perturbations

Metric V 0 + V = perturbations H Geometrical interpretation eliminating V 0 and V gives Gauge dependence Particular gauges 2 Conformal c00 + c0 =0 Newtonian/Longitudinal H r gauge Uniform-density gauge Using = and Poisson equation Equating gauge-invariant variables 2 3 2 Einstein equations = c Einstein equations in an r 2H arbitrary gauge Recovering Newtonian fluid gives linear second-order di↵erential equation equations Redshift-space distortions Recovering Newtonian 3 2 fluid equations 00 + 0 =0 c H c 2H c Theoretical Growth of structure cosmology In absence of any pressure perturbations, e.g., ⇤CDM David Wands 3 Homogeneous 2 cosmology c00 + c0 c =0 H 2H Perturbation theory Growth of structure is independent of scale in ⇤CDM Scalar perturbations Fourier transforms cosmology. Power spectrum Vector perturbations k (⌧)=D+(⌧)k,0 + D (⌧)˜k,0 Tensor perturbations Metric perturbations Geometrical interpretation Gauge dependence I Growing mode D+(⌧)normalisedsothatD+(⌧0)=1 Particular gauges Conformal today Newtonian/Longitudinal gauge I Decaying mode D (⌧) assumed negligible at late times Uniform-density gauge Equating gauge-invariant variables I In matter-dominated cosmology (⌦m = 1) then D+ a / Einstein equations Einstein equations in an d ln D+ arbitrary gauge f =1 Recovering Newtonian fluid ⌘ d ln a equations Redshift-space distortions Recovering Newtonian I In ⇤CDM fluid equations 6/11 f ⌦m '