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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 Einstein equations Einstein equations in an arbitrary gauge Recovering Newtonian fluid equations Redshift-space distortions Recovering Newtonian fluid equations Theoretical Outline cosmology David Wands 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 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 ✓dφ Newtonian/Longitudinal 1 Kr 2 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 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 ✓dφ 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 ✓dφ 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)+ " δ2⇢(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)+ " δ2⇢(t, ~x)+... Tensor perturbations 2 Metric 1 2 perturbations ~v(t, ~x)= "1~v(t, ~x)+ " δ2~v(t, ~x)+... Geometrical interpretation 2 Gauge dependence Particular gauges substitute into Eq. (1) Conformal Newtonian/Longitudinal gauge Uniform-density gauge @ 1 2 ⇢¯ + " ⇢ + " δ ⇢ + ... Equating gauge-invariant @t 1 2 2 variables ✓ ◆ Einstein equations 1 2 Einstein equations in an +~ ⇢¯ + " ⇢ + " δ ⇢ + ... arbitrary gauge 1 2 Recovering Newtonian fluid r · 2 equations ✓ ◆ Redshift-space distortions 1 2 Recovering Newtonian " ~v + " δ ~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 +" δ1⇢ + ~ (¯⇢1~v) Power spectrum @t r · Vector perturbations ⇢ Tensor perturbations 1 2 @ ~ Metric + " δ2⇢ + (¯⇢2~v +2δ1⇢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 δ1⇢ + C δ1~v =0 arbitrary gauge @t r · Recovering Newtonian fluid equations @ ~ ~ Redshift-space distortions δ2⇢ + C δ2~v = 2C (δ1⇢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⇡) δ k1 k2 Equating gauge-invariant − variables Z ⇣ ⌘ Einstein equations Coefficient 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
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