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Cosmological Perturbations Lecture 3 With Big Bang nucleosynthesis theory and observations we are confident of the theory of the early Universe at temperatures up to T 1 MeV, age t 1 second ! ! With the LHC, we hope to be able to go up to temperatures T 100 GeV, age t 10 10 second ∼ ∼ − Are we going to have a handle on even earlier epoch? Key: cosmological perturbations Our Universe is not exactly homogeneous. How are they measured? Inhomogeneities: density perturbations and associated $ gravitational potentials (3d scalar), observed; Cosmic microwave background: photographic picture of the gravitational waves (3d tensor), Universe at age 370 000 yrs, T = 3000 K $ not observed (yet?). Temperature anisotropy Polarization Today: inhomogeneities strong and non-linear Deep surveys of galaxies and quasars, cover good part of In the past: amplitudes small, entire visible Universe δρ 4 5 Gravitational lensing, etc. = 10− 10− ρ − Linear analysis appropriate. Overall consistency We have already learned a number of fundamental things Extrapolation back in time with known laws of physics and known elementary particles and fields = hot Universe, starts from Big Bang singularity (infinite temperature,⇒ infinite expansion rate) We know that this is not the whole story! NB: density perturbations = random field. k =wavenumber P(k) =powerspectrumtransferedtopresentepoch using linear theory Wavelength of perturbation grows as a(t). Properties of perturbations in conventional (“hot”) Universe. E.g., at radiation domination ˆ Friedmann–Lemaitre–Robertson–Walker metric: 1/2 λ(t) ∝ t while lH,t ∝ t ds2 = dt2 a2(t)d!x 2 − Today λ < lH,subhorizonregime a(t) ∝ t1/2 at radiation domination stage (before T 1 eV, Early on λ(t) > lH,superhorizonregime. t 60 thousand years) ! ! a(t) ∝ t2/3 at matter domination stage (until recently). Cosmological horizon at time t (assuming that nothing preceeded hot epoch): distance that light travels from Big Bang moment, 1 l H− (t) t H,t ∼ ∼ && superhorizon mode subhorizon mode Regimes at radiation (and matter) domination In other words, physical wavenumber (momentum) gets redshifted, 2π k q(t)= = , k = const = coordinate momentum λ(t) a(t) H(t) Today a˙ q > H ≡ a Early on q2(t) q(t) < H(t) q1(t) Very different regimes of evolution. superhorizon subhorizon NB: Horizon entry occured after Big Bang Nucleosynthesis epoch t t for modes of all relevant wavelengths no guesswork at this × point. ⇐⇒ q2 > q1 Major issue: origin of perturbations Perturbations in baryon-photon plasma = sound waves. Causality = perturbations can be generated only when they are If they were superhorizon, they started off with one and the same subhorizon.⇒ phase. Off-hand possibilities: Prototype example: wave equation in expanding Universe (not exactly the same as equation for sound waves, but captures Perturbations were never superhorizon, they were generated main properties). at the hot cosmological epoch by some causal mechanism. Massless scalar field φ in FLRW spacetime: action E.g., seeded by topological defects (cosmic strings, etc.) 1 4 µν The only possibility, if expansion started from hot Big Bang. S = d x √ gg ∂µ φ∂ν φ 2 − ! 2 2 2 No longer an option! gµν =(1, a , a , a ):spacetimemetric; µν − 2− −2 2 g =(1, a− , a− , a− ):itsinverse; − −6 − g = det (gµν)=a :itsdeterminant Hot epoch was preceeded by some other epoch. (d4x√ g:invariant4-volumeelement). Perturbations were generated then. − Wave equation in momentum space: 1 1 S = d3xdt a3(t) φ˙2 !∂φ !∂φ 2 − a2 · k2 ! " # ¨ ˙ φ + 3H(t)φ + 2 φ = 0 Field equation a (t) a˙ 1 φ¨ + 3 φ˙ Δφ = 0 Redshift effect: frequency ω(t)=k/a(t). a − a2 Hubble friction: the second term. NB. a˙/a = H: Hubble parameter. As promised, evoltion is different for k/a > H (subhorizon regime) Fourier decomposition in 3d space and k/a < H (superhorizon regime). Subhorion regime (late times): damped oscillations 3 i!k!x φ(!x,t)= d k e φ!k(t) t ! A! k φ (t)= k cos dt + ψ ! !k a(t) a(t) NB. k: coordinate momentum, constant in time. "!0 # Physical momentum q = k/a(t) gets redshifted. NB. Subhorizon sound waves in baryon-photon plasma: –Amplitudeofδρ/ρ does not decrease –Soundwavevs different from 1 (vs 1/√3). All the rest is the same ≈ Solution to wave equation in superhorizon regime (early times) at radiation domination Perturbations come to the time of photon last scattering (=recombination)atdifferentphases,dependingonwavevector: const φ = const and φ = t t3/2 δρ r vs δ(tr) (tr) ∝ cos k dt = cos(krs) ≡ ρ 0 a(t) Constant and decaying modes. " ! # NB: decaying mode is sometimes called growing, it grows as t 0. → rs: sound horizon at recombination. Same story for density perturbations. Waves with k = π/r have large δρ ,whilewaveswith s | | k =(π + 1/2)/r have δρ = 0 in baryon-photon component. δρ/ρ ∝ t 3/2: very inhomogeneous Universe at early times = s − This translates into oscillations| | in CMB angular spectrum inconsistency ⇒ Fourier decomposition of temperatue fluctuations: Under assumption that modes were superhorizon, the initial condition is unique (up to overall amplitude), δT (θ,ϕ)=∑almYlm(θ,ϕ) l,m δρ d δρ = const = = 0 ρ ⇒dt ρ a a = C ,temperatureangularspectrum; - lm∗ lm/ l Acoustic oscillations start after entering the horizon at zero velocity larger l smaller angular scales, shorter wavelengths ⇐⇒ of medium = phase of oscillations uniquely defined; ψ = 0. ⇒ Furthermore, there are perturbations which were superhorizon at the time of photon last scattering (low multipoles, l ! 50) These properties would not be present if perturbations were generated at hot epoch in causal manner: phase ψ would be random function of k,nooscillationsinCMBangularspectrum. 6000 WMAP 5000 Abelian Higgs strings Semilocal strings ] Textures 2 4000 K µ [ π 3000 / 2 l l(l+1)C 2000 1000 0 1 2 3 10 10 10 multipole moment: l Primordial perturbations were generated at some yet unknown epoch before the hot expansion stage. Physical wave number and Hubble parameter at inflation and later: That epoch must have been long and unusual: perturbations were subhorizon early at that epoch, our visible part of the Universe was in a causally connected region. Excellent guess: inflation Starobinsky’79; Guth’81; Linde’82; Albrecht and Steinhardt’82 Exponential expansion with almost constant Hubble rate, a(t)=e Hdt , H const $ ≈ Perturbations subhorizon early at inflation: k q(t)= H a(t) 0 CMB angular spectra Alternatives to inflation: Contraction — Bounce — Expansion, Start up from static state. Difficult, but not impossible. Other suggestive observational facts about density perturbations (valid within certain error bars!) Perturbations in overall density, not in composition (jargon: “adiabatic”) baryon density dark matter density = = const in space entropy density entropy density homogeneous chemical composition inhomogeneous chemical composition Consistent with generation of baryon asymmetry and dark matter at hot stage. NB: even weak variation of composition over space would mean Perturbation in chemical composition (jargon: “isocurvature” exotic mechanism of baryon asymmetry and/or dark matter generation = watch out Planck! or “entropy”) = wrong initial condition for acoustic ⇒ oscillations = ⇒wrong prediction for CMB angular spectrum. ⇒ Primordial perturbations are Gaussian. Gaussian random field δ(k): correlators obey Wick’s theorem, Inflation does the job very well: fluctuations of all light fields δ(k )δ(k )δ(k ) = 0 - 1 2 3 / get enhanced greatly due to fast expansion of the Universe. δ(k )δ(k )δ(k )δ(k ) = δ(k )δ(k ) δ(k )δ(k ) - 1 2 3 4 / - 1 2 /·- 3 4 / + permutations of momenta Including the field that dominates energy density (inflaton) = perturbations in energy density. ⇒ δ(k)δ ∗(k1) means averaging over ensemble of Universes. Mukhanov, Chibisov’81; Hawking’82; Starobinsky’82; Realization- in/ our Universe is intrinsically unpredictable. Guth, Pi’82; Bardeen et.al.’83 Hint on the origin: enhanced vacuum fluctuations of free quantum field Free quantum field Enhancement of vacuum fluctuations is less automatic in alternative scenarios 3 ikx (+) † ikx ( ) φ(x,t)= d ke− fk (t)ak + e fk− (t)ak ! % & ( ) iωkt In vacuo fk± (t)=e± ( ) Enhanced perturbations: large fk± . But in any case, Wick’s theorem valid Primordial power spectrum is flat (or almost flat). Non-Gaussianity: big issue Homogeneity and anisotropy of Gaussian random field: 1 Very small in the simplest inflationary theories δ(!k)δ(!k1) = P(k)δ(!k +!k1) - / 4πk3 Sizeable in more contrived inflationary models and in alternatives to inflation.Oftenbeginswithbispectrum P(k)=power spectrum, gives fluctuation in logarithmic (3-point function; vanishes for Gaussian field) interval of momenta, 2 δ(!k )δ(!k )δ(!k ) = δ(!k +!k +!k ) G(k2; !k !k ; !k !k ) δρ ∞ dk - 1 2 3 / 1 2 3 i 1 · 2 1 · 3 (!x) = P(k) - ρ / 0 k " # ! Shape of G(k2; !k !k ; !k !k ) different in different models i 1 · 2 1 · 3 = potential discriminator. Flat spectrum: P is independent of k ⇒ Harrison’ 70; Zeldovich’ 72 In some models bispectrum vanishes, e.g., due to some symmetries. But trispectrum (connected 4-point function) Parametrization ns 1 may be measurable. k − P(k)=A k Non-Gaussianity has not been detected yet " ∗ # A = amplitude, (ns 1)=tilt, k = fiducial momentum (matter of convention). Flat− spectrum ∗ n = 1. ⇐⇒ s Statistical anisotropy There must be some symmetry behind flatness of spectrum ! P P !uk kik j Inflation: symmetry of de Sitter space-time (k)= 0(k) 1 + + wij(k) 2 + ... ' k k ( ds2 = dt2 e2Htd!x 2 − !u, wij: fundamental vector, tensor in our part of the Symmetry: spatial dilatations supplemented by time Universe. translations Anisotropy of the Universe at pre-hot stage 1 !x λ!x , t t logλ Possible in inflation with strong vector fields (rather → → − 2H contrived). Inflation automatically generates nearly flat spectrum.
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