modern cosmology ingredient 2: fluid mechanics Björn Malte Schäfer Fakultät für Physik und Astronomie, Universität Heidelberg May 16, 2019
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation outline inflation 1 random processes 2 CMB 3 secondary anisotropies 4 random processes 5 large-scale structure 6 CDM spectrum 7 structure formation 8 modern cosmology Björn Malte Schäfer
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation expansion history of the universe expansion history of the universe modern cosmology Björn Malte Schäfer
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation Planck-scale • at a = 0, z = ∞ the metric diverges, and H ( a ) becomes infinite • description of general relativity breaks down, quantum effects become important • relevant scales: • quantum mechanics: de Broglie-wave length: λ QM = 2π ℏ mc • general relativity: Schwarzschild radius: r s = 2Gm c 2 • setting λ QM = r s defines the Planck mass ℏ c √ G ≃ 10 19 GeV / c 2 m P = (1) question how would you define the corresponding Planck length and the Planck time? what are their numerical values? modern cosmology Björn Malte Schäfer
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation flatness problem • construct a universe with matter w = 0 and curvature w = − 1 / 3 • Hubble function H 2 ( a ) = Ω m a 3 + Ω K (2) a 2 H 2 0 • density parameter associated with curvature H 2 H 2 Ω K ( a ) 0 0 (3) = a 3 ( 1 + w ) H 2 ( a ) = Ω K a 2 H 2 ( a ) • Ω K increases always and was smaller in the past ) − 1 1 + Ω m 1 ≃ Ω K ( Ω K ( a ) = a (4) a Ω K Ω m • we know (from CMB observations) that curvature is very small today, typical limits are Ω K < 0 . 01 → even smaller in the past modern cosmology Björn Malte Schäfer • at recombination Ω K ≃ 10 − 5 • at big bang nucleosynthesis Ω K 10 − 12
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation horizon problem • horizon size: light travel distance during the age of the universe da ∫ χ H = c (5) a 2 H ( a ) • assume Ω m = 1, integrate from a min = a rec . . . a max = 1 χ H = 2 c √ √ Ω m a rec = 175 Ω m Mpc / h (6) H 0 • comoving size of a volume around a point at recombination inside which all points are in causal contact • angular diameter distance from us to the recombination shell: d rec ≃ 2 c a rec ≃ 5Mpc / h (7) H 0 • angular size of the particle horizon at recombination: θ rec ≃ 2 ◦ • points in the CMB separated by more than 2 ◦ have never modern cosmology Björn Malte Schäfer been in causal contact → why is the CMB so uniform if there is no possibility of heat exchange?
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation inflation: phenomenology • curvature Ω K ∝ to the comoving Hubble radius c / ( aH ( a )) • if by some mechanism, c / ( aH ) could decrease, it would drive Ω K towards 0 and solve the fine-tuning required by the flatness problem • shrinking comoving Hubble radius: ( c d a = − c ¨ ) a 2 < 0 → ¨ a > 0 → q < 0 (8) dt aH ˙ • equivalent to the notion of accelerated expansion • accelerated expansion can be generated by a dominating fluid with sufficiently negative equation of state w = − 1 / 3 • horizon problem: fast expansion in inflationary era makes the universe grow from a small, causally connected region question what’s the relation between deceleration q and equation of modern cosmology Björn Malte Schäfer state w?
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation inflaton-driven expansion • analogous to dark energy, one postulates an inflaton field φ, with a small kinetic and a large potential energy, for having a sufficiently negative equation of state for accelerated expansion • pressure and energy density of a homogeneous scalar field φ 2 φ 2 p = ˙ ρ = ˙ 2 − V ( φ ) , 2 + V ( φ ) (9) • Friedmann equation φ 2 H 2 ( a ) = 8πG ˙ 2 + V ( φ ) (10) 3 • continuity equation φ = − dV φ + 3H ˙ (11) ¨ dφ modern cosmology Björn Malte Schäfer
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation slow roll conditions • inflation can only take place if ˙ φ 2 ≪ V ( φ ) • inflation needs to keep going for a sufficiently long time: d φ 2 ≪ d φ ≪ d dtV ( φ ) → ¨ dφV ( φ ) (12) dt ˙ • in this regime, the Friedmann and continuity equations simplify: H 2 = 8πG φ = − d 3 V ( φ ) , 3H ˙ dφV ( φ ) (13) • conditions are fulfilled if ) 2 1 ( V ′ 1 ( V ′′ ) ≡ ε ≪ 1 , ≡ η ≪ 1 (14) 24πG V 8πG V • ε and η are called slow-roll parameters modern cosmology Björn Malte Schäfer
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation stopping inflation • flatness problem: shrinkage by ≃ 10 30 ≃ exp ( 60 ) → 60 e -folds • due to the slow-roll conditions, the energy density of the inflaton field is almost constant • all other fluid densities drop by huge amounts, ρ m by 10 90 , ρ γ by 10 120 • eventually, the slow roll conditions are not valid anymore, the effective equation of state becomes less negative, acclerated expansion stops • but energy is stored in φ as kinetic energy ˙ φ 2 • reheating: couple φ to other particle fields, and generate particles from the inflaton’s kinetic energy • how exactly reheating occurs, is largely unknown modern cosmology Björn Malte Schäfer
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation generation of fluctuations • fluctuations of the inflation field can perturb the distribution of all other fluids • mean fluctuation amplitude is related to the variance of φ • fluctuations in φ perturb the metric, and all other fluids feel a perturbed potential • relevant quantity ⟨ δΦ 2 ⟩ ≃ H 2 √ (15) V which is approximately constant during slow-roll • Poisson-equation in Fourier-space k 2 Φ ( k ) = − δ ( k ) • variance of density perturbations: � 2 ∝ k 4 � � 2 ∝ k 3 P ( k ) � δ ( k ) � δΦ (16) � � � � � � � • defines spectrum P ( k ) of the initial fluctuations, P ( k ) ∝ k n with n ≃ 1 modern cosmology Björn Malte Schäfer • fluctuations are Gaussian, because of the central limit theorem
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation random fields • random process → probability density p ( δ ) dδ of event δ • alternatively: all moments ⟨ δ n ⟩ = dδ δ n p ( δ ) ∫ • in cosmology: • random events are values of the density field δ • outcomes for δ ( ⃗ x ) form a statistical ensemble at fixed ⃗ x • ergodic random processes: one realisation is consistent with p ( δ ) dδ • special case: Gaussian random field • only variance relevant modern cosmology Björn Malte Schäfer
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation characteristic function φ ( t ) • for a continuous pdf, all moments need to be known for reconstructing the pdf • reconstruction via characteristic function φ ( t ) (Fourier transform) ( itx ) n ( it ) n ∫ ∫ ⟨ x n ⟩ p φ ( t ) = dxp ( x ) exp ( itx ) = dxp ( x ) ∑ ∑ = n ! n ! n n (17) with moments ⟨ x n ⟩ = dxx n p ( x ) ∫ • Gaussian pdf is special: • all moments exist! (counter example: Cauchy pdf) • all odd moments vanish • all even moments are expressible as products of the variance • σ is enough to statistically reconstruct the pdf • pdf can be differentiated arbitrarily often (Hermite polynomials) • funky notation: φ ( t ) = ⟨ exp ( itx ) ⟩ modern cosmology Björn Malte Schäfer
inflation random processes CMB secondary anisotropies random processes large-scale structure CDM spectrum structure formation cosmic microwave background • inflation has generated perturbations in the distribution of matter • the hot baryon plasma feels fluctuations in the distribution of (dark) matter by gravity • at the point of (re)combination: • hydrogen atoms are formed • photons can propagate freely • perturbations can be observed by two effects: • plasma was not at rest, but flowing towards a potential well → Doppler-shift in photon temperature, depending to direction of motion • plasma was residing in a potential well → gravitational redshift • between the end of inflation and the release of the CMB, the density field was growth homogeneously → all statistical properties of the density field are conserved • testing of inflationary scenarios is possible in CMB modern cosmology Björn Malte Schäfer observations
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