Primordial non-Gaussianity in the cosmological perturbations - CBPF

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6.7.2 Small-field models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6.7.3 Hybrid models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 IV Inflation and the cosmological perturbations 59 7 Quantum fluctuations of a generic massless scalar field during inflation 62 7.1 Quantum fluctuations of a generic massless scalar field during a de Sitter stage . . . 62 7.2 Quantum fluctuations of a generic massive scalar field during a de Sitter stage . . . 65 7.3 Quantum to classical transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.4 The power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 7.5 Quantum fluctuations of a generic scalar field in a quasi de Sitter stage . . . . . . . 67 8 Quantum fluctuations during inflation 69 8.1 The metric fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 8.2 Perturbed affine connections and Einstein’s tensor . . . . . . . . . . . . . . . . . . . 73 8.3 Perturbed stress energy-momentum tensor . . . . . . . . . . . . . . . . . . . . . . . . 76 8.4 Perturbed Klein-Gordon equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 8.5 The issue of gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 8.6 The comoving curvature perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . 82 8.7 The curvature perturbation on spatial slices of uniform energy density . . . . . . . . 83 8.8 Scalar field perturbations in the spatially flat gauge . . . . . . . . . . . . . . . . . . . 84 8.9 Comments about gauge invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 8.10 Adiabatic and isocurvature perturbations . . . . . . . . . . . . . . . . . . . . . . . . 85 8.11 The next steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.12 Computation of the curvature perturbation using the longitudinal gauge . . . . . . . 88 8.13 A proof of time-independence of the comoving curvature perturbation for adiabatic modes: linear level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 8.14 A proof of time-independence of the comoving curvature perturbation for adiabatic modes: linear level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8.15 A proof of time-independence of the comoving curvature perturbation for adiabatic modes: all orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 9 Comoving curvature perturbation from isocurvature perturbation 96 9.1 Gauge-invariant computation of the curvature perturbation . . . . . . . . . . . . . . 99 10 Transferring the perturbation to radiation during reheating 103 11 The initial conditions provided by inflation 106 4
12 Symmetries of the de Sitter geometry 109 12.1 Killing vectors of the de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 13 Non-Gaussianity of the cosmological perturbations 114 13.1 The generation of non-Gaussianity in the primordial cosmological perturbations: generic considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 13.2 A brief Review of the in-in formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 13.3 The shapes of non-Gaussianity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 13.4 Theoretical Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 13.4.1 Single-Field Slow-Roll Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . 124 13.4.2 Models with Large Non-Gaussianity . . . . . . . . . . . . . . . . . . . . . . . 127 13.4.3 Multiple Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 13.4.4 A test of multi-field models of inflation . . . . . . . . . . . . . . . . . . . . . . 131 13.4.5 Non-Standard Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 V The impact of the non-Gaussianity on the CMB anisotropies132 13.5 Why do we expect NG in the cosmological perturbations? . . . . . . . . . . . . . . . 134 13.6 Primordial non-Gaussianity and the CMB anisotropies . . . . . . . . . . . . . . . . . 138 13.7 Non-Gaussianity in the CMB anisotropies at recombination in the squeezed limit . . 145 VI Matter perturbations 148 14 Spherical collapse 153 15 The dark matter halo mass function and the excursion set method 158 15.1 The computation of the halo mass function as a stochastic problem . . . . . . . . . . 159 16 The bias 165 VII The impact of the non-Gaussianity on the halo mass function 167 VIII The impact of the non-Gaussianity on the halo clustering172 IX Exercises 177 5
Page 1 and 2: Abstract Primordial non-Gaussianity
Page 3: Contents I Introduction 6 1 The Fri
Page 7 and 8: Figure 2: The CMB radiation project
Page 9 and 10: ds 2 = −dt 2 + a 2 dr2 (t) 1 −
Page 11 and 12: Now, let us introduce the polar coo
Page 13 and 14: It should be noted that the assumpt
Page 15 and 16: 1.3 Particle kinematics of a partic
Page 17 and 18: δt0 δt = . (37) a(t0) a(t) Theref
Page 19 and 20: For relativistic particles, radiati
Page 21 and 22: Because of isotropy, there are only
Page 23 and 24: the present epoch or at a generic i
Page 25 and 26: 4. For sufficiently large a, a nonz
Page 27 and 28: Figure 3: Qualitative behaviour of
Page 29 and 30: where µ is the chemical potential
Page 31 and 32: The total energy density and pressu
Page 33 and 34: s = S V = ρ + P T . (126) It is do
Page 35 and 36: H = ˙a a a′ H = = a2 a , ˙a = a
Page 37 and 38: curvature Ω − 1 = k H2 , (143)
Page 39 and 40: On the other hand, during the MD pe
Page 41 and 42: Figure 6: The CMBR anisotropy as fu
Page 43 and 44: Figure 7: An illustration of the ho
Page 45 and 46: Notice that is equivalent to requir
Page 47 and 48: Figure 9: The solution of the horiz
Page 49 and 50: always remain flat or closed, indep
Page 51 and 52: If we require that ˙ φ 2 0 ≪ V
Page 53 and 54: where ∆Nλ indicates the number o
Page 55 and 56:
as Bose condensates, and they behav
Page 57 and 58:
Figure 11: Large field models of in
Page 59 and 60:
V (φ) ∝ φ −p used in intermed
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averaged over some macroscopically
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we can write the equation for the f
Page 65 and 66:
In fact we can do much better, sinc
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the power spectrum can be defined a
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with 3 − 2νχ ηχ. Using Eq. (
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8.1 The metric fluctuations The mat
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Neglecting the terms − 2 Φ · X
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for which the first-order perturbat
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In components we have δT ij = δT0
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Figure 15: In the reference unpertu
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Φ = Φ − ξ 0′ − a′ a ξ0
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is the comoving curvature perturbat
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8.9 Comments about gauge invariance
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ackground solution and are therefor
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to slow-roll parameters), that is
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which reproduces our previous findi
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we get Similarly, we obtain u µ =
Page 95 and 96:
From (325) we find therefore −Ψ(
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where the ∗ stands for the value
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9.1 Gauge-invariant computation of
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∇ 2 H ΦGI = 4 π GN a 2 ΦGI H
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with nR − 1 = 3 − 2ν = 2η −
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perturbations ζφ and ζγ ζ ′
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∆ + ΦMD (τls) = − 4 SW 2 + 1
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Take for instance a model of chaoti
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The de Sitter algebra SO(1,4) has t
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which, for de Sitter space, they pr
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isocurvature mode associated with t
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that are not accessible to any coll
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our study of NG require going beyon
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• at second order 〈W (τ)〉 (2
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7.0 3.5 0 0.0 S local (1,x2,x3) 0.5
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S SR (k1, k2, k3) ∝ (ɛ − 2η)
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long-wavelength fluctuations ζL is
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defines the relative contribution o
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13.4.4 A test of multi-field models
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angular scales as density inhomogen
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to the higher-order statistics of t
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or ln ρ m = ln ρ 3/4 γ . (569) N
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where we have used the identity:
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Thus, we define the Fisher matrix F
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S(ℓ, k z τ0 , τ) = 0 dτ (τ0
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Figure 21: Signal-to-Noise ratio fo
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is the transformation for modes whi
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theory breaks down either when we g
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where in the second equality we hav
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function (sometimes called filter f
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universe so that ˙ Rm,i = HIRm,i,
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We thus obtain δL(π) 1.063 for t
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15.1 The computation of the halo ma
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S. We may refer to the evolution of
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so the two-point correlator is give
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16 The bias In order to make full u
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where ν = δc/S 1/2 = δc/σ(M) as
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and we follow them for a “time”
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can expand the three-point correlat
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Figure 26: The ratio between the ha
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ρ → n 1 + δn (1 + δℓ) (774)
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so that d ln ν/d ln M does not dep
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Primordial non-Gaussianity in the cosmological perturbations - CBPF

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