Primordial non-Gaussianity in the cosmological perturbations - CBPF

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we learn that during the de Sitter phase ρ = constant, HI = constant, where we have indicated by HI the value of the Hubble rate during inflation. Correspondingly, we obtain a = aI e HI(t−tI) , (164) where tI denotes the time at which inflation starts. Let us now see how such a period of exponential expansion takes care of the shortcomings of the standard Big Bang Theory. 3 6.1 Inflation and the horizon Problem During the inflationary (de Sitter) epoch the horizon scale H −1 I is constant. If inflation lasts long enough, all the physical scales that have left the Hubble radius during the RD or MD phase can re-enter the Hubble radius in the past: this is because such scales are exponentially reduced. Indeed, while during inflation the particle horizon grow exponential t RH(t) = a(t) while the Hubble radius remains constant tI dt ′ a(t ′ ) = aI e HI(t−tI) − 1 e HI −HI(t−tI) t tI a(t) , (165) HI HUBBLE RADIUS = a = H−1 I , (166) ˙a and points that our causally disconnected today could have been in contact during inflation. Notice that in comoving coordinates the comoving Hubble radius shrink exponentially COMOVING HUBBLE RADIUS = H −1 I e−HI(t−tI) , (167) while comoving length scales remain constant. An illustration of the solution to the horizon problem can therefore be visualized as in Fig. 9. As we have seen in the previous section, this explains both the problem of the homogeneity of CMB and the initial condition problem of small cosmological perturbations. Once the physical length is within the horizon, microphysics can act, the universe can be made approximately homogeneous and the primaeval inhomogeneities can be created. 3 Despite the fact that the growth of the scale factor is exponential and the expansion is superluminal, this is not in contradiction with what dictated by relativity. Indeed, it is the space-time itself which is progating so fast and not a light signal in it. 46
Figure 9: The solution of the horizon problem by inflation in comoving coordinates Let us see how long inflation must be sustained in order to solve the horizon problem. Let tI and tf be, respectively, the time of beginning and end of inflation. We can define the corresponding number of e-foldings N N = ln [HI(te − tI)] . (168) A necessary condition to solve the horizon problem is that the largest scale we observe today, the present horizon H −1 0 , was reduced during inflation to a value λH0 (tI) smaller than the value of horizon length H −1 I during inflation. This gives atI λH0 (tI) = H −1 0 atf at0 atf = H −1 0 T0 Tf e −N < ∼ H −1 I , where we have neglected for simplicity the short period of MD and we have called Tf the temperature at the end of inflation (to be indentified with the reheating temperature TRH at the beginning of the RD phase after inflation, see later). We get T0 N > ∼ ln − ln H0 Tf Apart from the logarithmic dependence, we obtain N > ∼ 70. 6.2 Inflation and the flateness problem HI Tf ≈ 67 + ln . HI Inflation solves elegantly the flatness problem. Since during inflation the Hubble rate is constant Ω − 1 = k a2 1 ∝ . H2 a2 On the other end the condition (150) tells us that to reproduce a value of (Ω0 − 1) of order of unity today the initial value of (Ω − 1) at the beginning of the RD phase must be |Ω − 1| ∼ 10 −60 . Since we identify the beginning of the RD phase with the beginning of inflation, we require |Ω − 1| t=tf ∼ 10−60 . 47
Page 1 and 2: Abstract Primordial non-Gaussianity
Page 3 and 4: Contents I Introduction 6 1 The Fri
Page 5 and 6: 12 Symmetries of the de Sitter geom
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: Notice that is equivalent to requir
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
Page 61 and 62: averaged over some macroscopically
Page 63 and 64: we can write the equation for the f
Page 65 and 66: In fact we can do much better, sinc
Page 67 and 68: the power spectrum can be defined a
Page 69 and 70: with 3 − 2νχ ηχ. Using Eq. (
Page 71 and 72: 8.1 The metric fluctuations The mat
Page 73 and 74: Neglecting the terms − 2 Φ · X
Page 75 and 76: for which the first-order perturbat
Page 77 and 78: In components we have δT ij = δT0
Page 79 and 80: Figure 15: In the reference unpertu
Page 81 and 82: Φ = Φ − ξ 0′ − a′ a ξ0
Page 83 and 84: is the comoving curvature perturbat
Page 85 and 86: 8.9 Comments about gauge invariance
Page 87 and 88: ackground solution and are therefor
Page 89 and 90: to slow-roll parameters), that is
Page 91 and 92: which reproduces our previous findi
Page 93 and 94: we get Similarly, we obtain u µ =
Page 95 and 96: From (325) we find therefore −Ψ(
Page 97 and 98:
where the ∗ stands for the value
Page 99 and 100:
9.1 Gauge-invariant computation of
Page 101 and 102:
∇ 2 H ΦGI = 4 π GN a 2 ΦGI H
Page 103 and 104:
with nR − 1 = 3 − 2ν = 2η −
Page 105 and 106:
perturbations ζφ and ζγ ζ ′
Page 107 and 108:
∆ + ΦMD (τls) = − 4 SW 2 + 1
Page 109 and 110:
Take for instance a model of chaoti
Page 111 and 112:
The de Sitter algebra SO(1,4) has t
Page 113 and 114:
which, for de Sitter space, they pr
Page 115 and 116:
isocurvature mode associated with t
Page 117 and 118:
that are not accessible to any coll
Page 119 and 120:
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
Page 125 and 126:
S SR (k1, k2, k3) ∝ (ɛ − 2η)
Page 127 and 128:
long-wavelength fluctuations ζL is
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defines the relative contribution o
Page 131 and 132:
13.4.4 A test of multi-field models
Page 133 and 134:
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
Page 139 and 140:
where we have used the identity:
Page 141 and 142:
Thus, we define the Fisher matrix F
Page 143 and 144:
S(ℓ, k z τ0 , τ) = 0 dτ (τ0
Page 145 and 146:
Figure 21: Signal-to-Noise ratio fo
Page 147 and 148:
is the transformation for modes whi
Page 149 and 150:
theory breaks down either when we g
Page 151 and 152:
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,
Page 157 and 158:
We thus obtain δL(π) 1.063 for t
Page 159 and 160:
15.1 The computation of the halo ma
Page 161 and 162:
S. We may refer to the evolution of
Page 163 and 164:
so the two-point correlator is give
Page 165 and 166:
16 The bias In order to make full u
Page 167 and 168:
where ν = δc/S 1/2 = δc/σ(M) as
Page 169 and 170:
and we follow them for a “time”
Page 171 and 172:
can expand the three-point correlat
Page 173 and 174:
Figure 26: The ratio between the ha
Page 175 and 176:
ρ → n 1 + δn (1 + δℓ) (774)
Page 177 and 178:
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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