Primordial non-Gaussianity in the cosmological perturbations - CBPF

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The energy (number) density scales like the inverse of the volume whose size is ∼ a 3 On the other hand, if the universe is dominated by, say, radiation, then one has the equation of state P = ρ/3, then ρr a 4 = constant (RADIATION). (53) The energy density scales the like the inverse of the volume (whose size is ∼ a 3 ) and the energy which scales like 1/a because of the red-shift: photon energies scale like the inverse of their wavelenghts which in turn scale like 1/a. More generally, for matter with equation of state parameter w, one finds ρ a 3(1+w) = constant. (54) In particular, for w = −1, ρ is constant and corresponds, as we will see more explicitly below, to a cosmological constant vacuum energy ρΛ = constant (VACUUM ENERGY). (55) The early universe was radiation dominated, the adolescent universe was matter dominated and the adult universe is dominated, as we shall see, by the cosmological constant. If the universe underwent inflation, there was again a very early period when the stress-energy was dominated by vacuum energy. As we shall see next, once we know the evolution of ρ and P in terms of the scale factor a(t), it is straightforward to solve for a(t). Before going on, we want to emphasize the utility of describing the stress energy in the universe by the simple equation of state P = wρ. This is the most general form for the stress energy in a FRW space-time and the observational evidence indicates that on large scales the RW metric is quite a good approximation to the space-time within our Hubble volume. This simple, but often very accurate, approximation will allow us to explore many early universe phenomena with a single parameter. 2.2 The Friedmann equations After these preliminaries, we are now prepared to tackle the Einstein equations. We allow for the presence of a cosmological constant and thus consider the equations It will be convenient to rewrite these equations in the form Rµν = 8πGN Gµν + Λgµν = 8πGNTµν. (56) Tµν − 1 2 gµνT λ λ 20 + Λgµν. (57)
Because of isotropy, there are only two independent equations, namely the 00-component and any one of the non-zero ij-components. Using Eqs. (5) we find ä a −3 ä a = 4πGN(ρ + 3P ) − Λ , ˙a2 k + 2 + 2 a2 a2 = 4πGN(ρ − P ) + Λ. (58) Using the first equation to eliminate ä from the second, one obtains the set of equations for the Hubble rate for the acceleration H 2 + k 8πGN = ρ + a2 3 Λ 3 ä a = −4πGN (ρ + 3P ) − 3 Λ 3 , (59) . (60) Together, this set of equation is known as the Friedman equations. They are supplemented this by the conservation equation (51). Note that because of the Bianchi identities, the Einstein equations and the conservation equations should not be independent, and indeed they are not. It is easy to see that (59) and (51) imply the second order equation (60) so that, a pleasant simplification, in practice one only has to deal with the two first order equations (59) and (51). Sometimes, however, (60) is easier to solve than (59), because it is linear in a(t), and then (59) is just used to fix one constant of integration. Notice that Eqs. (59) and (60) can be obtained, in the non-relativistic limit P = 0 from Newtonian physics. Imagine that the distribution of matter is uniform and its matter density is ρ. Put a test particle with mass m on a surface of a sphere of radius a and let gravity act. The total energy is constant and therefore Ekin + Epot = 1 2 m ˙a2 mM − GN a = κ = constant. (61) Since the mass M contained in a sphere of radius a is M = (4πρa 3 /3), we obtain 1 2 m ˙a2 − 4πGN 3 ma2 = κ = constant. (62) By divinding everything by (ma 2 /2) we obtain Eq. (59) with of course no cosmological constant and after setting k = 2κ/m. Eq. (60) can be analogously obtained from Newton’s law relating the gravitational force and the acceleration (but still with P = 0). The expansion rate of the universe is determined by the Hubble rate H which is not a constant and generically scales like t −1 . The Friedmann equation (59) can be recast as 21
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: For relativistic particles, radiati
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
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
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Neglecting the terms − 2 Φ · X
Page 75 and 76:
for which the first-order perturbat
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In components we have δT ij = δT0
Page 79 and 80:
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
Page 87 and 88:
ackground solution and are therefor
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to slow-roll parameters), that is
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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
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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
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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:
Page 141 and 142:
Thus, we define the Fisher matrix F
Page 143 and 144:
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
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
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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
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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
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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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