Primordial non-Gaussianity in the cosmological perturbations - CBPF

More documents

Recommendations

Info

For this reason, one can use the words horizon and Hubble radius interchangeably for standard cosmology. As we shall see, in inflationary models the horizon and Hubble radius are drastically different as the horizon distance grows exponentially relative to the Hubble radius; in fact, at the end of inflation they differ by e N , where N is the number of e-folds of inflation. The horizon sets the length scale for which two points separated by a distance larger than RH(t) they could never communicate, while the Hubble radius sets the scale at which these two points could not comunicate at the time t. Note also that a physical length scale λ is within the Hubble radius if λ < H −1 . Since we can identify the length scale λ with its wavenumber k, λ = 2πa/k, we will have the following rule k aH k aH Notice that in standard cosmology ≪ 1 =⇒ SCALE λ OUTSIDE THE HORIZON ≫ 1 =⇒ SCALE λ WITHIN THE HORIZON λ λ = = λ H ∼ PARTICLE HORIZON RH aH . (142) k This shows once more that Hubble radius and particle horizon can be used interchangeably in standard cosmology. 5 The shortcomings of the Standard Big-Bang Theory By now the shortcomings of the standard cosmology are well appreciated: the horizon or large-scale smoothness problem; the small-scale inhomogeneity problem (origin of density perturbations); and the flatness or oldness problem. We will only briefly review them here. They do not indicate any logical inconsistencies of the standard cosmology; rather, that very special initial data seem to be required for evolution to a universe that is qualitatively similar to ours today. Nor is inflation the first attempt to address these shortcomings: over the past two decades cosmologists have pondered this question and proposed alternative solutions. Inflation is a solution based upon well-defined, albeit speculative, early universe microphysics describing the post-Planck epoch. 5.1 The Flatness Problem Let us make a tremendous extrapolation and assume that Einstein equations are valid until the Plank era, when the temperature of the universe is TPl ∼∼ 10 19 GeV. From the equation for the 36
curvature Ω − 1 = k H2 , (143) a2 we read that if the universe is perfectly flat, then (Ω = 1) at all times. On the other hand, if there is even a small curvature term, the time dependence of (Ω − 1) is quite different. During a RD period, we have that H 2 ∝ ρr ∝ a −4 and During MD, ρNR ∝ a −3 and Ω − 1 ∝ 1 a 2 a −4 ∝ a2 . (144) Ω − 1 ∝ 1 a2 ∝ a. (145) a−3 In both cases (Ω − 1) decreases going backwards with time. Since we know that today (Ω0 − 1) is of order unity at present, we can deduce its value at tPl (the time at which the temperature of the universe is TPl ∼ 10 19 GeV) | Ω − 1 |T =TPl | Ω − 1 |T =T0 a2 Pl T 2 0 ≈ ≈ ≈ O(10 −64 ). (146) a 2 0 where 0 stands for the present epoch, and T0 ∼ 10 −13 GeV is the present-day temperature of the CMB radiation. If we are not so brave and go back simply to the epoch of nucleosynthesis when light elements abundances were formed, at TN ∼ 1 MeV, we get | Ω − 1 |T =TN | Ω − 1 |T =T0 a2 N T 2 0 ≈ ≈ ≈ O(10 −16 ). (147) a 2 0 In order to get the correct value of (Ω0 − 1) ∼ 1 at present, the value of (Ω − 1) at early times have to be fine-tuned to values amazingly close to zero, but without being exactly zero. This is the reason why the flatness problem is also dubbed the ‘fine-tuning problem’. 5.2 The Entropy Problem Let us now see how the hypothesis of adiabatic expansion of the universe is connected with the flatness problem. From the Friedman equations we know that during a RD period from which we deduce H 2 ρR T 2 Pl T 2 N T 4 M 2 Pl , (148) Ω − 1 = kM 2 Pl a4T 4 = kM 2 Pl S 2 . (149) 3 T 2 Under the hypothesis of adiabaticity, S is constant over the evolution of the universe and therefore |Ω − 1| t=tPl = M 2 Pl T 2 Pl 1 S 2/3 U 37 = 1 S 2/3 U ≈ 10 −60 , (150)
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: H = ˙a a a′ H = = a2 a , ˙a = a
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
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
Page 121 and 122:
• at second order 〈W (τ)〉 (2
Page 123 and 124:
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
Page 129 and 130:
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
Page 135 and 136:
to the higher-order statistics of t
Page 137 and 138:
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
Page 153 and 154:
function (sometimes called filter f
Page 155 and 156:
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
show all

Primordial non-Gaussianity in the cosmological perturbations - CBPF

Create successful ePaper yourself

Delete template?

Save as template?