Primordial non-Gaussianity in the cosmological perturbations - CBPF

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1.4 The cosmological redshift Without explicitly solving Einstein’s equations for the dynamics of the expansion, it is still possible to understand many of the kinematic effects of the expansion upon light from distant galaxies. The light emitted by a distant object can be viewed quantum mechanically as freely-propagating photons, or classically as propagating plane waves. In the quantum mechanical description, the wavelength of light is in- inversely proportional to the photon momentum λ = h/p. If the momentum changes, the wavelength of the light must change. It was shown in the previous section that the momentum of a photon changes in proportion to a −1 . Since the wavelength of a photon is inversely proportional to its momentum, the wavelength at time t0, denoted as λ0, will differ from that at time t1, denoted as λ1, by λ1 λ0 = a(t1) . (33) a(t0) As the universe expands, the wavelength of a freely-propagating photon increases, just as all physical distances increase with the expansion. This means that the red shift of the wavelength of a photon is due to the fact that the universe was smaller when the photon was emitted. It is also possible to derive the same result by considering the propagation of light from a distant galaxy as a classical wave phenomenon. Let us again place ourselves at the origin r = 0. We consider a radially travelling electro-magnetic wave (a light ray) and consider the equation ds 2 = 0 or dt 2 = a 2 (t) dr2 . (34) 1 − kr2 Let us assume that the wave leaves a galaxy located at r at time t. Then it will reach us at time t0 given by t0 t r dt = f(r) = a(t) 0 dr √ 1 − kr 2 = ⎧ ⎪⎨ ⎪⎩ sin −1 r = r + r 3 /6 + · · · (k = +1), r (k = 0), sinh −1 r = r − r 3 /6 + · · · (k = −1). As typical galaxies will have constant coordinates, f(r) (which can of course be given explicitly, but this is not needed for the present analysis) is time-independent. If the next wave crest leaves the galaxy at r at time (t + δt), it will arrive at time (t0 + δt0) given by r f(r) = 0 dr √ 1 − kr2 = t0+δt0 t+δt (35) dt . (36) a(t) Subtracting these two equations and making the (eminently reasonable) assumption that the cosmic scale factor a(t) does not vary significantly over the period δt given by the frequency of light, we obtain 16
δt0 δt = . (37) a(t0) a(t) Therefore the observed frequency ν0 is related to the emitted frequency ν by Astronomers like to express this in terms of the red-shift parameter which implies ν0 ν a(t) = . (38) a(t0) 1 + z = λ0 , (39) λ z = a(t0) a(t) − 1 . (40) Thus if the universe expands one has z > 0 and ther is a red-shift while in a contracting universe with a(t0) < a(t) the light of distant glaxies would be blue-shifted. A few remarks: 1. This cosmological red-shift has nothing to do with the stars own gravitational field - that contribution to the red-shift is completely negligible compared to the effect of the cosmological red-shift. 2. Unlike the gravitational red-shift i GR, this cosmological red-shift is symmetric between re- ceiver and emitter, .e. light sent from the earth to the distant galaxy would likewise be red-shifted if we observe a red-shift of the distant galaxy. 3. This red-shift is a combined effect of gravitational and Doppler red-shifts and it is not very meaningful to interpret this only in terms of, say, a Doppler shift. Nevertheless, as mentioned before, astronomers like to do just that, calling v = zc the recessional velocity. 4. Nowadays, astronomers tend to express the distance of a galaxy not in terms of light-years or megaparsecs, but directly in terms of the observed red-shift factor z, the conversion to distance then following from some version of Hubbles law. 5. The largest observed redshift of a galaxy is currently z ∼ 10, corresponding to a distance of the order of 13 billion light-years, while the cosmic microwave background radiation, which originated just a couple of 10 5 years after the Big Bang, has z ∼ 10 3 . 17
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: 1.3 Particle kinematics of a partic
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
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
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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 =
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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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