Primordial non-Gaussianity in the cosmological perturbations - CBPF

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Let us emphasize a subtle distinction between the particle horizon and the Hubble: if particles are separated by distances greater than RH(t) they never could have communicated with one another; if they are separated by distances greater than the Hubble radius H −1 , they cannot talk to each other at the time t. We shall see that the standard cosmology the distance to the horizon is finite, and up to numerical factors, equal to the Hubble radius, H −1 , but during inflation, for instance, they are drastically different. One can also define a comoving particle horizon distance t τH = 0 dt ′ a(t ′ ) = a da 0 ′ H(a ′ )a ′ 2 = a 0 d ln a ′ 1 Ha ′ (COMOVING PARTICLE HORIZON). Here, we have expressed the comoving horizon as an integral of the comoving Hubble radius, 1 aH (24) (COMOVING HUBBLE RADIUS), (25) which will play a crucial role in inflation. We see that the comoving horizon then is the logarithmic integral of the comoving Hubble radius (aH) −1 . The Hubble radius is the distance over which particles can travel in the course of one expansion time, i.e. roughly the time in which the scale factor doubles. So the Hubble radius is another way of measuring whether particles are causally connected with each other: if they are separated by distances larger than the Hubble radius, then they cannot communicate at a given time t (or τ). Let us reiterate that there is a subtle distinction between the comoving horizon τH and the comoving Hubble radius (aH) −1 . If particles are separated by comoving distances greater than τH, they never could have communicated with one another; if they are separated by distances greater than (aH) −1 , they cannot talk to each other at some time τ. It is therefore possible that τH could be much larger than (aH) −1 now, so that particles cannot communicate today but were in causal contact early on. As we shall see, this might happen if the comoving Hubble radius early on was much larger than it is now so that τH got most of its contribution from early times. We will see that this could happen, but it does not happen during matter-dominated or radiation-dominatd epochs. In those cases, the comoving Hubble radius increases with time, so typically we expect the largest contribution to τH to come from the most recent times. future A similar concept is the event horizon, or the maximum distance we can probe in the infinite ∞ Re(t) = a(t) t which is clearly infinite if the universe expands a a ∼ t n (n < 1). 14 dt ′ a(t ′ , (26) )
1.3 Particle kinematics of a particle Consider the geodesic motion of a particle that is not necessarily massless. The four-velocity u µ of a particle with respect to the comoving frame is referred to as the peculiar velocity. The geodesic equation of motion in terms of the affine parameter chosen to be the proper length is du µ The µ = 0 component of the geodesic equation is ds + Γµ νσu ν u σ = 0, u µ = dxµ ds . (27) du0 ds + Γ0νσu ν u σ = du0 ds + Γ0iju i u j = du0 ˙a + ds a giju i u j = 0. (28) Denoting by |u| 2 = giju i u j and recalling that −(u 0 ) 2 + |u| 2 = −1, it follows that u 0 du 0 = |u|d|u|, the geodesic equation becomes du 0 ds ˙a + a |u|2 = 1 u0 d|u| ds Finally, since u 0 = dt/ds, this equation reduces to 1 d|u| |u| dt ˙a + |u| = 0. (29) a ˙a = − . (30) a It implies that |u| ∼ a −1 . recalling that the four-momentum p µ = mu µ , we see that the magnitude of the three-momentum of a freely propagating particle red-shifts away as a −1 . Note that in eq. (29) the factors of ds cancel. That implies that the above discussion also applies to massless particles, where ds = 0 (formally by the choice of a different affine parameter). In terms of the ordinary three-velocity v i for which u µ = (u 0 , u i ) = (γ, γv i ) and γ = 1/ 1 − |v| 2 , we have |u| = |v| 1 − |v| 2 1 ∼ . (31) a We again see why the comoving frame is the natural frame. Consider an observer initially (t = t1) moving non-relativistically with respect to the comoving frame with physical three velocity of magnitude |v|1 . At a later time t2 the magnitude of the observer’s physical three-velocity |v|2 will be a(t1) |v|2 = |v|1 . (32) a(t2) In an expanding universe, the freely-falling observer is destined to come to rest in the comoving frame even if he has some initial velocity with respect to it. 15
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: It should be noted that the assumpt
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
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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