CHAPTER 6

QUANTUM INFLATIONARY PERTURBATIONS

The career of a young theoretical physicist consists of treating the harmonic oscillator in ever-increasing levels of abstraction. Sidney Coleman

The procedure till now has been completely classical. In this Chapter we present the standard tools for the quantization of a scalar field in a non-trivial background. We will first apply the general treatment to the limiting case of a scalar field living in a rigid de Sitter spacetime. Finally we will quantize the curvature and tensor inflationary perturbations computed in the previous chapter.

6.1 Canonical quantization

Consider the action for a scalar field with time-dependent mass Z Z 3 1 3  02 2 2 2 S = dτ d x = dτ d x v (∂iv) m (τ)v , (6.1) L 2 − − in some coordinates (τ, x). The first step in the canonical quantization procedure is to define the canonically conjugated momentum to the field v ∂ π = L = v0 , (6.2) ∂v0 and to construct the Hamiltonian Z Z Z 3 3 0
1 3  2 2 2 2 H = d x = d x πv = d x π + (∂iv) + m (τ)v . (6.3) H − L 2 In the quantum version of the theory the (classical) variables v and π become (quantum) operators satisfiying the equal-time conmutation relations

[ˆv (x, τ) , πˆ (y, τ)] = i δ(x y) , [ˆv (x, τ) , vˆ (y, τ)] = [ˆπ (x, τ) , πˆ (y, τ)] = 0 , (6.4) − 6.1 Canonical quantization 64

and H(v, π) Hˆ (ˆv, πˆ). The operatorv ˆ obeys the same equation of motion as the classical → variable v, namely vˆ00 2vˆ + m2(τ)v ˆ = 0 . (6.5) − ∇ The general solution of this partial differential equation can be written as an infinite super- position of Fourier modes

Z 3 d k ik·x vˆ (x, τ) = vke (2π)3/2 Z 3 d k  − + ∗  ik·x = aˆ vk(τ) +a ˆ v (τ) e (6.6) (2π)3/2 k −k k Z 3 d k  − ik·x + ∗ −ik·x = aˆ vk(τ)e +a ˆ v (τ)e . (2π)3/2 k k k

The temporal mode functions vk(τ) is this expression satisfy harmonic oscillator equations with time-dependent frequency

00 2 2 2 2 vk + ωk(τ)vk = 0 , ωk(τ) = k + m (τ) . (6.7)

Note that the mode functions vk(τ) depends only of k = k due to the isotropy of the FLRW 2 | 2| Universe and the associated form of the frequency ωk k . ± ∼ The operatorsa ˆk in Eq. (6.6) are taken to be time-independent and to obey the bosonic commutation relations

 − +  0  − −   + +  aˆ , aˆ 0 = δ(k k ), aˆ , aˆ 0 = aˆ , aˆ 0 = 0 . (6.8) k k − k k k k The consistency of these commutation relations with those in (6.4) requires the mode func- tions vk(τ) to satisfy the normalization condition

0 ∗ ∗ 0 0 ∗
v v vkv = 2 i Im v v = i . (6.9) k k − k k k −

Exercise Derive Eq. (6.9). Note that left hand side of this equation is nothing else than the ∗ 0 ∗ ∗ 0 Wronskian of the ordinary differential equation (6.7), W [vk, v ] v v vkv . k ≡ k k − k

6.1.1 Bogolyubov transformations

± The creation and annihilation operatorsa ˆk can be used to construct an orthonormal basis in the Hilbert space

m n − 1 h +   +  i aˆ 0 a = 0 , mk1 , nk2 , ... aˆ aˆ ... 0 a , (6.10) k | i | ia ≡ √m!n!... k1 k2 | i with 0 a the a-vacuum state for all k and mk , nk , ... denoting a set of excited states with | i | 1 2 ia occupation numbers m, n, . . . in the modes k1, k2,... Note, however, that an unambiguous interpretation of these states arises only after selecting the precise mode functions vk(τ) in 6.1 Canonical quantization 65

Eq. (6.6). Each choice of mode functions translates into a different set of creation and an- nihilation operators and consequently into different vacuum and excited states. To illustrate this, consider the linear combination

∗ uk(τ) αkvk(τ) + βkv (τ) , (6.11) ≡ k with αk and βk time-independent complex coefficients. Requiring the modes uk to satisfy the normalization condition 0 ∗ ∗ 0 0 ∗
u u uku = 2 i Im u u = i . (6.12) k k − k k k − we otain the following relation between the complex coefficients αk and βk

2 2 αk βk = 1. (6.13) | | − | |

Exercise Show this.

The uk(τ) modes satisfy, by construction, the wave equation (6.7). This allows us to use them as an alternative basis to expand the fieldv ˆ, namely

Z 3 d k  − ik·x + ∗ −ik·x vˆ (x, τ) = ˆb uk(τ)e + ˆb u (τ)e , (6.14) (2π)3/2 k k k

ˆ± with bk a different set of time-independent creation and annihilation operators satisfying the commutation relations

h − + i 0 h − − i h + + i ˆb , ˆb 0 = δ(k k ), ˆb , ˆb 0 = ˆb , ˆb 0 = 0 . (6.15) k k − k k k k ˆ± As before, the creation and annihilation operators bk can be used to construct and orthonor- mal basis in the Hilbert space

m n ˆ− 1 hˆ+  ˆ+  i b 0 b = 0 , mk1 , nk2 , ... b b ... 0 b . (6.16) k | i | ib ≡ √m!n!... k1 k2 | i with 0 b the b-vacuum for all k and mk , nk , ... a multiparticle state of b-particles. | i | 1 2 ia ± ˆ± The relation between the operatorsa ˆk amd bk can be determined by substituting the linear combination (6.11) into the mode expansion (6.14) and comparing the result with Eq. (6.6). Doing this we obtain

− ∗ ˆ− ˆ+ + ˆ+ ∗ ˆ− aˆk = αk bk + βk b−k , aˆk = αk bk + βk b−k . (6.17)

The expressions in (6.17) are called Bogolyubov transformations. The complex coefficients αk and βk are Bogolyubov coefficients.

Exercise Derive (6.17). 6.1 Canonical quantization 66

The relation (6.17) can be used to compute the expectation number of the a-particle number operator Nˆ a aˆ+aˆ− in the b-vacuum k ≡ k k a + ∗ − ∗ − + 2 b 0 Nˆ 0 b = b 0 (αk ˆb + β ˆb )(α ˆb + βk ˆb ) 0 = βk δ(0) . (6.18) h | k| i h | k k −k k k −k | ib | | The factor δ(0) is just the infinite spatial volume. It arises from the commutation relations when evaluating δ(k k0) at k = k0. Dividing by δ(0) we obtain the number density of − particles in a given mode, 2 nk βk . (6.19) ≡ | | This expression shows that the b-vacuum is generically a state without b-particles but with an a-particle number density nk in each mode. Natural modes and Killing vectors In Minkowski spacetime there exist a natural set of modes which are closely related to the Poincar´egroup. Any other mode function mixing positive and negative frequency natural modes is not Poincar´einvariant. Bogolubov transformations that do not mix positive and negative energy modes are called trivial or non-mixing. The existence of natural modes in curved spacetimes is associated to the presence of time-like Killing vectors.

6.1.2 Choice of physical vacuum

In the previous section we have seen that each vacuum state has a non-zero number density of particles for any other particle. Which is then the “true” physical vacuum? The usual Minkowski spacetime definition of the vacuum as the “lowest energy eigen- state” cannot be applied to the problem at hand. The Hamiltonian (6.3) involves an explicit function of time, meaning that the concept of “lowest energy eigenstate” depends on the particular instant τ0 at which it is defined. However, it is always possible to define an instan- taneous vacuum state at a given time τ0. To see this consider the expectation value of the Hamiltonian for an arbitrary mode function vk(τ). Inserting the mode expansion (6.6) into the quantized version of Eq. (6.3), we get Z ˆ 3  ∗ − − + + + −
 H = d k Fk aˆk aˆ−k + Fk aˆk aˆ−k + Ek 2ˆak aˆk + δ(0) , (6.20) with 1 0 2 2 2
1 02 2 2
Ek v + ω vk ,Fk v + ω v . (6.21) ≡ 2 | k| k| | ≡ 2 k k k Taking the expectation value of Eq. (6.20) removes the first three terms since each of them annihilates the vacuum on either the left- of the right-hand side. We are left with Z 3 v 0 Hˆ 0 v = δ(0) d k Ek . (6.22) h | | i Dividing by the δ(0)-term accounting for the infinite volume of space, we obtain the energy density Z 3 ε d k Ek. (6.23) ≡ 6.1 Canonical quantization 67

The minimum of this quantity at a given time τ0 can be obtained by minimizing the individual contributions Ek at that time. To this end, let us consider the polar representation

iαk vk = rke , (6.24) with rk and αk real functions. Replacing this expression into Eq. (6.21) we get   1 02 2 02 2 2
1 02 1 2 2 Ek(τ0) = rk + rkαk + ωkrk = rk + 2 + wkrk , (6.25) 2 2 4rk where in the last step we have made use of the Wronskian condition (6.9) in its polar form

0 ∗ ∗ 0 0 1 vkvk vkvk = i , = αk = 2 . (6.26) − − ⇒ −2rk Equation (6.25) is minimized for

0 1 rk(τ0) = 0 , rk(τ0) = , (6.27) √2ωk which corresponds a choice of instantaneous mode functions

1 −iωk(τ0)τ0 0 vk (τ0) = p e , vk(τ0) = iωkvk(τ0) . (6.28) 2ωk(τ0) −

± These modes functions define a particular set of operators ak and the associated basis of vacuum and excited states. In this basis, and at time τ = τ0, 1 E (τ0) = ω (τ0) ,F (τ0) = 0 , (6.29) k 2 k k and the Hamiltonian becomes diagonal1 Z   3 + − 1 Hˆ (τ0) = d k ω (τ0) aˆ aˆ + δ(0) . (6.31) k k k 2 We remark that the instantaneous mode function (6.28) would not give rise to the lowest energy eigenstate at times τ > τ0. In order for this to happen the quantiy Fk(τ) must be remain zero for all τ, i.e.  Z  1 02 2
Fk vk(τ) + ω (τ)vk(τ) = 0 , = vk(τ) = C exp i ωk(τ)dτ , (6.32) ≡ 2 k ⇒ ± which does not satisfy the mode equation (6.7) is ωk(τ) is time dependent.

Exercise Check this.

1Note the vacuum energy density 1 Z d3k ω (τ )δ(0) , (6.30) 2 k 0 is ultraviolet divergent and must be renormalized. 6.2 de Sitter limit 68

6.2 de Sitter limit

After discussing the generalities of quantum field theory in time dependent backgrounds and before dealing with the computation of the curvature perturbations generated during inflation let us consider a simplified case to get some intuition. The minimal value of the Hubble flow parameter  during inflation is zero. In this case the Hubble rate H is constant and the Universe expands exponentially fast, a(t) = eHt. This limit motivates the study of the so-called de Sitter spacetime.

6.2.1 de Sitter spacetime

The de Sitter spacetime dS4 can be represented as a 4-dimensional hyperboloid extrinsically A embedded in a d=5 Minkowski spacetime ηAB = diag( 1, 1, 1, 1, 1) with coordinates z − A B 2 2 2 2 2 2 ηABz z = z + z + z + z + z = l . (6.33) − 0 1 2 3 4 The quantity l 1/H is the so-called de Sitter radius.2 This representation makes explicit ≡ the symmetries of the de Sitter space: rotations and Lorentz transformations in the 10 planes formed by pairs of the five coordinates zA. This ten parameter SO(4, 1) group plays the same instrumental role than the Poincare group in Minkowski spacetime. In particular, it greatly facilitates computations as far as quantum field theory is concerned.

The dS4 spacetime can be also described in an intrinsic way. Consider the transformation

1 H Ht i j z0 = sinh(Ht) + e δijx x , (6.34) H 2 Ht zi = xie , (6.35)

1 H Ht i j z4 = cosh(Ht) e δijx x , (6.36) H − 2 with i = 1, 2, 3 and < t < and < xi < . In this coordinate system the line −∞ ∞ −∞ ∞ element (6.33) becomes a special case of the flat FLRW spacetime 2 2 2 i j ds = dt + a (t)δijdx dx , (6.37) − with a(t) = eHt. Note however that Eq. (6.37) is not completely equivalent to (6.33), since the coordinates t, xi cover only half of de Sitter manifold. This can be easily seen by adding { } the z0 and z4 coordinates (see also Fig. (6.1)),

1 Ht z0 + z4 = e 0 . (6.38) H ≥

Exercise 1. Derive Eq. (6.37) from the 5-dimensional embedding (6.33).

2. Other choices of coordinates leading to FLRW metrics with open and closed spatial sections can be also considered. Find these sets of coordinates.

2The choice of notation will become clear soon. 6.2 de Sitter limit 69

Figure 6.1: The embedding of de Sitter space into a five dimensional flat space-time with two spatial coordinates suppressed. The flat coordinates in (6.34)-(6.36) cover only half of de Sitter manifold. The surfaces (lines) of constant t and constant x are also indicated.

It is interesting to recast (6.37) in terms of the conformal time (3.35). Taking into account that 1 1 τ = e−Ht = a(τ) = , (6.39) −H ⇒ −Hτ the line element takes the manifestly conformally flat form

2 1 2 i j
ds = dτ + δijdx dx , (6.40) H2τ 2 − with τ ranging between and 0. Note that Eq. (6.40) is manifestly invariant under the −∞ rescaling τ λτ , xi λxi . (6.41) → → As we will see in the next sections, this symmetry plays a central role in the properties of the primordial perturbations generated during inflation.

6.2.2 Bunch-Davies vacuum

Consider an almost massless scalar field Z Z 1 4 µν 1 3 2  2 2  S √ g d x [g ∂µφ∂νφ + ...] = dτ d x a(τ) (∂τ φ) (∂iφ) ... , (6.42) ' −2 − 2 − leaving in a rigid de Sitter spacetime, with τ the conformal time. This action can be written as Z 1 3  002 2 2 2 S = dτd x v (∂iv) m (τ)v , (6.43) 2 − − dS 6.2 de Sitter limit 70

with a00 m2 (τ) , v a φ . (6.44) dS ≡ − a ≡ For a de Sitter expansion 1 a(τ) = , (6.45) −Hτ and   00 2 2 v + k vk = 0 . (6.46) k − τ 2 The general solution of Eq. (6.46) can be written as

e−ikτ  i  eikτ  i  vk(τ) = α 1 + β 1 + , (6.47) √2k − kτ √2k kτ with α and β some parameters to be fixed by the boundary conditions. Exercise Show that the Wronskian normalization (6.9) constrain the constants α and β

α 2 β 2 = 1 (6.48) | | − | | In the subhorizon limit (k aH, k τ ), the frequencies in Eq. (6.46) become time-  | | → ∞ independent. We are therefore in an adiabatic limit in which the expansion of the Universe is small as compared to the typical oscillation frequencies of the modes. Any distinction between de Sitter and Minkowski spacetime becomes suppressed and the vacuum can be naturally identified with the positive frequency mode

e−ikτ lim vk(τ) = , (6.49) k|τ|→∞ √2k

0 which is properly normalized and satisfies v = iωkvk. This boundary condition fixes the k − coefficient α and β in Eq. (6.47) to α = 1, β = 0, yielding the Bunch-Davies state

−ikτ   e i (aH) −ikτ vk(τ) = 1 = i e (1 + ikτ) . (6.50) √2k − kτ √2k3 The superhorizon limit of this equation is particularly interesting

(aH) lim vk = i . (6.51) k|τ|→0 √2k3

This expression represents a growing solution vk a, meaning that the physical scalar ∝ perturbation vk/a freezes and remains constant outside the horizon (see also Eq. (5.70)). Using the mode function (6.50) we can compute the power spectrum (i.e. two-point correla- tion function in Fourier space)

2 2 0 (aH) 2 2 0 0 vkvk0 0 vkvk0 0 = vk(τ) δ(k+k ) = (1+k τ )δ(k+k ) Pv(k)δ(k+k ) , (6.52) h i ≡ h | | i | | 2k3 ≡ 6.2 de Sitter limit 71

Figure 6.2: In a de Sitter spacetime the correlations are much larger than those in Minkowski and on extend over large distances ∆x a(t) eHt. | | ∼ ∼ which is diagonal in momenta and depends only on the norm of k due to the homogeneity and isotropy of the metric. On superhorizon scales Eq. (6.52) becomes 1 1 (aH)2 Pv = = . (6.53) 2k3 τ 2 2k3

6.2.3 A surprising result

Note that for a scalar field in de Sitter space φk = vk/a (cf. Eq. (6.44)) and 1 H2 constant P = Pv = = . (6.54) φ a2 2k3 k3

A spectrum with this k−3 dependence is said to be scale-invariant. The name comes from the fact that the Fourier transform of the power spectrum (i.e. the two-point correlation function in real space) becomes approximately constant (d3k k3) ∼ 0 2 φ(x, t)φ(x , t) dS H , (6.55) h i ∼ up to a slowly varying logarithmic factor. Note that this a surprising difference with respect to two-point correlations functions in Minkowski spacetime, which decay fast as we increase the distance between the points,

0 1 φ(x, t)φ(x , t) Mink . (6.56) h i ∼ ∆x 2 | | In a de Sitter spacetime quantum effects are not longer restricted to small distances! Indeed, starting from the pure quantum mechanical state (6.49) we get correlations much larger than those in Minkowski and on HUGE distances! (cf. Fig. 6.2).

6.2.4 A not so surprising result

As we saw in Section 6.2.1, the de Sitter spacetime is highly symmetric. Correlations functions should respect these symmetries. For instance, translation invariance gives rise to momentum 6.3 Quasi-de Sitter spacetime 72

conservation and this is reflected in the correlation function (6.52) via a delta function δ(k + k0). Also, rotation invariance tell us that there is no a privileged direction in the power spectrum. In the same way, the symmetry (6.41) restricts fixes the k−3 dependence of the power-spectrum. To see this note that dilations x λx act on the scalar field φ(x, τ) as →

φ(x) φλ(x) = φ(λx) . (6.57) → In Fourier space this correspond to a scaling Z Z 3 −ik·λx −3 3 −ip·x φ(λx) = d k e φk = λ d p e φp/λ , (6.58) with p = λk, or equivalently to −3 φk λ φ . (6.59) → k/λ To be compatible with the dilatation symmetry, the two-point function for the vk field is constrained to the form 3 (3) 0 F (kτ) φkφk0 = (2π) δ (k + k ) , (6.60) h i k3 in such a way that3

−6 −6 3 (3) 0 F (k τ) φkφk0 λ φ φ 0 = λ (2π) δ (k/λ + k /λ) h i → h k/λ k /λi (k/λ)3 F (k τ) = (2π)3δ(3)(k + k0) . (6.61) k3

Since the perturbations φk = vk/a becomes time independent when they leave the horizon, the function F must be a constant, leading to a scale invariant spectrum. Note that this result is independent of the details of the Lagrangian.

6.3 Quasi-de Sitter spacetime

6.3.1 Curvature perturbations

As explained in Section 5.3.1, the action for the curvature perturbation ζ is suppressed by the Hubble flow parameter . Strightly speaking, in a pure de Sitter expansion  = 0 and the curvature perturbation becomes “pure gauge“. This reflects the fact that inflation never ends for an exact de Sitter expansion. In a realistic situation, inflation must end for the standard hot Big Bang picture to begin. This graceful inflationary exit translates into a small but non-zero value for the flow parameter  which allows the curvature perturbation to exist. The power-spectrum of curvature perturbations for a quasi de-Sitter expansion can be easily computed by taking into account (5.71) together with the de Sitter spectrum (6.53)

1 1 H2 1 H4 Pζ = Pv = = . (6.62) z2 k=aH 4k3  k=aH 2k3 φ˙2 k=aH

3Note that the argument of F is dilatation invariant. The delta function transform with a factor λ3. 6.3 Quasi-de Sitter spacetime 73

Note that Pζ is evaluated at the time at which the mode k crosses the horizon. The structure of this expression can be easily understood. In the absence of quantum fluctuations (~ = 0), surfaces of constant φ coincide with surfaces of constant t and inflation ends at the same time φ = φend in every location. This is no longer true in the presence of quantum fluctuations (~ = 0) since quantum jumps will slightly move the inflaton field up or down the classical 6 inflationary trajectory. The overall expansion of the Universe will change from place to place, i.e. different locations will have a different scale factor, which leads to curvature perturbations aeff a(1 + 2ζ), ≈ δφ δa H δt , ζ Hδt δφ . (6.63) ∼ φ˙ ∼ a ∼ ∼ φ˙ Combining this with the de Sitter result (6.54) we get

H2 H2 1 H4 H2 Pζ . (6.64) 2 3 3 2 2 ∼ φ˙ 2k ∼ k φ˙ ∼ MP Curvature perturbations are generated by the time delay induced by inflation fluctuations!

6.3.2 Tensor perturbations

Consider the second-order action (5.74) Z (2) 1 3  0 2 2 2 2  δS = dτdx (v ) (∂vij) m¯ (τ)v , (6.65) t 2 ij − − ij where the variable vij related to the tensor perturbation hij by (cf. Eq. (5.75)) 2κ hij vij . (6.66) ≡ a Performing a Fourier transform

Z 3 d k X λ λ ik·x vij(τ, x) =  (k)v (τ)e , (6.67) (2π)3 ij k λ=+,×

λ with ij(k) two polarization tensors satisfying the symmetric, tranverse and traceless cond- tions λ λ γ i γ γ γ0 ij(k) = ji(k) , ii = k ij = 0 , ij(k)ij (k) = 2δγγ0 . (6.68) λ we get the following action for the functions vk(τ) describing the two-polarization modes of graviational waves (+ and ) × Z   00   (2) 1 X a δS = dτd3k (vλ0)2 k2 (vλ)2 . (6.69) t 2 k − − a k λ Note that for a de Sitter background

a00 2 = , (6.70) a τ 2 6.3 Quasi-de Sitter spacetime 74

λ and the equation of motion for vk(τ) coincides formally with that in Eq. (6.46). Taking into λ account the two polarization modes in vk(τ), the tensor spectrum for the gravitational wave perturbation hij can be written as  2 2 2 4κ 2 Pt = 2Ph = 2 Pv = 3 H . (6.71) k=aH × aMP k=aH k k=aH

6.3.3 Inflationary observables

Given the (dimensionfull) primordial spectra of scalar and tensor perturbations in Eqs. (6.62) and (6.71), we can construct the dimensionless quantities4 3  2 3  2 2 k 1 H 2 k 2 H ∆s(k) Pζ (k) = , ∆t (k) Pt(k) = 8κ . ≡ 2π2 2 2π k=aH ≡ 2π2 2π k=aH (6.72) 2 2 Note that while ∆s(k) depends on both  and H, the gravitational waves spectrum ∆t (k) 2 is only sensitive to the Hubble rate during inflation. A measure of ∆t (k) would be a direct 2 2 measure of the energy scale of inflation. The ratio of ∆t (k) to ∆s(k) is the so-called tensor- to-scalar ratio 2 ∆t (k) r 2 = 16 . (6.73) ≡ ∆s (k) The  suppression of this quantity makes the detection of gravitational waves difficult. The slow variation of the Hubble parameter H in a quasi-de Sitter expansion translates into sligth deviations from the scale-invariant spectra ∆s, ∆t = constant. These deviations are parametrized in terms of the so-called spectral tilts of scalar and tensor perturbations 2 2 d ln ∆s(k) d ln ∆t (k) ns 1 , nt , (6.74) − ≡ d ln k ≡ d ln k to the lowest order in the scale dependence. A spectrum of curvature perturbations with 5 ns < 1 is said to be red-tilted, while a spectrum with ns > 1 is said to be blue tilted (see Fig. 6.3)

Slightly different definitions Note the slight difference in the definitions of these two quantities. A scale-invariant spectrum of scalar perturbations corresponds to a spectral tilt ns = 1, while a scale- invariant spectrum of primordial gravitational waves corresponds to nt = 0.

Taking into account that dN d ln k −1  d ln H −1 k = aH ln k = N + ln H = = 1 + , (6.75) ⇒ ⇒ d ln k dN dN the spectral-tilts ns and nt can be written as 2    −1 d ln ∆s dN d ln H d ln  d ln H −1 ns 1 = = 2 1 + = ( 2 η)(1 ) , (6.76) − d ln N d ln k dN − dN dN − − −

4 2 2 Note that, due to the evaluation at horizon exit k = aH, both ∆s(k) and ∆t (k) are functions of k only. 5Higher orders on the scale dependence, usually called running of the tilts, could be also defined. 6.4 Workout examples 75

Figure 6.3: .

2    −1 d ln ∆t dN d ln H d ln H −1 nt = = 2 1 + = 2(1 ) , (6.77) d ln N d ln k dN dN − − where we have made use of the Hubble flow parameters’ definition (cf. Eqs. (3.48) and (3.50)). At first order in the flow parameters we have

ns 1 = 2 η , nt = 2 . (6.78) − − − − Combining the last expression with Eqs. (6.73) we get the consistency relation

r = 8nt . (6.79) −

6.4 Workout examples

For illustration purposes we provide now specific computations of the spectral tilt and the tensor-to-scalar ratio in inflationary models.

Monomial potential

Consider the monomial potential V (x) = M 4xp , (6.80) 6.4 Workout examples 76

with x κφ a dimensionless field variable, M a parameter with the dimensions of mass and ≡ p a positive constant. The Hubble flow functions for this model turns out to be independent of the mass scale M,

 2 2 " 2 # 1 V,φ p 2 V,φ V,φφ 2p  V = = , η 4V 2ηV = = . (6.81) ' 2κ2 V 2x2 ' − κ2 V − V x2

Inflation ends when (xend) V (xend) = 1, which, along the x > 0 branch of the potential, ' leads to p xend = . (6.82) √2 The relation between this field value and the number of e-folds of inflation is given by

Z φend  2  κ dφ 1 2 2
1 2 p N p | | , = N = xi xend = xi . (6.83) ' φi 2V(φ) ⇒ 2p − 2p − 2

Solving for xi in this expression,  p x2 = 2p N + , (6.84) i 4 and evaluating the slow-roll parameters (11.7) at x = xi, we get p 4  = , η =  . (6.85) 4 (N + p/4) p Taking into account Eqs. (6.73) and (6.78), we obtain 2(2 + p) 16p ns 1 = , r = . (6.86) − − 4N + p 4N + p

The inflationary predictions for monomial models lay on straight lines in a ns r plane − 2 + p ns = 1 r . (6.87) − 8p

Starobinsky/Higgs inflation potential

The potential of the Starobinsky/Higgs inflation model in the Einstein-frame reads  √ 2 V (x) = M 4 1 e− αx , (6.88) − with x κφ a dimensionless field variable, M a parameter with the dimensions of mass and p ≡ α = 2/3. The Hubble-flow parameters associated with this potential read

2α2 4α2eαx  V = , η 4V 2ηV = , (6.89) ' (eαx 1)2 ' − (eαx 1)2 − − The number of e-folds of inflation can be written as αx e αx xi 3 αxi N = −2 e . (6.90) 2α xend ' 4 6.4 Workout examples 77

with 1  2  xend = log 1 + , (6.91) α √3 the value of the field at the end of inflation. Evaluating the slow-roll parameters (11.15) at x = xi, we get 2 12 ns 1 , r , (6.92) − ' −N ' N 2 at N 1.  Recap:

1. The comoving horizon (aH)−1 decreases during inflation.

2. All relevant fluctuations are created quantum mechanically at subhorizon scales k−1 < (aH)−1.

3. Since the comoving scales k−1 are constant during and after inflation, the relevant fluctuations start inside the horizon and eventually cross it.

4. On superhorizon scales k−1 < (aH)−1 the perturbations freeze and cannot be affected by any physics taking place inside the horizon.

5. When inflation ends, (aH)−1 increases and the perturbations reenter the horizon with the same properties they had when they first cross it.