Baumann's Cosmology (8)

One of the most remarkable features of inflation is that it provides a natural mechanism for creating the primordial density fluctuations that seeded the structure in the universe. Recall that the evolution of the inflaton field 𝜙(𝑡) governs the energy density of the early universe 𝜌(𝑡) and hence, controls the end of inflation (see Fig. 8.1). Essentially, the field 𝜙 plays the role of a "clock" telling us the amount of inflationary expansion still remaining. However, by the uncertainty principle, arbitrary precise timing is impossible in quantum mechanics. Instead, quantum mechanical clocks necessarily have some variance, so the inflaton will therefore varaying fluctuations 𝛿𝜙(𝑡, 𝐱). The time at which inflation ends will therefore vary across space. Although these fluctuations are a small quantum effect, they trigger large differences in the local expansion histories, which lead to significant differences in the local densities after inflation, 𝛿𝜌(𝑡, 𝐱). Regions of space that end inflation first will start diluting and therefor end up having smaller densities, while regions that inflate for a longer time will have higher densities. It is worth emphasizing that the theory was not engineered to produce these fluctuations, but that their origin is instead a natural consequence of treating inflation quantum mechanically.
   In this chapter we will derive the spectrum of quantum fluctuations produced during inflation. We will begins, in Section 8.1 by deriving the classical equations of motion of the inflation fluctuations in spatially flat gauge. We will find that each Fourier mode satisfies the equation of a harmonic oscillator with a time-dependent frequency. In Section 8.2 we will show how quantum zero-point fluctuations after inflation, becoming the seeds for all structure in the universe. We will derive the forms of the primordial scalar and tensor power spectra, and relate these predictions to the shape of the inflation potential in slow-roll models. In section 8.4 we will present current observational constraints on inflationary models and comment on the prospects for future tests.
   For pedagogical reasons the presentation will focus on single-field slow-roll models of inflation. It is important to appreciate that the landscape of inflationary models is much larger and there are many different mechanisms that can produce the inflationary expansion of the spacetime (see e.g. [1] for a review). An interesting way of describing a large class of models is in terms of an effective theory of the inflationary fluctuations. Given a suitable Hubble parameter 𝐻(𝑡) (whose origin doesn't have to be specified), we can construct an effective theory for the associated Goldston boson of spontaneously broken time translations. On superhorizon scales, this Goldstone boson is closely related to the primordial curvature perturbations and adiabatic density fluctuations.^{[verification needed]} By  focusing directly on the inflationary fluctuations, this EFT of inflation provides an effective way of describing the prediction of inflation [2]. The quantization procesure described inthis chapter for slow-roll inflation applies straightfowardly in broader context.

          8.1 Inflationary Perturbations
The dynamics of the inflaton field is determined by the following action
    (8.1)   𝑆 = ∫ d⁴𝑥√-𝑔 [-1/2 𝑔^𝜇𝜈∂_𝜇𝜙∂_𝜈𝜙 - 𝑉(𝜙)
]. where 𝑔 ≡ det(𝑔^𝜇𝜈) (see Appendix A). The coupling to the metric will lead to a nontrivial mixing between the inflaton fluctuations 𝛿𝜙 and the metric fluctuations 𝛿𝑔^𝜇𝜈. The form of this mixing is gauge dependent. Foe example in comoving gauge the inflaton fluctuation vanish, 𝛿𝜙 ≡ 0. and all fluctuations arein metric. To make contact with the intuitive picture of inflaton fluctuation shown in Fig. 8.1, however, it will instead be convenient to work in the spatially flat gauge, where the line element (with scalar fluctuations) takes form
    (8.2)   d𝑠² = 𝑎(𝜂) [-(1 + 2𝛢)d𝜂² + 2∂_𝑖𝛣d𝑥^𝑖d𝜂 + 𝛿_𝑖𝑗d𝑥^𝑖d𝑥^𝑗].
The metric fluctuation 𝛢 and 𝛣 are related to the inflaton fluctuations 𝛿𝜙 through the Einstein equations, leaving a single degree of freedom to describe the inflationary fluctuations. We will begin by deriving the classical equation of motion for 𝛿𝜙 and then discuss its quantization.

          8.1.1 Equation of Motion
Varying of the action (8.1) leads to Klein-Gordon equation for the inflation field.
    (8.3)   1/√-𝑔 ∂_𝜇(√-𝑔 𝑔^𝜇𝜈∂_𝜈𝜙) = 𝑉_,𝜙,
where 𝑉_,𝜙 ≡ d𝑉/d𝜙.

Exercise 8.1   Show that (8.3) follows from (8.1).
[Solution]   Varying the action with respect to 𝜙, we get
   (a)   𝛿𝑆 = ∫ d⁴𝑥√-𝑔 (-𝑔^𝜇𝜈∂_𝜇𝛿𝜙∂_𝜈𝜙 - 𝑉_,𝜙𝛿𝜙) = ∫ d⁴𝑥√-𝑔 [1/√-𝑔 ∂_𝜈(√-𝑔 𝑔^𝜇𝜈 ∂_𝜇𝜙) - 𝑉_,𝜙]𝛿𝜙.
Since the principle of least action states 𝛿𝑆 = 0, for arbitrary 𝛿𝜙 the expression in the square brackets must vanish. So we find
      (b)   1/√-𝑔 ∂_𝜈(√-𝑔 𝑔^𝜇𝜈 ∂_𝜇𝜙) - 𝑉_,𝜙 = 0.   ⇒   1/√-𝑔 ∂_𝜈(√-𝑔 𝑔^𝜇𝜈 ∂_𝜇𝜙) = 𝑉_,𝜙.  ▮

   At first order in the fluctuations, the components of the inverse metric are
    (8.4)   𝑔⁰⁰ = -𝑎^-2(1 - 2𝛢),     𝑔^0𝑖 = 𝑎^-2∂^𝑖𝛣,     𝑔^𝑖𝑗 = 𝑎^-2𝛿^𝑖𝑗, and √-𝑔 = 𝑎⁴(1 + 𝛢). Substituting the perturbed field 𝜙 = 𝜙̄ + 𝛿𝜙 and the perturbed metric (8.2) into the equation of motion (8.3), we get
    (8.5)   𝛿𝜙ʺ + 2𝓗𝛿𝜙ʹ - ∇²𝛿𝜙 = (𝛢ʹ + ∇²𝛣)𝜙̇ʹ - 2𝑎²𝑉_,𝜙𝛢 - 𝑎²𝑉_,𝜙𝜙𝛿𝜙, where 𝑉_,𝜙𝜙 ≡ d²𝑉/d𝜙².

Exercise 8.2   Show that (8.5) follows from (8.3).
[Solution]   The Klein-Gordon equation (8.3) can be written as
   (a)   ∂_𝜇(√-𝑔 𝑔^𝜇𝜈∂_𝜈𝜙) - 1/√-𝑔 𝑉_,𝜙 = 0
The first term can be written as
   (b)   ∂_𝜇(√-𝑔 𝑔^𝜇𝜈∂_𝜈𝜙) = ∂₀(√-𝑔 𝑔⁰⁰∂₀𝜙) + ∂_𝑖(√-𝑔 𝑔^0𝑖∂₀𝜙) + ∂₀(√-𝑔 𝑔^𝑖0∂_𝑖𝜙) + ∂_𝑗(√-𝑔 𝑔^𝑖𝑗∂_𝑖𝜙)
Since ∂_𝑖𝜙 = ∂_𝑖𝜙𝛿𝜙 can be evaluated using 𝜙 = 𝜙̄ + 𝛿𝜙. Then 𝑔̄^𝑖0 = 0, 𝑔̄^𝑖𝑗 = 𝑎^-2 and √-𝑔̄ = 𝑎⁴. So we find
   (b)   ∂_𝜇(√-𝑔 𝑔^𝜇𝜈∂_𝜈𝜙) = ∂₀(√-𝑔 𝑔⁰⁰∂₀𝜙) + ∂_𝑖(√-𝑔 𝑔^0𝑖∂₀𝜙) + 𝑎²∇²𝛿𝜙
Since 𝑔^0𝑖 = 𝑎^-2∂^𝑖𝛣, ∂₀𝜙 = 𝜙̄ʹ + 𝛿𝜙ʹ and if we neglect 𝑔^0𝑖𝛿𝜙ʹ term which is so small, we then have
   (c)   ∂_𝜇(√-𝑔 𝑔^𝜇𝜈∂_𝜈𝜙) = ∂₀(√-𝑔 𝑔⁰⁰∂₀𝜙) + 𝑎²∇²𝛣𝜙̄ʹ + 𝑎²∇²𝛿𝜙.
Substituting (8.4) into the first term in the right-hand side if we also neglect -2𝛢² term here, we then get
   (d)   ∂₀(√-𝑔 𝑔⁰⁰∂₀𝜙) = ∂₀[𝑎⁴)(1 + 𝛢)(-𝑎^-2(1 - 2𝛢)∂₀𝜙] = - ∂₀[𝑎²(1 - 𝛢)(𝜙̄ʹ + 𝛿𝜙ʹ)] = -∂₀(𝑎²𝜙̄ʹ) - ∂₀(𝑎²𝛿𝜙ʹ) + ∂₀(𝑎²𝜙̄ʹ)𝛢 + 𝑎²𝜙̄ʹ𝛢ʹ,
   (e)   ∂_𝜇(√-𝑔 𝑔^𝜇𝜈∂_𝜈𝜙) =  -∂₀(𝑎²𝜙̄ʹ) - ∂₀(𝑎²𝛿𝜙ʹ) + 𝑎²∇²𝛿𝜙 + ∂₀(𝑎²𝜙̄ʹ)𝛢 + 𝑎²(𝛢ʹ + ∇²𝛣)𝜙̄ʹ.
The second term of (a) with its variation, if we neglect 𝛢𝑉_,𝜙𝜙𝛿𝜙 term, can be written
   (f)   √-𝑔 𝑉_,𝜙 = 𝑎⁴(1 + 𝛢)(𝑉_,𝜙 + 𝑉_,𝜙𝜙𝛿𝜙) = 𝑎⁴𝑉_,𝜙 + 𝑎⁴𝑉_,𝜙𝛢 + 𝑎⁴𝑉_,𝜙𝜙𝛿𝜙.
Now (a) becomes
   (g)   0 = -∂₀(𝑎²𝜙̄ʹ) - ∂₀(𝑎²𝛿𝜙ʹ) + 𝑎²∇²𝛿𝜙 + ∂₀(𝑎²𝜙̄ʹ)𝛢 + 𝑎²(𝛢ʹ + ∇²𝛣)𝜙̄ʹ -𝑎⁴𝑉_,𝜙 - 𝑎⁴𝑉_,𝜙𝛢 - 𝑎⁴𝑉_,𝜙𝜙𝛿𝜙
            = -[∂₀(𝑎²𝜙̄ʹ) + 𝑎⁴𝑉_,𝜙]- ∂₀(𝑎²𝛿𝜙ʹ) + 𝑎²∇²𝛿𝜙 - 𝑎⁴𝑉_,𝜙𝜙𝛿𝜙 + [∂₀(𝑎²𝜙̄ʹ) - 𝑎⁴𝑉_,𝜙]𝛢 + 𝑎²(𝛢ʹ + ∇²𝛣)𝜙̄ʹ.
Since the first line is zeroth order in fluctuations and and implies the background equation of motion, we can find
   (h)   ∂₀(𝑎²𝜙̄ʹ) + 𝑎⁴𝑉_,𝜙 = 0     ⇒      𝜙̄ʺ + 2𝓗𝜙̄ʹ + 𝑎²𝑉_,𝜙 = 0.
Finally to find we get
   (i)   𝛿𝜙ʺ + 2𝓗𝛿𝜙ʹ - ∇²𝛿𝜙 = (𝛢ʹ + ∇²𝛣)𝜙̄ʹ - 2𝑎²𝑉_,𝜙𝛢 - 𝑎²𝑉_,𝜙𝜙𝛿𝜙.  ▮

   Next, we will use the Einstein equations to eliminate 𝛢 and 𝛣 in favor of 𝛿𝜙. One finds
    (8.6-7)  𝛢 = 𝜀 𝓗/𝜙̄ʹ 𝛿𝜙,     ∇²𝛣 = -𝜀 𝓗/𝜙̄ʹ [𝛿𝜙ʹ + (𝛿 - 𝜀)𝓗𝛿𝜙].
    (8.8-9)  𝜀 ≡ - Ḣ/𝐻² = 1 - 𝓗ʹ/𝓗² = 4π𝐺 𝜙̄ʹ²/𝓗²,     𝛿 ≡ - (d𝜙̇/d𝑡)/𝐻𝜙̄̇ = 1 - 𝜙̄ʺ/𝓗𝜙̄ʹ.
Despite of the similar appearance we did not make any slow-roll approximation in (8.6-7). Nevertheless it is woth highlighting that the metric perturbations are proportional to the slow-roll parameters, so that the mixing with the inflation fluctuation vanishes at leading order in the slow-roll limit, 𝜀, 𝛿 → 0. This is a special feature of spatially flat gauge. In a general gauge this mixing would be order one in the slow-roll limit. Even in spatially flat gauge, the leading slow-roll correction will play an important role on superhorizon scales.

Derivation   Now we will provide some of the intermediate steps of derivation of (8.6-7). First, we consider the 0𝑖 Einstein equation. It takes the form
    (8.10)   𝛿𝐺⁰_𝑖 = - 2𝓗/𝑎² ∂_𝑖𝛢 = 8π𝐺 𝛿𝛵⁰ _𝑖 = -8π𝐺 𝜙̄ʹ/𝑎² ∂_𝑖𝛿𝜙,
where the final equality follows from the perturbed energy-momentum tensor of a scalar field (4.53). This implies the result in (8.6):
    (8.11)   𝛢 = 4π𝐺 𝜙̄ʹ/𝓗 𝛿𝜙 = 𝜀 𝓗/𝜙̄ʹ 𝛿𝜙.
Next, we look at the 00 Einstein equation,
    (8.12)  𝛿𝐺⁰₀ = 2𝓗/𝑎² (3𝓗𝛢 + ∇²𝛣) = 8π𝐺 𝛿𝛵⁰ ₀ = -8π𝐺 [𝑎^-2(𝜙̄ʹ 𝛿𝜙 - 𝜙̄ʹ²𝛢) + 𝑉_,𝜙𝛿𝜙].
Using the background equation of motion
    (8.13)  𝜙̄ʺ + 2𝓗𝜙̄ʹ + 𝑎²𝑉_,𝜙 = 0,
and the solution (8.11) for 𝛢, we obtain the result (8.7).  ▮

   Substituting (8.6-7) into (8.5) leads to a closed form equation for the inflation fluctuations:
    (8.14)  𝛿𝜙̄ʺ + 2𝓗𝛿𝜙ʹ - ∇²𝛿𝜙 = [2𝜀(3 + 𝜀 - 2𝛿) - 𝑎²𝑉_,𝜙𝜙/𝓗²]𝓗²𝛿𝜙.
Note that (𝛢ʹ + ∇²𝛣)𝜙̄ʹ term was canceled so that the mixing with the metric fluctuation only contributes to the effective mass of the inflaton fluctuation. Using the background equation of motion to write 𝑉_,𝜙𝜙 in terms of the slow-roll parameters, we find
    (8.15)  𝛿𝜙̄ʺ + 2𝓗𝛿𝜙ʹ - ∇²𝛿𝜙 = [(3 + 2𝜀 - 𝛿)(𝜀 - 𝛿) - 𝛿ʹ/𝓗]𝓗²𝛿𝜙.
   The friction term 2𝓗𝛿𝜙ʹ can be eliminated by defining the variable
    (8.16)  𝑓 ≡ 𝑎𝛿𝜙,
so that
    (8.17)  𝛿𝜙̄ʺ + 2𝓗𝛿𝜙ʹ = 1/𝑎 [𝑓ʺ - (2 - 𝜀)𝓗²𝑓].
Collecting all the terms, we find a beautiful equation, the so-called Mukhanov-Sasaki equation:
    (8.18)  𝑓ʺ + (𝑘² - 𝑧ʺ/𝑧)𝑓 = 0,     where    𝑧 ≡ 𝑎𝜙̄ʹ/𝓗.
This equation is quite remarkable: it contains the coupling matter and metric fluctuations, does not make any slow-roll approximation, and is valid on all scales. It will be our master equation for the rest of the chaprer.

Exercise 8.4   Fill in the gaps in the derivation of (8.18)
   Hints: Show that
    (8.19)   - 𝑎²𝑉_,𝜙𝜙/𝓗² = (𝛿 - 3)(𝜀 + 𝛿) - 𝛿ʹ/𝓗, 
    (8.20)   𝑧ʺ/𝑧 = [2 + 2𝜀 - 3𝛿 + (2𝜀 - 𝛿)(𝜀 - 𝛿) - 𝛿ʹ/𝓗]𝓗².
[Solution]   We start from equations (8.5-9), we write again them
   (a)   𝛿𝜙ʺ + 2𝓗𝛿𝜙ʹ - ∇²𝛿𝜙 = (𝛢ʹ + ∇²𝛣)𝜙̇ʹ - 2𝑎²𝑉_,𝜙𝛢 - 𝑎²𝑉_,𝜙𝜙𝛿𝜙,     where 𝑉_,𝜙𝜙 ≡ d²𝑉/d𝜙².
   (b)   𝛢 = 𝜀 𝓗/𝜙̄ʹ 𝛿𝜙,     ∇²𝛣 = -𝜀 𝓗/𝜙̄ʹ [𝛿𝜙ʹ + (𝛿 - 𝜀)𝓗𝛿𝜙].
   (c)   𝜀 = 1 - 𝓗ʹ/𝓗² = 4π𝐺 𝜙̄ʹ²/𝓗²,     𝛿 = 1 - 𝜙̄ʺ/𝓗𝜙̄ʹ.
   (d)   𝛢ʹ + ∇²𝛣 = (𝜀 𝓗/𝜙̄ʹ)ʹ𝛿𝜙 - 𝜀 𝓗/𝜙̄ʹ(𝜀 𝓗/𝜙̄ʹ,
   (e)   (𝜀 𝓗/𝜙̄ʹ)ʹ𝛿𝜙 = (4π𝐺 𝜙̄ʹ²/𝓗²)ʹ 𝛿𝜙 = 4π𝐺 (𝜙̄ʺ/𝓗 - 𝜙̄ʹ𝓗ʹ/𝓗²) 𝛿𝜙 = 4π𝐺 𝜙̄ʹ/𝓗 (𝜙̄ʺ/𝜙̄ʹ𝓗 - 𝓗ʹ/𝓗²) 𝓗𝛿𝜙
                    = 𝜀 𝓗/𝜙̄ʹ (1 - 𝛿 + 𝜀 - 1) 𝓗𝛿𝜙 = 𝜀 𝓗/𝜙̄ʹ (𝜀 - 𝛿) 𝓗𝛿𝜙.
The first term on the right-hand side of (a) is
   (f)   (𝛢ʹ + ∇²𝛣)𝜙̄ʹ = 2𝜀(𝜀 - 𝛿) 𝓗²𝛿𝜙.
The second term there is
   (g)   -2𝑎²𝑉_,𝜙𝛢 = 2(𝜙̄ʺ + 2𝓗𝜙̄ʹ) 𝜀 𝓗/𝜙̄ʹ 𝛿𝜙 = 2𝜀 (𝜙̄ʺ/𝓗𝜙̄ʹ + 2) 𝓗²𝛿𝜙 = 2𝜀 (3 - 𝛿) 𝓗²𝛿𝜙.
   (h)   (𝛢ʹ + ∇²𝛣)𝜙̄ʹ - 2𝑎²𝑉_,𝜙𝛢 = 2𝜀 (3 + 𝜀 - 2𝛿) 𝓗²𝛿𝜙.
So equation (a) can be written as
   (i)   𝛿𝜙ʺ + 2𝓗𝛿𝜙ʹ - ∇²𝛿𝜙 = [2𝜀 (3 + 𝜀 - 2𝛿) - 𝑎²𝑉_,𝜙𝜙/𝓗²] 𝓗²𝛿𝜙.
Then we wish to write 𝑉_,𝜙𝜙 in term of 𝜀 and 𝛿
   (j)   𝑉_,𝜙 = 1/𝑎² (-𝜙̄ʺ - 2𝓗𝜙̄ʹ) = 1/𝑎² [-𝓗𝜙̄ʹ(1 - 𝛿) - 2𝓗𝜙̄ʹ] = (𝛿 -3) 𝓗𝜙̄ʹ/𝑎²
Taking a time derivative on both sides we get    (k)  𝑉_,𝜙𝜙𝜙̄ʹ = 𝛿ʹ 𝓗𝜙̄ʹ/𝑎² + (𝛿 -3)(𝓗ʹ𝜙̄ʹ/𝑎² + 𝓗𝜙̄ʺ/𝑎² - 2 𝓗²𝜙̄ʹ/𝑎²) = 𝓗²𝜙̄ʹ/𝑎² [𝛿ʹ/𝓗 + (𝛿 -3)(𝓗ʹ/𝓗² + 𝜙̄ʺ/𝓗𝜙̄ʹ - 2)]
              = 𝓗²𝜙̄ʹ/𝑎² [𝛿ʹ/𝓗 -(𝛿 -3)(𝜀 + 𝛿)]     ⇒     -𝑎²𝑉_,𝜙𝜙/𝓗² = (𝛿 -3)(𝜀 + 𝛿) - 𝛿ʹ/𝓗.
Substituting this into (a) we obtain
   (l)  𝛿𝜙ʺ + 2𝓗𝛿𝜙ʹ - ∇²𝛿𝜙 = [(3 + 2𝜀 - 𝛿)(𝜀 - 𝛿) - 𝛿ʹ/𝓗] 𝓗²𝛿𝜙.
Finally, since we define 𝛿𝜙 ≡ 𝑓/𝑎, we can find
   (m)  𝛿𝜙ʹ = 1/𝑎 (𝑓ʹ - 𝓗𝑓),     𝛿𝜙ʺ = 1/𝑎 (𝑓ʺ - 2𝓗𝑓ʹ + 𝜀𝓗²𝑓)
   (o)  𝛿𝜙ʺ + 2𝓗𝛿𝜙ʹ = 1/𝑎 [𝑓ʺ - (2 - 𝜀)𝓗²𝑓].
So equation (a)-(8.5) becomes
   (p)  𝑓ʺ - ∇²𝑓 = [2 + 2𝜀 -3𝛿 + (2𝜀 - 𝛿)(𝜀 - 𝛿) -  𝛿ʹ/𝓗] 𝓗²𝑓.
Since we can change ∇²𝑓 to -𝑘²𝑓 for Fourier mode, we only need to prove the right-hand side is 𝑧ʺ/𝑧 𝑓.
   (q)  𝑧 ≡ 𝑎𝜙̄ʹ/𝓗,     𝑧ʹ = 𝑎ʹ𝜙̄ʹ/𝓗 + 𝑎𝜙̄ʺ/𝓗 - 𝑎𝜙̄ʹ/𝓗² 𝓗ʹ = 𝑎𝜙̄ʹ(1 + 𝜙̄ʺ/𝓗𝜙̄ʹ - 𝓗ʹ/𝓗²) = 𝑎𝜙̄ʹ(1 + 𝜀 - 𝛿).
   (r)   𝑧ʺ = (𝑎ʹ𝜙̄ʹ + 𝑎ʹ𝜙̄ʺ)(1 + 𝜀 - 𝛿) + 𝑎𝜙̄ʹ(𝜀ʹ - 𝛿ʹ) = 𝑎𝜙̄ʹ/𝓗 [(1 + 𝜙̄ʺ/𝓗𝜙̄ʹ)(1 + 𝜀 - 𝛿) + 𝜀ʹ/𝓗 - 𝛿ʹ/𝓗] 𝓗² = 𝑧 [(2 - 𝛿)(1 + 𝜀 - 𝛿) + 𝜀ʹ/𝓗 - 𝛿ʹ/𝓗] 𝓗².
   (s)   𝜀ʹ = 4π𝐺 [(𝜙̄ʹ)²/𝓗²]ʹ = 8π𝐺 [𝜙̄ʹ𝜙̄ʺ/𝓗² - (𝜙̄ʹ)²𝓗ʹ/𝓗³] = 8π𝐺 (𝜙̄ʹ)²/𝓗² (𝜙̄ʺ/𝓗𝜙̄ʹ - 𝓗ʹ/𝓗²) 𝓗 = 2𝜀(𝜀 - 𝛿) 𝓗
Substituting this into (r), we prove
   (t)   𝑧ʺ/𝑧 = [2 + 2𝜀 -3𝛿 + (2𝜀 - 𝛿)(𝜀 - 𝛿) -  𝛿ʹ/𝓗] 𝓗².
At last, we find
   (x)   𝑓ʺ + (𝑘² - 𝑧ʺ/𝑧)𝑓 = 0.  ▮

   To quantize the theory the equation of motion will not be sufficient, but we need to know the quadratic action from which it arises. Given (8.18), this action is fixed up to an overall normalization
    (8.21)   𝑆₂ = 𝑁/2 ∫ d𝜂 d³𝑥 [(𝑓ʹ)² - (∇𝑓)² + 𝑧ʺ/𝑧 𝑓²].
However, the conjugate momentum π ≡ 𝛿𝑆₂/𝛿𝑓ʹ = 𝑁𝑓ʹ depends on the unfixed constant 𝑁, and unless we determine 𝑁, the size of quantum fluctuations will not be fully specified. Fortunately there is simple way to find 𝑁. Consider the early-time limit, in which the perturbations are deep inside the horizon and the metric perturbations and potential are dynamically irrelevant in any reasonable gauge. In this limit, the action for inflaton fluctuations is simply given by the kinetic term in the unperturbed spacetime.
    (8.22)   𝑆₂^{(𝑘≫𝓗)} ≈ 1/2 ∫ d𝜂 d³𝑥 𝑎²[(𝛿𝜙ʹ)² - (∇𝛿𝜙)²] = ≈ 1/2 ∫ d𝜂 d³𝑥 𝑎²[(𝑓ʹ)² - (∇𝑓)²].
This shows that  𝑁 = 1 in (8.21) and conjugate momentum is π = 𝑓ʹ.

          8.1.2 From Micro to Macro
We note that (8.18) is the equation of a harmonic oscillator with a time-dependent frequency
    (8.23)   𝜔²(𝜂, 𝑘) ≡ 𝑘² - 𝑧ʺ/𝑧.
During the slow-roll inflation, 𝐻 and d𝜙̄/d𝑡 (and hence 𝜙̄ʹ/𝓗) are approximately constant and
    (8.24)   𝑧ʺ/𝑧 ≈ 𝑎ʺ/𝑎 ≈ 2𝓗²,
where we used that 𝑎ʹ = 𝑎²𝐻. We see that the inverse of 𝑧ʺ/𝑧 is a measure of the comoving Hubble radius 𝓗^-1, which for simplicity, will be referred to as the "horizon" (see discussion in Section 4.2).

•  At early times, 𝓗^-1 must be large and all modes are inside the horizon. Taking the limits 𝑘² ≫ ∣𝑧ʺ/𝑧∣, equation (8.18) reduces to
    (8.25)   𝑓ʺ + 𝑘²𝑓 = 0     (subhorizon).
   This is the equation of motion of a harmonic oscillator with a fixed frequency, 𝜔 = 𝑘, which has the solutions 𝑓 ∝ 𝑒^{±𝑖𝑘𝜂}. The amplitude of these oscillations will experience quantum mechanical zero-point fluctuations. We will study this in the next section.
•  A the comoving horizon shrinks, modes will eventually exit the horizon and heir evolution changes drastically (see Fig. 8.2). This can be seen by the frequency in (8.23) becoming imaginary on superhorizon scales. In physical coordinates, this evolution corresponds to microscopic scales being stretched outside of the constant Hubble radius 𝐻^-1 to be macroscopic cosmological fluctuations. Taking the limit 𝑘² ≪ ∣𝑧ʺ/𝑧∣, (8.18) now reads
    (8.26)   𝑓ʺ - 𝑧ʺ/𝑧 𝑓 = 0     (superhorizon).
   This has the growing mode solution 𝑓 ∝𝑧 and the decaying mode solution 𝑓 ∝𝑧^-2. As we will see now, the growing mode corresponds to a frozen curvature perturbation.
   Recall the definition of comoving perturbation (6.55), which in spatially flat gauge becomes
    (8.27)   𝓡 = -𝓗(𝑣 + 𝛣),
where the peculiar velocity was defined through the perturbed momentum density 𝛿𝛵^𝑖₀ = -(𝜌̄ + 𝑃̄)∂^𝑖𝑣. Given the energy-momentum tensor of a scalar field in (4.53), we infer that
    (8.28)   𝛿𝛵^𝑖₀ = 𝑔^𝑖𝜇∂_𝜇𝜙∂₀𝜙 = 𝑔^𝑖0(𝜙̄ʹ)² + 𝑔^𝑖𝑗𝜙̄ʹ∂_𝑗𝛿𝜙 = 𝑎^-2(𝜙̄ʹ)²∂^𝑖(𝛣 + 𝛿𝜙/𝜙̄ʹ).
Since (𝜌̄ + 𝑃̄) = 𝑎^-2(𝜙̄ʹ)², we have 𝛿𝛵^𝑖₀ = 𝑎^-2(𝜙̄ʹ)²∂^𝑖𝑣, so that 𝑣 + 𝛣 = - 𝛿𝜙/𝜙̄ʹ and hence     (8.29)   𝓡 = 𝓗/𝜙̄ʹ 𝛿𝜙 = 𝑓/𝑧   ^{(𝑘≪𝓗)}→   const,
which proves that the growing mode of 𝓡 is indeed frozen on large scales. Notice that (8.29) takes from 𝓡 = 𝐻𝛿𝑡, confirming the intuition that the curvature perturbation is induced by the time delay to the end  of inflation. moreover, since 𝓡 stays well-defined after inflation, while 𝛿𝜙 loses its meaning, it is the natural variable to track in the transition from inflation to the hot Big Bang.  

         8.2 Quantum Fluctuations
As we have just seen, the Fourier modes of the field fluctuations satisfy the equation of motion of a harmonic oscillator. In this section the quantization of these oscillators will be described.

         8.2.1 Quantum Harmonic Oscillators
We first study the simple case of a 1-dimensional harmonic oscillator in quantum mechanics. Consider a mass 𝑚 attached to a spring with spring constant 𝜅. Let 𝑞 be the deviation from the equilibrium point, whose equation is d²𝑞/dt² = - 𝜔²𝑞, where 𝜔 ≡ √(𝜅/𝑚)^* is the oscillation frequency. [^*correction: 𝜔² ≡ √(𝜅/𝑚) in the text → 𝜔 ≡ √(𝜅/𝑚)]. The quantization of the oscillator is a textbook problem in quantum mechani
   The Hamiltonian of the oscillator is
    (8.30)  𝐻 = 1/2 𝑝² + 1/2 𝜔²𝑞²,
where we rescaled the amplitude to set 𝑚 ≡ 1and introduced the conjugate momentum 𝑝 ≡ 𝑞̇. Finding the spectrum energy eigenstates of this Hamiltonian, we promote 𝑝 and 𝑞 to quantum operator 𝑝̂ and 𝑞̂ and impose the canonical commutation relation
    (8.31)  [𝑞̂, 𝑝̂] ≡ 𝑞̂𝑝̂ - 𝑝̂𝑞̂ = 𝑖ℏ,
where ℏ is the reduced Planck constant. An elegant way to solve for the spectrum of states is to express 𝑞̂ and 𝑝̂ in term of annihilation and creation operators (ladder operators in some textbooks):
    (8.32)  𝑞̂ = √(ℏ/2𝜔) (𝑎̂ + 𝑎̂^†),     𝑝̂ = -𝑖√(ℏ𝜔/2) (𝑎̂ - 𝑎̂^†),
where the commutation relation (8.31) is satisfied if
    (8.33)  [𝑎̂, 𝑎̂^†] = 1.
The Hamiltonian (8.30) can then be written as
    (8.34)  𝐻̂ = 1/2 𝑝̂² + 1/2 𝜔²𝑞̂² = 1/2 ℏ𝜔(𝑎̂𝑎̂^† + 𝑎̂^†𝑎̂) = ℏ𝜔(𝑎̂𝑎̂^† + 1/2).
The ground state of the oscillator is defined by
    (8.35)  𝑎̂∣0⟩ = 0,
and has energy 𝐸₀ ≡ ⟨0∣𝐻̂∣0⟩ = 1/2 ℏ𝜔. Excited states are construct by repeated application of the creation operator
    (8.36)  𝑎̂∣𝑛⟩ = 1/√𝑛! (𝑎̂^†)^𝑛∣0⟩,
and have energy 𝐸_𝑛 ≡ ⟨𝑛∣𝐻̂∣𝑛⟩ = ℏ𝜔(𝑛 +
1/2).    The expectation value of the position operator 𝑞̂ in the ground states ∣0⟩ vanishes,
    (8.37)  ⟨𝑞̂⟩ ≡ ⟨0∣𝑞̂∣0⟩ = √(ℏ/2𝜔) ⟨0∣𝑎̂ + 𝑎̂^†∣0⟩ = 0,
because 𝑎̂ annihilates ∣0⟩ when acting on it from the left, and 𝑎̂^† annihilates ⟨0∣ when acting on it from the right. However the expectation value of the square of the position operator receives finite zero-point fluctuations:
    (8.38)  ⟨∣𝑞̂∣²⟩ ≡ ⟨0∣𝑞̂^†𝑞̂∣0⟩ = ℏ/2𝜔⟨0∣(𝑎̂^† + 𝑎̂)(𝑎̂ + 𝑎̂^†)∣0⟩ = ℏ/2𝜔⟨0∣𝑎̂𝑎̂^†∣0⟩ = ℏ/2𝜔⟨0∣[𝑎̂, 𝑎̂^†] + 𝑎̂^†𝑎̂∣0⟩ = ℏ/2𝜔 ⟨0∣[𝑎̂, 𝑎̂^†]∣0⟩ = ℏ/2𝜔.
These zero-point fluctuations play an important role in many areas of physics and, in particular, seem to be the origin of all structure in the universe.

Uncertainty principle   A heuristic way to derive the result (8.38) is from the uncertainty principle     (8.39)  ∆𝑞̂∆𝑝̂ = ℏ/2,
where ∆𝑞̂ and ∆𝑝̂ are the uncertainties in the position and momentum of the oscillator, respectively. The energy of the quantum harmonic oscillator due to these quantum fluctuation the is
    (8.40)  𝐸 = ℏ²/8(∆𝑞̂)² + 1/2 𝜔²(∆𝑞̂)².
This energy is minimized for
    (8.41)  d𝐸/d(∆𝑞̂)²) = - ℏ²/8(∆𝑞̂)⁴ + 1/2 𝜔² = 0    ⇒      ∆𝑞̂ = √(ℏ/2𝜔),
reproducing our previous result (8.38). Substituting his back into (8.40) gives 𝐸₀ = 1/2 ℏ𝜔, the zero-point energy of the oscillator in the ground state.  ▮

   The only subtlety for applying it to inflation will be that the frequency of the oscillations is time dependent, 𝜔 = 𝜔(𝑡). It is most convenient to describe this time dependence in the Heisenberg picture where operators vary in time , while states are time independent. An operator 𝑂̂ in the Heisenberg picture satisfies the following equation:
    (8.42)   d𝑂̂/d𝑡 = 𝑖/ℏ [𝐻̂, 𝑂̂],
where 𝐻̂ is the Hamiltonian.

Exercise 8.5   Applying (8.42) to the operators 𝑞̂ and 𝑝̂, show that
    (8.42)   d²𝑞̂/d𝑡² + 𝜔²𝑞̂ = 0,
   i.e. the quantum operator 𝑞̂ obeys the same equation of motion as classical variable 𝑞.
[Solution]   According to Heisenberg picture equation we have
   (a)   d𝑞̂/d𝑡 = 𝑖/ℏ [𝐻̂, 𝑞̂],     d𝑝̂/d𝑡 = 𝑖/ℏ [𝐻̂, 𝑝̂]
Since 𝐻̂ = 1/2 𝑝̂² + 1/2 𝜔²𝑞̂
   (b)   [𝐻̂, 𝑞̂] = 1/2 [𝑝̂², 𝑞̂] = [𝑝̂, 𝑞̂] 𝑝̂ = -𝑖ℏ 𝑝̂,
   (c)   [𝐻̂, 𝑝̂] = 1/2 𝜔²[𝑞̂², 𝑝̂] = 𝜔²[𝑞̂, 𝑝̂] 𝑞̂ = 𝑖ℏ𝜔² 𝑞̂,    So we get
   (d)   d𝑞̂/d𝑡 = 𝑝̂,     d𝑝̂/d𝑡 = -𝜔²𝑞̂,
Hence we find
   (e)   d²𝑞̂/d𝑡² + 𝜔²𝑞̂ = 0.  ▮

   We now write the position operator as
    (8.44)   𝑞̂ = 𝑞(𝑡)𝑎̂(𝑡_𝑖) + 𝑞^*(𝑡)𝑎̂^†(𝑡_𝑖),
where the complex mode function 𝑞(𝑡) satisfies d²𝑞/d𝑡² + 𝜔²(𝑡)𝑞 = 0. The annihilation operator 𝑎̂(𝑡_𝑖)-and hence the associated vacuum state ∣0⟩-depends on the time 𝑡_𝑖 at which it is defined. In the inflationary application, there will be preferred moment at which we define the vacuum state, namely the beginning of inflation.
   Substituting (8.44) into (8.31), we get
    (8.45)   [𝑞̂, 𝑝̂] = (𝑞𝑞̇^* - 𝑞̇𝑞^*) = 𝑖ℏ.
In order for this to give [𝑎̂, 𝑎̂^†] = 1, we require the mode function  to obey the normalization
    (8.46)   𝑞𝑞̇^* - 𝑞̇𝑞^* = 𝑖ℏ.
To completely fix the mode function and hence give a unique meaning to the operator 𝑎̂(𝑡_𝑖) and the corresponding vacuum state ∣0⟩, we need to impose a second condition on the solution 𝑞(𝑡).
   As we saw in (8.25), at early times, the inflaton fluctuations satisfy the equation of motion of a harmonic oscillator with a fixed frequency. To model the situation during inflation, we consider a harmonic oscillator whose frequency initially a constant and then slowly develops a time dependence. We take the initial condition to be the ground  state of the fixed -frequency oscillator (which will include the zero-point fluctuations discussed above) and then use the equation of motion with a time-dependent frequency to evolve these fluctuations forward in time.
   To see how a preferred mode function arises for a harmonic oscillator with a fixed frequency, let us write the most general solution as
    (8.47)   𝑞(𝑡) = 𝑟(𝑡)𝑒^{𝑖𝑠(𝑡)},
where 𝑟(𝑡) and 𝑠(𝑡) are real functions of time, and (8.46) implies
    (8.48)   𝑠̇ = - ℏ/2𝑟².
We would like to determine the solution that minimizes the vacuum expectation value of the Hamiltonian
    (8.49)   ⟨0∣𝐻̂∣0⟩ = 1/2 (∣𝑞̇∣² + 𝜔²∣𝑞∣²) = 1/2 (𝑟̇² + 𝑟²𝑠̇² + 𝜔²𝑟²).
Substituting (8.48), we get
    (8.50)   ⟨0∣𝐻̂∣0⟩ = 1/2 (𝑟̇² + ℏ²/4𝑟² + 𝜔²𝑟²).
This is minimized for 𝑟̇ = 0 and [If we apply d/d𝑟² at both sides of equality, then we get]
    (8.51)   0 = d/d𝑟² (ℏ²/4𝑟² + 𝜔²𝑟²) = -ℏ²/4(𝑟²)² + 𝜔²      ⇒     𝑟 = √(ℏ/2𝜔)
Substituting this back into (8.48), we find
    (8.52)   𝑠̇ = -𝜔     ⇒     𝑠 = -𝜔𝑡 + const.
Up to an irrelevant phase, we therefore get
    (8.53)   𝑞(𝑡) = √(ℏ/2𝜔) 𝑒^{-𝑖𝜔𝑡}    (𝜔 = const).
We see that the ground state corresponds to the positive-frequency solution, 𝑒^{-𝑖𝜔𝑡}, while adding any amount of the negative frequency solution, 𝑒^{+𝑖𝜔𝑡}, would raise the energy.
   Using the solution (8.53) as an initial condition at 𝑡 = 𝑡_𝑖 uniquely fixes the solution to d²𝑞/d𝑡² = 𝜔²(𝑡)𝑞. Of course the explicit solution depends on 𝜔(𝑡), which in the case of inflation is determined by the evolution of the inflationary background. Repeating the manipulations that led to (8.38), we get that the variance of the position operator is simply the square of the mode function:
    (8.54)   ⟨∣𝑞̂∣²⟩ = ∣𝑞(𝑡)∣².
This describes how the initial zero-point fluctuations, specified in (8.38), evolve in time.

         8.2.2 Inflationary Vacuum Fluctuations
We are now ready to apply the quantization procedure to the field fluctuations during inflation. for simplicity, we will work in the slow-roll approximation, taking the background spacetime to be de Sitter space, with 𝐻 = const and 𝑎 = -(𝐻𝜂)^-1,, The Mukhanov-Sasaki equation (8.18) then takes the form
    (8.55)   𝑓_𝐤ʺ + (𝑘² - 2/𝜂²)𝑓_𝐤 = 0.
In spatially flat gauge, this is a good approximation before horizon crossing. However on superhorizon scales slow-roll corrections are the leading effect. For example, ignoring the time dependence of expansion rate will lead to a fictitious evolution of the curvature perturbation 𝓡 on large scales. To minimize this error, we will evaluate all fluctuations at the moment of horizon crossing, 𝑘 = 𝑎𝐻, where we match them to the conserved curvature perturbation. Includinf a time-dependent expansion rate 𝐻(𝑡), in the horizon crossing condition then extends our results to a quasi-de Sitter background. We also include the weak time dependence in the conversion from 𝛿𝜙 to 𝓡 in (8.29). In turns out that this simple prescription gives the right answer, as we will prove later by a more rigorous treatment.

          Canonical quantization
The quantization of the field fluctuation s is essentially the same as for the harmonic oscillator. We first promote the field 𝑓(𝜂, 𝐱) and its conjugate momentum 𝜋(𝜂, 𝐱) to quantum operators 𝑓̂(𝜂, 𝐱) and 𝜋̂(𝜂, 𝐱), and then impose
    (8.56)   [𝑓̂(𝜂, 𝐱), 𝜋̂(𝜂, 𝐱ʹ)] = 𝑖 𝛿_𝐷(𝐱 - 𝐱ʹ),
where we set ℏ ≡ 1. This is the field theory equivalent of (8.31). The delta function enforces locality: modes at different points in space are independent and the corresponding operators therefore commute. In Fourier space, we find
    (8.57)   [𝑓̂(𝜂, 𝐱), 𝜋̂(𝜂, 𝐱ʹ)] = ∫ d³𝑥 ∫ d³𝑥ʹ [𝑓̂(𝜂, 𝐱), 𝜋̂(𝜂, 𝐱ʹ)] 𝑒^{-𝑖𝐤⋅𝐱}𝑒^{-𝑖𝐤ʹ⋅𝐱ʹ} = 𝑖 ∫ d³𝑥 𝑒^{-𝑖(𝐤+𝐤ʹ)⋅ 𝐱} = 𝑖(2π)³𝛿_𝐷(𝐤 + 𝐤ʹ),
where delta function implies that modes with different wavelength commute.
   As before we are working in the Heisenberg picture where operators vary in time, while states are time independent . The operator solution 𝑓̂_𝐤(𝜂) is determined by two initial conditions 𝑓̂_𝐤(𝜂_𝑖) and 𝜋̂_𝐤(𝜂_𝑖) = ∂_𝜂𝑓̂_𝐤(𝜂_𝑖), and since the evolution equation is linear, the solution is linear in the operators. It is convenient to trade 𝑓̂_𝐤(𝜂_𝑖) and 𝜋̂_𝐤(𝜂_𝑖) for a single non-Hermitian operator 𝑎̂_𝐤(𝜂_𝑖), so that
    (8.58)   𝑓̂_𝐤(𝜂) = 𝑓_𝑘(𝜂)𝑎̂_𝐤 + 𝑓_𝑘^*(𝜂)𝑎̂^†_-𝐤,
where the complex mode function 𝑓_𝑘(𝜂) satisfies the classical equation of motion. This is the same as the mode expansion in (8.44) with some Fourier labels added. Since the operator 𝑓̂(𝜂, 𝐱) is Hermitian, its Fourier transform must satisfy 𝑓̂_𝐤(𝜂)^† = 𝑓̂_-𝐤(𝜂). This explains the -𝐤 on 𝑎̂^†_-𝐤. We can then write 𝑓̂(𝜂, 𝐱) as
    (8.59)   𝑓̂(𝜂, 𝐱) = ∫ d³𝑘/(2π)³ 𝑓̂_𝐤(𝜂) 𝑒^{𝑖𝐤⋅𝐱} = ∫ d³𝑘/(2π)³ [𝑎̂_𝐤𝑓_𝑘(𝜂) 𝑒^{𝑖𝐤⋅𝐱} + 𝑎̂^†_𝐤𝑓_𝑘^*(𝜂) 𝑒^{-𝑖𝐤⋅𝐱}],
which is manifestly Hermitian. The mode function 𝑓_𝑘(𝜂) depends only on the magnitude of the wavevector, 𝑘 ≡ ∣𝐤∣.¹
   As in (8.46), we choose the normalization of the mode functions, so that
    (8.60)   𝑓_𝑘𝑓_𝑘ʹ^* - 𝑓_𝑘ʹ𝑓_𝑘^* ≡ 𝑖.
Substituting (8.58) into (8.57), we then get
    (8.61)   [𝑎̂_𝐤, 𝑎̂^†_𝐤ʹ] = (2π)³𝛿_𝐷(𝐤 - 𝐤ʹ),
which is the field theory generalization of (8.33).

          Choice of vacuum
The vacuum state is defined in the usual way
    (8.62)   𝑎̂_𝐤(𝜂_𝑖∣0⟩ = 0.
However, because the Hamiltonian is time dependent, the choice of the vacuum state depends on the time 𝜂_𝑖 at which it is defined. It is natural to define that  vacuum as the ground state of the Hamiltonian as the beginning of inflation. Formally, we take the limit 𝜂_𝑖 → -∞, but any time satisfying ∣𝑘𝜂_𝑖∣ ≫ 1 would work. At this time , all modes were deep inside the horizon and satisfy the equation of a harmonic oscillator with a fixed frequency 𝜔_𝑘 → 𝑘, cf. (8.25). For each Fourier mode, we choose the natural vacuum state of the simple harmonic oscillator. Above we derived  the preferred mode function corresponding to a harmonic oscillator in the ground state, cf. (8.53). We use this mode function as the initial condition for each Fourier mode:
    (8.63)   lim_{𝑘𝜂→-∞} 𝑓_𝑘(𝜂) = 1/√(2𝑘) 𝑒^{-𝑖𝑘𝜂}.
Solving the Mukhanov-Sasaki equation (8.55)  with this boundary condition leads to a unique mode function called the  Bunch-Davies mode function:
    (8.64)   𝑓_𝑘(𝜂) = 1/√(2𝑘) (1 - 𝑖/𝑘𝜂) 𝑒^{-𝑖𝑘𝜂}.
The corresponding state is the Bunch-Davies vacuum.    Figure 8.3 shows the evolution of the imaginary part of the comoving curvature perturbation, 𝓡 = 𝑓/𝑧, given the solution (8.64). As expected, the solution oscillates at early times and freezes after horizon crossing. The initial amplitude of the oscillations is set by quantum zero-point fluctuations.

Exercise 8.6   Show that (8.64) implies
    (8.65-7)  Re[𝓡] = 𝛢[sin(𝑥) - 𝑥 cos(𝑥)],    Im[𝓡] = 𝛢[cos(𝑥) + 𝑥 sin(𝑥)],     [𝓡] = 𝛢√(1 + 𝑥²),
   where 𝑥 ≡ 𝑘𝜂.
[Solution]   According to (8.8), since in (8.55) 𝐻 = const and 𝑎 = -(𝐻𝜂)^-1,
   (a)   𝑧 ≡ 𝑎𝜙̄ʹ/𝓗 = 𝑎√(𝜀/4π𝐺) = -√(𝜀/4π𝐺𝐻²) 1/𝜂.
   (b)   𝓡 = 𝑓/𝑧 = -√[(4π𝐺𝐻²)/(2𝜀𝑘³)] 𝑘𝜂 (1 - 𝑖/𝑘𝜂) 𝑒^-𝑘𝜂 = √[(4π𝐺𝐻²)/(2𝜀𝑘³)] (𝑖 - 𝑘𝜂) 𝑒^-𝑘𝜂   ⇒    𝓡 = 𝛢(𝑖 - 𝑥) 𝑒^-𝑖𝑥
where 𝛢 ≡ √[(4π𝐺𝐻²)/(2𝜀𝑘³)] = const.
   (c)   Re[𝓡] = 1/2 [𝓡 + 𝓡^*] = 𝛢/2[(𝑖 - 𝑥) 𝑒^-𝑖𝑥 + (-𝑖 - 𝑥)𝑒^𝑖𝑥] = 𝛢 [(𝑒^𝑖𝑥 - 𝑒^-𝑖𝑥)/2𝑖 - 𝑥 (𝑒^𝑖𝑥 + 𝑒^-𝑖𝑥)/2] = 𝛢[sin(𝑥) - 𝑥 cos(𝑥)].
   (d)   Im[𝓡] = 1/2 [𝓡 - 𝓡^*] = 𝛢/2𝑖[(𝑖 - 𝑥) 𝑒^-𝑖𝑥 - (-𝑖 - 𝑥)𝑒^𝑖𝑥] = 𝛢 [(𝑒^𝑖𝑥 + 𝑒^-𝑖𝑥)/2 + 𝑥 (𝑒^𝑖𝑥 - 𝑒^-𝑖𝑥)/2𝑖] = 𝛢[cos(𝑥) + 𝑥 sin(𝑥)].
   (e)   ∣𝓡∣ = √(𝓡^*𝓡) = 𝛢√[(-𝑖 - 𝑥)((𝑖 - 𝑥)] = 𝛢√(1 + 𝑥²).  ▮

          Zero-point fluctuation
Finally, we can predict the quantum statistics of operator
    (8.68)   𝑓̂(𝜂, 𝐱) = ∫ d³𝑘/(2π)³ [𝑓_𝑘(𝜂)𝑎̂_𝐤 + 𝑓_𝑘^*(𝜂)𝑎̂^†_-𝐤] 𝑒^{𝑖𝐤⋅𝐱}.
The expectation value of 𝑓̂ vanishes, i.e. ⟨𝑓̂⟩ ≡ ⟨0∣𝑓̂∣0⟩ =0. However, the variance of inflaton fluctuations receives nonzero quantum fluctuations:
    (8.69)   ⟨∣𝑓̂∣²⟩ ≡ ⟨0∣𝑓̂(𝜂, 𝟬)𝑓̂(𝜂, 𝟬)∣0⟩ = ∫ d³𝑘/(2π)³ ∫ d³𝑘ʹ/(2π)³ ⟨0∣(𝑓_𝑘^*(𝜂)𝑎̂^†_-𝐤 + 𝑓_𝑘(𝜂)𝑎̂_𝐤)(𝑓_𝑘ʹ(𝜂)𝑎̂_𝐤ʹ + 𝑓_𝑘ʹ^*(𝜂)𝑎̂^†_-𝐤ʹ)∣0⟩
                  = ∫ d³𝑘/(2π)³ ∫ d³𝑘ʹ/(2π)³ 𝑓_𝑘(𝜂)𝑓_𝑘ʹ^*(𝜂) ⟨0∣𝑎̂_𝐤, 𝑎̂^†_-𝐤ʹ∣0⟩ = ∫ d³𝑘/(2π)³ ∣𝑓_𝑘(𝜂)∣² = ∫ d ln 𝑘 𝑘³/2π² ∣𝑓_𝑘(𝜂)∣².
We see that the variance of the quantum fluctuations is determined by the (dimension less) power spectrum of the classical solution:
    (8.70)   ∆²_𝑓(𝑘, 𝜂) ≡ 𝑘³/2π² ∣𝑓_𝑘(𝜂)∣².
Substituting the Bunch-Davis mode function (8.64) for 𝑓 = 𝑎 𝛿𝜙, we find
    (8.71)   ∆²_𝛿𝜙(𝑘, 𝜂) = ∆_𝑓²(𝑘, 𝜂)/𝑎²(𝜂) = (𝐻/2π)² [1 + (𝑘𝜂)²] ^𝑘𝜂→0→ (𝐻/2π)².
Note that in the superhorizon limit, 𝑘𝜂 → 0, the dimensionless power spectrum ∆_𝛿𝜙² approaches the same constant for all momenta, which is the characteristic of a scale-invariant spectrum.
   Although the time dependence of the Hubble rate during inflation is small, it is crucial that it is nonzero. Otherwise, there wouldn't be any evolution and inflation would never end. Moreover, curvature perturbations would be ill-defined since 𝓡 ∝ Ḣ^-1 → ∞. To incorporate the effect of a varying 𝐻(𝑡), we evaluate the power spectrum (8.71) at horizon crossing, 𝑘 = 𝑎𝐻, which for each Fourier mode, corresponds to a different moment in time. Since 𝐻 is evolving, this leads to a slight scale dependence of the spectrum
    (8.72)   ∆²_𝛿𝜙(𝑘) ≈ [𝐻(𝑡)/2π]²∣_{𝑘=𝑎𝐻(𝑡)}.
Because 𝐻(𝑡) decreases during inflation, the amplitude of fluctuations will be slightly larger for long-wavelength fluctuations which exit the horizon at the beginning of inflation.

          From quantum to classical
An important consequence of the stretching of fluctuations from microscopic to macroscopic scales is the fact that the initial quantum fluctuations become classical fluctuations after horizon crossing. To see this, note from (8.64) that the mode function 𝑓_𝑘(𝜂) becomes purely imaginary on the superhorizon scales:
    (8.73)   Re[𝑓_𝑘] → 0,     [𝑓_𝑘] ∝ 𝑎(𝜂).
The field operator and its conjugate momentum then becomes proportional to each other:
    (8.74)   𝑓̂_𝐤(𝜂) ^𝑘𝜂→0→ 𝑓_𝑘(𝜂)(𝑎̂_𝐤 - 𝑎̂^†_-𝐤),
    (8.75)   𝜋̂_𝐤(𝜂) ^𝑘𝜂→0→ 𝑓_𝑘ʹ(𝜂)(𝑎̂_𝐤 - 𝑎̂^†_-𝐤) ≈ 𝑓_𝑘ʹ(𝜂)/𝑓_𝑘(𝜂) 𝑓̂_𝐤(𝜂).
While both 𝑓̂ ∝ 𝑎(𝜂) and ∆ ∝ 𝑎²(𝜂) grow outside the horizon, the commutator [𝑓̂, 𝜋̂] stays constant. Since the uncertainty ∆𝑓̂∆𝜋̂ is proportional to [𝑓̂, 𝜋̂], a state ∣𝑓_𝑘⟩ with a definite field value has an almost definite momentum. Using (8.64), we can compute the ratio
    (8.76)   𝑅 ≡ ∆𝑓̂∆𝜋̂/∣𝑓̂∣∣𝜋̂∣ = (ℏ/2)/[ℏ/2 √{1 + (𝑘𝜂)^-6}] = 1/√[1 + (𝑘𝜂)^-6] ^𝑘𝜂→0→  (𝑘𝜂)³ ≪ 1.
We see that the quantum uncertainty is large (𝑅 ~ 1) at early times, but becomes very small (𝑅 ≪ 1) at late times and on large scales.

         8.3 Primordial Power Spectra
Our final task is to relate the fluctuations in the inflaton field to the observable fluctuations after inflation. As we described in Section 6.2.4, the bridge between inflation and the late universe is provided by the primordial curvature perturbations. In this section, we will map the spectrum of inflaton fluctuations 𝛿𝜙 to the spectrum of the comoving curvature perturbation 𝓡. We will also derive the spectrum of primordial tensor fluctuations.

         8.3.1 Curvature Perturbations
Recall that, in spatially flat gauge, 𝓡 and 𝛿𝜙 are related by a simple conversion factor, cf. (8.29). The power spectrum of 𝓡 can therefore be written as
    (8.77)   ∆²_𝓡 = [𝐻/(d𝜙̄/d𝑡)]²∆_𝛿𝜙,
and, using (8.72), we get
    (8.78)   ∆²_𝓡(𝑘) = [𝐻²/{2π(d𝜙̄/d𝑡)²}]²∣_{𝑘=𝑎𝐻} =  [1/8π²𝜀][𝐻²/𝑀_𝑃𝑙²]∣_{𝑘=𝑎𝐻}.
The time dependence of the conversion factor-captured by the slow-roll parameter 𝜀(𝑡)-introduces an additional source of scale dependence. The scalar spectral index is defined as
    (8.79)   𝑛_𝑠 - 1 ≡ d ln ∆²_𝓡(𝑘)/d ln 𝑘,
where the shift by -1 is a historical accident, cf. Section 5.3.3. Using 𝑘 = 𝑎𝐻, we get
    (8.80-1)   𝑛_𝑠 - 1 = d ln ∆²_𝓡(𝑘)/d ln (𝑎𝐻) ≈ d ln ∆²_𝓡(𝑘)/d ln 𝑎 = d ln ∆²_𝓡(𝑘)/𝐻 d𝑡 = (2 d ln 𝐻)/(𝐻 d𝑡) - d ln 𝜀/(𝐻 d𝑡) = 2 Ḣ/𝐻² - 𝜀̇/𝐻𝜀,
which in terms of the Hubble slow-roll parameters defined in (4.34) and (4.36) becomes
    (8.82)   𝑛_𝑠 - 1 = -2𝜀 - 𝜅.
In the following insert, it will be shown how the same result ca be also be derived from a systematic slow-roll expansion of the Mukhanov-Sasaki equation.

Slow-roll expansion   The derivation above involved elements that didn't look completely rigorous. So we will show how the same result can be obtained from a systematic slow-roll expansion. Let us begin with Mukhanov-Sasaki equation (8.18):
    (8.83)   𝑓ʺ + (𝑘² - 𝑧ʺ/𝑧) 𝑓 = 0,     where     𝑧 ≡ 𝑎𝜙̄ʹ/𝓗 = 𝑎√(2𝜀) 𝑀_𝑃𝑙,
    (8.84)   𝑧ʹ/𝑧 = 𝓗 [1 + 1/2 𝜅],
    (8.85)   𝑧ʺ/𝑧 = 𝓗² [2 - 𝜀 + 3/2 𝜅],
where (8.84) is exact and (8.85) is valid to first order in slow-roll parameters. Since in (8.8) 𝜀 = 1 - 𝓗ʹ/𝓗² ≈ const, we get
    (8.86)   𝓗 = -1/𝜂 (1 + 𝑧),
    (8.87)   𝑧ʺ/𝑧 = 1/𝜂² [2 + 3(𝜀 + 1/2 𝜅)].
    (8.88)   𝑓ʺ + (𝑘² - (𝜈² - 1/4)/𝜂²) 𝑓 = 0,     where     𝜈 ≡ 3/2 + 𝜀 + 1/2 𝜅.
Imposing the normalization (8.60) and the Bunch-Davis initial condition (8.63), we get the solution
    (8.89)   𝑓_𝑘(𝜂) = √π/2 √-𝜂 𝐻_𝜈⁽¹⁾(-𝑘𝜂),
where 𝐻_𝜈⁽¹⁾ is a Hankel function of the first kind (see Appendix D). This has the correct early-time limit because
    (8.90)   lim_{𝑘𝜂→-∞} ∣𝐻_𝜈^(1,2)(-𝑘𝜂)∣ = √(2/π) 1/√(-𝑘𝜂) 𝑒^{∓𝑖𝑘𝜂}.
To compute the power spectrum of 𝓡 = 𝑓/𝑧, we need 𝑧(𝜂). Substituting (8.86) into (8.84), and integrating the resulting expression, we find
    (8.91)   𝑧(𝜂) =number  𝑧_*(𝜂/𝜂_*)^{1/2 - 𝜈},
where 𝜂_* is some reference time. It will be convenient to take 𝜂_* = -𝑘_*^-1, i.e. the moment of horizon crossing of a mode of wavenumber 𝑘_*. The power spectrum then is
    (8.92)   ∆²_𝓡(𝑘) = 𝑘³/2π² 1/𝑧²(𝜂) ∣𝑓_𝑘(𝜂)∣² = 𝑘³/2π² 1/(2𝜀_*𝑀_𝑃𝑙²𝑎_*²) (-𝑘_*𝜂)^{2𝜈 - 1} π/4 (-𝜂)∣𝐻_𝜈(1)(-𝑘𝜂)²∣.
Using 𝑎_* = 𝑘_*/𝐻_* and the late-time limit of the Hankel function,
    (8.93)   lim_{𝑘𝜂→-∞} ∣𝐻_𝜈⁽¹⁾(-𝑘𝜂)∣² =  [2^2𝜈𝛤(𝜈)²]/π² (-𝑘𝜂)^-2𝜈 ≈ 2/π (-𝑘𝜂)^-2𝜈,
we get
    (8.94)   ∆²_𝓡(𝑘) = 1/(8π²𝜀_*) 𝐻_*²/𝑀_𝑃𝑙² (𝑘/𝑘_*)^{3 - 2𝜈},
which is equivalent to our previous result (8.78). Note that the time dependence in (8.92) have canced, as it had to because 𝓡 is conserved on superhorizon scales. The exponent in (8.94) is precisely the spectral index we found before
    (8.95)   𝑛_𝑠 - 1 ≡ 3 - 2𝜈 = -2𝜀 - 𝜅,
where we used the definition of 𝜈 in (8.88).  ▮

   In summary, we have found that the power spectrum of the curvature perturbation 𝓡 takes a power law form:
    (8.96)   ∆²_𝓡(𝑘) = 𝛢_𝑠 (𝑘/𝑘_*)^{𝑛_𝑠 - 1},
where the amplitude and the spectral index are
    (8.97)   𝛢_𝑠 ≡ 1/(8π²𝜀_*) 𝐻_*²/𝑀_𝑃𝑙²,
    (8.98)   𝑛_𝑠 ≡ 1 - 2𝜀_* - 𝜅_*.
All scale quantities on the right side are evaluated at the time when the reference 𝑘_* excited the horizon. The observational constraints on the scalar amplitude and spectral index are 𝛢_𝑠 = (2.098 ± 0.023) × 10^-9 and 𝑛_𝑠 = 0.965 ± 0.004 (for 𝑘_* = 0.05 Mpc^-1) [3]. The observed percent-level deviation from the scale-invariant value, 𝑛_𝑠 = 1, is the first direct measurement of the time dependence during inflation.
         8.3.2 Gravitational Waves
Arguably the most robust and model-independent prediction of inflation is a stochastic background of gravitational waves (GWs), or tensor metric perturbations 𝛿𝑔_𝑖𝑗 = 𝑎²𝘩_𝑖𝑗. In Section 6.5 we reviewed the mathematical properties of cosmological GWs, and in Chapter 7, we discussed their effect on the CMB anisotropies. We will now complete the story by deriving the tensor power spectrum predicted by inflation, which we define that ⟨𝘩²_𝑖𝑗⟩ = ∫ d ln 𝑘 ∆²_𝑖𝑗(𝑘).
  In Problem 6.4 of Chapter 6 we showed that tensor fluctuations in an FRW background satisfy the following equation:
    (8.99)   𝘩ʺ_𝑖𝑗 + 2𝓗𝘩ʹ_𝑖𝑗 - ∇²𝘩_𝑖𝑗 = 0.
This equation arises from the action
    (8.100)  𝑆₂ = 𝑁/2 ∫ d𝜂 d³𝑥 𝑎² [(𝘩ʹ_𝑖𝑗)² - (∇²𝘩_𝑖𝑗)²].
Expanding the Einstein Hilbert action to second order in the tensor leads to 𝑁 = 𝑀_𝑃𝑙²/4.

Derivation   We first note that the Einstein-Hilbert action can be written as [4]-(P. Dirac General Theory of Relativity)
    (8.101)  𝑆 = 𝑀_𝑃𝑙²/2 ∫ d⁴ 𝑥 √-𝑔 𝑅
    (8.102)    = 𝑀_𝑃𝑙²/2 ∫ d⁴ 𝑥 √-𝑔 𝑔^𝜇𝜈𝛤^𝛼_𝜇[𝛽𝛤^𝛽_𝜈]𝛼 + boundary term,
where the square brackets denote antisymmetrization of the indice (i.e. 𝛤^𝛼_𝜇[𝛽𝛤^𝛽_𝜈]𝛼 = 𝛤^𝛼_𝜇𝛽𝛤^𝛽_𝜈𝛼 - 𝛤^𝛼_𝜇𝜈𝛤^𝛽_𝛽𝛼). The first term in the second line  is called the Schrödinger form fo the action. A nice feature of writing the action in this way is that it contains only first derivatives of the metric. To find the overall normalization of the quadratic action for tensors, we evaluate (8.102) in the subhorizon limit. In this limit, we can ignore the time dependence of the scale factor and the nonzero Christoffel symbols are
    (8.103)  𝛤⁰_𝑖𝑗 = 𝛤^𝑖_0𝑗 = 𝛤^𝑖_𝑗0 = 1/2 𝘩ʹ_𝑖𝑗,     𝛤^𝑘_𝑖𝑗 = 1/2 (∂_𝑖𝘩^𝑘_𝑗 + ∂_𝑗𝘩^𝑘_𝑖 - ∂^𝑘𝘩_𝑖𝑗). Substituting these into (8.102), we find
    (8.104)   𝑆^{(𝑘≫𝓗)}₂ = 𝑀_𝑃𝑙²/8 ∫ d𝜂d³𝑥 𝑎² [(𝘩ʹ_𝑖𝑗)² - (∇²𝘩_𝑖𝑗)²],
where we have used that 𝘩_𝑖𝑗 is transverse and traceless, and dropped the total derivative term 2∂^𝑘𝘩_𝑖𝑗∂^𝑖𝘩_𝑘𝑗 = 2∂^𝑘(𝘩_𝑖𝑗∂^𝑖𝘩_𝑘𝑗). Although (8.104) was derived in the subhorizon limit, we see from (8.99) that it is valid on all scales. Comparison with (8.100) shows that 𝑁 = 𝑀_𝑃𝑙²/4.  ▮

   It is convenient to use rotational symmetry to align the 𝑧-axis of the coordinate system with the momentum of the mode, i.e. 𝐤 ≡ (0, 0, 𝑘), and write
    (8.105)   𝑀_𝑃𝑙²/√2 𝑎 𝘩_𝑖𝑗 ≡ ⌈ 𝑓₊  𝑓_×  0 ⌉
                                            ┃ 𝑓_× -𝑓₊  0 ┃
                                             ⌊ 0    0    0 ⌋
where 𝑓₊ and 𝑓_× describe the two polarization modes of the gravitational wave. Note that 𝘩²_𝑖𝑗 = (4/𝑀_𝑃𝑙²)(𝑓²₊ + 𝑓²_×)/𝑎². The action (8.100) then becomes
    (8.106)   𝑆₂ = 1/2 {_𝜆=+,× ∫ d𝜂 d³𝑥 [(𝑓ʹ_𝜆)² - (∇𝑓_𝜆)² + 𝑎ʺ/𝑎ʹ 𝑓²_𝜆],
which is just two copies of the action of a massless scalar field, cf (8.21). The power spectrum of tensor fluctuations is simply a rescaling of the result for the inflation fluctuations
    (8.107)   ∆²_𝘩(𝑘) = 2 × 4/𝑀_𝑃𝑙² × ∆²_𝛿𝜙(𝑘, 𝜂)∣_{𝑘=𝑎𝐻},
where the factor of 2 accounts for the sum over the two polarization modes. Using (8.72), we get
    (8.108)  ∆²_𝘩(𝑘) = 2/π² (𝐻/𝑀_𝑃𝑙)²∣_{𝑘=𝑎𝐻}.
Notice that the tensor power spectrum is a direct measure of the expansion rate 𝐻 during inflation, while the scalar power spectrum depends on both 𝐻 and 𝜀. The time dependence of the expansion rate 𝐻(𝑡) determines the tensor spectral index:
    (8.109)  𝑛_𝑡 ≡ (d ln ∆²_𝘩)/(d ln 𝑘) = (d ln ∆²_𝘩)/(d ln 𝑎𝐻) ≈ (d ln ∆²_𝘩)/(d ln 𝑎) = (d ln ∆²_𝘩)/𝐻d𝑡 = 2Ḣ/𝐻² = -2𝜀.

Slow-roll expansion   The equation of motion for each polarization mode is
    (8.110)  𝑓ʺ_𝑘 + (𝑘² - 𝑎ʺ/𝑎)𝑓_𝑘 = 0,
where the effective mass can be written as
    (8.111)  𝑎ʺ/𝑎 = (𝜈² - 1/4)/𝜂²,      with     𝜈 ≈ 3/2 + 𝜀.
The Bunch-Davis mode function is again given by (8.89). The superhorizon limit of the power spectrum of tensor fluctuation then is
    (8.112)   ∆²_𝘩(𝑘) = 2 × (2/𝑎𝑀_𝑃𝑙)² lim_𝑘𝜂→0 𝑘³/2π² ∣𝑓_𝑘(𝜂)∣² = 2/π² 𝐻²_*/𝑀_𝑃𝑙)² (𝑘/𝑘_*)^3-2𝜈,
where 𝐻_* is the Hubble parameter evaluated at the time when the reference scale 𝑘_* exited the horizon. This is equivalent to our previous result (8.108). In particular, the tensor spectral index is he same as before
    (8.113)   𝑛_𝑡 ≡ 3 - 2𝜈 = -2𝜀.   [RE (8.111)]

   In summary, we have found that the tensor spectrum is:
    (8.114)  ∆²_𝘩(𝑘) = 𝛢_𝑡 (𝑘/𝑘_*)^𝑛_𝑡,
    (8.115-6)  𝛢_𝑡 ≡ 2/𝜋² 𝐻²_*/𝑀_𝑃𝑙²,     𝑛_𝑡 ≡ -2𝜀_*.
Observational constraints on the tensor amplitude are ususally expressed in terms of the tensor-to-scalar ratio,
    (8.117)  𝑟 ≡ 𝛢_𝑡/𝛢_𝑠 = 16𝜀_*.
Since the amplitude of scalar fluctuations has been measured, the tensor-to-scalar ratio quantities the tensor fluctuations. Tensor modes will be observable by CMB polarization experiments if 𝑟 > 0.01 (see Section 8.4.2).

          Energy scale of inflation
If primordial gravitational waves ere detected, it would tell us the energy at which inflation occurred. Using (8.115), we can write
    (8.118)   𝐻/𝑀_𝑃𝑙 = π√𝛢_𝑠 √(𝑟/2) ≈ 10^-5 (𝑟/0.01)^1/2,
where we used 𝛢_𝑠 ≈ 2.1 × 10^-9 for the scalar amplitude. Detecting inflationary tensor perturbations at the level 𝑟 ≥ 0.01 would imply that the expansion rate during inflation was about 10^-5 𝑀_𝑃𝑙. This can also be expressed in terms of the energy scale of inflation,   [RE (3.2) 𝑀_𝑃𝑙 = 2.4 × 10¹⁸ GeV]
    (8.119)   𝛦_inf ≡ (3𝐻²𝑀_𝑃𝑙²)^1/4 = 5 × 10^-3 (𝑟/0.01)^1/4 𝑀_𝑃𝑙.
Note that the tensor-to-scalar ratio depends sensitively on the energy scale of inflation, 𝑟 ∝ 𝐸_inf⁴. Decreasing 𝛦_inf by a factor of 10 reduces the tensor-to-scalar ratio by a factor of 10⁴. Gravitational waves from inflation are therefore only observable if inflation occurred near GUT scale, 𝛦_inf ~ 10⁶ GeV.

          The Lyth bound
Another interesting feature of inflationary models the  with observable gravitational waves is the fact that the field moves over a super-Planckian distance, ∆𝜙 ≡ ∣𝜙* - 𝜙𝑒∣ > 𝑀𝑃𝑙. The derivation of this result is simple, but the theoretical consequences are propound.
   Using the definition of the slow-roll parameter 𝜀 in (8.8), the tensor-to-scalar ratio (8.117) can be written as
    (8.120)   𝑟 = 8/𝑀𝑃𝑙2 (d𝜙/d𝑁)2,
where d𝑁 = 𝐻d𝑡. We see that the tensor-to-scalar ratio is proportional to the inflation speed measured in terms of the 𝑒-folding time. The perturbations which eventually created the CMB fluctuations exited the horizon at early times, about 𝑁* = 50 𝑒-folds before the end of inflation. The total distance that the field moved between this time and the end of inflation can then be written as
    (8.121)   ∆𝜙/𝑀𝑃𝑙 = ∫𝑁*0 d𝑁 √(𝑟/8) ≈ √(𝑟*/0.01) ∫𝑁*0 d𝑁/50 √(𝑟/𝑟*),
where 𝑟* ≡ 𝑟(𝑁*) is the tensor-to-scalar ratio on CMB scales. During the slow-roll evolution, 𝑟(𝑁) doesn't change much and we obtain the so-called Lyth bound [5]:
    (8.122)   ∆𝜙/𝑀𝑃𝑙 = 𝑂(1) × √ (𝑟*/0.01).
Large values of the tensor-to-scalar ratio, 𝑟 > 0.0.1, therefore correlate with super-Planckian field excursions, ∆𝜙
𝑀𝑃𝑙. It is quite a remarkable coincidence that the level of tensors that is experimentally accessible in the near future is tied to the fundamental scale of quantum gravity, 𝑀𝑃𝑙 = 2.4 × 1018 GeV.

          Ultraviolet sensitivity
How scared should we be of these super-Plakian field excursions? From (8.119) we see that the super-Planckian field values are not a problem of the energy density becoming larger than the Planck scale. It is also not a problem of radiative stavility of the potential. A theory with only the inflaton and the graviton is radiatively stable, since quantum corrections are proportional to the parameters that break a shift symmetry for the inflation. The small values of these parameters are technically natural even if the field moves overs over a super-Planckian range. The essence of the problem of large-field inflation is instead that gravity requires an ultraviolet completion, and coupling of the inflaton to the additional degrees of freedom that provide this ultraviolet completion do not necessarily respect the symmetry structures needed to protect the inflaton potential in the low-energy theory.
   From our general discussion of effective field theory in Chapter 4, we know that intergrating outfields of mass 𝛬 with order-unity couplings to the inflaton 𝜙 will lead to corrections to the inflaton potential of the form
    (8.123)   ∆𝑉(𝜙) = ∑^∞_𝑛=1 (𝑐_𝑛/𝛬^2𝑛 𝜙^4+2𝑛 + ∙ ∙ ∙ ),
where 𝑐_𝑛 are dimensionless Wilson coefficients that are typically of order one. We expect that the functional form of the potential will change when the field moves a distance of order 𝛬: in other orders, there is "structure" in the potential on scales of order 𝛬 (see Fig. 8.4). Even under the optimistic assumption that 𝛬 ≈ 𝑀_𝑃𝑙, the potential (8.123) will not support large-field inflation unless one effectively finetunes the infinite set of Wilson coefficients 𝑐_𝑛.
   The leading idea foe implementing large-field inflation is to use a symmetry to suppress the dangerous higher-dimension contributions in (8.123). For example, an unbroken shift symmetry,
    (8.124)   𝜙  ↦  𝜙 + const,
forbids all non-derivative interactions, including the desirable parts of the inflaton potential, while a suitable weakly broken shift symmetry² can give rise to a radiatively stable model of large-filed inflation. Whether such a shift symmetry can be UV-completed is a subtle and important question for a Planckian-scale theory like string theory. This is discussed in more detail in D. Baumann's book with L. McAllister [1] and explicit examples of large-field inflation in string theory were developed in [7, 8].

         8.3.3 Slow-Roll Predictions
So far, we have expressed the inflatinary predictions in terms of the Hubble rate during inflation, evaluated at the time when the modes of interest exited the horizon. We would like to related this to the shape of the shape of the potential in slow-roll inflation. Using the relation between the Hubble and potential slow-roll parameters derived in (4.66), we can write the amplitude of the scalar spectrum (8.97) and its tilt (8.98) as
    (8.125)   𝛢𝑠 = 1/24π2 1/𝜀𝑉, 𝑉*/𝑀𝑃𝑙4,
    (8.126)   𝑛𝑠 - 1 = -6𝜀𝑉,* + 2𝜂𝑉,*,
where 𝜀𝑉 and 𝜂𝑉 were defined in (4.65). Similarly, the amplitude of the tensor spectrum (8.115) and its tilt (8.116) become
    (8.127)   𝛢𝑡 = 2/3π2 𝑉*/𝑀𝑃𝑙4,
    (8.128)   𝑛𝑡 = -2𝜀𝑉,*.
Observations near the reference scale 𝑘* then probe the shape of the inflaton potential around 𝜙* ≡ 𝜙(𝑡*), where 𝑡* is the moment of horizon crossing of the fluctuation with wavenumber 𝑘* (see Fig. 8.5). The largest observable fluctuations are probed in the CMB and exit the horizon at early times. Short-wavelength fluctuations exit the horizon later as the field has rolled further down its potential.
   It is often useful to define the moment of horizon exit by the number of 𝑒-folds remaining until the end of inflation
    (8.129)   𝑁* ≡ ∫𝜙𝑒𝜙* 𝐻/𝜙̇ d𝜙 ≈ ∫𝜙𝑒𝜙* 1/√(2𝜀𝑉) ∣d𝜙∣/𝑀𝑃𝑙,
where 𝜙𝑒 is the field value at which 𝜀𝑉 = 1. CMB observations probe a range of about 7 𝑒-folds centered around 𝑁* ~ 50 𝑒-folds before the end of inflation.

          Case study: quadratic inflation
Let us illustrate the slow-roll predictions for 𝑚²𝜙² inflation. In Chapter 4, we showed that the slow-roll parameters of this model are
    (8.130)   𝜀_𝑉(𝜙) = 𝜂_𝑉(𝜙) = 2𝑀_𝑃𝑙²/𝜙²,
and the number of 𝑒-folds before the end of inflation is
    (8.131)   𝑁(𝜙) = 𝜙²/4𝑀_𝑃𝑙² - 1/2 ≈ 𝜙²/4𝑀_𝑃𝑙².
At  the time when the fluctuation which eventually created the CMB anisotripoies crossed the horizon, we have
    (8.132)   𝜀_𝑉,* = 𝜂_𝑉,* ≈ 1/𝑁_* ≈ 0.01,
where the final equality is for a fiducial value of 𝑁_* ≈ 50. The spectral tilt and the tensor-to-scalar ratio then are
    (8.133)   𝑛_𝑠 ≡ 1 - 6𝜀_𝑉,* + 2𝜂_𝑉,* = 1 -2/𝑁_* ≈ 0.96,
    (8.134)   𝑟 ≡ 16𝜀_𝑉,* = 8/𝑁_* ≈ 0.16.
As we will see in Section 8.4, the relatively large value of the tensor-to-scalar ratio is inconsistent with the data, so the model is ruled out. We can explore predictions of other slow-roll models in the problems in this chapter.

         8.4 Observational Constraints^*
In the previous sections, we showed that inflation contains a built-in mechanism to produce a nearby scale-invariant spectrum of primordial curvature perturbations. In this section the current observational evidence for inflation will be described and future observational tests will be discussed.

         8.4.1 Current Results
It is remarkable that we now have observations that probe the physical conditions just fractions of a second after the Big Bang, when the energies were many orders of magnitude higher than the highest energies accessible to particle colliders. In this following, what we have learned from these observations will be explained.

          Scale invariance
As we saw in Chapter 7, the main feature of the CMB spectrum arise from the evolution of coupled photon-baryon fluctuations on subhorizon scales. We found that a nearby featureless spectrum of initial fluctuations evolves into the peaks and troughs of the CMB anisotropy spectrum (see Fig. 8.6. Since the evolution of the fluctuations after inflation is well understood, we can use the CMB observations to put constraonts on the primordial power spectrum and hence on the physics of inflation.
   A key prediction of inflation is the fact that the primordial correlations are approximately scale invariant, corresponding to the approximate time-translation invariance of the the inflationary dynamics. Inflation also predicts a percent-level deviation from perfect scale invariance arising from the small time dependence that is required because inflation has to end. This breaking of perfect scale invariance was measured by the WMAP and Plank satellites. The latest measurement of the scalar spectral index is [3]
    (8.135)   𝑛_𝑠 = 0.9652 ± 0.0042,
which deviates at a stastically significant level from 𝑛_𝑠 =1. The sign of the deviation, 𝑛_𝑠 < 1, is also what would be expected from a decreasing energy density during inflation.

          Coherent phases
Possibly the most remarkable evidence for inflation come from the phase coherence of the primordial fluctuations. If the perturbations in photon-baryon fluid didn't have coherent phases, then the sound waves in the plasma wouldn't interfere constructively at recombination and the CMB power spectrum would not have its famous peak structure. It will now be explained why this phase cohereence arises naturally in inflationary models. A more detailed version of the smae argument has appeared in a nice article by Dodelson [9].
   The CMB is a snapshot of sound waves at the time of recombination (or more precisely photon decoupling). Each wave is captured at a certain phase  in its evolution (see Fig. 8.7). Since the oscillation frequency is set by the wavelength of the wave, 𝜔 = 𝑐_𝑠𝑘, waves with different wavelengths are observed at different moments in their evolution. This is the reason why the angular power spectrum of the temperature fluctuations goes up and down as a function of scale. While this cartoon is essentially correct, it doesn't highlight the fact that the initial phases must be synchronized. Even a single Fourier mode should be of as an ensemble of many waves with fixed wavelength, but amplitudes that are drawn from a Gaussian probability distribution. These waves need to be added to get the real space density distribution and hence the CMB anisotropies. The modes will only have the same phase at recombination if their initial phases are the same (see Fig. 8.8). Thhis coherence of the phase is necessary in order for the superposition of many modes to lead to the peak structure seen in the CMB spectrum. If the initial phase were random, then the peaks of the spectrum would be washed out.
   In inflation, the initial phases are set by the fact that the modes are frozen noutside of the horizon. When the modes re-enter the horizon, the time derivative of the amplitude, 𝓡ʹ, is small. All modes with the same wavenumber 𝑘, but distinct wavevectors 𝐤, therefore start their evolution at the sametime. Thinking of the Fourier modes as a combination of sine and cosine solutions, inflation only excites the cosine solution. What is remarkable is that the coherence of the phases extends to superhorizon scales at recombination. This is seen most convincingly in the ctross correlation between fluctuations in the CMB temperature and E-mode polarization. Since polarization is generated only by scattering of the photons off the free electrons just before decoupling, this signal cannot be created after recombination. Without inflation (or smething very similar), such a signal would causality [10]. The large-scale TE(temperature-E mode) correlation signal was detected for the first time by the WMAP satellite [11].

          Adiabaticity
In Section 6.2 we showed that the most general initial conditions for the fluid components of the hot Big bang may include isocurvature contributions. However, these isocurvature fluctuations are not a natural outcome of the simplest inflationary models. Instead, single-field inflation has only one fluctuating scalar degree of freedom, which induces purely adiabatic initial conditions. Moreover, thermal equilibrium after inflation tends to wash out any isocurvature modes.
   The peak structure of the CMB spectrum is evidence that the dominant contribution to the primordial perturbations is adiabatic. As we have seen, adiabatic initial conditions produce a cosine oscillation in the photon-baryon plasma. Isovurvature initial conditions, on the other hand, would create a sine oscillation which is not consistent with the observed peaks of the CMB. The possibility of a dominant isocurvature perturbation is ruled out and even a subdominant level of isocurvature fluctuations is highly constrained.
   Observational constraints on isocurvature modes depend on whether they are correlated or uncorrelated with the adiabatic mode. In the totally correlated case, all fluctuations are still proportional to 𝓡and we cawrite the matter isocurvature mode as 
      (8.136)   𝑆_𝑚 = √𝛼 𝓡.
The Planck limit on the proportionality constant is [12]
    (8.137)   𝛼 < 0.0003^+0.0016_-0.0012.
A more detailed analysis for CDM, neutrino density and neutrino velocity isocurvature modes gives similar results. There is, hence, no evidence for any isocurvature contribution to the initial perturbations, in agreement  with the expectation from single-field inflation.

          Gaussianity
So far, we have only described constraints arising from the two-point function of the primordial correlations. This two-point function, in fact, contains all the information about the initial conditions if the perturbations are drwan from a Gaussian probability distribution. The simplestinflationary models indeed predict Gaussanity to be a good approximation. It is easy to see why. Slow-roll inflation only occurs on the flat part of the potential where the self-interactions of the field are small. The linearized equation of motion is then a good apporoximation for the evolution of the inflaton fluctuations. This equation took the form of a harmonic oscillator equation, which we used to compute the expectation value of quantum fluctuations in the ground states. Since the ground state wave function of the harmonic oscillator is a Gaussian, we expect the initial perturbations created by inflation also to obey Gaussian statistics. All odd 𝑁-point functions then vanish, while all even 𝑁-point functions are related to the power spectrum (which contains all the information).
   Small amounts of non-Gaussianity may nevertheless still exist and would teach us a lot about the physics of inflation. In particular, extentions of slow-roll modes can produce non-Gaussian fluctuations from enhanced self-interactions of the inflaton or couplings to additional fields. Below the current constraints and future tests of these types of non-Gaussianity.

         8.4.2 Future Tests
Although the evidence for inflation is intrigging, we cannot yet claim that it is part of the standard model of cosmology at the same level as, for example, BBN and/or recombination are. Since unlike them inflation requires physics beyond the Standard Model of particle physics. We need further observational tests. In this section, two avenues that are considered particularly promising: tensor mode and non-Gaussianity will be mentioned.

          Tensor modes A major goal of current efforts in observational cosmology is to detect the tensor component of the primordial fluctuations. As we saw, the tensor amplitude depends on the energy scale of inflation and is therefore nor predicted (i.e. it varies between models). While this makes the search for primordial tensor modes difficult, it is also what makes it exciting. Detecting tensors would reveal the energy scale at which inflation occurred, providing an importan clue about the physics driving the inflationary expansion. As in Section 8.3,2 the tensor signal is also unusually sensitive to high-energy corrections and therefore probes important aspects the microphysics of inflation and the nature of quantum gravity [1].
   Most searches for tensor modes focus on their imprint on the polarization of the CMB. As in Chapter 7 polarization is generatedthrough the scattering of the anisotropic radiation field off the free electons just before decoupling. The presence of a gravitational wave background creates an anisotropic stretching of the spacetime which induces a special type of polarization pattern, the so-called B-mode pattern ( a pattern whose "curl" doesn't vanish). Such a pattern cannot be created by scalar fluctuations and is therefore a unique signature of primordial tensors. Because tensors are such a clean prediction of inflation, a B-mode detection would be a milestone towards establishing that inflation really occurred in the early universe.
   The latest observational constraints are shown in Fig. 8.9. implying an upper bound on the tensor-to-scalar ratio of 𝑟 < 0.035 [13]. A large number of ground-based, balloon and satellite experiments are searching for B-mode signal predicted by inflation (see e.g. [14]). The main challenge for these experiments is to isolate the primordial signal from secondary B-modes created by astrophysical foregrounds. This can be done by using the frequency dependence of the signal (the primordial signal is a blackbody) and the spatial dependence (the primordial signal is statistically isotropic).

          Non-Gaussianity We have seen that inflation predicts that the initial correlations are highly Gaussian and therefore described well by the power spectrum. Nevertheless, in principle, a significant amount of information about the physics of inflation can be coded in small levels of non-Gaussianity. In particular, the higher-order correlations associate with non-Gaussian initial conditions are sensitive to nonlinear interactions, while the power spectrum only probes the free theory. Measurements of primordial non-Gaussian would probe the particle content and the interactions during inflation.³
   The leading signature of non-Gaussianity is a nonero thre-point correlation function, or its Fourier equivalent, the bispectrum:  
    (8.138)   ⟨𝓡_𝐤1 𝓡_𝐤2 𝓡_𝐤3⟩ = (2π)³ 𝛿_𝐷(𝐤₁ + 𝐤₂ + 𝐤₃) (2π²)²/(𝑘₁𝑘₂𝑘₃)² 𝛣_𝓡(𝑘₁, 𝑘₂, 𝑘₃),
where the delta function is a consequence of the homogeneity of the background. The sum of the wavevectors must form a closed triangle and the strength of the signal will depend on the shape of the triangle. The bispectrum is a function of the magnitudes of the wavevectors, 𝑘_𝑛=1,2,3, as required by the isotropy of the background. The factor of (𝑘₁𝑘₂𝑘₃)^-2 has been extracted in (8.138) to make the function 𝛣_𝓡(𝑘₁, 𝑘₂, 𝑘₃) dimensionless. The amplitude of the non-Gaussianity is then defined as the size of the bispectrum in the equilateral configuration:
    (8.139)   𝑓_𝛮𝐿(𝑘) ≡ - 5/18 𝛣_𝓡(𝑘₁, 𝑘₂, 𝑘₃)/∆⁴_𝓡(𝑘),
where the numerical factor is a historical accident and the overall minus sign is a consequence of Baumann's convenstion for sign of 𝓡 (which is different from that used in the Planck and WMAP analysis). In general, the parameter 𝑓_𝛮𝐿 can depend on the overall wavenumber (or the size of the triangle), but for scale-invariant initial conditions it would be a constant. Then the bispectrum can be written as
    (8.140)   𝛣_𝓡(𝑘₁, 𝑘₂, 𝑘₃) ≡ -18/5 𝑓_𝛮𝐿(𝑘) × 𝑆(𝑥₂, 𝑥₃) × ∆⁴_𝓡,
where 𝑥₂ ≡ 𝑘₂/𝑘₁ and 𝑥₃ ≡ 𝑘₃/𝑘₁.⁴ The shape function 𝑆(𝑥₂, 𝑥₃) is normalized so that 𝑆(1, 1) ≡ 1. The shape of the non-Gaussianity contains lots of information about the microphysics of inflation. This is to be contrasted with the power spectrum, which described by just two numbers, 𝛢_𝑠 and 𝑛_𝑠, and not a whole function.
   In the following, a mostly qualitative discussion of popular forms of non-Gaussianinty.  Further details and explicit computations can be found in [16, 17].

  • Local non-Gaussianity  One way to create a non-Gaussian field is to take a Gaussian one and add to it the square of itself [18, 19]:⁵
    (8.141)   𝓡(𝐱) = 𝓡_𝑔(𝐱) - 3/5 𝑓^local_𝛮𝐿 [𝓡²_𝑔(𝐱) - ⟨𝓡²_𝑔 ⟩].
   This local field redefinition creates the so-called local non-Gaussianity. The corresponding bispectrum, and the associated shape function, are
    (8.142)   𝛣^local_𝓡 ∝ (𝑘₁𝑘₂𝑘₃)² (𝒫₁𝒫₂ + 𝒫₁𝒫₃ + 𝒫₂𝒫₃),     𝑆_local = 1/3 (𝑘₃²/𝑘₁𝑘₂ + 𝑘₂²/𝑘₁𝑘₃ + 𝑘₁²/𝑘₂𝑘₃),
   where 𝒫_𝑛 ≡ 𝒫_𝓡(𝑘_𝑛) and the result for 𝑆_local is a scale-invariant power spectrum, 𝒫_𝑛 ∝ 𝑘_𝑛^-3. The shape of the local bispectrum is shown in Fig. 8.10. We see the signal is peaked in the "squeezed limit." where one side of the triangle is much smaller than the other two (say 𝑘₃ ≪ 𝑘₂ ≈ 𝑘₁). Interestingly, this feature cnnot arise in single-field inflation because of the single-field consistency relation [20, 21]. This theorem states that, for any sigle-field model of inflation, the signal in the squeezed limit must satisfy
    (8.143)   lim_𝑘3→0 𝛣^local_𝓡(𝑘₁, 𝑘₂, 𝑘₃) = (𝑛_𝑠 - 1) (𝑘₁𝑘₂𝑘₃)²/(2π²)² 𝒫_𝓡(𝑘₃)𝒫_𝓡(𝑘₁),
   which vanishes for scale-invariant fluctuations. Therefore detecting a signal in the squeezed limit would rule out all models of single-field  inflation. with Bunch Davis initial conditions), not just slow-roll models. Conversely, the signal in the squeezed limit can be used as a particular clean diagnostic for extra fields during inflation. Because inflation occurred at very high energies, even very massive particles (𝑚 ≲ 𝐻) can be produced by the rapid expansion of the spacetime [15, 22]. When these particles decay into the inflaton, they can find to a distinct signature in the squeezed limit of the bispectrum.Planck constraint on local non-Gaussianity is [23]
    (8.144)   𝑓^local_𝛮𝐿 = -0.9 ± 5.1.
   Future galaxy survey may improve the bound by an order of magnitude [24], providing a sensitive probe of the particle spectrum during inflation. These measurements exploit the fact that local non-Gaussianity leads to a scale-dependent bias [25] on large scales, which can be measured in the galaxy power spectrum.

  • Equilaterial non-Gaussianity  In slow-roll inflation the flatness of the inflation potential constrains the size of the inflation self-interactions. However interesting models of inflation have been suggested in which  higher derivative corrections-such as (∂𝜙)⁴-play an important role during inflation [26, 27]. These inflation leads to cubic interactions of the curvature perturbations-such as (d𝓡/d𝑡)³ and (d𝓡/d𝑡)(∂_𝑡𝓡)²-and hence a nonzero bispectrum.⁶ A key characteristic of derivative interactions is that they are suppressed when any individual mode is far outside the horizon. This suggests that the bispectrum is small in the squeezed limit and maximal when all three modes have wavelengths equal to the horizon size. Therefore the bispectrum has a shape that peaks for an equilaterial triangle configuration (𝑘₁ ≈ 𝑘₂ ≈ 𝑘₃).
   The precise shape of this equilaterial non-Gaussianity depends on the specific cubic interactions from which it arises; for example, the shapes coming from (d𝓡/d𝑡)³ and (d𝓡/d𝑡)(∂_𝑡𝓡)² are slightly different away from the equilateral limit. The CMB data is typically analyzed using the following template shape [28]:
    (8.145)   𝑆_equil = - (𝑘₁²/𝑘₂𝑘₃ + 2 perms) + (𝑘₁/𝑘₂ + 5 perms) - 2,
   which is plotted in Fig. 8.11. This template is a good approximation to all forms of equilateral non-Gaussianity near the equilateral limit, which is where most of the constraining power of the CMB data comes from. When using other data sets, however, it's important to remember that the template  (8.145) is only an approximation.  For example, the exact bispectrum coming from the interaction  (d𝓡/d𝑡)³ is [27]
    (8.146)   𝑆_{(d𝓡/d𝑡)³} = 27𝑘₁𝑘₂𝑘₃/(𝑘₁ + 𝑘₂ + 𝑘₃)³,
   which agrees with (8.145) in the equilaterial configuration, but differs away from it.
   Using the template shape (8.145), the planck collaboration derived the following constraint on equilateral non-Gaussianity [23]the secondary non-Gaussianity produced by late
    (8.147)   𝑓^local_𝛮𝐿 = -26 ± 47.
   It will be hard (but not impossible) to improve on this constraint with future galaxy surveys, because gravitational nonlinearities also produce an equilateral signal. This makes it challenging to distinguish the primordial non-Gaussianity from the secondary non-Gaussianity produced by the late-time evolution of the perturbations.

  • Folded non-Gaussianity  The Gaussianity of slow-roll inflation also relies on the fact that we evaluated the quantum fluctuations in the Bunch-Davis vacuum (corresponding to the ground stae of the harmonic oscillator). In contrast, starting from an excited initial state would lead to non-Gaussianity. The detailed shape of this non-Gaussianity, and  hence the onstraint on 𝑓_𝛮𝐿, depends on the model for the excited initial state [27, 29]. A universal feature is that the correlations are enhanced for "folded" triangles where two of the wavevectors become colinear.
   A template that captures the essential features of non-Gaussianity from excited initial states is [16]
    (8.148)   𝑆_folded ∝ 𝑘₁²𝑘₃ (-𝑘₁ + ² + 𝑘₃)/(𝑘_𝑐 - 𝑘¹ + 𝑘₂ + 𝑘₃)⁴ + 2 perms,
   where 𝑘_𝑐 is a cutoff that regulates the singularity in the folded limit, 𝑘₂ + 𝑘₃ → 𝑘¹. This shape is plotted in Fig. 8.12. Ignoring the cutoff 𝑘_𝑐, the function in (8.148) is almost the same as that in (8.146), except for a flipped sign in front of one of the wavenumbers ("energies"). This sign flip arise because an excited initial state contains an admixture of the negative-frequency solution 𝑒^{+𝑖𝑘𝑛}. The alternating signs in (8.148) arises when correlating positive- and negative-frequency modes. It is these alternating signs that are the origin of the enhanced signal in the folded limit.
   The signal in the folded configuration also provides an interesting test of the quantum origin of the fluctuations [30] While classical fluctuations would generally have non-vanishing correlations in the folded limit, quantum fluctuations in the Bunch-Davis vacuum are characterized by the absence of such a signal.

         8.5 Summary In this chapter, we have studied the initial fluctuations created by inflation. We showed how quantum zero-point fluctuations get stretched outside the horizon during inflation, becoming the seeds for the large-scale structures of the universe.
   We stared by deriving the equation for the inflation fluctuations. In spatially flat gauge, the equation for 𝑓 ≡ 𝑎𝛿𝜙 takes the form
    (8.149)   𝑓ʺ + (𝑘² - 𝑧ʺ/𝑧) 𝑓 = 0,     where     𝑧 ≡ 𝑎𝜙̄ʹ/
𝓗. The comoving curvature perturbation is 𝓡 = 𝑓/𝑧. The field 𝑓 and its conjugate momentum 𝜋 ≡ 𝑓ʹ are promoted to quantum operators which satisfy the equal time commutation relation
    (8.150)   [𝑓̂(𝜂, 𝐱), 𝜋̂(𝜂, 𝐱ʹ)] = 𝑖 𝛿_𝐷(𝐱 - 𝐱ʹ).
The field operator is then written as
    (8.151)   𝑓̂(𝜂, 𝐱) = ∫ d³𝑘/(2π)³ [𝑓_𝑘(𝜂) 𝑎̂_𝐤 𝑒^{𝑖𝐤⋅𝐱} + 𝑓_𝑘^*(𝜂) 𝑎̂^†_𝐤 𝑒^{-𝑖𝐤⋅𝐱}],
where 𝑎̂_𝐤 and 𝑎̂^†_𝐤 are annihilation and creation operators, which obey
    (8.152)   [𝑎̂_𝐤, 𝑎̂^†_𝐤ʹ] = (2π)³𝛿_𝐷(𝐤 - 𝐤ʹ).
The vacuum state ∣0⟩ is the state annihilated by 𝑎̂_𝐤. A preferred initial state-the Bunch-Davis boundary vacuum-is defined by the ground state of a harmonic oscillator. With this boundary condition, and assuming a de Sitter background, the mode function in (8.151) becomes
    (8.153)   𝑓_𝑘(𝜂) = 𝑒^{-𝑖𝑘𝜂}/√(2𝑘) (1 - 𝑖/𝑘𝜂).
Using this, the dimensionless power spectrum of the inflaton fluctuation is
    (8.154)   ∆²_𝛿𝜙(𝑘, 𝜂) = 𝑘³/2π² ∣𝑓_𝑘(𝜂)∣²/𝑎²(𝜂) = (𝐻/2π)² [1 + (𝑘𝜂)²] ^𝑘𝜂→0→ (𝐻/2π)².
The power spectrum of the curvature perturbation 𝓡 takes a power law form
    (8.155)   ∆²_𝓡(𝑘) = 𝛢_𝑠 (𝑘/𝑘_*)^{𝑛_𝑠 - 1},
where the amplitude and the spectral index are
    (8.156)   𝛢_𝑠 ≡ 1/8π² 1/𝜀_* 𝐻_*²/𝑀_𝑃𝑙²,
    (8.157)   𝑛_𝑠 ≡ 1 - 2𝜀_* - 𝜅_*.
All quantities on the right side are evaluated at the time when the reference scale 𝑘_* exited the horizon, 𝑘_* = (𝑎𝐻)_*. Computing the power spectrum at horizon crossing mitigates the error that is being made in (8.153) by assuming a de Sitter background. The measured value of the scalar spectral index, 𝑛_𝑠 = 0.965 ± 0.004 [3], shows a deviation from a scale-invariant spectrum, as expected from the time evolution during inflation.
   A similar calculation for tensor fluctuations during inflation gives
    (8.158)   ∆²_𝘩(𝑘) = 𝛢_𝑡 (𝑘/𝑘_*)^𝑛_𝑡,
where the amplitude and the spectral index are
    (8.159)   𝛢_𝑡 ≡ 2/π² 𝐻_*²/𝑀_𝑃𝑙²,
    (8.160)   𝑛_𝑡 ≡ -2𝜀_*.
Observational constraints on the tensor amplitude are usually expressed in terms of the tensor-to-scalar ratio,
    (8.161)   𝑟 ≡ 𝛢_𝑡/𝛢_𝑠 = 16𝜀_*.
A large value of the tensor-to scalar ratio, 𝑟 > 0.01, correspons to the inflaton field moving over a super-Planckian distance during inflation, ∆𝜙 > 𝑀_𝑃𝑙. The current upper bound on the tensor-to-scalar ratio is 𝑟 < 0.035 [13]. Future observations of CMB polarization will improve this bound by at least an order of magnitude.

³ An analogy with particle physics may be informative: the measurements of the power spectrum are the analog of measuring the propagators of particles. While these propagators are important features of long-lived particles, their form is completely fixed by Lorentz symmetry and doesn't contain dynamical information. To learn about the dynamics of the theory we collide particles and study the resulting interactions. Measurements of higher-order correlations (or non-Gaussianity) are the analog of measuring collisions in particle physics. The study of non-Gaussianity also goes by the name of cosmological collider physics [15].
⁴ We will order the wavenumbers such that 𝑥₃ ≤  𝑥₂ ≤  1. Since the wavenumbers are side lengths of a triangle, we have constraint 𝑥₂ + 𝑥₃ > 1.
⁵ The factor of 3/5 appears because this field redefinition was originally introduced for the gravitational potential 𝛷, which during the matter era is related to 𝓡 by a factor of 3/5. This also explains the normalization factor in (8.139).
⁶ A systematic way to classify these derivative interactions is in terms of an EFT for the inflationary fluctuations. [2].

9 Outlook

   We have learned a combination of theoretical advances and precision observations have transformed cosmology into a quantitative science. Nowadays cosmology has become part of mainstream science. So cosmologist have been able to reconstruct much of the history of the universe from a limited amount of observational clues. This culminated in the 𝛬CDM model which describes the cosmological evolution from 1 second after Big bang until today. The details of cosmological structure formation, through the gravitational clustering of small density variations in the primordial universe, are well understood. There is now even evidence for the rather spectacular proposal that quantum fluctuations during a period of exponential expansion about 10^-34 seconds after the Big Bang provided the seed fluctuations for the large-scale structure of the universe.

   At the same time, many fundamental questions in cosmology remain unanswered. We do not know what drove the inflationary expansion, what kind of matter filled the universe after inflation, and how the asymmetry between matter and antimatter was created. The nature of dark matter remains unknown and how dark energy fits into quantum field theory is still a complete mystery. Of course, we can only hypothesize what happened at the Big Bang singularity, and if there was any spacetime before that. We hope that future observations of the CMB and of the large-scale structure of the universe will help to unlock these mysteries. We are awaiting new theoretical ideas to be validated for them.