Scaling Relation

The power spectrum of R is defined through the expectation value of ˆR²(x, t),

h ˆR²(x, t)i = Z dk

k P (k). (3.25)

ExpandingR in terms of the creation and annihilation operators,

R(x, t) = Z

d³kh

a_kRk(t)e^ik·x+ a^†_kR^⇤k(t)e ^ik·xi

, (3.26)

and using the commutator

[ak, a^†_k0] = ³(k k⁰), (3.27)

one finds that

P (k) = 4⇡k³|Rk|². (3.28)

Recalling that R = u/A, we find that, for w 6= 1 (so ↵ 6= 1), the power spectrum given by the solution (3.24) is

P = H_i²²|1 + ↵⇠| ^2/↵

For the slow-roll case, we approximate A by the slow-roll limit (3.19), and the mode function in that limit is then given by (3.24) with w' 1. In terms of the slow-roll parameter,

the power spectrum at the slow-roll limit is given by P = H_i²²|1 + ↵⇠| ^2/↵

Using the small-argument expansion of the Whittaker function [96], the super-horizon limits of the power spectrum are found for di↵erent ranges of w. For w <

1/3 and w > 1 except w = 1, one has

For w' 1, the slow-roll power spectrum (3.31) recovers the familiar scale-invariant spectrum

P⌧1 = H_i²

⇡✏. (3.33)

For 1/3 < w < 1, the super-horizon power spectrum decays with time, given by P⌧1 = H_i^{2 2}(2µ)

with

where is the Euler-Mascheroni constant.

We can summarize the power-law relations of the super-horizon power spectrum with respect to the normalized wavenumber  by the scaling relation⁵

P /  ^2µ+3, (3.38)

where the correspondence between the case of w ' 1 and the slow-roll limit is understood. For the convenience of readers, we also state this result in terms of the more familiar parameter, w. For w < 1/3 and w > 1,

P / 6(1+w)/(1+3w). (3.39)

For 1/3 < w < 1,

P / ^12w/(1+3w). (3.40)

The super-horizon behavior of the spectrum can be divided into three types according to the scaling relation. Some typical cases are plotted in Figure 3.1. For µ = 3/2, the spectrum is scale-invariant. This is the case for w ' 1 (slow-roll) and w = 0. When µ < 3/2, the spectrum is blue-tilted and the power is lower than the scale-invariant spectrum at super-horizon scales. This is attainable from the positive-pressure (w > 0) or superinflation era (w < 1). In the third case, the super-horizon spectrum is red-tilted, which is achieved when µ > 3/2, or equivalently an era with 1 < w < 0 except w = 1/3 (Figure 3.2). Here we reiterate that the spectra obtained are based on the adiabatic vacuum (3.22), where the corresponding initial condition for the decelerating universe (w > 1/3) is acausal.

5This relation is also derived in [97] as the approximation to the super-horizon spectrum at the end of the multi-stage inflationary evolution. The authors focus on the recursive matrix formalism of the multi-stage preinflationary era, with the assumption that every preinflation era is an accelerating expansion (or decelerating contraction in the bounce inflation scenario).

Sub !horizon Super !horizon

w " !1.2 w ! !1

w " !2!3

w " 0

w " 1!3

w " 1

w " 5

!10 !5 0 5 10

!10

!5 0 5 10

log

Κ log

"Π # 1%Α $ P ! H

%

Figure 3.1 The power spectra (3.29) with sample parameters w = 1.2, 2/3, 0, 1/3, 1, 5, and at the slow-roll limit w ' 1. They are plotted with normalizations such that they have the same magnitude at the sub-horizon limit. For w ' 1 the power spectrum is given by (3.31), and the normalization is instead ⇡✏P/H_i². The complete relationship between the slope of the super-horizon spectrum and the equation-of-state parameter is described by the scaling relation (3.38), and is plotted in Figure 3.2.

blue !tilted

!suppression"

red !tilted

!enhancement"

super ! inflation accelerating decelerating negative potential

!5 !4 !3 !2 !1 0 1 2 3 4 5

!4

!2 0 2 4

w

exponent

Figure 3.2 The plot of 2µ+3, the exponent of the power-law scaling relation (3.38) with respect to the equation-of-state parameter, w.

Chapter 4

Evolution of the Power Spectrum

As the universe evolves from the preinflationary era to the inflationary era, the equation of state changes accordingly. To understand how the initial spectrum of the preinflationary era is constantly shaped by the expansion of the background geometry, eventually evolving into the super-horizon spectrum, we develop a use-ful technique to analyze this process: the spectrum evolution. By analyzing the evolution of the spectrum, we find that the character of the long-wavelength spec-trum at the end of inflation actually reflects the nature of the initial state in the preinflationary era. If at early times before the onset of inflation, the universe is initially described by some adiabatic vacuum in the background with constant equa-tion of state, the power spectrum is given by (3.29) during that era. As the universe evolves with time, although the power spectrum transforms continuously, the spec-tral shape at the long-wavelength limit is nevertheless not a↵ected by the evolution;

it is preserved in the late-time spectrum.

We demonstrate in this chapter that the spectra can be radically di↵erent at large scales for distinct initial vacua. Particularly, if the universe transits from a kinetic era into the inflation era, the large-scale spectrum is suppressed because of the initial adiabatic vacuum assumed in the kinetic era. If the initial vacuum is di↵erent—for example, changed by an earlier accelerating era before the kinetic era,

as discussed in this section—the large-scale spectrum may even become enhanced.

We compare the evolution of the power spectrum in three cases, modeled phe-nomenologically by the single field dynamics. The first one is a single slow-roll era (denoted as era C) with Hubble parameter H_C. The second one is the slow-roll era (era C) preceded by a kinetic era (era B). In the third case we add one more slow-roll era (era A), with Hubble parameter HA, before the kinetic era (era B), which is again followed by the slow-roll era (era C). When applicable, the quantities at the transition from era A to B are denoted by subscript 1 (so, for example, the scale factor at the transition is equal to a1), and those at the transition from B to C are by subscript 2.

In view of the acausal character of the adiabatic vacuum in the initially deceler-ating era (the second case with only era B and C), we also analyze the case of having a superinflation era (era S) before the slow-roll era (era C), which also implies power suppression at large scales but is free from the acausal property. Analogously, the quantities at the transition from era S to C are denoted by subscript 2.

4.1 Slow-Roll

In the slow-roll era (era C), the general solution to the mode function at the slow-roll limit w ' 1 is viewed as quantities at some reference time, ⌘2. The notations are chosen for the convenience of later comparison and should not cause confusion. Note that in the slow-roll era, one can approximate HC ⇡ H2 as a constant. The normalized power

spectrum is given by

are the normalized power spectrum and the slow-roll parameter evaluated in era C, respectively.

In the adiabatic vacuum (3.22), only the second term in (4.1) remains, and the mode function reduces to

which recovers the well-known form in the super-horizon limit, P = 1 The comoving horizon size decreases with time, moving toward the right to the small scales in Figure 4.1. After the mode exits the horizon, lying on the left-hand side of the horizon scale, the power stays scale-invariant.