Consider the case when ⌘ < ⌘2 the universe is in the superinflation era (era S), which is modeled by the Lagrangian (3.20). With w < 1, the mode function is
uS =S+M0,µ
2i˜kS
↵ ˜a↵S
!
+ S W0,µ 2i˜kS
↵ ˜a↵S
!
, (4.19)
where ˜kS = k/a1H1, ˜aS = a/a1 = [1 + ↵a1H1(⌘ ⌘1)]1/↵, and ↵ and µ are given by (3.14) and (3.17), respectively. Here ⌘1 < ⌘2 also denotes the reference time for era
First slow !roll
Kinetic
Second slow !roll
"CDM Present
Hubble radius
Wavelengths
Scales causally connected IR scale!invariant Kink near IR plateau Joining segment Kink near UV plateau UV scale!invariant
N Physical
length
!log scale"
Figure 4.7 Illustration of the evolution of the physical wavelengths of the modes and the Hubble radius with respect to the number of e-fold, N . The five parallel straight lines denote the modes with di↵erent wavelengths, long to short from top to bottom (color online). The corresponding features they generate in the power spectrum, Figure 4.6, are labeled in the legends (top to bottom corresponding to long to short wavelengths). The black piecewise-connected lines denote the Hubble radius evolving from the first roll era to the kinetic era, then to the second slow-roll era, finally into the ⇤CDM era. The shaded region denotes the scales within which are causally connected.
S. The normalized power spectrum is
The universe goes into the slow-roll era (era C) at ⌘2, with mode function given by (4.1). Using the same matching conditions, the coefficients ˜C± are found to be
C˜+=
with all Whittaker functions evaluated at 2i˜kC/↵. The slow-roll parameter in era C is still given by ✏, and NS is the number of e-folds from ⌘1 to ⌘2.
The adiabatic vacuum (3.24) in era S corresponds to the coefficients
S˜+ =0, (4.24)
!2 !1 0 1 2 3 4 5
!4
!3
!2
!1 0 1 2
log
10k "
S
log
10P
S"
Figure 4.8 Time evolution of the power spectrum in era S in the case of two-stage evolution (era S and C), with w = 1.2. The solid curves are the spectra ofR from early time to late time (from light to dark). The vertical dashed lines denote the comoving horizon size at the corresponding instants (also from light to dark).
where we have used the relation ˜kS = ˜kCe ↵NS. The power spectrum in era S is plotted in Figure 4.8, taking w = 1.2 as an example. Similar to the case of the kinetic era (era B), the super-horizon spectrum is blue-tilted, but in the superinflation era the comoving Hubble radius decreases, and the super-horizon spectrum remains constant in time.
As the universe enters the slow-roll era (era C), the super-horizon modes have two di↵erent types of history (Figure 4.9). The modes with longer wavelengths exit the horizon in era S, retaining the blue-tilted super-horizon spectrum and staying constant in era C. The modes with shorter wavelengths exit the horizon in era C, acquiring the scale-invariant spectrum at horizon crossing.
0 5 10
!10
!9
!8
!7
!6
log
10k "
C
log
10P
C"
Figure 4.9 Time evolution of the power spectrum in era C in the case of two-stage evolution (era S and C), with w = 1.2, NS = 6, and ✏ = 0.1. The solid curves are the spectra of R from early time to late time (from light to dark). The vertical dashed lines denote the comoving horizon size at the corresponding instants (also from light to dark).
Chapter 5
Two-field cascade inflation
After establishing the understanding of the spectrum evolution in multi-stage infla-tion using the ad hoc single-field analysis, in this chapter we calculate the spectrum generated by a two-stage inflation model from the given potential. The purposes are to show that the evolution pattern we obtain in the single-field analysis is also reflected in the two-field dynamics, and to check whether the large-scale power is suppressed due to the coupling between the two fields.
5.1 Background Evolution and the Attractor So-lution
We investigate a simple two-field cascade inflation, which is driven by a heavy scalar field and a light scalar field with the action
S = Z
d4xp g
1
2@↵ @↵ 1
2@↵ @↵ V ( , ) . (5.1) In large field inflation, the field operates at the super-Plankian scale when the coeffi-cient of the kinetic term is normalized to 1/2. If the field value is of the order of the Planck mass, MP = 1/p
G, the energy scale of inflation is determined by the mass of the field. The typical evolution can therefore be divided into four stages. The first
stage is the inflationary era driven by the heavy field, with fields starting far away from the potential minimum and rolling down alone the hillside of the potential. At the second stage the heavy field falls into the potential minimum and oscillates with damping amplitude. As the energy drops to the scale of the light field mass, the universe enters the third stage in which the light field initiates the other inflation era. Finally at the fourth stage the light field decays, ending the inflation, and the standard ⇤CDM evolution begins.
Consider the cascade inflation realized by the potential V ( , ) =
0
2
2 2+ 1
2m2 2. (5.2)
The heavy-field inflation is driven by the coupling term 0 2 2/2, and the light-field inflation is driven by the mass term m2 2/2. At the stage of heavy-field inflation, the fields follow the attractor solutions, which can be obtained through expressing the equations of motion in terms of the number of e-folds, N ⌘ ln(a/ai), where ai
is some initial scale factor. The Friedmann equation is H2 =
where H = ˙a/a. The dots denote the derivatives with respect to t. The equations of motion can be casted into the form
d2
It can be shown that the attractor solutions satisfy d
provided (d /dN )2+ (d /dN )2 3/4⇡ < 0 as constrained by (5.3).1 With potential dominated by 0 2 2/2, the attractor solutions are
(N ) =
where the subscripts i denote the initial values.
At the second stage, exits the slow-roll regime and oscillates at the minimum of the potential with its amplitude damped with time. The potential is still dominated by the coupling term before the next inflation begins. During the oscillatory stage, remains slow roll, while acquires a large kinetic energy that is of the same order of the potential energy, ˙2 ⇠ 0 2 2 ⇠ H2, and the energy density evolves e↵ectively according to w = 0 (zero pressure). Ignoring the kinetic energy of in the Friedmann equation, we have
✓d
Combining (5.5) and (5.10), we obtain a set of di↵erential equations of A and ✓ dA
1For , the type of the attractor considered here is the horizontal trajectory on the –d /dN plane; that is, the one with (d/d )(d /dN ) = 0. Since d2 /dN2= (d /dN )⇥ (d/d )(d /dN), we obtain the condition for the attractor solution with non-vanishing d /dN as d2 /dN2= 0, which in turn leads to (5.6). Similar argument applies to .
Among the two terms contributing to the frequency d✓/dN , the first term 1/A is of the order of 1/ , since A = H/p
0 ⇡ p
0 /p
0 = . The second term is of order unity as slow rolls. After drops below the Planck mass, 1/A dominates and ✓ oscillates rapidly, so we can approximate cos2✓ in (5.13) by 1/2. With given by the slow-roll solution (5.8), A and ✓ can then be integrated to yield
A(N ) = Ai
As the field attenuates, gradually the energy density from the coupling term is taken over by the m2 2/2 term. The universe enters the third stage, in which the other inflation driven by the light-field begins. This stage is e↵ectively described by the single-field inflation, with the same attractor solution (5.8) for .