# Hopfions of massive gauge bosons in early universe

M. Bousder<EMAIL_ADDRESS>Laboratory of Condensend Matter and
Interdisplinary Sciences, Department of physics,
Faculty of Sciences, Mohammed V University in Rabat, Morocco  H. Ez-Zahraouy
<EMAIL_ADDRESS>Laboratory of Condensend Matter and Interdisplinary
Sciences, Department of physics,
Faculty of Sciences, Mohammed V University in Rabat, Morocco

###### Abstract

This letter presents a novel model that characterizes the curvature of space-
time, influenced by a massive gauge field in the early universe. This
curvature can lead to a multitude of observations, including the Hubble
tension issue and the isotropic stochastic gravitational-wave background. We
introduce, for the first time, the concept of gauge field Hopfions, which
exist in the space-time. We further investigate how hopfions can influence
Hubble parameter values. Our findings open the door to utilizing hopfions as a
topological source which links both gravitation and the gauge field.

neutron stars – equation of state – sound speed

††preprint: APS/123-QED

Introduction. During the inflationary period, the interplay between coupled
axion and SU(2) gauge fields [1, 2] offers a diverse range of phenomena,
distinguishing itself from the characteristics observed in conventional single
scalar field inflation models [3, 4]. Notably, this coupling gives rise to
intriguing outcomes, such as the generation of a stochastic background
comprising chiral gravitational waves [5, 6], characterized by their non-
Gaussian nature [7, 8]. Moreover, recent studies [9, 10, 11] have demonstrated
that the production of these chiral gravitational waves results in a non-zero
parity-violating gravitational anomaly. This anomaly breaks lepton number
symmetry and helps to explain the observed baryon asymmetry in the universe.
Massless gauge bosons generate gravitational wave signals, as studied in [12,
13]. Gravitational waves from tensor perturbations are a common outcome of
simple inflation models. In [14], they show how massive gauge bosons also
produce gravitational wave signals through tensor fluctuations during
inflation. Massive particles are usually hard to produce during inflation due
to a Boltzmann-like factor $exp(-\pi m/H)$, where $m$ is the particle’s mass
and $H$ is the Hubble rate, the inflationary scale. This reduces the signals
at the cosmological collider. Within the realm of topological solitons,
magnetic skyrmions are two-dimensional entities resembling particles,
characterized by a continuous twist of magnetization. On the other hand,
magnetic Hopfions are three-dimensional structures that can be constructed
from a closed loop of intertwined skyrmion strings. Theoretical frameworks
propose that magnetic Hopfions can achieve stability in frustrated or chiral
magnetic systems, and that target skymions can be converted into Hopfions by
adjusting their perpendicular magnetic anisotropy. However, experimental
confirmation of these theories remains elusive to date [15, 16, 17].
Recently, there has been a significant orientation towards the study of
magnetic Hopfions [18, 19, 20, 21, 22]. Our aim in this letter is to present,
for the first time, the concept of gauge field Hopfions. These exist in space-
time in the early universe and could potentially resolve the tension issue of
Hubble and the signal from pulsar timing arrays.

Inflation with massive gauge bosons. We study how to generate a force field
with mass using the Chern-Simons interaction. We use a U(1) gauge field
$A_{\mu}$ and add a term $\phi F_{\mu\nu}\tilde{F}^{\mu\nu}$ that connects it
with the inflaton field $\phi$ [23, 14]

$\displaystyle S$ $\displaystyle=$ $\displaystyle\int
d^{4}x\sqrt{-g}(-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V\left(\phi\right)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}.$
$\displaystyle-\frac{\sigma}{4\Lambda}\phi
F_{\mu\nu}\tilde{F}^{\mu\nu}+\frac{1}{2}m_{A}^{2}A^{\mu}A_{\mu}).$

Here, $\Lambda$ represents the mass scale that suppresses higher-dimensional
operators in the theory, while $\sigma$ is the dimensionless coupling of the
inflaton-gauge field. Here $A_{\mu}$ is U(1) gauge field $A_{\mu}$ and its
field strength tensor
$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. We also have its
dual
$\tilde{F}^{\mu\nu}=\frac{1}{2}\frac{\epsilon^{\mu\nu\alpha\beta}}{\sqrt{-g}}F_{\alpha\beta}$.
We focus on the key role of $\frac{1}{2}m_{A}^{2}A^{\mu}A_{\mu}$, which
connects the gauge field and the gravitational waves through the mass $m_{A}$.
The energy of the Chern-Simons interaction is $-\frac{\sigma}{8\Lambda}\int
d^{3}xe^{-i\mathbf{k\cdot
x}}\delta\phi\epsilon^{\mu\nu\alpha\beta}F_{\mu\nu}F_{\alpha\beta}$. We
consider a homogeneous and isotropic universe described by the Friedmann-
Lemaître-Robertson-Walker (FLRW) metric
$ds^{2}=-dt^{2}+a^{2}(t)g_{ij}dx^{i}dx^{j}$ where $a(t)$ is the scale factor
and $H=da/dt$ is the Hubble parameter. We also assume the inflaton $\phi$ that
only depends on time $t$. This field is similar to an axion, a light
pseudoscalar particle with a shift symmetry $\phi\rightarrow\phi+\phi_{0}$. We
take a simple quadratic potential for the inflaton,
$V\left(\phi\right)=\frac{1}{2}m^{2}\phi^{2}$, where $m=1.9\times
10^{13}GeV\left(\propto\Lambda^{2}\right)$ [24]. The background dynamics of
$\phi\left(t\right)$ and $A(t,x)$ [25] are given by

$\ddot{\phi}+3H\dot{\phi}+\frac{dV\left(\phi\right)}{d\phi}-\frac{\nabla^{2}\phi}{a^{2}\left(t\right)}=\frac{\sigma}{\Lambda}\left\langle\mathbf{E\cdot
B}\right\rangle,$ (2)
$3M_{p}^{2}H^{2}-\frac{1}{2}\dot{\phi}^{2}-V\left(\phi\right)-\frac{\left(\nabla\phi\right)^{2}}{2a^{2}\left(t\right)}=\frac{1}{2}\left\langle\mathbf{E}^{2}+\mathbf{B}^{2}+\frac{m_{A}^{2}}{a^{2}}\mathbf{A}^{2}\right\rangle,$
(3)

where
$\mathbf{E}=-\frac{\mathbf{A}\prime}{a^{2}}=-\frac{1}{a^{2}}\frac{\partial\mathbf{A}}{\partial\tau}$
and $\mathbf{B}=\frac{\mathbf{\nabla}\times\mathbf{A}}{a^{2}}$ are the
electric and magnetic fields corresponding to the gauge field respectively.
Here $M_{p}^{2}\approx 2.4\times 10^{18}GeV$ is the reduced Planck mass. We
now consider that the evolution of the scale factor is written as a power law
$a=a_{0}t^{q}$. That implies the Hubble parameter can rewrite as
$H=\frac{1}{a\tau}$, where $\tau$ is the conformal time with
$dt=a\left(\tau\right)d\tau$. The inflaton perturbations follow the equation
of motion

$\delta\ddot{\phi}+3\beta
H\delta\dot{\phi}-\left(\frac{\nabla^{2}}{a^{2}}-\frac{d^{2}V\left(\phi\right)}{d\phi^{2}}\right)\delta\phi=\frac{\sigma}{\Lambda}\left(\mathbf{E\cdot
B-}\left\langle\mathbf{E\cdot B}\right\rangle\right).$ (4)

We define the curvature perturbation on hypersurfaces characterized by uniform
density as
$\zeta\left(\tau,\mathbf{x}\right)=-\frac{H}{\dot{\phi}}\delta\phi\left(\tau,\mathbf{x}\right)$.
Without the presence of gauge fields, the inflaton perturbation’s classical
vacuum solution can be formulated as follows:
$\delta\phi\left(\tau,\mathbf{x}\right)=\frac{H}{\sqrt{2k^{3}}}\left(1+ik\tau\right)e^{-ik\tau}$,
and we take

$\zeta\left(\tau,\mathbf{x}\right)=-\frac{H^{2}}{\dot{\phi}\sqrt{2k^{3}}}\left(1+ik\tau\right)e^{-ik\tau}.$
(5)

The decoupling of modes can be achieved by breaking down the quantum field
$\mathbf{A}\left(\tau,\mathbf{x}\right)$ into components:

$\displaystyle\mathbf{A}\left(\tau,\mathbf{x}\right)$ $\displaystyle=$
$\displaystyle\sum_{\lambda=\pm,0}\int\frac{d^{3}k}{\left(2\pi\right)^{3}}\mathbf{\epsilon}_{\lambda}\left(\mathbf{k}\right)e^{i\mathbf{k\cdot
x}}(a_{\lambda}\left(\mathbf{k}\right)A_{\lambda}\left(\tau,k\right)$
$\displaystyle+a_{-\lambda}^{\dagger}\left(\mathbf{k}\right)A_{\lambda}^{\ast}\left(\tau,k\right).)$

The polarization vector, $\mathbf{\epsilon}_{\lambda}\left(\mathbf{k}\right)$,
and the annihilation operator, $a_{\lambda}\left(\mathbf{k}\right)$, adhere to
the standard commutation and orthonormality relations. The commutation
relation of the creation ($a_{-\lambda}^{\dagger}$) and annihilation
($a_{\lambda}\left(\mathbf{k}\right)$) operators can be expressed as
$\left[a_{\lambda}\left(\mathbf{k}\right),a_{-\lambda^{\prime}}^{\dagger}\left(\mathbf{k}^{\prime}\right)\right]=\left(2\pi\right)^{3}\delta_{\lambda\lambda^{\prime}}\delta^{\left(3\right)}\left(\mathbf{k}-\mathbf{k}^{\prime}\right).$
The generation of the dominant vector field is controlled by the field
equations governing the transverse Fourier modes

$\frac{\partial^{2}A_{\mathbf{\pm}}\left(\tau,k\right)}{\partial\tau^{2}}+\left(k^{2}+a^{2}\left(\mathbf{\tau}\right)m_{A}^{2}\pm\frac{\sigma
k\dot{\phi}}{\Lambda H\tau}\right)A_{\pm}\left(\tau,k\right)=0.$ (7)

where $\dot{\phi}=\frac{\partial\phi}{\partial\tau}$,
$A_{\pm}\left(\tau,k\right)$ represents the mode function, with the
longitudinal mode denoted as $\lambda=0$ and the two transverse modes as
$\lambda=\pm$, respectively. The expression for the two helicity modes
function of the vector field can be formulated using the $W_{\mp
i\xi,i\omega}$ as

$A_{\mathbf{\pm}}\left(\tau,k\right)=\frac{1}{\sqrt{2k}}e^{\pm\frac{\pi\xi}{2}}W_{\mp
i\xi,i\omega}\left(2ik\tau\right).$ (8)

where $W_{\mp i\xi,i\omega}$ is the Whittaker W function. Here we have
introduced a convenient notation

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{\sigma\dot{\phi}}{2\Lambda H},$ (9)
$\displaystyle\omega^{2}$ $\displaystyle=$
$\displaystyle\frac{m_{A}^{2}}{H^{2}}-\frac{1}{4}.$ (10)

The Taylor expansion of the Whittaker function around a point $2ik\tau_{0}$
can be expressed as follows:

$\displaystyle W_{\mp i\xi,i\omega}\left(2ik\tau\right)$ $\displaystyle=$
$\displaystyle
e^{-ik\tau}z^{i\omega+\frac{1}{2}}\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{n!}$
$\displaystyle\times\frac{\Gamma\left(i\omega\mp
i\xi+\frac{1}{2}-n\right)}{\Gamma\left(i\omega\mp
i\xi+\frac{1}{2}\right)}\left(2ik\tau-2ik\tau_{0}\right)^{n}.$

In this case, we can deduce the expression for $A_{\mathbf{\pm}}$ as

$\displaystyle A_{\mathbf{\pm}}$ $\displaystyle=$
$\displaystyle\frac{z^{i\omega+\frac{1}{2}}}{\sqrt{2k}}e^{\pm\frac{\pi\xi}{2}-ik\tau}\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}\left(2ik\right)^{n}}{n!}$
$\displaystyle\times\frac{\Gamma\left(i\omega\mp
i\xi+\frac{1}{2}-n\right)}{\Gamma\left(i\omega\mp
i\xi+\frac{1}{2}\right)}\left(\tau-\tau_{0}\right)^{n}.$

We can use this approximation
$A_{\mathbf{\pm}}\sim\frac{z^{i\omega+\frac{1}{2}}}{\sqrt{2k}}e^{\pm\frac{\pi\xi}{2}-ik\tau}\left[1-2ik\left(\tau-\tau_{0}\right)\frac{\Gamma\left(i\left(\omega\mp\xi\right)-\frac{1}{2}\right)}{\Gamma\left(i\left(\omega\mp\xi\right)+\frac{1}{2}\right)}\right]$
to plot Figs. (1)-(2)-(3). For $\tau=\tau_{0}$, the gauge field can be
expressed as

$A_{\mathbf{\pm}}\left(\tau_{0}\right)\sim\frac{1}{\sqrt{2k}}z^{i\omega+\frac{1}{2}}e^{\pm\frac{\pi\xi}{2}-ik\tau_{0}}.$
(13)

We can relate the gauge field to the Hubble parameter with this ratio
$\frac{A_{\mathbf{+}}\left(\tau_{0},k\right)}{A_{\mathbf{-}}\left(\tau_{0},k\right)}\sim
e^{\pi\xi}$. In the Figs. (1)-(2)-(3) we will only study the evolution of
$A_{\mathbf{+}}$ in term of $\left(\omega,\xi\right)$.

Figure 1: The plot shows how the gauge field $A_{\mathbf{+}}$ changes with
respect to $\left(\omega,\xi\right)$. Parametersare defined. as $z=k=\tau=1$
and $\tau_{0}=1.$ Figure 2: The plot shows how the gauge field
$A_{\mathbf{+}}$ changes with respect to $\left(\omega,\xi\right)$.
Parametersare defined. as $z=k=\tau=1$ and $\tau_{0}=1.5$. Figure 3: The plot
shows how the gauge field $A_{\mathbf{+}}$ changes with respect to
$\left(\omega,\xi\right)$. Parametersare defined. as $z=k=\tau=1$ and
$\tau_{0}=3$.

We generate a grid of $\omega$ and $\xi$ values to compute $A_{\mathbf{+}}$
over an interval in Fig. 1. As shown in Figs. (1)-(2), the parameter
$\tau_{0}$ is crucial for describing how $A_{\mathbf{+}}$ changes. Moreover,
the diagram in Fig. (3) not only shows the case for $\tau_{0}=3$, but also
illustrates how $A_{\mathbf{+}}$ evolves for all other values of $\tau_{0}\geq
3$. The goal of this study is to examine how the mass of the gauge field
$m_{A}\sim\omega$ affects the Hubble parameter $H\sim\frac{1}{\xi}$. Perhaps
the discrepancy in the measurements of the universe’s expansion rate, known as
the Hubble tension [27, 28, 29], is related to how the Hubble parameter $H$
depends on the mass of the gauge bosons in the early universe. The value of
$\tau_{0}=3$ signifies the critical point at which the mass of the gauge boson
begins to affect the Hubble parameter. This effect is primarily due to the
space-time curvature induced by these massive bosons.

Hopf fibration. In the study [26], they showed the existence of gravitational
type gravitational hopfions. Likewise, we will consider that the additional
term $M_{\mp i\xi,-i\omega}$ which is added to $A_{\mathbf{\pm}}$ describes
the hopfions in the early universe. In this section, our objective is to
demonstrate the presence of hopfions in the early universe. The average energy
density of the gauge field $A_{\mu}$ can be written as
$\rho_{A}=\frac{1}{2}\left\langle\mathbf{E}^{2}+\mathbf{B}^{2}+\frac{m_{A}^{2}}{a^{2}}\mathbf{A}^{2}\right\rangle.$
We express the term $\mathbf{E\cdot B}$ in the form given by [24]

$\left\langle\mathbf{E\cdot
B}\right\rangle=-\frac{1}{a^{4}}\int\frac{d^{3}k}{\left(2\pi\right)^{3}}\frac{k}{2}\frac{d}{d\tau}\left(\left|A_{\mathbf{+}}\right|^{2}-\left|A_{\mathbf{-}}\right|^{2}\right).$
(14)

Next, we introduce the conjugate term:

$\left\langle\mathbf{E\cdot
B}\right\rangle^{\ast}\equiv-\frac{1}{a^{4}}\int\frac{d^{3}k}{\left(2\pi\right)^{3}}\frac{k}{2}\frac{d}{d\tau}\left(2A_{\mathbf{+}}A_{\mathbf{-}}^{\ast}\right).$
(15)

The spherically equivariant map to compactifiy $\mathbb{R}^{3}\rightarrow
S_{A_{\mathbf{+}},A_{\mathbf{-}}}^{3}\subset\mathbb{C}^{2}$ is

$S_{A_{\mathbf{+}},A_{\mathbf{-}}}^{3}\cong\left\\{\left(A_{\mathbf{+}}A_{\mathbf{-}}\right)\in\mathbb{C}^{2},\left|A_{\mathbf{+}}\right|^{2}+\left|A_{\mathbf{-}}\right|^{2}=1\right\\},$
(16)

where $S_{A_{\mathbf{+}},A_{\mathbf{-}}}^{3}\simeq SU(2)$. The Hopf fibration
$\mathbf{P}_{h}$ is then defined as [30, 31, 32]

$\mathbf{P}_{h}\left(A_{\mathbf{+}},A_{\mathbf{-}}\right)=\left(2A_{\mathbf{+}}A_{\mathbf{-}}^{\ast},\left|A_{\mathbf{+}}\right|^{2}-\left|A_{\mathbf{-}}\right|^{2}\right),$
(17)

which satisfies
$\frac{d\mathbf{P}_{h}\left(A_{\mathbf{+}},A_{\mathbf{-}}\right)}{d\tau}=-3\left(2\pi\right)^{3}\left(\frac{2a}{k}\right)^{4}\left(\left\langle\mathbf{E\cdot
B}\right\rangle^{\ast},\left\langle\mathbf{E\cdot B}\right\rangle\right).$ The
Hopf index $\mathcal{H}$ emerges as the most appropriate topological invariant
for characterizing these phenomena

$\mathcal{H}=\frac{1}{\left(8\pi\right)^{2}}\int d^{3}x\mathbf{A\cdot B.}$
(18)

The spacetime Hopf index $\mathcal{H}$ emerges as the most appropriate
topological invariant for characterizing these phenomena
$\frac{d\mathcal{H}}{d\tau}=-\frac{\pi a^{2}}{\left(2\pi\right)^{3}}\int
d^{3}x\mathbf{E\cdot B-}\frac{\pi a}{\left(2\pi\right)^{3}}\int
d^{3}x\mathbf{A\cdot}\nabla\times\mathbf{E,}$ where
$\frac{\partial\mathbf{B}}{\partial
t}=\frac{\partial\mathbf{B}}{a\partial\tau}=-\nabla\times\mathbf{E}$. This
perturbation can be expressed as

$\frac{d\mathcal{H}}{d\tau}\mathbf{\approx}-\frac{\pi
a^{2}}{\left(2\pi\right)^{3}}\int d^{3}x\left\langle\mathbf{E\cdot
B}\right\rangle.$ (19)

Substituting Eq. (14) into Eq. (19), we ascertain that

$\frac{d\mathcal{H}}{d\tau}\mathbf{=}\frac{\pi}{2\left(2\pi\right)^{3}a^{2}}\int
d^{3}x\int\frac{d^{3}k}{\left(2\pi\right)^{3}}k\frac{d}{d\tau}\left(\left|A_{\mathbf{+}}\right|^{2}-\left|A_{\mathbf{-}}\right|^{2}\right)e^{ikx}e^{-ikx},$

or equivalently

$\frac{d\mathcal{H}}{d\tau}\mathbf{=}\frac{\pi
k}{2\left(2\pi\right)^{3}a^{2}}\frac{d}{d\tau}\left(\left|A_{\mathbf{+}}\right|^{2}-\left|A_{\mathbf{-}}\right|^{2}\right).$
(20)

Subsequently we assume that $a^{2}=\frac{\pi k}{2\left(2\pi\right)^{3}}$ and
we find

$\mathcal{H}\mathbf{=}\left|A_{\mathbf{+}}\right|^{2}-\left|A_{\mathbf{-}}\right|^{2}.$
(21)

The terms $\left|A_{\mathbf{+}}\right|$ and $\left|A_{\mathbf{-}}\right|^{2}$
can be linked to energy or load density associated with these fields. From
this relationship we notice for $\mathcal{H}=0$, the magnitudes of
$\left|A_{\mathbf{+}}\right|$ and $\left|A_{\mathbf{-}}\right|$ are equal. On
the other hand we can calculate the conjugate of $\mathcal{H}$ as follows

$\frac{d\mathcal{H}^{\ast}}{d\tau}=-\frac{\pi
a^{2}}{\left(2\pi\right)^{3}}\int d^{3}x\left\langle\mathbf{E\cdot
B}\right\rangle^{\ast}.$ (22)

Replacing the expression from Eq. (15) into Eq. (22), we establish that

$\frac{d\mathcal{H}^{\ast}}{d\tau}=\frac{\pi
k}{2a^{2}\left(2\pi\right)^{3}}\frac{d}{d\tau}\left(2A_{\mathbf{+}}A_{\mathbf{-}}^{\ast}\right).$

Setting $a^{2}=\frac{\pi k}{2\left(2\pi\right)^{3}}$, we derive that

$\mathcal{H}^{\ast}=2A_{\mathbf{+}}A_{\mathbf{-}}^{\ast}.$ (23)

Figure 4: This figure represents the Hopf fibration $\mathbf{P}_{h}$ about the
gauge field modes $A_{\mathbf{+}}$ and $A_{\mathbf{-}}$. Each point on the
plot signifies the outcome of the Hopf fibration for a specific pair of
$A_{\mathbf{+}}$ and $A_{\mathbf{-}}$ values.

Fig. (4) illustrates the outcome of our research, demonstrating that the
massive gauge field can produce a Hopf fibration, representing regions in
space-time where Hopfions are applicable. These Hopfions encompass points that
form lines, adhering to the space-time curvature induced by the mass $m_{A}$.
From Eqs. (17)-(21)-(23) we get

$\mathbf{P}_{h}\left(A_{\mathbf{+}},A_{\mathbf{-}}\right)=\left(\mathcal{H}^{\ast},\mathcal{H}\right).$
(24)

We have shown that the Hopf fibration is articulated through the presence of
two Hopf index components Fig. (4). This revelation enables the
characterization of the Hopfion state in the early universe. Following the
inflationary epoch characterized by the inflation field, the radiation era
begins, orchestrated by the electromagnetic field. However, the emergence of
topological defects in spacetime, specifically hopfions, occurs during this
phase. Within these regions, novel fluctuations manifest, potentially serving
as the source of the pulsar timing arrays signal [33, 34, 35, 36]. On the
other hand, the $\mathbf{P}_{h}\left(A_{\mathbf{+}},A_{\mathbf{-}}\right)$
resides on the unit 2-sphere in $\mathbb{C}\times\mathbb{R}$. Conclusion. This
study introduces a new method for examining the dynamics of inflation fields
in conjunction with a massive gauge boson. We developed a new model that
characterizes the curvature of space-time, influenced by a massive gauge field
in the early universe. This curvature can lead to numerous observations, such
as the Hubble tension issue and the isotropic stochastic gravitational-wave
background. To address this, we are introducing the concept of gauge field
Hopfions. The solution for the gauge field is represented by the Whittaker
function. This framework allows for the direct determination of the
relationships between the mode functions of the gauge field
$A_{\mathbf{\pm}}$, the Hubble parameter $H$, and the mass of the gauge field
$m_{A}$. Additionally, our results show a significant correlation between the
Hubble parameter and the mass $m_{A}$, as shown in the curves Figs.
(1)-(2)-(3), which illustrate that the parameter $\tau_{0}$ is crucial for
determining the variation of mode $A_{\mathbf{+}}$. This correlation could
potentially offer a comprehensive explanation for the Hubble tension.
Moreover, Fig. (3) presents the case for $\tau_{0}=3$ and also the evolution
of mode $A_{\mathbf{+}}$ for all other values of $\tau\geq 3$. The goal of
this research is to investigate how the Hubble parameter is affected by the
mass of the gauge field. Furthermore, we have shown that the mode functions of
the gauge field can be expressed within the context of a component of the Hopf
fibration. After our calculations, we confirmed the accuracy of this
proposition by demonstrating that the Hopf fibration of the gauge field is
articulated in terms of the Hopf index $\mathcal{H}$ and their combination. We
have demonstrated that the Hopf fibration is expressed via the existence of
two components of the Hopf index as shown in Fig. (4). This discovery allows
for the description of the Hopfion state in the early universe. After the
inflationary period defined by the inflation field, the era of radiation
begins, governed by the electromagnetic field. However, during this phase,
topological defects in space-time, particularly hopfions, emerge. Within these
areas, new fluctuations appear, which could potentially be the origin of the
pulsar timing arrays.

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# Power spectrum with $k^{6}$ growth for primordial black holes

Rongrong<EMAIL_ADDRESS>Hongwei Yu1,2222Corresponding
author<EMAIL_ADDRESS>and Puxun Wu1,2333Corresponding author:
<EMAIL_ADDRESS>1Department of Physics and Synergetic Innovation Center for
Quantum Effects and Applications, Hunan Normal University, Changsha, Hunan
410081, China
2Institute of Interdisciplinary Studies, Hunan Normal University, Changsha,
Hunan 410081, China

###### Abstract

The decrease of both the rolling speed of the inflaton and the sound speed of
the curvature perturbations can amplify the curvature perturbations during
inflation so as to generate a sizable amount of primordial black holes. In the
ultraslow-roll inflation scenario, it has been found that the power spectrum
of curvature perturbations has a $k^{4}$ growth. In this paper, we find that
when the speed of sound decreases suddenly, the curvature perturbations
becomes scale dependent in the infrared limit and the power spectrum of the
curvature perturbation only has a $k^{2}$ growth. Furthermore, by studying the
evolution of the power spectrum in the inflation model, in which both the
sound speed of the curvature perturbations and the rolling speed of the
inflaton are reduced, we find that the power spectrum is nearly scale
invariant at the large scales to satisfy the constraint from the cosmic
microwave background radiation observations, and at the same time can be
enhanced at the small scales to result in an abundant formation of primordial
black holes. In the cases of the simultaneous changes of the sound speed and
the slow-roll parameter $\eta$ and the change of the sound speed preceding
that of the slow-roll parameter $\eta$, the power spectrum can possess a
$k^{6}$ growth under certain conditions, which is the steepest growth of the
power spectrum reported so far.

##  I Introduction

During the standard slow-roll inflation, the solution of the Sasaki-Mukhanov
equation for the evolution of the curvature perturbations $\mathcal{R}$
contains, in the infrared limit, a constant term and a time-decaying one, and
this solution results in a nearly scale-invariant power spectrum of the
curvature perturbations Bruce A. Bassett ; Antonio Riotto , which is well-
consistent with the cosmic microwave background (CMB) radiation observations.
The CMB observations have limited the amplitude of the power spectrum to the
order of $\mathcal{O}(10^{-9})$ at the CMB scale Smoot1997 ; Spergel2003 ;
Komatsu2011 ; Akrami2020 ; Aghanim2020 . It has been found that, if the
amplitude of the power spectrum of the curvature perturbations can be enhanced
for about seven orders at the scales smaller than the CMB one S. Chongchitnan
; K. Kohri ; C. T. Byrnes ; E. Bugaev , a sizable amount of primordial black
holes can be generated when these enhanced perturbations reenter the horizon
during the radiation- or matter-dominated era Zeldovich ; Hawking ; Carr ;
Meszaros ; Carr1975 ; Khlopov ; Ozsoy2023 ; Bhattacharya2023b .

The amplitude of the power spectrum of the curvature perturbations in the
standard slow-roll inflation can be expressed as
$\mathcal{P_{R}}=\frac{H^{2}}{8\pi^{2}\epsilon c_{s}}$ when the mode exits the
horizon during inflation, where $H$ is the Hubble parameter which is
approximately constant during inflation, $\epsilon$ the slow-roll parameter
and $c_{s}$ the sound speed of the curvature perturbations. Thus, a natural
way to amplify the curvature perturbations is to reduce the rolling speed of
the inflaton which is proportional to $\epsilon$ or to suppress the sound
speed. Decreasing the inflaton’s rolling speed can be realized in the
ultraslow-roll inflation Yokoyama1998 ; Choudhury2014 ; Germani2017 ;
Motohashi2017 ; Ezquiaga2017 ; H. Di2018 ; Ballesteros2018 ; Dalianis2019 ;
Gao2018 ; Tada2019 ; Mishra2020 ; Atal2020 ; Ragavendra2021 ; Bhaumik2020 ;
Drees2021 ; C.Fu2020 ; Xu2020 ; Lin2020 ; Dalianis2021 ; Yi2021 ; Gao2021 ;
Yi2021b ; TGao2021 ; Solbi2021 ; Gao2021b ; Solbi2021b ; Zheng2021 ;
Teimoori2021a ; Cai2021 ; Wang2021 ; Fuchengjie2019 ; fuchengjie2020 ;
Dalianis2020 ; Teimoori2021 ; Karam2022 ; Heydari2022 ; Heydari2022b ;
Bellido2017 ; Ezquiaga2018 ; Pi2022 ; Choudhury2023 ; Meng2022 ; Mu2022 ;
Kawaguchi2022 ; Fu2022 ; Chen2022 ; Gu2023 ; Yi2022 ; Choudhury2023c ;
Choudhury2023e ; Choudhury2023b ; Cai2023 ; Ghoshal2023 ; Franciolini2023 , in
which the slow-roll parameter $\eta$, which is defined to be
$\eta=\frac{\dot{\epsilon}}{\epsilon H}$ with an overdot denoting a derivative
with respect to the time $t$, equals approximately $-6$. During the transition
of $\eta$ from $\eta\simeq 0$, which corresponds to the slow-roll inflation,
to $-6$, the Israel junction conditions Israel1966 ; Deruelle1995  are used to
obtain the solution of the curvature perturbations in the ultraslow-roll
phase. Expanding this solution in the infrared limit, one can see that the
decaying term in the solution of the Sasaki-Mukhanov equation for the
evolution of the curvature perturbations becomes a growing one. This growing
term gradually dominates if the ultraslow-roll inflation persists a
sufficiently long time and it results in the enhancement of the curvature
perturbations to meet the requirement of formation of a sizable amount of
primordial black holes. It has been found that the power spectrum of the
curvature perturbations displays a $k^{4}$ growth and has a dip preceding the
$k^{4}$ dependence Garrilho2019 ; Liu2020 ; Byrnes2019 . The steeper
$k^{5}(\log k)^{2}$ growth of the power spectrum can be obtained if an
$\eta=-1$ middle phase between the slow- and ultraslow-roll inflations
Garrilho2019 ; Cole2022  is added.

When the amplification of the curvature perturbations results from the
decrease of the sound speed G. Ballesteros ; Kamenshchik ; Gorji2022 ; Romano3
; Ballesteros2022 , which equals $1$ in the canonical scalar field inflation
model, the solution of the curvature perturbations, although does not contain
growing terms, still has a constant component and a decay part. However, the
constant component becomes scale variant at small scales, which makes the
power spectrum become enhanced. If the Israel junction conditions are utilized
to match the curvature perturbation and its derivative at the time when the
sound speed decreases suddenly, the power spectrum has a $k^{4}$ growth
Ballesteros2022 ; Zhai2022 . However, this junction conditions is inapplicable
in the case that the sound speed suddenly decreases due to the appearance of
the square of the delta function Nakashima2011 . Therefore, in this paper we
will employ an improved junction conditions Gorji2022  to restudy the
evolution of the power spectrum when the sound speed decreases suddenly and
find that the growth of the power spectrum is $k^{2}$ rather than $k^{4}$.

Furthermore, in the Dirac-Born-Infeld-inspired nonminimal kinetic coupling
inflation model Qiu2022 , both $\epsilon$ and $c_{s}^{2}$ are closely related
to the concrete form of the inflationary potential, which indicates that this
inflation model may accommodate both a small sound speed of the curvature
perturbations and a small rolling speed of the inflaton at the same time. When
both the inflaton’s rolling speed and the sound speed of the curvature
perturbations are suppressed during inflation, will the growth of the power
spectrum of the curvature perturbations be steeper than $k^{4}$? This is an
interesting problem, which we are also going to address in this paper.

The rest of this paper is organized as follows: In Sec. II, the evolution of
the power spectrum is investigated when the speed of sound decreases suddenly.
In Sec. III, we study the evolution of the power spectrum in the case that
both the sound speed and the slow-roll parameter are suppressed and present
our conclusions in Sec. IV. Throughout this paper, we set
$c=\hbar=M_{\mathrm{Pl}}=1$.

##  II Growth of power spectrum from the sudden decrease of the sound speed

The evolution of the curvature perturbations $\mathcal{R}$ satisfies the
Sasaki-Mukhanov equation, which in the Fourier space takes the form

$\displaystyle
v^{\prime\prime}_{k}+\left(c_{s}^{2}k^{2}-\frac{z^{\prime\prime}}{z}\right)v_{k}=0\;,$
(1)

where $v_{k}=z\mathcal{R}_{k}$, a prime indicates a derivative with respect to
the conformal time $\tau$, and $z$ is defined as

$\displaystyle z^{2}\equiv\frac{2a^{2}\epsilon}{c_{s}^{2}}$ (2)

with $a$ being the cosmic scale factor. From the definition of $z$, one can
obtain

$\displaystyle\frac{z^{\prime\prime}}{z}=(aH)^{2}\left(2-\epsilon+\frac{3}{2}\eta-3s+s^{2}+s\epsilon-s\eta+\frac{1}{4}\eta^{2}-\frac{1}{2}\eta\epsilon\right)\;.$
(3)

Here $s=\frac{\dot{c}_{s}}{c_{s}H}$. During the (ultra)slow-roll inflation,
one has $\epsilon\ll 1$, and thus $aH\simeq-\frac{1}{\tau}$. Then Eq. (1) can
be rewritten as

$\displaystyle
v^{\prime\prime}_{k}+\left(c_{s}^{2}k^{2}-\frac{\nu^{2}-1/4}{(-\tau)^{2}}\right)v_{k}=0\;,$
(4)

where

$\displaystyle\nu\simeq\frac{3}{2}+\frac{1}{2}\eta-s\;.$ (5)

If $\eta$ and $c_{s}$ are constants, Eq. (4) has a general solution

$\displaystyle
v_{k}(\tau)=\alpha\sqrt{-\tau}H_{\nu}^{(1)}(-c_{s}k\tau)+\beta\sqrt{-\tau}H_{\nu}^{(2)}(-c_{s}k\tau)\;.$
(6)

Here $H_{\nu}^{(1)}$ and $H_{\nu}^{(2)}$ are the first and second Hankel
functions, respectively, and $\alpha$ and $\beta$ are two constants.

Figure 1:  The sound speed $c_{s}$ varies suddenly at $\tau_{1}$.

Now, we discuss the scenario of a sudden decrease of the sound speed, as shown
in Fig. 1. In the first stage ($\tau<\tau_{1}$), which corresponds to the
canonical slow-roll inflation, the sound speed $c_{s}$ is equal to one and the
slow-roll parameter $\eta$ is near zero. Thus, one has $\nu\simeq 3/2$ and
$\epsilon=\epsilon_{0}(\frac{\tau}{\tau_{0}})^{-\eta}\simeq\epsilon_{0}$.
Imposing that the solution of the Sasaki-Mukhanov equation matches the plane-
wave form in the ultraviolet regime ($-k\tau\gg 1$), we can derive the
evolution of the curvature perturbations

$\displaystyle\mathcal{R}^{(0)}_{k}(\tau)=i\frac{H}{2\sqrt{\epsilon_{0}k^{3}}}e^{-ik\tau}(1+ik\tau)\;.$
(7)

Apparently, the solution of the curvature perturbations contains a constant
term and a time-decaying one since $|\tau|$ decreases with the cosmic
expansion during inflation, which results in a nearly scale-invariant power
spectrum of the curvature perturbations in the superhorizon scales
($-k\tau\rightarrow 0$) with the amplitude of the power spectrum being

$\displaystyle\mathcal{P}_{0}=\frac{H^{2}}{8\pi^{2}\epsilon_{0}}\;.$ (8)

The CMB observations have limited $\mathcal{P}_{0}$ to be $\sim 10^{-9}$
Smoot1997 . At the moment $\tau_{1}$, we assume that the sound speed decreases
suddenly from 1 to a very tiny constant value $c_{s_{1}}$ and the inflation
enters the second stage. In this stage, the general solution of the curvature
perturbations has the form

$\displaystyle\mathcal{R}^{(1)}_{k}(\tau)=\frac{ic_{s_{1}}H\left(-\tau\right)^{3/2}}{\sqrt{2\epsilon_{0}}}\left[\alpha_{1}H_{3/2}^{(1)}(c_{s_{1}}k\tau)+\beta_{1}H_{3/2}^{(2)}(c_{s_{1}}k\tau)\right]\;,$
(9)

where $\alpha_{1}$ and $\beta_{1}$ are two constants. Usually, the Israel
junction conditions
$\mathcal{R}^{(0)}_{k}(\tau_{1})=\mathcal{R}^{(1)}_{k}(\tau_{1})$ and
$\mathcal{R}^{\prime(0)}_{k}(\tau_{1})=\mathcal{R}^{\prime(1)}_{k}(\tau_{1})$
are used to determine the values of $\alpha_{1}$ and $\beta_{1}$
Ballesteros2022 ; Zhai2022 . However, it has been found, in Ref. Nakashima2011
, that there is a square term of the delta function $\delta(\tau-\tau_{1})$
arising form $(c_{s}^{\prime}/c_{s})^{2}$ in $z^{\prime\prime}/z$ when a
sudden variation of the sound velocity occurs, which makes the analysis
impossible, and thus the Israel junction conditions need to be modified or a
new variable has to be introduced. In Nakashima2011 , a new variable is
defined, which satisfies the Israel junction conditions at $\tau_{1}$. Here,
we do not use this new variable Nakashima2011  but consider an improved
junction conditions, i.e. $\mathcal{R}_{k}$ and its conjugate momentum
$A\mathcal{R}^{\prime}_{k}$ are continuous at $\tau_{1}$, where
$A\equiv\frac{2\epsilon}{c_{s}^{2}}$ Gorji2022 . We have checked that the
introduction of the new viable and the improved junction conditions can give
the same results. Considering the improved junction conditions

$\displaystyle\mathcal{R}^{(0)}_{k}(\tau_{1})=\mathcal{R}^{(1)}_{k}(\tau_{1})\;,\quad
A_{0}\mathcal{R}^{\prime(0)}_{k}(\tau_{1})=A_{1}\mathcal{R}^{\prime(1)}_{k}(\tau_{1})\;$
(10)

with

$\displaystyle A_{0}=2\epsilon_{0}\;,\quad
A_{1}=\frac{2\epsilon_{0}}{c_{s_{1}}^{2}}\;,$ (11)

we can obtain that

$\displaystyle\alpha_{1}$ $\displaystyle=$
$\displaystyle-\frac{\left(1-c_{s_{1}}\right)\sqrt{\pi}e^{-i\left(1+c_{s_{1}}\right)k\tau_{1}}}{4\sqrt{c_{s_{1}}}},$
$\displaystyle\beta_{1}$ $\displaystyle=$
$\displaystyle-\frac{\left(1+c_{s_{1}}\right)\sqrt{\pi}e^{-i\left(1-c_{s_{1}}\right)k\tau_{1}}}{4\sqrt{c_{s_{1}}}}\;.$
(12)

Substituting $\alpha_{1}$ and $\beta_{1}$ into Eq. (9), we find the expression
of the curvature perturbation in the second phase, and then we can derive the
corresponding power spectrum, which is shown in Fig. 2. From it, one can find
that the power spectrum only has a $k^{2}$ growth. Thus, the result of a
$k^{4}$ growth, which is obtained from the standard Israel junction
conditions, should be incorrect.

To figure out the physical reason behind the $k^{2}$ growth of the power
spectrum, we expand the expression of the curvature perturbations (Eq. (9)) in
the infrared limits: $-c_{s_{1}}k\tau\rightarrow 0$ and
$-c_{s_{1}}k\tau_{1}\rightarrow 0$, and obtain

$\displaystyle\mathcal{R}^{(1)}_{k}(\tau)$ $\displaystyle=$
$\displaystyle\frac{iHe^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k^{3}}}-\frac{H\tau_{1}e^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k}}-\frac{ic_{s_{1}}^{2}H\tau_{1}^{2}k^{1/2}e^{-ik\tau_{1}}}{4\sqrt{\epsilon_{0}}}+\frac{ic_{s_{1}}^{2}Hk^{1/2}e^{-ik\tau_{1}}}{4\sqrt{\epsilon_{0}}}(-\tau)^{2}+\dots\;.$
(13)

Apparently, the wave number $k$ in Eq. (13) must satisfy the condition
$k\ll{k}_{c}\equiv-1/(c_{s_{1}}\tau_{1})$. It is easy to see that in the
infrared limit the leading part of the curvature perturbations is independent
of $\tau$, which contains three different $k$-dependent terms, and the
subleading part decays with time since $|\tau|$ decreases during inflation.
These characters are different from that in the case of the transition from
the slow-roll inflation to the ultraslow-roll one, where there is an
appearance of the growing term.

From Eq. (13), we obtain the power spectrum of the curvature perturbations

$\displaystyle\dfrac{\mathcal{P}_{\mathcal{R}^{(1)}_{k}}}{\mathcal{P}_{0}}\simeq
1+\left(1-c_{s_{1}}^{2}\right)\tau_{1}^{2}k^{2}+\frac{1}{4}c_{s_{1}}^{4}\tau_{1}^{4}k^{4}\;$
(14)

after neglecting all decaying terms. If the $k^{2}$ term becomes comparable to
the constant one, the wave number needs to be equal to about

$\displaystyle k_{1}\simeq-\frac{1}{\sqrt{1-c_{s_{1}}^{2}}\tau_{1}}\simeq
c_{s_{1}}k_{c}\;.$ (15)

The wave number at which the $k^{4}$ term becomes comparable with the $k^{2}$
one is

$\displaystyle
k_{2}\simeq-\frac{2\sqrt{1-c_{s_{1}}^{2}}}{c_{s_{1}}^{2}\tau_{1}}\simeq\frac{2}{c_{s_{1}}}k_{c}\;.$
(16)

It is obvious that $k_{1}\ll k_{c}$, but $k_{2}\gg k_{c}$ since $c_{s_{1}}\ll
1$. Thus, $k_{2}$ is beyond the infrared condition $k\ll k_{c}$, which means
that the power spectrum has no $k^{4}$ growth, and the steepest growth of the
power spectrum is only $k^{2}$. At the CMB scale, the first term in Eq. (14)
dominates, which leads to a scale-invariant spectrum consistent with the CMB
observations. Going to the scales which are smaller than the CMB one, the
second term begins to play a dominant role. The power spectrum becomes scale
dependent and has a $k^{2}$ growth. These results are shown clearly in Fig. 2,
in which the approximate result given in Eq. (14) is very consistent with the
numerical one. There is no dip in the power spectrum since no term cancels the
constant one, which is different from the case of the ultraslow-roll
inflation.

Figure 2:  The evolution of the power spectrum as a function of wave number
$k$. The solid-gray and dashed-blue lines represent the numerical and
approximate results, respectively. The dashed-red line indicates the $k^{2}$
growth.

##  III Growth behavior of power spectrum when both the sound speed and the
slow-roll parameter $\eta$ are changed suddenly

We have known that the enhancement of the power spectrum can be realized by
decreasing the sound speed $c_{s}$ or reducing the slow-roll parameter
$\epsilon$. In the following, we will study the growth of the power spectrum
when both $\epsilon$ and $c_{s}$ are suppressed. For simplicity, we will
consider that the sound speed changes suddenly from $1$ to a constant much
less than one, and the slow-roll parameter $\eta$ suddenly from a constant
near zero to a negative constant. A negative $\eta$ will lead to the decrease
of $\epsilon$ since $\epsilon\propto\tau^{-\eta}$. We first consider the case
that the variations of $c_{s}$ and $\eta$ occur simultaneously.

###  III.1 Simultaneous changes of sound speed and slow-roll parameter $\eta$

The scenario considered in this subsection is shown in Fig. 3. Initially, the
Universe undergoes a standard slow-roll inflation, in which $c_{s}=1$,
$\epsilon\simeq\epsilon_{0}\ll 1$ and $\eta\sim 0$. At time $\tau_{1}$, the
sound speed $c_{s}$ and the slow-roll parameter $\eta$ change suddenly from
$1$ and $\sim 0$ to a small value $c_{s_{1}}$ and a negative constant
$\eta_{1}$, respectively. Thus, during the second phase, the slow-roll
parameter $\epsilon$ decays as
$\epsilon(\tau)=\epsilon_{0}(\tau/\tau_{1})^{-\eta_{1}}$ and the sound speed
is a small constant $c_{s_{1}}$.

Figure 3:  The slow-roll parameter $\eta$ (red line) and the sound speed
$c_{s}$ (blue line) vary simultaneously at $\tau_{1}$.

From Eq. (6), we obtain the general solution of the curvature perturbations

$\displaystyle\overline{\mathcal{R}}^{(1)}_{k}(\tau)=-\frac{c_{s_{1}}H\tau_{1}^{-\frac{\eta_{1}}{2}}\tau^{\frac{3+\eta_{1}}{2}}}{\sqrt{2\epsilon_{0}}}\left[\alpha_{2}H_{\nu}^{(1)}(c_{s_{1}}k\tau)+\beta_{2}H_{\nu}^{(2)}(c_{s_{1}}k\tau)\right]\;$
(17)

in the second phase, where $\alpha_{2}$ and $\beta_{2}$ are two constants, and
$\nu=(3+\eta_{1})/2$. Matching $\mathcal{R}^{(0)}_{k}$ and
$\overline{\mathcal{R}}^{(1)}_{k}$ at $\tau=\tau_{1}$ by using the improved
junction conditions,
$\mathcal{R}^{(0)}_{k}(\tau_{1})=\overline{\mathcal{R}}^{(1)}_{k}(\tau_{1})$
and
$A_{0}\mathcal{R}^{\prime(0)}_{k}(\tau_{1})=A_{1}\overline{\mathcal{R}}^{\prime(1)}_{k}(\tau_{1})$,
one can achieve that

$\displaystyle\alpha_{2}$ $\displaystyle=$ $\displaystyle\frac{\pi
e^{-ik\tau_{1}}}{4\sqrt{2c_{s_{1}}^{2}\tau_{1}^{3}k^{3}}}\bigg{[}\left(\left(3+\eta_{1}\right)\left(1+ik\tau_{1}\right)-(c_{s_{1}}k\tau_{1})^{2}\right)H_{\nu}^{(2)}(c_{s_{1}}k\tau_{1})$
$\displaystyle\qquad\qquad\qquad-
c_{s_{1}}k\tau_{1}\left(1+ik\tau_{1}\right)H_{\nu+1}^{(2)}(c_{s_{1}}k\tau_{1})\bigg{]}\;,$

and

$\displaystyle\beta_{2}$ $\displaystyle=$ $\displaystyle-\frac{\pi
e^{-ik\tau_{1}}}{4\sqrt{2c_{s_{1}}^{2}\tau_{1}^{3}k^{3}}}\bigg{[}\left(\left(3+\eta_{1}\right)\left(1+ik\tau_{1}\right)-(c_{s_{1}}k\tau_{1})^{2}\right)H_{\nu}^{(1)}(c_{s_{1}}k\tau_{1})$
$\displaystyle\qquad\qquad\qquad-
c_{s_{1}}k\tau_{1}\left(1+ik\tau_{1}\right)H_{\nu+1}^{(1)}(c_{s_{1}}k\tau_{1})\bigg{]}\;.$

Substituting $\alpha_{2}$ and $\beta_{2}$ into Eq. (17) gives the expression
of the curvature perturbations during the second phase. Expanding this
expression in the infrared limit ($-c_{s_{1}}k\tau\rightarrow 0$ and
$-c_{s_{1}}k\tau_{1}\rightarrow 0$), we arrive at

$\displaystyle\overline{\mathcal{R}}^{(1)}_{k}(\tau)$ $\displaystyle=$
$\displaystyle\frac{iHe^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k^{3}}}-\frac{H\tau_{1}e^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k}}-\left(\frac{3ic_{s_{1}}^{2}H\tau_{1}^{2}e^{-ik\tau_{1}}}{4\sqrt{\epsilon_{0}}(3+\eta_{1})}+\frac{ic_{s_{1}}^{2}H\eta_{1}\left(-\tau_{1}\right)^{-1-\eta_{1}}e^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}}(1+\eta_{1})(3+\eta_{1})}(-\tau)^{3+\eta_{1}}\right)k^{1/2}$
(18) $\displaystyle+$
$\displaystyle\left(\frac{c_{s_{1}}^{2}H\tau_{1}^{3}e^{-ik\tau_{1}}}{4\sqrt{\epsilon_{0}}(3+\eta_{1})}-\frac{c_{s_{1}}^{2}He^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}}(1+\eta_{1})(3+\eta_{1})\left(-\tau_{1}\right)^{\eta_{1}}}(-\tau)^{3+\eta_{1}}\right)k^{3/2}$
$\displaystyle+$ $\displaystyle\dots\;.$

It is easy to see that the solution contains a time-independent part and a
time-dependent one, which will decay with time when $\eta_{1}>-3$, and grow
when $\eta_{1}<-3$.

Now, we assume that the second phase lasts for $N_{2}$ number of e-folds,
which means that $\tau_{2}=\tau_{1}e^{-N_{2}}$ if this phase ends at
$\tau=\tau_{2}$. The power spectrum of the curvature perturbations can be
obtained from Eq. (18), which takes the form

$\displaystyle\dfrac{\mathcal{P}_{\overline{\mathcal{R}}^{(2)}_{k}}}{\mathcal{P}_{0}}$
$\displaystyle\simeq$ $\displaystyle
1+\left(1+\frac{2c_{s_{1}}^{2}\eta_{1}}{(1+\eta_{1})(3+\eta_{1})}e^{-\left(3+\eta_{1}\right)N_{2}}\right)\tau_{1}^{2}k^{2}$
$\displaystyle-$
$\displaystyle\left(\begin{split}&\frac{1}{3+\eta_{1}}+\frac{2}{(1+\eta_{1})(3+\eta_{1})}e^{-\left(3+\eta_{1}\right)N_{2}}-\frac{c_{s_{1}}^{2}\eta_{1}^{2}}{(1+\eta_{1})^{2}(3+\eta_{1})^{2}}e^{-2\left(3+\eta_{1}\right)N_{2}}\end{split}\right)c_{s_{1}}^{2}\tau_{1}^{4}k^{4}$
$\displaystyle+$
$\displaystyle\left(\begin{split}&\frac{1}{4(3+\eta_{1})^{2}}+\frac{1}{(1+\eta_{1})(3+\eta_{1})^{2}}e^{-\left(3+\eta_{1}\right)N_{2}}+\frac{1}{(1+\eta_{1})^{2}(3+\eta_{1})^{2}}e^{-2\left(3+\eta_{1}\right)N_{2}}\end{split}\right)c_{s_{1}}^{4}\tau_{1}^{6}k^{6}\;.$

Here we have kept the $k^{6}$ term in Eq. (III.1) since the $k^{6}$ growth may
occur under certain conditions. We need to compare the $k^{n}$ term with the
$k^{n-2}$ one to determine whether a $k^{n}$ growth will appear. We set
$k_{1}$, $k_{2}$, and $k_{3}$ to denote the wave number at which the scale-
invariant term becomes comparable with the $k^{2}$ term, the $k^{2}$ term
becomes comparable with the $k^{4}$ term, and the $k^{4}$ term becomes
comparable with the $k^{6}$ term, respectively.

Let us first study the $\eta_{1}>-3$ case, which means that there are no
growing terms in the solution of the curvature perturbations and all time-
dependent terms in Eq. (18) decay with the cosmic expansion. Thus, all terms
containing $N_{2}$ in Eq. (III.1) can be neglected. Then, we find that

$\displaystyle k_{1}\simeq-\frac{1}{\tau_{1}}=c_{s_{1}}k_{c}\;,$ (20)

and

$\displaystyle
k_{2}\simeq-\frac{\sqrt{3+\eta_{1}}}{c_{s_{1}}\tau_{1}}=\sqrt{3+\eta_{1}}k_{c}\;.$
(21)

The condition $k_{2}\ll k_{c}$ for a $k^{4}$ growth requires that
$\sqrt{3+\eta_{1}}\ll 1$. This is hard to satisfy since $\eta_{1}$ must be
fine-tuned to be very close to $-3$. Thus, usually the highest growth of the
power spectrum can only reach $k^{2}$. Furthermore, since the power spectrum
only has the $k^{2}$ growth, the dip phenomenon does not appear. These
characters can be found clearly in Fig. 4, where the power spectrums from
numerical and approximate analyses are plotted in the $\eta_{1}=-2$ case.

Figure 4:  The power spectrum in the $\eta_{1}=-2$ case. The gray-solid and blue-dashed lines represent the numerical and approximate results, respectively. The dashed-red line indicates the $k^{2}$ growth. Table 1:  The predominant terms of $k^{n}$ in three different situations.  |  $c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}\ll 1$ |  $c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}=1$ |  $c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}\gg 1$
---|---|---|---
$\tau_{1}^{2}k^{2}$ |  $1$ |  $1+\frac{2\eta_{1}}{(1+\eta_{1})(3+\eta_{1})}$ |  $\frac{2c_{s_{1}}^{2}\eta_{1}e^{-(3+\eta_{1})N_{2}}}{(1+\eta_{1})(3+\eta_{1})}$
$c_{s_{1}}^{2}\tau_{1}^{4}k^{4}$ |  $-\frac{2e^{-\left(3+\eta_{1}\right)N_{2}}}{(1+\eta_{1})(3+\eta_{1})}$ |  $-\frac{(\eta_{1}^{2}+8\eta_{1}+6)e^{-\left(3+\eta_{1}\right)N_{2}}}{(1+\eta_{1})^{2}(3+\eta_{1})^{2}}$ |  $\frac{c_{s_{1}}^{2}\eta_{1}^{2}e^{-2\left(3+\eta_{1}\right)N_{2}}}{(1+\eta_{1})^{2}(3+\eta_{1})^{2}}$
$c_{s_{1}}^{4}\tau_{1}^{6}k^{6}$ |  $\frac{e^{-2\left(3+\eta_{1}\right)N_{2}}}{(1+\eta_{1})^{2}(3+\eta_{1})^{2}}$ |  $\frac{e^{-2\left(3+\eta_{1}\right)N_{2}}}{(1+\eta_{1})^{2}(3+\eta_{1})^{2}}$ |  $\frac{e^{-2\left(3+\eta_{1}\right)N_{2}}}{(1+\eta_{1})^{2}(3+\eta_{1})^{2}}$

When $\eta_{1}<-3$, since the coefficient of the $k^{n}$ term is very tedious,
we only consider three special cases ($c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}\ll
1$, $c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}=1$ and
$c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}\gg 1$), as shown in Table 1 in which the
predominant terms of $k^{n}$ are given, to investigate analytically the growth
of the power spectrum. In the case of $c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}\ll
1$, we find that

$\displaystyle k_{1}$ $\displaystyle\simeq$
$\displaystyle-\frac{1}{\tau_{1}}=c_{s_{1}}k_{c}\;,$ $\displaystyle k_{2}$
$\displaystyle\simeq$
$\displaystyle-\frac{\sqrt{(1+\eta_{1})(3+\eta_{1})}}{\sqrt{2}c_{s_{1}}\tau_{1}}e^{\frac{(3+\eta_{1})N_{2}}{2}}=\frac{\sqrt{(1+\eta_{1})(3+\eta_{1})}}{\sqrt{2}}e^{\frac{(3+\eta_{1})N_{2}}{2}}k_{c}\;,$
$\displaystyle k_{3}$ $\displaystyle\simeq$
$\displaystyle-\frac{\sqrt{2(1+\eta_{1})(3+\eta_{1})}}{c_{s_{1}}\tau_{1}}e^{\frac{(3+\eta_{1})N_{2}}{2}}=\sqrt{2(1+\eta_{1})(3+\eta_{1})}e^{\frac{(3+\eta_{1})N_{2}}{2}}k_{c}\;.$
(22)

Apparently, the condition $k_{3}<k_{c}$ is easy to be satisfied since
$\eta_{1}<-3$. Thus the power spectrum can have a $k^{6}$ growth. Since the
coefficient of the $k_{4}$ term is negative, the power spectrum has a dip
preceding the $k^{6}$ growth.

When $c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}=1$, we have

$\displaystyle k_{1}$ $\displaystyle\simeq$
$\displaystyle\sqrt{\frac{\eta_{1}^{2}+4\eta_{1}+3}{(\eta_{1}^{2}+6\eta_{1}+3)}}c_{s_{1}}k_{c}\;,$
$\displaystyle k_{2}$ $\displaystyle\simeq$
$\displaystyle\sqrt{\frac{(\eta_{1}^{2}+4\eta_{1}+3)(\eta_{1}^{2}+6\eta_{1}+3)}{(\eta_{1}^{2}+8\eta_{1}+6)}}c_{s_{1}}k_{c}\;,$
$\displaystyle k_{3}$ $\displaystyle\simeq$
$\displaystyle\sqrt{\eta_{1}^{2}+8\eta_{1}+6}~{}c_{s_{1}}k_{c}\;.$ (23)

We find that $k_{1}\simeq k_{2}\simeq k_{3}<k_{c}$, which means that the power
spectrum will go directly to the $k^{6}$ growth after the scale-invariant
spectrum.

In the case of $c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}\gg 1$, one can obtain

$\displaystyle k_{1}$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{(1+\eta_{1})(3+\eta_{1})}}{\sqrt{-2\eta_{1}}}e^{\frac{1}{2}(3+\eta_{1})N_{2}}k_{c}\;,$
$\displaystyle k_{2}$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{(1+\eta_{1})(3+\eta_{1})}}{\sqrt{-\eta_{1}}}e^{\frac{1}{2}(3+\eta_{1})N_{2}}k_{c}\;,$
$\displaystyle k_{3}$ $\displaystyle\simeq$
$\displaystyle|\eta_{1}|c_{s_{1}}k_{c}\;.$ (24)

Since $k_{2}\ll k_{c}$, the power spectrum will grow with $k^{4}$ when the
scales become smaller than the CMB one. It eventually goes into $k^{6}$ growth
due to $k_{3}<k_{c}$. A negative $\eta_{1}$ leads to a negative coefficient of
the $k^{2}$ term, which indicates that the power spectrum has a dip preceding
the $k^{4}$ growth. These characters can be seen in Fig. 5, where the
numerical (gray -solid lines) and approximate (blue-dashed lines) results of
the power spectrum with $\eta_{1}=-4,-5,-6$ are plotted. One can see that the
approximate results are consistent with the numerical ones.

Figure 5:  The power spectrums in the $\eta_{1}=-4$, $-5$, and $-6$ cases. The
gray-solid and blue-dashed lines represent the numerical and approximate
results, respectively. The first, middle and last columns correspond to the
$c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}\ll 1$,
$c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}=1$, and
$c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{2}}\gg 1$ cases, respectively.

Moreover, Eq. (18) is clearly inapplicable for the cases of $\eta_{1}=-1$ and
$\eta_{1}=-3$ due to the appearance of singularity. These two cases need to be
treated separately.

  * •

$\eta_{1}=-1$

We find that in the infrared limit, the solution of the curvature
perturbations [Eq. (17)] has the form

$\displaystyle\overline{\mathcal{R}}^{(1)}_{k}(\tau)=\frac{iHe^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k^{3}}}-\frac{H\tau_{1}e^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k}}-\dfrac{ic_{s_{1}}^{2}Hk^{1/2}e^{-ik\tau_{1}}}{8\sqrt{\epsilon_{0}}}\left[3\tau_{1}^{2}-(-\tau)^{2}\right]+\dots\;.$
(25)

The solution of the curvature perturbations consists of the constant part and
the decaying one, which is similar to the solution in the case of
$\eta_{1}>-3$. The power spectrum of the curvature perturbations has the form

$\displaystyle\dfrac{\mathcal{P}_{\overline{\mathcal{R}}^{(1)}_{k}}}{\mathcal{P}_{0}}\simeq
1+\tau_{1}^{2}k^{2}+\frac{9}{16}c_{s_{1}}^{4}\tau_{1}^{4}k^{4}\;.$ (26)

We can see that the $k^{2}$ and $k^{4}$ terms become comparable at

$\displaystyle
k_{2}\approx-\frac{4}{3c_{s_{1}}^{2}\tau_{1}}=\frac{4}{3c_{s_{1}}}k_{c}\;.$
(27)

Since $k_{2}>k_{c}$, the power spectrum has no $k^{4}$ growth. The
corresponding numerical and approximate results of the power spectrum are
shown in Fig. 6. Apparently at the small scales the power spectrum grows with
a $k^{2}$ slope.

Figure 6:  The power spectrum in the $\eta_{1}=-1$ case .The gray-solid and
blue-dashed lines represent the numerical and approximate results,
respectively. The dashed-red line indicates the $k^{2}$ growth.

  * •

$\eta_{1}=-3$

In the infrared limit, Eq. (17) can be simplified to be

$\displaystyle\overline{\mathcal{R}}^{(1)}_{k}(\tau)$ $\displaystyle=$
$\displaystyle\frac{iHe^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k^{3}}}-\frac{H\tau_{1}e^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k}}+\dfrac{ic_{s_{1}}^{2}H\tau_{1}^{2}k^{1/2}e^{-ik\tau_{1}}}{8\sqrt{\epsilon_{0}}}\left[1+6\log\left(\dfrac{\tau}{\tau_{1}}\right)\right]$
(28) $\displaystyle-$
$\displaystyle\dfrac{ic_{s_{1}}^{2}H\tau_{1}^{3}k^{3/2}e^{-ik\tau_{1}}}{8\sqrt{\epsilon_{0}}}\left[1+2\log\left(\dfrac{\tau}{\tau_{1}}\right)\right]+\dots\;.$

The solution for the curvature perturbation consists of the constant term and
the logarithmically growing one. This characteristic is different from the
power-law growth in the $\eta_{1}<-3$ case. Thus, we obtain that the power
spectrum has the expression

$\displaystyle\dfrac{\mathcal{P}_{\overline{\mathcal{R}}^{(1)}_{k}}}{\mathcal{P}_{0}}$
$\displaystyle\simeq$ $\displaystyle
1+\frac{1}{2}\left(2-c_{s_{1}}^{2}\left(6N_{2}-1\right)\right)\tau_{1}^{2}k^{2}+\frac{1}{16}c_{s_{1}}^{2}\left(8+c_{s_{1}}^{2}\left(6N_{2}-1\right)^{2}-16N_{2}\right)\tau_{1}^{4}k^{4}$
(29) $\displaystyle+$
$\displaystyle\frac{1}{16}c_{s_{1}}^{4}\left(2N_{2}-1\right)^{2}\tau_{1}^{6}k^{6}\;.$

Since the maximum value of $N_{2}$ is about $30-40$ and
$c_{s_{1}}\sim\mathcal{O}(10^{-4})$, $N_{2}c_{s_{1}}^{2}$ is significantly
less than 1. Therefore, the expression of the power spectrum given in Eq. (29)
can be simplified to be

$\displaystyle\dfrac{\mathcal{P}_{\overline{\mathcal{R}}^{(1)}_{k}}}{\mathcal{P}_{0}}\simeq
1+\tau_{1}^{2}k^{2}+\frac{1}{2}(1-2N_{2})c_{s_{1}}^{2}\tau_{1}^{4}k^{4}+\frac{1}{16}(2N_{2}-1)^{2}c_{s_{1}}^{4}\tau_{1}^{6}k^{6}\;.$
(30)

From the above expression, we obtain

$\displaystyle k_{1}$ $\displaystyle\simeq$ $\displaystyle c_{s_{1}}k_{c}\;,$
$\displaystyle k_{2}$ $\displaystyle\simeq$
$\displaystyle\frac{1}{\sqrt{N_{2}-1/2}}k_{c}\;,$ $\displaystyle k_{3}$
$\displaystyle\simeq$ $\displaystyle\frac{2}{\sqrt{N_{2}-1/2}}k_{c}\;.$ (31)

Obviously, $k_{1}\ll k_{c}$ and $k_{2}\approx k_{3}$, which are less than
$k_{c}$ if $N_{2}>9/2$. Thus, the power spectrum will have an era with a
$k^{2}$ growth and eventually with a $k^{6}$ one at scales smaller than the
CMB one when $N_{2}>9/2$. Since the coefficient of the $k^{4}$ term is
negative, there is a dip preceding the $k^{6}$ growth. These characters can be
seen clearly in Fig. 7.

Figure 7:  The power spectrum in the $\eta_{1}=-3$ case. The gray-solid and
blue-dashed lines represent the numerical and approximate results,
respectively.

###  III.2 Changes of sound speed previous to slow-roll parameter $\eta$

Figure 8:  The sound speed $c_{s}$ (blue line) varies at $\tau_{1}$, and the
slow-roll parameter $\eta$ (red line) changes at $\tau_{2}$
$(|\tau_{2}|<|\tau_{1}|$).

The simultaneous change of $c_{s}$ and $\eta$ is a harsh requirement. In the
following, we abandon it, and first consider that the change of the sound
speed is followed by that of $\eta$. We assume that the sound speed changes
suddenly from $1$ to a small constant $c_{s_{1}}$ at time $\tau_{1}$, and
$\eta$ varies from $\sim 0$ to a negative constant $\eta_{1}$ at $\tau_{2}$
($|\tau_{2}|<|\tau_{1}|$), as is shown in Fig. 8. The $|\tau|>|\tau_{1}|$ era,
which represents the first phase, is the standard slow-roll inflation. In the
second phase ($|\tau_{2}|<|\tau|<|\tau_{1}|$), the sound speed of the
curvature perturbations is a small value $c_{s_{1}}$. When
$|\tau|<|\tau_{2}|$, which corresponds to the third phase, the slow-roll
parameter $\epsilon$ decreases with the power-law:
$\epsilon(\tau)=\epsilon_{0}(\tau/\tau_{2})^{-\eta_{1}}$ and the sound speed
keeps the small value $c_{s_{1}}$.

When $|\tau|>|\tau_{1}|$, the solution of $\mathcal{R}_{k}^{(0)}$ is given in
Eq. (7). In the second phase, the solution of the curvature perturbations has
the form

$\displaystyle\widetilde{\mathcal{R}}^{(1)}_{k}(\tau)=\frac{ic_{s_{1}}H\left(-\tau\right)^{3/2}}{\sqrt{2\epsilon_{0}}}\left[\alpha_{3}H_{3/2}^{(1)}(c_{s_{1}}k\tau)+\beta_{3}H_{3/2}^{(2)}(c_{s_{1}}k\tau)\right]\;,$
(32)

where $\alpha_{3}$ and $\beta_{3}$ are two constants. Using the matching
condition:
$\mathcal{R}^{(0)}_{k}(\tau_{1})=\widetilde{\mathcal{R}}^{(1)}_{k}(\tau_{1})$
and
$A_{0}\mathcal{R}^{\prime(0)}_{k}(\tau_{1})=A_{1}\widetilde{\mathcal{R}}^{\prime(1)}_{k}(\tau_{1})$,
one can obtain that

$\displaystyle\alpha_{3}$ $\displaystyle=$
$\displaystyle-\frac{\left(1-c_{s_{1}}\right)\sqrt{\pi}e^{-i\left(1+c_{s_{1}}\right)k\tau_{1}}}{4\sqrt{c_{s_{1}}}},$
$\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle-\frac{\left(1+c_{s_{1}}\right)\sqrt{\pi}e^{-i\left(1-c_{s_{1}}\right)k\tau_{1}}}{4\sqrt{c_{s_{1}}}}\;.$
(33)

If $|\tau|<|\tau_{2}|$, the solution of the curvature perturbations becomes

$\displaystyle\widetilde{\mathcal{R}}^{(2)}_{k}(\tau)=-\frac{c_{s_{1}}H\tau_{2}^{-\frac{\eta_{1}}{2}}\tau^{\frac{3+\eta_{1}}{2}}}{\sqrt{2\epsilon_{0}}}\left[\alpha_{4}H_{\nu}^{(1)}(c_{s_{1}}k\tau)+\beta_{4}H_{\nu}^{(2)}(c_{s_{1}}k\tau)\right],$
(34)

where $\alpha_{4}$ and $\beta_{4}$ are two constants and $\nu=(3+\eta_{1})/2$.
Using the matching condition:
$\widetilde{\mathcal{R}}^{(1)}_{k}(\tau_{2})=\widetilde{\mathcal{R}}^{(2)}_{k}(\tau_{2})$
and
$\widetilde{\mathcal{R}}^{\prime(1)}_{k}(\tau_{2})=\widetilde{\mathcal{R}}^{\prime(2)}_{k}(\tau_{2})$,
one finds

$\displaystyle\alpha_{4}$ $\displaystyle=$
$\displaystyle\frac{i\pi^{3/2}c_{s_{1}}^{1/2}\tau_{2}k}{16}e^{-i(1+c_{s_{1}})k\tau_{1}}\left[~{}\xi
H_{\frac{3+\eta_{1}}{2}}^{(2)}(c_{s_{1}}k\tau_{2})-\lambda
H_{\frac{1+\eta_{1}}{2}}^{(2)}(c_{s_{1}}k\tau_{2})\right],$ (35)
$\displaystyle\beta_{4}$ $\displaystyle=$
$\displaystyle-\frac{i\pi^{3/2}c_{s_{1}}^{1/2}\tau_{2}k}{16}e^{-i(1+c_{s_{1}})k\tau_{1}}\left[~{}\xi
H_{\frac{3+\eta_{1}}{2}}^{(1)}(c_{s_{1}}k\tau_{2})-\lambda
H_{\frac{1+\eta_{1}}{2}}^{(1)}(c_{s_{1}}k\tau_{2})\right]\;,$ (36)

where

$\displaystyle\xi$
$\displaystyle=(1-c_{s_{1}})H_{\frac{1}{2}}^{(1)}(c_{s_{1}}k\tau_{2})+(1+c_{s_{1}})e^{2ic_{s_{1}}k\tau_{1}}H_{\frac{1}{2}}^{(2)}(c_{s_{1}}k\tau_{2})\;,$
$\displaystyle\lambda$
$\displaystyle=(1-c_{s_{1}})H_{\frac{3}{2}}^{(1)}(c_{s_{1}}k\tau_{2})+(1+c_{s_{1}})e^{2ic_{s_{1}}k\tau_{1}}H_{\frac{3}{2}}^{(2)}(c_{s_{1}}k\tau_{2})\;.$
(37)

In the infrared region ($-c_{s_{1}}k\tau\rightarrow 0$,
$-c_{s_{1}}k\tau_{1}\rightarrow 0$ and $-c_{s_{1}}k\tau_{2}\rightarrow 0$ ),
we obtain that

$\displaystyle\widetilde{\mathcal{R}}^{(2)}_{k}(\tau)$ $\displaystyle=$
$\displaystyle\frac{iHe^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k^{3}}}-\frac{H\tau_{1}e^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k}}$
(38) $\displaystyle-$
$\displaystyle\left(\frac{ic_{s_{1}}^{2}H\left((3+\eta_{1})\tau_{1}^{2}-\eta_{1}\tau_{2}^{2}\right)e^{-ik\tau_{1}}}{4\sqrt{\epsilon_{0}}(3+\eta_{1})}+\frac{ic_{s_{1}}^{2}H\eta_{1}\tau_{2}^{-1-\eta_{1}}e^{i\pi\eta_{1}-ik\tau_{1}}}{2\sqrt{\epsilon_{0}}(1+\eta_{1})(3+\eta_{1})}(-\tau)^{3+\eta_{1}}\right)k^{1/2}$
$\displaystyle+$
$\displaystyle\left(\begin{split}&\frac{c_{s_{1}}^{2}H\left((3+\eta_{1})\tau_{1}^{3}-3\eta_{1}\tau_{1}\tau_{2}^{2}+2\eta_{1}\tau_{2}^{3}\right)e^{-ik\tau_{1}}}{12\sqrt{\epsilon_{0}}(3+\eta_{1})}\\\
+&\frac{c_{s_{1}}^{2}H\left(\eta_{1}\tau_{1}-\tau_{2}-\eta_{1}\tau_{2}\right)\tau_{2}^{-1-\eta_{1}}e^{i\pi\eta_{1}-ik\tau_{1}}}{2\sqrt{\epsilon_{0}}(1+\eta_{1})(3+\eta_{1})}(-\tau)^{3+\eta_{1}}\end{split}\right)k^{3/2}$
$\displaystyle+$ $\displaystyle\dots\;.$

Clearly, the solution given in Eq. (38) contains a time-independent part and a
time-dependent one, which will grow with time when $\eta_{1}<-3$.

Using $N_{2}$ and $N_{3}$ to denote the number of $e$-folds during the second
and third phases, respectively, i.e. $\tau_{2}=\tau_{1}e^{-N_{2}}$ and
$\tau=\tau_{1}e^{-N_{2}-N_{3}}$, we obtain, from Eq. (38), the expression of
the power spectrum of the curvature perturbations

$\displaystyle\dfrac{\mathcal{P}_{\widetilde{\mathcal{R}}^{(2)}_{k}}}{\mathcal{P}_{0}}$
$\displaystyle\simeq$ $\displaystyle
1+\left(1-c_{s_{1}}^{2}+\frac{c_{s_{1}}^{2}\eta_{1}}{3+\eta_{1}}e^{-2N_{2}}+\frac{2c_{s_{1}}^{2}\eta_{1}}{(1+\eta_{1})(3+\eta_{1})}e^{-2N_{2}-\left(3+\eta_{1}\right)N_{3}}\right)\tau_{1}^{2}k^{2}$
(39) $\displaystyle-$
$\displaystyle\left(\begin{split}&\frac{1}{3}+\frac{2\eta_{1}e^{-3N_{2}}}{9+3\eta_{1}}-\frac{3\eta_{1}e^{-2N_{2}}}{9+3\eta_{1}}+\frac{2+2\eta_{1}-2\eta_{1}e^{N_{2}}}{(1+\eta_{1})(3+\eta_{1})}e^{-3N_{2}-\left(3+\eta_{1}\right)N_{3}}\\\
-&c_{s_{1}}^{2}\left(\frac{1}{2}-\frac{\eta_{1}}{6+2\eta_{1}}e^{-2N_{2}}-\frac{\eta_{1}}{(1+\eta_{1})(3+\eta_{1})}e^{-2N_{2}-\left(3+\eta_{1}\right)N_{3}}\right)^{2}\end{split}\right)c_{s_{1}}^{2}\tau_{1}^{4}k^{4}$
$\displaystyle+$
$\displaystyle\left(\frac{1}{6}+\frac{\eta_{1}e^{-3N_{2}}}{9+3\eta_{1}}-\frac{\eta_{1}e^{-2N_{2}}}{6+2\eta_{1}}+\frac{1+\eta_{1}-\eta_{1}e^{N_{2}}}{(1+\eta_{1})(3+\eta_{1})}e^{-3N_{2}-\left(3+\eta_{1}\right)N_{3}}\right)^{2}c_{s_{1}}^{4}\tau_{1}^{6}k^{6}\;.$

We first study the $\eta_{1}>-3$ case, which means that there are no growing
terms in the solution of the curvature perturbations and all time-dependent
terms in Eq. (38) decay with the cosmic expansion. Thus, all terms containing
$N_{2}$ and $N_{3}$ in Eq. (39) can be neglected and the power spectrum can be
simplified as

$\displaystyle\dfrac{\mathcal{P}_{\widetilde{\mathcal{R}}^{(2)}_{k}}}{\mathcal{P}_{0}}$
$\displaystyle\simeq$ $\displaystyle
1+\tau_{1}^{2}k^{2}-\frac{1}{3}c_{s_{1}}^{2}\tau_{1}^{4}k^{4}+\frac{1}{36}c_{s_{1}}^{4}\tau_{1}^{6}k^{6}\;.$
(40)

It can be found that the wave numbers at which the scale-invariant term is
comparable to the $k^{2}$ term and the $k^{2}$ term is comparable to $k^{4}$
one happen, respectively, at

$\displaystyle k_{1}$ $\displaystyle\simeq$ $\displaystyle c_{s_{1}}k_{c}\;,$
$\displaystyle k_{2}$ $\displaystyle\simeq$ $\displaystyle\sqrt{3}k_{c}\;.$
(41)

The power spectrum has a growth rate of $k^{2}$ since $k_{2}>k_{c}$. These
results can be found clearly in Fig. 9, where the evolutions of the power
spectrum from numerical and approximate analyses are plotted in the
$\eta_{1}=-2$ case. At the CMB scale, the power spectrum is scale invariant
which is consistent with the CMB observations. At scales smaller than the CMB
scale, the power spectrum becomes scale-dependent with a $k^{2}$ growth.

Figure 9:  The power spectrum in the $\eta_{1}=-2$ case. The gray-solid and
blue-dashed lines represent the numerical and approximate results,
respectively. The red-dashed line indicates the $k^{2}$ growth.

When $\eta_{1}<-3$, since the coefficient of $k^{n}$ is very tedious, we only
consider three special cases ($N_{2}\gg N_{3}$, $N_{2}\ll N_{3}$, and
$N_{2}=N_{3}$) to investigate analytically the growth of the power spectrum.

First of all, when $N_{2}\gg N_{3}$, all terms containing $N_{2}$ in Eq. (39)
can be neglected. Thus, the dominant term of $k^{n}$ is the same as the one in
the case of $\eta_{1}>-3$. Therefore, the evolution of the power spectrum is
the same as in the case of $\eta_{1}>-3$, and only has the $k^{2}$ growth.

When $N_{2}=N_{3}$, the expression for the power spectrum given in Eq. (39)
can be simplified to be

$\displaystyle\frac{\mathcal{P}_{\widetilde{\mathcal{R}}^{(2)}_{k}}}{\mathcal{P}_{0}}$
$\displaystyle\simeq$ $\displaystyle
1+\tau_{1}^{2}k^{2}-\left(\frac{1}{3}-\frac{2\eta_{1}}{(1+\eta_{1})(3+\eta_{1})}e^{-\left(5+\eta_{1}\right)N_{3}}\right)c_{s_{1}}^{2}\tau_{1}^{4}k^{4}$
(42) $\displaystyle+$
$\displaystyle\left(\frac{1}{36}-\frac{\eta_{1}}{3(1+\eta_{1})(3+\eta_{1})}e^{-\left(5+\eta_{1}\right)N_{3}}+\frac{\eta_{1}^{2}}{(1+\eta_{1})^{2}(3+\eta_{1})^{2}}e^{-2\left(5+\eta_{1}\right)N_{3}}\right)c_{s_{1}}^{4}\tau_{1}^{6}k^{6}\;.$

Since different values of $\eta_{1}$ will give different results, we will
discuss this situation by considering following different cases:

$\displaystyle\frac{\mathcal{P}_{\widetilde{\mathcal{R}}^{(2)}_{k}}}{\mathcal{P}_{0}}\simeq\left\\{\begin{array}[]{ll}1+\tau_{1}^{2}k^{2}-\dfrac{1}{3}c_{s_{1}}^{2}\tau_{1}^{4}k^{4}+\dfrac{1}{36}c_{s_{1}}^{4}\tau_{1}^{6}k^{6}&-5<\eta_{1}<-3\vspace{0.4cm}\\\
1+\tau_{1}^{2}k^{2}-\dfrac{19}{12}c_{s_{1}}^{2}\tau_{1}^{4}k^{4}+\dfrac{361}{576}c_{s_{1}}^{4}\tau_{1}^{6}k^{6}&~{}~{}~{}~{}\eta_{1}=-5\vspace{0.4cm}\\\
1+\tau_{1}^{2}k^{2}+\dfrac{2\eta_{1}e^{-\left(5+\eta_{1}\right)N_{3}}}{(1+\eta_{1})(3+\eta_{1})}c_{s_{1}}^{2}\tau_{1}^{4}k^{4}+\dfrac{\eta_{1}^{2}e^{-2\left(5+\eta_{1}\right)N_{3}}}{(1+\eta_{1})^{2}(3+\eta_{1})^{2}}c_{s_{1}}^{4}\tau_{1}^{6}k^{6}&~{}~{}~{}~{}\eta_{1}<-5\
.\\\ \end{array}\right.$

When $-5\leq\eta_{1}<-3$, we find that $k_{2}\simeq k_{c}$, which means that
the power spectrum only has the $k^{2}$ growth. In the $\eta_{1}<-5$ case, we
obtain

$\displaystyle k_{1}$ $\displaystyle\simeq$ $\displaystyle c_{s_{1}}k_{c}\;,$
$\displaystyle k_{2}$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{(1+\eta_{1})(3+\eta_{1})}}{\sqrt{-2\eta_{1}}}e^{\frac{1}{2}(5+\eta_{1})N_{3}}k_{c}\;,$
$\displaystyle k_{3}$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{2(1+\eta_{1})(3+\eta_{1})}}{\sqrt{-\eta_{1}}}e^{\frac{1}{2}(5+\eta_{1})N_{3}}k_{c}\;.$
(46)

Apparently, $k_{3}$ is less than $k_{c}$ if
$\frac{\sqrt{2(1+\eta_{1})(3+\eta_{1})}}{\sqrt{-\eta_{1}}}e^{\frac{1}{2}(5+\eta_{1})N_{3}}\ll
1$, which can be realized easily since $\eta_{1}<-5$. Thus, the power spectrum
can have the $k^{6}$ growth. Since there is a negative coefficient in the
$k^{4}$ term, the power spectrum has a dip preceding the $k^{6}$ growth. In
Fig. 10, the power spectrums in the cases of $\eta_{1}=-4,-5$ and $-6$ for the
$N_{2}\gg N_{3}$ and $N_{2}=N_{3}$ cases are plotted. We can see that when
$N_{2}\gg N_{3}$, the highest growth slope of the power spectrum is $k^{2}$.
In the case of $N_{2}=N_{3}$, the highest growth rate for $\eta_{1}=-4$ and
$-5$ is still $k^{2}$, while for $\eta_{1}=-6$, the highest growth rate can
reach up to $k^{6}$.

Figure 10:  The power spectrums in the $\eta_{1}=-4$, $-5$ and $-6$ cases. The
gray-solid and blue-dashed lines represent the numerical and approximate
results, respectively.

When $N_{2}\ll N_{3}$, the power spectrum can be approximated as

$\displaystyle\frac{\mathcal{P}_{\widetilde{\mathcal{R}}^{(2)}_{k}}}{\mathcal{P}_{0}}$
$\displaystyle\simeq$ $\displaystyle
1+\left(1+\frac{2c_{s_{1}}^{2}\eta_{1}}{(1+\eta_{1})(3+\eta_{1})}e^{-\left(3+\eta_{1}\right)N_{3}}\right)\tau_{1}^{2}k^{2}$
(47) $\displaystyle+$
$\displaystyle\left(\frac{2\eta_{1}}{(1+\eta_{1})(3+\eta_{1})}e^{-\left(3+\eta_{1}\right)N_{3}}+\frac{c_{s_{1}}^{2}\eta_{1}^{2}}{(1+\eta_{1})^{2}(3+\eta_{1})^{2}}e^{-2\left(3+\eta_{1}\right)N_{3}}\right)c_{s_{1}}^{2}\tau_{1}^{4}k^{4}$
$\displaystyle+$
$\displaystyle\frac{\eta_{1}^{2}}{(1+\eta_{1})^{2}(3+\eta_{1})^{2}}e^{-2\left(3+\eta_{1}\right)N_{3}}c_{s_{1}}^{4}\tau_{1}^{6}k^{6}\;.$

If $c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{3}}\ll 1$, the wave numbers when the
$k^{n}$ and $k^{n-2}$ terms become comparable happen at

$\displaystyle k_{1}$ $\displaystyle\simeq$ $\displaystyle c_{s_{1}}k_{c}\;,$
$\displaystyle k_{2}$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{(1+\eta_{1})(3+\eta_{1})}}{\sqrt{-2\eta_{1}}}e^{\frac{1}{2}(3+\eta_{1})N_{3}}k_{c}\;,$
$\displaystyle k_{3}$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{2(1+\eta_{1})(3+\eta_{1})}}{\sqrt{-\eta_{1}}}e^{\frac{1}{2}(3+\eta_{1})N_{3}}k_{c}\;.$
(48)

Apparently, $k_{3}<k_{c}$ can be satisfied easily since $\eta_{1}<-3$ and thus
the power spectrum can have a $k^{6}$ growth. The dip will appear preceding
the $k^{6}$ growth since the $k^{4}$ term has a negative coefficient. If
$c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{3}}\gg 1$, we obtain

$\displaystyle k_{1}$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{(1+\eta_{1})(3+\eta_{1})}}{\sqrt{-2\eta_{1}}}e^{\frac{1}{2}(3+\eta_{1})N_{3}}k_{c}\;,$
$\displaystyle k_{2}$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{2(1+\eta_{1})(3+\eta_{1})}}{\sqrt{-\eta_{1}}}e^{\frac{1}{2}(3+\eta_{1})N_{3}}k_{c}\;,$
$\displaystyle k_{3}$ $\displaystyle\simeq$ $\displaystyle c_{s_{1}}k_{c}\;.$
(49)

It can be found that $k_{3}<k_{c}$ and $k_{1}\approx k_{2}$, which means that
the power spectrum will enter $k^{4}$ growth directly after the scale-
invariant spectrum, and will enter finally the $k^{6}$ growth. The power
spectrum has the dip preceding the $k^{4}$ growth due to the negative
coefficient in the $k^{2}$ term. These characters can be found in Fig. 11,
where the power spectrums in the cases of $\eta_{1}=-4,-5$ and $-6$ for the
$N_{2}\ll N_{3}$ are plotted.

Figure 11:  The power spectrums for $N_{2}\ll N_{3}$ in the $\eta_{1}=-4,-5$
and $-6$ cases. The gray-solid and blue-dashed lines represent the numerical
and approximate results, respectively. The first line is corresponding to the
$c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{3}}\ll 1$ case and the second line to the
$c_{s_{1}}^{2}e^{-(3+\eta_{1})N_{3}}\gg 1$ case.

Furthermore, when $\eta_{1}=-1$ and $-3$ there are singularities in Eq. (38)
and thus these cases need to be studied separately. To avoid the problem, we
first set the value of $\eta_{1}$, and then expand Eq. (34) in the infrared
limit to obtain

$\displaystyle\widetilde{\mathcal{R}}^{(2)}_{k}(\tau)=\left\\{\begin{array}[]{ll}\dfrac{iHe^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k^{3}}}-\dfrac{H\tau_{1}e^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k}}-\dfrac{iHc_{s_{1}}^{2}e^{-ik\tau_{1}}k^{1/2}}{8\sqrt{\epsilon_{0}}}\left[2\tau_{1}^{2}+\tau_{2}^{2}-(-\tau)^{2}\right]+\dots&\eta_{1}=-1\vspace{0.4cm}\\\
\dfrac{iHe^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k^{3}}}-\dfrac{H\tau_{1}e^{-ik\tau_{1}}}{2\sqrt{\epsilon_{0}k}}-\dfrac{iHc_{s_{1}}^{2}e^{-ik\tau_{1}}k^{1/2}}{8\sqrt{\epsilon_{0}}}\left[2\tau_{1}^{2}-3\tau_{2}^{2}-6\tau_{2}^{2}\log\left(\dfrac{\tau}{\tau_{2}}\right)\right]+\dots&\eta_{1}=-3\
.\\\ \end{array}\right.$

In the case of $\eta_{1}=-1$, the solution consists of constant and decaying
terms, and thus it is dominated by the constant terms. When $\eta_{1}=-3$,
except for the constant terms, the solution of the curvature perturbations
contains a logarithmic-growing term. This character is different from that in
the $\eta_{1}<-3$ case, where the growth of the solution is power law. Since
the coefficient of the logarithmic-growing term depends on $c_{s_{1}}^{2}$,
which is much less than one in our analysis, the contribution of the growth
term in the solution is negligible. Thus, from Eq. (52), we obtain that the
power spectrum has the same expression

$\displaystyle\dfrac{\mathcal{P}_{\widetilde{\mathcal{R}}^{(2)}_{k}}}{\mathcal{P}_{0}}\simeq
1+\tau_{1}^{2}k^{2}+\dfrac{1}{4}c_{s_{1}}^{4}\tau_{1}^{4}k^{4}$ (52)

for $\eta_{1}=-1$ and $-3$. The steepest growth is $k^{2}$ apparently. The
corresponding numerical and approximate results of the power spectrum are
shown in Fig. 12.

Figure 12:  The power spectrums in the $\eta_{1}=-1$ and $\eta_{1}=-3$ cases.
The gray-solid and blue-dashed lines represent the numerical and approximate
results, respectively.

###  III.3 Changes of slow-roll parameter $\eta$ previous to the sound speed

This scenario is shown in Fig. 13. The slow-roll parameter $\eta$ changes from
$\sim 0$ to a negative constant $\eta_{1}$ at $\tau_{1}$, and the sound speed
decreases from $1$ to a small constant $c_{s_{1}}$ at $\tau_{2}$
($|\tau_{2}|<|\tau_{1}|$).

Figure 13:  The sound speed $c_{s}$ (blue line) varies at $\tau_{2}$, and the
slow-roll parameter $\eta$ (red line) changes at $\tau_{1}$
$(|\tau_{2}|<|\tau_{1}|$).

Since the method for analytical solution in this case is similar to that of
the preceding subsection, we do not give the details here. The general
expression of the curvature perturbations in the infrared region
($-c_{s_{1}}k\tau\rightarrow 0$, $-c_{s_{1}}k\tau_{2}\rightarrow 0$ and
$-k\tau_{1}\rightarrow 0$) is so complex, and we do not show it here. We only
consider the integer $\eta_{1}$ cases. Different from the results obtained in
the preceding subsection, we find that the analytical results do not coincide
with the numerical ones when $\eta_{1}=-1$ and $-2$. Therefore, we only give
the infrared expressions ($-k\tau_{1}\rightarrow
0,-c_{s_{1}}k\tau_{2}\rightarrow 0$ and $-c_{s_{1}}k\tau\rightarrow 0$) of the
curvature perturbations in the $\eta_{1}\leq-3$ case:

$\displaystyle\hat{\mathcal{R}}^{(2)}_{k}(\tau)=\left\\{\begin{array}[]{ll}\dfrac{iH}{2\sqrt{\epsilon_{0}k^{3}}}\\\
+\left(\dfrac{iH\left(3\tau_{1}^{2}-\left(1-c_{s_{1}}^{2}\right)\tau_{2}^{2}\right)}{8\sqrt{\epsilon_{0}}}+\dfrac{3iH\tau_{1}^{2}}{4\sqrt{\epsilon_{0}}}\left(c_{s_{1}}^{2}\log\left[\dfrac{\tau}{\tau_{2}}\right]-\log\left[\dfrac{\tau_{1}}{\tau_{2}}\right]\right)\right)k^{1/2}\\\
+\left(\dfrac{H\tau_{1}^{3}}{6\sqrt{\epsilon_{0}}}-\dfrac{H\tau_{1}^{3}}{2\sqrt{\epsilon_{0}}}\log\left[\dfrac{\tau_{1}}{\tau_{2}}\right]+\dfrac{c_{s_{1}}^{2}H\tau_{1}^{3}}{2\sqrt{\epsilon_{0}}}\log\left[\dfrac{\tau}{\tau_{2}}\right]\right)k^{3/2}+\dots&\eta_{1}=-3\vspace{0.4cm}\\\
\dfrac{iH}{2\sqrt{\epsilon_{0}k^{3}}}\\\
-\left(\dfrac{iH\left(\left(1-c_{s_{1}}^{2}\right)\left(8\tau_{1}^{3}+\tau_{2}^{3}\right)-12\tau_{1}^{2}\tau_{2}\right)}{12\sqrt{\epsilon_{0}}\tau_{2}}-\dfrac{2ic_{s_{1}}^{2}H\tau_{1}^{3}}{3\sqrt{\epsilon_{0}}}(-\tau)^{-1}\right)k^{1/2}.\\\
-\left(\dfrac{H\tau_{1}^{3}\left(3\left(1-c_{s_{1}}^{2}\right)\tau_{1}-4\tau_{2}\right)}{6\sqrt{\epsilon_{0}}\tau_{2}}-\dfrac{c_{s_{1}}^{2}H\tau_{1}^{4}}{2\sqrt{\epsilon_{0}}}\left(-\tau\right)^{-1}\right)k^{3/2}+\dots&\eta_{1}=-4\vspace{0.4cm}\\\
\dfrac{iH}{2\sqrt{\epsilon_{0}k^{3}}}\\\
-\left(\dfrac{iH\left(\left(1-c_{s_{1}}^{2}\right)\left(5\tau_{1}^{4}+\tau_{2}^{4}\right)-10\tau_{1}^{2}\tau_{2}^{2}\right)}{16\sqrt{\epsilon_{0}}\tau_{2}^{2}}+\dfrac{5ic_{s_{1}}^{2}H\tau_{1}^{4}}{16\sqrt{\epsilon_{0}}}(-\tau)^{-2}\right)k^{1/2}\\\
-\left(\dfrac{H\tau_{1}^{3}\left(3\left(1-c_{s_{1}}^{2}\right)\tau_{1}^{2}-5\tau_{2}^{2}\right)}{12\sqrt{\epsilon_{0}}\tau_{2}^{2}}+\dfrac{c_{s_{1}}^{2}H\tau_{1}^{5}}{4\sqrt{\epsilon_{0}}}\left(-\tau\right)^{-2}\right)k^{3/2}+\dots&\eta_{1}=-5\vspace{0.4cm}\\\
\dfrac{iH}{2\sqrt{\epsilon_{0}k^{3}}}\\\
-\left(\dfrac{iH\left(\left(1-c_{s_{1}}^{2}\right)\left(4\tau_{1}^{5}+\tau_{2}^{5}\right)-10\tau_{1}^{2}\tau_{2}^{3}\right)}{20\sqrt{\epsilon_{0}}\tau_{2}^{3}}-\dfrac{5ic_{s_{1}}^{2}H\tau_{1}^{5}}{5\sqrt{\epsilon_{0}}}(-\tau)^{-3}\right)k^{1/2}\\\
-\left(\dfrac{H\tau_{1}^{3}\left(\left(1-c_{s_{1}}^{2}\right)\tau_{1}^{3}-2\tau_{2}^{3}\right)}{6\sqrt{\epsilon_{0}}\tau_{2}^{3}}-\dfrac{c_{s_{1}}^{2}H\tau_{1}^{6}}{6\sqrt{\epsilon_{0}}}\left(-\tau\right)^{-3}\right)k^{3/2}+\dots&\eta_{1}=-6\
\;.\end{array}\right.$ (64)

From the above expression, one can obtain the power spectrum:

$\displaystyle\frac{\mathcal{P}_{\hat{\mathcal{R}}^{(2)}_{k}}}{\mathcal{P}_{0}}\simeq\left\\{\begin{array}[]{ll}1+\frac{1}{2}\left(3-e^{-2N_{2}}-6N_{2}-6c_{s_{1}}^{2}N_{3}\right)\tau_{1}^{2}k^{2}\\\
~{}+\frac{1}{16}\left(3-e^{-2N_{2}}-6N_{2}-6c_{s_{1}}^{2}N_{3}\right)^{2}\tau_{1}^{4}k^{4}\\\
~{}+\frac{1}{9}\left(1-3N_{2}-3c_{s_{1}}^{2}N_{3}\right)^{2}\tau_{1}^{6}k^{6}&\eta_{1}=-3\vspace{0.4cm}\\\
1+\frac{1}{3}\left(12-e^{-2N_{2}}-8e^{N_{2}}+8c_{s_{1}}^{2}e^{N_{2}}-8c_{s_{1}}^{2}e^{N_{2}+N_{3}}\right)\tau_{1}^{2}k^{2}\\\
~{}+\frac{1}{36}\left(12-e^{-2N_{2}}-8e^{N_{2}}+8c_{s_{1}}^{2}e^{N_{2}}-8c_{s_{1}}^{2}e^{N_{2}+N_{3}}\right)^{2}\tau_{1}^{4}k^{4}\\\
~{}+\frac{1}{9}\left(4-3e^{N_{2}}+3c_{s_{1}}^{2}e^{N_{2}}-3c_{s_{1}}^{2}e^{N_{2}+N_{3}}\right)^{2}\tau_{1}^{6}k^{6}&\eta_{1}=-4\vspace{0.4cm}\\\
1+\frac{1}{4}\left(10-e^{-2N_{2}}-5e^{2N_{2}}+5c_{s_{1}}^{2}e^{2N_{2}}-5c_{s_{1}}^{2}e^{2N_{2}+2N_{3}}\right)\tau_{1}^{2}k^{2}\\\
~{}+\frac{1}{64}\left(10-e^{-2N_{2}}-5e^{2N_{2}}+5c_{s_{1}}^{2}e^{2N_{2}}-5c_{s_{1}}^{2}e^{2N_{2}+2N_{3}}\right)^{2}\tau_{1}^{4}k^{4}\\\
~{}+\frac{1}{36}\left(5-3e^{2N_{2}}+3c_{s_{1}}^{2}e^{2N_{2}}-3c_{s_{1}}^{2}e^{2N_{2}+2N_{3}}\right)^{2}\tau_{1}^{6}k^{6}&\eta_{1}=-5\vspace{0.4cm}\\\
1+\dfrac{1}{5}\left(10-e^{-2N_{2}}-4e^{3N_{2}}+4c_{s_{1}}^{2}e^{3N_{2}}-4c_{s_{1}}^{2}e^{3N_{2}+3N_{3}}\right)\tau_{1}^{2}k^{2}\\\
~{}+\frac{1}{100}\left(10-e^{-2N_{2}}-4e^{3N_{2}}+4c_{s_{1}}^{2}e^{3N_{2}}-4c_{s_{1}}^{2}e^{3N_{2}+3N_{3}}\right)^{2}\tau_{1}^{4}k^{4}\\\
~{}+\frac{1}{9}\left(2-e^{3N_{2}}+c_{s_{1}}^{2}e^{3N_{2}}-c_{s_{1}}^{2}e^{3N_{2}+3N_{3}}\right)^{2}\tau_{1}^{6}k^{6}&\eta_{1}=-6\
\;.\end{array}\right.$ (77)

Here $N_{2}$ and $N_{3}$ are the number of $e$-folds during the second and
third phase, respectively. The maximum wave number in the infrared limit is
$k|\tau_{1}|$. So, in Eq. (77), the wave number must satisfy
$k\ll\bar{k}_{c}\equiv-1/\tau_{1}$. When the $k^{6}$ dominant term becomes
comparable to the $k^{4}$ one, the wave number should be equal to about

$\displaystyle
k_{3}\simeq\left\\{\begin{array}[]{ll}-\dfrac{3}{2\tau_{1}}=\dfrac{3}{2}\bar{k}_{c}{}{}{}{}&\eta_{1}=-3\vspace{0.2cm}\\\
-\dfrac{4}{3\tau_{1}}=\dfrac{4}{3}\bar{k}_{c}&\eta_{1}=-4\vspace{0.2cm}\\\
-\dfrac{5}{4\tau_{1}}=\dfrac{5}{4}\bar{k}_{c}&\eta_{1}=-5\vspace{0.2cm}\\\
-\dfrac{6}{5\tau_{1}}=\dfrac{6}{5}\bar{k}_{c}&\eta_{1}=-6\
.\end{array}\right.$ (82)

It is obvious that all wave number $k_{3}$ are larger than $\bar{k}_{c}$ and
thus are beyond the infrared region, which means that the steepest growth of
the power spectrum is $k^{4}$. Furthermore, we find that the power spectrum
will dip before the $k^{4}$ growth. The corresponding numerical and
approximate results are shown in Fig. 14.

Figure 14:  The power spectrums for different constant values of $\eta_{1}$.
The gray-solid and blue-dashed lines represent the numerical and approximate
results, respectively.

##  IV conclusions

The generation of a significant amount of primordial black holes requires a
sufficiently large power spectrum of the curvature perturbations of the order
of about $\mathcal{O}(10^{-2})$ at the scales smaller than the CMB one. There
are two natural ways to amplify the curvature perturbations. One is to reduce
the rolling speed of the inflaton and the other to suppress the sound speed
$c_{s}$ of the curvature perturbations. In the ultraslow-roll inflation
scenario, it has been found that the power spectrum of the curvature
perturbations has the $k^{4}$ growth. In this paper, we use the improved
junction conditions to find that the power spectrum of the curvature
perturbation has a $k^{2}$ growth when the speed of sound decreases suddenly.
Furthermore, by investigating the evolution of the power spectrum in the
inflation model, which can realize decrease of both the sound speed and the
rolling speed of the inflaton, we find that the power spectrum at the large
scales is nearly scale invariant to satisfy the constraint from the CMB
observations, and at the same time it will be enhanced at the small scales to
achieve an abundant formation of primordial black holes. In the cases that the
change of the slow-roll parameter $\eta$ precedes that of the sound speed
$c_{s}$, the power spectrum of the curvature perturbations only has a $k^{4}$
growth. While if $\eta$ and $c_{s}$ changes simultaneously or the change of
$c_{s}$ precedes that of $\eta$, the power spectrum can possess a $k^{6}$
growth under certain conditions, which is the steepest growth of the power
spectrum reported analytically so far. In the multifield inflation, numerical
results show that the growth larger than $k^{6}$ is possible due to the
violent tachyonic instability Braglia2020 ; Palma2020 ; Fumagalli2023 ;
Braglia2021 ; Braglia2023 .

###### Acknowledgements.

We appreciate very much the insightful comments and helpful suggestions by the
anonymous referee. This work is supported by the National Key Research and
Development Program of China Grant No. 2020YFC2201502, and by the National
Natural Science Foundation of China under Grants No. 12275080 and No.
12075084.

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