aainstitutetext: Department of Astronomy, Key Laboratory of Astroparticle
Physics of Yunnan Province,
School of Physics and Astronomy, Yunnan University,
No.2 Cuihu North Road, Kunming, 650091 Chinabbinstitutetext: Department of
Physics, Physics College, Nanjing University,
No.22 Hankou Road Nanjing, 650091 Chinaccinstitutetext: Institute of Nuclear
and Particle Physics, Demokritos National Research Centre, Athens, Greece.

# Kinetically stabilized inflation

Changhong Li111Corresponding Author<EMAIL_ADDRESS>a , Hao Gong b,c
and Yeuk-kwan Edna Cheung<EMAIL_ADDRESS>
###### Abstract

In this work, we propose a string-inspired two fields inflation model to
address the fine-tuning problem that the standard inflation model suffers. The
fast-rolling tachyon $\mathcal{T}$ originated from the D-brane and anti-D-
brane pair annihilation locks the inflaton $\varphi$ slowly rolling on a
Higgs-like potential
$V(\varphi)=-m_{\varphi}^{2}\varphi^{2}+\lambda\varphi^{4}$ and drives a
kinetically stabilized (KS) inflation. Our numerical simulation confirms such
a solution is a dynamic attractor. In particular, for $\lambda<0.8\times
10^{-3}$, the e-folding number contributed by the KS inflation phase can be
larger than $62$ to solve the horizon and flatness problems of Big Bang
theory. Notably, this KS inflation generates a nearly scale-invariant
primordial curvature perturbations spectrum consistent with current cosmic
microwave background (CMB) observations. It predicts a low tensor-to-scalar
ratio, which the current primordial gravitational wave background (the B-modes
in CMB) searches favor.

##  1 Introduction

The canonical single-field slow-roll inflation (standard inflation) is a
paradigm for solving the initial problems of the Big Bang Theory, the flatness
and horizon problems Guth:1980zm ; Starobinsky:1980te ; Sato:1980yn ;
Linde:1981mu ; Albrecht:1982wi , and also for generating a nearly scale-
invariant primordial curvature perturbation spectrum Mukhanov:1990me , which
is consistent with current cosmic microwave background (CMB) observations
WMAP:2010qai ; Planck:2015fie ; Planck:2018vyg . However, the essential part
of the standard inflation, the slow-roll condition, is challenging to realize
as it is not a dynamic attractor (see Ref.Baumann:2009ds  for a review and
citations to the original reference). Additionally, the current primordial
gravitational wave (PGW) (or primordial B-modes in CMB Kamionkowski:2015yta )
searches imply that the tensor-to-scalar ratio could be smaller than the
conventional expectation in the standard inflation BICEP:2021xfz ;
Tristram:2021tvh .

This work attempts to realize the inflation process as a dynamic attractor by
introducing an auxiliary field. In practice, we adopt a string-inspired open
string tachyon field $\mathcal{T}$ to couple with the scalar inflaton
$\varphi$ in the Dirac-Born-Infeld (DBI) form. After the open string tachyon
condenses following a D-brane and anti-D-brane pair annihilating, the tachyon
field locks the inflaton on the not-so-flat inflaton potential to provide a
locked slow-roll inflation phase. This proposal has two merits: 1) the D brane
and anti-D brane pair annihilation is dynamically irreversible in an expanding
universe Sen:1999nx ; Shiu:2002xp . Therefore, the attractor is stable against
various potential shapes and cosmic red-shifts; and 2) after the tachyon
condenses into cold matter form Sen:2002nu ; Sen:2002in , tachyon matter gets
red-shifted to sub-dominated without ruining the desired inflation phase.

According to our analysis, this newly proposed model consists of three phases
before the cosmic reheating.

  1. 1.

Phase I: Annihilating D-brane and anti-D-brane pair dominated inflation. In
this phase, an initial intact D-brane and anti-D-brane pair starts
annihilating. Their residual tension drives the Universe to inflate222This
phase is an analogy to the well-known D-brane inflation models (for instance,
see Refs. Dvali:1998pa ; Maartens:1999hf ; Burgess:2001fx ; Kachru:2003sx ;
Maartens:2003tw  and $k$ inflation models (for instance, see Refs. Armendariz-
Picon:1999hyi ; Scherrer:2004au ; Babichev:2007dw ; Li:2012vta ; Cai:2016ngx ;
Odintsov:2019ahz ). The difference between them is the brane inflation needs a
prolonged annihilation process to get a large e-folding number. But in our
setup, D-brane and anti-D-brane pair annihilation is very swift as the energy
scale is very high and serves as an initial condition for the newly proposed
kinetically stabilized inflation.. Ultimately, the D-brane and anti-D-brane
pair releases all energy into the tachyon field and condenses into a cold
matter form (the condensed tachyon matter).

  2. 2.

Phase II: Tachyon matter dominated expansion. In this phase, the condensed
tachyon matter dominates the cosmic background to expand. As a dynamic
attractor (confirmed by our numerical simulation), the fast-rolling tachyon
locks the inflaton to roll slowly on a Higgs-like potential,
$V(\varphi)=-m_{\varphi}^{2}\varphi^{2}+\lambda\varphi^{4}$ 333Without loss of
generality, we adopt such a Higgs-like potential to facilitate our analysis in
this work. Note that this choice differs from the well-known Higgs inflation
Bezrukov:2007ep ; Barbon:2009ya ; Bezrukov:2010jz ; Germani:2010gm  , which
field rolls from a large value to a small value. In this work, we assume
$\varphi$ rolls down from the false vacuum at $\varphi=0$ (a small value) to
the true vacuum $\varphi_{v}=m_{\varphi}/\sqrt{2\lambda}$ (a large value).,
behaving like a slow-varying cosmological constant. At the end of this
expanding phase, tachyon matter is red-shifted to be sub-dominated, and the
slow-roll inflation field $\varphi$ becomes dominant.

  3. 3.

Phase III: Kinetically stabilized (KS) inflation. In this phase, the fast-
rolling tachyon continues to lock the inflaton to roll down slowly, and the
inflation field dominates the evolution of the Universe. We call this
inflation phase kinetically stabilized (KS) inflation to distinguish it from
standard slow-roll inflation. After a long slow-roll process, the KS inflation
eventually attains the true vacuum at
$\varphi_{v}=\sqrt{2}m_{\varphi}/2\lambda$, and the Universe reheats into
standard model (SM) particles dominated era444In some Higgs warm inflation
like Ref. Dymnikova:2001jy , the reheating process can happen earlier. In some
kinetic axion model $F(R)$ gravity theory like Ref. Oikonomou:2022tux , the
axion field can oscillate around its vacuum during inflation rather than
reheating. Bassett:2005xm ; Allahverdi:2010xz ; Amin:2014eta . As we will
show, taking a generic initial condition for $\varphi$, $\varphi_{i}\sim
m_{\varphi}$, the e-folding number ($N_{e}\equiv\ln(a/a_{i})$) contributed by
the KS inflation (with $\lambda<0.8\times 10^{-3}$ and $m_{s}=10^{-3}M_{p}$)
can be greater than $62$ to solve the horizon and flatness problems, where
$m_{s}$ being the string mass and $M_{p}$ the reduced Planck mass. Moreover,
by fine-tuning the initial value of $\varphi$ to be $\varphi_{i}\simeq
10^{-3}m_{\varphi}$, one can even obtain $\lambda=\mathcal{O}(1)$. We expect
many effective field theories to satisfy the requirement of $\lambda$,
including the Higgs boson in the SM of particle physics (for a short review,
see the Higgs boson section in ParticleDataGroup:2018ovx ).

After elucidating the evolution of the cosmic background, we compute the
primordial curvature perturbation for the KS inflation. Its spectrum is nearly
scale-invariant and consistent with the current observation of E-modes in CMB,
similar to the standard inflation, except having a larger pre-factor
$c_{s}^{-1}\gg 1$. Such a large factor implies a more significant gap between
the scalar and tensor modes of primordial metric perturbation for KS
inflation. We then compute the tensor-to-scalar ratio for KS inflation and
find that compared to the standard inflation, the current PGW searches’
results favor KS inflation.

##  2 The kinetically coupled tachyon and inflation fields model

In string theory, the D-brane and anti-D-brane pair annihilation (or the
unstable D-branes decay) corresponds to a phase transition from the false
vacuum (the open string vacuum) to the true vacuum (the closed string vacuum)
Sen:1999nx . In this tachyonic process, the D brane and anti-D brane pair
releases all energy into the tachyon field $\mathcal{T}$ and make it fully
condense into a cold matter form (called tachyon matter) Sen:2002nu ;
Sen:2002in . In the effective theory, one can use the following DBI action of
the single tachyon field $\mathcal{T}$ to describe a spatially filling $D_{3}$
brane and anti-$D_{3}$ brane pair annihilation Sen:2004nf ,

$\mathcal{L}_{\mathcal{T}}=-V(\mathcal{T})\sqrt{1+\partial^{\mu}\mathcal{T}\partial_{\mu}\mathcal{T}}~{},$
(1)

where the hill-like potential of the tachyon field taking
$V(\mathcal{T})=V_{0}/\cosh(\mathcal{T}/\sqrt{2})$, $V_{0}=m_{s}^{3}M_{p}$
being the tension of D brane and anti-D brane pair, and $m_{s}$ and $M_{p}$
being the string mass and the reduced Planck mass respectively. Throughout
this paper, we take $m_{s}=10^{-3}M_{p}$ and the convention $m_{s}=1$.

In the homogeneous and isotropic cosmic background, which the Friedmann -
Lemaitre - Robertson - Walker (FLRW) metric
($g_{\mu\nu}=\\{-1,a^{2}(t)\delta_{ij}\\}$) describes, the equation of motion
of the tachyon field takes

$\ddot{\mathcal{T}}-3H\dot{\mathcal{T}}w_{st}-\frac{d\ln
V(\mathcal{T})}{d\mathcal{T}}w_{st}=0~{},$ (2)

and the Friedmann equation takes

$H^{2}=\frac{8\pi G}{3}\rho_{st},$ (3)

where $w_{st}=-(1-\dot{\mathcal{T}}^{2})$ and
$\rho_{st}=V(\mathcal{T})/\sqrt{1-\dot{\mathcal{T}}^{2}}$ being the equation
of state (EoS) and the energy density of the single tachyon field
respectively.

Solving Eq.(2) and Eq.(3), one can obtain the cosmic evolution of the tachyon
field. In particular, with a small initial displacement, the tachyon field
accelerates to roll down from the top of the hill-like potential. This
corresponds to the initial intact $D_{3}$ brane and anti-$D_{3}$ brane pair
($\rho_{\mathcal{T}}\simeq V_{0}$ at $\mathcal{T}\simeq 0$ and
$\dot{\mathcal{T}}\simeq 0$ ) starts to annihilate. At the end
($\mathcal{T}\gg 1$), tachyon field attains the maximal velocity

$\dot{\mathcal{T}}\rightarrow 1,\ddot{\mathcal{T}}\rightarrow 0~{},$ (4)

which reflects that the D brane and anti-D brane pair releases all energy into
the tachyon field and make it condense into a cold dark matter form Sen:2004nf
,

$w_{st}=-(1-\dot{\mathcal{T}}^{2})\rightarrow 0,\quad\rho_{st}\propto a^{-3},$
(5)

where $\rho_{st}$ is the energy density of the single tachyon field.

Encouragingly, such a tachyon condensation process is a dynamic attractor.
Figure 1 illustrates tachyon condensation in the phase space.
$(\dot{\mathcal{T}},\ddot{\mathcal{T}})=(1,0)$ is a dynamic attractor for the
tachyon field evolution, which reflects D-brane and anti-D-brane pair
annihilating into tachyon matter is inevitable and irreversible. In the
D-brane/k inflation and the matter bounce cosmology (for instance, see Refs.
Dvali:1998pa ; Maartens:1999hf ; Burgess:2001fx ; Kachru:2003sx ;
Maartens:2003tw ; Armendariz-Picon:1999hyi ; Scherrer:2004au ; Babichev:2007dw
; Li:2012vta ; Cai:2016ngx ; Odintsov:2019ahz  and Refs. Sen:2003mv ;
Li:2011nj ; Li:2013bha ; Zhang:2019tct  respectively), such a single tachyon
condensation process has been extensively studied.

Figure 1:  The parametric plot of $\dot{T}$ and $\ddot{T}$ illustrates that
$(\dot{\mathcal{T}},\ddot{\mathcal{T}})=(1,0)$ is a dynamic attractor for the
single tachyon field model. The longest curve is plotted by solving Eq.(2)
numerically with initial condition
$(\mathcal{T}(0),\dot{\mathcal{T}}(0))=(0.001,0.001)$ while other curves are
plotted with $(0.05,0.1)$, $(0.1,0.1)$, $(0.2,0.1)$, and $(0.3,0.1)$
respectively. All curves converge into the fixed point
$(\dot{\mathcal{T}},\ddot{\mathcal{T}})=(1,0)$. For simplicity, we have
ignored the red-shifted term, $3H$, in this numerical simulation.

In this work, we adopt this dynamic attractor ($\dot{\mathcal{T}}\rightarrow
1$, $\ddot{\mathcal{T}}\rightarrow 0$) to lock the inflaton $\varphi$ to
slowly roll ($\dot{\varphi}\simeq 0$) on a not-so-flat potential and avoid the
fine-tuning problem of the standard inflation. In practice, we consider the
following simple extension,

$\mathcal{L}_{\text{kinetic}}=-V(\mathcal{T})\sqrt{1+\partial^{\mu}\mathcal{T}\partial_{\mu}\mathcal{T}+\partial^{\mu}\varphi\partial_{\mu}\varphi}~{}.$
(6)

Without loss of generality, the not-so-flat potential of $\varphi$ takes a
Higgs-like form,

$V(\varphi)=-m_{\varphi}^{2}\varphi^{2}+\lambda\varphi^{4}~{},$ (7)

where $m_{\varphi}$ being the mass of $\varphi$, and $\lambda$ being a
dimensionless coupling constant. In this work, we assume that $m_{\varphi}\ll
m_{s}\ll M_{p}$.

In this work, we consider that the Universe is dominated by these coupled
tachyon and scalar fields
($\mathcal{L}_{\text{tot}}=\mathcal{L}_{\text{kinetic}}-V(\varphi)$). The
tachyon field part describes a spatially filling $D_{3}$ brane and
anti-$D_{3}$ brane pair, which will annihilate soon and condense into a form
of massive cold tachyon matter. The scalar field part, which could be
identified with inflaton in this model, is assumed to couple with the tachyon
field in the DBI form by its kinetic term. Initially, the scalar field can be
viewed as an extra external field weakly acting on the $D_{3}$ brane and
anti-$D_{3}$ brane pair. After the $D_{3}$ brane and anti-$D_{3}$ brane pair
annihilating, the massive tachyon matter locks it to realize a slow-roll
inflation 555This model can be viewed as one of the specific realizations in
the generalized multi-fields Horndeski model Horndeski:1974wa ; Nicolis:2008in
; Deffayet:2009wt ; Deffayet:2009mn ; Deffayet:2011gz ; Kobayashi:2011nu ;
Charmousis:2011bf ; Gleyzes:2014dya ; BenAchour:2016fzp ; Crisostomi:2016tcp ;
deRham:2016wji ; Kobayashi:2019hrl . We thank the anonymous referee for
pointing this out to us. . Varying the action of this newly proposed model,

$S=\int dx^{4}\sqrt{-g}\left[\frac{1}{16\pi
G}R+\mathcal{L}_{\text{kinetic}}-V(\varphi)\right],$ (8)

with respect to $g^{\mu\nu}$, $\mathcal{T}$ and $\varphi$ respectively, we
obtain

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi
G\left[\left(\partial_{\mu}\mathcal{T}\partial_{\nu}\mathcal{T}+\partial_{\mu}\varphi\partial_{\nu}\varphi\right)\rho_{\mathcal{T}}+g_{\mu\nu}\left(p_{\mathcal{T}}-V(\varphi)\right)\right],$
(9)
$\partial_{\mu}\left[\sqrt{-g}\rho_{\mathcal{T}}\partial^{\mu}\mathcal{T}\right]+\sqrt{-g}p_{\mathcal{T}}\frac{d\ln
V(\mathcal{T})}{d\mathcal{T}}=0,$ (10)

and

$\partial_{\mu}\left[\sqrt{-g}\rho_{\mathcal{T}}\partial^{\mu}\varphi\right]-\sqrt{-g}\left(\frac{dV(\varphi)}{d\varphi}\right)=0,$
(11)

where $\rho_{\mathcal{T}}\equiv
V(\mathcal{T})/\sqrt{1+\partial^{\mu}\mathcal{T}\partial_{\mu}\mathcal{T}+\partial^{\mu}\varphi\partial_{\mu}\varphi}$
and
$p_{\mathcal{T}}\equiv-V(\mathcal{T})\sqrt{1+\partial^{\mu}\mathcal{T}\partial_{\mu}\mathcal{T}+\partial^{\mu}\varphi\partial_{\mu}\varphi}$
being the energy density and the pressure of the coupled tachyon field.

##  3 Kinetically stabilized inflation

At the background order, Eq.(9), Eq.(10) and Eq.(11), respectively, take

$H^{2}=\frac{8\pi G}{3}\left[\rho_{\mathcal{T}}+V(\varphi)\right]~{},$ (12)

$\ddot{\mathcal{T}}-3H\dot{\mathcal{T}}\frac{w_{\mathcal{T}}}{1-\dot{\varphi}^{2}}-\frac{d\ln[V(\mathcal{T})]}{dt}\dot{\mathcal{T}}^{-1}w_{\mathcal{T}}-\dot{\varphi}\ddot{\varphi}\dot{\mathcal{T}}\frac{1}{1-\dot{\varphi}^{2}}=0~{},$
(13)

and 666Note that, for the coupled tachyon field, the attractor is at
$\dot{\mathcal{T}}^{2}=1-\dot{\varphi}^{2}$ in contrast to the single tachyon
field at $\dot{\mathcal{T}}^{2}=1$. Therefore, the last term of Eq.(14) is not
divergent at the attractor.

$\ddot{\varphi}-3H\dot{\varphi}\frac{w_{\mathcal{T}}}{1-\dot{\mathcal{T}}^{2}}-\frac{d\ln[V(\mathcal{T})]}{dt}\dot{\varphi}\frac{w_{\mathcal{T}}}{1-\dot{\mathcal{T}}^{2}}-\dot{\mathcal{T}}\ddot{\mathcal{T}}\dot{\varphi}\frac{1}{1-\dot{\mathcal{T}}^{2}}-\frac{w_{\mathcal{T}}}{1-\dot{\mathcal{T}}^{2}}\rho_{\mathcal{T}}^{-1}\frac{dV(\varphi)}{d\varphi}=0~{}.$
(14)

where $w_{\mathcal{T}}\equiv
p_{\mathcal{T}}/\rho_{\mathcal{T}}=-(1-\dot{\mathcal{T}}^{2}-\dot{\mathcal{\varphi}}^{2})$
being the EoS of the coupled tachyon field. Afterward, we use $\mathcal{T}$
and $\varphi$ to denote the background field and $\delta\mathcal{T}$ and
$\delta\varphi$ to denote their perturbations.

Solving Eq.(12), Eq.(13) and Eq.(14) with the initial condition
($\mathcal{T}=\mathcal{T}(0)\ll 1$, $\dot{\mathcal{T}}=0$,
$\varphi=\varphi(0)\simeq m_{\varphi}$, $\dot{\varphi}=0$, $a=a(0)$, and
$\dot{a}=0$ at $t=0$), we obtain the following three cosmic phases.

  1. 1.

Annihilating D-brane and anti-D-brane pair dominated inflation. Before fully
annihilating, the tension of the D-brane and anti-D-brane pair
($\rho_{\mathcal{T}}\simeq V_{0}$ at $t\simeq 0$) dominates the Universe to
inflate for a short period, $\Delta\tilde{t}\simeq\mathcal{O}(1)\times
m_{s}^{-1}$ 777By fine-tuning the initial values of $\mathcal{T}$ to be very
tiny, for instance, $\mathcal{T}(0)=10^{-20}m_{s}$, the annihilating D-brane
and anti-D-brane pair dominated inflation phase can last into a much longer
period Li:2011nj . But such a setup is too artificial. In this work, we
consider a generic initial condition $\mathcal{T}(0)\simeq 10^{-3}m_{s}$.,

$a(t)\simeq a(0)e^{\sqrt{\frac{8\pi G}{3}V_{0}}t}~{}.$ (15)

During this process, the coupled tachyon field accelerates to its maximal
velocity,

$\dot{\mathcal{T}}\rightarrow\sqrt{1-\dot{\varphi}^{2}}~{},\quad\ddot{\mathcal{T}}\rightarrow
0,$ (16)

and condensates into cold matter, which is analogy to Eq.(5),

$w_{\mathcal{T}}=-(1-\dot{\mathcal{T}}^{2}-\dot{\mathcal{\varphi}}^{2})\rightarrow
0~{}.$ (17)

Note that the D-brane and anti-D-brane pair annihilates very swiftly
($\Delta\tilde{t}\simeq\mathcal{O}(1)\times m_{s}^{-1}$ as aforementioned), so
the e-folding number contributed by this phase is small,
$N_{e}=\ln{a(\Delta\tilde{t})/a(0)}$ $\simeq\sqrt{m_{s}/M_{p}}\ll 1$, and the
value of $\varphi$ is almost unchanged (This is because the slop of $\varphi$
potential is much smaller than tachyon field, $m_{\varphi}^{2}\ll m_{s}^{2}$.
),

$\varphi(\Delta\tilde{t})\simeq\varphi(0)e^{\mathcal{O}(1)\times\frac{m_{\varphi}^{2}}{m_{s}\sqrt{M_{p}m_{s}}}}\simeq\varphi(0)\simeq
m_{\varphi}~{},\dot{\varphi}(\Delta\tilde{t})\simeq\frac{m_{\varphi}^{2}}{\sqrt{m_{s}M_{p}}}\varphi(\Delta\tilde{t})\ll
m_{s}^{2}=1$ (18)

where $m_{\varphi}\ll m_{s}\ll M_{p}$ being used. Substituting Eq.(18) into
Eq.(16), we obtain a dynamic attractor solution, which is an analogy to
Eq.(4),

$\dot{\mathcal{T}}\rightarrow 1~{},\quad\ddot{\mathcal{T}}\rightarrow 0.$ (19)

In Figure 2 and Figure 3, we numerically solve Eq.(13) and Eq.(14) to
illustrate this dynamic attractor for $\mathcal{T}$ part and $\varphi$ part
respectively. For simplicity, we have neglected the red-shifted term, $3H$, in
the expanding phase and neglected the $\lambda\varphi^{4}$ term near
$\varphi\simeq 0$ region888We take $\varphi(0)\simeq 0.01m_{\varphi}$ and
$m_{\varphi}=10^{-6}m_{s}$ ($\varphi(0)\sim 10^{-8}$) to perform this
simulation. One has plenty of computing resources that can reproduce similar
numerical results for other initial values.. Clearly, the coupled tachyon
field $\mathcal{T}$ is also bound to move toward the dynamic attractor,
$(\dot{\mathcal{T}},\ddot{\mathcal{T}})=(1,0)$, as shown in Figure 2. And the
inflation field $\varphi$ is locked around the slow-roll condition,
$\dot{\varphi}=0$, as shown in Figure 3. Note that without such a dynamic
attractor, $\varphi$ will increase exponentially and destroy the slow-roll
condition. After the D-brane and anti-D-brane pair fully annihilates, the
Universe evolves into the next phase.

Figure 2:  The parametric plot of $\dot{\mathcal{T}}$ and $\ddot{\mathcal{T}}$
obtained by numerically solving Eq.(13) and Eq.(14), which illustrates that
$(\dot{\mathcal{T}},\ddot{\mathcal{T}})=(1,0)$ is a dynamic attractor for the
coupled tachyon field. The longest curve is plotted with the initial condition
$(\mathcal{T}(0),\dot{\mathcal{T}}(0),\varphi(0),\dot{\varphi}(0))=(0.001,0.001,10^{-8},0)$
and the other two curves are plotted with $(0.1,0.05,3\times 10^{-8},0)$ and
$(0.05,0.1,2\times 10^{-8},0)$ respectively. In this numerical simulation, we
have neglected the red-shifted term ($3H$) for the expanding phase and
neglected the $\lambda\varphi^{4}$ term near $\varphi\simeq 0$ region for
simplicity.  Figure 3:  The parametric plot of $\dot{\varphi}$ and
$\ddot{\varphi}$ obtained by numerically solving Eq.(13) and Eq.(14), which
illustrates that the dynamic attractor of the coupled tachyon field
($(\dot{\mathcal{T}},\ddot{\mathcal{T}})=(1,0)$) can lock $\varphi$ around the
slow-roll condition $\dot{\varphi}=0$. The smallest curve is plotted with the
initial condition
$(\mathcal{T}(0),\dot{\mathcal{T}}(0),\varphi(0),\dot{\varphi}(0))=(0.001,0.001,10^{-8},0)$
and the other two curves are plotted with $(0.1,0.05,3\times 10^{-8},0)$ and
$(0.05,0.1,2\times 10^{-8},0)$ respectively. In this numerical simulation, we
have neglected the red-shifted term ($3H$) for the expanding phase and
neglected the $\lambda\varphi^{4}$ term near $\varphi\simeq 0$ region for
simplicity.

  2. 2.

Tachyon matter dominated expansion. After the D-brane and anti-D-brane pair
annihilation completing, the condensed coupled tachyon matter dominates the
Universe,

$H^{2}\simeq\frac{8\pi
G}{3}\left[V_{0}\left(\frac{a(\Delta\tilde{t})}{a(t)}\right)^{3}+\rho_{\varphi}\right]\simeq\frac{8\pi
GV_{0}}{3}\left(\frac{a(\Delta\tilde{t})}{a(t)}\right)^{3}~{},$ (20)

where the energy density of $\varphi$ taking 999Taking $V(\varphi_{v})=0$ at
true vacuum ($\varphi=\varphi_{v}$), the energy density of $\varphi$ around
the false vacuum ($\varphi=0$) takes
$\rho_{\varphi}=\frac{1}{2}\dot{\varphi_{i}}^{2}+\frac{m_{\varphi}^{4}}{4\lambda}+\lambda\varphi_{i}^{4}\simeq\frac{m_{\varphi}^{4}}{4\lambda},$
(21) where $\frac{m_{\varphi}^{4}}{4\lambda}$ being the $V(\varphi)$
difference between true and false vacuum, and $\lambda<1$ and Eq.(18) being
used.,

$\rho_{\varphi}\simeq\frac{m_{\varphi}^{4}}{4\lambda}.$ (22)

At the beginning of this phase, $\rho_{\varphi}$ is sub-dominated,
$\rho_{\varphi}\ll\rho_{\mathcal{T}}(t)\simeq
V_{0}\left[a(\Delta\tilde{t})/a(t)\right]^{3}$. However, the aforementioned
dynamic attractor locks $\varphi$ to roll slowly, $\dot{\varphi}\simeq 0$.
Thus $\rho_{\varphi}$ behaves as a slow-varying cosmological constant against
the red-shift effect and eventually dominates at the end of this phase,
$\rho_{\varphi}\geq\rho_{\mathcal{T}}(t_{f})\simeq
V_{0}\left[a(\Delta\tilde{t})/a_{f}\right]^{3}$, where the subscript $~{}_{f}$
denoting the end of this phase and the beginning of KS inflation.

  3. 3.

Kinetically stabilized inflation. In this phase, $\rho_{\varphi}$ is dominant,

$H^{2}=\frac{1}{3M_{p}^{2}}\frac{m_{\varphi}^{4}}{4\lambda}~{},$ (23)

which solution takes

$a=a_{f}e^{\frac{1}{2\sqrt{3}}\frac{m_{\varphi}^{2}}{\sqrt{\lambda}M_{p}}(t-t_{f})}~{}.$
(24)

The dynamic attractor locks $\varphi$ around $\dot{\varphi}\simeq 0$ to
realize slow-roll inflation, the kinetically stabilized (KS) inflation. As
shown in Figure 2 and Figure 3, KS inflation does not suffer the fine-tuning
problem of the standard inflation.

During this phase, $\varphi$ rolls very slowly, so it takes a long time to
attain its true vacuum, $\varphi_{v}=m_{\varphi}/\sqrt{2\lambda}$. Then the
Universe reheats into the SM particles-dominated era. In Appendix A, we
estimate the e-folding number contributed by the KS inflation phase and obtain

$N_{e}=\ln\left(\frac{a_{r}}{a_{f}}\right)\simeq\frac{1}{4\sqrt{2}}\frac{1}{\lambda}\sqrt{\frac{m_{s}}{M_{p}}}\left(\frac{m_{\varphi}}{\varphi_{i}}\right)\simeq\frac{1}{4\sqrt{2}}\frac{1}{\lambda}\sqrt{\frac{m_{s}}{M_{p}}}~{},$
(25)

where $a_{r}$ denotes the end of KS inflation and the onset of cosmic
reheating. We adopt a generic initial value for $\varphi$, $\varphi_{i}\simeq
m_{\varphi}$, rather than fine-tuning it. Note that this generic condition
($\varphi_{i}\simeq m_{\varphi}$) coincidentally cancels other $m_{\varphi}$
factors in Eq.(25). This fact reflects our analysis neglecting the possibility
of $\varphi(0)$ being suspiciously small.

For solving the horizon and flatness problems, the e-folding number should be
larger than $62$ Dodelson:2003ft . In Figure 4, we use Eq.(25) to plot the
parameter region of $(m_{s},\lambda)$ that allows $N_{e}\geq 62$ for the KS
inflation. In particular, to ensure $N_{e}\geq 62$, we obtain $\lambda\leq
0.8\times 10^{-3}$ with $m_{s}/M_{p}=10^{-3}$. Note that according to the
third term in Eq.(25), we can even obtain $\lambda=\mathcal{O}(1)$ by fine-
tuning the initial value of $\varphi$ to be $\varphi_{i}\simeq
10^{-3}m_{\varphi}$. We expect many effective field theories can satisfy the
requirement of $\lambda$, including the Higgs boson in the SM of particle
physics ParticleDataGroup:2018ovx . In the next section, we are computing the
power spectrum of primordial curvature perturbation generated during this KS
inflation phase.

Figure 4:  The contour plot of $N_{e}$ in $(m_{s},\lambda)$-plane for
$N_{e}\geq 62$ in the KS inflation, using Eq.(25).

##  4 The nearly scale-invariant primordial curvature perturbation spectrum of
kinetically stabilized inflation

As presented in Appendix B, adopting the perturbative Friedmann-Lemaitre-
Robertson-Walker (FLRW) metric in conformal Newtonian gauge,

$g_{\mu\nu}=\\{-1-2\Psi(\vec{x},t),a^{2}(t)\delta_{ij}\left[1+2\Phi(\vec{x},t)\right]\\}~{},$
(26)

and expanding Eq.(9) with $\mathcal{T}=\mathcal{T}+\delta\mathcal{T}$ and
$\varphi=\varphi+\delta\varphi$ to the first order, we obtain

$\ddot{\Phi}_{k}+\left(H-\frac{\ddot{H}}{\dot{H}}\right)\dot{\Phi}_{k}+\left(2\dot{H}-H\frac{\ddot{H}}{\dot{H}}\right)\Phi_{k}+\frac{c_{s}^{2}k^{2}}{a^{2}}\Phi_{k}=0~{},$
(27)

where $\Phi_{k}$ being the Fourier modes of $\Phi(\vec{x},t)$ and
$c_{s}^{2}=1-\dot{\mathcal{T}}^{2}-\dot{\varphi}^{2}$ being the sound speed
parameter. Note that Eq.(27) is the same as the standard inflation (c.f. Eq.),
except $c_{s}\ll 1$ for the KS inflation. It implies that the KS inflation
also can generate a nearly scale-invariant power spectrum.

Adopting the gauge-invariant variable,
$\zeta=\Phi-\frac{H}{\dot{H}}\left(\dot{\Phi}+H\Phi\right)$, and a canonical
variable, $v=z\zeta$ with $z\equiv a\sqrt{2\epsilon}c_{s}^{-1}$ and
$\epsilon\equiv-\dot{H}/H^{2}$ Mukhanov:1990me , we simplifies Eq.(27) to be

$v_{k}^{\prime\prime}+\left(c_{s}^{2}k^{2}-\frac{z^{\prime\prime}}{z}\right)v_{k}=0$
(28)

where the superscript $~{}^{\prime}$ denoting the derivative with respect to
the conformal time $\eta=\int a^{-1}dt$. During KS inflation,
$\eta=(aH)^{-1}\propto a^{-1}$ and $z^{\prime\prime}/z=2\eta^{-2}$, which
leads to

$v_{k}^{\prime\prime}+\left(c_{s}^{2}k^{2}-\frac{2}{\eta^{2}}\right)v_{k}=0~{}.$
(29)

Solving Eq.(29) within the Bunch-Davies vacuum
($v_{k}=\frac{e^{-ic_{s}k\eta}}{\sqrt{2c_{s}k}}$ for
$\eta\rightarrow-\infty$), we obtain

$v_{k}=\frac{e^{ic_{s}k\eta}}{\sqrt{2c_{s}k}}\left(1-\frac{i}{c_{s}k\eta}\right)~{},$
(30)

and its spectrum for KS inflation,

$\mathcal{P}_{\zeta}\equiv\frac{k^{3}}{2\pi^{2}M_{p}^{2}}|\zeta_{k}|^{2}=\frac{k^{3}}{2\pi^{2}}\left|\frac{v_{k}}{z}\right|^{2}=c_{s}^{-1}\times\frac{H^{2}}{8\pi^{2}M_{p}^{2}|\epsilon|},\quad
k\eta\rightarrow 0.$ (31)

Eq.(31) indicates that the KS inflation also has a nearly scale-invariant
primordial curvature spectrum and is consistent to current observations of the
E-modes in CMB, similar with the standard inflation, except $c_{s}^{-1}\gg 1$.
As we will show in the next section, the larger pre-factor $c_{s}^{-1}$
implies that the KS inflation has a more significant gap between the scalar
and the tensor modes spectra, i.e. a smaller tensor-to-scalar ratio compared
to the standard inflation.

##  5 The tensor-to-scalar ratio of KS inflation

During KS inflation (Eq.(24)), solving the equation of motion for the Fourier
tensor modes of metric perturbation $h_{+/\times}$,

$h^{\prime\prime}_{k}+2\frac{a^{\prime}}{a}h^{\prime}_{k}+k^{2}h_{k}=0~{}$
(32)

we obtain the solution,

$h_{k}=\frac{\sqrt{16\pi
G}}{a}\frac{e^{-ik\eta}}{\sqrt{2k}}\left(1-\frac{i}{k\eta}\right)~{},$ (33)

and the spectrum,

$\mathcal{P}_{h}\equiv\frac{k^{3}}{2\pi^{2}}|h_{k}|^{2}=\frac{H^{2}}{2\pi^{2}M_{p}^{2}}~{},\quad
k\eta\rightarrow 0,$ (34)

which is identical to the well-known expression in the standard inflation
Dodelson:2003ft .

Using Eq.(31) and Eq.(34), we obtain the tensor-to-scalar ratio for the KS
inflation,

$r\equiv\frac{\mathcal{P}_{h}}{\mathcal{P}_{\xi}}=c_{s}\times 4|\epsilon|,$
(35)

which is the same as the standard inflation, except $c_{s}\ll 1$. Using the
data $\epsilon\simeq(1-n_{s})/4\simeq 0.01$ and $n_{s}\simeq 0.96$ from WMAP
and Planck observations WMAP:2010qai ; Planck:2015fie ; Planck:2018vyg , we
find that compared to the standard inflation
($r_{\textbf{standard}}=4\epsilon\simeq 0.04$), the newly proposed KS
inflation ($r_{\textbf{KSI}}=c_{s}\times 4|\epsilon|\ll 0.04$) is favored by
the current PGW searches ($r_{\textbf{BICEP+}}<0.03$) BICEP:2021xfz ;
Tristram:2021tvh .

##  6 Conclusion

We proposed the kinetically stabilized (KS) inflation model in this paper. In
this model, the fast-rolling tachyon field originated from the D-brane and
anti-D-brane pair annihilation locks the inflaton slowly rolling on a Higgs-
like potential ($V(\varphi)=-m_{\varphi}^{2}\varphi^{2}+\lambda\varphi^{4}$)
and drives a long inflation process ($N_{e}>62$ for $\lambda<0.8\times
10^{-3}$). Moreover, by fine-tuning the initial value of $\varphi$ to be
$\varphi_{i}\simeq 10^{-3}m_{\varphi}$, we can even obtain
$\lambda=\mathcal{O}(1)$. We expect many effective field theories can satisfy
the requirement of $\lambda$, including the Higgs boson in the SM of particle
physics. We performed a numerical simulation, confirming that such an
inflation process is a dynamic attractor that avoids the fine-tuning problem
of standard inflation.

We computed the primordial curvature perturbation spectrum of this KS
inflation model. We found it is nearly scale-invariant and consistent with the
current observation of E-modes in CMB, similar to the standard inflation,
except the factor $c_{s}^{-1}\gg 1$. This large factor implies a more
significant gap between the scalar and tensor modes of primordial metric
perturbation for KS inflation. More specifically, by computing the tensor-to-
scalar ratio, we found that compared to the standard inflation
($r_{\textbf{standard}}\simeq 0.04$), the current PGW searches
($r_{\textbf{BICEP+}}<0.03$) favors the newly proposed KS inflation
($r_{\textbf{KSI}}\ll 0.04$).

We want to emphasize that our estimation of the e-folding number of KS
inflation is rough (we adopt a constant rolling velocity for $\varphi$ rather
than the small quasi-oscillating rolling velocity to facilitate our
estimation), which should be improved in the future. Furthermore, although the
tensor-to-scalar ratio $r$ predicted by the standard inflation seems too large
to be compatible with current PGW searches’ results, there are still some
mechanisms that can suppress a large primordial $r$ to be smaller during the
(post-)reheating phase, for example, the perturbative resonance mechanism
proposed in Ref.Li:2019std . Therefore, a small $r$ implied by current
observations may have multiple physics origins, which are worthy of further
study. And the dynamic attractor is robust against various $V(\varphi)$ and
$a(t)$. We expect it to apply multiple inflation and bouncing universe models
in future studies (for an extensive review of cosmological models, see
Nojiri:2017ncd  and citations to the original works), not only for KS
inflation.

###### Acknowledgements.

C.L. is supported by the NSFC under Grants No.11963005 and No.11603018, by
Yunnan Provincial Foundation under Grants No.2016FD006 and No.2019FY003005, by
Reserved Talents for Young and Middle-aged Academic and Technical Leaders in
Yunnan Province Program, by Yunnan Provincial High-level Talent Training
Support Plan Youth Top Program, and by the NSFC under Grant No.11847301 and by
the Fundamental Research Funds for the Central Universities under Grant No.
2019CDJDWL0005. Y.C. is supported by the NSFC under Grants No.11775110.

## References

  * (1) A. H. Guth, “The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems,” Phys. Rev. D 23 (1981), 347-356
  * (2) A. A. Starobinsky, “A New Type of Isotropic Cosmological Models Without Singularity,” Phys. Lett. B 91, 99-102 (1980)
  * (3) K. Sato, “First Order Phase Transition of a Vacuum and Expansion of the Universe,” Mon. Not. Roy. Astron. Soc. 195, 467-479 (1981) NORDITA-80-29.
  * (4) A. D. Linde, “A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems,” Phys. Lett. B 108, 389-393 (1982)
  * (5) A. Albrecht and P. J. Steinhardt, “Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking,” Phys. Rev. Lett. 48, 1220-1223 (1982)
  * (6) V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, “Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions,” Phys. Rept. 215, 203-333 (1992)
  * (7) E. Komatsu et al. [WMAP], “Seven-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation,” Astrophys. J. Suppl. 192, 18 (2011)
  * (8) P. A. R. Ade et al. [Planck], “Planck 2015 results. XIII. Cosmological parameters,” Astron. Astrophys. 594, A13 (2016)
  * (9) N. Aghanim et al. [Planck], “Planck 2018 results. VI. Cosmological parameters,” Astron. Astrophys. 641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)] [arXiv:1807.06209 [astro-ph.CO]].
  * (10) D. Baumann, “Inflation,” [arXiv:0907.5424 [hep-th]].
  * (11) M. Kamionkowski and E. D. Kovetz, “The Quest for B Modes from Inflationary Gravitational Waves,” Ann. Rev. Astron. Astrophys. 54, 227-269 (2016) [arXiv:1510.06042 [astro-ph.CO]].
  * (12) P. A. R. Ade et al. [BICEP and Keck], “Improved Constraints on Primordial Gravitational Waves using Planck, WMAP, and BICEP/Keck Observations through the 2018 Observing Season,” Phys. Rev. Lett. 127, no.15, 151301 (2021) [arXiv:2110.00483 [astro-ph.CO]].
  * (13) M. Tristram, A. J. Banday, K. M. Górski, R. Keskitalo, C. R. Lawrence, K. J. Andersen, R. B. Barreiro, J. Borrill, L. P. L. Colombo and H. K. Eriksen, et al. Phys. Rev. D 105, no.8, 083524 (2022) [arXiv:2112.07961 [astro-ph.CO]].
  * (14) A. Sen and B. Zwiebach, “Tachyon condensation in string field theory,” JHEP 03, 002 (2000) [arXiv:hep-th/9912249 [hep-th]].
  * (15) G. Shiu, S. H. H. Tye and I. Wasserman, “Rolling tachyon in brane world cosmology from superstring field theory,” Phys. Rev. D 67, 083517 (2003) [arXiv:hep-th/0207119 [hep-th]].
  * (16) A. Sen, “Rolling tachyon,” JHEP 04, 048 (2002) [arXiv:hep-th/0203211 [hep-th]].
  * (17) A. Sen, “Tachyon matter,” JHEP 07, 065 (2002) [arXiv:hep-th/0203265 [hep-th]].
  * (18) G. R. Dvali and S. H. H. Tye, “Brane inflation,” Phys. Lett. B 450, 72-82 (1999) [arXiv:hep-ph/9812483 [hep-ph]].
  * (19) R. Maartens, D. Wands, B. A. Bassett and I. Heard, “Chaotic inflation on the brane,” Phys. Rev. D 62, 041301 (2000) [arXiv:hep-ph/9912464 [hep-ph]].
  * (20) C. P. Burgess, M. Majumdar, D. Nolte, F. Quevedo, G. Rajesh and R. J. Zhang, JHEP 07, 047 (2001) doi:10.1088/1126-6708/2001/07/047 [arXiv:hep-th/0105204 [hep-th]].
  * (21) S. Kachru, R. Kallosh, A. D. Linde, J. M. Maldacena, L. P. McAllister and S. P. Trivedi, “Towards inflation in string theory,” JCAP 10, 013 (2003) [arXiv:hep-th/0308055 [hep-th]].
  * (22) R. Maartens, “Brane world gravity,” Living Rev. Rel. 7, 7 (2004) [arXiv:gr-qc/0312059 [gr-qc]].
  * (23) C. Armendariz-Picon, T. Damour and V. F. Mukhanov, “k - inflation,” Phys. Lett. B 458, 209-218 (1999) [arXiv:hep-th/9904075 [hep-th]].
  * (24) R. J. Scherrer, “Purely kinetic k-essence as unified dark matter,” Phys. Rev. Lett. 93, 011301 (2004) [arXiv:astro-ph/0402316 [astro-ph]].
  * (25) E. Babichev, V. Mukhanov and A. Vikman, “k-Essence, superluminal propagation, causality and emergent geometry,” JHEP 02, 101 (2008) [arXiv:0708.0561 [hep-th]].
  * (26) S. Li and A. R. Liddle, “Observational constraints on K-inflation models,” JCAP 10, 011 (2012) [arXiv:1204.6214 [astro-ph.CO]].
  * (27) Y. F. Cai, J. O. Gong, D. G. Wang and Z. Wang, “Features from the non-attractor beginning of inflation,” JCAP 10, 017 (2016) [arXiv:1607.07872 [astro-ph.CO]].
  * (28) S. D. Odintsov and V. K. Oikonomou, “Constant-roll $k$-Inflation Dynamics,” Class. Quant. Grav. 37, no.2, 025003 (2020) [arXiv:1912.00475 [gr-qc]].
  * (29) F. L. Bezrukov and M. Shaposhnikov, “The Standard Model Higgs boson as the inflaton,” Phys. Lett. B 659, 703-706 (2008) [arXiv:0710.3755 [hep-th]].
  * (30) J. L. F. Barbon and J. R. Espinosa, “On the Naturalness of Higgs Inflation,” Phys. Rev. D 79, 081302 (2009) [arXiv:0903.0355 [hep-ph]].
  * (31) F. Bezrukov, A. Magnin, M. Shaposhnikov and S. Sibiryakov, “Higgs inflation: consistency and generalisations,” JHEP 01, 016 (2011) [arXiv:1008.5157 [hep-ph]].
  * (32) C. Germani and A. Kehagias, “New Model of Inflation with Non-minimal Derivative Coupling of Standard Model Higgs Boson to Gravity,” Phys. Rev. Lett. 105, 011302 (2010) [arXiv:1003.2635 [hep-ph]].
  * (33) I. Dymnikova and M. Khlopov, “Decay of cosmological constant in selfconsistent inflation,” Eur. Phys. J. C 20, 139-146 (2001)
  * (34) V. K. Oikonomou, “Kinetic axion F(R) gravity inflation,” Phys. Rev. D 106, no.4, 044041 (2022) [arXiv:2208.05544 [gr-qc]].
  * (35) B. A. Bassett, S. Tsujikawa and D. Wands, “Inflation dynamics and reheating,” Rev. Mod. Phys. 78, 537-589 (2006) [arXiv:astro-ph/0507632 [astro-ph]].
  * (36) R. Allahverdi, R. Brandenberger, F. Y. Cyr-Racine and A. Mazumdar, “Reheating in Inflationary Cosmology: Theory and Applications,” Ann. Rev. Nucl. Part. Sci. 60, 27-51 (2010) [arXiv:1001.2600 [hep-th]].
  * (37) M. A. Amin, M. P. Hertzberg, D. I. Kaiser and J. Karouby, “Nonperturbative Dynamics Of Reheating After Inflation: A Review,” Int. J. Mod. Phys. D 24, 1530003 (2014) [arXiv:1410.3808 [hep-ph]].
  * (38) M. Tanabashi et al. [Particle Data Group], “Review of Particle Physics,” Phys. Rev. D 98, no.3, 030001 (2018)
  * (39) A. Sen, “Tachyon dynamics in open string theory,” Int. J. Mod. Phys. A 20, 5513-5656 (2005) [arXiv:hep-th/0410103 [hep-th]].
  * (40) A. Sen, “Remarks on tachyon driven cosmology,” Phys. Scripta T 117, 70-75 (2005) [arXiv:hep-th/0312153 [hep-th]].
  * (41) C. Li, L. Wang and Y. K. E. Cheung, “Bound to bounce: A coupled scalar–tachyon model for a smooth bouncing/cyclic universe,” Phys. Dark Univ. 3, 18-33 (2014) [arXiv:1101.0202 [gr-qc]].
  * (42) C. Li and Y. K. E. Cheung, “The scale invariant power spectrum of the primordial curvature perturbations from the coupled scalar tachyon bounce cosmos,” JCAP 07, 008 (2014) [arXiv:1401.0094 [gr-qc]].
  * (43) N. Zhang and Y. K. E. Cheung, “Primordial Gravitational Waves Spectrum in the Coupled-Scalar-Tachyon Bounce Universe,” Eur. Phys. J. C 80, no.2, 100 (2020) [arXiv:1901.06423 [gr-qc]].
  * (44) G. W. Horndeski, “Second-order scalar-tensor field equations in a four-dimensional space,” Int. J. Theor. Phys. 10, 363-384 (1974)
  * (45) A. Nicolis, R. Rattazzi and E. Trincherini, “The Galileon as a local modification of gravity,” Phys. Rev. D 79, 064036 (2009) [arXiv:0811.2197 [hep-th]].
  * (46) C. Deffayet, G. Esposito-Farese and A. Vikman, “Covariant Galileon,” Phys. Rev. D 79, 084003 (2009) [arXiv:0901.1314 [hep-th]].
  * (47) C. Deffayet, S. Deser and G. Esposito-Farese, “Generalized Galileons: All scalar models whose curved background extensions maintain second-order field equations and stress-tensors,” Phys. Rev. D 80, 064015 (2009) [arXiv:0906.1967 [gr-qc]].
  * (48) C. Deffayet, X. Gao, D. A. Steer and G. Zahariade, “From k-essence to generalised Galileons,” Phys. Rev. D 84, 064039 (2011) [arXiv:1103.3260 [hep-th]].
  * (49) T. Kobayashi, M. Yamaguchi and J. Yokoyama, “Generalized G-inflation: Inflation with the most general second-order field equations,” Prog. Theor. Phys. 126, 511-529 (2011) [arXiv:1105.5723 [hep-th]].
  * (50) C. Charmousis, E. J. Copeland, A. Padilla and P. M. Saffin, “General second order scalar-tensor theory, self tuning, and the Fab Four,” Phys. Rev. Lett. 108, 051101 (2012) [arXiv:1106.2000 [hep-th]].
  * (51) J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, “New Class of Consistent Scalar-Tensor Theories(Healthy theories beyond Horndeski),” Phys. Rev. Lett. 114, no.21, 211101 (2015) [arXiv:1404.6495 [hep-th]].
  * (52) J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui and G. Tasinato, “Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order,” JHEP 12, 100 (2016) [arXiv:1608.08135 [hep-th]].
  * (53) M. Crisostomi, M. Hull, K. Koyama and G. Tasinato, “Horndeski: beyond, or not beyond?,” JCAP 03, 038 (2016) [arXiv:1601.04658 [hep-th]].
  * (54) C. de Rham and A. Matas, “Ostrogradsky in Theories with Multiple Fields,” JCAP 06, 041 (2016) [arXiv:1604.08638 [hep-th]].
  * (55) T. Kobayashi, “Horndeski theory and beyond: a review,” Rept. Prog. Phys. 82, no.8, 086901 (2019) [arXiv:1901.07183 [gr-qc]].
  * (56) S. Dodelson, “Modern Cosmology,” Academic Press, 2003,
  * (57) C. Li, “Perturbative resonance in WIMP paradigm and its cosmological implications on cosmic reheating and primordial gravitational wave detection,” Phys. Dark Univ. 38, 101129 (2022) [arXiv:1902.08963 [astro-ph.CO]].
  * (58) S. Nojiri, S. D. Odintsov and V. K. Oikonomou, “Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution,” Phys. Rept. 692, 1-104 (2017) [arXiv:1705.11098 [gr-qc]].

##  Appendix A The e-folding number

Using Eq.(24), we obtain the e-folding number of KS inflation,

$N_{e}=\ln\left(\frac{a_{r}}{a_{f}}\right)=\frac{1}{2\sqrt{3}}\frac{m_{\varphi}^{2}}{\sqrt{\lambda}M_{p}}\Delta
t~{},$ (36)

where $\Delta t\equiv t_{r}-t_{f}$ being the duration of KS inflation. On the
other hand, the change of $\varphi$ during KS inflation takes

$\Delta\varphi=\varphi_{v}-\varphi_{i}\simeq\varphi_{v}=m_{\varphi}/\sqrt{2\lambda},$
(37)

where $\varphi_{i}$ being the value of $\varphi$ at the onset of KS inflation,
and $\varphi_{v}$ being the expectation value of $\varphi$ at the true vacuum.

According to our analysis and numerical result (Figure 3), during KS
inflation, the dynamic attractor locks $\varphi$ at a small velocity
$\dot{\varphi}(t)\ll m_{\phi}^{2}$ rather than exponentially increasing.
Before the tachyon fully condenses, $\varphi$ can freely roll along the
potential and accelerates to $\dot{\varphi}_{i}$. After that, the fully
condensed tachyon matter locks $\varphi$ around $\dot{\varphi}_{i}$,
performing a quasi-oscillating rolling, as shown in Figure 3. In this work, we
approximate $\varphi(t)$ to be $\dot{\varphi}_{i}$ for simplicity. Thus we
have

$\Delta\varphi=\int_{t_{i}}^{t_{r}}\dot{\varphi}(t)dt\simeq\dot{\varphi}_{i}\Delta
t=\dot{\varphi}_{i}\times
2\sqrt{3}\frac{\sqrt{\lambda}M_{p}}{m_{\varphi}^{2}}N_{e}=\frac{m_{\varphi}}{\sqrt{2\lambda}}~{},$
(38)

which leads to

$N_{e}=\frac{1}{2\sqrt{6}}\frac{1}{\lambda}\frac{m_{\varphi}}{M_{p}}\frac{m_{\varphi}^{2}}{\dot{\varphi}_{i}}~{},$
(39)

where Eq.(36) being used.

Substituting the locking condition $\ddot{\varphi}_{i}=0$ into the equation of
motion of $\varphi$ during the annihilating D-brane and anti-D-brane pair
dominated inflation phase (Phase II),

$\ddot{\varphi}+3H\dot{\varphi}+\frac{m_{s}^{4}}{V_{0}}\left(-2m_{\varphi}^{2}\varphi\right)=0~{},$
(40)

we obtain

$\dot{\varphi}_{i}\simeq\frac{2}{3}\frac{m_{\varphi}^{2}m_{s}^{4}}{HV_{0}}\varphi=\frac{2}{\sqrt{3}}\frac{m_{\varphi}^{2}}{\sqrt{m_{s}M_{p}}}\varphi_{i}~{},$
(41)

which is the same order of Eq.(18). Substituting Eq.(41) into Eq.(39), we
obtain

$N_{e}=\frac{1}{4\sqrt{2}}\frac{1}{\lambda}\sqrt{\frac{m_{s}}{M_{p}}}\left(\frac{m_{\varphi}}{\varphi_{i}}\right)~{},$
(42)

which indicates that by fine-tuning the initial value of $\varphi$, one can
obtain an arbitrarily large e-folding number for KS inflation
($N_{e}\propto\varphi_{i}^{-1}$). However, in generic, we expect the initial
value of $\varphi$ is the same order of its mass,

$\varphi_{i}\simeq m_{\varphi}.$ (43)

Substituting Eq.(43) into Eq.(42), we obtain

$N_{e}=\frac{1}{4\sqrt{2}}\frac{1}{\lambda}\sqrt{\frac{m_{s}}{M_{p}}}~{}.$
(44)

Eq.(44) reflects that a small $\lambda$ corresponds to a large
$\rho_{\varphi}$ and a large $\varphi_{v}$, suggests a large e-folding number
for KS inflation.

##  Appendix B The equation of motion of primordial curvature perturbation
during KS inflation

In this section, we use the perturbative Einstein equation

$\delta G_{\nu}^{\mu}=8\pi G\delta T_{\nu}^{\mu}$ (45)

to derive the equation of motion of primordial curvature perturbation (PCP)
during the KS inflation,

$\ddot{\Phi}_{k}+\left(H-\frac{\ddot{H}}{\dot{H}}\right)\dot{\Phi}_{k}+\left(2\dot{H}-H\frac{\ddot{H}}{\dot{H}}\right)\Phi_{k}+\frac{c_{s}^{2}k^{2}}{a^{2}}\Phi_{k}=0~{},$
(46)

where $c_{s}^{2}=\left(1-\dot{\mathcal{T}}^{2}-\dot{\varphi}^{2}\right)$ being
the sound speed of coupled tachyon matter, $\Phi_{k}$ being the Fourier modes
of metric perturbation $\Phi(x,t)$, and $\Psi=-\Phi$ being used.

Using FLRW metric (Eq.(26)), we obtain the terms on the left-hand side (LHS)
of Eq.(45)

$\displaystyle\delta
G^{0}_{0}=-2\frac{\nabla^{2}}{a^{2}}\Phi+6H(\dot{\Phi}+H\Phi),$ (47)
$\displaystyle\delta G^{0}_{i}=-2\partial_{i}(\dot{\Phi}+H\Phi),$ (48)
$\displaystyle\delta G^{i}_{j}=0,$ (49) $\displaystyle\delta
G^{i}_{i}=2\ddot{\Phi}+8H\dot{\Phi}+2(2\dot{H}+3H^{2})\Phi~{}.$ (50)

Expanding the right-hand side (RHS) term of Eq.(9) to the first order and
using $\delta T_{0}^{0}=-\delta\rho$, $\delta T_{i}^{0}=-\partial_{i}\delta
q$, $\delta T_{i}^{j}=\partial_{i}\partial^{j}\delta s$ and $\delta
T_{i}^{i}=\delta p$, respectively, we obtain

$\displaystyle\delta\rho=\rho_{\mathcal{T}}\left(\frac{\dot{\varphi}\delta\dot{\varphi}+\dot{\mathcal{T}}\delta\dot{\mathcal{T}}-\left(\dot{\varphi}^{2}+\dot{\mathcal{T}}^{2}\right)\Phi}{1-\dot{\mathcal{T}}^{2}-\dot{\varphi}^{2}}+\frac{d\ln
V(\mathcal{T})}{d\mathcal{T}}\delta\mathcal{T}\right)+\frac{dV(\varphi)}{d\varphi}\delta\varphi~{},$
(53) $\displaystyle\delta
q=\rho_{\mathcal{T}}\left(\dot{\mathcal{T}}\delta\mathcal{T}+\dot{\varphi}\delta\varphi\right)~{},$
$\displaystyle\delta s=0~{},$ $\displaystyle\delta
p=p_{\mathcal{T}}\left(-\frac{\dot{\varphi}\delta\dot{\varphi}+\dot{\mathcal{T}}\delta\dot{\mathcal{T}}-\left(\dot{\varphi}^{2}+\dot{\mathcal{T}}^{2}\right)\Phi}{1-\dot{\mathcal{T}}^{2}-\dot{\varphi}^{2}}+\frac{d\ln
V(\mathcal{T})}{d\mathcal{T}}\delta\mathcal{T}\right)-\frac{dV(\varphi)}{d\varphi}\delta\varphi$
(54)

where $\rho_{\mathcal{T}}$ and $p_{\mathcal{T}}$ are the background energy
density and pressure of the coupled tachyon matter, respectively.

Using Eq.(53), Eq.(47), Eq.(50), Eq.(53), and Eq.(54), we have

$\displaystyle\frac{1}{8\pi G}\left(\delta G_{i}^{i}+c_{s}^{2}\delta
G_{0}^{0}\right)$ $\displaystyle=2\frac{d\ln
V(\mathcal{T})}{\dot{\mathcal{T}}d\mathcal{T}}\left(-c_{s}^{2}\right)\delta
q-2\frac{\ln
V(\mathcal{T})}{d\mathcal{T}}p_{\mathcal{T}}\frac{\dot{\varphi}}{\dot{\mathcal{T}}}\delta\varphi-\frac{dV(\varphi)}{d\varphi}\delta\varphi~{},$
(55) $\displaystyle=2\frac{d\ln
V(\mathcal{T})}{\dot{\mathcal{T}}d\mathcal{T}}\left(-c_{s}^{2}\right)\delta
q-\frac{\ln
V(\mathcal{T})}{d\mathcal{T}}p_{\mathcal{T}}\frac{\dot{\varphi}}{\dot{\mathcal{T}}}\delta\varphi~{}~{},$
(56) $\displaystyle=2\frac{d\ln
V(\mathcal{T})}{\dot{\mathcal{T}}d\mathcal{T}}\left(-c_{s}^{2}\right)\delta
q~{}.$ (57)

In the second line, we have used $-\frac{dV(\varphi)}{d\varphi}=\frac{d\ln
V(\mathcal{T})}{d\mathcal{T}}p_{\mathcal{T}}\frac{\dot{\varphi}}{\dot{\mathcal{T}}}$,
which is derived from the background part of Eq.(10) and Eq.(11) during the KS
inflation ($\ddot{T}\simeq 0$ and $\ddot{\varphi}\simeq 0$). In the third
line, we further neglect the term $\frac{\ln
V(\mathcal{T})}{d\mathcal{T}}p_{\mathcal{T}}\frac{\dot{\varphi}}{\dot{\mathcal{T}}}\delta\varphi$
as it is suppressed by the factor $\dot{\varphi}/\dot{\mathcal{T}}\ll
m_{\varphi}^{2}/m_{s}^{2}\simeq 0$.

Substituting Eq.(47), Eq.(48), Eq.(50) and Eq.(53) into Eq.(57), we obtain

$\displaystyle\ddot{\Phi}+\alpha\dot{\Phi}+\beta\Phi-
c_{s}^{2}\frac{\nabla^{2}}{a^{2}}\Phi=0~{},$ (58)

where

$\displaystyle\alpha$ $\displaystyle\equiv H+(3H+3Hc_{s}^{2})+2\frac{d\ln
V(\mathcal{T})}{\dot{\mathcal{T}}d\mathcal{T}}c_{s}^{2}\simeq H+3H~{};$ (59)
$\displaystyle\beta$ $\displaystyle\equiv
2\dot{H}+H\left(3H+3Hc_{s}^{2}+2\frac{d\ln
V(\mathcal{T})}{\dot{\mathcal{T}}d\mathcal{T}}c_{s}^{2}\right)\simeq
2\dot{H}+H\cdot 3H~{}.$ (60)

Submitting

$\displaystyle\frac{\ddot{H}}{\dot{H}}=\frac{\dot{\rho}+\dot{p}}{\rho+p}=\frac{\dot{\rho}_{\mathcal{T}}+\dot{p}_{\mathcal{T}}}{\rho_{\mathcal{T}}+p_{\mathcal{T}}}\simeq\frac{\dot{\rho}_{\mathcal{T}}}{\rho_{\mathcal{T}}}=-3H~{},$
(61)

into Eq.(58), we obtain

$\displaystyle\ddot{\Phi}+\left(H-\frac{\ddot{H}}{\dot{H}}\right)\dot{\Phi}+\left(2\dot{H}-H\frac{\ddot{H}}{\dot{H}}\right)\Phi-
c_{s}^{2}\frac{\nabla^{2}}{a^{2}}\Phi=0~{},$ (62)

which becomes Eq.(27) after Fourier transformation.

Note that in the standard single field slow-roll inflation model
($\mathcal{L}=-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi+V(\phi)$), one
has $c_{s}^{2}(\phi)=1$ and

$\displaystyle\alpha_{\phi}$
$\displaystyle=H+(3H+3Hc_{s}^{2}(\phi))+2\frac{dV(\mathcal{\phi})}{\dot{\phi}d\phi}c_{s}^{2}(\phi)=H+6H+2\frac{dV(\mathcal{\phi})}{\dot{\phi}d\phi}~{};$
(63) $\displaystyle\beta_{\phi}$
$\displaystyle=2\dot{H}+H\left(3H+3Hc_{s}^{2}(\phi)+2\frac{dV(\mathcal{T})}{\dot{\phi}d\phi}c_{s}^{2}(\phi)\right)=2\dot{H}+H\cdot\left(6H+2\frac{dV(\mathcal{\phi})}{\dot{\phi}d\phi}\right).$
(64)

for Eq.(58) Mukhanov:1990me . Using

$\frac{\ddot{H}}{\dot{H}}=-6H-2\frac{dV(\mathcal{\phi})}{\dot{\phi}d\phi},$
(65)

derived in the standard inflation Mukhanov:1990me , one can re-obtain

$\displaystyle\ddot{\Phi}+\left(H-\frac{\ddot{H}}{\dot{H}}\right)\dot{\Phi}+\left(2\dot{H}-H\frac{\ddot{H}}{\dot{H}}\right)\Phi-\frac{\nabla^{2}}{a^{2}}\Phi=0~{},$
(66)

which leads to the desired nearly scale-invariant perturbation power spectrum.

Comparing Eq.(61) and Eq.(65), we find that, although $\ddot{H}/\dot{H}$ are
different between the standard inflation model and the newly proposed KS
model, Eq.(62) and Eq.(66) are the same for them, expect $c_{s}^{2}\ll 1$ for
KS inflation. This observation can explain why KS inflation also produces a
nearly scale-invariant PCP from a mathematical viewpoint.