# Intermittent null energy condition violations during inflation and
primordial gravitational waves

Yong<EMAIL_ADDRESS>Yun-Song<EMAIL_ADDRESS>1
School of Physics and Microelectronics, Zhengzhou University, Zhengzhou, Henan
450001, China  2 School of Physics, University of Chinese Academy of Sciences,
Beijing 100049, China  3 School of Fundamental Physics and Mathematical
Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China
4 International Center for Theoretical Physics Asia-Pacific, Beijing/Hangzhou,
China  5 Institute of Theoretical Physics, Chinese Academy of Sciences, P.O.
Box 2735, Beijing 100190, China

###### Abstract

Primordial null energy condition (NEC) violation would imprint a blue-tilted
spectrum on gravitational wave background (GWB). However, its implications on
the GWB might be far richer than expected. We present a scenario, in which
after a slow-roll (NEC-preserving) inflation with Hubble parameter $H\simeq
H_{inf1}$, the Universe goes through an NEC-violating period and then enters
subsequent slow-roll inflation with a higher $H$ ($=H_{inf2}\gg H_{inf1}$).
The resulting primordial gravitational wave spectrum is nearly flat at the
cosmic microwave background band, as well as at the frequency $f\sim 1/{\rm
yr}$ but with higher amplitude (compatible with the recent NANOGrav result).
It is also highlighted that for the multi-stage inflation if the NEC
violations happened intermittently, we might have a Great Wall-like spectrum
of the stochastic GWB at the corresponding frequency band.

###### Contents

  1. I Introduction
  2. II Our scenario
    1. II.1 Intermittent NEC violation during inflation
    2. II.2 Primordial GW spectrum
    3. II.3 Primordial GWB at low-frequency band
    4. II.4 Multi-stage inflation with NEC violations
  3. III Conclusion
  4. A On stability of scalar perturbations

##  I Introduction

The primordial gravitational wave background (GWB) Starobinsky:1979ty ;
Rubakov:1982df  with a broad frequency-band ($10^{-18}-10^{10}$ Hz) carries
rich information about the early Universe. It is usually thought that its
detection will not only solidify our confidence in inflation, but also offer
us an unparalleled probe to the physics related to the cosmological
(non)singularity, in which the null energy condition (NEC) violation might
play a significant role Rubakov:2014jja ; Libanov:2016kfc ; Kobayashi:2016xpl
; Ijjas:2016vtq ; Dobre:2017pnt ; Cai:2016thi ; Creminelli:2016zwa ;
Cai:2017tku ; Cai:2017dyi ; Kolevatov:2017voe ; Ye:2019sth , and the UV-
complete gravity theory.

The primordial gravitational waves (GWs) at the ultra-low frequency band
($10^{-18}-10^{-16}$ Hz) would induce the B-mode polarization in the cosmic
microwave background (CMB). The search for the primordial GWs with CMB has
been still in progress. The Pulsar Timing Array (PTA) experiments focus on GWB
at frequencies $f\sim 1/{\rm yr}$ ($\sim 10^{-8}$ Hz). Recently, based on the
$12.5$-yr data analysis, the NANOGrav Collaboration reported evidence for a
stochastic common-spectrum process Arzoumanian:2020vkk , which might be
interpreted as a stochastic GWB with a spectrum tilt $-1.5\lesssim
n_{T}\lesssim 0.5$, see Vagnozzi:2020gtf ; Li:2020cjj ; Tahara:2020fmn ;
Kuroyanagi:2020sfw  for the implications of NANOGrav’s result in inflation.
The current bound on GWB at CMB band indicates a tensor-to-scalar ratio
$r\lesssim 0.06$ Ade:2018gkx . Therefore, only if the primordial GWs have a
blue-tilted spectrum, it is able to be detected by the experiments and
detectors at other frequency bands.

It is well-known that for inflation, if initially the GW modes sit in the
Bunch-Davis state (or e.g., Li:2020cjj ; Ashoorioon:2014nta ), a blue-tilted
spectrum suggests that the corresponding inflation is inevitably NEC-
violating, i.e., $T_{\mu\nu}n^{\mu}n^{\nu}<0$, which corresponds to
${\dot{H}}>0$, namely, super-inflation333In this paper, the term super-
inflation is used for the accelerated expansion in which
$\epsilon=-\dot{H}/H^{2}<0$, since $\ddot{a}/a=H^{2}(1-\epsilon)$. Piao:2003ty
; Piao:2004tq ; Baldi:2005gk ; Piao:2006jz . The NEC-violating inflation may
be performed stably with the Galileon theory Kobayashi:2010cm ;
Kobayashi:2011nu  and the effective field theory (EFT) of inflation
Creminelli:2006xe ; Capurri:2020qgz . If initially the NEC is violated
drastically (${\dot{H}}\gg H^{2}$), it is also possible that our Universe is
asymptotically Minkowskian and slowly expanding in infinite past Piao:2003ty .
In such scenarios, the hot “big bang” evolution or inflation starts after the
end of the slow expansion or Genesis Creminelli:2010ba ; Liu:2011ns ;
Wang:2012bq ; Liu:2012ww ; Creminelli:2012my ; Hinterbichler:2012fr ;
Hinterbichler:2012yn ; Liu:2014tda ; Pirtskhalava:2014esa ; Nishi:2015pta ;
Kobayashi:2015gga ; Cai:2016gjd ; Nishi:2016ljg ; Mironov:2019qjt ;
Ageeva:2020gti ; Ilyas:2020zcb . Based on the beyond-Horndeski EFT (see e.g.
Langlois:2018dxi ; Kobayashi:2019hrl  for reviews), the Genesis could be
implemented without pathologies (including instabilities and superluminality)
Cai:2016thi ; Creminelli:2016zwa ; Cai:2017tku ; Cai:2017dyi ;
Kolevatov:2017voe ; Ye:2019sth .

Inspired by current (and upcoming) experiments searching for GWB, it is
significant to resurvey the imprints of NEC violation on stochastic GWB.
Recently, a scenario in which the super-inflation is followed by a slow-roll
(NEC-preserving) inflation has been proposed in Tahara:2020fmn , which yields
a large stochastic GWB with $n_{T}\simeq 0.9$ at the PTA band. But the
spectrum of scalar perturbations, i.e., $P_{s}$, is highly blue-tilted too,
since $n_{s}-1\simeq n_{T}$. Consequently, other fields must be responsible
for the density perturbation at the CMB band. However, it is possible that a
slow-roll (NEC-preserving) inflation with $H=H_{inf1}$, which results in
$P_{s}\sim H^{2}_{inf1}/\epsilon\sim 10^{-9}$ at the CMB band, happened before
the NEC-violating phase, which is subsequently followed by a slow-roll
inflation with a higher scale $H_{inf2}\gg H_{inf1}$. In Refs. Liu:2013xt ;
Alberte:2016izw , such a low-scale inflation is regarded as current
accelerated expansion with $H^{2}_{inf1}\sim\Lambda$. The consistent joint of
an NEC-preserving spacetime to an NEC-violating phase is also explored in Ref.
Rubakov:2013kaa , see also Elder:2013gya .

In this paper, we investigate the possibility of a short NEC violation during
the NEC-preserving inflation. In this scenario, the low-scale inflation prior
to NEC violation is responsible for the density perturbation on large scales.
We calculate the corresponding primordial GW spectrum. Specially, we
highlighted that for the multi-stage (NEC-preserving) inflation, if the NEC
violations happened intermittently, a Great Wall-like landscape of primordial
GW spectrum at the full frequency-band will present.

##  II Our scenario

###  II.1 Intermittent NEC violation during inflation

In our scenario (see Fig. 1), initially the field $\phi$ (canonical scalar
field) slowly rolls down a nearly flat potential, i.e., ${\dot{\phi}}^{2}\ll
V(\phi)\approx V_{inf1}$, which results in the slow-roll (NEC-preserving)
inflation. In the NEC-violating phase, $\phi$ climbs up the potential rapidly
so that ${\dot{H}}>0$. After $\phi$ arrives at another nearly flat region of
the potential but with higher energy, i.e., $V_{inf2}\gg V_{inf1}$, the slow-
roll inflation restarts again.

We present a model as follows:

$\displaystyle S=\int d^{4}x\sqrt{-g}\Big{[}{M_{p}^{2}\over
2}R-M_{p}^{2}g_{1}(\phi)X/2+g_{2}(\phi)X^{2}/4-M_{p}^{4}V(\phi)\Big{]}\,,$ (1)

where $X=\nabla_{\mu}\phi\nabla^{\mu}\phi$. Here, the Galileon operator
$\Box\phi=\nabla_{\mu}\nabla^{\mu}\phi$ is not required,444See e.g., Ref.
Tahara:2020fmn , for the case in which $X\Box\phi$ instead of $X^{2}$ is used
to realize the NEC violation. see also e.g. Cai:2017dyi ; Buchbinder:2007ad ;
Koehn:2015vvy . The corresponding background equations are

$\displaystyle
3H^{2}M_{p}^{2}=\frac{M_{p}^{2}}{2}g_{1}\dot{\phi}^{2}+\frac{3}{4}g_{2}\dot{\phi}^{4}+M_{p}^{4}V\,,$
(2)
$\displaystyle\dot{H}M_{p}^{2}=-\frac{M_{p}^{2}}{2}g_{1}\dot{\phi}^{2}-\frac{1}{2}g_{2}\dot{\phi}^{4}\,,$
(3) $\displaystyle
0=\left(g_{1}+\frac{3g_{2}\dot{\phi}^{2}}{M_{p}^{2}}\right)\ddot{\phi}+3g_{1}H\dot{\phi}+\frac{1}{2}g_{1,\phi}\dot{\phi}^{2}+\frac{3g_{2}H\dot{\phi}^{3}}{M_{p}^{2}}+\frac{3g_{2,\phi}\dot{\phi}^{4}}{4M_{p}^{2}}+M_{p}^{2}V_{,\phi}\,,$
(4)

where “${}_{,\phi}=d/d\phi$.” Only two of Eqs. (2)-(4) are independent.

In the NEC-preserving regimes, we require $g_{1}(\phi)=1$, $g_{2}(\phi)=0$ and
the potential is nearly flat (see Fig. 1), so that the scalar filed $\phi$ is
canonical and the slow-roll inflation ($0<\epsilon=-{\dot{H}}/H^{2}\ll 1$) can
happen. In Fig. 1, $V_{inf1}\simeq m^{2}\phi^{2}/2$,
$V_{inf2}\simeq\lambda[1-\frac{(\phi-\phi_{1})^{2}}{\sigma^{2}}]^{2}$ and
$V_{inf2}\gg V_{inf1}$.

In the NEC-violating regime, we set $g_{1}(\phi)\approx-{f_{1}e^{2\phi}\over
1+f_{1}e^{2\phi}}<0$ and $g_{2}(\phi)=f_{2}$ with $f_{1,2}$ being
dimensionless constants. The coefficient of $\ddot{\phi}$ in Eq. (4), i.e.,
$g_{1}+3g_{2}\dot{\phi}^{2}/M_{p}^{2}$, is positive throughout so that there
is no ghost instability, see Appendix A. The scalar field $\phi$ will climb up
the potential rapidly ($H\ll\dot{\phi}<M_{p}$) and arrive at the flat region
$V=V_{inf2}$, as long as the condition
$\frac{1}{2}g_{1,\phi}\dot{\phi}^{2}+M_{p}^{2}V_{,\phi}<0$ lasts for
sufficiently long time. We require $\epsilon\ll-1$, i.e., $H^{2}\ll{\dot{H}}$.
According to Eq. (2), considering ${\dot{\phi}}^{2}\gg V,\,H^{2}$, we have
$\frac{M_{p}^{2}}{2}g_{1}\dot{\phi}^{2}+\frac{3}{4}g_{2}\dot{\phi}^{4}\approx
0$, which suggests $e^{2\phi}\sim\dot{\phi}^{2}$ for $\phi<0$. Thus,
$\dot{\phi}$ is approximately

$\dot{\phi}\simeq{1\over(t_{*}-t)}\,,\quad t<t_{*}\,.$ (5)

According to Eq. (3), we have $\dot{H}\sim\dot{\phi}^{4}$; hence,

$H\sim{1\over(t_{*}-t)^{3}}+{\rm const..}$ (6)

When $t\ll t_{*}$, we have $H\simeq{\rm const.}=H_{inf1}$, which suggests that
the NEC-violating phase has the chance to start after a slow-roll inflation.

As a phenomenological example, we set

$\displaystyle g_{1}(\phi)$ $\displaystyle=$ $\displaystyle{1\over
1+e^{q_{2}(\phi-\phi_{3})}}-{f_{1}e^{2\phi}\over 1+f_{1}e^{2\phi}}+{2\over
1+e^{-q_{1}(\phi-\phi_{0})}}\,,$ (7) $\displaystyle g_{2}(\phi)$
$\displaystyle=$ $\displaystyle{f_{2}\over 1+e^{-q_{2}(\phi-\phi_{3})}}{1\over
1+e^{q_{3}(\phi-\phi_{0})}}\,,$ (8) $\displaystyle V(\phi)$ $\displaystyle=$
$\displaystyle{1\over 2}m^{2}\phi^{2}{1\over
1+e^{q_{2}(\phi-\phi_{2})}}+\lambda\left[1-\frac{(\phi-\phi_{1})^{2}}{\sigma^{2}}\right]^{2}{1\over
1+e^{-q_{4}(\phi-\phi_{1})}}\,,$ (9)

where $\lambda$, $m$, $f_{1,2}$ and $q_{1,2,3,4}$ are positive constants. We
require that $\phi_{3}<\phi_{2}<0<\phi_{1}<\phi_{0}$. Here, for
$\phi\ll\phi_{3}$, we have $g_{1}=1$, $g_{2}=0$ and $V=V_{inf1}\simeq
m^{2}\phi^{2}/2$, while for $\phi\gg\phi_{0}$, we have $g_{1}=1$, $g_{2}=0$
and $V=V_{inf2}\simeq\lambda[1-\frac{(\phi-\phi_{1})^{2}}{\sigma^{2}}]^{2}$.

We solve Eqs. (3) and (4) numerically. The initial value of $H$ is set as
$H_{\rm ini}\simeq H_{inf1}=1.29\times 10^{-5}M_{p}$ at $t=t_{\rm ini}=0$, so
that the “$inf1$” is responsible for the scalar perturbations on the CMB band,
which indicates that
$P_{s}\simeq\frac{1}{2M_{p}^{2}\epsilon_{inf1}}\left(\frac{H_{inf1}}{2\pi}\right)^{2}\approx
2.1\times 10^{-9}$ for $\epsilon_{inf1}=0.001$.

We plot the evolutions of $\phi$ and $\dot{\phi}$ in Fig. 2. We can see that
in the slow-roll (NEC-preserving) regimes, ${\dot{\phi}}\ll H$, the field
$\phi$ rolls slowly. In the NEC-violating regime, $H\ll{\dot{\phi}}<M_{p}$, so
that the field can rapidly climb up the potential $V=V_{inf2}$. We plot the
evolutions of $H$ and $\epsilon$ in Fig. 3. During the slow-roll (NEC-
preserving) phases ($0<\epsilon\ll 1$), we have $H\simeq{\rm const.}$, which
is intervened by an NEC-violating phase (${\dot{H}}>0$ and $\epsilon\ll-1$).
Due to the NEC-violating evolution, we have

${H_{inf2}/H_{inf1}}\simeq 10^{3}\gg 1.$ (10)

Figure 1: A sketch of our scenario: a slow-roll (NEC-preserving) inflation
occurred before the NEC-violating phase, which is subsequently followed by the
slow-roll (NEC-preserving) inflation with a higher scale $H_{inf2}\gg
H_{inf1}$. The potential $V(\phi)$ given by Eq. (9) is plotted with
logarithmic coordinates on the vertical axis.

(a)

(b)

Figure 2: Left: the evolution of $\phi$ with respect to $t$. Here, $\phi_{\rm
bottom}$ and $\phi_{\rm top}$ correspond to the bottom and top of the
potential in Fig. 1, respectively. Right: the evolution of $\dot{\phi}/M_{p}$
and $\dot{\phi}/H$. During the NEC-violating phase, which approximately
corresponds to $730084<t<730328$, $H\ll\dot{\phi}<M_{p}$ is satisfied. We set
$\phi(t_{\rm ini})=-7$, $\dot{\phi}(t_{\rm ini})=0$, $t_{\rm ini}=0$,
$\phi_{0}=3.2$, $\phi_{1}=2$, $\phi_{2}=-4$, $\phi_{3}=-4.38$, $q_{1}=10$,
$q_{2}=6$, $q_{3}=10$, $q_{4}=4$, $f_{1}=1$, $f_{2}=40$, $\lambda=0.01$,
$\sigma=23$ and $m=-4.5\times 10^{-6}$.

(a)

(b)

Figure 3: Left: the evolution of $H$. Right: the evolution of
$\epsilon=-\dot{H}/H^{2}$. We have $0<\epsilon\ll 1$ during slow-roll
inflations and $\epsilon\ll-1$ during NEC-violating super-inflation.

###  II.2 Primordial GW spectrum

In this subsection, we calculate the spectrum of primordial GWs. Generally,
for the tensor perturbation $\gamma_{ij}$, we have555Here, the propagating
speed of GWs is $c_{T}=1$ (or see e.g., Cai:2015yza ; Cai:2016ldn ;
Giovannini:2015kfa ; Giovannini:2018zbf ; Giare:2020vss ; Giare:2020plo ).

$S^{(2)}_{\gamma}={M_{p}^{2}\over 8}\int
d^{4}xa^{3}\left[\dot{\gamma}_{ij}^{2}-{(\partial_{k}\gamma_{ij})^{2}\over
a^{2}}\right]\,.$ (11)

In the momentum space, we have

$\gamma_{ij}(\tau,\mathbf{x})=\int\frac{d^{3}k}{(2\pi)^{3}}e^{-i\mathbf{k}\cdot\mathbf{x}}\sum_{\lambda=+,\times}\hat{\gamma}_{\lambda}(\tau,\mathbf{k})\epsilon^{(\lambda)}_{ij}(\mathbf{k}),$
(12)

where
$\hat{\gamma}_{\lambda}(\tau,\mathbf{k})=\gamma_{\lambda}(\tau,k)a_{\lambda}(\mathbf{k})+\gamma_{\lambda}^{*}(\tau,-k)a_{\lambda}^{{\dagger}}(-\mathbf{k})$,
$\epsilon_{ij}^{(\lambda)}(\mathbf{k})$ satisfies
$k_{j}\epsilon_{ij}^{(\lambda)}(\mathbf{k})=0$,
$\epsilon_{ii}^{(\lambda)}(\mathbf{k})=0$,
$\epsilon_{ij}^{(\lambda)}(\mathbf{k})\epsilon_{ij}^{*(\lambda^{\prime})}(\mathbf{k})=\delta_{\lambda\lambda^{\prime}}$
and
$\epsilon_{ij}^{*(\lambda)}(\mathbf{k})=\epsilon_{ij}^{(\lambda)}(-\mathbf{k})$;
$a_{\lambda}(\mathbf{k})$ and $a^{{\dagger}}_{\lambda}(\mathbf{k}^{\prime})$
satisfy
$[a_{\lambda}(\mathbf{k}),a_{\lambda^{\prime}}^{{\dagger}}(\mathbf{k}^{\prime})]=\delta_{\lambda\lambda^{\prime}}\delta^{(3)}(\mathbf{k}-\mathbf{k}^{\prime})$.
The equation of motion for $\gamma_{\lambda}(\tau,k)$ is

$\frac{d^{2}u_{k}}{d\tau^{2}}+\left(k^{2}-\frac{a^{\prime\prime}}{a}\right)u_{k}=0\,,$
(13)

where $u_{k}=\gamma_{\lambda}(\tau,k)aM_{p}/2$ and $\tau=\int a^{-1}dt$.

In Sec. II.1, we have shown that a model of our scenario can be constructed.
Here, for simplicity, we assume that $\epsilon\approx{\rm const.}\ll-1$ during
the NEC-violating phase, which requires some delicate design of $g_{1}$,
$g_{2}$ and $V$. Around the beginning or the end of the NEC-violating phase,
the detailed variation of $\epsilon$ may make some model-dependent
contributions. However, for our purpose, the simplification will not make a
qualitative difference.

Henceforth, we assume that the epoch of “inflation” consists of different
phases with $\epsilon_{j}=-{\dot{H}}_{j}/H^{2}_{j}={3\over
2}(1+w_{j})\simeq{\rm const.}$, where $w_{j}$ is the state parameter. We have
Cai:2015nya

$a_{j}(\tau)\sim\left({\tau}_{R,j}-\tau\right)^{\frac{1}{\epsilon_{j}-1}},$
(14)

for the $j$-th phase, where
${\tau}_{R,j}=\tau_{j}-(\epsilon_{j}-1)^{-1}\mathcal{H}^{-1}(\tau_{j})$ and
$a(\tau_{j})$ is set by requiring the continuity of $a$ at the end of phase
$j$ (i.e., $\tau=\tau_{j}$). As a result, we have

$\frac{a_{j}^{\prime\prime}}{a_{j}}={\nu_{j}^{2}-{1/4}\over{(\tau-\tau_{R,j})^{2}}}\,,$
(15)

where $\nu_{j}={3\over 2}\left|{1-w_{j}\over 1+3w_{j}}\right|$. Regarding the
phases $j$ and $j+1$ as adjacent phases, we have the solutions to Eq. (13) as

$\displaystyle u_{k,j}(\tau)$ $\displaystyle=$
$\displaystyle{\sqrt{\pi(\tau_{R,j}-\tau)}\over
2}\left\\{\alpha_{j}H_{\nu_{j}}^{(1)}[k(\tau_{R,j}-\tau)]+\beta_{j}H_{\nu_{j}}^{(2)}[k(\tau_{R,j}-\tau)]\right\\},\,\,(\tau<\tau_{j})\,,$
(16) $\displaystyle u_{k,{j+1}}(\tau)$ $\displaystyle=$
$\displaystyle{\sqrt{\pi(\tau_{R,{j+1}}-\tau)}\over
2}\Big{\\{}\alpha_{j+1}H_{\nu_{j+1}}^{(1)}[k(\tau_{R,j+1}-\tau)]$ (17)
$\displaystyle\qquad\qquad\qquad\quad+\beta_{j+1}H_{\nu_{j+1}}^{(2)}[k(\tau_{R,j+1}-\tau)]\Big{\\}},\qquad\qquad\qquad(\tau>\tau_{j})\,,$

respectively, where $\alpha_{j(j+1)}$ and $\beta_{j(j+1)}$ are $k$-dependent
coefficients. Using the matching conditions
$u_{k,j}(\tau_{j+1})=u_{k,j+1}(\tau_{j+1})$ and
$u_{k,j}^{\prime}(\tau_{j+1})=u_{k,j+1}^{\prime}(\tau_{j+1})$, we have

$\displaystyle\left(\begin{array}[]{ccc}\alpha_{j+1}\\\
\beta_{j+1}\end{array}\right)$ $\displaystyle=$ $\displaystyle{\cal
M}^{(j)}\left(\begin{array}[]{ccc}\alpha_{j}\\\
\beta_{j}\end{array}\right)\,,\quad{\rm where}\quad{\cal
M}^{(j)}=\left(\begin{array}[]{ccc}{\cal M}^{(j)}_{11}&{\cal M}^{(j)}_{12}\\\
{\cal M}^{(j)}_{21}&{\cal M}^{(j)}_{22}\end{array}\right)\,,$ (24)

see Refs. Cai:2015nya ; Cai:2019hge  for the matrix elements of ${\cal
M}^{(j)}$. The information of the $1,2\cdots j$-th phases of the Universe has
been encoded fully in the Bogoliubov coefficients $\alpha_{j+1}$ and
$\beta_{j+1}$. We set the initial state as the Bunch-Davies vacuum (see also
Piao:2003zm ; Piao:2005ag ; Liu:2013kea ; Qiu:2015nha ; Cai:2017pga  for pre-
inflationary bounce), i.e., $u_{k}=\frac{1}{\sqrt{2k}}e^{-ik\tau}$. Thus,
$|\alpha_{1}|=1$, $|\beta_{1}|=0$.

In the following, we focus on the scenario in Fig. 1. Regarding “$inf1$”, NEC-
violating and “$inf2$” phases as the $j=1,2,3$-th phases, respectively, we
have

$u_{k,3}(\tau)={\sqrt{\pi(\tau_{R,3}-\tau)}\over
2}\left\\{\alpha_{3}H_{3/2}^{(1)}[k(\tau_{R,3}-\tau)]+\beta_{3}H_{3/2}^{(2)}[k(\tau_{R,3}-\tau)]\right\\},$
(25)

where $\nu_{3}\simeq 3/2$ for de Sitter expansion. On super-horizon scale, we
have $H^{(1)}_{3/2}(-k\tau)=-H_{3/2}^{(2)}(-k\tau)\overset{-k{\tau}\rightarrow
0}{\approx}-i\sqrt{2/(-\pi k^{3}\tau^{3})}$. The resulting spectrum of
primordial GWs is

$P_{T}=\frac{4k^{3}}{\pi^{2}M_{p}^{2}}\cdot\frac{\left|u_{k,3}\right|^{2}}{a^{2}}=P_{T,inf2}\left|\alpha_{3}-\beta_{3}\right|^{2}\,,$
(26)

where

$\displaystyle\left(\begin{array}[]{ccc}\alpha_{3}\\\
\beta_{3}\end{array}\right)$ $\displaystyle=$ $\displaystyle{\cal
M}^{(2)}{\cal M}^{(1)}\left(\begin{array}[]{ccc}\alpha_{1}\\\
\beta_{1}\end{array}\right)\,,$ (31)

$P_{T,inf2}={2H_{inf2}^{2}\over M_{p}^{2}\pi^{2}}$, ${\cal M}^{(1)}$ and
${\cal M}^{(2)}$ are given by Eq. (24). The information of “$inf1$”, NEC-
violating and “$inf2$” phases has been encoded in the Bogoliubov coefficients
$\alpha_{3}$ and $\beta_{3}$.

###  II.3 Primordial GWB at low-frequency band

It is interesting to connect $P_{T}$ in (26) with the observations of
stochastic GWB at low-frequency bands. The BICEP/Keck+Planck bound at CMB band
is $r\lesssim 0.06$ Ade:2018gkx , which corresponds to $P_{T}\lesssim
10^{-10}$. The analysis result of NANOGrav 12.5-yr data Arzoumanian:2020vkk ,
if regarded as the stochastic GWB (see inspired studies e.g. Ellis:2020ena ;
Blasi:2020mfx ; DeLuca:2020agl ; Buchmuller:2020lbh ; Addazi:2020zcj ;
Kohri:2020qqd ; Ratzinger:2020koh ; Samanta:2020cdk ; Bian:2020bps ;
Vagnozzi:2020gtf ; Neronov:2020qrl ; Li:2020cjj ; Sugiyama:2020roc ;
Liu:2020mru ; Domenech:2020ers ; Bhattacharya:2020lhc ; Wong:2020yig ;
Kuroyanagi:2020sfw ), suggests $\Omega_{GW}\sim 10^{-9}$ with the tilt
$-1.5\lesssim n_{T}\lesssim 0.5$, where

$\Omega_{GW}(\tau_{0})=\frac{k^{2}}{12a_{0}^{2}H^{2}_{0}}P_{T}(k)\left[\frac{3\Omega_{{m}}j_{1}(k\tau_{0})}{k\tau_{0}}\sqrt{1.0+1.36\frac{k}{k_{\text{eq}}}+2.50\left(\frac{k}{k_{\text{eq}}}\right)^{2}}\right]^{2},$
(32)

is the energy density spectrum of GWs, see e.g. Turner:1993vb  (see also
Boyle:2005se ; Zhao:2006mm ; Kuroyanagi:2014nba ; Liu:2015psa ). Here,
$1/k_{eq}$ is the comoving Hubble scale at matter-radiation equality,
$\Omega_{m}=\rho_{m}/\rho_{c}$ and $\rho_{{c}}=3H^{2}_{0}/\big{(}8\pi
G\big{)}$ is the critical energy density.

According to (26), we plot $P_{T}$ and $\Omega_{GW}$ in Fig. 4 ($f=k/(2\pi
a_{0})$). We set $H_{1}=1.29\times 10^{-5}$, which corresponds to $P_{s}\sim
2.1\times 10^{-9}$ for $\epsilon_{1}=0.001$ and $H_{3}\sim 10^{-2}$;
$w_{1}\gtrsim-1$ and $w_{3}\gtrsim-1$ for the slow-roll inflations, while
$w_{2}\lesssim-10$ for the NEC-violating phase. We see that the yielded power
spectrum of primordial GWs has a nearly flat amplitude at the CMB band and
also a higher nearly flat amplitude at the PTA band. Here, the NEC-violating
regime contributes the upward section of $P_{T}$, in which the spectrum has a
blue tilt $n_{T}\simeq 2$ (since $\epsilon\ll-1$). Therefore, our scenario not
only explains the result reported by the NANOGrav Collaboration, but also has
a detectable signal $r\sim 0.01$ in the CMB.

(a) (b)

Figure 4: Left: $P_{T}$. Right: $\Omega_{\rm GW}h^{2}$, where $h=H_{0}/({\rm
km/s/Mpc})$.

###  II.4 Multi-stage inflation with NEC violations

The multi-stage inflation model (see e.g., earlier Refs. Adams:1997de ;
Burgess:2005sb ; Ashoorioon:2006wc ; Ashoorioon:2008qr ; Liu:2009pk ; see also
e.g., DAmico:2020euu  for recent study), in which a sequence of short
inflations are interrupted by short periods of decelerated expansions with
$w>-1/3$, is interesting, since it helps to make the EFT of inflation UV-
complete Obied:2018sgi ; Bedroya:2019snp ; Li:2019ipk ; Berera:2019zdd ;
Dhuria:2019oyf ; Torabian:2019zms . Usually, in such a scenario, a high-scale
inflation is followed by a sequence of low-scale inflations. However, it might
be also possible that a sequence of short inflations ($w\gtrsim-1$) are
interrupted by short periods of not only decelerated expansions with $w>-1/3$
but also super-inflation or Genesis with $w<-1$, so that the scales of
subsequent short (NEC-preserving) inflations might be higher, see e.g. Fig. 1.

According to Eqs. (17), (24) and (26), for a multi-stage scenario of inflation
in which a sequence of short slow-roll inflations ($w\gtrsim-1$) are
interrupted by lots of short periods of expansions with $w>-1/3$ and $w<-1$
(NEC violation), we can write the spectrum $P_{T}$ of primordial GWs as

$P_{T}=P_{T,l}^{\rm inf}\left|\alpha_{l}-\beta_{l}\right|^{2}={2H_{l}^{2}\over
M_{p}^{2}\pi^{2}}\left|\alpha_{l}-\beta_{l}\right|^{2}\,,$ (33)

where

$\displaystyle\left(\begin{array}[]{ccc}\alpha_{l}\\\
\beta_{l}\end{array}\right)$ $\displaystyle=$
$\displaystyle\prod_{j=1}^{l}{\cal
M}^{(j)}\left(\begin{array}[]{ccc}\alpha_{1}\\\
\beta_{1}\end{array}\right)\,,$ (38)

and ‘$l$’ labels the last short slow-roll inflations. The frequency band of
stochastic GWB yielded is

$10^{-18}\,{\rm Hz}\lesssim
f\lesssim\exp{\left(\sum_{j=1}^{l}N_{j}\right)}10^{-18}\,{\rm Hz},$ (39)

where $N_{j}\equiv\ln{a_{j,e}H_{j,e}\over a_{j,ini}H_{j,ini}}$ is the
$e$-folds number of the perturbation modes passing through the $j$-th phase.
Note that $N_{j}<0$ for the decelerated expansion ($w>-1/3$).

According to (33), we plot $P_{T}/P_{T,l}^{inf}$ in Fig. 5(a) for a multi-
stage scenario of inflation with short periods of slow-roll inflations
($j=1,3,5,7,9$) with different $H_{j}$. The panorama of $P_{T}$ at
corresponding GW frequency band looks like the Great Wall, see Fig. 5(b), in
which the nearly flat roads correspond to GWB yielded by short slow-roll (NEC-
preserving) inflations, the upward and downward slopes correspond to the NEC-
violating expansions ($w<-1$) and decelerated expansions ($w>-1/3$),
respectively. It is well-known that each section of the Great Wall records a
unique history.

(a) (b)

Figure 5: The spectrum $P_{T}/P_{T,l}^{inf}$ of primordial GWB yielded in a
multi-stage scenario of inflation, in which a sequence of short slow-roll
inflations ($j=1,3,5,7,9$) are interrupted by short periods of decelerated
expansion ($j=6$) and NEC-violating expansion ($j=2,4,8$). We set the equation
of state parameters $w_{1,3,5,7,9}\simeq-1$, $w_{2}=-15$, $w_{4,8}=-10$ and
$w_{6}=1/3$. The frequency band of GW spans about $28$ orders. The lower panel
is the Great Wall (sketched by Yu-Ze Piao). The panorama of $P_{T}$ looks like
the Great Wall. When we climb up the Great Wall, we would see the beacon
towers of different physics.

##  III Conclusion

The NEC violation in the primordial Universe will bring a blue-tilted GWB.
However, its implications to the GWB might be far richer than expected. We
presented a scenario, in which after a slow-roll (NEC-preserving) inflation
with $H\simeq H_{inf1}$ (responsible for the density perturbation on large
scales), the Universe goes through an NEC-violating period, which is followed
again by the slow-roll inflation but with $H_{inf2}\gg H_{inf1}$. We
calculated the power spectrum of the yielded primordial GWs. As expected, the
spectrum has an observable amplitude $P_{T}\sim H_{inf1}^{2}$ ($n_{T}\simeq
0$) at the CMB band and a higher amplitude $P_{T}\sim H_{inf2}^{2}$
($n_{T}\simeq 0$) at the PTA band (compatible with recent NANOGrav result).
Here, the NEC violation responsible for the upward tilt of $P_{T}$ played an
indispensable role.

It is well-known that the detection of stochastic GWB will not only solidify
our confidence in inflation but also offer us a probe to the physics of the
early Universe. Though the model we consider is simplified, it highlights an
unexpected point that the GWB yielded in the primordial Universe might have a
unique landscape. We explore the observable imprints of short NEC violations
on primordial GWB. It is especially highlighted that for the multi-stage
inflation, consisting of a sequence of short slow-roll inflations
($w\gtrsim-1$) interrupted by lots of short period of expansions with $w>-1/3$
and $w<-1$, we will have a Great Wall spectrum of stochastic GWB, which might
be detectable.

Acknowledgments

We thank Gen Ye for helpful discussion. Y. C. is funded by the China
Postdoctoral Science Foundation (Grant No. 2019M650810) and the NSFC (Grant
No. 11905224). Y. S. P. is supported by NSFC Grants No. 12075246 and No.
11690021.

##  Appendix A On stability of scalar perturbations

In the unitary gauge, for (1), we have

$S^{(2)}_{\zeta}=\int
d^{4}xa^{3}Q_{s}\left[\dot{\zeta}^{2}-c_{s}^{2}{(\partial\zeta)^{2}\over
a^{2}}\right]\,,$ (40)

where

$\displaystyle Q_{s}=\epsilon M_{p}^{2}+{g_{2}\dot{\phi}^{4}\over
H^{2}}={M_{p}^{2}\dot{\phi}^{2}\over
2H^{2}}\left(g_{1}+3g_{2}{\dot{\phi}^{2}\over M_{p}^{2}}\right)\,,$ (41)
$\displaystyle c_{s}^{2}={\epsilon M_{p}^{2}\over Q_{s}}.$ (42)

Around the NEC violation, though $c_{s}^{2}<0$ (see Fig. 6), $c_{s}^{2}=1$ can
be set with the higher-order derivative (beyond-Horndeski) operators, see
e.g., Refs. Cai:2016thi ; Cai:2017tku ; Cai:2017dyi ; Cai:2017dxl ; Ye:2019frg
for related details. Here, $Q_{s}>0$ throughout. Therefore, the ghost and
gradient instabilities can be cured in the EFT regime. It is interesting to
ask whether there is a danger of instabilities from the higher-derivative
terms in the perspective of the UV theory. However, it is still unknown how to
embed the EFT with such terms in a UV complete theory so far. The related
issues require further investigation in the future.

(a)

(b)

Figure 6: Left: The evolution of $Q_{s}$, which is positive throughout in our
model. Right: The evolution of $c_{s}^{2}$, which is negative during the NEC-
violating phase (approximately corresponding to $730084<t<730328$).

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