# Generating enhanced parity-violating gravitational waves during inflation
with violation of the null energy condition

Yong Cai<EMAIL_ADDRESS>School of Physics and Microelectronics,
Zhengzhou University, Zhengzhou, Henan 450001, China

###### Abstract

A violation of the null energy condition (NEC) during inflation in a single-
field inflation model will naturally enhance the amplitude of the parity
violation effect (defined by $\Delta\chi$) of inflationary primordial
gravitational waves (GWs), provided the inflaton is non-minimally coupled to a
gravitational Chern-Simons term. After going through the NEC-violating phase,
the Universe enters subsequent slow-roll inflation with a higher energy scale
(i.e., a greater Hubble parameter $H$), which results in an enhanced nearly
scale-invariant power spectrum (i.e., $P_{\rm T}$) of inflationary primordial
GWs in the high-frequency band, while $P_{\rm T}$ remains consistent with
observations in the frequency band of the cosmic microwave background.
Therefore, the violation of NEC during inflation will amplify the
observability (i.e., $P_{\rm T}\cdot\Delta\chi$) of the parity violation
effect on small scales. Intriguingly, our model has particular oscillatory
features on $\Delta\chi$ that may not be mimicked by others.

###### Contents

  1. I Introduction
  2. II NEC violation and Parity violation of inflationary gravitational waves
    1. II.1 Setup
    2. II.2 Model and numerical solution
  3. III Parity-violating gravitational wave background and observations
  4. IV Conclusion

##  I Introduction

The inflation scenario Guth:1980zm ; Linde:1981mu ; Albrecht:1982wi ;
Starobinsky:1980te  has achieved tremendous success in simultaneously solving
several puzzles of the hot big bang cosmology. The nearly scale-invariant
power spectrum of the primordial scalar perturbations predicted by inflation
has been confirmed by observations of the temperature anisotropy of cosmic
microwave background (CMB) Planck:2018vyg ; Planck:2018jri , while the
primordial gravitational waves (GWs) predicted by inflation remains
undetermined. For a scale-invariant power spectrum of the primordial GWs, the
tensor-to-scalar ratio is bounded to be $r_{0.002}<0.035$ at $95\%$ confidence
level BICEP:2021xfz  in the observational window of CMB. In recent years, the
detections of GWs from binary black holes LIGOScientific:2016aoc  and a binary
neutron star inspiral LIGOScientific:2017vwq  by the LIGO and Virgo
collaborations have opened a new window for gravitational physics.

The inflationary primordial GWs background Starobinsky:1979ty ; Rubakov:1982df
, which spans a broad frequency band (about $10^{-18}-10^{10}$ Hz), may bring
us a wealth of information about gravity and the early Universe in light of
the recent and future observations in a wide multi-frequency range, including
those of the Pulsar Timing Array (PTA) NANOGrav:2020bcs ; Goncharov:2021oub ;
Hobbs:2009yy ; Kramer:2013kea , SKA Weltman:2018zrl , LISA LISA:2017pwj ,
Taiji Ruan:2018tsw , TianQin TianQin:2020hid , DECIGO Kawamura:2011zz  and BBO
Crowder:2005nr . In contrast to primordial scalar perturbations, primordial
GWs possess a distinct characteristic, namely, chirality, which can be
manifested in parity-violating theories of gravity in the primordial Universe,
see e.g. Lue:1998mq ; Balaji:2003sw ; Alexander:2004wk ; Satoh:2007gn ;
Satoh:2008ck ; Li:2009rt ; Cai:2016ihp ; Smith:2016jqs ; Gubitosi:2016yoq ;
Obata:2016oym ; Bartolo:2017szm ; Yagi:2017zhb ; Yoshida:2017cjl ;
Bartolo:2018elp ; Nishizawa:2018srh ; Gao:2019liu ; Nojiri:2019nar ;
Mylova:2019jrj ; Fujita:2020iyx ; Nojiri:2020pqr ; Chu:2020iil ; Li:2020xjt ;
Fu:2020tlw ; Cai:2021uup ; Li:2022mti ; Li:2022vtn ; Martinovic:2021hzy ;
Kamada:2021kxi ; Zhang:2022xmm ; Odintsov:2022hxu ; Peng:2022ttg ; Bastero-
Gil:2022fme ; Jiang:2022uxp ; Sulantay:2022sag ; Qiao:2022mln  for some of the
recent studies, see also Alexander:2004us ; Maleknejad:2011sq ;
Maleknejad:2011jw ; Kawai:2017kqt ; Odintsov:2019mlf ; Bombacigno:2022lcx ;
Odintsov:2021kup ; Boudet:2022nub ; Bombacigno:2022naf ; Odintsov:2022cbm . A
dynamical coupling of the inflaton or a spectator field to a parity-violating
term, such as the gravitational Chern-Simons (gCS) term Jackiw:2003pm ;
Alexander:2009tp , which is well motivated Green:1984sg , is able to produce
asymmetric right- and left-handed circularly polarised isotropic GWs, see also
Cai:2016ihp  for the chirality oscillation. Notably, a hint of parity-
violating physics in the CMB (i.e., the cosmic birefringence) was recently
extracted from the polarization data of Planck and WMAP Minami:2020odp ;
Diego-Palazuelos:2022dsq ; Eskilt:2022cff , which may require further
verification. This signal of birefringence (if confirmed by more evidence)
must have been generated by some mechanism beyond the slow-roll inflation,
see, e.g., Fujita:2022qlk .

In single-field inflationary models, the effect of parity violation is
proportional to $\dot{\phi}\equiv{\rm d}\phi/{\rm d}t$ (where $\phi$ is the
inflaton), which is suppressed by the slow-roll condition ($\dot{\phi}\ll H$),
unless a delicate design of the coupling is introduced. Furthermore, the
stringent bound on the tensor-to-scalar ratio in the observational window of
CMB makes the situation even worse. Consequently, the parity violation of
primordial GWs generated in single-field models of slow-roll inflation can
hardly be observed in general situations. However, the approximate $60$
$e-$folds inflationary expanding history could be more complicated than that
described by the slow-roll inflation, see e.g., Ramirez:2011kk ; Liu:2013kea ;
Cai:2015nya ; Mishima:2019vlh ; Cai:2020qpu ; Bernal:2020ywq ;
Kuroyanagi:2020sfw ; Giare:2020vss ; DAmico:2020euu ; DAmico:2021fhz ;
Lewicki:2021xku ; Giare:2022wxq . As can be inferred, the scale invariance of
the GW power spectrum could be broken in such a broad frequency band through
many different ways Kuroyanagi:2014nba ; Cai:2015dta ; Cai:2015ipa ;
Cai:2015yza ; Cai:2016ldn ; Li:2016awk ; Cai:2020ovp ; Liu:2020mru ;
Benetti:2021uea . Recently, the NANOGrav Collaboration reported evidence for a
stochastic common-spectrum process NANOGrav:2020bcs , which might be
interpreted as a stochastic GW background with a spectrum tilt $-1.5\lesssim
n_{T}\lesssim 0.5$. If this result could be attributed to the primordial GWs,
it would mean that we may need new physics beyond the slow-roll inflation, see
e.g., Vagnozzi:2020gtf .

One possibility is when there is a violation of the null energy condition
(NEC) during inflation, which might play a significant role in the evolution
history of the Universe, see Rubakov:2014jja  for a review. Implementing a
fully stable NEC violation in cosmology is a challenging task Rubakov:2014jja
; Libanov:2016kfc ; Kobayashi:2016xpl ; Ijjas:2016vtq ; Dobre:2017pnt , see
also Dubovsky:2005xd ; Creminelli:2006xe ; Nicolis:2009qm ; Rubakov:2013kaa ;
Elder:2013gya ; Battarra:2014tga ; Koehn:2015vvy ; Qiu:2015nha ; Cai:2022ori .
It has be demonstrated with the effective field theory method that a fully
stable NEC violation can be realized in “beyond Horndeski” theory Cai:2016thi
; Creminelli:2016zwa ; Cai:2017tku ; Cai:2017dyi ; Kolevatov:2017voe , see
also Cai:2017dxl ; Cai:2017pga ; Mironov:2018oec ; Qiu:2018nle ; Ye:2019sth ;
Ye:2019frg ; Mironov:2019qjt ; Akama:2019qeh ; Tahara:2020fmn ; Ilyas:2020qja
; Ilyas:2020zcb ; Zhu:2021whu ; Zhu:2021ggm ; Ageeva:2021yik ; Mironov:2022ffa
; Mironov:2022quk ; Wolf:2022yvd  for recent studies. The violation of NEC is
able to enhance the power spectrum of primordial GWs in the higher frequency
band Cai:2020qpu ; Cai:2022nqv , where the scale invariance is recovered due
to the recovery of slow-roll condition in the subsequent inflationary phase,
which has a larger Hubble parameter $H$. Furthermore, the slow-roll condition
will be inevitably violated around the NEC-violating phase, which is able to
naturally enhance the amplitude of the parity violation effect of inflationary
primordial GWs in single-field inflationary models, as we will show in the
following. To our knowledge, the enhanced parity-violating GWs produced by a
NEC-violating phase during inflation has not been investigated so far.

In this paper, we illustrate the ability of a NEC violation in generating
enhanced parity-violating primordial GWs during inflation with a (non-
canonical) single-field model of inflation, in which the inflaton is non-
minimally coupled to the gCS term. We show that the violation of NEC during
inflation will naturally amplify the observability of the parity violation
effect, which might be detectable at scales much smaller than the CMB scale in
the future.

##  II NEC violation and Parity violation of inflationary gravitational waves

The gCS term is the leading-order parity-violating correction to general
relativity motivated by the anomaly cancellation in particle physics and
string theory, see e.g. Alexander:2009tp  for a review. In this section, we
consider a non-minimal coupling of the inflaton to the gCS term, while setting
the background evolution to the same as that introduced in Cai:2020qpu .

Initially, the canonical scalar field $\phi$ (i.e., the inflaton) slowly rolls
down a nearly flat potential $V(\phi)$, so that the Universe undergoes a
period of slow-roll inflation, which is NEC-preserving. The primordial scalar
perturbation and GWs generated in this period should be consistent with
observations in the window of the CMB. As the field $\phi$ becomes non-
canonical and climbs up the potential rapidly such that ${\dot{H}}>0$, the NEC
and the slow-roll condition are violated. After exiting the NEC-violating
phase, the Universe gradually enters a subsequent slow-roll inflationary phase
(NEC-preserving) with higher energy scale (i.e., a greater Hubble parameter),
during which the inflaton $\phi$ becomes canonical again.

###  II.1 Setup

The action of our single-field model can be given as follows,

$S=\int d^{4}x\sqrt{-g}\Big{[}{M_{p}^{2}\over 2}R-M_{p}^{2}{g_{1}(\phi)\over
2}X+{g_{2}(\phi)\over 4}X^{2}-M_{p}^{4}V(\phi)+{g_{3}(\phi)\over 8}R\wedge
R+L_{\rm HD}\Big{]}\,,$ (1)

where the inflaton $\phi$ is dimensionless,
$X=\nabla_{\mu}\phi\nabla^{\mu}\phi$, $g_{n}(\phi)$ are dimensionless
functions; the gCS term $R\wedge
R=\epsilon^{\alpha\beta\gamma\delta}R_{\alpha\beta\mu\nu}R_{\gamma\delta}^{~{}~{}\mu\nu}$,
$\epsilon^{\alpha\beta\gamma\delta}$ is the four dimensional Levi-Civita
tensor with $\epsilon^{0123}=-1/\sqrt{-g}$; $L_{\rm HD}$ represents some
higher derivative terms which are assumed to have negligible contributions to
the background evolution and the tensor perturbations at quadratic order, see
e.g. Cai:2017dyi  for $L_{\delta g^{00}R^{(3)}}$. In this paper, we will focus
on the primordial tensor perturbations. The dangerous ghost or gradient
instabilities (see, e.g., Battarra:2014tga ; Koehn:2015vvy ; Libanov:2016kfc ;
Kobayashi:2016xpl ) of the scalar perturbations around the NEC-violating phase
are assumed to be cured by $L_{\rm HD}$ (see, e.g., Cai:2016thi ;
Creminelli:2016zwa ; Cai:2017tku ; Cai:2017dyi ; Kolevatov:2017voe , for
$L_{\rm HD}=L_{\delta g^{00}R^{(3)}}$), for simplicity. A detailed study of
the scalar sector of this theory will be carried out separately.

With the flat Friedmann-Robertson-Walker metric, i.e.,
$ds^{2}=-dt^{2}+a^{2}(t)d\vec{x}^{2}$, the background equations can be found
as

$\displaystyle
3H^{2}M_{p}^{2}=\frac{M_{p}^{2}}{2}g_{1}\dot{\phi}^{2}+\frac{3}{4}g_{2}\dot{\phi}^{4}+M_{p}^{4}V\,,$
(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)

where “${}_{,\phi}\equiv d/d\phi$”. The background evolution is unaffected by
the gCS term or $L_{\rm HD}$. By choosing the functions $g_{1}(\phi)$,
$g_{2}(\phi)$ and $V(\phi)$ appropriately, we can realize the background
evolution of intermittent NEC violation during inflation, see Cai:2020qpu .

In the unitary gauge, we will set the tensor perturbation as
$h_{ij}=a^{2}\left(\mathrm{e}^{\gamma}\right)_{ij}$, where
$\gamma_{ii}=0=\partial_{i}\gamma_{ij}$. The quadratic action of tensor
perturbation can be given as

$\displaystyle S_{\gamma}^{(2)}={M_{p}^{2}\over 8}\int d\tau
d^{3}x\left\\{a^{2}\Big{[}{\gamma_{ij}^{\prime}}^{2}-(\partial_{k}\gamma_{ij})^{2}\Big{]}-{g_{3}^{\prime}\over
M_{p}^{2}}\epsilon^{ijk}\left[(\partial_{i}\gamma_{jl})^{\prime}(\gamma_{k}^{l})^{\prime}-\partial_{i}\partial_{l}\gamma_{jq}\partial^{l}\gamma_{k}^{q}\right]\right\\}\,,$
(4)

where ${}^{\prime}=d/d\tau$ and $\tau=\int dt/a$,
$\epsilon^{ijk}=\epsilon^{0ijk}$ is the three dimensional Levi-Cevita symbol,
the higher derivative terms proportional to $g_{3}^{\prime}$ are the parity-
violating corrections contributed by the gCS term. During the slow-roll
inflation, the effect of parity violation could be suppressed by $\dot{\phi}$
since $g_{3}^{\prime}=ag_{3,\phi}\dot{\phi}$, unless a delicate design of
$g_{3}(\phi)$ is adopted. However, the parity violation during inflation could
be greatly enhanced by an intermediate NEC violation, during which
$\dot{\phi}$ can naturally and significantly violate the slow-roll condition.

In the Fourier space, we have $\gamma_{ij}(\tau,\mathbf{x})=\sum_{s={\rm
L},{\rm
R}}\int{d^{3}\mathbf{k}\over(2\pi)^{3}}\gamma^{(s)}_{\mathbf{k}}(\tau)p_{ij}^{(s)}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{x}}$,
where $p_{ij}^{(s)}$ is the circular polarization tensor which satisfies
$p_{ij}^{({\rm R})}{p^{ij}}^{({\rm R})}=p_{ij}^{({\rm L})}{p^{ij}}^{({\rm
L})}=0$, $p_{ij}^{({\rm R})}{p^{ij}}^{({\rm L})}=2$ and
$ik_{l}\epsilon^{nlj}{p_{ij}}^{(s)}=k\lambda^{(s)}{p^{n}_{i}}^{(s)}$. The
parameters $\lambda^{({\rm L})}=-1$ and $\lambda^{({\rm R})}=1$ correspond to
the left- and the right-handed modes, respectively. For convenience, we also
define $\lambda^{({\rm N})}=0$, which corresponds to the situation where there
is no violation of parity.

From Eq. (4), we can obtain the equation of motion of
$\gamma_{\mathbf{k}}^{(s)}$ as

${u^{(s)}_{\mathbf{k}}}^{\prime\prime}+\left[\left(c_{{\rm
T}\mathbf{k}}^{(s)}\right)^{2}k^{2}-{{z_{\rm T}^{(s)}}^{\prime\prime}\over
z_{\rm T}^{(s)}}\right]u^{(s)}_{\mathbf{k}}=0\,,$ (5)

where we have defined ${u^{(s)}_{\mathbf{k}}}\equiv z_{\rm
T}^{(s)}{\gamma_{\mathbf{k}}^{(s)}}$ and $\quad z_{\rm T}^{(s)}={a\over
2}\sqrt{1-\lambda^{(s)}{k\over a^{2}}{g_{3}^{\prime}\over M_{p}^{2}}}$, and
the effective sound speed is $c_{{\rm T}\mathbf{k}}^{(s)}=1$ for both the
left- and the right-handed GW modes, see e.g., Nishizawa:2018srh . In order to
avoid the appearance of the ghost modes, we require that ${k\over
a^{2}}{g_{3}^{\prime}\over M_{p}^{2}}<1$, see e.g., Dyda:2012rj .

The perturbation modes are deep inside their horizon initially, i.e.,
$k^{2}\gg{z_{\rm T}^{(s)}}^{\prime\prime}/{z_{\rm T}^{(s)}}$, which indicates
${u^{(s)}_{\mathbf{k}}}\simeq e^{-ik\tau}{/\sqrt{2k}}$ for
$\tau\rightarrow-\infty$. The power spectrum of primordial GWs is $P_{\rm
T}^{(s)}={k^{3}\over 2\pi^{2}}\left|\gamma_{\mathbf{k}}^{(s)}\right|^{2}$ for
$k\ll aH$, where $s=$ L, R and N.111Note that $P_{\rm T}^{(N)}=P_{\rm T}/2$ in
the situation where there is no parity violation. To evaluate the intensity of
parity violation of primordial GWs, it is convenient to define the chiral
parameter

$\Delta\chi={P_{\rm T}^{({\rm L})}-P_{\rm T}^{({\rm R})}\over P_{\rm T}^{({\rm
L})}+P_{\rm T}^{({\rm R})}}\,.$ (6)

From an observational point of view, what we really care about is the
difference between the power spectra of the left- and right-handed GW modes,
i.e., $P_{\rm T}^{({\rm L})}-P_{\rm T}^{({\rm R})}$. With a sufficient large
$P_{\rm T}=P_{\rm T}^{({\rm L})}+P_{\rm T}^{({\rm R})}$, a $\Delta\chi$ as
small as a few percent in the GW background might be detectable for future
observations. Furthermore, the peculiar features on $\Delta\chi$ may encode
the characteristic of our model.

###  II.2 Model and numerical solution

We adopt the phenomenological model proposed in Cai:2020qpu , in which we set

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

where $m$, $\lambda$, $f_{1,2}$ and $q_{1,2,3,4}$ are some positive constants.
We set $\phi_{3}<\phi_{2}<0<\phi_{1}<\phi_{0}$ for convenience.

In the limit $\phi\ll\phi_{3}$, we have $g_{1}=1$, $g_{2}=0$ and $V=V_{\rm
inf1}\simeq m^{2}\phi^{2}/2$, which is responsible for the first stage of
slow-roll inflation. Apparently, the inflaton is canonical when
$\phi\ll\phi_{3}$. For $\phi\gg\phi_{0}$, we have $g_{1}=1$, $g_{2}=0$ and
$V=V_{\rm inf2}\simeq\lambda[1-\frac{(\phi-\phi_{1})^{2}}{\sigma^{2}}]^{2}\gg
V_{\rm inf1}$, which is responsible for the subsequent slow-roll inflation
with a higher energy scale (or equivalently, a greater Hubble parameter $H$).
Again, the inflaton becomes canonical in this stage. In these two NEC-
preserving regimes (i.e., $\phi\ll\phi_{3}$ and $\phi\gg\phi_{0}$), the
potential is nearly flat, so that the predicted primordial scalar
perturbations are consistent with the observations in the window of CMB.

We numerically solve Eqs. (2) and (3) using (7) to (9), and show the
evolutions of $H$, $\dot{\phi}$ and $\ddot{\phi}$ around the NEC-violating
phase in Fig. 1. Apparently, the slow-roll condition is strongly violated
around the NEC-violating phase. More details about the background evolution
can be found in Cai:2020qpu . The evolution of
${z_{\text{T}}^{(s)}}^{\prime\prime}/{z_{\text{T}}^{(s)}}$ is displayed in
Fig. 2. Actually, it is mainly the nontrivial evolutions of $\dot{\phi}$ and
$\ddot{\phi}$ that cause ${z_{\rm T}^{(s)}}^{\prime\prime}/{z_{\rm T}^{(s)}}$
to deviate greatly from $a^{\prime\prime}/a$, where $a^{\prime\prime}/a$
corresponds to the situation in which parity is preserved (i.e.,
$\lambda^{(s)}=0$). Therefore, the NEC violation during inflation is able to
naturally enhance the parity violation, compared with the case where the slow-
roll condition is preserved, as can inferred.

We also numerically solve Eq. (5) and plot the evolutions of perturbation
modes $|u|_{\mathbf{k}}^{(s)}$ and $|\gamma|_{\mathbf{k}}^{(s)}$ for $s=$L, R
and N in Fig. 3, in which we set the comoving wavenumber $k=8.8\times 10^{6}$
$\text{Mpc}^{-1}$. Basically, the chirality asymmetry of GW modes becomes
relatively apparent when $|\dot{\phi}|$ and $|\ddot{\phi}|$ become sufficient
large, as we can see from Figs. 1 to 3. The left- and right-handed GW modes
oscillate with each other and eventually arrive at different values after
exiting their horizon. As a result, the parity violation of primordial GWs
enhanced by NEC violation during inflation will be encoded in the power
spectra, see also Cai:2016ihp  for the chirality oscillation.

The power spectra and the chiral parameter $\Delta\chi$ are displayed in Figs.
4 and 5, respectively. We can see that the parity violation becomes evident in
the range $10^{5}{\rm Mpc}^{-1}\lesssim k\lesssim 10^{8}{\rm Mpc}^{-1}$. This
range corresponds to those GW modes exiting horizon around the end of the NEC-
violating phase, around which $|\dot{\phi}|$ and $|\ddot{\phi}|$ reach their
maximum values. It should be pointed out that the specific relation between
the comoving wavenumber $k$ and the time $t$ depends on the choice of
parameters in the numerical calculation. For example, with a different
normalization of the scale factor, we can obtain a shift of the power spectra
as well as $\Delta\chi$ with respect to $k$.

Notably, we can see in Fig. 5 that there are some peculiar oscillatory
features on $\Delta\chi$ at small scales. These features together with the
peculiar power spectra (as shown in Fig. 4) make our model characteristic with
respect to others in the literature (e.g., Obata:2016oym ; Cai:2021uup ;
Zhang:2022xmm ; Peng:2022ttg ; Bastero-Gil:2022fme ).

(a) $\dot{\phi}$ and $H$ (b) $\ddot{\phi}$

Figure 1: Evolution of the background. Approximately, $|\dot{\phi}|$ and
$|\ddot{\phi}|$ reach their maximum values around the end of the NEC violation
phase. Figure 2: Comparison of the evolutions of
${z_{\text{T}}^{(s)}}^{\prime\prime}/{z_{\text{T}}^{(s)}}/a^{2}$ for $s={\rm
R}$ ($\lambda_{s}=1$), $s={\rm L}$ ($\lambda_{s}=-1$) and $s={\rm N}$
($\lambda_{s}=0$), where $s={\rm N}$ corresponds to $a^{\prime\prime}/a^{3}$.
We have set $k=8.8\times 10^{5}$ $\text{Mpc}^{-1}$.

(a) $|u|_{\mathbf{k}}^{(s)}$ (b)
$|u|_{\mathbf{k}}^{{(s)}}/|u|_{\mathbf{k}}^{\text{(N)}}$ (c)
$|\gamma|_{\mathbf{k}}^{(s)}$ (d)
$|\gamma|_{\mathbf{k}}^{(s)}/|\gamma|_{\mathbf{k}}^{\text{(N)}}$

Figure 3: Evolution of the GW mode with comoving wavenumber $k=8.8\times
10^{6}$ $\text{Mpc}^{-1}$.

(a) $P_{\rm T}^{({\rm s})}/P_{\rm T,inf2}^{({\rm s})}$ (b) $P_{\rm T}^{({\rm
s})}/P_{\rm T,inf2}^{({\rm s})}$ (c) $P_{\rm T}^{(s)}/P_{\rm T}^{({\rm N})}$
(d) $\left(P_{\rm T}^{({\rm L})}+P_{\rm T}^{({\rm R})}\right)/P_{\rm T}^{({\rm
N})}/2$

Figure 4: Comparisons of the power spectra of the left-handed
($\lambda^{(s)}=-1$, $s={\rm L}$), right-handed ($\lambda^{(s)}=1$, $s={\rm
R}$) and parity preserving ($\lambda^{(s)}=0$, $s={\rm N}$) primordial GWs
modes which exited their horizon around the NEC-violating phase (part a) and
that around the end of the NEC-violating phase (part b). The ratios of the
power spectra of the chiral GW modes to that of the parity preserving ones
(parts c and d). Here, the power spectra are nearly scale-invariant on both
large and small scales, which may be distinctive from that produced by other
mechanism. Figure 5: The chiral parameter $\Delta\chi$ is defined by Eq. (6).
These nontrivial oscillations of $\Delta\chi$ become evident in the range
$10^{5}{\rm Mpc}^{-1}\lesssim k\lesssim 10^{8}{\rm Mpc}^{-1}$, which
corresponds to those GW modes exiting horizon around the end of the NEC-
violating phase.

##  III Parity-violating gravitational wave background and observations

The energy density spectrum of GWs is defined as Turner:1993vb ; Boyle:2005se
(see also Kuroyanagi:2014nba )

$\Omega_{\rm GW}=\frac{1}{\rho_{\text{c}}}\frac{d\rho_{\rm GW}}{d\ln
k}=\frac{k^{2}}{12a^{2}_{0}H^{2}_{0}}P_{\rm T}{\cal T}^{2}(k,\tau_{0})\,,$
(10)

where the critical energy density of the Universe is
$\rho_{\text{c}}=3M_{p}^{2}H^{2}_{0}$, $H_{0}=67.8\,{\rm km/s/Mpc}$,
$\tau_{0}=1.41\times 10^{4}$ Mpc, $a_{0}=1$, the reduced Hubble parameter
$h=H_{0}/(100\,{\rm km/s/Mpc})$, $\rho_{\text{GW}}$ is the energy density of
GWs at present, and the wavenumber relates to the frequency as $k=2\pi f$. The
transfer function can be given as

${\cal
T}(k,\tau_{0})=\frac{3\Omega_{\text{m}}j_{1}(k\tau_{0})}{k\tau_{0}}\sqrt{1.0+1.36\frac{k}{k_{\text{eq}}}+2.50\left(\frac{k}{k_{\text{eq}}}\right)^{2}},$
(11)

where $k_{\text{eq}}=0.073\,\Omega_{\text{m}}h^{2}$ Mpc-1 is the comoving
wavenumber of the perturbation mode that entered the horizon at the equality
of matter and radiation, $\Omega_{\rm m}$ is the density fraction of matter
today.

Currently, the bound on the tensor to scalar ratio put by data of CMB
BICEP:2021xfz  indicates that $\Omega_{\rm GW}<10^{-15}$ at $f\sim 10^{-16}$
Hz. The 12.5-yr data of NANOGrav NANOGrav:2020bcs  suggests $\Omega_{\rm
GW}\sim 10^{-9}$ at $f\sim 10^{-8}$ Hz, provided the stochastic common-
spectrum process can be interpreted as a stochastic GW background. The
observations in the frequency band of LIGO and Virgo LIGOScientific:2016aoc ;
LIGOScientific:2017vwq  put the bound $\Omega_{\rm GW}<10^{-7}$ at $f\simeq
30$ Hz.

We plot $\Omega_{\rm GW}^{(s)}h^{2}\sim P_{\rm T}^{(s)}$ with respect to the
physical frequency $f$ for left-, right-handed and parity-preserving GW modes
in Fig. 6. We use the 12.5-yr data of NANOGrav and the future sensitivity
curves of SKA, LISA, Taiji, TianQin, DECIGO and BBO to compare the theoretical
predictions with experimental projections. In Fig. 6, the largest parity
violation effect appears in the observational window of PTA. We also choose a
different normalization of the scale factor in our numerical calculation so
that we can shift the largest parity violation effect to the band of space-
based interferometers, as displayed in Fig 7.

Additionally, with a different choice of the parameter space, we should obtain
a larger effect of parity violation, i.e., a larger maximum value of
$|\Delta\chi|$. Here, for simplicity and to show that the NEC violation during
inflation can naturally enhance the effect of parity violation in primordial
GWs, we have chosen the same model parameters in the numerical calculation as
in Cai:2020qpu . Furthermore, since the GW power spectra $P_{\rm T}$ (or
$\Omega_{\rm GW}h^{2}$) are greatly enhanced for more than seven orders by the
NEC violation, the absolute value of $\left|P_{\rm T}^{({\rm L})}-P_{\rm
T}^{({\rm R})}\right|$ (or $\Omega_{\rm GW}^{({\rm L})}h^{2}-\Omega_{\rm
GW}^{({\rm R})}h^{2}$) could become quite substantial, even though the maximum
value of $|\Delta\chi|$ is only a few percent. Therefore, the NEC violation is
able to amplify the observability of parity violation in the GW background. It
is also interesting to note that observations of an enhanced $P_{\rm T}$ and
the peculiar oscillatory features on $\Delta\chi$ might distinguish our model
from others in the literature (e.g., Obata:2016oym ; Cai:2021uup ;
Zhang:2022xmm ; Peng:2022ttg ; Bastero-Gil:2022fme ).

We did not extend the curve to higher frequencies in Figs. 6 and 7 because
there is a cutoff to avoid the appearance of ghost modes Dyda:2012rj  in gCS
theory, as can be seen from the expression of $z_{\text{T}}^{(s)}$. It is also
because we did not deal with the exit of the second inflationary phase, which
is an open question. Interestingly, there might be multistage inflation with
intermittent NEC violations and short periods of decelerated expansion, as has
been discussed in Cai:2020qpu . As a result, the GW spectra might look like
the Great Wall while satisfying constraints in the band of CMB.

(a) (b)

Figure 6: The energy density spectrum of GWs $\Omega_{\rm GW}^{(s)}h^{2}$ with
respect to the physical frequency $f$, where $h=H_{0}/(100{\rm km/s/Mpc})$;
$s=$L and R correspond to the left- and right-handed GW modes, respectively;
$s=$N corresponds to the parity-preserving mode. The largest parity violation
effect appears in the band of PTA.

(a) (b)

Figure 7: The energy density spectrum of GWs $\Omega_{\rm GW}^{(s)}h^{2}$ with
respect to the physical frequency $f$, where $h=H_{0}/(100{\rm km/s/Mpc})$;
$s=$L and R correspond to the left- and right-handed GW modes, respectively;
$s=$N corresponds to the parity-preserving mode. The largest parity violation
effect appears in the band of space-based interferometers.

##  IV Conclusion

Chirality might be an intrinsic characteristic of GWs. We have shown that a
violation of the NEC during inflation is able to naturally enhance the parity-
violating primordial GWs such that the chirality asymmetry of GWs can be
observable at scales much smaller than the CMB scale. We have illustrated this
new mechanism with a single-field inflationary model, in which the inflaton
$\phi$ is non-minimally coupled to a gravitational Chern-Simons term.

The enhancement of parity-violating GWs in our model is twofold. First,
because of the strong violation of the slow-roll condition induced by the
violation of NEC, $\Delta\chi$ is naturally enhanced compared with that in the
case where the slow-roll condition is preserved, though $\Delta\chi$ cannot
approach $|\Delta\chi|\sim 1$ in the current model. Second, the power spectrum
of parity-violating GWs (i.e., $P_{\rm T}$) can be enhanced for more than
seven orders in higher frequency band by the NEC violation during inflation,
which might be detectable at small scales in the future, while the power
spectrum remains consistent with constrains in the observational window of
CMB. Therefore, the violation of NEC will amplify the observability of the
parity violation effect in the GW background, since
$P_{\text{T}}\cdot\Delta\chi$ is the difference between the power spectrum of
the left- and right-handed GWs.

An enhanced nearly scale-invariant power spectrum of the parity-violating
inflationary primordial GWs in the high-frequency band with peculiar
oscillatory features on $\Delta\chi$ could be distinguishable from that
generated by other mechanism (e.g. Obata:2016oym ; Cai:2021uup ; Zhang:2022xmm
; Peng:2022ttg ; Bastero-Gil:2022fme ). A wide range of multi-frequency
observations aimed at searching for GW signals will bring us rich information
about the gravity and the early Universe. A signal of parity-violating GW
background found by future observations could be attributed to that generated
by our mechanism.

###### Acknowledgements.

We would like to thank Yun-Song Piao, Gen Ye, Hao-Hao Li, Mian Zhu and Chao
Chen for helpful discussions. Y. C. is supported in part by the National
Natural Science Foundation of China (Grant No. 11905224), the China
Postdoctoral Science Foundation (Grant No. 2021M692942) and Zhengzhou
University (Grant No. 32340282). We acknowledge the use of the computing
server Arena317@ZZU.

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