# Tensor non-Gaussianity
in chiral scalar-tensor theories of gravity

Nicola Bartolo  Luca Caloni  Giorgio Orlando  and Angelo Ricciardone

###### Abstract

Violation of parity symmetry in the gravitational sector, which manifests into
unequal left and right circular polarization states of primordial
gravitational waves, represents a way to test high-energy modifications to
general relativity. In this paper we study inflation within recently proposed
chiral scalar-tensor theories of gravity, that extend Chern-Simons gravity by
including parity-violating operators containing first and second derivatives
of the non-minimally coupled scalar (inflaton) field. Given the degeneracy
between different parity-violating theories at the level of the power spectrum
statistics, we make a detailed analysis of the parity violation on primordial
tensor non-Gaussianity. We show, with an explicit computation, that no new
contributions arise in the graviton bispectra if the couplings in the new
operators are constant in a pure de Sitter phase. On the other hand, if the
coupling functions are time-dependent during inflation, the tensor bispectra
acquire non-vanishing contributions from the parity-breaking operators even in
the exact de Sitter limit, with maximal signal in the squeezed and equilateral
configurations. We also comment on the consistency relation of the three-point
function of tensor modes in this class of models and discuss prospects of
detecting parity-breaking signatures through Cosmic Microwave Background
$B$-mode bispectra.

##  1 Introduction

Within the Einstein gravity framework, single (scalar) field slow-roll models
of inflation [1, 2, 3, 4, 5, 6] are in accordance with measurements of the
Cosmic Microwave Background (CMB), in particular with the recent data provided
by WMAP and Planck missions [7, 8, 9, 10]. Due to the invariance under parity
symmetry of general relativity, the two circular polarized right (R) and left
(L)-handed polarization modes of primordial gravitational waves (PGWs) share
exactly the same statistical properties and give the same contribution to the
total tensor power spectrum. However, at very high energies, when inflation
takes place, it might be that signatures of modification to Einstein gravity
are left imprinted on the inflationary quantum fluctuations. In particular,
several candidates of quantum gravity admit the presence of additional parity
breaking gravitational terms yielding to the violation of parity in the
gravitational interaction.

One example of these terms is the so-called gravitational four-dimensional
Chern-Simons term coupled to a scalar field, proposed for the first time in
[11], and which naturally appears in the context of anomaly cancellation in
string theory (via the so-called Green-Schwarz mechanism [12, 13, 14, 15, 16,
17]) and in loop quantum gravity [18, 19, 20, 21, 22]. The four-dimensional
Chern-Simons operator is also commonly introduced as a low energy effective
field theory in an expansion in the curvature invariants [23]. Indeed, it
represents the fully covariant operator that breaks parity with the lowest
number of derivatives. Another example is the so-called three-dimensional
Chern-Simons term which arises from Horava-Lifshitz gravity [24, 25].

In slow-roll inflation with the additional presence of these terms, the parity
violation generates a different behavior in the propagation of the R and
L-handed polarization modes of PGWs. At linear level this parity breaking is
quantified by the relative difference between the super-horizon R and L-handed
tensor power spectra, which in the literature is usually referred to chirality
of PGWs. However, both three [26, 27] and four [28, 29, 30, 31, 32]
dimensional Chern-Simons terms are expected to give a low level of chirality
assuming adiabaticity for PGWs. In fact, the natural production of a large
level of chirality in the tensor power spectrum is typical of other scenarios,
such as the so-called Chromonatural inflation scenario [33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45, 46, 47], where an additional SU(2) gauge boson
is coupled to a pseudo-scalar (axion-like) field through a Chern-Simons like
operator.

Recently, based on the Chern-Simons four-dimensional term, another ghost-free
parity-breaking theory of gravity has been proposed in [48] by including terms
with higher order derivatives of the non-minimally coupled scalar field. The
level of parity breaking induced by these theories in the primordial tensor
power spectrum has been explored in [49]. With respect to the four dimensional
Chern-Simons scenario one of the distinguishable features of higher
derivatives of the coupling scalar field is that they lead to the velocity
birefringence phenomenon, i.e. they induce a difference in the propagation
speed of the two circular polarizations. However, despite this distinct
feature, the final amount of circular polarization (chirality) produced at the
end of inflation is degenerate with respect to e.g. four-dimensional Chern-
Simons gravity, yielding a small level of chirality in the tensor power
spectrum. The limited amount of parity breaking operators quadratic in PGWs
[23, 50, 51] is also responsible for such a degeneracy.

From the observational point of view, it is well known that parity breaking
signatures in the primordial power spectra leave distinct imprints in the CMB
$TB$ and $EB$ angular correlators (see, e.g., the original Ref. [11]).
However, it has been recently shown that CMB angular power spectra, even in
the ideal case, are able to probe parity breaking in the primordial Universe
only for models with maximum chirality [52].

Parity breaking from the primordial Universe can be detected also by ground-
based and space-based interferometers (see, e.g., [53] and Refs. therein). In
fact, even if in principle the co-planar geometry of the planned space-based
interferometers makes these unable to detect such signatures, the dipolar
anisotropy kinematically induced by the motion of the solar system with
respect to the cosmic reference frame allows for a possible measurement even
in absence of other relevant anisotropies (see, e.g., [54] for more details).
However, given the current constraints on tensor power spectrum from CMB
experiments, the forthcoming interferometers (like, e.g., LISA or Einstein
Telescope experiments [55, 56]) are expected to be able to detect the
primordial background of gravitational waves only for a subset of inflationary
scenarios where PGWs have a blue tensor tilt, i.e. under the condition that
their power spectrum can grow at scales smaller than those probed by CMB
experiments [57].

Another proposal for measuring at high precision the chirality of PGWs can be
found in [58], where it has been claimed that, with futuristic measurements of
the 21 cm surveys, one can constrain the chirality at the few percentage
level. However, the effective ability of galaxy surveys in building precision
maps of the 3D galaxy shape makes this proposal still at a very preliminary
stage. Also in [59] possible detection prospects of parity breaking signatures
in the primordial Universe from the 2D galaxy shear power spectrum have been
considered: similarly to the case of CMB observations, detection of parity
breaking in the near future with this method seems challenging due to
instrumental and observational noise.

Thus, given the current experiments, measuring the effects of parity breaking
modifications of Einstein gravity just relying on the power spectrum
statistics seems very challenging. As we already mentioned above with a
specific example, another important aspect to keep in mind is the high
degeneracy regarding the effects of modified gravity operators on the tensor
power spectrum. In fact, the final predictions for the level of chirality turn
out to be proportional to a ratio $H/M_{\text{PV}}$, with $H$ denoting the
Hubble parameter during inflation and $M_{\text{PV}}$ the characteristic
energy scale of the parity-violating theories, which is however unknown in the
absence of a complete theory of quantum gravity. This makes all these models
indistinguishable from the observational point of view just taking into
consideration the power spectrum statistics.

Therefore, it is interesting and crucial to investigate parity breaking
signatures that arise in higher-order correlators, such as the primordial
bispectra. In fact, it is well known that the latter contain features (like
the shape function) that may be characteristic of a given inflationary model
(see, e.g., [60, 61, 62, 10]), removing any kind of degeneracy. For instance,
in [63, 64, 65, 66, 67, 68, 69, 70, 71] possible parity violating signatures
in tensor non-Gaussianity due to modified gravity operators have already been
considered.

In this paper we will extend these studies making an original analysis of the
effects of new parity violating operators explored in [48] on the graviton
bispectrum. In particular we show that, in the de Sitter limit and assuming
constant couplings, the graviton self-interactions contribute to the non-
linear graviton wave-function only via a pure phase, thus not affecting the
graviton bispectrum. On the other hand we show that these tensor bispectra get
interesting features only when we take into account the time dependence of the
coupling functions, giving rise to equilateral and squeezed-type bispectra. We
discuss the implication of these operators on the so-called “consistency
relation” for tensor bispectra and the observational prospects for detecting
these parity-breaking signatures in the primordial bispectra through
experiments focused on the detection of the CMB polarization (like, e.g., the
LiteBIRD experiment [72, 73]).

The paper is organized as follows: in Sec. 2 we will introduce the parity
breaking operators that we will study for the rest of the paper. In Sec. 3 we
will make a review of the effects of these operators on the tensor power
spectrum statistics. In Sec. 4, which contains our original computations, we
will make an analysis of the parity breaking signatures of these operators in
the graviton bispectrum. We will also discuss their original features and
observational prospects in light of the recent forecasts on tensor non-
Gaussianity from CMB polarization data. Finally, in Sec. 5, we will draw our
conclusions.

##  2 Chiral scalar-tensor theories with higher-order derivatives

In this section we introduce chiral scalar-tensor theories with higher-order
derivatives as proposed in [48]. These consist in parity breaking covariant
terms having more derivatives with respect to both the Einstein-Hilbert term
and the so-called gravitational Chern-Simons term, the latter being the full
covariant parity breaking operator with less derivatives (see, e.g., [23,
74]). The action of these parity-breaking theories has the following form

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{M_{Pl}^{2}}{2}R+\mathcal{L}_{\rm
PV}+\mathcal{L}_{\phi}\right]\,,$ (2.1)

where, as usual, $g=det\left[g_{\mu\nu}\right]$, $M_{Pl}^{2}=(8\pi G)^{-1}$ is
the reduced Planck mass, $R$ is the Ricci scalar, $\mathcal{L}_{\rm PV}$ is a
Lagrangian containing parity-violating operators, and $\mathcal{L}_{\phi}$ is
the Lagrangian for a scalar degree of freedom, which is assumed to be non-
minimally coupled to gravity. Since we are interested in the effects of these
parity breaking theories during inflation, we will assume $\phi$ to play the
role of the inflaton field, with $\mathcal{L}_{\phi}$ being the following
Lagrangian

$\displaystyle\mathcal{L}_{\phi}=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\,,$
(2.2)

where $V(\phi)$ denotes the (slow-roll) potential of the inflaton field. The
parity-violating Lagrangian of the theory can be written as the sum of two
pieces

$\displaystyle\mathcal{L}_{\rm PV}=\mathcal{L}_{\rm PV1}+\mathcal{L}_{\rm
PV2}\,,$ (2.3)

where $\mathcal{L}_{\rm PV1}$ contains up to only first derivatives of the
scalar field and is given by [48]

$\displaystyle\mathcal{L}_{\rm PV1}=$
$\displaystyle\sum_{A=1}^{4}a_{A}L_{A}\,,$ (2.4)

where111Here we are implicitly assuming that the higher derivative operators
are suppressed by corresponding powers of the Planck mass, $M_{Pl}$. We will
reintroduce these factors explicitly in the next sections.

$\displaystyle L_{1}$
$\displaystyle=\varepsilon^{\mu\nu\alpha\beta}R_{\alpha\beta\rho\sigma}R_{\mu\nu\;\lambda}^{\;\;\;\rho}\phi^{\sigma}\phi^{\lambda}\,,\qquad\qquad\qquad\qquad
L_{3}=\varepsilon^{\mu\nu\alpha\beta}R_{\alpha\beta\rho\sigma}R^{\sigma}_{\;\;\nu}\phi^{\rho}\phi_{\mu}\,,$
$\displaystyle L_{2}$
$\displaystyle=\varepsilon^{\mu\nu\alpha\beta}R_{\alpha\beta\rho\sigma}R_{\mu\lambda}^{\;\;\;\rho\sigma}\phi_{\nu}\phi^{\lambda}\,,\qquad\qquad\qquad\qquad
L_{4}=\varepsilon^{\mu\nu\rho\sigma}R_{\rho\sigma\alpha\beta}R^{\alpha\beta}_{\;\;\;\;\mu\nu}\phi^{\lambda}\phi_{\lambda}\,,$
(2.5)

where $\varepsilon_{\rho\sigma\alpha\beta}$ is the covariant Levi-Civita
tensor defined in terms of the antisymmetric Levi-Civita symbol
$\epsilon^{\rho\sigma\alpha\beta}$ as
$\varepsilon^{\rho\sigma\alpha\beta}=\epsilon^{\rho\sigma\alpha\beta}/\sqrt{-g}$,
and $\phi^{\mu}=\nabla^{\mu}\phi$ with $\nabla^{\mu}$ denoting the covariant
derivative. Notice that the couplings $a_{A}$ in (2.4) are generic functions
of the scalar field and its kinetic term, i.e.
$a_{A}=a_{A}(\phi,\phi^{\mu}\phi_{\mu})$.

In [48] it has been shown that in the so-called unitary gauge, where the
scalar field is homogeneous, these operators do not introduce the Ostrogradsky
(unstable) modes under the constraint

$4a_{1}+2a_{2}+a_{3}+8a_{4}=0\,,$ (2.6)

that leaves only 3 independent coefficients.

On the other hand, the term $\mathcal{L}_{\rm PV2}$, which includes also
second-order derivatives of the scalar field, reads [48]

$\displaystyle\mathcal{L}_{\rm PV2}$
$\displaystyle=\sum_{A=1}^{7}b_{A}M_{A}\,,$ (2.7)

where

$\displaystyle M_{1}$
$\displaystyle=\varepsilon^{\mu\nu\alpha\beta}R_{\alpha\beta\rho\sigma}\phi^{\rho}\phi_{\mu}\phi^{\sigma}_{\nu}\,,\qquad\qquad\qquad\qquad
M_{4}=\varepsilon^{\mu\nu\alpha\beta}R_{\alpha\beta\rho\sigma}\phi_{\nu}\phi_{\mu}^{\rho}\phi^{\sigma}_{\lambda}\phi^{\lambda},$
$\displaystyle M_{2}$
$\displaystyle=\varepsilon^{\mu\nu\alpha\beta}R_{\alpha\beta\rho\sigma}\phi^{\rho}_{\mu}\phi^{\sigma}_{\nu}\,,\qquad\qquad\qquad\qquad\,\,\,\,\,\,M_{5}=\varepsilon^{\mu\nu\alpha\beta}R_{\alpha\rho\sigma\lambda}\phi^{\rho}\phi_{\beta}\phi^{\sigma}_{\mu}\phi^{\lambda}_{\nu},$
$\displaystyle M_{3}$
$\displaystyle=\varepsilon^{\mu\nu\alpha\beta}R_{\alpha\beta\rho\sigma}\phi^{\sigma}\phi^{\rho}_{\mu}\phi^{\lambda}_{\nu}\phi_{\lambda}\,,\qquad\qquad\qquad\,\,\,\,\,M_{6}=\varepsilon^{\mu\nu\alpha\beta}R_{\beta\gamma}\phi_{\alpha}\phi^{\gamma}_{\mu}\phi^{\lambda}_{\nu}\phi^{\lambda},$
$\displaystyle M_{7}$ $\displaystyle=(\Box\phi)M_{1}\,,$ (2.8)

where in this case $\phi^{\sigma}_{\nu}=\nabla^{\sigma}\nabla_{\nu}\phi$ and
$b_{A}=b_{A}(\phi,\phi^{\mu}\phi_{\mu})$.

Here, in order to avoid the Ostrogradsky modes in the unitary gauge, the
following conditions have to be imposed [48]

$b_{7}=0\,,\qquad b_{6}=2(b_{4}+b_{5})\,,\qquad
b_{2}=-A_{*}^{2}(b_{3}-b_{4})/2\,,$ (2.9)

where $A_{*}=\dot{\phi}(t)/N$ and $N$ is the so-called lapse function of the
spacetime. In this case, we are left with 4 independent coefficients.

In the following sections we will analyze signatures of these operators on
tensor perturbations during slow-roll inflation, reviewing results about power
spectra statistics and making an original analysis about tensor non-
Gaussianity.

In order to do the computations, we will adopt the so-called Arnowitt-Deser-
Misner (ADM) formalism for the perturbed Friedmann-Lemaitre-Robertson-Walker
(FLRW) metric (see e.g. [75, 76, 77]), where the metric reads

$ds^{2}=-(N^{2}-N_{i}N^{i})\,dt^{2}+N_{i}\,dx^{i}dt+h_{ij}\,dx^{i}dx^{j}\,,$
(2.10)

where $N$ and $N_{i}$ are the so-called lapse and shift functions respectively
and $h_{ij}$ is the three-dimensional spatial metric. Focusing only on
transverse and traceless tensor perturbations $\gamma_{ij}$, the (non-linear)
perturbed spatial metric reads [78, 77]

$\displaystyle
h_{ij}=a^{2}\left[\exp\gamma\right]_{ij}\,,\qquad{\gamma_{i}}^{i}=0\,,\quad\partial_{i}\gamma^{ij}=0\,,$
$\displaystyle\left[\exp\gamma\right]_{ij}=\delta_{ij}+\gamma_{ij}+\frac{1}{2!}\gamma_{ik}{\gamma^{k}}_{j}+...\,,$
(2.11)

where $a(t)$ denotes the scale factor of the Universe. Moreover, in standard
gravity $N$ and $N_{i}$ are non-dynamical fields that can be removed in the
final action after solving their algebraic Euler-Lagrange equations in terms
of dynamical fields. In general, in a modified gravity setting, the equations
of motion for $N$ and $N_{i}$ are modified, with the possibility to have an
increased number of dynamical degrees of freedom. However, it has been shown
in [48] that in the unitary gauge and under the constraints (2.6) and (2.9) no
new degrees of freedom are introduced in the theory with respect to standard
gravity.

In App. A we report the expressions of the Lagrangians (2.4)-(2.7) in the
unitary gauge within the ADM formalism, as derived in [48]. We will assume
these to be the fundamental Lagrangians defining the theories that we will
study. Indeed, as commented in [48], because of the presence of the
Ostrogradsky modes in the original theories (2.4)-(2.7), there are two
possible approaches to follow: the first one is to consider these theories as
low energy effective field theories valid up to the energy scale at which the
Ostrogradsky modes appear; the second possibility is to restrict the theories
to the unitary gauge, with the additional constraints (2.6)-(2.9), and treat
them as new Lorentz breaking (and parity violating) theories. The latter is
the approach adopted in Ref. [48] and which we will also follow in the rest of
the paper. Notice that the two Lagrangians (A.1)-(A.8) do not contain any
higher order time derivative of the metric, but only higher order space
derivatives. Because of this fact, these theories break Lorentz invariance
similarly to what happens in Horava-Lifshitz gravity [24, 25]. As we will see
in the rest of the paper, this feature will have important consequences on the
phenomenology of the models under scrutiny, both in the propagation of PGWs
(leading to a speed of propagation of tensor modes different from the speed of
light during inflation) and in the predictions for primordial bispectra.

Before analyzing in details the effects of these parity-breaking operators on
primordial tensor modes, some comments are in order regarding inflation within
these modified gravity theories. The first thing to consider when introducing
some modifications to gravity is how the dynamics at the background level is
affected. In our specific case, it is easy to check that the new operators
have no effects on the background dynamics of inflation, that is the same as
in single field slow-roll models within general relativity. At the
perturbation level, instead, rotational invariance implies that $N$-point
correlation functions of scalar perturbations can have parity-odd signals only
for $N\geq 4$, i.e. the trispectrum ($N=4$) is the lowest order correlator
involving only scalars that can manifest parity-breaking signatures [79].
Thus, the power spectrum of scalar perturbations has the usual expression that
is obtained within general relativity.

Finally let us recall a more technical point, which is however important for
the following computations: if one is not interested in expanding the actions
beyond cubic order in the perturbations, it is sufficient to find the
expressions of the lapse function $N$ and the shift vector $N_{i}$ at first
order only. However, since we are focusing only on tensor fluctuation modes
and it is not possible to have first order perturbations in $N$ and $N_{i}$
including only tensor perturbations, we are left solely with their zero-order
value, namely

$\displaystyle N=1\,,\qquad N_{i}=0\,.$ (2.12)

##  3 Chirality in primordial tensor power-spectra

In this section we are going to review the effects of higher order operators
introduced in [48] on the dynamics of PGWs, with a particular focus on the
parity breaking signatures that arise in the primordial tensor power spectrum.
This has been studied in [49] (see also [80, 81, 82] for an analysis of the
propagation of gravitational waves in the late-time Universe within these
parity-breaking theories). We first start by recalling the Fourier expansion
of PGWs, that is given by

$\gamma_{ij}(\mathbf{x},t)=\int\frac{d^{3}k}{(2\pi)^{3}}\sum_{s=L,R}\gamma_{s}(\mathbf{k},t)\epsilon^{(s)}_{ij}(\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{x}}\,,$
(3.1)

where $\gamma_{s}(\mathbf{k},t)$ are the mode function of primordial tensor
modes, $s$ being the polarization index, and $\epsilon^{(s)}_{ij}(\mathbf{k})$
are the polarization tensor in the chiral basis, i.e. in the basis of L and R
circular polarization states. This is defined in terms of the more common $+$
and $\times$ basis as

$\epsilon^{R}_{ij}(\mathbf{k})=\frac{\epsilon^{+}_{ij}(\mathbf{k})+i\epsilon^{\times}_{ij}(\mathbf{k})}{\sqrt{2}}\,,\quad\epsilon^{L}_{ij}(\mathbf{k})=\frac{\epsilon^{+}_{ij}(\mathbf{k})-i\epsilon^{\times}_{ij}(\mathbf{k})}{\sqrt{2}}\,,$
(3.2)

and the mode functions in the two different bases are related by

$\gamma_{R}(\mathbf{k},t)=\frac{\gamma_{+}(\mathbf{k},t)-i\gamma_{\times}(\mathbf{k},t)}{\sqrt{2}}\,,\quad\gamma_{L}(\mathbf{k},t)=\frac{\gamma_{+}(\mathbf{k},t)+i\gamma_{\times}(\mathbf{k},t)}{\sqrt{2}}\,.$
(3.3)

This decomposition into circular polarization states is particularly useful
when studying parity violating theories. Indeed, while the $+$ and $\times$
polarization states are mixed by the parity violating terms, the equations of
motion for the L and R polarization modes are decoupled. One can prove that
the following relations hold (see, e.g., [28])

$\displaystyle\epsilon_{ij}^{L}(\mathbf{k})\epsilon_{L}^{ij}(\mathbf{k})$
$\displaystyle=\epsilon_{ij}^{R}(\mathbf{k})\epsilon_{R}^{ij}(\mathbf{k})=0\,,$
$\displaystyle\epsilon_{ij}^{L}(\mathbf{k})\epsilon_{R}^{ij}(\mathbf{k})$
$\displaystyle=2\,,$ $\displaystyle\epsilon_{ij}^{R}(-\mathbf{k})$
$\displaystyle=\epsilon_{ij}^{L}(\mathbf{k})\,,$
$\displaystyle\epsilon^{(s)*}_{ij}(-\mathbf{k})$
$\displaystyle=\epsilon^{(s)}_{ij}(\mathbf{k}),$
$\displaystyle\gamma_{L}(-\mathbf{k})$
$\displaystyle=\gamma_{R}^{*}(\mathbf{k})\,,$ $\displaystyle
k_{l}\epsilon^{mlj}{\epsilon_{j}^{(s)j}}(\mathbf{k})$
$\displaystyle=-i\lambda_{s}k\epsilon^{(s)im}(\mathbf{k})\,,$ (3.4)

where $\lambda_{R}=+1$ and $\lambda_{L}=-1$, and $\epsilon^{mlj}$ with 3 Latin
indices denotes the Levi-Civita anti-symmetric symbol. We will make an
extensive use of these relations throughout all of the paper.

The first step required to compute the power spectrum of PGWs is to expand the
Lagrangians introduced in the previous section at second order in tensor
perturbations. Working at leading order in slow-roll parameters, the action
derived by Lagrangian PV1 (A.1) (including the contribution from standard
gravity) at quadratic order in tensor perturbations is given by

$S_{\gamma\gamma}^{\text{PV1}}=\sum_{s=L,R}\int
d\tau\int\frac{d^{3}k}{(2\pi)^{3}}\Big{[}A_{T,s}^{2}|\gamma^{\prime}_{s}(\mathbf{k},\tau)|^{2}-B_{T,s}^{2}k^{2}|\gamma_{s}(\mathbf{k},\tau)|^{2}\Big{]}\,,$
(3.5)

where the prime denotes a derivative with respect to the conformal time $\tau$
and we have defined

$A_{T,s}^{2}\equiv\frac{M_{Pl}^{2}}{2}a^{2}\left(1-\lambda_{s}\frac{k_{phys}}{M_{\text{PV1}}}\right),\quad
B_{T,s}^{2}\equiv\frac{M_{Pl}^{2}}{2}a^{2}\left[1-\frac{4}{M_{Pl}^{6}}\frac{\dot{\phi}^{2}}{a}(\dot{f}+\dot{g})\lambda_{s}k\right]\,,$
(3.6)

with

$M_{\text{PV1}}\equiv\frac{M_{Pl}^{6}}{8}\frac{1}{\dot{\phi}^{2}}\frac{1}{(f+g)H}\,,$
(3.7)

and

$f\equiv a_{1}+\frac{a_{2}}{2}+2a_{4}\,,\qquad
g\equiv\frac{a_{2}}{2}+2a_{4}\,,$ (3.8)

where the dot denotes a derivative with respect to cosmic time. From Eq. (3.6)
we realize that the right-handed graviton modes ($\lambda_{R}=+1$) with a
physical wavenumber, $k_{phys}=k/a$, larger than $M_{\text{PV1}}$ get a
negative kinetic term, thus becoming unstable. At the quantum level this
instability may result in severe problems, since it leads either to a
violation of unitarity or to the propagation of negative energy modes forward
in time. Since unitarity has to be preserved in order for the theory to make
sense, we must admit the presence of particles with negative energies, which
means that the energy spectrum is unbounded from below. However, in such a
case, the vacuum state would be highly unstable under the decay into particles
of positive and negative energies [83]. So, to avoid to deal with this kind of
problem we introduce a UV cut-off $\Lambda\leq M_{\text{PV1}}$ in the theory
and consider only gravitons with $k_{phys}<\Lambda$ at the inset of inflation.
Then, since deep inside the horizon the condition $k_{phys}\gg H$ holds, it
follows that we must assume $M_{\text{PV1}}\gg H$ during inflation in order
for the theory to make sense222Notice that assuming just
$M_{\text{PV1}}\gtrsim H$ is not enough, as in this case a given physical mode
$k_{phys}$ encounters issues near the horizon crossing. Thus, this model can
be considered well-defined only when $M_{\text{PV1}}\gg H$, i.e. in the
“effective field theory” limit, with $M_{\text{PV1}}$ playing the role of
scale of new physics..

As explained in Sec. 2, the couplings $a_{i}$ that enter in the definitions
(3.8) of $f$ and $g$ are functions of the scalar field and its kinetic term
($=\dot{\phi}^{2}/2$ in the unitary gauge). This allows us to reabsorb the
$\dot{\phi}^{2}$ terms in Eqs. (3.6)-(3.7) by defining two new
couplings333Notice that, despite the fact that operators in (2.4) contain 6
derivatives, only some of them act on the perturbations, while the others act
on the background. Thus, since these theories make sense only in unitary
gauge, we are allowed to reabsorb some of the (time) derivatives through a
coupling redefinition.

$f_{1}\equiv\frac{\dot{\phi}^{2}}{M_{Pl}^{4}}f\,,\quad
g_{1}\equiv\frac{\dot{\phi}^{2}}{M_{Pl}^{4}}g\,,$ (3.9)

that are still dimensionless like $f$ and $g$. If we now define the graviton
speed as

$c_{T,s}^{2}\equiv\frac{B_{T,s}^{2}}{A_{T,s}^{2}}\,,$ (3.10)

the action (3.5) can be rewritten as

$S_{\gamma\gamma}^{\text{PV1}}=\sum_{s=L,R}\int
d\tau\int\frac{d^{3}k}{(2\pi)^{3}}A^{2}_{T,s}\Big{[}|\gamma^{\prime}_{s}(\mathbf{k},\tau)|^{2}-c_{T,s}^{2}k^{2}|\gamma_{s}(\mathbf{k},\tau)|^{2}\Big{]}\,.$
(3.11)

By making the field redefinition

$\mu_{s}\equiv A_{T,s}\gamma_{s}\,,$ (3.12)

we can then rewrite the action for the new field as

$S_{\gamma\gamma}^{\text{PV1}}=\sum_{s=L,R}\int
d\tau\int\frac{d^{3}k}{(2\pi)^{3}}\left[|\mu^{\prime}_{s}(\mathbf{k},\tau)|^{2}-c^{2}_{T,s}k^{2}|\mu_{s}(\mathbf{k},\tau)|^{2}+\frac{A^{\prime\prime}_{T,s}}{A_{T,s}}|\mu_{s}(\mathbf{k},\tau)|^{2}\right]\,.$
(3.13)

Varying this action yields the equations of motion for the fields $\mu_{s}$,
which read

$\mu^{\prime\prime}_{s}+\left(c^{2}_{T,s}k^{2}-\frac{A^{\prime\prime}_{T,s}}{A_{T,s}}\right)\mu_{s}=0\,,$
(3.14)

where the time-dependent effective mass is

$\frac{A_{T,s}^{\prime\prime}}{A_{T,s}}=\frac{d}{d\tau}\left(\frac{A_{T,s}^{\prime}}{A_{T,s}}\right)+\left(\frac{A_{T,s}^{\prime}}{A_{T,s}}\right)^{2}=\frac{2+3\epsilon}{\tau^{2}}-\frac{\lambda_{s}k}{\tau}\frac{H}{M_{\text{PV1}}}+\mathcal{O}\left(\epsilon^{2},\frac{H^{2}}{M_{\text{PV1}}^{2}},\epsilon\frac{H}{M_{\text{PV1}}}\right)\,,$
(3.15)

and $\epsilon$ is a slow-roll parameter, defined as

$\epsilon=\frac{M_{Pl}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2}\simeq\frac{1}{2}\frac{\dot{\phi}^{2}}{H^{2}M_{Pl}^{2}}\,.$
(3.16)

The first term in Eq. (3.15) is the usual contribution present in slow-roll
models of inflation with standard gravity. The second term, instead, is a new
contribution that arises similarly in inflationary models with the
gravitational Chern-Simons coupling (see [28, 29, 68, 32]), with the Chern-
Simons mass replaced in our case by $M_{\text{PV1}}$. Thus, the equations of
motion for the fields $\mu_{s}$ are

$\mu^{\prime\prime}_{s}+\left(c_{T,s}^{2}k^{2}-\frac{\nu_{T}^{2}-\frac{1}{4}}{\tau^{2}}+\lambda_{s}\frac{k}{\tau}\frac{H}{M_{\text{PV1}}}\right)\mu_{s}=0\,,$
(3.17)

with

$\nu_{T}\simeq\frac{3}{2}+\epsilon\,,$ (3.18)

and

$c_{T,s}^{2}\simeq 1-\lambda_{s}k\frac{H}{M_{\text{PV1}}}\tau$ (3.19)

at leading order in slow-roll dynamics and in the ratio $H/M_{\text{PV1}}$.

From Eq. (3.17) we realize that, due to the presence of the higher derivative
operators, the speed of propagation of tensor modes is modified in this model,
since $c_{T,s}\neq 1$; this is actually already evident from the action (3.5),
since the time and space derivatives of the field are multiplied by different
functions. In particular, from (3.19) we notice that the two circular
polarization states propagate with a different speed during inflation, being
this dependent on the polarization index $s$: this is the so-called velocity
birefringence phenomenon and it is the main difference at quadratic level with
respect to the case with Chern-Simons gravity. We also stress that the speed
of tensor modes is not constant, but varies with time during inflation. This
is a peculiar feature of this kind of models. Notice that in the limit where
$f_{1}=g_{1}=0$ ($M_{\text{PV1}}=\infty$) we recover the usual result,
$c_{T,s}=1$, as expected.

There is a further important feature that arises from Eq. (3.19): during
inflation one of the two polarization states is superluminal (i.e.
$c_{T,s}>1$), while the other one is subluminal. This was already noticed e.g.
in [81] and is a phenomenological manifestation of the breaking of Lorentz
invariance that occurs in this model. Notice that such an invariance is
recovered at the end of inflation.

We can now canonically quantize the fields $\mu_{s}$ by expanding them in
terms of the creation and annihilation operators

$\hat{\mu}_{s}(\mathbf{k},\tau)=u_{s}(k,\tau)\hat{a}_{s}(\mathbf{k})+u^{*}_{s}(k,\tau)\hat{a}^{\dagger}_{s}(-\mathbf{k})\,.$
(3.20)

The creation and annihilation operators satisfy the equal time commutation
relations

$[\hat{a}_{s}(\mathbf{k}),\hat{a}^{\dagger}_{s^{\prime}}(\mathbf{k}^{\prime})]=(2\pi)^{3}\delta^{(3)}(\mathbf{k}-\mathbf{k}^{\prime})\delta_{ss^{\prime}}\,,\quad[\hat{a}_{s}(\mathbf{k}),\hat{a}_{s^{\prime}}(\mathbf{k}^{\prime})]=0=[\hat{a}_{s}^{\dagger}(\mathbf{k}),\hat{a}_{s^{\prime}}^{\dagger}(\mathbf{k}^{\prime})]\,,$
(3.21)

and act on the vacuum state as

$\hat{a}_{s}|0\rangle=0\,,\quad\langle 0|\hat{a}_{s}^{\dagger}=0\,.$ (3.22)

The equations of motion for the mode functions $u_{s}$ follow
straightforwardly from Eq. (3.17) and read

$u_{s}^{\prime\prime}+\left[k^{2}\left(1-\lambda_{s}k\frac{H}{M_{\text{PV1}}}\tau\right)-\frac{\nu_{T}^{2}-\frac{1}{4}}{\tau^{2}}+\lambda_{s}\frac{k}{\tau}\frac{H}{M_{\text{PV1}}}\right]u_{s}=0\,.$
(3.23)

Now, this equation has the same form as Eq. (4.11) of [49] when taking
$c_{2}=0$ and $c_{1}\epsilon_{*}=H/M_{\text{PV1}}$. As shown in [49],
equations of the kind of (3.23) admit an approximate analytical solution in
terms of Airy functions [84]

$\displaystyle
u_{s}(y)=\alpha\left(\frac{\xi(y)}{g(y)}\right)^{1/4}\textrm{Ai}(\xi)+\beta\left(\frac{\xi(y)}{g(y)}\right)^{1/4}\textrm{Bi}(\xi)\,,$
(3.24)

where $\alpha$ and $\beta$ are two integration constants, $y=-k\tau$ and the
functions $\xi(y)$ and $g(y)$ in our conventions are given by

$\displaystyle
g(y)=\frac{\nu_{T}^{2}}{y^{2}}-1-\lambda_{s}y\frac{H}{M_{\text{PV1}}}+\lambda_{s}\frac{H}{M_{\text{PV1}}}\frac{1}{y}\,,$
(3.25)

and

$\displaystyle\xi(y)=\begin{cases}\left(-\frac{3}{2}\int_{y_{0}^{s}}^{y}\sqrt{g(y^{\prime})}\,dy^{\prime}\right)^{2/3}\qquad\thinspace
y\leq y_{0}^{s}\,,\\\
-\left(\frac{3}{2}\int_{y_{0}^{s}}^{y}\sqrt{g(y^{\prime})}\,dy^{\prime}\right)^{2/3}\qquad
y\geq y_{0}^{s}\,,\end{cases}$ (3.26)

with

$\displaystyle
y_{0}^{s}=-\frac{1-2^{1/3}\left[1+3\left(\frac{H}{M_{\text{PV1}}}\right)^{2}\right]/Y-2^{-1/3}Y}{3\lambda_{s}\frac{H}{M_{\text{PV1}}}}\,,$
(3.27)

where

$\displaystyle Y=$
$\displaystyle\left(Y_{1}+\sqrt{-4\left[1+3\left(\frac{H}{M_{\text{PV1}}}\right)^{2}\right]^{3}+Y_{1}^{2}}\right)^{1/3}\,,$
(3.28) $\displaystyle Y_{1}=$
$\displaystyle-2+27\nu_{T}^{2}\left(\frac{H}{M_{\text{PV1}}}\right)^{2}-9\left(\frac{H}{M_{\text{PV1}}}\right)^{2}\,.$
(3.29)

The final solution for $u_{s}(y)$ is found by matching the sub-horizon limit
($y\rightarrow\infty$) of (3.24) with the following initial condition

$\displaystyle\lim_{y\to+\infty}u_{s}(y)=\sqrt{\frac{1}{2\omega_{k}}}\exp\left(-i\int^{\tau}_{\tau_{i}}\omega_{k}\,d\tau^{\prime}\right)\,,$
(3.30)

that physically corresponds to the assumption that the Universe was initially
in an adiabatic vacuum state. Here $\omega_{k}=k\sqrt{-g(-k\tau)}$ denotes the
dispersion relation of PGWs that can be read off by Eq. (3.23). Notice that
(3.30) corresponds to the usual Bunch-Davies initial vacuum state in the limit
in which $\omega_{k}=k$. By doing this matching we find [49]

$\displaystyle\alpha=\frac{\pi}{2k}e^{i\pi/4}\,,\qquad\qquad\beta=i\frac{\pi}{2k}e^{i\pi/4}\,,$
(3.31)

that fixes our solution. In [49] it has been shown that this analytical
approach is in optimal agreement with exact numerical solutions.

We can now derive the super-horizon power spectra for the two circular
polarization modes of tensor perturbations, which are defined as

$P_{T}^{L}=2\frac{|u_{L}(y)_{y\ll 1}|^{2}}{A_{T,L}^{2}}\,,\qquad
P_{T}^{R}=2\frac{|u_{R}(y)_{y\ll 1}|^{2}}{A_{T,R}^{2}}\,.$ (3.32)

At leading order in slow-roll the final result reads [49]

$P_{T}^{L}=\frac{P_{T}}{2}\exp\left[\frac{\pi}{16}\frac{H}{M_{\text{PV1}}}\right]\,,\qquad
P_{T}^{R}=\frac{P_{T}}{2}\exp\left[-\frac{\pi}{16}\frac{H}{M_{\text{PV1}}}\right]\,,$
(3.33)

where here $P_{T}$ denotes the total tensor power spectrum as predicted in
general relativity.

Now, the level of parity violation in the power spectra of primordial tensor
modes can be quantified by means of the chirality parameter $\chi$, which is
defined as the relative difference between the power spectra of right and left
polarization modes. The leading-order contributions to the chirality can be
computed by Taylor-expanding the exponentials in Eqs. (3.33), since
$H/M_{\text{PV1}}\ll 1$. Thus, we find

$\chi\equiv\frac{P_{T}^{R}-P_{T}^{L}}{P_{T}^{R}+P_{T}^{L}}=-\frac{\pi}{16}\frac{H}{M_{\text{PV1}}}\,.$
(3.34)

From Eq. (3.34) it is clear that, just like it happens with the gravitational
Chern-Simons coupling [28, 29, 68, 32], the chirality is suppressed by the
requirement of dealing with an effective field theory, i.e. $H/M_{PV1}\ll 1$.
This indeed does not come as a surprise, since, as already mentioned, in this
case the equations of motion (3.17) are similar to the case of a Chern-Simons
coupling.

Another interesting quantity to compute is the modifications to the tensor-to-
scalar ratio with respect to the result obtained within standard gravity. The
total dimensionless tensor power spectrum can be written as

$\Delta_{T}^{\text{PV1}}=\Delta_{T}^{R}+\Delta_{T}^{L}=\Delta_{T}\left[1+\frac{\pi^{2}}{256}\left(\frac{H}{M_{\text{PV1}}}\right)^{2}\right]=\Delta_{T}\left(1+\chi^{2}\right)\,.$
(3.35)

Since, as discussed in Sec. 2, the scalar power spectrum does not receive any
contribution from the parity-violating operators, the tensor-to-scalar ratio
can be readily computed as

$r_{\text{PV1}}\equiv\frac{\Delta_{T}^{\text{PV1}}}{\Delta_{S}}=r\left(1+\chi^{2}\right)\,,$
(3.36)

where $r$ is the tensor-to-scalar ratio obtained in slow-roll models without
the parity-breaking operators. We can clearly see that, since the chirality
$\chi$ is $\ll 1$ in this model, the correction to $r$ is suppressed.

The presence of the new operators induces some corrections also to the
spectral index of tensor perturbations, that quantifies how the amplitude of
the fluctuations varies with the scale. This has the following expression

$n_{T}\equiv\frac{d\ln\Delta_{T}^{\text{PV1}}}{d\ln
k}\simeq-2\epsilon+\frac{\pi^{2}}{128}\left(\frac{H}{M_{\text{PV1}}}\right)\left[-2\epsilon\left(\frac{H}{M_{\text{PV1}}}\right)-\frac{\dot{M}_{\text{PV1}}}{M_{\text{PV1}}^{2}}\right]\,.$
(3.37)

Thus, even in this case the corrections to the standard result
($n_{T}=-2\epsilon$) are in general small444 Indeed, it is easy to show that
$\frac{\dot{M}_{\text{PV1}}}{M_{\text{PV1}}^{2}}\simeq\epsilon\left(\frac{H}{M_{\text{PV1}}}\right)-\sqrt{2\epsilon}M_{Pl}\left(\frac{H}{M_{\text{PV1}}}\right)\frac{(\partial
f_{1}/\partial\phi)+(\partial g_{1}/\partial\phi)}{f_{1}+g_{1}}\,.$ (3.38) ,
making PGWs predicted within this model far from the reach of GW
interferometers.

Now we want to make a similar analysis for the Lagrangian PV2, given by Eq.
(A.8). At second order in tensor perturbations it has the following form

$S_{\text{PV2}}^{\gamma\gamma}=\sum_{s=L,R}\int
d\tau\int\frac{d^{3}k}{(2\pi)^{3}}\left[\tilde{A}_{T,s}^{2}|\gamma^{\prime}_{s}(\mathbf{k},\tau)|^{2}-\frac{M_{Pl}^{2}}{2}a^{2}k^{2}|\gamma_{s}(\mathbf{k},\tau)|^{2}\right]\,,$
(3.39)

where we have defined

$\tilde{A}_{T,s}^{2}\equiv\frac{M_{Pl}^{2}}{2}a^{2}\left(1-\lambda_{s}\frac{k_{phys}}{M_{\text{PV2}}}\right)\,,$
(3.40)

with

$M_{\text{PV2}}\equiv\frac{M_{Pl}}{2}\left(\tilde{b}_{1}-b\frac{H}{M_{Pl}}\right)^{-1}\,.$
(3.41)

Just like in the previous case, we have reabsorbed the powers of
$\dot{\phi}^{2}$ by defining two new couplings $\tilde{b}_{1}$ and $b$ as

$\tilde{b}_{1}\equiv\frac{\dot{\phi}^{3}}{M_{Pl}^{6}}b_{1}\,,\quad
b\equiv\frac{\dot{\phi}^{4}}{M_{Pl}^{8}}(b_{4}+b_{5}-b_{3})\,,$ (3.42)

where $b_{1},b_{3},b_{4}$ and $b_{5}$ are the independent couplings of the
model. The energy scale $M_{\text{PV2}}$ has been introduced for the same
reason as $M_{\text{PV1}}$ in the case with only first derivatives of the
scalar field: it represents the energy scale at which the right-handed
graviton modes acquire a negative kinetic term, thus becoming unstable.
Proceeding as in the previous case, we introduce a UV cut-off $\Lambda\leq
M_{\text{PV2}}$ imposing that $k_{phys}<\Lambda$. By requiring also that the
modes started deep inside the horizon, it follows that $H/M_{\text{PV2}}\ll
1$.

By defining the new field $\mu_{s}\equiv\tilde{A}_{T,s}\gamma_{s}$ and
repeating the same steps as in the previous case, we obtain the following
equations of motion

$\mu_{s}^{\prime\prime}+\left(\tilde{c}_{T,s}^{2}k^{2}-\frac{\nu_{T}^{2}-\frac{1}{4}}{\tau^{2}}+\lambda_{s}\frac{k}{\tau}\frac{H}{M_{\text{PV2}}}\right)\mu_{s}=0\,,$
(3.43)

where also in this case

$\nu_{T}\simeq\frac{3}{2}+\epsilon$ (3.44)

holds at leading order in slow-roll parameters. The speed of propagation of
tensor modes during inflation can be written at leading order in slow-roll and
in the ratio $H/M_{\text{PV2}}$ as

$\tilde{c}_{T,s}^{2}\simeq 1-\lambda_{s}k\frac{H}{M_{\text{PV2}}}\tau\,.$
(3.45)

If we then canonically quantize the field $\mu_{s}$ as

$\hat{\mu}_{s}(\mathbf{k},\tau)=u_{s}(k,\tau)\hat{a}_{s}(\mathbf{k})+u^{*}_{s}(k,\tau)\hat{a}^{\dagger}_{s}(-\mathbf{k})\,,$
(3.46)

we can immediately write down the equations of motion for the mode functions
$u_{s}$, which read

$u_{s}^{\prime\prime}+\left[k^{2}\left(1-\lambda_{s}k\frac{H}{M_{\text{PV2}}}\tau\right)-\frac{\nu_{T}^{2}-\frac{1}{4}}{\tau^{2}}+\lambda_{s}\frac{k}{\tau}\frac{H}{M_{\text{PV2}}}\right]u_{s}=0\,.$
(3.47)

This is basically the same equation as Eq. (3.23), with $M_{\text{PV2}}$
replacing $M_{\text{PV1}}$. We can thus write the solution of Eq. (3.47) in
terms of the Airy functions and the integration constants are again fixed by
imposing the adiabatic initial condition (3.30). We can then compute the
leading order contribution to the chirality parameter $\chi$, which takes the
following form

$\chi=-\frac{\pi}{16}\frac{H}{M_{\text{PV2}}}\,.$ (3.48)

As in the PV1 model, the chirality of PGWs is suppressed since
$H/M_{\text{PV2}}\ll 1$. Analogous considerations hold for the tensor-to-
scalar ratio and the spectral index of tensor perturbations, that have the
same expressions as in Eqs. (3.36)-(3.37) with $M_{\text{PV2}}$ replacing
$M_{\text{PV1}}$.

At this point, we want to make a brief comment about the observability of this
signature: as emphasized in the introduction, the CMB $EB$ and $TB$ angular
cross-correlators are able to probe parity breaking in the primordial Universe
only for models predicting maximum chirality, i.e. $\chi\simeq 1$ [52], while
the models under considerations predict $\chi\ll 1$. Moreover, as we have
already discussed, in general we do not expect a significant modification of
the standard slow-roll models tensor tilt (see Eq. (3.37)), making in general
difficult to probe these models with forthcoming interferometers.

Thus, measuring the linear effects of these parity breaking operators seems
very challenging. Another important aspect to keep in mind is the high
degeneracy regarding their signatures on the primordial power spectra. In
fact, the final predictions for the level of chirality (3.34) and (3.48) are
equivalent apart for a redefinition of the $M_{\text{PV}}$ scale, which is
unknown in the absence of a more fundamental theory able to predict its value.
This makes these models indistinguishable just taking into consideration the
power spectrum statistics.

Therefore, it is interesting and crucial to investigate the kind of parity
breaking signatures that arise in higher-order correlators, like, e.g., the
graviton bispectrum. In fact, the latter contains features that may be
characteristic of a given inflationary model, removing any kind of degeneracy.
For this reason, in the next section we are going to make an original study of
the chirality in the primordial tensor bispectra provided by the set of
operators in (2.4) and (2.7).

##  4 Chirality in primordial tensor bispectra

Before entering into the details of the computations, let us first recall the
basic definition of the bispectrum, which is the Fourier transform of the
three-point correlation function. Given three perturbation fields
$\delta_{1}(\mathbf{x},t)$, $\delta_{2}(\mathbf{x},t)$ and
$\delta_{3}(\mathbf{x},t)$, the bispectrum $B(k_{1},k_{2},k_{3})$ is defined
through the relation

$\langle\delta_{1}(\mathbf{k_{1}})\delta_{2}(\mathbf{k_{2}})\delta_{3}(\mathbf{k_{3}})\rangle=(2\pi)^{3}\delta^{(3)}(\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}})B(k_{1},k_{2},k_{3})\,.$
(4.1)

The Dirac delta enforces the invariance under spatial translations and implies
that the wave vectors $\mathbf{k_{1}}$, $\mathbf{k_{2}}$ and $\mathbf{k_{3}}$
must form a closed triangle in Fourier space. Because of rotational
invariance, instead, $B(k_{1},k_{2},k_{3})$ depends only on the magnitude of
the three wave vectors. The bispectrum $B(k_{1},k_{2},k_{3})$ can further be
rewritten as [61, 62, 10]

$B(k_{1},k_{2},k_{3})=f_{\text{NL}}\frac{S(k_{1},k_{2},k_{3})}{(k_{1}k_{2}k_{3})^{2}}\,,$
(4.2)

where $f_{\text{NL}}$ is a dimensionless parameter quantifying the amplitude
of the bispectrum555The exact definition of $f_{\rm NL}$ is fixed except for a
constant normalization that may vary depending on the literature and the kind
of primordial bispectrum under consideration. As an example, within the Planck
mission it has been chosen to normalize the $f_{\text{NL}}$ coefficient of the
scalar bispectrum such that [10]
$f^{\zeta}_{\text{NL}}=\frac{5}{18}\,\frac{B_{\zeta}(k,k,k)}{P^{2}_{\zeta}(k)}\,.$
(4.3) Here $B_{\zeta}(k_{1},k_{2},k_{3})$ and $P_{\zeta}(k)$ denote
respectively the scalar bispectrum and power spectrum from inflation. and the
shape function $S(k_{1},k_{2},k_{3})$ encodes the functional dependence of the
bispectrum on the specific triangle configurations. Typically, the shape
function is normalized such that $S(k,k,k)=1$ in the equilateral limit, where
the three momenta are equal. Notice that, due to the fact that the momenta
form a closed triangle, once we specify two of the three momenta, the last one
is automatically fixed. As a consequence, the shape function depends only on
the ratios between two of the three momenta and the third one (at least for
almost scale invariant bispectra), e.g. $x_{2}=k_{2}/k_{1}$ and
$x_{3}=k_{3}/k_{1}$.

The bispectrum of primordial tensor modes (evaluated at a given conformal time
$\tau$) can be computed by means of the in-in formalism (see, e.g., [77, 85,
86, 87]) as

$\langle\gamma_{s_{1}}(\mathbf{k_{1}},\tau)\gamma_{s_{2}}(\mathbf{k_{2}},\tau)\gamma_{s_{3}}(\mathbf{k_{3}},\tau)\rangle=-i\int_{-\infty}^{\tau}d\tau^{\prime}\langle
0|\left[\gamma_{s_{1}}(\mathbf{k_{1}},\tau)\gamma_{s_{2}}(\mathbf{k_{2}},\tau)\gamma_{s_{3}}(\mathbf{k_{3}},\tau),H_{\text{int}}^{\gamma\gamma\gamma}(\tau^{\prime})\right]|0\rangle\,,$
(4.4)

where
$H_{\text{int}}^{\gamma\gamma\gamma}=-L_{\text{int}}^{\gamma\gamma\gamma}$
denotes the interaction Hamiltonian at cubic order in tensor perturbations. In
the computation of the in-in integrals we will adopt the usual
$i\epsilon$-prescription, which amounts to deform the contour of integration
by making the rotation $-\infty\rightarrow-\infty(1-i\epsilon)$ in the complex
plane (see, e.g., [77]). The $i\epsilon$ contribution turns off the
interactions in the far past and projects onto the vacuum state of the free
theory. We are in particular interested in evaluating the primordial
bispectrum on super-horizon scales, i.e. taking the limit $\tau\rightarrow 0$
of Eq. (4.4).

###  4.1 Graviton bispectra for constant coupling functions

We start with the computation of the bispectrum from the Lagrangian PV1. The
explicit expressions of the operators of $\mathcal{L}_{\text{PV1}}$ at cubic
order in tensor perturbations, as well as the interaction Hamiltonian, can be
found in App. A. We can then plug the interaction Hamiltonian (A.7) into the
in-in formula (4.4) and use the Wick theorem, with the contractions between
the fields that are given by definition by

$\langle
0|\gamma_{s_{1}}(\mathbf{k_{1}},0)\gamma_{s_{2}}(\mathbf{k},\tau)\epsilon^{(s_{2})}_{ij}(\mathbf{k})|0\rangle=(2\pi)^{3}\delta_{s_{1}s_{2}}\delta^{(3)}(\mathbf{k_{1}}+\mathbf{k})u_{s_{1}}(\mathbf{k_{1}},0)u^{*}_{s_{2}}(\mathbf{k},\tau)\epsilon^{(s_{2})*}_{ij}(\mathbf{k})\,,$
(4.5) $\langle
0|\gamma_{s_{1}}(\mathbf{k_{1}},0)\gamma^{\prime}_{s_{2}}(\mathbf{k},\tau)\epsilon^{(s_{2})}_{ij}(\mathbf{k})|0\rangle=(2\pi)^{3}\delta_{s_{1}s_{2}}\delta^{(3)}(\mathbf{k_{1}}+\mathbf{k})u_{s_{1}}(\mathbf{k_{1}},0)u^{*\prime}_{s_{2}}(\mathbf{k},\tau)\epsilon^{(s_{2})*}_{ij}(\mathbf{k})\,.$
(4.6)

The bispectrum of primordial tensor perturbations can then be written as666The
delta functions in the momenta lead to terms like
$\epsilon^{(s)*}_{ij}(-\mathbf{k})$, but
$\epsilon^{(s)*}_{ij}(-\mathbf{k})=\epsilon^{(s)}_{ij}(\mathbf{k})$ (3.4). The
delta functions in the polarization indices instead reduce the initial nine
polarizations (3+6 in the interaction Hamiltonian) down to only three.

$\begin{split}\langle\gamma_{s_{1}}(\mathbf{k_{1}})\gamma_{s_{2}}(\mathbf{k_{2}})\gamma_{s_{3}}(\mathbf{k_{3}})\rangle_{\text{PV1}}&=(2\pi)^{3}\delta^{(3)}(\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}})\,4\,\text{Im}\Big{[}I_{1}\,C_{1}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})\\\
&+I_{2}\,C_{2}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})+I_{3}\,C_{3}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})\\\
&+I_{4}\,C_{4}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})+I_{5}\,C_{5}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})\\\
&+I_{6}\,C_{6}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})+I_{7}\,C_{7}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})\\\
&+I_{8}\,C_{8}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})+I_{9}\,C_{9}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})\Big{]}+\text{perm.}\,(k_{i})\,,\end{split}$
(4.7)

where we have defined

$\displaystyle I_{1}$ $\displaystyle\equiv-
u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}\frac{d\tau^{\prime}}{\tau^{\prime}}\,(f_{1}+g_{1})\,\frac{du_{s_{1}}^{*}}{d\tau^{\prime}}(k_{1},\tau^{\prime})\frac{du_{s_{2}}^{*}}{d\tau^{\prime}}(k_{2},\tau^{\prime})u_{s_{3}}^{*}(k_{3},\tau^{\prime})\,,$
$\displaystyle I_{2}$ $\displaystyle\equiv-
u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}d\tau^{\prime}\,\frac{f_{1}}{2}\,\frac{du_{s_{1}}^{*}}{d\tau^{\prime}}(k_{1},\tau^{\prime})\frac{du_{s_{2}}^{*}}{d\tau^{\prime}}(k_{2},\tau^{\prime})\frac{du_{s_{3}}^{*}}{d\tau^{\prime}}(k_{3},\tau^{\prime})\,,$
$\displaystyle I_{3}$
$\displaystyle\equiv\,u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}d\tau^{\prime}\,\frac{(f_{1}+g_{1})}{2}\,\frac{du_{s_{1}}^{*}}{d\tau^{\prime}}(k_{1},\tau^{\prime})u_{s_{2}}^{*}(k_{2},\tau^{\prime})u_{s_{3}}^{*}(k_{3},\tau^{\prime})\,,$
$\displaystyle I_{4}$
$\displaystyle\equiv\,u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}d\tau^{\prime}\,\frac{g_{1}}{2}\,u_{s_{1}}^{*}(k_{1},\tau^{\prime})u_{s_{2}}^{*}(k_{2},\tau^{\prime})\frac{du_{s_{3}}^{*}}{d\tau^{\prime}}(k_{3},\tau^{\prime})\,,$
$\displaystyle I_{5}$ $\displaystyle\equiv-
u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}d\tau^{\prime}\,\left(f_{1}+\frac{g_{1}}{2}\right)\,\frac{du_{s_{1}}^{*}}{d\tau^{\prime}}(k_{1},\tau^{\prime})u_{s_{2}}^{*}(k_{2},\tau^{\prime})u_{s_{3}}^{*}(k_{3},\tau^{\prime})\,,$
$\displaystyle I_{6}$
$\displaystyle\equiv\,u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}d\tau^{\prime}\,\frac{f_{1}}{2}\,\frac{du_{s_{1}}^{*}}{d\tau^{\prime}}(k_{1},\tau^{\prime})u_{s_{2}}^{*}(k_{2},\tau^{\prime})u_{s_{3}}^{*}(k_{3},\tau^{\prime})\,,$
$\displaystyle I_{7}$ $\displaystyle\equiv-
u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}\frac{d\tau^{\prime}}{\tau^{\prime}}\,(f_{1}+g_{1})\,u_{s_{1}}^{*}(k_{1},\tau^{\prime})\frac{du_{s_{2}}^{*}}{d\tau^{\prime}}(k_{2},\tau^{\prime})\frac{du_{s_{3}}^{*}}{d\tau^{\prime}}(k_{3},\tau^{\prime})\,,$
$\displaystyle I_{8}$
$\displaystyle\equiv\,u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}d\tau^{\prime}\,\frac{(f_{1}+g_{1})}{2}\,u_{s_{1}}^{*}(k_{1},\tau^{\prime})u_{s_{2}}^{*}(k_{2},\tau^{\prime})\frac{du_{s_{3}}^{*}}{d\tau^{\prime}}(k_{3},\tau^{\prime})\,,$
$\displaystyle I_{9}$ $\displaystyle\equiv-
u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}d\tau^{\prime}\,\frac{g_{1}}{2}\,\frac{du_{s_{1}}^{*}}{d\tau^{\prime}}(k_{1},\tau^{\prime})\frac{du_{s_{2}}^{*}}{d\tau^{\prime}}(k_{2},\tau^{\prime})\frac{du_{s_{3}}^{*}}{d\tau^{\prime}}(k_{3},\tau^{\prime})\,,$
(4.8)

where the
$C_{i}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$, whose
full expressions can be found in App. B, are defined in terms of contractions
between the wave vectors and the polarization tensors.

As a first approximation we assume that $a(\tau)\simeq-1/(H\tau)$, which holds
at leading order in slow-roll, and take the Hubble parameter $H$ as constant
during inflation. In the same spirit, we approximate the exact graviton mode
function with the mode function in a de Sitter space-time. For tensor modes,
this is given by (see, e.g., [77, 87])

$u_{s}(k,\tau)=\frac{iH}{M_{Pl}\sqrt{k^{3}}}\thinspace(1+ik\tau)\thinspace
e^{-ik\tau}\,.$ (4.9)

This is justified because the corrections to the mode function (4.9) that
arise in this model are proportional to $\epsilon$ and $H/M_{\text{PV1}}$,
which are both very small during inflation. So, using the de Sitter mode
function gives the leading order contribution to the bispectrum. For the
moment we also take the coupling functions $f_{1}$ and $g_{1}$ to be constant,
leaving the more general case of time dependent couplings for the next
section.
We can now solve analytically the integrals in Eq. (4.8), which give

$\displaystyle
I_{1}=(f_{1}+g_{1})\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)k_{1}^{2}k_{2}^{2}\left[\frac{1}{k_{T}^{2}}+2\frac{k_{3}}{k_{T}^{3}}\right]\,,$
(4.10) $\displaystyle
I_{2}=\frac{f_{1}}{2}\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)k_{1}^{2}k_{2}^{2}k_{3}^{2}\frac{6}{k_{T}^{4}}\,,$
(4.11) $\displaystyle
I_{3}=\frac{f_{1}+g_{1}}{2}\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)k_{1}^{2}\left[\frac{1}{k_{T}^{2}}+2\frac{k_{2}+k_{3}}{k_{T}^{3}}-6\frac{k_{2}k_{3}}{k_{T}^{4}}\right]\,,$
(4.12) $\displaystyle
I_{4}=\frac{g_{1}}{2}\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)k_{3}^{2}\left[\frac{1}{k_{T}^{2}}+2\frac{k_{1}+k_{2}}{k_{T}^{3}}-6\frac{k_{1}k_{2}}{k_{T}^{4}}\right]\,,$
(4.13) $\displaystyle
I_{5}=-\left(f_{1}+\frac{g_{1}}{2}\right)\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)k_{1}^{2}\left[\frac{1}{k_{T}^{2}}+2\frac{k_{2}+k_{3}}{k_{T}^{3}}-6\frac{k_{2}k_{3}}{k_{T}^{4}}\right]\,,$
(4.14) $\displaystyle
I_{6}=\frac{f_{1}}{2}\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)k_{1}^{2}\left[\frac{1}{k_{T}^{2}}+2\frac{k_{2}+k_{3}}{k_{T}^{3}}-6\frac{k_{2}k_{3}}{k_{T}^{4}}\right]\,,$
(4.15) $\displaystyle
I_{7}=(f_{1}+g_{1})\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)k_{2}^{2}k_{3}^{2}\left[\frac{1}{k_{T}^{2}}+2\frac{k_{1}}{k_{T}^{3}}\right]\,,$
(4.16) $\displaystyle
I_{8}=\frac{f_{1}+g_{1}}{2}\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)k_{3}^{2}\left[\frac{1}{k_{T}^{2}}+2\frac{k_{1}+k_{2}}{k_{T}^{3}}-6\frac{k_{1}k_{2}}{k_{T}^{4}}\right]\,,$
(4.17) $\displaystyle
I_{9}=\frac{g_{1}}{2}\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)k_{1}^{2}k_{2}^{2}k_{3}^{2}\frac{6}{k_{T}^{4}}\,,$
(4.18)

where we have defined the total momentum $k_{T}\equiv k_{1}+k_{2}+k_{3}$.
Notice that all the previous integrals give real results. Since all the
$C_{i}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$ terms
in Eq. (4.7) are real (this is true also for the contributions proportional to
$i\epsilon^{ijl}$ in Eqs. (B)-(B)-(B.6)-(B), as can be easily checked by
direct computations), the full bispectrum vanishes:

$\langle\gamma_{s_{1}}(\mathbf{k_{1}})\gamma_{s_{2}}(\mathbf{k_{2}})\gamma_{s_{3}}(\mathbf{k_{3}})\rangle_{\text{PV1}}=0\,.$
(4.19)

Hence, no parity-violating signatures arise in the graviton bispectrum if the
couplings $f_{1}$ and $g_{1}$ are taken to be constant. We have found that our
cubic interactions, even if not trivial, contribute to the non-linear graviton
wave-function only via a pure (field-dependent) phase (see [63, 88, 71] for
more details on this regards), which does not affect super-horizon
correlators. Notice that this is true under the approximation of using the de
Sitter mode function for tensor modes. We expect that, using the exact
solutions from the equation of motion of PGWs, may lead to a non-vanishing
result, but with contributions that are suppressed by slow-roll parameters and
the ratio $H/M_{\text{PV1}}$ (see [64] for a detailed analysis of this issue
in the case of slow-roll inflation with the parity violating Weyl cubic term).

However, since there are no reasons for the coupling functions to be constant
throughout all inflation, it is interesting to study the more general scenario
in which the couplings are free to vary with time. In this case, using the de
Sitter mode function will give the leading order non-vanishing contribution.

Before doing this, let us make a similar analysis for the Lagrangian PV2. Once
again, we refer the reader to App. A for the full expression of the
interaction Hamiltonian at third order in tensor perturbations, which is
reported in Eq. (A.11). By plugging this into the in-in formula (4.4), we find

$\begin{split}\langle\gamma_{s_{1}}(\mathbf{k_{1}})\gamma_{s_{2}}(\mathbf{k_{2}})\gamma_{s_{3}}(\mathbf{k_{3}})\rangle_{\text{PV2}}&=(2\pi)^{3}\delta^{(3)}(\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}})\,\text{Im}\bigg{[}\frac{\lambda_{s_{1}}}{2}\tilde{I}_{1}\,k_{1}\epsilon_{(s_{1})}^{mi}(\mathbf{k_{1}})\epsilon^{(s_{2})}_{mr}(\mathbf{k_{2}})\epsilon^{r}_{(s_{3})i}(\mathbf{k_{3}})\\\
&-\frac{i}{2}(\tilde{I}_{2}+\tilde{I}_{3})\epsilon^{ijl}k_{1r}\epsilon_{l}^{(s_{1})m}(\mathbf{k_{1}})\epsilon_{j}^{(s_{2})r}(\mathbf{k_{2}})\epsilon^{(s_{3})}_{mi}(\mathbf{k_{3}})\bigg{]}+\text{perm.}\,(k_{i})\,,\end{split}$
(4.20)

where we have defined

$\displaystyle\tilde{I}_{1}\equiv-
u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}\frac{d\tau^{\prime}}{\tau^{\prime}}\left[\tilde{b}_{1}\frac{M_{Pl}}{H}-b\right]\frac{du_{s_{1}}^{*}}{d\tau^{\prime}}(k_{1},\tau^{\prime})\frac{du_{s_{2}}^{*}}{d\tau^{\prime}}(k_{2},\tau^{\prime})u_{s_{3}}^{*}(k_{3},\tau^{\prime})\,,$
$\displaystyle\tilde{I}_{2}\equiv-
u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}\frac{d\tau^{\prime}}{\tau^{\prime}}\left[\tilde{b}_{1}\frac{M_{Pl}}{H}-b\right]u_{s_{1}}^{*}(k_{1},\tau^{\prime})\frac{du_{s_{2}}^{*}}{d\tau^{\prime}}(k_{2},\tau^{\prime})\frac{du_{s_{3}}^{*}}{d\tau^{\prime}}(k_{3},\tau^{\prime})\,,$
$\displaystyle\tilde{I}_{3}\equiv
u_{s_{1}}(k_{1},0)u_{s_{2}}(k_{2},0)u_{s_{3}}(k_{3},0)\int_{-\infty}^{0}d\tau^{\prime}\,\frac{b}{2}\,\frac{du_{s_{1}}^{*}}{d\tau^{\prime}}(k_{1},\tau^{\prime})\frac{du_{s_{2}}^{*}}{d\tau^{\prime}}(k_{2},\tau^{\prime})\frac{du_{s_{3}}^{*}}{d\tau^{\prime}}(k_{3},\tau^{\prime})\,.$
(4.21)

Also in this case, we first work under the approximation of constant coupling
functions and using de Sitter mode functions given in Eq. (4.9) for gravitons,
taking the Hubble parameter $H$ to be constant during inflation. The integrals
in (4.21) can then be easily solved and give

$\displaystyle\tilde{I}_{1}=\left[\tilde{b}_{1}\frac{M_{Pl}}{H}-b\right]\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)k_{1}^{2}k_{2}^{2}\left[\frac{1}{k_{T}^{2}}+2\frac{k_{3}}{k_{T}^{3}}\right]\,,$
(4.22)
$\displaystyle\tilde{I}_{2}=\left[\tilde{b}_{1}\frac{M_{Pl}}{H}-b\right]\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)k_{2}^{2}k_{3}^{2}\left[\frac{1}{k_{T}^{2}}+2\frac{k_{1}}{k_{T}^{3}}\right]\,,$
(4.23)
$\displaystyle\tilde{I}_{3}=-3b\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)\frac{k_{1}^{2}k_{2}^{2}k_{3}^{2}}{k_{T}^{4}}\,.$
(4.24)

Just like in the PV1 case, all the integrals give real contributions and thus
none of the parity-breaking operators contributes to the bispectrum of
primordial tensor modes:

$\langle\gamma_{s_{1}}(\mathbf{k_{1}})\gamma_{s_{2}}(\mathbf{k_{2}})\gamma_{s_{3}}(\mathbf{k_{3}})\rangle_{\text{PV2}}=0\,.$
(4.25)

###  4.2 Graviton bispectra for time dependent coupling functions

So far, we have demonstrated that the parity-breaking operators introduced in
[48] give no contributions to the primordial tensor bispectrum, assuming the
de Sitter mode function for gravitons and in the case of constant coupling
functions. However, as already discussed, there are no theoretical reasons why
the couplings should be really constant during all the inflationary phase.
Indeed, the couplings in Eqs. (3.9)-(3.42) depend on the scalar field and its
kinetic term in full generality. Moreover, they could acquire a non-trivial
time evolution because they might also depend on fields that are different
from the inflaton field, being therefore not necessarily limited to a slow-
roll evolution. Furthermore, it has been shown, e.g. in Ref. [65] for the
parity-violating Weyl cubic terms, that, even if these operators do not
contribute to the graviton bispectrum for constant couplings in the de Sitter
limit [64], they can instead leave non-vanishing signatures if the couplings
are free to vary with time.

Motivated by these reasons, we now extend the analysis of the previous section
to the more general scenario where the coupling functions are allowed to
evolve with time during inflation, and investigate whether parity-breaking
signatures may arise in the primordial graviton bispectrum.

We start by analyzing the PV1 model (defined by the Lagrangian (A.8)). We
restrict in particular to the interesting case where

$f_{1}+g_{1}=0\,,$ (4.26)

such that the model is free from instabilities (see Eqs. (3.6),(3.7), (3.9),
and the discussion after Eq. (3.8)). Indeed, when this additional constraint
is satisfied, the $|\gamma^{\prime}|^{2}$ term in the quadratic action
receives no contributions from the parity-violating operators (see Eqs.
(3.5)-(3.6)) and hence the kinetic term is always positive. Notice that in
such a case we can remove the UV cut-off $\Lambda$ and the theory makes sense
also without requiring an effective field theory treatment. If we renounce to
(4.26), then the couplings have to obey $H/M_{PV1}\ll 1$, which will
significantly limit also the amplitude of tensor bispectra.

In order to make explicit computations, we need to assume a specific form of
the coupling functions. In the rest of the paper we consider the case of a
dilaton-like coupling, that naturally arises in theories with extra
dimensions, like string theory. This can be written as
$f_{1}=e^{(\phi-\phi_{*})/M}$, where $M$ is some arbitrary energy scale. In
slow-roll inflation this leads to a coupling that is simply given by a power
of the conformal time777 Indeed, in slow-roll inflation the equation of motion
for the inflaton field is
$\phi^{\prime}\simeq\pm\sqrt{2\epsilon}M_{Pl}\tau^{-1}\,,$ (4.27) where the
$+$ and $-$ signs are for $\partial V/\partial\phi>0$ and $\partial
V/\partial\phi<0$ respectively. This can be integrated to give
$\phi=\phi_{*}\pm\sqrt{2\epsilon}M_{Pl}\ln\left(\frac{\tau}{\tau_{*}}\right)\,.$
(4.28) Substituting this into the exponential dilaton coupling, we end up with
Eq. (4.29).  (see [65] for more details)

$f_{1}(\tau)=\left(\frac{\tau}{\tau_{*}}\right)^{A}\,,\qquad
A=\pm\sqrt{2\epsilon}\,\frac{M_{Pl}}{M}\,,$ (4.29)

where $\epsilon$ is the usual slow-roll parameter, defined as in Eq. (3.16).
The value of $\tau_{*}$ is fixed by the initial condition of Eq. (4.27). For
example, we can take $\tau_{*}$ to be the time when the scale corresponding to
present observable Universe crosses the horizon during inflation, such that
$|\tau_{*}|=k_{*}^{-1}\sim 14$ Gpc [65]. In this case, $f_{1}$ would be of
order unity for the current cosmological scales.

Thanks to the constraint (4.26), the integrals $I_{1}$, $I_{3}$, $I_{7}$,
$I_{8}$ in Eq. (4.8) vanish,

$I_{1}=I_{3}=I_{7}=I_{8}=0\,.$ (4.30)

As regarding the other integrals, once we have replaced the explicit
expression of the mode function (4.9) for gravitons in (4.8), we end up with
integrals of this kind

$\displaystyle
I_{(n,A)}=\tau_{*}^{-A}\int_{-\infty}^{0}d\tau\,\tau^{n+A}e^{ik_{T}\tau}\,,$
(4.31)

with $n+A>-1$. These can be analytically performed (with the usual $i\epsilon$
prescription) and give

$\displaystyle I_{(n,A)}$
$\displaystyle=\tau_{*}^{-A}(-1)^{(A+n)}\,(ik_{T})^{(-1-A-n)}\,\Gamma(n+1+A)$
$\displaystyle=(-1)^{n}\,\,(-i)^{(-n-1)}\,(n+A)!\,(-k_{T}\,\tau_{*})^{-A}(-k_{T})^{(-n-1)}\left[\cos\left(\frac{\pi}{2}A\right)+i\sin\left(\frac{\pi}{2}A\right)\right]\,,$
(4.32)

where $\Gamma(x)=(x-1)!$ is the Gamma function and in the second equality we
have used the Euler’s formula. Using Eq. (4.2) we find

$\displaystyle
I_{2}=-I_{9}=\frac{(3+A)!}{2}\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)\frac{k_{1}^{2}k_{2}^{2}k_{3}^{2}}{k_{T}^{4}}\,(-k_{T}\tau_{*})^{-A}\left[\cos\left(\frac{\pi}{2}A\right)+i\,\sin\left(\frac{\pi}{2}A\right)\right]\,,$
(4.33)
$\displaystyle\begin{split}&I_{4}=\frac{1}{2}\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)(-k_{T}\tau_{*})^{-A}\,k_{3}^{2}\left[-\frac{(1+A)!}{k_{T}^{2}}-(2+A)!\,\frac{k_{1}+k_{2}}{k_{T}^{3}}+(3+A)!\,\frac{k_{1}k_{2}}{k_{T}^{4}}\right]\\\\[5.0pt]
&\quad\times\left[\cos\left(\frac{\pi}{2}A\right)+i\,\sin\left(\frac{\pi}{2}A\right)\right]\,,\end{split}$
(4.34)
$\displaystyle\begin{split}&I_{5}=-I_{6}=\frac{1}{2}\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)(-k_{T}\tau_{*})^{-A}\,k_{1}^{2}\left[-\frac{(1+A)!}{k_{T}^{2}}-(2+A)!\,\frac{k_{2}+k_{3}}{k_{T}^{3}}+(3+A)!\,\frac{k_{2}k_{3}}{k_{T}^{4}}\right]\\\\[5.0pt]
&\qquad\qquad\times\left[\cos\left(\frac{\pi}{2}A\right)+i\,\sin\left(\frac{\pi}{2}A\right)\right]\,.\end{split}$
(4.35)

Notice that for $A=0$ we recover the result of the previous section, since the
imaginary parts of the integrals (4.33)-(LABEL:eq:PV1-integral-5-dilaton)
vanish and the bispectrum receives no contributions from the PV1 operators.
When instead $A\neq 0$ (i.e. for time dependent couplings), the imaginary
parts of these integrals switch on parity-breaking signatures in the
primordial tensor bispectrum. This can be computed by plugging Eqs.
(4.33)-(LABEL:eq:PV1-integral-5-dilaton) into Eq. (4.7). By doing so, we find

$\begin{split}\langle\gamma_{s_{1}}(\mathbf{k_{1}})\gamma_{s_{2}}(\mathbf{k_{2}})\gamma_{s_{3}}(\mathbf{k_{3}})\rangle_{\text{PV1}}&=(2\pi)^{3}\delta^{(3)}(\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}})\left(\frac{H}{M_{Pl}}\right)^{6}\left(\frac{\tau_{*}}{\bar{\tau}}\right)^{-A}\sin\left(\frac{\pi}{2}A\right)\\\
&\times
2B_{s_{1}s_{2}s_{3}}^{\text{PV1}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})+\text{perm.}\,(k_{i})\,,\end{split}$
(4.36)

where we have defined

$\begin{split}B_{s_{1}s_{2}s_{3}}^{\text{PV1}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=\frac{1}{k_{1}^{3}k_{2}^{3}k_{3}^{3}}&\bigg{\\{}(3+A)!\,\frac{k_{1}^{2}k_{2}^{2}k_{3}^{2}}{k_{T}^{4}}\,T_{1}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})+k_{3}^{2}\left[-\frac{(1+A)!}{k_{T}^{2}}\right.\\\
&\left.\;\;-(2+A)!\,\frac{k_{1}+k_{2}}{k_{T}^{3}}+(3+A)!\,\frac{k_{1}k_{2}}{k_{T}^{4}}\right]T_{2}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})\\\
&\;\;+k_{1}^{2}\left[-\frac{(1+A)!}{k_{T}^{2}}-(2+A)!\,\frac{k_{2}+k_{3}}{k_{T}^{3}}+(3+A)!\,\frac{k_{2}k_{3}}{k_{T}^{4}}\right]\\\
&\;\;\times
T_{3}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})\bigg{\\}}\,.\end{split}$
(4.37)

The $T_{i}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$
are again defined in terms of contractions between the wave vectors and the
polarization tensors. Their expressions, which can be written as linear
combinations of the
$C_{i}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$, can
be found in App. B.

In Eq. (4.36) we have evaluated the bispectrum at the horizon-crossing time of
the total momentum $k_{T}$, $\bar{\tau}=-1/k_{T}$ . In fact, it is well known
that, when performing in-in integrals, the main contributions arise around the
horizon crossing of the overall momentum $k_{T}$ in the case of derivative
interactions, as in the models under study. Notice that, because of the term
$(\tau_{*}/\bar{\tau})^{-A}$ arising due to the time dependence of the
coupling, the amplitude of the bispectrum is scale-dependent. In particular,
for values of $A<0$ the amplitude increases going to small scales. For $A>0$,
instead, the amplitude of the graviton bispectrum increases going to large
scales.

In Fig. 1 we plot the shape functions for two different polarization
configurations, the first having $s_{1}=s_{2}=s_{3}=R$ and the second with
$s_{1}=s_{2}=R$ and $s_{3}=L$. The other cases differ from these only by a
minus sign, thus giving the same shape function. In particular, from the
definition (4.2), we show the quantities
$B_{RRR}(1,x_{2},x_{3})x_{2}^{2}x_{3}^{3}$ and
$B_{RRL}(1,x_{2},x_{3})x_{2}^{2}x_{3}^{3}$ as functions of $x_{2}=k_{2}/k_{1}$
and $x_{3}=k_{3}/k_{1}$. The plots are done assuming $A=1$, but we have
checked that the qualitative behaviour of the shapes is independent from the
value of $A$, which thus affects only the amplitude (and, in particular, its
scale-dependence) of the primordial bispectra.

Figure 1: Shapes of the $RRR$ and $RRL$ graviton bispectra in the PV1 model,
with $A=1$. The quantities $B_{RRR}(1,x_{2},x_{3})x_{2}^{2}x_{3}^{3}$ and
$B_{RRL}(1,x_{2},x_{3})x_{2}^{2}x_{3}^{3}$ are plotted as functions of
$x_{2}=k_{2}/k_{1}$ and $x_{3}=k_{3}/k_{1}$. They are both normalized to unity
in the equilateral limit, $x_{2}=x_{3}=1$.

As already remarked in the previous sections, the shape function of the
bispectrum is a powerful tool to discriminate among the various inflationary
models. Indeed, different models of inflation contain different interaction
terms between the dynamical fields of the theory, and thus leave distinctive
signatures in the shapes of the primordial bispectra [10].

To quantify how much a shape $S_{1}$ is similar to a reference shape $S_{2}$,
it is common to introduce the cosine of the two shapes

$\cos(S_{1},S_{2})\equiv\frac{S_{1}\cdot S_{2}}{(S_{1}\cdot
S_{1})^{1/2}(S_{2}\cdot S_{2})^{1/2}}\,,$ (4.38)

where the scalar product is defined as (see, e.g., [89, 61, 65])

$S_{1}\cdot
S_{2}\equiv\sum_{k_{i}}\frac{S_{1}(k_{1},k_{2},k_{3})S_{2}(k_{1},k_{2},k_{3})}{P(k_{1})P(k_{2})P(k_{3})}\,,$
(4.39)

where the summation runs over all the wave vectors that form a triangle in the
momentum space and $P(k)$ denotes the (tensor) power spectrum. By definition,
the cosine is equal to 1 when $S_{1}=S_{2}$. Apart for an overall coefficient
(that eventually cancels out when computing (4.38)), the quantity (4.39) can
be expressed as an integral over $x_{2}$ and $x_{3}$ as888Notice that the
ratio between two scales that enter in a given bispectrum can never be exactly
$0$ or $1$, as in both cases we would deal with unphysical infinite-wavelength
modes. This motivates the presence of the ratios $k_{\rm min}/k_{\rm max}$ in
the extremes of integration in (4.40), so that the cosine between shapes
depends on the ratio between the minimum and maximum scales that a given
experiment can probe.

$S_{1}\cdot S_{2}\propto\int_{\frac{k_{\rm min}}{k_{\rm max}}}^{1-\frac{k_{\rm
min}}{k_{\rm max}}}dx_{2}\int_{1-x_{2}}^{1-\frac{k_{\rm min}}{k_{\rm
max}}}dx_{3}\,x_{2}^{4}x_{3}^{4}\,S_{1}(1,x_{2},x_{3})\,S_{2}(1,x_{2},x_{3})\,.$
(4.40)

As far as CMB experiments are concerned, in the following we evaluate the
cosine between the shapes by summing over all the corresponding configurations
in multipole space (using, as a first approximation, that $\ell\propto k$) for
multipoles $\ell$ ranging, in an ideal case, from $\ell_{\text{min}}=2$ up to
$\ell_{\text{max}}=1000$999In such a case, the choice of
$\ell_{\text{max}}=1000$ is just indicative. Notice that this choice is rather
optimistic if one considers bispectra involving the $B$-mode polarization
field of the CMB (see, e.g., [90, 91]). However, we have explicitly checked
that the cosines between shapes are not very sensitive to $\ell_{\text{max}}$
beyond $\ell_{\text{max}}\sim 100$.. In the case of the shapes plotted in Fig.
1, we find

$\cos\left(B_{RRR}^{\text{PV1}},B_{S}\right)\simeq\begin{cases}0.006\,,\qquad
S=\text{equilateral}\\\ 0.795\,,\qquad S=\text{local}\\\ 0.328\,,\qquad
S=\text{orthogonal}\end{cases}$ (4.41)

for the $RRR$ case, while for the $RRL$ polarizations we have

$\cos\left(B_{RRL}^{\text{PV1}},B_{S}\right)\simeq\begin{cases}0.046\,,\qquad
S=\text{equilateral}\\\ 0.546\,,\qquad S=\text{local}\\\ 0.299\,,\qquad
S=\text{orthogonal}\,.\end{cases}$ (4.42)

From these results and Fig. 1, we realize that the maximum contributions come
mainly from the squeezed configuration (corresponding to, e.g., $k_{3}\ll
k_{1}\simeq k_{2}$).

We now compute the bispectrum from the PV2 Lagrangian (A.8), assuming time
dependent couplings. Analogously to what we have done for the PV1 model, we
restrict to the case with

$\tilde{b}_{1}M_{Pl}/H-b=0\,,$ (4.43)

such that the model is free from instabilities. As a time dependent dilaton-
like coupling we choose

$b(\tau)=\left(\frac{\tau}{\tau_{*}}\right)^{A}\,,\qquad
A=\pm\sqrt{2\epsilon}\,\frac{M_{Pl}}{M}\,.$ (4.44)

The integrals $\tilde{I}_{1}$ and $\tilde{I}_{2}$ in Eq. (4.21) vanish, while
for $\tilde{I}_{3}$ we find

$\displaystyle\tilde{I}_{3}=-(3+A)!\left(\frac{H^{6}}{M_{Pl}^{6}k_{1}^{3}k_{2}^{3}k_{3}^{3}}\right)\frac{k_{1}^{2}k_{2}^{2}k_{3}^{2}}{k_{T}^{4}}\,(-k_{T}\tau_{*})^{-A}\left[\text{cos}\left(\frac{\pi}{2}A\right)+i\,\text{sin}\left(\frac{\pi}{2}A\right)\right]\,.$
(4.45)

The bispectrum in Eq. (4.20) can thus be rewritten as

$\begin{split}\langle\gamma_{s_{1}}(\mathbf{k_{1}})\gamma_{s_{2}}(\mathbf{k_{2}})\gamma_{s_{3}}(\mathbf{k_{3}})\rangle_{\text{PV2}}&=(2\pi)^{3}\delta^{(3)}(\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}})\,\frac{(3+A)!}{2}\left(\frac{H}{M_{Pl}}\right)^{6}\left(\frac{\tau_{*}}{\bar{\tau}}\right)^{-A}\sin\left(\frac{\pi}{2}A\right)\\\
&\times
B_{s_{1}s_{2}s_{3}}^{\text{PV2}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})+\text{perm.}\,(k_{i})\,,\end{split}$
(4.46)

where we have defined

$B_{s_{1}s_{2}s_{3}}^{\text{PV2}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=\frac{1}{k_{1}^{3}k_{2}^{3}k_{3}^{3}}\bigg{\\{}\frac{k_{1}^{2}k_{2}^{2}k_{3}^{2}}{k_{T}^{4}}\left[i\epsilon^{ijl}k_{1r}\,\epsilon_{l}^{(s_{1})m}(\mathbf{k_{1}})\epsilon_{j}^{(s_{2})r}(\mathbf{k_{2}})\epsilon^{(s_{3})}_{mi}(\mathbf{k_{3}})\right]\bigg{\\}}\,.$
(4.47)

Thus, also the PV2 operators give a non-vanishing contribution to the three
graviton bispectrum in the case of time-dependent couplings. Notice that in
the limit where $A=0$ we recover the result of the previous section and the
parity-breaking signatures are not present anymore.

In Fig. 2 we plot the shape functions for the polarization configurations
$RRR$ and $RRL$.

Figure 2: Shapes of the $RRR$ and $RRL$ graviton bispectra in the PV2 model,
with $A=1$. The quantities $B_{RRR}(1,x_{2},x_{3})x_{2}^{2}x_{3}^{3}$ and
$B_{RRL}(1,x_{2},x_{3})x_{2}^{2}x_{3}^{3}$ are plotted as functions of
$x_{2}=k_{2}/k_{1}$ and $x_{3}=k_{3}/k_{1}$. They are both normalized to unity
in the equilateral limit, $x_{2}=x_{3}=1$.

As in the previous case, we can compute the cosine of the shape functions. For
$\ell_{\text{max}}=1000$, we find

$\cos\left(B_{RRR}^{\text{PV2}},B_{S}\right)\simeq\begin{cases}0.999\,,\qquad
S=\text{equilateral}\\\ 0.421\,,\qquad S=\text{local}\\\ 0.205\,,\qquad
S=\text{orthogonal}\end{cases}$ (4.48)

for the $RRR$ case, and

$\cos\left(B_{RRL}^{\text{PV2}},B_{S}\right)\simeq\begin{cases}0.700\,,\qquad
S=\text{equilateral}\\\ 0.429\,,\qquad S=\text{local}\\\ 0.095\,,\qquad
S=\text{orthogonal}\end{cases}$ (4.49)

for the $RRL$ polarizations. The $RRR$ shape peaks in the equilateral
configuration (corresponding to $k_{1}\simeq k_{2}\simeq k_{3}$), as can also
be seen directly from Fig. 2. The $RRL$ shape receives instead a non-
negligible contribution also in the squeezed configuration.

###  4.3 Comments and observational prospects

An interesting feature of the bispectra just derived is that they potentially
lead to the breaking of the single field slow-roll consistency relation for
tensor bispectra [77, 92, 93, 94]101010It has been argued (see, e.g., [95, 96,
97, 98]) that in the context of single field slow-roll models of inflation
primordial bispectra in the squeezed limit (when one of the $k_{i}$ modes is
much smaller than the others) correspond to a gauge artifact: in such a case
one can perform a residual gauge transformation, passing from global
coordinates to Conformal Fermi Coordinates (CFC), the latter being the
coordinate frame of an observer that follows inflation in the background
perturbed by this long-wavelength mode $k_{L}$. After performing this
coordinate transformation, the squeezed limit bispectrum vanishes at leading
order in the $k_{L}/k_{S}$ ratio, with corrections that, by the virtue of the
equivalence principle, must be quadratic in $k_{L}/k_{S}$. However, recently
in [99] it has been shown that at least for the case of the scalar bispectrum
this gauge artifact is valid only in the unphysical exactly infinite-
wavelength limit where the long mode $k_{L}=0$, while for physical modes
$k_{L}\neq 0$ we lose this residual gauge freedom so that the consistency
relations are indeed physical and measurable.. These relations allow to
predict the strict squeezed limit behaviour of the primordial bispectra in
terms of the primordial power spectra [77, 100, 95, 101, 102, 103, 97, 96, 92,
93, 94, 98]. According to these and neglecting the small scale dependence of
the power spectra, the squeezed limit expression of the 3-graviton bispectrum
is predicted to be

$\displaystyle\langle\gamma_{s_{1}}(\mathbf{k_{1}})\gamma_{s_{2}}(\mathbf{k_{2}})\gamma_{s_{3}}(\mathbf{k_{3}})\rangle|_{k_{1}\rightarrow
0}=(2\pi)^{3}\delta^{(3)}(\mathbf{k_{1}}+\mathbf{k_{2}}+\mathbf{k_{3}})\,\frac{3}{2}\,P^{s_{1}}_{T}(k_{1})P^{s_{2}}_{T}(k_{2})\,\epsilon_{ij}^{(s_{1})}\frac{k_{2}^{i}k_{2}^{j}}{k_{2}^{2}}\,\delta_{s_{2}s_{3}}\,.$
(4.50)

In the literature we already find some examples of models breaking the tensor
consistency relation, such as scenarios where we have either additional
particle content (see, e.g., [104, 94]), violation of the adiabatic evolution
of tensor perturbations (see, e.g., [105, 106, 107]), or violation of spatial
diffeomorphisms (see, e.g., [108, 109]).

In our case, we have found that both $RRR$ ($LLL$) and $RRL$ ($LLR$) tensor
bispectra originated by the new PV1 parity breaking operators give a nonzero
contribution in the squeezed limit (see Fig. 1), which is proportional to the
couplings $f_{1}$ and $g_{1}$. On the contrary, primordial tensor power
spectra, being the same of general relativity due to the constraint (4.26), do
not depend by either $f_{1}$ or $g_{1}$. The result is that this squeezed
signal is not predicted by Eq. (4.50), leading to the violation of the
3-graviton consistency relation. A similar pattern is also shared by the $RRL$
($LLR$) bispectrum originated by the PV2 parity breaking operators111111The
reason why the $RRR$ ($LLL$) bispectrum originated by PV2 operators does not
follow this behavior seems to be that, in the case where one correlates the
same polarizations states, the squeezed contributions arise from the operators
with more spatial derivatives, e.g. those in Eq. (A.1) which contain the
Riemann or Ricci tensors in the PV1 case. However, in the PV2 model these
operators do not give any contribution to tensor perturbations (see App. A),
as they also contain the derivative of the lapse function, which vanishes
since $N=1$. Because of this, in the PV2 model we find a signal that peaks in
the equilateral shape when the three polarizations are equal, but peaks in
between the equilateral and squeezed limits for bispectra involving mixed
polarization states. A similar feature has been found in the bispectra of PGWs
in Horava-Lifshitz gravity for some of the operators, as noticed in [66]. We
expect however that squeezed contributions may be present in the PV2 model in
mixed correlators between scalar and tensor perturbations, since in this case
also the operators with more spatial derivatives contribute to the final
result. . Notice that a similar pattern has been found previously in [66, 67]
in the context of studies of primordial non-Gaussianities in Horava–Lifshitz
gravity: in such a case we get squeezed parity violating tensor bispectra that
violate the tensor consistency relation, as they get a nonzero contribution in
the exact squeezed limit that is not completely predicted by how primordial
tensor power spectra in the right-hand side of (4.50) are modified by
Horava–Lifshitz gravity.

We argue that this analogy is not accidental, but related to the fact that
both the theories introduce parity violating operators that violate also the
Lorentz symmetry. In this regards, notice that this signature can be found
only in particular late-time observables sensitive to the parity violation. In
fact, for instance the modification introduced to the overall tensor bispectra
(when summing over all the tensor polarizations) is equal to zero, due to the
fact that the graviton bispectra under consideration have odd parity. Namely,
they obey

$\displaystyle\Braket{\gamma_{R}(\mathbf{k_{1}})\gamma_{R}(\mathbf{k_{2}})\gamma_{R}(\mathbf{k_{3}})}=-\Braket{\gamma_{L}(-\mathbf{k_{1}})\gamma_{L}(-\mathbf{k_{2}})\gamma_{L}(-\mathbf{k_{3}})}\,,$
(4.51)

and analogously for $RRL$ bispectrum versus $LLR$.

Finally, let us discuss the observational prospects for detecting these
parity-breaking signatures in the primordial bispectra through CMB
experiments. In order to find out CMB bispectra sensitive to our parity
breaking signature, we follow the same reasoning as in [65, 110, 70, 111]. The
starting point is the expression of the following spherical harmonic
coefficients of the temperature ($X=T$) and E/B-mode polarization ($X=E/B$)
anisotropies from the two circular polarizations of tensor perturbations [112,
113]

$\displaystyle a_{\ell
m}^{(t)X}=4\pi(-i)^{\ell}\int\frac{d^{3}k}{(2\pi)^{3}}{\cal
T}_{\ell(t)}^{X}(k)\sum_{s=\pm
2}\left(\frac{s}{2}\right)^{x}\gamma_{s}(\mathbf{k})\,{}_{-s}Y_{\ell
m}^{*}(\hat{k})\,,$ (4.52)

where for convenience of the subsequent notation we have defined $\gamma_{\pm
2}(\mathbf{k})\equiv\gamma_{R/L}(\mathbf{k})$. Here, ${}_{s}Y_{\ell
m}(\hat{k})$ denotes a spin-weighted spherical harmonic, ${\cal
T}_{\ell(t)}^{X}(k)$ is the tensor transfer function, and $x\equiv 0\,(1)$ for
$X=T,E\,(B)$. Using (4.52), the CMB bispectra sourced by the primordial
graviton bispectra can be written as

$\displaystyle\Braket{a_{\ell_{1}m_{1}}^{(t)X_{1}}a_{\ell_{2}m_{2}}^{(t)X_{2}}a_{\ell_{3}m_{3}}^{(t)X_{3}}}=$
$\displaystyle\prod_{n=1}^{3}4\pi(-i)^{\ell_{n}}\int\frac{d^{3}k_{n}}{(2\pi)^{3}}{\cal
T}_{\ell_{n}(t)}^{X_{n}}(k_{n})\sum_{s_{n}=\pm
2}\left(\frac{s_{n}}{2}\right)^{x_{n}}{}_{-s_{n}}Y_{\ell_{n}m_{n}}^{*}(\hat{k}_{n})$
$\displaystyle\times\,\Braket{\gamma_{s_{1}}(\mathbf{k_{1}})\gamma_{s_{2}}(\mathbf{k_{2}})\gamma_{s_{3}}(\mathbf{k_{3}})}\,.$
(4.53)

Using the following well-known property of the weighted spherical harmonics

$\displaystyle{}_{-s}Y_{\ell m}(-\hat{k})=(-1)^{\ell}\,{}_{s}Y_{\ell
m}(\hat{k})\,,$ (4.54)

we can rewrite (4.53) as

$\displaystyle\Braket{a_{\ell_{1}m_{1}}^{(t)X_{1}}a_{\ell_{2}m_{2}}^{(t)X_{2}}a_{\ell_{3}m_{3}}^{(t)X_{3}}}=$
$\displaystyle\prod_{n=1}^{3}4\pi(-i)^{\ell_{n}}\int\frac{d^{3}\vec{k}_{n}}{(2\pi)^{3}}{\cal
T}_{\ell_{n}(t)}^{X_{n}}(k_{n})\sum_{s_{n}=\pm
2}\left(\frac{s_{n}}{2}\right)^{x_{n}}{}_{-s_{n}}Y_{\ell_{n}m_{n}}^{*}(\hat{k}_{n})$
$\displaystyle\times\,(-1)^{x_{1}+x_{2}+x_{3}+\ell_{1}+\ell_{2}+\ell_{3}}\Braket{\gamma_{-s_{1}}(-\mathbf{k_{1}})\gamma_{-s_{2}}(-\mathbf{k_{2}})\gamma_{-s_{3}}(-\mathbf{k_{3}})}\,.$
(4.55)

Now, matching Eq. (4.53) with (4.55) under the parity-odd condition (4.51), we
find that

$\Braket{a_{\ell_{1}m_{1}}^{(t)X_{1}}a_{\ell_{2}m_{2}}^{(t)X_{2}}a_{\ell_{3}m_{3}}^{(t)X_{3}}}[1+(-1)^{x_{1}+x_{2}+x_{3}+\ell_{1}+\ell_{2}+\ell_{3}}]=0$
(4.56)

must always hold, independently by the kind of CMB modes that we are cross-
correlating. Thus, a non-vanishing contribution to the CMB angular bispectrum
is confined to the following multipole configurations

$x_{1}+x_{2}+x_{3}+\ell_{1}+\ell_{2}+\ell_{3}={\rm odd}\,.$ (4.57)

It is worth stressing that these combinations are not realized by the usual
parity-conserving theories like Einstein gravity (for which the sum defined in
(4.57) has to be even) and therefore they can become robust indicators of
parity breaking models with odd tensor bispectra if they are detected.

According to the latest forecasts and previsions (see, e.g., [90, 111, 114,
91]), forthcoming CMB experiments focusing on the polarization field (like,
e.g., the LiteBIRD mission) will be able to probe order 1 amplitudes of tensor
squeezed non-Gaussianities through the measurement of the CMB angular
bispectra involving the $B$ modes. This would justify a more detailed analysis
of the detection prospects of these models in CMB angular bispectra in view of
next experiments focusing on the search for the $B$-mode polarization field of
the CMB.

##  5 Conclusions

In this work, we have firstly reviewed the parity breaking signatures in
primordial tensor power spectrum arising from chiral scalar-tensor theories
with higher order derivatives proposed in [48]. We have shown that, due to the
possible appearance of instabilities of one of the two circular polarizations,
the final amount of chirality in PGWs is theoretically constrained to be very
small in a way that it is going to be challenging to measure with current and
forthcoming experiments. We have also remarked that the final prediction on
the level of chirality (3.34) and (3.48) is expected to be degenerate with
that of other parity violating operators previously studied in the context of
inflationary models.

Thus, with the idea of breaking this degeneracy, we have made an original
analysis of the parity breaking effects of these operators on the graviton
bispectrum statistics. We have shown that, taking de Sitter mode functions and
assuming constant couplings, the graviton self-interactions contribute to the
non-linear graviton wave-function only via a pure phase, thus not affecting
the graviton bispectrum.

Therefore, we have fixed a particular setup where the effects of the operators
under study are vanishing on the tensor power spectrum, removing the need for
an effective field theory treatment of the theory. We thus have assumed time
dependent couplings and computed the corresponding signature in the graviton
bispectra for all the possible kind of polarization combinations. In such a
case, we got non-vanishing tensor bispectra which peak in the squeezed and
equilateral configurations (see Figs. 1 and 2), and are also characterized by
odd parity (4.51). As also previously shown, both these peculiarities can
improve the Signal-to-Noise ratio detection of primordial bispectra in CMB
experiments focusing on the $B$-mode detection, making our case study relevant
for the forthcoming CMB experiments aiming to improve the measurements of the
CMB polarization fields (like, e.g., the LiteBIRD satellite). This would
motivate a subsequent forecast on the detection prospects of the models under
consideration using CMB bispectra.

We have finally noticed that the computed bispectra, written in the circular
basis, lead to the breaking of tensor consistency relations. We argue that
this is a manifestation of the simultaneous violation of parity and Lorentz
symmetries, leaving a more general study in this direction for future works.

#### Acknowledgements

We thank Matteo Biagetti, Giovanni Cabass, Alexander Ganz, Sabino Matarrese
and Gianmassimo Tasinato for useful comments and valuable discussions on this
work. G.O. acknowledges partial financial support by ASI Grant No.
2016-24-H.0. N.B. and L.C. acknowledge support from the COSMOS network
(www.cosmosnet.it) through the ASI (Italian Space Agency) Grants 2016-24-H.0
and 2016-24-H.1-2018.

##  Appendix A Interaction Hamiltonians at cubic order in tensor perturbations

###  A.1 PV1 interaction Hamiltonian

Assuming $N=1$ and $N_{i}=0$ for the reason explained in Sec. 2, the PV1
Lagrangian in the ADM formalism under the constraints (2.6) can be written as
[48]

$\displaystyle\sqrt{-g}\,\mathcal{L}_{\text{PV1}}=\frac{2\dot{\phi}^{2}}{M_{Pl}^{4}}\epsilon^{ijl}$
$\displaystyle\left[2(2a_{1}+a_{2}+4a_{4})\left(KK_{mi}D_{l}K_{j}^{\thinspace
m}+\prescript{(3)}{\noindent}{R_{mi}}D_{l}K_{j}^{\thinspace
m}-K_{mi}K^{mn}D_{l}K_{jn}\right)\right.$
$\displaystyle\left.-(a_{2}+4a_{4})\left(2K_{mi}K_{j}^{\thinspace
n}D_{n}K_{l}^{\thinspace
m}+\prescript{(3)}{\noindent}{R_{jlm}}^{n}D_{n}K_{i}^{\thinspace
m}\right)\right]\,,$ (A.1)

where $K_{ij}=\dot{h}_{ij}/2$ is the extrinsic curvature tensor,
$K=h^{ij}K_{ij}$ its trace, $D_{i}$ denotes the three-dimensional covariant
derivative, $\prescript{(3)}{\noindent}{R_{mi}}$ and
$\prescript{(3)}{\noindent}{R_{jlm}}^{n}$ are the three-dimensional Ricci and
Riemann tensors respectively. Notice that we have reintroduced the Planck mass
by dimensional analysis. By expanding $\gamma_{ij}$ up to third order, the
operators present in (A.1) at cubic order in tensor perturbations take the
following forms

$\displaystyle\epsilon^{ijl}KK_{mi}D_{l}K_{j}^{\thinspace m}$
$\displaystyle=\frac{3}{4}\dot{a}a\,\epsilon^{ijl}\left[(\partial_{l}\dot{\gamma}_{j}^{\thinspace
m})\dot{\gamma}_{mr}\gamma^{r}_{\thickspace
i}+(\partial_{r}\gamma_{l}^{\thinspace m})\dot{\gamma}_{j}^{\thinspace
r}\dot{\gamma}_{mi}\right],$ (A.2)
$\displaystyle\epsilon^{ijl}\prescript{(3)}{\noindent}{R_{mi}}D_{l}K_{j}^{\thinspace
m}$
$\displaystyle=\frac{1}{4}\epsilon^{ijl}\left[-\frac{1}{2}(\partial_{r}\partial^{r}\gamma_{mi})(\partial_{l}\gamma_{jk})\dot{\gamma}^{km}-\frac{1}{2}(\partial_{r}\partial^{r}\gamma_{mi})\gamma_{jk}(\partial_{l}\dot{\gamma}^{km})\right.$
$\displaystyle\left.\qquad\quad\;\;\;-\frac{1}{2}(\partial_{r}\partial^{r}\gamma_{mi})(\partial_{k}\gamma_{l}^{\thinspace
m})\dot{\gamma}_{j}^{\thickspace
k}+\frac{1}{2}(\partial_{r}\partial^{r}\gamma_{mi})(\partial^{m}\gamma_{lk})\dot{\gamma}_{j}^{\thinspace
k}\right.$
$\displaystyle\left.\qquad\quad\;\;\;+(\partial_{l}\dot{\gamma}_{j}^{\thinspace
m})\gamma^{kr}(\partial_{k}\partial_{r}\gamma_{mi})+(\partial_{l}\dot{\gamma}_{j}^{\thinspace
m})(\partial_{m}\gamma^{kr})(\partial_{k}\gamma_{ir})\right.$
$\displaystyle\left.\qquad\quad\;\;\;-(\partial_{l}\dot{\gamma}_{j}^{\thinspace
m})\gamma^{kr}(\partial_{k}\partial_{i}\gamma_{mr})-\frac{1}{2}(\partial_{l}\dot{\gamma}_{j}^{\thinspace
m})\gamma^{r}_{\thickspace i}(\partial_{k}\partial^{k}\gamma_{mr})\right.$
$\displaystyle\left.\qquad\quad\;\;\;-\frac{1}{2}(\partial_{l}\dot{\gamma}_{j}^{\thinspace
m})(\partial_{i}\gamma_{r}^{\thinspace k})(\partial_{m}\gamma_{k}^{\thickspace
r})-(\partial_{l}\dot{\gamma}_{j}^{\thinspace
m})(\partial^{k}\gamma_{ri})(\partial^{r}\gamma_{km})\right],$ (A.3)
$\displaystyle\epsilon^{ijl}K_{mi}K^{mn}D_{l}K_{jn}$
$\displaystyle=\epsilon^{ijl}\left[-\frac{1}{2}\dot{a}a\,(\partial_{r}\gamma_{l}^{\thinspace
m})\dot{\gamma}_{jm}\dot{\gamma}_{i}^{\thinspace
r}+\frac{1}{2}\dot{a}a(\partial_{l}\dot{\gamma}_{jr})\gamma_{mi}\dot{\gamma}^{mr}+\frac{1}{8}a^{2}(\partial_{l}\dot{\gamma}_{jr})\dot{\gamma}_{mi}\dot{\gamma}^{mr}\right],$
(A.4) $\displaystyle\epsilon^{ijl}K_{mi}K_{j}^{\thinspace
n}D_{n}K_{l}^{\thinspace m}$
$\displaystyle=\frac{1}{4}\epsilon^{ijl}\left[\dot{a}a\,(\partial_{j}\dot{\gamma}_{l}^{\thinspace
m})\dot{\gamma}_{mr}\gamma^{r}_{\thickspace
i}+\dot{a}a\,(\partial_{r}\gamma_{j}^{\thinspace
m})\dot{\gamma}_{l}^{\thinspace
r}\dot{\gamma}_{mi}+\frac{1}{2}a^{2}(\partial_{r}\dot{\gamma}_{l}^{\thinspace
m})\dot{\gamma}_{j}^{\thinspace r}\dot{\gamma}_{mi}\right],$ (A.5)
$\displaystyle\epsilon^{ijl}\prescript{(3)}{\noindent}{R_{jlm}}^{n}D_{n}K_{i}^{\thinspace
m}$
$\displaystyle=\frac{1}{4}\epsilon^{ijl}\bigg{[}-\gamma_{lr}(\partial_{m}\partial_{j}\gamma^{rn})(\partial_{n}\dot{\gamma}_{i}^{\thinspace
m})+\gamma_{lr}(\partial^{n}\partial_{j}\gamma^{r}_{\thickspace
m})(\partial_{n}\dot{\gamma}_{i}^{\thinspace m})$
$\displaystyle\qquad\quad\;\;\;\thinspace-2\gamma^{pn}(\partial_{m}\partial_{l}\gamma_{pj})(\partial_{n}\dot{\gamma}_{i}^{\thinspace
m})-2\gamma^{pn}(\partial_{p}\partial_{j}\gamma_{lm})(\partial_{n}\dot{\gamma}_{i}^{\thinspace
m})$
$\displaystyle\qquad\quad\;\;\;\thinspace+(\partial_{m}\gamma_{j}^{\thickspace
r})(\partial_{r}\gamma_{l}^{\thickspace
n})(\partial_{n}\dot{\gamma}_{i}^{\thinspace
m})-(\partial_{m}\gamma_{j}^{\thickspace
r})(\partial^{n}\gamma_{lr})(\partial_{n}\dot{\gamma}_{i}^{\thinspace m})$
$\displaystyle\qquad\quad\;\;\;\thinspace-(\partial^{n}\gamma_{j}^{\thickspace
r})(\partial_{r}\gamma_{lm})(\partial_{n}\dot{\gamma}_{i}^{\thinspace
m})-(\partial^{n}\gamma_{mr})(\partial_{l}\gamma^{r}_{\thickspace
j})(\partial_{n}\dot{\gamma}_{i}^{\thinspace m})$
$\displaystyle\qquad\quad\;\;\;\thinspace+(\partial_{r}\gamma_{mj})(\partial_{l}\gamma^{nr})(\partial_{n}\dot{\gamma}_{i}^{\thinspace
m})-(\partial_{r}\gamma_{mj})(\partial^{r}\gamma_{l}^{\thickspace
n})(\partial_{n}\dot{\gamma}_{i}^{\thinspace m})$
$\displaystyle\qquad\quad\;\;\;\thinspace-(\partial_{j}\gamma_{rm})(\partial_{l}\gamma^{nr})(\partial_{n}\dot{\gamma}_{i}^{\thinspace
m})+(\partial_{j}\gamma_{rm})(\partial^{r}\gamma_{l}^{\thickspace
n})(\partial_{n}\dot{\gamma}_{i}^{\thinspace m})$
$\displaystyle\qquad\quad\;\;\;\thinspace+2(\partial_{m}\partial_{l}\gamma^{n}_{\thickspace\,j})\dot{\gamma}_{ik}(\partial_{n}\gamma^{km})+(\partial_{m}\partial_{l}\gamma^{n}_{\thickspace\,j})\gamma_{ik}(\partial_{n}\dot{\gamma}^{km})$
$\displaystyle\qquad\quad\;\;\;\thinspace+(\partial^{n}\partial_{j}\gamma_{lm})\gamma_{ik}(\partial_{n}\dot{\gamma}^{km})-(\partial_{m}\partial_{l}\gamma^{n}_{\thickspace\,j})(\partial_{i}\gamma_{n}^{\thickspace
k})\dot{\gamma}_{k}^{\thickspace m}$
$\displaystyle\qquad\quad\;\;\;\thinspace-(\partial^{n}\partial_{j}\gamma_{lm})(\partial_{i}\gamma_{n}^{\thickspace
k})\dot{\gamma}_{k}^{\thickspace
m}+(\partial_{m}\partial_{l}\gamma^{n}_{\thickspace\,j})(\partial^{k}\gamma_{ni})\dot{\gamma}_{k}^{\thickspace
m}$
$\displaystyle\qquad\quad\;\;\;\thinspace+(\partial^{n}\partial_{j}\gamma_{lm})(\partial^{k}\gamma_{ni})\dot{\gamma}_{k}^{\thickspace
m}-(\partial_{m}\partial_{l}\gamma^{n}_{\thickspace\,j})(\partial^{m}\gamma_{nk})\dot{\gamma}_{i}^{\thinspace
k}\bigg{]}.$ (A.6)

From these, we can then compute the interaction Hamiltonian. In Fourier space,
this reads

$\begin{split}H_{\gamma\gamma\gamma}^{\text{PV1}}(\tau)&=\sum_{s_{1},s_{2},s_{3}}\int
d^{3}k\int d^{3}p\int d^{3}q\
\frac{\delta^{(3)}(\mathbf{k}+\mathbf{p}+\mathbf{q})}{(2\pi)^{6}}2\Bigg{\\{}\lambda_{s_{1}}(f_{1}+g_{1})aHk\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{\prime
s_{2}}_{\mathbf{p}}\gamma^{s_{3}}_{\mathbf{q}}\\\
&\times\epsilon_{(s_{1})}^{mi}(\mathbf{k})\epsilon_{(s_{2})}^{mr}(\mathbf{p})\epsilon_{(s_{3})i}^{r}(\mathbf{q})+\lambda_{s_{2}}\frac{(f_{1}+g_{1})}{2}k^{2}p\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{s_{3}}_{\mathbf{q}}\epsilon^{(s_{1})}_{mi}(\mathbf{k})\epsilon_{k}^{(s_{2})i}(\mathbf{p})\epsilon_{(s_{3})}^{km}(\mathbf{q})\\\
&+\lambda_{s_{3}}(f_{1}+g_{1})qk_{k}q_{r}\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{s_{3}}_{\mathbf{q}}\epsilon_{j}^{(s_{1})m}(\mathbf{k})\epsilon_{(s_{2})}^{kr}(\mathbf{p})\epsilon_{m}^{(s_{3})j}(\mathbf{q})-\lambda_{s_{1}}\left(f_{1}+\frac{g_{1}}{2}\right)kp_{m}q_{k}\\\
&\times\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{s_{3}}_{\mathbf{q}}\epsilon_{(s_{1})}^{mi}(\mathbf{k})\epsilon_{(s_{2})}^{kr}(\mathbf{p})\epsilon^{s_{3}}_{ir}(\mathbf{q})+\lambda_{s_{1}}\frac{f_{1}}{2}kq^{2}\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{s_{3}}_{\mathbf{q}}\epsilon_{(s_{1})}^{mi}(\mathbf{k})\epsilon^{r}_{(s_{2})i}(\mathbf{p})\epsilon^{(s_{3})}_{mr}(\mathbf{q})\\\
&+\lambda_{s_{1}}\frac{f_{1}}{2}kp_{i}q_{m}\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{s_{3}}_{\mathbf{q}}\epsilon_{(s_{1})}^{mi}(\mathbf{k})\epsilon_{r}^{(s_{2})k}(\mathbf{p})\epsilon_{k}^{(s_{3})r}(\mathbf{q})+\lambda_{s_{2}}(f_{1}+g_{1})pp^{k}q^{r}\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{s_{3}}_{\mathbf{q}}\\\
&\times\epsilon_{(s_{1})}^{mi}(\mathbf{k})\epsilon_{r}^{(s_{2})j}(\mathbf{p})\epsilon^{(s_{3})}_{km}(\mathbf{q})-\lambda_{s_{1}}\frac{f_{1}}{2}k\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{\prime s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon_{r}^{(s_{1})i}(\mathbf{k})\epsilon_{mi}^{(s_{2})}(\mathbf{p})\epsilon_{(s_{3})}^{mr}(\mathbf{q})\\\
&+\lambda_{s_{1}}\frac{g_{1}}{2}kk_{n}p_{m}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon_{r}^{(s_{1})i}(\mathbf{k})\epsilon_{(s_{2})}^{rn}(\mathbf{p})\epsilon_{i}^{(s_{3})m}(\mathbf{q})+\lambda_{s_{2}}\frac{g_{1}}{2}k^{2}p\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\\\
&\times\epsilon^{(s_{1})}_{mr}(\mathbf{k})\epsilon_{(s_{2})}^{ri}(\mathbf{p})\epsilon_{i}^{(s_{3})m}(\mathbf{q})+\lambda_{s_{1}}\frac{g_{1}}{2}kk_{m}q_{n}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon_{(s_{1})}^{ni}(\mathbf{k})\epsilon^{(s_{2})}_{ik}(\mathbf{p})\epsilon_{(s_{3})}^{km}(\mathbf{q})\\\
&+\lambda_{s_{1}}\frac{g_{1}}{2}k(\mathbf{k}\cdot\mathbf{p})\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon_{m}^{(s_{1})i}(\mathbf{k})\epsilon^{(s_{2})}_{ik}(\mathbf{p})\epsilon_{(s_{3})}^{km})(\mathbf{q})-\lambda_{s_{1}}\frac{g_{1}}{2}kk_{m}p_{i}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\\\
&\times\epsilon_{(s_{1})}^{ni}(\mathbf{k})\epsilon_{n}^{(s_{2})k}(\mathbf{p})\epsilon_{k}^{(s_{3})m}(\mathbf{q})+\lambda_{s_{1}}\frac{g_{1}}{2}kk^{n}p_{i}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon_{m}^{(s_{1})i}(\mathbf{k})\epsilon_{n}^{(s_{2})k}(\mathbf{p})\epsilon_{k}^{(s_{3})m}(\mathbf{q})\\\
&+\lambda_{s_{1}}\frac{g_{1}}{2}kk_{m}p^{k}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon_{(s_{1})}^{ni}(\mathbf{k})\epsilon^{(s_{2})}_{ni}(\mathbf{p})\epsilon_{k}^{(s_{3})m}(\mathbf{q})-\lambda_{s_{1}}\frac{g_{1}}{2}kk^{n}p^{k}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\\\
&\times\epsilon_{m}^{(s_{1})i}(\mathbf{k})\epsilon^{(s_{2})}_{ni}(\mathbf{p})\epsilon_{k}^{(s_{3})m}(\mathbf{q})+\epsilon^{ijl}\bigg{[}-i(f_{1}+g_{1})aHk_{r}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{\prime
s_{2}}_{\mathbf{p}}\gamma^{\prime s_{3}}_{\mathbf{q}}\\\
&\times\epsilon_{l}^{(s_{1})m}(\mathbf{k})\epsilon_{j}^{(s_{2})r}(\mathbf{p})\epsilon^{(s_{3})}_{mi}(\mathbf{q})-i\frac{(f_{1}+g_{1})}{2}k^{2}q_{l}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon^{(s_{1})}_{mi}(\mathbf{k})\epsilon^{(s_{2})}_{jk}(\mathbf{p})\epsilon_{(s_{3})}^{km}(\mathbf{q})\\\
&-i\frac{(f_{1}+g_{1})}{2}k^{2}p_{k}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon^{(s_{1})}_{mi}(\mathbf{k})\epsilon_{l}^{(s_{2})m}(\mathbf{p})\epsilon_{j}^{(s_{3})k}(\mathbf{q})+i\frac{(f_{1}+g_{1})}{2}k^{2}p^{m}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\\\
&\times\epsilon^{(s_{1})}_{mi}(\mathbf{k})\epsilon^{(s_{2})}_{lk}(\mathbf{p})\epsilon_{j}^{(s_{3})k}(\mathbf{q})+i\left(f_{1}+\frac{g_{1}}{2}\right)k_{k}p_{l}q_{r}\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{s_{3}}_{\mathbf{q}}\epsilon_{j}^{(s_{1})m}(\mathbf{k})\epsilon_{(s_{2})}^{kr}(\mathbf{p})\epsilon^{(s_{3})}_{mi}(\mathbf{q})\\\
&-i\left(f_{1}+\frac{g_{1}}{2}\right)k_{n}p_{l}q_{i}\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{s_{3}}_{\mathbf{q}}\epsilon_{j}^{(s_{1})m}(\mathbf{k})\epsilon_{(s_{2})}^{nr}(\mathbf{p})\epsilon^{(s_{3})}_{mr}(\mathbf{q})+i\left(f_{1}+\frac{g_{1}}{2}\right)p^{k}q_{l}q^{r}.\\\
&\times\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{s_{3}}_{\mathbf{q}}\epsilon_{j}^{(s_{1})m}(\mathbf{k})\epsilon_{ri}^{(s_{2})}(\mathbf{p})\epsilon^{(s_{3})}_{km}(\mathbf{q})+i\frac{g_{1}}{2}k_{r}\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{\prime s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon_{l}^{(s_{1})m}(\mathbf{k})\epsilon_{j}^{(s_{2})r}(\mathbf{p})\epsilon^{(s_{3})}_{mi}(\mathbf{q})\\\
&-i\frac{g_{1}}{2}(\mathbf{p}\cdot\mathbf{q})p_{j}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon^{(s_{1})}_{lr}(\mathbf{k})\epsilon_{(s_{2})m}^{r}(\mathbf{p})\epsilon_{i}^{(s_{3})m}(\mathbf{q})-i\frac{g_{1}}{2}k_{m}p_{r}q_{n}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\\\
&\times\epsilon_{j}^{(s_{1})r}(\mathbf{k})\epsilon_{l}^{(s_{2})n}(\mathbf{p})\epsilon_{i}^{(s_{3})m}(\mathbf{q})-i\frac{g_{1}}{2}(\mathbf{k}\cdot\mathbf{p})k_{m}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon_{j}^{(s_{1})r}(\mathbf{k})\epsilon_{lr}^{(s_{2})}(\mathbf{p})\epsilon_{i}^{(s_{3})m}(\mathbf{q})\bigg{]}\Bigg{\\}},\end{split}$
(A.7)

where the prime denotes a derivative with respect to conformal time.

###  A.2 PV2 interaction Hamiltonian

Assuming $N=1$ and $N_{i}=0$, the PV2 Lagrangian in the ADM formalism under
the constraints (2.9) can be written as [48]

$\sqrt{-g}\,\mathcal{L}_{\text{PV2}}=2\frac{\dot{\phi}^{3}}{M_{Pl}^{5}}\epsilon^{ijl}\left[b_{1}K_{mi}D_{l}K_{j}^{\thinspace
m}+\frac{(b_{4}+b_{5}-b_{3})}{M_{Pl}^{3}}\dot{\phi}K_{mi}K_{j}^{\thinspace
n}D_{n}K_{l}^{\thinspace m}\right]\,.$ (A.8)

Notice that the operators in the second line of the action (3.23) in [48] do
not give any contribution to tensor perturbations, since $D_{i}N=0$. At cubic
order in tensor perturbations, the operators in (A.8) read

$\displaystyle\epsilon^{ijl}K_{mi}D_{l}K_{j}^{\thinspace m}$
$\displaystyle=\frac{1}{4}a^{2}\,\epsilon^{ijl}\left[(\partial_{l}\dot{\gamma}_{j}^{\thinspace
m})\dot{\gamma}_{mr}\gamma^{r}_{\thickspace
i}+(\partial_{r}\gamma_{l}^{\thinspace m})\dot{\gamma}_{j}^{\thinspace
r}\dot{\gamma}_{mi}\right],$ (A.9)
$\displaystyle\epsilon^{ijl}K_{mi}K_{j}^{\thinspace n}D_{n}K_{l}^{\thinspace
m}$
$\displaystyle=\frac{1}{4}\epsilon^{ijl}\left[\dot{a}a\,(\partial_{j}\dot{\gamma}_{l}^{\thinspace
m})\dot{\gamma}_{mr}\gamma^{r}_{\thickspace
i}+\dot{a}a\,(\partial_{r}\gamma_{j}^{\thinspace
m})\dot{\gamma}_{l}^{\thinspace
r}\dot{\gamma}_{mi}+\frac{1}{2}a^{2}(\partial_{r}\dot{\gamma}_{l}^{\thinspace
m})\dot{\gamma}_{j}^{\thinspace r}\dot{\gamma}_{mi}\right].$ (A.10)

The interaction Hamiltonian in Fourier space is thus given by

$\begin{split}H_{\gamma\gamma\gamma}^{\text{PV2}}=&\sum_{s_{1},s_{2},s_{3}}\int
d^{3}k\int d^{3}p\int d^{3}q\
\frac{\delta^{(3)}(\mathbf{k}+\mathbf{p}+\mathbf{q})}{(2\pi)^{6}}\left\\{\frac{1}{2}\left[\tilde{b}_{1}M_{Pl}-bH\right]\Big{[}\lambda_{s_{1}}ak\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{\prime
s_{2}}_{\mathbf{p}}\gamma^{s_{3}}_{\mathbf{q}}\right.\\\
&\left.\times\epsilon_{(s_{1})}^{mi}(\mathbf{k})\epsilon_{mr}^{(s_{2})}(\mathbf{p})\epsilon^{r}_{(s_{3})i}(\mathbf{q})+i\epsilon^{ijl}ak_{r}\gamma^{s_{1}}_{\mathbf{k}}\gamma^{\prime
s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon_{l}^{(s_{1})m}(\mathbf{k})\epsilon_{j}^{(s_{2})r}(\mathbf{p})\epsilon_{mi}^{(s_{3})}(\mathbf{q})\Big{]}\right.\\\
&\left.+i\frac{b}{4}\epsilon^{ijl}k_{r}\gamma^{\prime
s_{1}}_{\mathbf{k}}\gamma^{\prime s_{2}}_{\mathbf{p}}\gamma^{\prime
s_{3}}_{\mathbf{q}}\epsilon_{l}^{(s_{1})m}(\mathbf{k})\epsilon_{j}^{(s_{2})r}(\mathbf{p})\epsilon_{mi}^{(s_{3})}(\mathbf{q})\right\\}.\end{split}$
(A.11)

##  Appendix B Explicit expressions of
$C_{i}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$ and
$T_{i}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$

We report here the complete expressions of the contributions that appear in
the PV1 bispectrum (4.7):

$\displaystyle
C_{1}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=C_{2}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=\lambda_{s_{1}}k_{1}\epsilon_{(s_{1})}^{mi}(\mathbf{k_{1}})\epsilon^{(s_{2})}_{mr}(\mathbf{k_{2}})\epsilon_{(s_{3})i}^{r}(\mathbf{k_{3}})\,,$
(B.1) $\displaystyle
C_{3}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=\lambda_{s_{2}}k_{2}\left[k_{1}^{2}\epsilon_{mi}^{(s_{1})}(\mathbf{k_{1}})\epsilon_{k}^{(s_{2})i}(\mathbf{k_{2}})\epsilon_{(s_{3})}^{km}(\mathbf{k_{3}})+2k_{2}^{k}k_{3}^{r}\,\epsilon_{j}^{(s_{1})m}(\mathbf{k_{1}})\epsilon_{r}^{(s_{2})j}(\mathbf{k_{2}})\epsilon_{km}^{(s_{3})}(\mathbf{k_{3}})\right]$
$\displaystyle\qquad\qquad\qquad\qquad\,+2\lambda_{s_{3}}k_{3}k_{1k}k_{3r}\,\epsilon_{j}^{(s_{1})m}(\mathbf{k_{1}})\epsilon_{(s_{2})}^{kr}(\mathbf{k_{2}})\epsilon_{m}^{(s_{3})j}(\mathbf{k_{3}})\,,$
(B.2) $\displaystyle
C_{4}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=\left[\lambda_{s_{2}}k_{1}^{2}k_{2}+\lambda_{s_{1}}(\mathbf{k_{1}}\cdot\mathbf{k_{2}})k_{1}\right]\epsilon_{ri}^{(s_{1})}(\mathbf{k_{1}})\epsilon_{(s_{2})}^{im}(\mathbf{k_{2}})\epsilon_{m}^{(s_{3})r}(\mathbf{k_{3}})$
$\displaystyle\qquad\qquad\qquad\qquad+\lambda_{s_{1}}k_{1}k_{1n}k_{2m}\Big{[}\epsilon_{r}^{(s_{1})i}(\mathbf{k_{1}})\epsilon_{(s_{2})}^{rn}(\mathbf{k_{2}})\epsilon_{i}^{(s_{3})m}(\mathbf{k_{3}})-\epsilon_{(s_{1})}^{rm}(\mathbf{k_{1}})\epsilon_{r}^{(s_{2})i}(\mathbf{k_{2}})\epsilon_{i}^{(s_{3})n}(\mathbf{k_{3}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;+\epsilon_{r}^{(s_{1})m}(\mathbf{k_{1}})\epsilon_{(s_{2})}^{ni}(\mathbf{k_{2}})\epsilon_{i}^{(s_{3})r}(\mathbf{k_{3}})+\epsilon_{(s_{1})}^{ri}(\mathbf{k_{1}})\epsilon_{ri}^{(s_{2})}(\mathbf{k_{2}})\epsilon_{(s_{3})}^{mn}(\mathbf{k_{3}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\;\;-\epsilon_{r}^{(s_{1})i}(\mathbf{k_{1}})\epsilon^{n}_{(s_{2})i}(\mathbf{k_{2}})\epsilon_{(s_{3})}^{mr}(\mathbf{k_{3}})\Big{]}$
$\displaystyle\qquad\qquad\qquad\qquad+\lambda_{s_{1}}k_{1}k_{1n}k_{3m}\,\epsilon_{(s_{1})}^{mi}(\mathbf{k_{1}})\epsilon^{(s_{2})}_{ir}(\mathbf{k_{2}})\epsilon_{(s_{3})}^{rn}(\mathbf{k_{3}})$
$\displaystyle\qquad\qquad\qquad\qquad+i\epsilon^{ijl}\Big{[}(\mathbf{k_{2}}\cdot\mathbf{k_{3}})k_{2j}\,\epsilon^{(s_{1})}_{lr}(\mathbf{k_{1}})\epsilon_{(s_{2})m}^{r}(\mathbf{k_{2}})\epsilon_{i}^{(s_{3})m}(\mathbf{k_{3}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+k_{1m}k_{2r}k_{3n}\,\epsilon_{j}^{(s_{1})r}(\mathbf{k_{1}})\epsilon_{l}^{(s_{2})n}(\mathbf{k_{2}})\epsilon_{i}^{(s_{3})m}(\mathbf{k_{3}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad+(\mathbf{k_{1}}\cdot\mathbf{k_{2}})k_{1m}\,\epsilon_{j}^{(s_{1})r}(\mathbf{k_{1}})\epsilon^{(s_{2})}_{lr}(\mathbf{k_{2}})\epsilon_{i}^{(s_{3})m}(\mathbf{k_{3}})\Big{]}\,,$
(B.3) $\displaystyle
C_{5}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=\lambda_{s_{1}}k_{1}k_{2m}k_{3k}\,\epsilon_{(s_{1})}^{mi}(\mathbf{k_{1}})\epsilon_{(s_{2})}^{kr}(\mathbf{k_{2}})\epsilon_{ir}^{(s_{3})}(\mathbf{k_{3}})$
$\displaystyle\qquad\qquad\qquad\qquad\,+i\epsilon^{ijl}\Big{[}k_{1k}k_{2l}k_{3r}\,\epsilon_{j}^{(s_{1})m}(\mathbf{k_{1}})\epsilon_{(s_{2})}^{kr}(\mathbf{k_{2}})\epsilon_{mi}^{(s_{3})}(\mathbf{k_{3}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\,-k_{1k}k_{2l}k_{3i}\epsilon_{j}^{(s_{1})m}(\mathbf{k_{1}})\epsilon_{(s_{2})}^{kr}(\mathbf{k_{2}})\epsilon^{(s_{3})}_{mr}(\mathbf{k_{3}})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\quad\,+k_{2}^{k}k_{3l}k_{3}^{r}\,\epsilon_{j}^{(s_{1})m}(\mathbf{k_{1}})\epsilon^{(s_{2})}_{ri}(\mathbf{k_{2}})\epsilon^{(s_{3})}_{km}(\mathbf{k_{3}})\Big{]}\,,$
(B.4) $\displaystyle
C_{6}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=\lambda_{s_{1}}k_{1}\left[k_{2i}k_{3m}\,\epsilon_{(s_{1})}^{mi}(\mathbf{k_{1}})\epsilon_{r}^{(s_{2})k}(\mathbf{k_{2}})\epsilon_{k}^{(s_{3})r}(\mathbf{k_{3}})+k_{3}^{2}\epsilon_{(s_{1})}^{mi}(\mathbf{k_{1}})\epsilon_{(s_{2})i}^{r}(\mathbf{k_{2}})\epsilon_{mr}^{(s_{3})}(\mathbf{k_{3}})\right]\,,$
(B.5) $\displaystyle
C_{7}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=C_{9}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=i\epsilon^{ijl}k_{1r}\,\epsilon_{l}^{(s_{1})m}(\mathbf{k_{1}})\epsilon_{j}^{(s_{2})r}(\mathbf{k_{2}})\epsilon_{mi}^{(s_{3})}(\mathbf{k_{3}})\,,$
(B.6) $\displaystyle
C_{8}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=ik_{1}^{2}\epsilon^{ijl}\left[k_{3l}\,\epsilon^{(s_{1})}_{mi}(\mathbf{k_{1}})\epsilon^{(s_{2})}_{jk}(\mathbf{k_{2}})\epsilon_{(s_{3})}^{km}(\mathbf{k_{3}})+k_{2k}\,\epsilon^{(s_{1})}_{mi}(\mathbf{k_{1}})\epsilon_{l}^{(s_{2})m}(\mathbf{k_{2}})\epsilon_{j}^{(s_{3})k}(\mathbf{k_{3}})\right.$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\,\left.-k_{2}^{m}\epsilon^{(s_{1})}_{mi}(\mathbf{k_{1}})\epsilon^{(s_{2})}_{lk}(\mathbf{k_{2}})\epsilon_{j}^{(s_{3})k}(\mathbf{k_{3}})\right]\,.$
(B.7)

The $T_{i}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$
that enter in the final expression of the PV1 bispectrum (4.36) with time-
dependent couplings can be written in terms of the
$C_{i}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})$ as

$\displaystyle
T_{1}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=C_{2}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})-C_{7}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})\,,$
(B.8) $\displaystyle
T_{2}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=C_{4}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})\,,$
(B.9) $\displaystyle
T_{3}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})=C_{5}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})-C_{6}^{s_{1}s_{2}s_{3}}(\mathbf{k_{1}},\mathbf{k_{2}},\mathbf{k_{3}})\,.$
(B.10)

##  Appendix C Polarization tensors

In this section we set our conventions for the polarization tensors by fixing
an explicit representation for them. We can first use the momentum
conservation, $\mathbf{k_{1}}+\mathbf{k_{1}}+\mathbf{k_{3}}=0$, and the
invariance under rotations to make the three wave vectors lying on the same
plane, that we choose to be the $(x,y)$ plane:

$\mathbf{k_{1}}=k_{1}(1,0,0)\,,\quad\mathbf{k_{2}}=k_{2}(\cos\theta,\sin\theta,0)\,,\quad\mathbf{k_{3}}=k_{3}(\cos\varphi,\sin\varphi,0)\,,$
(C.1)

where $\theta$ and $\varphi$ are the angles that $\mathbf{k_{1}}$ forms with
$\mathbf{k_{2}}$ and $\mathbf{k_{3}}$ respectively. Without loss of
generality, we can choose $0\leq\theta\leq\pi$ and $\pi\leq\varphi\leq 2\pi$,
such that

$\cos\theta=\frac{k_{3}^{2}-k_{1}^{2}-k_{2}^{2}}{2k_{1}k_{2}}\,,\quad\sin\theta=\frac{\lambda}{2k_{1}k_{2}}\,,\quad\cos\varphi=\frac{k_{2}^{2}-k_{3}^{2}-k_{1}^{2}}{2k_{3}k_{1}}\,,\quad\sin\varphi=-\frac{\lambda}{2k_{3}k_{1}}\,,$
(C.2)

with

$\lambda=\sqrt{2k_{1}^{2}k_{2}^{2}+2k_{2}^{2}k_{3}^{2}+2k_{3}^{2}k_{1}^{2}-k_{1}^{4}-k_{2}^{4}-k_{3}^{4}}\,.$
(C.3)

With this representation of the wave vectors we can then write explicitly the
polarization tensors as [115, 64]

$\epsilon^{(s)}(\mathbf{k_{1}})=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0\\\
0&1&i\lambda_{s}\\\ 0&i\lambda_{s}&-1\\\ \end{pmatrix},$ (C.4)
$\epsilon^{(s)}(\mathbf{k_{2}})=\frac{1}{\sqrt{2}}\begin{pmatrix}\sin^{2}\theta&-\sin\theta\cos\theta&-i\lambda_{s}\sin\theta\\\
-\sin\theta\cos\theta&\cos^{2}\theta&i\lambda_{s}\cos\theta\\\
-i\lambda_{s}\sin\theta&i\lambda_{s}\cos\theta&-1\\\ \end{pmatrix},$ (C.5)
$\epsilon^{(s)}(\mathbf{k_{3}})=\frac{1}{\sqrt{2}}\begin{pmatrix}\sin^{2}\varphi&-\sin\varphi\cos\varphi&-i\lambda_{s}\sin\varphi\\\
-\sin\varphi\cos\varphi&\cos^{2}\varphi&i\lambda_{s}\cos\varphi\\\
-i\lambda_{s}\sin\varphi&i\lambda_{s}\cos\varphi&-1\\\ \end{pmatrix},$ (C.6)

where $\lambda_{s}=\pm 1$ for $s=R$ and $s=L$ respectively.

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