aainstitutetext: Department of Physics, The Hong Kong University of Science
and Technology,
Clear Water Bay, Kowloon, Hong Kong, P.R. Chinabbinstitutetext: The HKUST
Jockey Club Institute for Advanced Study,
The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong, P.R. Chinaccinstitutetext: Department of
Physics, Tsinghua University,
Beijing 100084, China

# Large Spin-2 Signals at the Cosmological Collider

Xi Tong c and Zhong-Zhi Xianyu<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the theory and phenomenology of massive spin-2 fields during the
inflation with nonzero background chemical potential, and extend the
cosmological collider physics to tensor modes. We identify a unique
dimension-5 and parity-violating chemical potential operator for massive
spin-2 fields, which leads to a ghost-free linear theory propagating one
scalar mode and two tensor modes. The chemical potential greatly boosts the
production of one tensor mode even for very heavy spin-2 particles, and
thereby leads to large and distinct cosmological collider signals for massive
spin-2 particles. The large signals show up at the tree-level in both the
curvature trispectrum and the tensor-curvature mixed bispectrum.

##  1 Introduction

It is widely believed that the physics of our world can be described by a
quantum theory of fields at energy scales well below the Planck scale
Weinberg:1995mt ; Weinberg:1996kw . The notion of particles with integer or
half-integer spins then emerges from quantizing the fluctuating modes of
fields Fierz1939 ; Pauli:1940zz . Although only lower-spin particles with
spin-$0,1/2,1$ are found as fundamental particles in the Standard Model,
massive higher-spin particles do exist as composite particles. Also, massive
higher-spin particles are expected to play an important role in many Beyond
Standard Model (BSM) models, with massive spin-2 particle as a notable
example. Models involving massive spin-2 states include large-$N$ gauge
theories tHooft:1973alw ; Witten:1979kh , massive gravity Fierz:1939ix ;
Hinterbichler:2011tt ; deRham:2014zqa , extra-dimension models Kaluza:1921tu ;
Klein:1926tv ; Aragone:1987dtt ; Arkani-Hamed:1998jmv ; Randall:1999ee ;
Randall:1999vf , and, perhaps most notably, string theory Rahman:2012thy . The
mass scale of such BSM spin-2 particles covers a broad range, leaving us with
a vast landscape for phenomenological exploration.

In this work, focusing on the high-energy side of the massive spin-2
landscape, we initiate a study of massive spin-2 particles in the context of
Cosmological Collider (CC) physics. The correlators of curvature fluctuations
and tensor fluctuations encode valuable information about the particle
spectrum during the inflation. The program of CC physics aims at utilizing the
inflationary spacetime as a test ground of particle physics Chen:2009zp ;
Baumann:2011nk ; Noumi:2012vr ; Gong:2013sma ; Arkani-Hamed:2015bza ;
Lee:2016vti ; Chen:2016nrs ; Chen:2016uwp ; Chen:2016hrz . The soft limits of
the inflationary correlators could exhibit an oscillatory behavior known as
the CC signal or clock signal Chen:2015lza . The CC signal reflects the mass,
spin, coupling and CP properties of the interacting particle. With an
inflationary energy scale reaching up to $H\lesssim 10^{14}$ GeV, the CC is an
ideal probe of heavy spin-2 modes such as that from KK compactification
Kumar:2018jxz  and string theory Kim:2019wjo ; Noumi:2019ohm .

It has been shown that one of the most salient feature of the corresponding CC
signal in the curvature bispectrum is a characteristic $P_{2}(\cos\theta)$
angular dependence Arkani-Hamed:2015bza ; Lee:2016vti ;
MoradinezhadDizgah:2018ssw , due to the mixing between the longitudinal
component of the spinning particle and the inflaton. However, the overall
amplitude of the CC signal is exponentially suppressed by the particle mass
$m$. This is because the non-local type CC signal comes from on-shell particle
production Tong:2021wai . But a pure gravitational production is subjected to
a Boltzmann suppression $e^{-m/T_{dS}}$ with the intrinsic de Sitter (dS)
temperature $T_{dS}=\frac{H}{2\pi}$. The Boltzmann suppression makes it
difficult to probe particles with a large mass. This deficiency necessitates a
mechanism to amplify the CC signal strength of massive spin-2 particles, or in
other words, a mechanism to enhance their production.

Although the inflationary geometry resembles that of a dS spacetime with ten
isometries, the rolling speed of the inflaton $\dot{\phi}_{0}(t)$
spontaneously breaks four of them Cheung:2007st , including three dS boost
transformations ($i.e.$, the special conformal transformations in the boundary
CFT). The rolling inflaton automatically selects a preferred frame in which
the background is uniform. Any dS boost would change this uniformity and bring
in space dependence in the inflaton background. In this special frame,
particle distribution needs not to take the dS-invariant Boltzmann form.
Instead, the symmetry-breaking scale $|\dot{\phi}_{0}|^{1/2}$ may bring
dramatic enhancement to particle production. Such is the case if the inflaton
background plays the role of an external chemical potential Chen:2018xck ;
Hook:2019zxa ; Wang:2019gbi ; Liu:2019fag ; Wang:2020ioa ; Bodas:2020yho ;
Sou:2021juh ; Wang:2021qez . It has been shown that in the presence of this
symmetry-breaking chemical potential, the production rate of massive particles
with lower spins ($S=0,\frac{1}{2},1$) deviate from the dS-invariant Boltzmann
form $e^{-m_{S}/T_{dS}}$ with exponential enhancement, leading to sizable
observational effects in the CC signals Chen:2018xck ; Hook:2019zxa ;
Wang:2019gbi ; Liu:2019fag ; Wang:2020ioa ; Bodas:2020yho . It is therefore
tantalizing to generalize the chemical potential to particles with higher
spins, among which $S=2$ is a first step. We will show that the lowest
dimensional chemical potential operator capable of enhancing particle
production is uniquely fixed by the consistency of the theory. An inspection
of the linear theory in dS shows that the massive spin-2 field is ghost-free,
but only propagates three degrees of freedom (DoF) instead of five if there is
a non-zero chemical potential. Among the three propagating degrees, the two
tensor modes receive helicity-dependent modulation of particle production
whereas the longitudinal mode is not affected. Once coupled to the “visible”
inflaton and graviton, the enhanced mode can lead to large CC signals in the
mixed bispectra and curvature trispectrum, both at the tree level. The
resulting non-Gaussian signal strengths can be as large as
$f_{NL,\zeta\zeta\gamma}\sim\mathcal{O}(10)$ and
$g_{NL,\zeta^{4}}\sim\mathcal{O}(10^{4})$, respectively. Therefore, they are
promising targets for the next generation of Cosmic Microwave Background (CMB)
measurements such as LiteBIRD, as well as future 21-cm interferometry
experiments.

In the existing literature, works on such large CC signals Flauger:2016idt ;
Chen:2018xck ; Tong:2018tqf ; Chua:2018dqh ; Lu:2019tjj ; Kumar:2019ebj ;
Wang:2019gbi  mostly focuses on scalar non-Gaussianities, while those on large
mixed/tensor non-Gaussianities Agrawal:2017awz ; Goon:2018fyu ;
Dimastrogiovanni:2018uqy ; Fujita:2021flu ; Dimastrogiovanni:2021cif ;
Ozsoy:2021onx ; Cabass:2021iii ; Cabass:2021fnw  rarely discuss the non-
analytic oscillatory signatures from on-shell effects. As a result, our model
serves as a first extension of cosmological collider phenomenology to external
tensor modes with potentially large and observable signals.

This paper is structured as follows. In Sect. 2, we start with a brief review
of the chemical potential, and then move on to the introduction of a chemical
potential to a massive spin-2 field in dS in Sect. 3. After working out the
linear theory, we turn on interactions between the spin-2 field and the
“visible” sector fields, and study its impact on the inflationary observables
in Sect. 4. Finally, we conclude in Sect. 5.

##  2 Chemical potential briefly reviewed

In this section, we briefly review the chemical potential effects for lower-
spin states previously studied in the context of cosmological collider
physics. Such chemical potential has its origin from statistical mechanics,
where the chemical potential $\kappa$ is introduced as a Lagrange multiplier
to control the total particle number in a grand canonical ensemble. The
partition function $Z$ then picks up a dependence on the particle number $N$
in the system,

$Z=e^{-(H-\kappa N)/T}~{}.$ (1)

This partition function can be directly adapted to a field theory, in the form
of a path integral over the phase space weighted by a phase factor containing
the Hamiltonian and the particle number,

$Z=\int\mathcal{D}\Phi\mathcal{D}\Pi e^{i\int dtd^{3}x\Pi\dot{\Phi}}e^{-i\int
dt\left(H[\Phi,\Pi]-\kappa N[\Phi]\right)}~{},$ (2)

where $\Phi$ represents a general matter field and $\Pi$ is its canonical
conjugate. For simplicity, we have assumed a particle number independent of
the canonical momentum $\Pi$, which turns out be the case in most of the
scenarios we are interested in. We then perform the integration over $\Pi$ to
obtain the Lagrangian $L=\int\Pi\dot{\Phi}-H$, since it is more convenient to
develop a perturbation theory with a Lagrangian formalism. Evoking locality
and (local) Lorentz covariance, we can promote the chemical potential to a
local vector field $\kappa_{\mu}=\kappa_{\mu}(x)$. Then the partition function
reads

$Z=\int\mathcal{D}\Phi e^{i\int
d^{4}x\sqrt{-g}\left[\mathcal{L}(\Phi,\partial\Phi)+\kappa_{\mu}J^{\mu}(\Phi,\partial\Phi)\right]}~{},$
(3)

where $J^{\mu}(\Phi,\partial\Phi)$ is the current density whose integral over
a spatial slice $\Sigma$ gives the particle number,
$N=\int_{\Sigma}d^{3}x_{\mu}J^{\mu}$. The second term $\kappa_{\mu}J^{\mu}$ in
(3) represents the effect of an external field $\kappa$ on the matter, in the
form of a chemical potential. We note that the matter current does not have to
be conserved. In fact, it can be shown that, for the chemical potential term
to nontrivially impact on the particle production, a necessary condition is
that either the one-form field $\kappa=\kappa_{\mu}(x)dx^{\mu}$ is not closed,
or the current is not conserved Sou:2021juh :

$d\kappa\neq 0~{}~{}~{}\text{or}~{}~{}~{}\nabla\cdot J\neq 0~{}.$ (4)

For spin-0 fields, the $U(1)$ current is conserved and we need a non-closed
chemical potential with a non-zero field strength,
$\nabla_{\mu}\kappa_{\nu}-\nabla_{\nu}\kappa_{\mu}\neq 0$. This essentially
provides a background electromagnetic field whose Schwinger effect gives an
anisotropic enhancement of particle production.

For spin-$1/2$ and spin-1 fields, there are natural choices of non-conserved
currents quadratic in the field, namely, the axial current and the Chern-
Simons current. In such cases, one can introduce a closed 1-form field as the
chemical potential, without breaking the background rotation symmetry. This
1-form field is most naturally realized by the gradient of a scalar field,
usually the inflaton itself,

$\kappa_{\mu}=\frac{\nabla_{\mu}\phi(t)}{\Lambda_{c}}~{},$ (5)

where $\Lambda_{c}$ is a cutoff scale for the chemical potential operator
$\kappa_{\mu}J^{\mu}$.

More explicitly, we take a massive spin-1 Proca field $A_{\mu}$ in the dS
spacetime ($g_{\mu\nu}=a^{2}(\tau)\eta_{\mu\nu}$, $a(\tau)=-1/H\tau$) as an
example. In addition to the conventional kinetic term and mass term,

$S_{\text{Proca}}[A]=\int
d^{4}x\sqrt{-g}\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-\frac{1}{2}m^{2}A_{\mu}A^{\mu}\right]~{},$
(6)

we turn on the chemical potential by adding a dimension-5 operator

$S_{c}[A,\phi]=\frac{1}{2}\int d^{4}x\sqrt{-g}\kappa_{\mu}J^{\mu}_{CS}=-\int
d^{4}x\sqrt{-g}\frac{\phi}{4\Lambda_{c}}\mathcal{E}^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}~{},$
(7)

where $\phi$ is the inflaton field that depends only on time $t$ at the
background level,
$J_{CS}^{\mu}=\mathcal{E}^{\mu\nu\rho\sigma}A_{\nu}F_{\rho\sigma}$ is the
Abelian Chern-Simons current, and
$\mathcal{E}^{\mu\nu\rho\sigma}\equiv\frac{\epsilon^{\mu\nu\rho\sigma}}{\sqrt{-g}}$
is the Levi-Civita tensor density. Notice that the shift symmetry of the
inflaton $\phi\to\phi+C$ is respected, since the constant inflaton background
contributes to a total derivative. However, with a rolling inflaton
background, the chemical potential term $S_{c}[A,\phi(t)]$ gives non-trivial
effects on the dynamics of the spin-1 field. Varying the action with respect
to $A$, we obtain the Equation-of-Motion (EoM) for the Proca field,

$0=\frac{\delta(S_{\text{Proca}}+S_{c})}{\sqrt{-g}\delta A_{\mu}}=\Box
A^{\nu}-\nabla_{\mu}\nabla^{\nu}A^{\mu}-m^{2}A^{\nu}-\frac{2\nabla_{\mu}\phi}{\Lambda_{c}}\mathcal{E}^{\mu\nu\rho\sigma}\nabla_{\rho}A_{\sigma}~{}.$
(8)

Taking the divergence and utilizing a few useful identities given in Appendix
C, we arrive at the constraint $\nabla^{\mu}A_{\mu}=0$. Insert this back into
the EoM, and we obtain a modified Klein-Gordon equation with a constraint

EoM:
$\displaystyle\left(\Box-(m^{2}+3H^{2})\right)A_{\nu}-\frac{\nabla_{\alpha}\phi}{\Lambda_{c}}\mathcal{E}^{\alpha\kappa\rho\sigma}g_{\kappa\nu}\nabla_{\rho}A_{\sigma}=0~{},$
(9) Constraint: $\displaystyle\nabla^{\mu}A_{\mu}=0~{}.$ (10)

The constraint relates the temporal mode to the longitudinal mode and reduces
the DoF to $4-1=3$. This can be seen more clearly under a helicity
decomposition,

$A_{\mu}=\sum_{\lambda=-1}^{1}A_{\mu}^{(\lambda)}~{}.$ (11)

Without loss of generality, we go to momentum space and focus on a mode
propagating in the $+z$ direction,

$\displaystyle A_{0}^{(0)}=v_{0}~{},~{}$ $\displaystyle A_{0}^{(\pm 1)}=0~{},$
(12) $\displaystyle A_{i}^{(0)}=f_{0}e_{i}^{0}~{},$ $\displaystyle A_{i}^{(\pm
1)}=v_{\pm 1}e_{i}^{\pm 1}~{}.$ (13)

The polarization vectors are chosen as

$e_{i}^{0}=\left(\begin{array}[]{c}0\\\ 0\\\
1\end{array}\right)~{}~{}~{},~{}~{}~{}e_{i}^{\pm
1}=\left(\begin{array}[]{c}1\\\ \pm i\\\ 0\end{array}\right)~{}.$ (14)

Denoting the chemical potential by
$\kappa\equiv\frac{\dot{\phi}_{0}}{\Lambda_{c}}$, the mode functions satisfy

$\displaystyle
v_{0}^{\prime\prime}+2aHv_{0}^{\prime}+\left(k^{2}+(m^{2}+2H^{2})a^{2}\right)v_{0}$
$\displaystyle=0~{},$ (15) $\displaystyle v_{\pm
1}^{\prime\prime}+\left(k^{2}\mp 2k\kappa a+m^{2}a^{2}\right)v_{\pm 1}$
$\displaystyle=0~{}.$ (16)

The longitudinal component is solved from the temporal component via

$f_{0}=-\frac{i}{k}(v_{0}^{\prime}+2aHv_{0})~{}.$ (17)

As a result, the temporal mode $v_{0}$ is not affected by the chemical
potential, whereas the transverse modes $v_{\pm 1}$ receive helicity-dependent
corrections. This dS-symmetry-breaking correction is further reflected in the
detailed particle production history. Using the Stokes-line method
Berry:1989zz ; Sou:2021juh , one can show that mode functions satisfying the
EoM (16) develop a negative frequency part at the conformal time111(18) only
applies when $|\tilde{\kappa}|<\tilde{m}$. For a chemical potential greater
than the mass, (16) acquires a period of tachyonic instability in the region
$-k\tau\in\left(\tilde{\kappa}-\sqrt{\tilde{\kappa}^{2}-\tilde{m}^{2}},\tilde{\kappa}+\sqrt{\tilde{\kappa}^{2}-\tilde{m}^{2}}\right)$,
indicating copious classical production with $\langle n_{\pm
1}(\mathbf{k})\rangle^{\prime}>1$.

$\tau_{*,\pm 1}\simeq-\frac{0.66\tilde{m}\mp 0.34\tilde{\kappa}}{k}~{},$ (18)

where $\tilde{m}\equiv\frac{m}{H}$ and $\tilde{\kappa}\equiv\frac{\kappa}{H}$.
This emergent negative frequency is physically interpreted as real particle
production through the Bogoliubov transformation. The production amount is
given by the amplitude of the negative frequency mode as

$\langle
n_{\lambda}(\mathbf{k})\rangle^{\prime}=\frac{e^{2\pi(\tilde{m}-\lambda\tilde{\kappa})}+1}{e^{4\pi\tilde{m}}-1}\xrightarrow{\tilde{m}\gg
1}e^{-2\pi(\tilde{m}+\lambda\tilde{\kappa})}~{}.~{}~{}~{}~{}~{}(\lambda=0,\pm
1)$ (19)

In this way, we see that the dS-invariant Boltzmann factor is altered in the
presence of the chemical potential. Assuming $\tilde{\kappa}>0$, the
production of $\lambda=-1$ mode is exponentially enhanced while the
$\lambda=+1$ mode is exponentially suppressed. Such is a general feature of a
chemical potential, that it gives a linear and biased shift in the effective
mass of a bosonic particle. Particles in the amplified mode, once produced,
will subsequently decay into massless inflatons and gravitons, producing
sizable CC signals in the non-Gaussian correlators of curvature fluctuations
and tensor fluctuations.

##  3 Linear theory of a massive spin-2 field in dS with chemical potential

###  3.1 Introducing the chemical potential

Now, we move on to the discussion of a massive spin-2 field. A consistent
theory of a free massive spin-2 field $h_{\mu\nu}$ in dS is described by the
quadratic part of the Einstein-Hilbert action,

$\displaystyle S_{EH}=\int d^{4}x\sqrt{-g}\Bigg{[}$
$\displaystyle\frac{1}{2}\nabla_{\mu}h^{\mu\lambda}\nabla^{\nu}h_{\nu\lambda}-\frac{1}{4}\nabla_{\mu}h_{\nu\lambda}\nabla^{\mu}h^{\nu\lambda}+\frac{1}{4}\nabla^{\mu}h\nabla_{\mu}h-\frac{1}{2}\nabla^{\nu}h^{\mu}_{~{}~{}\nu}\nabla_{\mu}h$
$\displaystyle-\frac{1}{2}H^{2}\left(h_{\mu\nu}h^{\mu\nu}+\frac{1}{2}h^{2}\right)\Bigg{]}~{},$
(20)

and a Fierz-Pauli mass term to ensure the absence of ghosts Fierz:1939ix ,

$S_{FP}=\int
d^{4}x\sqrt{-g}\left[\frac{1}{4}m^{2}\left(h_{\mu\nu}h^{\mu\nu}-h^{2}\right)\right]~{}.$
(21)

where $h\equiv h^{\mu}_{~{}\mu}$. Here the covariant derivatives and index
contractions are computed under the dS metric $g$. In dS, the absence of ghost
states further require that the mass $m$ appeared in (21) satisfies the
Higuchi bound $m^{2}>2H^{2}$ Higuchi:1986py . Then, the combination
$S_{EH}+S_{FP}$ gives rise to five DoFs without negative-norm states. In this
work, we are more interested in the states in the principal series which could
give rise to oscillatory signals. For spin-2, they correspond to the mass
range $m^{2}>9H^{2}/4$.

Inspired by the chemical potential for the spin-1 field $A_{\mu}$, we wish to
introduce a chemical potential for massive spin-2 field $h_{\mu\nu}$ through
its coupling with a rolling inflaton background $\phi(t)$. _A priori_ there
could be multiple choices for introducing this coupling operator. Thus we
further introduce a set of conditions that the operator should meet:

  1. (i)

The chemical potential should be from a local operator with the lowest
possible mass dimension.

  2. (ii)

The operator must take the form of $\kappa_{\mu}J^{\mu}(h)$, where
$\kappa_{\mu}\propto\partial_{\mu}\phi$ and $J$ must be non-conserved, to have
nontrivial effects of the particle production.

  3. (iii)

The operator must be quadratic in $h_{\mu\nu}$ so that it contributes to the
linear theory.

  4. (iv)

The operator should (spontaneously) break dS boosts while respecting spatial
translations, spatial rotations and dilation.

  5. (v)

The operator must give rise to a consistent linear theory without ghosts.

These conditions uniquely fix the chemical potential operator to be

$\displaystyle S_{c}=\int
d^{4}x\sqrt{-g}\left[\frac{\phi}{2\Lambda_{c}}\mathcal{E}^{\mu\nu\rho\sigma}\nabla_{\mu}h_{\nu\lambda}\nabla_{\rho}h_{\sigma}^{~{}\lambda}\right]~{}.$
(22)

This is the lowest dimensional parity-odd operator one can write down that is
quadratic in $h_{\mu\nu}$. To see that it takes the form of
$\kappa_{\mu}J^{\mu}(h)$, one can perform an integration-by-part,

$S_{c}=\int d^{4}x\sqrt{-g}\kappa_{\mu}(\phi)J^{\mu}(h)=-\int
d^{4}x\sqrt{-g}\frac{\nabla_{\mu}\phi}{2\Lambda_{c}}\mathcal{E}^{\mu\nu\rho\sigma}h_{\nu\lambda}\nabla_{\rho}h_{\sigma}^{~{}\lambda}~{}.$
(23)

This form also makes clear the symmetry-breaking pattern. Namely, the rolling
inflaton background defines a time-like vector
$\kappa_{\mu}=\frac{\nabla_{\mu}\phi(t)}{\Lambda_{c}}$ normal to the spatial
hypersurface $t=$ const, breaking dS boosts while preserving spatial
isometries. In the slow-roll phase, $\dot{\phi}_{0}\approx$ const, and
dilation is also preserved. Note also that (22) is shift-symmetric,
$\phi\to\phi+C$, hence it does not back-react to the inflation potential. The
P-odd current $J$ only has one spatial derivative and no time derivative on
$h$, which ensures the absence of Ostrogradski ghosts or singularities
Ostrogradsky:1850fid ; Woodard:2015zca .

A notable feature of the chemical potential term (22) is that it is not
invariant under the linear gauge transformation for the spin-2 field $\delta
h_{\mu\nu}=-(\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu})$, which would be
problematic in massless or partially massless cases. For massive spin-2 fields
satisfying the Higuchi bound, we know that gauge invariance must be realized
non-linearly, and (22) can be understood as the unitary gauge result. At high
enough energy scales, the spin-2 field becomes strongly-coupled and new DoFs
must be introduced to maintain unitarity. Thus we need to make sure our
relevant physical processes are under perturbative control. We will elaborate
more on this in Sect. 4.1.

Before moving on to further developments, we point out that experienced
readers might have in mind a more familiar alternative to the operator (22).
Namely, we could have assigned a gravitational Chern-Simons term $\phi
R\tilde{R}(h)$ constructed from the massive spin-2 field. This term is gauge-
invariant and quadratic in $h$ at leading order. Upon integration-by-parts, it
takes the required form $\kappa_{\mu}J_{CS}^{\mu}(h)$, with $J_{CS}^{\mu}(h)$
being the gravitational Chern-Simons current. It also satisfies the required
symmetry-breaking patterns. Its only deficiency compared to our chemical
potential operator seems to be having higher dimensionality, due to two more
derivatives. However, this turns out to be the crucial reason why we favor
(22) instead. We will show later in Sect. 3.4 that this deficiency of the
Chern-Simons term leads to pathologies that are non-negligible for the
parameter regime that we are interested in.

###  3.2 EoMs and DoF analysis

Now, let us take a closer look at the linear theory with chemical potential. A
variation of the quadratic action gives

$0=\frac{\delta\left(S_{EH}+S_{FP}+S_{c}\right)}{\sqrt{-g}\delta
h^{\mu\nu}}\equiv
X_{\mu\nu}-\frac{1}{2}m^{2}(h_{\mu\nu}-g_{\mu\nu}h)-Y_{\mu\nu}~{},$ (24)

where $X_{\mu\nu}$ comes from the linearized Einstein-Hilbert action, and
$Y_{\mu\nu}$ represents the terms from the chemical potential $S_{c}$,

$\displaystyle X_{\mu\nu}=$ $\displaystyle\frac{1}{2}\left(\Box
h_{\mu\nu}-\nabla_{\mu}\nabla_{\alpha}h^{\alpha}_{~{}\nu}-\nabla_{\nu}\nabla_{\alpha}h^{\alpha}_{~{}\mu}+\nabla_{\mu}\nabla_{\nu}h\right)+\frac{1}{2}g_{\mu\nu}\left(\nabla_{\alpha}\nabla_{\beta}h^{\alpha\beta}-\Box
h\right)$
$\displaystyle-H^{2}\left(h_{\mu\nu}+\frac{1}{2}g_{\mu\nu}h\right)~{},$ (25)
$\displaystyle Y_{\mu\nu}=$
$\displaystyle\frac{\nabla_{\alpha}\phi}{2\Lambda_{c}}\mathcal{E}^{\alpha\kappa\rho\sigma}\left(g_{\kappa\nu}\nabla_{\rho}h_{\mu\sigma}+g_{\kappa\mu}\nabla_{\rho}h_{\nu\sigma}\right)~{}.$
(26)

Take the divergence of the EoM (24) and use the background Bianchi identity
$\nabla^{\mu}X_{\mu\nu}=0$, we obtain

$\nabla^{\mu}Y_{\mu\nu}=-\frac{1}{2}m^{2}(\nabla^{\mu}h_{\mu\nu}-\nabla_{\nu}h)~{}.$
(27)

With a useful identity (236) due to the antisymmetric property of
$\mathcal{E}^{\alpha\kappa\rho\sigma}$ and the conformally flat dS metric,
(27) can be simplified to

$\frac{1}{\Lambda_{c}}\mathcal{E}^{\alpha\kappa\rho\sigma}\nabla_{\alpha}\phi
g_{\kappa\nu}\nabla_{\rho}\nabla^{\mu}h_{\mu\sigma}=-m^{2}(\nabla^{\mu}h_{\mu\nu}-\nabla_{\nu}h)~{},$
(28)

where we have assumed the spatial homogeneity of the inflaton background
$\phi=\phi(t)$. Take a further divergence and use the identity (236) again, we
arrive at a constraint on $h$ alone:

$\nabla^{\mu}\nabla^{\nu}h_{\mu\nu}=\Box h~{}.$ (29)

This eliminates one out of ten DoFs in $h$. However, there are more
constraints hidden in the EoM. This can be seen from taking the trace of (24),

$(m^{2}-2H^{2})h=0~{},$ (30)

where (29) is applied for simplification. Massive spin-2 particles in the
principal series satisfy $m^{2}>2H^{2}$ and is therefore traceless, $h=0$.
Inserting this zero trace back into (28) and (29), and denoting
$C_{\nu}\equiv\nabla^{\mu}h_{\mu\nu}$, we have

$\displaystyle
C_{\nu}+\frac{\nabla_{\alpha}\phi}{m^{2}\Lambda_{c}}\mathcal{E}^{\alpha\kappa\rho\sigma}g_{\kappa\nu}\nabla_{\rho}C_{\sigma}$
$\displaystyle=$ $\displaystyle 0~{},$ (31) $\displaystyle\nabla^{\nu}C_{\nu}$
$\displaystyle=$ $\displaystyle 0~{}.$ (32)

Taking $\nu=0$ in (31), it is easy to check that the temporal component
vanishes by the spatial homogeneity of $\phi$ and the antisymmetry of the
Levi-Civita tensor density:

$C_{0}=0~{}.$ (33)

The spatial components satisfy the constraint

$\epsilon_{ijk}\partial_{j}C_{k}=-\frac{m^{2}\Lambda_{c}a^{2}(\tau)}{\phi^{\prime}(\tau)}C_{i}~{}.$
(34)

Thus the 3-vector $C_{i}$ is the eigenfunction of the curl operator with a
time-dependent eigenvalue. In plasma physics, this equation describes a force-
free magnetic field and is well-studied in the context of planetary and
stellar quasistatic magnetic fields Chandrasekhar1956 ; Chandrasekhar1957 .
The general solution to $C_{i}$ consists of the toroidal and poloidal
components:

$C_{i}=T_{i}+P_{i}~{},~{}~{}\text{with}~{}~{}T_{i}=\epsilon_{ijk}\partial_{j}\psi\hat{n}_{k}~{},~{}~{}~{}P_{i}=-\frac{\phi^{\prime}}{m^{2}\Lambda_{c}a^{2}}\epsilon_{ijk}\partial_{j}T_{k}~{},$
(35)

where $\hat{n}$ is an arbitrary unit vector and $\psi$ satisfies the Helmholtz
equation

$\left(\partial_{i}^{2}+\frac{m^{4}\Lambda_{c}^{2}a^{4}}{\phi^{\prime
2}}\right)\psi=0~{}.$ (36)

Thus we see that the vector $C_{i}$ does not propagate any dynamical degree of
freedom (specifically, its momentum spectrum is of measure zero in
$\mathbf{k}\in\mathbb{R}^{3}$). One trivial solution to (34) is simply
$C_{i}=0$. In fact, there is a uniqueness theorem 1534-0392_2003_2_251 ,
which, when accompanied by suitable boundary conditions, guarantees that this
trivial solution is the only one. We therefore adopt the boundary condition
$C_{1}=C_{2}=0$ on the $x$-$y$ plane, then the uniqueness theorem dictates

$C_{i}=0$ (37)

throughout spacetime. Thus setting $C_{\mu}=0$, we arrive at the same set of
five constraints as the case without chemical potential. These eliminates five
out of ten DoFs in the massive spin-2 field. We summarize the simplified EoM
and the constraints as follows:

EoM:
$\displaystyle\left(\Box-(m^{2}+2H^{2})\right)h_{\mu\nu}-\frac{\nabla_{\alpha}\phi}{\Lambda_{c}}\mathcal{E}^{\alpha\kappa\rho\sigma}\left(g_{\kappa\nu}\nabla_{\rho}h_{\mu\sigma}+g_{\kappa\mu}\nabla_{\rho}h_{\nu\sigma}\right)=0~{},$
(38) Constraints: $\displaystyle\nabla^{\mu}h_{\mu\nu}=0~{},~{}~{}~{}h=0~{}.$
(39)

The last term in (38) further generalizes the linear EoM for spinning fields
in dS considered in Kehagias:2017cym  by incorporating the parity-violating
Levi-Civita tensor density. As we shall see below, just like the case of
spin-1, this chemical potential term introduces a biased shift of mass for the
spin-2 particle, thereby enhancing the production rate for one helicity while
suppressing the other.

It is tempting to conclude that $h_{\mu\nu}$ contains $10-5=5$ DoFs on-shell
at this stage, just as what one would intuitively imagine for a massive spin-2
particle. However, we will demonstrate below that there are two more (non-
local) constraints hidden in (38) and (39), further reducing the number of on-
shell DoF to $10-5-2=3$.

Since the dS boosts are now broken, particles with different helicities do not
linearly mix into each other, we can label them by their momentum $\mathbf{k}$
and their helicity $\lambda=\mathbf{\hat{k}}\cdot\mathbf{S}$. More explicitly,
we go to the momentum space and perform the helicity decomposition,

$h_{\mu\nu}(\tau,\mathbf{k})=\sum_{\lambda=-2}^{2}h^{(\lambda)}_{\mu\nu}(\tau,\mathbf{k})~{}.$
(40)

In component form, we have

$\displaystyle h_{00}^{(0)}=h_{0},$ $\displaystyle h_{00}^{(\pm 1)}=0,$
$\displaystyle h_{00}^{(\pm 2)}=0,$ $\displaystyle
h_{0i}^{(0)}=f_{0}e_{i}^{0},$ $\displaystyle h_{0i}^{(\pm 1)}=h_{\pm 1}e^{\pm
1}_{i},$ $\displaystyle h_{0i}^{(\pm 2)}=0,$ $\displaystyle
h_{ij}^{(0)}=g_{0}e_{ij}^{0}+\frac{1}{3}h_{0}\delta_{ij},$ $\displaystyle
h_{ij}^{(\pm 1)}=v_{\pm 1}e^{\pm 1}_{ij},$ $\displaystyle h_{ij}^{(\pm
2)}=h_{\pm 2}e^{\pm 2}_{ij}~{}.$ (41)

The polarization tensor is chosen to satisfy

$\displaystyle e_{i}^{0}$ $\displaystyle=\hat{k}_{i},$ $\displaystyle
k_{i}e_{i}^{\pm 1}=0,$ $\displaystyle e_{i}^{\pm 1}e_{i}^{\pm 1*}=2,$
$\displaystyle\hat{k}_{i}e_{ij}^{0}$ $\displaystyle=e_{j}^{0},$
$\displaystyle\hat{k}_{i}e_{ij}^{\pm 1}=\frac{3}{2}e_{j}^{\pm 1},$
$\displaystyle~{}~{}~{}~{}\hat{k}_{i}e_{ij}^{\pm 2}=0,$ $\displaystyle
e_{ij}^{\pm 2}=e_{ij}^{\mp 2*},$ $\displaystyle e_{ij}^{\pm 2}e_{ij}^{\pm
2*}=4~{}.$ (42)

For a plane wave propagating in the $+z$-direction, the component form of the
polarization tensor is

$\displaystyle e_{i}^{0}$ $\displaystyle=\left(\begin{array}[]{c}0\\\ 0\\\
1\end{array}\right),$ $\displaystyle e_{i}^{\pm
1}=\left(\begin{array}[]{c}1\\\ \pm i\\\ 0\end{array}\right),$ (49)
$\displaystyle e_{ij}^{0}$
$\displaystyle=\left(\begin{array}[]{ccc}-\frac{1}{2}&0&0\\\
0&-\frac{1}{2}&0\\\ 0&0&1\end{array}\right),$ $\displaystyle e_{ij}^{\pm
1}=\frac{3}{2}\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&\pm i\\\ 1&\pm
i&0\end{array}\right),$ $\displaystyle e_{ij}^{\pm
2}=\left(\begin{array}[]{ccc}1&\pm i&0\\\ \pm i&-1&0\\\
0&0&0\end{array}\right)~{}.$ (59)

Inserting them into (38), we obtain the EoMs for the five mode functions
$h_{\lambda}$,

$\displaystyle
h_{0}^{\prime\prime}+2aHh_{0}^{\prime}+\left[k^{2}+m^{2}a^{2}\right]h_{0}$
$\displaystyle=$ $\displaystyle 0~{},$ (60a) $\displaystyle h_{\pm
1}^{\prime\prime}+\left[k^{2}\pm k\kappa
a+\left(m^{2}-2H^{2}\right)a^{2}\right]h_{\pm 1}$ $\displaystyle=$
$\displaystyle 0~{},$ (60b) $\displaystyle h_{\pm 2}^{\prime\prime}-2aHh_{\pm
2}^{\prime}+\left[k^{2}\pm 2k\kappa
a+\left(m^{2}-2H^{2}\right)a^{2}\right]h_{\pm 2}$ $\displaystyle=$
$\displaystyle 0~{},$ (60c)

as well as those for other mode functions,

$\displaystyle
f_{0}^{\prime\prime}-2aHf_{0}^{\prime}+\left[k^{2}+(m^{2}-6H^{2})a^{2}\right]f_{0}$
$\displaystyle=$ $\displaystyle-2ikaH\left(g_{0}+\frac{8}{3}h_{0}\right)~{},$
(61a) $\displaystyle
g_{0}^{\prime\prime}-2aHg_{0}^{\prime}+\left[k^{2}+(m^{2}-2H^{2})a^{2}\right]g_{0}$
$\displaystyle=$ $\displaystyle-\frac{8ikaH}{3}f_{0}~{},$ (61b) $\displaystyle
v_{\pm 1}^{\prime\prime}-2aHv_{\pm 1}^{\prime}+\left[k^{2}\pm k\kappa
a+(m^{2}-2H^{2})a^{2}\right]v_{\pm 1}$ $\displaystyle=$
$\displaystyle-\frac{4ikaH}{3}h_{\pm 1}~{},$ (61c)

where we have again denoted
$\kappa=\frac{\dot{\phi}_{0}}{\Lambda_{c}}\sim\text{const}$. The constraints
(39) give relations among mode functions and their first-order derivatives,

$\displaystyle f_{0}$ $\displaystyle=$
$\displaystyle-\frac{i}{k}(h_{0}^{\prime}+2aHh_{0})~{}~{}~{},~{}~{}~{}g_{0}=-\frac{i}{k}(f_{0}^{\prime}+2aHf_{0})-\frac{1}{3}h_{0}~{},$
(62a) $\displaystyle v_{\pm 1}$ $\displaystyle=$
$\displaystyle-\frac{2i}{3k}(h_{\pm 1}^{\prime}+2aHh_{\pm 1})~{}.$ (62b)

By differentiating twice on the constraint (62b) and the repeated application
of the EoM (60b), one can arrive at an expression of $v_{\pm
1}^{\prime\prime}$ in terms of $h_{\pm 1}$ and $h_{\pm 1}^{\prime}$.
Alternatively, one can repeatedly apply the constraint (62b) and its
derivative on the EoM (61c). This leads to another expression of $v_{\pm
1}^{\prime\prime}$ in terms of $h_{\pm 1}$ and $h_{\pm 1}^{\prime}$.
Interestingly, a comparison of these two expressions gives an algebraic
equation on $h_{\pm 1}$:

$i\kappa a^{2}h_{\pm 1}=0~{}.$ (63)

Therefore, with a non-zero $\kappa$, the vector modes are required to vanish
by theory consistency. In contrast, there is no such requirement on the
$\lambda=0,\pm 2$ modes: Their EoMs and constraints allow non-trivial
solutions. As a result, we conclude that our theory (24) propagates only one
scalar DoF and two tensor DoFs.

Of course, physically speaking, in the limit of small chemical potential
$\kappa\to 0$ (for example, by setting $\Lambda_{c}\to\infty$), the effect of
the inflaton background must be negligible, and the theory must return to that
of a freely propagating massive spin-2 particle with five DoFs. This apparent
discontinuity is somewhat similar to the vDVZ discontinuity in massive gravity
vanDam:1970vg ; Zakharov:1970cc . The vDVZ discontinuity is resolved by
considering the non-linear theory and resumming the higher-order contributions
Vainshtein:1972sx ; Deffayet:2001uk . This leads to a suppressed scalar mode
at a distance scale smaller than the Vainshtein radius, effectively recovering
massless gravity. In our case, although the full non-linear theory is unknown,
we can see from the EoM (60b) that in the IR limit $\tau\to 0$, the chemical
potential term $\pm k\kappa a$ is negligible compared to the mass term
$m^{2}a^{2}$. This somehow suggests an explanation similar to the Vainshtein
mechanism, namely, that the two vector DoFs are frozen at distances smaller
than $\mathcal{O}\left(\frac{\kappa}{m^{2}}\right)$. But in the late-time
limit, with a longer physical wavelength, the vector modes start to emerge. We
leave a more detailed study of this question for future.

###  3.3 Mode functions and particle production

From (60a) and (60c), we see that similar to the case of spin-1, the chemical
potential influences the transverse modes by providing a linear and biased
shift of the effective mass. For $\kappa>0$, the effective mass of particles
with $\lambda=+2$ is raised up, thereby increasing adiabaticity and leading to
suppressed particle production. Particles with $\lambda=-2$, in contrast,
enjoy a smaller effective mass and enhanced particle production.

We can solve the EoMs under the Bunch-Davies initial condition. The
longitudinal mode function is in a Hankel form

$h_{0}(\tau,k)=\mathcal{N}_{0}(-\tau)^{3/2}H_{i\mu}^{(1)}(-k\tau)~{},$ (64)

while the tensor mode function is given by the Whittaker W function,

$h_{\pm 2}(\tau,k)=-\frac{\mathcal{N}_{\pm 2}}{\tau}W_{\pm
i\tilde{\kappa},i\mu}(2ik\tau)~{},~{}~{}~{}\mu\equiv\sqrt{\tilde{m}^{2}-\frac{9}{4}}~{},$
(65)

with $\mu\equiv\sqrt{\tilde{m}^{2}-\frac{9}{4}}$ and
$\tilde{\kappa}\equiv\frac{\kappa}{H}$. The normalization constants are given
by

$\displaystyle\mathcal{N}_{0}$
$\displaystyle=\sqrt{\frac{\pi}{3}}\frac{k^{2}}{H}\frac{e^{-\pi\mu/2}}{H\sqrt{\left(\mu^{2}+1/4\right)\left(\mu^{2}+9/4\right)}}~{},$
(66) $\displaystyle\mathcal{N}_{\pm 2}$
$\displaystyle=\frac{e^{\mp\pi\tilde{\kappa}/2}}{2H\sqrt{k}}~{}.$ (67)

Traditionally, these normalization constants are fixed by imposing canonical
commutation relations for different components of field operators (see
Higuchi:1986py  for instance). Here, for simplicity, we follow a cleaner
shortcut using the symplectic inner product method, which is also easier to
generalize to higher spins. First notice that from any two solutions
$A_{\mu\nu}$ and $B_{\mu\nu}$ of the simplified EoM (38), one can construct a
current

$K^{\rho}(A,B)\equiv
A_{\mu\nu}^{*}\nabla^{\rho}B^{\mu\nu}-B_{\mu\nu}\nabla^{\rho}A^{*\mu\nu}+\frac{\phi}{\Lambda_{c}}\mathcal{E}^{\alpha\nu\rho\sigma}\nabla_{\alpha}\left(A^{*}_{\mu\nu}B^{\mu}_{~{}\sigma}-B_{\mu\nu}A^{*\mu}_{~{}~{}~{}\sigma}\right)~{}.$
(68)

Using the EoM (38), it can be shown that this current is conserved on-shell:

$\nabla_{\rho}K^{\rho}=0~{}.$ (69)

As a result, we can define an inner product by integrating $K$ over a spatial
hyperspace $\Sigma(\tau)$ with induced metric $\hat{g}$,

$\Big{(}A_{\mu\nu},B_{\rho\sigma}\Big{)}_{\tau}\equiv\int_{\Sigma(\tau)}\sqrt{\hat{g}}d^{3}x_{\rho}K^{\rho}(A,B)~{}.$
(70)

The inner product is time-independent by virtue of the conservation of $K$:

$\Big{(}A_{\mu\nu},B_{\rho\sigma}\Big{)}_{\tau}-\Big{(}A_{\mu\nu},B_{\rho\sigma}\Big{)}_{\tau_{0}}=\int_{\Sigma(\tau)\cup\Sigma(\tau_{0})}\sqrt{\hat{g}}d^{3}x_{\rho}K^{\rho}=\int_{\mathcal{V}(\tau,\tau_{0})}\sqrt{-g}d^{4}x\nabla_{\rho}K^{\rho}=0~{}.$
(71)

Henceforth, we define the normalization condition as222Notice that this
definition differs from that in the literature Lee:2016vti ; Kehagias:2017cym
by a factor of $2$. This is because of our different convention of fundamental
field $h_{\mu\nu}$ in the action (20).

$\Bigg{(}h_{\mu\nu}^{(\lambda)}(\tau,\mathbf{k})e^{i\mathbf{k}\cdot\mathbf{x}},h_{\rho\sigma}^{(\lambda^{\prime})}(\tau,\mathbf{k}^{\prime})e^{i\mathbf{k}^{\prime}\cdot\mathbf{x}}\Bigg{)}=2\delta^{\lambda\lambda^{\prime}}\delta^{3}({\mathbf{k}-\mathbf{k}^{\prime}})~{}.$
(72)

In dS, this condition reduces to

$ia^{-2}\eta^{\mu\rho}\eta^{\nu\sigma}\mathcal{W}\left(h_{\mu\nu}^{(\lambda)*},h_{\rho\sigma}^{(\lambda)}\right)=2~{},$
(73)

where $\mathcal{W}$ stands for the Wronskian. Inserting the mode functions
into (73), we obtain the results of the normalization constants333Actually, an
alternative way to see the absence of vector modes is that after solving (60b)
and (62b) up to a normalization constant $\mathcal{N}_{\pm 1}$, the inner
product turns out to be time-dependent, $\Big{(}h_{\mu\nu}^{(\pm
1)},h_{\rho\sigma}^{(\pm 1)}\Big{)}\propto
ia^{-2}\eta^{\mu\rho}\eta^{\nu\sigma}\mathcal{W}\left(h_{\mu\nu}^{(\pm
1)*},h_{\rho\sigma}^{(\pm
1)}\right)=\frac{8H^{2}}{k}\left(\mu^{2}+\frac{9}{4}\pm\tilde{\kappa}k\tau\right)|\mathcal{N}_{\pm
1}|^{2}$. The only way to maintain theory consistency is to have
$\mathcal{N}_{\pm 1}=0$. (67).

Focusing on the $\lambda=\pm 2$ helicities, the mode functions exhibit Stokes
phenomenon and grow a negative-frequency part after the Stokes-line crossing
time

$\tau_{*,\pm 2}\simeq-\frac{0.66\mu\mp 0.34\tilde{\kappa}}{k}~{}.$ (74)

Due to the dilation symmetry of dS, the production time for modes with
different momenta all corresponds to a particular energy scale, defined by the
physical momentum at production,

$\mathcal{M}_{*,\pm 2}\equiv\frac{k}{a(\tau_{*,\pm 2})}\simeq 0.66\mu H\mp
0.34\kappa~{}.$ (75)

It is only below this energy scale are particles produced and begin their
subsequent interactions. The IR behavior of the mode function reads

$h_{\pm 2}(\tau,k)\xrightarrow{\tau\to 0}\alpha_{\pm
2}\frac{1}{2\sqrt{\mu}H}(-\tau)^{-\frac{1}{2}+i\mu}+\beta_{\pm
2}\frac{1}{2\sqrt{\mu}H}(-\tau)^{-\frac{1}{2}-i\mu}~{},$ (76)

where

$\displaystyle\alpha_{\pm 2}$ $\displaystyle=$
$\displaystyle(1-i)(2k)^{i\mu}e^{\frac{1}{2}\pi(\mu\mp\kappa)}\frac{\sqrt{\mu}~{}\Gamma(-2i\mu)}{\Gamma\left(\frac{1}{2}-i\mu\mp
i\tilde{\kappa}\right)}~{},$ (77) $\displaystyle\beta_{\pm 2}$
$\displaystyle=$
$\displaystyle(1-i)(2k)^{-i\mu}e^{-\frac{1}{2}\pi(\mu\pm\tilde{\kappa})}\frac{\sqrt{\mu}~{}\Gamma(2i\mu)}{\Gamma\left(\frac{1}{2}+i\mu\mp
i\tilde{\kappa}\right)}~{},$ (78)

are the Bogoliubov coefficients satisfying the condition $|\alpha_{\pm
2}|^{2}-|\beta_{\pm 2}|^{2}=1$. The average particle number density is

$\langle n_{\pm 2}(\mathbf{k})\rangle^{\prime}=|\beta_{\pm
2}|^{2}=\frac{e^{2\pi(\mu\mp\tilde{\kappa})}+1}{e^{4\pi\mu}-1}\xrightarrow{\mu\gg
1}e^{-2\pi(\mu\pm\tilde{\kappa})}~{}.$ (79)

Therefore, massive spin-2 particles with helicity $\lambda=-2$ receive an
exponential enhancement from the external chemical potential (22).

###  3.4 UV completions?

Before concluding this section, we make a few remarks on the UV completion
regarding the spin-2 gauge symmetry of our chemical potential operator.

  1. $\bullet$

We have constrained the chemical potential operator to take the lowest mass
dimension possible. On one hand, this follows from the effective field theory
logic, that operators with lower dimensions is more relevant in the IR. On the
other hand, this is because higher dimensionality implies more derivatives,
and the resulting EoM may incite the appearance of ghosts or singularities.
This is the case, for example, if one picks the dimension-7 operator derived
from the Chern-Simons term instead of (22),

$\displaystyle S_{CS}=$ $\displaystyle\int
d^{4}x\sqrt{-\bar{g}}\frac{\phi}{8\Lambda_{CS}}\mathcal{E}^{\mu\nu\rho\sigma}\bar{R}_{\alpha\beta\mu\nu}\bar{R}^{\alpha\beta}_{~{}~{}~{}\rho\sigma}$
$\displaystyle=$
$\displaystyle\int\sqrt{-g}\frac{\phi}{2\Lambda_{CS}^{3}}\mathcal{E}^{\mu\nu\rho\sigma}\nabla_{\mu}\nabla_{[\alpha}h_{\beta]\nu}\nabla_{\rho}\nabla^{[\alpha}h^{\beta]}_{~{}\sigma}+\mathcal{O}(h^{3})~{},$
(80)

where $\bar{R}[\bar{g}]$ is computed with the metric $\bar{g}\equiv
g+\frac{h}{\Lambda_{CS}}$. The massless version of this term leads to the so-
called Chern-Simons gravity Jackiw:2003pm ; Alexander:2009tp , and has been
proposed as a source of gravitational birefringence Lue:1998mq  as well as
leptogenesis Alexander:2004us . This action is invariant under the linear
gauge transformation $\delta
h_{\mu\nu}=-(\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu})$, suggesting a
possible completion when going from the unitary-gauge massive IR theory to the
UV picture with linear symmetry realization. However, this operator contains
higher-order derivatives, and leads to pathologies if we try to use this
operator to assist particle production in the IR. This can be seen by
inspecting the EoM of the helicity $\lambda=\pm 2$ tensor modes,

$\left(1\mp\frac{2\hat{\kappa}k}{aH}\right)h_{\pm 2}^{\prime\prime}-2(aH\mp
4\hat{\kappa}k)h_{\pm 2}^{\prime}+\left[k^{2}+(m^{2}-2H^{2})a^{2}\mp
4\hat{\kappa}kaH\left(1+\frac{k^{2}}{2a^{2}H^{2}}\right)\right]h_{\pm
2}=0~{},$ (81)

where $\hat{\kappa}\equiv\frac{\dot{\phi}_{0}H}{\Lambda_{CS}^{3}}$ is the
equivalent of our chemical potential $\tilde{\kappa}$. In the late-time limit
$\tau\to 0$, one can canonicalize this mode and find enhancement/suppression
of particle production for the negative-/positive-helicity mode, in a way
similar to our chemical potential $\tilde{\kappa}$. However, the coefficient
before the $h_{\pm 2}^{\prime\prime}$ term is not positive-definite. This
means that, at an early time, the $\lambda=+2$ mode first encounters a
singularity once the coefficient of the second-order time derivative term
vanishes, and then becomes a ghost when its physical momentum exceeds the
scale $k/a(\tau)\gtrsim H/\hat{\kappa}=\Lambda_{CS}^{3}/\dot{\phi}_{0}$. This
is essentially the vacuum instability problem in Chern-Simons gravity
Dyda:2012rj . In contrast, our chemical potential operator avoids such a
problem by leaving the second-order time derivative unchanged.

Interestingly, however, we note that it is possible to deform the problematic
equation (81) into a healthy one such as (60c). Generically in EFTs, higher-
order derivative terms are present in the field EoM. One can then evoke the
reduction-of-order procedure Solomon:2017nlh  to reduce these potentially
pathological terms. In our case, we can reorganize (81) into the form

$h_{\pm 2}^{\prime\prime}-2aHh_{\pm
2}^{\prime}+\left[k^{2}+(m^{2}-2H^{2})a^{2}\right]h_{\pm
2}=\pm\frac{2\hat{\kappa}k}{aH}\left[h_{\pm 2}^{\prime\prime}-4aHh_{\pm
2}^{\prime}+2a^{2}H^{2}\left(1+\frac{k^{2}}{2a^{2}H^{2}}\right)h_{\pm
2}\right]~{},$ (82)

and then replace the first term on the right-hand side by the free on-shell
EoM,

$h_{\pm 2}^{\prime\prime}\to 2aHh_{\pm
2}^{\prime}-\left[k^{2}+(m^{2}-2H^{2})a^{2}\right]h_{\pm 2}~{}.$ (83)

This changes the form of the EoM to

$\mathrm{h}_{\pm 2}^{\prime\prime}-2aH\mathrm{h}_{\pm
2}^{\prime}+\left[k^{2}\pm
2ka\frac{\hat{\kappa}(m^{2}-2H^{2})}{H}+\left(m^{2}-2H^{2}\right)a^{2}\right]\mathrm{h}_{\pm
2}+\mathcal{O}(\hat{\kappa}^{2})=0~{},$ (84)

where $\mathrm{h}_{\pm 2}\equiv e^{\mp 2k\hat{\kappa}/aH}h_{\pm 2}$. Comparing
this EoM to (60c), we find that they take the same form upon identification
$\tilde{\kappa}\leftrightarrow(m^{2}/H^{2}-2)\hat{\kappa}$. Of course, one
should be reminded that the reduction-of-order procedure is accurate only up
to $\mathcal{O}(\hat{\kappa}^{2})$ errors. The solution of (84) (analogous to
our case) only represents a resummation of a selected piece of the full effect
in the massive Chern-Simons gravity, which still has pathologies on its own.
The question is whether this selected piece reflects the correct UV physics.
After all, the Chern-Simons term (80) is only one of many non-renormalizable
operators ($e.g.$, $\phi\bar{R}^{2},\phi\bar{R}_{\mu\nu}^{2}$) that are also
capable of influencing the linear EoM. This question remains unclear to us. On
the other hand, even if we directly adopt (84), to have significant
enhancement of particle production, we expect
$\tilde{\kappa}\sim\mu^{2}\hat{\kappa}\sim\mu$ and thus
$\hat{\kappa}\sim\mu^{-1}$. But then the problematic scale
$\frac{H}{\hat{\kappa}}\sim m$ lies just near the particle production scale
$\mathcal{M}_{*,\pm 2}\sim m$. Therefore, the parameter regime that we are
interested in is exactly where the theory breaks down. In view of the above
difficulties, our dimension-5 chemical potential operator seems still more
reasonable than the Chern-Simons term.

  2. $\bullet$

On the other hand, it may be possible to embed the chemical potential operator
into something similar to the axion-$SU(2)$ model Dimastrogiovanni:2016fuu ;
Agrawal:2017awz ; Thorne:2017jft ; Agrawal:2018mrg ; Maleknejad:2018nxz ;
Dimastrogiovanni:2018xnn ; Fujita:2018vmv ; Fujita:2021flu , where a gauge
boson VEV introduces a mixing between spacetime indices and internal indices.
This leads to the emergence of a spin-2 particle with similar dynamics as that
under a chemical potential. We point out there are two key differences in our
setups and objectives. First, the underlying symmetries and what we mean by
spin-2 particles are different. We are focusing on spin-2 particles that
transform under $SO(3,1)$ in the high-energy limit. In the axion-$SU(2)$
model, the spin-2 particle originates from the VEV of the gauge bosons, and
transforms under the remaining $SO(3)$ subgroup of spacetime symmetries and
internal symmetries. In the high-energy limit, the gauge boson VEV becomes
negligible and Lorentz symmetry is approximately restored, then they should be
distinguishable by having different Lorentz transformation properties. Second,
in addition to the analytical non-Gaussianity shapes, we are targeting at the
non-analytic CC signals which provide direct and clear information about the
system.

  3. $\bullet$

Another possible UV completion may be similar to that of teleparallel gravity
with a Nieh-Yan term Li:2020xjt ; Li:2021wij . By coupling the torsion tensors
to an axion-like field, parity-violating production of chiral gravitational
waves can be produced Cai:2021uup . It is then interesting to see the effect
of including a graviton mass in such theories, which would be the equivalent
of massive spin-2 field with chemical potential.

##  4 Impact on observables

With a healthy theory of free massive spin-2 particle with chemical potential,
In this section, we turn on interactions and couple the massive spin-2 field
to the “visible” sector, namely the inflaton $\varphi$ and graviton
$\gamma_{ij}$. We will show that with the help of the chemical potential, the
enhanced helicity $\lambda=-2$ modes can leave sizable and distinctive CC
signals on the mixed bispectra and curvature trispectrum even when its mass is
much larger than the Hubble scale. Our model thus provides a first example of
CC physics generalized to tensor modes.

###  4.1 Model setup

Before turning on additional couplings, notice that the dimension-5 chemical
potential operator $S_{c}$ itself brings a non-linear interaction between
$h_{\mu\nu}$ and the inflaton fluctuation $\varphi\equiv\phi-\phi_{0}$. It
induces 1-loop corrections to the correlators $\langle\zeta^{n}\rangle$, with
$n=2,3,\cdots$. However, the loop integral introduces a loop factor
$(4\pi)^{-2}$, and suppresses the signal strength. This issue can be avoided
if we are able to observe the enhanced spin-2 particles at tree-level. We can
form tree-level amplitudes with spin-2 signals by the following set of
effective operators:

$\displaystyle S_{c}$ $\displaystyle=\int d\tau
d^{3}x\frac{\phi^{\prime}}{2\Lambda_{c}}a^{-2}\epsilon_{ijk}h_{il}\partial_{j}h_{kl}~{},$
(85) $\displaystyle S_{h\gamma}$ $\displaystyle=\int d\tau
d^{3}x\rho\phi^{\prime}\gamma_{ij}^{\prime}h_{ij}~{},$ (86) $\displaystyle
S_{h\phi^{2}}$ $\displaystyle=-\int d\tau
d^{3}x\frac{1}{M}h_{ij}\partial_{i}\phi\partial_{j}\phi~{}.$ (87)

Notice that these operators are all of the lowest possible dimension giving
the field content and the symmetry requirement, and are all dimension-5.
Therefore, our choice of these EFT operators is natural and does not require
any tuning a priori. The first term $S_{c}$ is just the chemical potential
operator (22) written in a non-covariant form. $S_{h\gamma}$ gives a linear
mixing between the massive spin-2 particle and the graviton Lee:2016vti ;
Dimastrogiovanni:2018uqy ; Dimastrogiovanni:2021cif ; Dimastrogiovanni:2021mfs
, and $S_{h\phi^{2}}$ is the massive spin-2 field’s coupling to the inflaton.
These coupling can induce non-Gaussianities444Notice that the mixing
$S_{h\gamma}$ also brings a parity-violating correction to the graviton power
spectrum $\langle\gamma^{2}\rangle$ at Gaussian level, producing helical
primordial gravitational waves. In this work, we will keep the mixing
perturbative, thus the primordial gravitational wave is still dominated by
vacuum fluctuations and is not maximally helical. at tree level via the
Feynman diagrams shown in Fig. 1.

Figure 1: Non-Gaussian correlators induced at tree level from (85), (86) and
(87).

As mentioned in Sect. 3.1, $S_{c}$ is not invariant under a linear gauge
transformation $\delta
h_{\mu\nu}=-(\nabla_{\mu}\xi_{\nu}+\nabla_{\nu}\xi_{\mu})$. Consequently, it
must be understood as an effective operator that is valid below the strong-
coupling scale. And the gauge invariance of the unknown UV theory is realized
non-linearly below this cutoff. Now we have introduced extra couplings of the
massive spin-2 field as EFT operators that break both the massive spin-2 gauge
symmetry and time diffeomorphism. As a result, for the sake of consistency, we
should explore the CC phenomenology of chemical potential with the EFT under
control. In particular, the energy scale $\mathcal{M}_{*,\pm 2}$ of chemical-
potential-assisted particle production must be lower than both the strong-
coupling scales $\Lambda_{sc}$ of the massive spin-2 field, and the non-linear
scales $\Lambda_{nl}$ of time diffeomorphism,

$\displaystyle\mathcal{M}_{*,\pm 2}<\Lambda_{sc},\Lambda_{nl}~{},$ (88)

where $\mathcal{M}_{*,\pm 2}$ is given by (75) and is not far away from the
mass scale $m$.

Both $\Lambda_{sc}$ and $\Lambda_{nl}$ can be found by using the Naive
Dimensional Analysis (NDM) method Weinberg:1978kz ; Manohar:1983md . Namely,
we perform a Stueckelberg trick and enforce perturbative unitarity on the
newly generated EFT operators to find their cutoffs. For example, introducing
a Stueckelberg field $A_{\mu}$, we can decompose the spin-2 field as

$h_{\mu\nu}=\hat{h}_{\mu\nu}+\frac{\nabla_{\mu}A_{\nu}+\nabla_{\nu}A_{\mu}}{m}~{}.$
(89)

Then the chemical potential term changes into

$\displaystyle S_{c}$ $\displaystyle=\frac{1}{2}\int
d^{4}x\epsilon^{\mu\nu\rho\sigma}\left[\frac{\phi}{\Lambda_{c}}\nabla_{\mu}\hat{h}_{\nu\lambda}\nabla_{\rho}\hat{h}_{\sigma}^{~{}\lambda}+\frac{2\phi}{m\Lambda_{c}}\nabla_{\mu}\hat{h}_{\nu\lambda}\nabla_{\rho}\nabla^{\lambda}A_{\sigma}+\frac{\phi}{m^{2}\Lambda_{c}}\nabla_{\mu}\nabla_{\lambda}A_{\nu}\nabla_{\rho}\nabla^{\lambda}A_{\sigma}\right]~{}.~{}~{}~{}$

These are dimension-5,6,7 operators, respectively, if we take the inflaton
field $\phi$ on its quantum fluctuations $\varphi$. Perturbative unitarity
then requires the cutoff energy scale being the smallest derived from them,

$\Lambda_{sc}^{(c)}\sim\min\left\\{4\pi\Lambda_{c},(4\pi
m\Lambda_{c})^{1/2},(4\pi m^{2}\Lambda_{c})^{1/3}\right\\}~{},$ (90)

where $4\pi$ comes from the loop factor. For $\Lambda_{c}>m$, the constraint
from the dimension-7 operator is the most strict one, giving
$\Lambda_{sc}^{(c)}=(4\pi m^{2}\Lambda_{c})^{1/3}$. Thus if the chemical
potential is not significantly larger than the mass scale, $\kappa\sim m$, the
weak-coupling constraint $\mathcal{M}_{*,\pm 2}\sim m<(4\pi
m^{2}\Lambda_{c})^{1/3}=\Lambda_{sc}^{(c)}$ is always satisfied. Therefore,
the consistency of the weakly coupled picture of a massive spin-2 particle
with enhanced particle production set up in Sect. 3 is reassured from an EFT
point of view. Repeating the procedure for the other two operators, we obtain
their strong coupling scales:

$\Lambda_{sc}^{(h\gamma)}=\sqrt{\frac{4\pi
mM_{p}}{\rho}}~{},~{}\Lambda_{sc}^{(h\phi^{2})}=\sqrt{4\pi mM}~{}.$ (91)

Figure 2: A schematic illustration of the life of a helicity $\lambda=-2$ mode
with constant comoving momentum $k$. The horizontal axis is time and the
vertical axis represents the redshifted physical momentum of this mode. The
blue line is dashed when the mode is at the vacuum and is solid when it is
occupied by particles. The characteristic events are $t_{0}$: the mode emerges
out of the cutoffs as the BD vacuum, $t_{*,-2}$: particle production with
chemical potential, $t_{H}$: horizon-crossing, $t_{\bullet}$: the particle
decays into inflatons and gravitons, with the mode going back to vacuum. The
whole CC process occurs well below the strong-coupling scale and the non-
linear scale of time diffeomorphism, and is under perturbative control.

For the time diffeomorphism, we perform the Stueckelberg trick $t\to
t+\varphi/\dot{\phi}_{0}$ and apply NDA to the operators above. The resulting
non-linear scales for time diffeomorphism are

$\Lambda_{nl}^{(h\gamma)}=\left(\frac{4\pi
M_{p}\dot{\phi}_{0}}{\rho}\right)^{1/3}~{},~{}\Lambda_{nl}^{(h\phi^{2})}=\left(4\pi
M\dot{\phi}_{0}\right)^{1/3}~{}.$ (92)

We do not expect any non-linear scale due to broken time diffeomorphism for
chemical potential operator (85), because it is already completed to a
covariant-form (22) in which the spacetime diffeomorphisms are made
explicit555One can also start with covariant completions, such as for example,
$S_{h\gamma}\subset\int
d^{4}x\sqrt{-g}\frac{\nabla_{\rho}\phi}{\Lambda_{mix}^{2}}\nabla^{\rho}R^{\mu\nu}h_{\mu\nu}$
and $S_{h\phi^{2}}\subset\int
d^{4}x\sqrt{-g}\frac{1}{M}h^{\mu\nu}\nabla_{\mu}\phi\nabla_{\nu}\phi$.
However, the size of the couplings may be small, limited by naive choices of
higher-dimensional operators as covariant completions. In this work, we would
like to keep the discussion general and adopt a bottom-up approach, while
paying attention to the EFT validity.. As a summary, we will work within the
energy scale below the lowest cutoff found above,

$\displaystyle\Lambda_{sc}$
$\displaystyle=\min\left\\{\Lambda_{sc}^{(c)},\Lambda_{sc}^{(h\gamma)},\Lambda_{sc}^{(h\phi^{2})}\right\\}~{},$
(93) $\displaystyle\Lambda_{nl}$
$\displaystyle=\min\left\\{\Lambda_{nl}^{(h\gamma)},\Lambda_{nl}^{(h\phi^{2})}\right\\}~{}.$
(94)

Physically speaking, the modes of the massive spin-2 field are assumed to stay
at the Bunch-Davies (BD) vacuum when they exit the cutoffs
$\Lambda_{sc},\Lambda_{nl}$. Then as they are redshifted to a lower energy
scale $\mathcal{M}_{*,\pm 2}$, chemical-potential-assisted particle production
happens, and the modes are occupied abundantly with pairs of spin-2 particles.
At a lower scale, these particle pairs decay into inflatons and gravitons via
the corresponding couplings, producing observable CC signals. This physical
picture is illustrated in Fig. 2.

###  4.2 Cosmological collider signals

With the interactions in hand, we can now perform a detailed computation of
the non-Gaussian correlators,

$\langle\mathcal{O}(\\{\mathbf{k}_{i}\\})\rangle\equiv(2\pi)^{3}\delta^{3}\left(\sum_{i}\mathbf{k}_{i}\right)\langle\mathcal{O}(\\{\mathbf{k}_{i}\\})\rangle^{\prime}~{},$
(95)

where $\mathcal{O}(\\{\mathbf{k}_{i}\\})$ represents products of curvature and
tensor perturbations $\zeta(\mathbf{k}_{i}),\gamma(\mathbf{k}_{j})|_{\tau\to
0}$ observed at the end of inflation. The Schwinger-Keldysh propagators of
various fields and corresponding Feynman rules are given in Appendix A. To
avoid cluttering of unnecessary lengthy equations, we also put the technical
expressions of various diagrams in Appendix A. See Chen:2017ryl  for a review
of our diagrammatic notations and Feynman rules. Our strategy is to apply the
recently developed cutting rule Tong:2021wai  (see a brief review in Appendix
B) to obtain analytical results of the leading-order CC signals. They are then
checked against the results from direct numerical integration.

####  4.2.1 $\langle\zeta\zeta\gamma\rangle$

The mixed correlator $\langle\zeta\zeta\gamma\rangle$ can be decomposed into
different helicity components,

$\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\gamma_{ij}(\mathbf{k}_{3})\rangle^{\prime}=\left(\frac{H}{\dot{\phi}_{0}}\right)^{2}\sum_{\lambda=\pm
2}e^{\lambda*}_{ij}\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\gamma_{\lambda}(\mathbf{k}_{3})\rangle^{\prime}~{}.$
(96)

More explicitly, the helicity-basis correlator can be written as the product
of the kinematic factor and the dynamical factor,

$\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\gamma_{\lambda}(\mathbf{k}_{3})\rangle^{\prime}=\Pi^{\lambda}\mathcal{I}_{\lambda}~{}.$
(97)

The kinematic factor reduces to that of an associated Legendre polynomial,

$\Pi^{\lambda}\equiv(\mathbf{k}_{1})_{i}(\mathbf{k}_{2})_{j}e^{\lambda}_{ij}(\mathbf{\hat{k}}_{3})=-\frac{k_{1}^{2}}{3}P_{2}^{2}(\cos\theta_{13})~{},$
(98)

reflecting the spin-2 nature of the field $h_{\mu\nu}$.

Figure 3: Left: The leading-order contribution of the amplified massive spin-2
field to $\langle\zeta\zeta\gamma\rangle$. Right: The kinematic configuration.

The technical expression for the dynamical factor $\mathcal{I}_{\lambda}$ is
given by (228). It can be evaluated analytically to leading order via the
cutting rule. The total bispectrum is mimicked by the oscillatory signal part
$S_{\lambda}$ and the analytic EFT666Note that this EFT stands for the
effective theory of inflatons and gravitons with the massive spin-2 field
integrated out, which is further in the IR than the EFT operators we are
considering. The characteristic energy scales for them are $H<m$ and
$\mathcal{M}_{*,\pm 2}<\Lambda_{sc},\Lambda_{nl}$, respectively. As a matter
of fact, much interesting physics can also be extracted from studying the
graviton non-Gaussianities from this pure massless EFT point-of-view
Bordin:2020eui ; Cabass:2021fnw . background $B$.

$\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\gamma_{\lambda}(\mathbf{k}_{3})\rangle^{\prime}\simeq\Pi^{\lambda}\left(S_{\lambda}+B_{\lambda}\right)~{}.$
(99)

The CC signal part can be analytically computed as

$\displaystyle S_{\pm 2}=\frac{\rho H^{5}}{MM_{p}^{2}\dot{\phi}_{0}}$
$\displaystyle\frac{\pi
e^{\mp\pi\tilde{\kappa}}\sinh\left(\pi\tilde{\kappa}\right)\Gamma\left(2\mp
i\tilde{\kappa}\right)\text{sech}^{2}(\pi\mu)}{4\tilde{\kappa}\left(1\mp
i\tilde{\kappa}\right)}\frac{1}{k_{1}^{3}k_{2}^{3}k_{3}^{2}}$
$\displaystyle\times\Bigg{\\{}\,{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{1}{2}-i\mu,\frac{1}{2}+i\mu\\\\[2.0pt]
1\mp i\tilde{\kappa}\end{array}\Bigg{|}\,\frac{k_{123}}{2k_{3}}\Bigg{]}$ (102)
$\displaystyle~{}~{}~{}~{}~{}-\frac{k_{12}}{2k_{3}}\left(\mu^{2}+\frac{1}{4}\right)\,{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{3}{2}-i\mu,\frac{3}{2}+i\mu\\\\[2.0pt]
2\mp i\tilde{\kappa}\end{array}\Bigg{|}\,\frac{k_{123}}{2k_{3}}\Bigg{]}$ (105)
$\displaystyle~{}~{}~{}~{}~{}+\frac{k_{1}k_{2}}{4k_{3}^{2}}\left(\mu^{2}+\frac{1}{4}\right)\left(\mu^{2}+\frac{9}{4}\right)\,{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{5}{2}-i\mu,\frac{5}{2}+i\mu\\\\[2.0pt]
3\mp
i\tilde{\kappa}\end{array}\Bigg{|}\,\frac{k_{123}}{2k_{3}}\Bigg{]}\Bigg{\\}}+\text{c.c.}$
(108) $\displaystyle+\mathcal{O}\Big{(}|\beta_{\pm 2}$
$\displaystyle|^{2}\Big{)}~{},$ (109)

Here we have used the abbreviation $k_{i_{1}\cdots i_{m}}\equiv
k_{i_{1}}+\cdots+k_{i_{m}}$, and the regularized hypergeometric function
${}_{2}\mathbf{F}_{1}$ is defined from the usual hypergeometric function
${}_{2}{F}_{1}$ as

$\displaystyle{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}a,b\\\\[2.0pt]
c\end{array}\Bigg{|}\,z\Bigg{]}\equiv{}_{2}{F}_{1}\Bigg{[}\begin{array}[]{c}a,b\\\\[2.0pt]
c\end{array}\Bigg{|}\,z\Bigg{]}/\Gamma(c)~{}.$ (114)

The leading order EFT background is generated from the effective vertex,

$\int d\tau
d^{3}x\frac{4\rho\dot{\phi}_{0}}{m^{2}M}a\gamma^{\prime}_{ij}\partial_{i}\varphi\partial_{j}\varphi~{},$
(115)

which leads to

$B_{\lambda}=-\frac{\rho
H^{7}}{m^{2}MM_{p}^{2}\dot{\phi}_{0}}\frac{2k_{12}^{2}+2k_{1}k_{2}+3k_{12}k_{3}+k_{3}^{2}}{k_{1}^{3}k_{2}^{3}k_{3}k_{123}^{3}}~{}.$
(116)

In Fig. 4, we plot the total mixed bispectrum (99) as well as the signals
$S_{\lambda}$ for different parameter choices. The numerical result is also
presented for comparison. It can be seen that they match very well in most
parameter regions. The numerical CC signals for $\lambda=+2$ becomes less
precise when the mass of the particle becomes large, because the heavy
Boltzmann suppression demands an exponentially increasing numerical accuracy.
This is not a problem for the enhanced mode $\lambda=-2$, where the
oscillatory features are apparent.

Figure 4: The CC signals in the $\langle\zeta\zeta\gamma_{\lambda}\rangle$
correlator for different mass $\mu$ and chemical potential $\tilde{\kappa}$.
The correlator is normalized by
$\left(k_{1}/k_{3}\right)^{9/2}k_{3}^{6}\langle\zeta(k_{1})\zeta(k_{1})\gamma_{\lambda}(k_{3})\rangle^{\prime}/\frac{\rho
H^{5}}{MM_{p}^{2}\dot{\phi}_{0}}$, with the upper panels showing the full
result and the lower panels showing the oscillatory CC signal only. The blue
solid lines correspond to the analytical cut result while the red dashed lines
correspond to numerical results (the signals are filtered from the total
numeric integration result using high-pass filters with a Blackman window
function).

The equilateral limit of the mixed bispectrum is dominated by the EFT
contribution, with the signal strength characterized by the $f_{NL}$
parameter777An alternative definition is given by Lee:2016vti
$f_{NL,\zeta\zeta\lambda}=\frac{6}{17}\frac{\langle\zeta(\mathbf{k})\zeta(\mathbf{k})\gamma_{\lambda}(\mathbf{k})\rangle^{\prime}}{\langle\zeta(\mathbf{k})^{2}\rangle^{\prime
2}}\simeq\frac{\sqrt{r}\rho\dot{\phi}_{0}H}{12m^{2}M}|\alpha_{\lambda}|^{2}$,
with single field inflation predicting $\sum_{\lambda=\pm
2}f_{NL,\zeta\zeta\lambda}=\sqrt{r}/16$..

$\displaystyle f_{NL,\zeta\zeta\lambda}$
$\displaystyle\equiv\frac{10}{9}\frac{\langle\zeta(k)\zeta(k)\gamma_{\lambda}(k)\rangle^{\prime}}{\langle\zeta(k)^{2}\rangle^{\prime
2}}\simeq
0.9\times|\alpha_{\lambda}|^{2}\left(\frac{\rho}{10}\right)\left(\frac{r}{0.05}\right)\left(\frac{m}{5H}\right)^{-2}\left(\frac{M}{20H}\right)^{-1}~{},$
(117)

where only the leading order EFT part (116) is used, and $r\lesssim 0.1$ is
the tensor-to-scalar ratio. For a large chemical potential with
$\tilde{\kappa}\gtrsim\mu$, the $f_{NL,\zeta\zeta-2}$ parameter receives
exponential amplification from $|\alpha_{-2}|^{2}$ due to copious particle
production. This can be much greater than that in the single field slow-roll
inflation scenario, where $f_{NL,\zeta\zeta\lambda}\sim 0.1r\lesssim 0.01$.
Furthermore, the squeezed-limit bispectrum is dominated by the signal part,
and the helicity $\lambda=-2$ mode exhibits enhanced characteristic
oscillations with the momentum ratio,

$\displaystyle\lim_{k_{3}\to
0}\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\gamma_{\lambda}(\mathbf{k}_{3})\rangle^{\prime}=$
$\displaystyle\frac{\pi\rho
H^{5}\mathcal{A}(\mu,\tilde{\kappa})}{12\sqrt{2}MM_{p}^{2}\dot{\phi}_{0}}\frac{P_{2}^{2}(\cos\theta_{13})}{k_{1}^{3}k_{3}^{3}}\left(\frac{k_{3}}{k_{1}}\right)^{3/2}|\alpha_{-2}^{*}\beta_{-2}|\sin\left(\mu\ln\frac{k_{1}}{k_{3}}+\chi(\mu,\tilde{\kappa})\right)$
$\displaystyle+\mathcal{O}\left(|\beta_{-2}|^{2}\right)~{}.$ (118)

Here the dimensionless factor is

$\mathcal{A}(\mu,\tilde{\kappa})\equiv\sqrt{\frac{\left(\mu^{2}+\frac{9}{4}\right)\left(\mu^{2}+\frac{81}{4}\right)}{\tilde{\kappa}\mu}}~{},$
(119)

and $\chi(\mu,\tilde{\kappa})$ is a phase dependent on the mass and chemical
potential, whose detailed form is not important. Thus we see that the dS-
invariant Boltzmann suppression of a massive spin-2 particle is largely
alleviated, as a direct consequence of enhanced particle production
$\beta_{-2}\sim e^{-\pi(\mu-\tilde{\kappa})}\gg e^{-\pi\mu}$. This is also
reflected in Fig. 4, where large $\beta_{-2}$-suppressed oscillations are
found for one helicity and smaller Boltzmann-suppressed oscillations are found
for the other. One can also observe this in the angular dependence of the
bispectrum (see Fig. 5). In the presence of chemical potential, the
characteristic $P_{2}^{2}(\cos\theta_{13})$ behavior of the massive spin-2
mode acquires helicity-dependent modulations. In practice, an observation of
such parity-violating angular dependence and CC signal oscillations will be a
clear signature of spin-2 chemical potential.

Figure 5: The angular dependence of the mixed bispectrum
$\langle\zeta\zeta\gamma\rangle$ for different helicities and chemical
potential choices. The horizontal axis is the angle $\theta_{13}$ between
momenta $\mathbf{k}_{1},\mathbf{k}_{3}$. The vertical axis is the mixed
bispectrum normalized by
$(k_{1}/k_{3})^{9/2}k_{3}^{6}\langle\zeta(k_{1})\zeta(k_{2})\gamma_{\lambda}(k_{3})\rangle^{\prime}/\frac{\rho
H^{5}}{MM_{p}^{2}\dot{\phi}_{0}}$. The momentum ratio is held fixed as
$k_{1}/k_{3}=5$, with the mass of the spin-2 particle taken to be $\mu=3$. We
observe the characteristic $P_{2}^{2}(\cos\theta_{13})$ dependence of the
massive spin-2 particle. In addition, the chemical potential introduces a bias
for the two helicities, and the $\lambda=-2$ mode is enhanced in amplitude.

####  4.2.2 $\langle\gamma\gamma\zeta\rangle$

Another type of mixed bispectrum $\langle\gamma\gamma\zeta\rangle$ also arises
in our model. It consists of two diagrams ((i) and (ii) in Fig. 6) whose
kinematics and dynamics differ from each other, thus we consider them
separately. We first decompose the correlator in terms of helicity components,

$\displaystyle\langle\gamma_{ij}(\mathbf{k}_{1})\gamma_{kl}(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\rangle^{\prime}$
$\displaystyle=\sum_{\lambda_{1},\lambda_{2}=\pm
2}e^{\lambda_{1}*}_{ij}(\hat{\mathbf{k}}_{1})e^{\lambda_{2}*}_{kl}(\hat{\mathbf{k}}_{2})\langle\gamma_{\lambda_{1}}(\mathbf{k}_{1})\gamma_{\lambda_{2}}(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\rangle^{\prime}~{}.$

Figure 6: Left: The two tree diagrams contributing to
$\langle\gamma\gamma\zeta\rangle$. Right: The kinematic configuration.

Thus the helicity-basis 3-point function can be written as

$\langle\gamma_{\lambda_{1}}(\mathbf{k}_{1})\gamma_{\lambda_{2}}(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\rangle^{\prime}=\Pi_{(\text{i})}^{\lambda_{1}\lambda_{2}}\mathcal{I}^{(\text{i})}_{\lambda_{1}\lambda_{2}}+\Pi_{(\text{ii})}^{\lambda_{1}\lambda_{2}}\mathcal{I}^{(\text{ii})}_{\lambda_{1}\lambda_{2}}~{}.$
(120)

The kinematic factors are

$\displaystyle\Pi_{(\text{i})}^{\lambda_{1}\lambda_{2}}$ $\displaystyle\equiv
e^{\lambda_{1}}_{ij}(\hat{\mathbf{k}}_{1})e^{\lambda_{2}}_{ij}(\hat{\mathbf{k}}_{2})~{},$
(121) $\displaystyle\Pi_{(\text{ii})}^{\lambda_{1}\lambda_{2}}$
$\displaystyle\equiv\epsilon_{ijk}(\mathbf{k}_{1}-\mathbf{k}_{2})_{i}e^{\lambda_{1}}_{jl}(\hat{\mathbf{k}}_{1})e^{\lambda_{2}}_{kl}(\hat{\mathbf{k}}_{2})/k_{3}~{}.$
(122)

Here, four combinations of helicities are all allowed. In the soft limit
$k_{1}\to 0$, the kinematic factors reduce to a characteristic angular
dependence

$\displaystyle\Pi_{(\text{i})}^{+2,+2}=\Pi_{(\text{i})}^{-2,-2}=i\Pi_{(\text{ii})}^{+2,+2}=-i\Pi_{(\text{ii})}^{-2,-2}=4\sin^{4}\frac{\theta_{13}}{2}~{},$
(123)
$\displaystyle\Pi_{(\text{i})}^{-2,+2}=\Pi_{(\text{i})}^{+2,-2}=i\Pi_{(\text{ii})}^{-2,+2}=-i\Pi_{(\text{ii})}^{+2,-2}=4\cos^{4}\frac{\theta_{13}}{2}~{}.$
(124)

The dynamical factors
$\mathcal{I}^{(\text{i}),(\text{ii})}_{\lambda_{1}\lambda_{2}}$ are again
given by (229) and (230) in the appendix. Evoking the cutting rule, we proceed
in a diagram-by-diagram fashion. The first diagram is approximated as

$\langle\gamma_{\lambda_{1}}(\mathbf{k}_{1})\gamma_{\lambda_{2}}(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\rangle^{\prime}_{(\text{i})}\simeq\Pi_{(\text{i})}^{\lambda_{1}\lambda_{2}}\left(S^{(\text{i})}_{\lambda_{1}\lambda_{2}}+B^{(\text{i})}_{\lambda_{1}\lambda_{2}}\right)~{},$
(125)

The CC signal in the $k_{3}/k_{1}$ channel is

$\displaystyle S^{(\text{i})}_{\pm 2,\lambda_{2}}=$
$\displaystyle\frac{i\pi\rho^{2}H^{4}}{32M_{p}^{4}}e^{\mp\pi\tilde{\kappa}}\sinh\left(\mp\pi\tilde{\kappa}\right)\Gamma\left(\mp
i\tilde{\kappa}\right)\text{sech}^{2}(\pi\mu)$
$\displaystyle\times\frac{1}{k_{1}^{4}k_{2}k_{3}}\left(\mu^{2}+\frac{1}{4}\right)\left(\mu^{2}+\frac{9}{4}\right)\,{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{5}{2}-i\mu,\frac{5}{2}+i\mu\\\\[2.0pt]
3\mp
i\tilde{\kappa}\end{array}\Bigg{|}\,\frac{k_{123}}{2k_{1}}\Bigg{]}+\mathcal{O}\left(|\beta_{\pm
2}|^{2}\right)+\text{c.c.}~{},$ (128)

and the CC signal in the $k_{3}/k_{2}$ channel is obtained via a simple
replacement $k_{1}\leftrightarrow k_{2}$. The background
$B^{(\text{i})}_{\lambda_{1}\lambda_{2}}$ is mimicked by the operator

$\int d\tau
d^{3}x\frac{\rho^{2}H^{2}}{m^{2}}a\varphi^{\prime}\gamma_{ij}^{\prime 2}~{},$
(129)

yielding

$B^{(\text{i})}_{\lambda_{1}\lambda_{2}}=\frac{\rho^{2}H^{6}}{4M_{p}^{4}m^{2}}\frac{1}{k_{1}k_{2}k_{3}k_{123}^{3}}~{}.$
(130)

The second diagram is slightly more complicated. According to the cutting
algorithm, to obtain the leading order CC signal, we can cut one massive
propagator at a time, and approximate the others by local EFT operators. For
instance, in the soft limit $k_{1}\ll k_{2},k_{3}$, the massive spin-2
propagator with momentum $\mathbf{k}_{2}$ can be integrated out to give an
effective operator

$\int d\tau
d^{3}x\frac{\rho\dot{\phi}_{0}}{2m^{2}\Lambda_{c}}a^{-1}\varphi^{\prime}\epsilon_{ijk}\left(h_{il}\partial_{j}\gamma_{kl}^{\prime}+\gamma_{il}^{\prime}\partial_{j}h_{kl}\right)~{}.$
(131)

Applying the cutting rule, we can write the correlator as

$\langle\gamma_{\lambda_{1}}(\mathbf{k}_{1})\gamma_{\lambda_{2}}(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\rangle^{\prime}_{(\text{ii})}\simeq\Pi_{(\text{ii})}^{\lambda_{1}\lambda_{2}}\left(S^{(\text{ii})}_{\lambda_{1}\lambda_{2}}+B^{(\text{ii})}_{\lambda_{1}\lambda_{2}}\right)~{}.$
(132)

The CC signal in this limit is then approximated by

$\displaystyle S^{(\text{ii})}_{\pm
2,\lambda_{2}}\simeq\frac{i\rho^{2}\dot{\phi}_{0}H^{3}}{64M_{p}^{4}m^{2}\Lambda_{c}}\Bigg{\\{}$
$\displaystyle\frac{\pi
e^{\mp\pi\tilde{\kappa}}\sinh\left(\mp\pi\tilde{\kappa}\right)\Gamma\left(\mp
i\tilde{\kappa}\right)\text{sech}^{2}(\pi\mu)}{k_{1}^{5}k_{2}}$
$\displaystyle\times\left(\mu^{2}+\frac{1}{4}\right)\left(\mu^{2}+\frac{9}{4}\right)\left(\mu^{2}+\frac{25}{4}\right)\,{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{7}{2}-i\mu,\frac{7}{2}+i\mu\\\\[2.0pt]
4\mp i\tilde{\kappa}\end{array}\Bigg{|}\,\frac{k_{123}}{2k_{1}}\Bigg{]}$ (135)
$\displaystyle+\text{c.c.}\Bigg{\\}}+\mathcal{O}\left(|\beta_{\pm
2}|^{2}\right)~{}.$ (136)

Interestingly, the background $B^{(\text{ii})}_{\lambda_{1}\lambda_{2}}$ in
this case receives no contribution from the large-mass EFT at all orders. One
might naively come up with a leading order large-mass EFT operator from
integrating out all massive spin-2 propagators as

$\int d\tau
d^{3}x\frac{\rho^{2}\dot{\phi}^{2}}{m^{4}\Lambda_{c}}\varphi^{\prime}\epsilon_{ijk}\gamma_{il}^{\prime}\partial_{j}\gamma_{kl}^{\prime}~{}.$
(137)

However, although this operator does affect the dynamics of the massless
$(\varphi,\gamma)$-EFT, it fails to enter the correlator
$\langle\gamma\gamma\zeta\rangle$ at tree-level. One can easily verify this by
either directly computing $B_{\lambda_{1}\lambda_{2}}$ from (137), or more
generally, from inspecting the diagram using the power-counting method
presented in Liu:2019fag . As a result, we conclude that the mixed-type non-
Gaussianity generated via diagram $(\text{ii})$ does not admit a perturbative
expansion in $\mu^{-1}$. Its leading order is already non-perturbatively
suppressed,

$B^{(\text{ii})}_{\lambda_{1}\lambda_{2}}=0+\mathcal{O}\left(e^{-\pi\mu}\right)~{}.$
(138)

Figure 7: The CC signals in the
$\langle\gamma_{\lambda_{1}}\gamma_{\lambda_{2}}\zeta\rangle_{(\text{i})}$
correlator for different choices of mass $\mu$ and chemical potential
$\tilde{\kappa}$. For the purpose of demonstration, here we have only shown
two helicity choices, namely $\lambda_{1}=-2,\lambda_{2}=+2$ (left) and
$\lambda_{1}=\lambda_{2}=+2$ (right). The other helicities can also be
calculated in a similar fashion. The correlator is normalized by
$\left(k_{3}/k_{1}\right)^{9/2}k_{1}^{6}\langle\gamma_{\lambda_{1}}(k_{1})\gamma_{\lambda_{2}}(k_{1})\zeta(k_{3})\rangle^{\prime}_{(\text{i})}/\frac{\rho^{2}H^{4}}{M_{p}^{4}}$,
with the upper panels showing the full result and the lower panels showing the
oscillatory CC signal only. The blue solid lines correspond to the analytical
cut result while the red dashed lines correspond to numerical results (the
signals are filtered from the total numeric integration result using high-pass
filters with a Blackman window function).

Figure 8: The CC signals in the
$\langle\gamma_{\lambda_{1}}\gamma_{\lambda_{2}}\zeta\rangle_{(\text{ii})}$
correlator for different choices of mass $\mu$ and chemical potential
$\tilde{\kappa}$. For the purpose of demonstration, here we have only shown
two helicity choices, namely $\lambda_{1}=-2,\lambda_{2}=+2$ (left) and
$\lambda_{1}=\lambda_{2}=+2$ (right). The other helicities can also be
calculated in a similar fashion. The correlator is normalized by
$\left(k_{3}/k_{1}\right)^{9/2}k_{1}^{6}\langle\gamma_{\lambda_{1}}(k_{1})\gamma_{\lambda_{2}}(k_{1})\zeta(k_{3})\rangle^{\prime}_{(\text{ii})}/\frac{\rho^{2}\dot{\phi}H^{3}}{MM_{p}^{4}\Lambda_{c}}$,
with the upper panels showing the full result and the lower panels showing the
oscillatory CC signal only. The blue solid lines correspond to the analytical
cut result while the red dashed lines correspond to numerical results (the
signals are filtered from the total numeric integration result using high-pass
filters with a Blackman window function).

We compare our cut results for both diagrams to numerical results in Fig. 7
and Fig. 8. Crucially, we see that for both diagrams, the oscillatory CC
signals are well-approximated by the analytical cut results. Diagram (i)
admits a $\mu^{-1}$ expansion, whose effect is captured very well with the
leading order EFT background. In contrast, diagram (ii) has a non-zero
background that is not covered by the cut result, as anticipated from the
discussion above. It also becomes apparent that this background must be non-
perturbative in $\mu^{-1}$, because the bispectrum amplitudes for $\mu=3$ and
$\mu=5$ differ drastically by two orders of magnitude. Henceforth, in the
equilateral shape limit, we can estimate the size of non-Gaussianity by

$\displaystyle f_{NL,\lambda_{1}\lambda_{2}\zeta}$
$\displaystyle\equiv\frac{10}{9}\frac{\langle\gamma_{\lambda_{1}}(k)\gamma_{\lambda_{1}}(k)\zeta(k)\rangle^{\prime}}{\langle\zeta(k)^{2}\rangle^{\prime
2}}$ $\displaystyle\simeq-1.3\times
10^{-4}\times|\alpha_{-2}|^{2}\left(\frac{\rho}{10}\right)^{2}\left(\frac{r}{0.05}\right)^{2}\left(\frac{m}{5H}\right)^{-2}$
$\displaystyle\quad+0.2\times\left(|\alpha_{\lambda_{1}}|^{2}|\alpha_{\lambda_{2}}^{*}\beta_{\lambda_{2}}|^{2}+|\alpha_{\lambda_{1}}^{*}\beta_{\lambda_{1}}|^{2}|\alpha_{\lambda_{2}}|^{2}\right)\tilde{\kappa}\left(\frac{\rho}{10}\right)^{2}\left(\frac{r}{0.05}\right)^{2}\left(\frac{m}{5H}\right)^{4}~{},$
(139)

where the two lines correspond to diagram (i) and (ii), respectively. Note
that although the empirical mass power $m^{4}$ for diagram (ii) is positive,
the implicit exponential suppression in the Bogoliubov coefficients ensures
its size decreases fast with an increasing mass. The CC signals sourced by the
enhanced $\lambda_{1}=-2$ mode take the form

$\displaystyle\lim_{k_{1}\to 0}\langle\gamma_{-2}(\mathbf{k}_{1})\gamma_{\pm
2}(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\rangle_{(\text{i})}^{\prime}=$
$\displaystyle\mp\frac{\pi\rho^{2}H^{2}\mathcal{B}(\mu,\tilde{\kappa})}{M_{p}^{4}}\frac{\cos^{4}\left(\frac{\theta_{13}}{2}+\frac{\pi}{4}\mp\frac{\pi}{4}\right)}{k_{1}^{3}k_{3}^{3}}\left(\frac{k_{1}}{k_{3}}\right)^{3/2}$
$\displaystyle\times\left[|\alpha_{-2}^{*}\beta_{-2}|\sin\left(\mu\ln\frac{k_{3}}{k_{1}}+\varsigma(\mu,\tilde{\kappa})\right)+\mathcal{O}\left(|\beta_{-2}|^{2}\right)\right]~{},$
(140) $\displaystyle\lim_{k_{1}\to
0}\langle\gamma_{-2}(\mathbf{k}_{1})\gamma_{\pm
2}(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\rangle_{(\text{ii})}^{\prime}=$
$\displaystyle\mp\frac{\pi\rho^{2}\dot{\phi}H^{3}\mathcal{C}(\mu,\tilde{\kappa})}{8\sqrt{2}MM_{p}^{4}\Lambda_{c}}\frac{\cos^{4}\left(\frac{\theta_{13}}{2}+\frac{\pi}{4}\mp\frac{\pi}{4}\right)}{k_{1}^{3}k_{3}^{3}}\left(\frac{k_{1}}{k_{3}}\right)^{3/2}$
$\displaystyle\times\left[|\alpha_{-2}^{*}\beta_{-2}|\sin\left(\mu\ln\frac{k_{3}}{k_{1}}+\iota(\mu,\tilde{\kappa})\right)+\mathcal{O}\left(|\beta_{-2}|^{2}\right)\right]~{}.$
(141)

Here

$\mathcal{B}(\mu,\tilde{\kappa})\equiv\sqrt{\frac{\left(\mu^{2}+\frac{1}{4}\right)\left(\mu^{2}+\frac{9}{4}\right)}{\tilde{\kappa}\mu}}~{},\quad\mathcal{C}(\mu,\tilde{\kappa})\equiv\sqrt{\frac{\left(\mu^{2}+\frac{1}{4}\right)\left(\mu^{2}+\frac{9}{4}\right)\left(\mu^{2}+\frac{25}{4}\right)}{\tilde{\kappa}\mu}}~{},$
(142)

and $\varsigma(\mu,\tilde{\kappa}),\iota(\mu,\tilde{\kappa})$ are parameter-
dependent phases whose detailed form is irrelevant here. In principle, by
observing the amplified oscillations as well as the angular dependence for
different helicities, we are able to extract ample physical information of the
massive spin-2 particle. However, (139) implies that the
$\langle\gamma\gamma\zeta\rangle$ signal in our mode may be too small for a
perturbative mixing diagram, and is inaccessible to current CMB observations
such as Planck (which roughly requires
$f_{NL,\gamma\gamma\zeta}\sim\mathcal{O}(100)$ Bartolo:2018elp ). Allowing a
larger chemical potential can increase $\alpha_{-2},\beta_{-2}$ exponentially,
yet the mixing between the graviton and the massive spin-2 particle will
become non-perturbative. This scenario deserves a separate analysis and
whether it can then be detected via future experiments is left for future
works.

####  4.2.3 $\langle\zeta^{4}\rangle$

In addition to the mixed bispectra, the spin-2 particle also leaves
characteristic imprints on the curvature trispectrum. As mentioned before,
with a non-zero chemical potential, the massive spin-2 particle only
propagates two tensor DoFs and a longitudinal scalar DoF. The trispectrum will
thus receive contributions from only three helicity channels, as opposed to
five in the ordinary case. Of these three helicities, the $\lambda=-2$ mode is
enhanced, the $\lambda=0$ mode is unaffected, and the $\lambda=+2$ mode is
suppressed. This special feature is again observable in the 4-pt correlator

$\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\zeta(\mathbf{k}_{4})\rangle^{\prime}=\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\zeta(\mathbf{k}_{4})\rangle^{\prime}_{s}+(\text{$t,u$-channels})~{},$
(143)

In the $s$-channel, the contributions from different helicities can be
separated,

$\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\zeta(\mathbf{k}_{4})\rangle^{\prime}_{s}=\sum_{\lambda=0,\pm
2}\Pi^{\lambda}\mathcal{J}_{\lambda}~{},$ (144)

with the kinematic factor

$\Pi^{\lambda}\equiv(\mathbf{k}_{1})_{i}(\mathbf{k}_{2})_{j}e^{\lambda}_{ij}(\mathbf{\hat{k}}_{I})(\mathbf{k}_{3})_{m}(\mathbf{k}_{4})_{n}e^{\lambda}_{mn}(-\mathbf{\hat{k}}_{I})~{},$
(145)

and the dynamical factor $\mathcal{J}_{\lambda}$ whose explicit form is given
by (231). Choosing $\mathbf{k}_{I}$ to align along the $+z$-axis, the
kinematic factor then reduces to that of a product of associated Legendre
polynomials,

$\displaystyle\Pi^{\pm 2}$ $\displaystyle=\frac{8k_{1}^{2}k_{3}^{2}}{3}e^{\mp
2i\psi_{I}}P_{2}^{2}(\cos\theta_{1I})P_{2}^{-2}(\cos\theta_{3I})~{},$ (146)
$\displaystyle\Pi^{0}$
$\displaystyle=k_{1}^{2}k_{3}^{2}P_{2}^{0}(\cos\theta_{1I})P_{2}^{0}(\cos\theta_{3I})+\mathcal{O}\left(k_{I}\right)~{},$
(147)

where $\psi_{I}$ is the dihedral angle between two planes spanned by
$\mathbf{k}_{1},\mathbf{k}_{2}$ and $\mathbf{k}_{3},\mathbf{k}_{4}$ (see Fig.
9).

Figure 9: Left: The leading contribution of the amplified massive spin-2 field
to $\langle\zeta^{4}\rangle$. Right: The kinematic configuration.

Using the cutting rule, we can evaluate the 4-pt correlator analytically at
the leading order as a combination of CC signals and an analytical background,

$\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\zeta(\mathbf{k}_{4})\rangle^{\prime}_{s}\simeq\sum_{\lambda=0,\pm
2}\Pi^{\lambda}S_{\lambda}+B~{}.$ (148)

The signal part again consists combinations of Hypergeometric functions,

$\displaystyle S_{\pm 2}$
$\displaystyle=-\frac{H^{10}}{M^{2}\dot{\phi}^{4}}\frac{\pi^{2}e^{\mp\pi\tilde{\kappa}}\text{sech}^{2}(\pi\mu)}{16(k_{1}k_{2}k_{3}k_{4})^{3}k_{I}}\mathcal{K}^{\pm
2}_{L}(k_{1},k_{2},k_{I})\mathcal{K}^{\pm
2}_{R}(k_{3},k_{4},k_{I})+\text{c.c.}+\mathcal{O}\left(|\beta_{\pm
2}|^{2}\right)~{},$ (149) $\displaystyle S_{0}$
$\displaystyle=-\frac{H^{10}}{M^{2}\dot{\phi}^{4}}\frac{\pi^{2}\text{sech}^{2}(\pi\mu)}{96(k_{1}k_{2}k_{3}k_{4})^{3}k_{I}}\frac{\mu^{2}+9/4}{\mu^{2}+1/4}\mathcal{K}^{0}_{L}(k_{1},k_{2},k_{I})\mathcal{K}^{0}_{R}(k_{3},k_{4},k_{I})+\text{c.c.}+\mathcal{O}\left(|\beta_{0}|^{2}\right)~{}.$
(150)

Here, we have abbreviated

$\displaystyle\mathcal{K}^{\pm 2}_{L}(k_{1},k_{2},k_{I})\equiv$
$\displaystyle\,{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{1}{2}-i\mu,\frac{1}{2}+i\mu\\\\[2.0pt]
1\mp i\tilde{\kappa}\end{array}\Bigg{|}\,\frac{k_{12I}}{2k_{I}}\Bigg{]}$ (153)
$\displaystyle-\frac{k_{12}}{2k_{I}}\left(\mu^{2}+\frac{1}{4}\right)\,{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{3}{2}-i\mu,\frac{3}{2}+i\mu\\\\[2.0pt]
2\mp i\tilde{\kappa}\end{array}\Bigg{|}\,\frac{k_{12I}}{2k_{I}}\Bigg{]}$ (156)
$\displaystyle+\frac{k_{1}k_{2}}{4k_{I}^{2}}\left(\mu^{2}+\frac{1}{4}\right)\left(\mu^{2}+\frac{9}{4}\right)\,{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{5}{2}-i\mu,\frac{5}{2}+i\mu\\\\[2.0pt]
3\mp i\tilde{\kappa}\end{array}\Bigg{|}\,\frac{k_{12I}}{2k_{I}}\Bigg{]}~{},$
(159) $\displaystyle\mathcal{K}^{0}_{L}(k_{1},k_{2},k_{I})\equiv$
$\displaystyle\left(\mu-\frac{i}{2}\right)\left(\mu+\frac{9i}{2}\right)\,{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{1}{2}-i\mu,\frac{1}{2}+i\mu\\\\[2.0pt]
1\end{array}\Bigg{|}\,\frac{k_{12I}}{2k_{I}}\Bigg{]}$ (162)
$\displaystyle-4i\left(\mu+\frac{i}{2}\right)\,{}_{2}\mathbf{F}_{1}\Bigg{[}\begin{array}[]{c}\frac{3}{2}-i\mu,\frac{1}{2}+i\mu\\\\[2.0pt]
1\end{array}\Bigg{|}\,\frac{k_{12I}}{2k_{I}}\Bigg{]}+\mathcal{O}\left(\frac{k_{I}}{k_{12}},\frac{k_{I}}{k_{34}}\right)~{},$
(165)

and

$\displaystyle\mathcal{K}^{\pm 2}_{R}(k_{3},k_{4},k_{I})$
$\displaystyle\equiv\mathcal{K}^{\pm
2}_{L}(k_{1},k_{2},k_{I})|_{k_{1,2}\to{\scalebox{0.75}[1.0]{$-$}}k_{3,4},\tilde{\kappa}\to{\scalebox{0.75}[1.0]{$-$}}\tilde{\kappa}}~{},$
(166) $\displaystyle\mathcal{K}^{0}_{R}(k_{3},k_{4},k_{I})$
$\displaystyle\equiv\mathcal{K}^{0}_{L}(k_{1},k_{2},k_{I})|_{k_{1,2}\to{\scalebox{0.75}[1.0]{$-$}}k_{3,4}}~{}.$
(167)

Due to the lack of vector modes, the analytic background cannot be fully
mimicked by local EFT operators. However, in special kinematic configurations,
namely, when $\mathbf{k}_{1},\cdots,\mathbf{k}_{4}$ are approximately
orthogonal to $\mathbf{k}_{I}$, it can be well-approximated by the effect of
the operator

$\int d\tau d^{3}x\frac{1}{m^{2}M^{2}}(\partial_{i}\varphi)^{4}~{}.$ (168)

In the $s$-channel, the analytic background reads

$\displaystyle B=$
$\displaystyle\frac{H^{12}}{m^{2}M^{2}\dot{\phi}^{4}}\frac{(\mathbf{k}_{1}\cdot\mathbf{k}_{3})(\mathbf{k}_{2}\cdot\mathbf{k}_{4})+(\mathbf{k}_{1}\cdot\mathbf{k}_{4})(\mathbf{k}_{2}\cdot\mathbf{k}_{3})-(\mathbf{k}_{1}\cdot\mathbf{k}_{2})(\mathbf{k}_{3}\cdot\mathbf{k}_{4})}{2(k_{1}k_{2}k_{3}k_{4})^{3}}$
$\displaystyle\times\left(\frac{1}{k_{1234}}+\frac{k_{1}k_{2}+k_{3}k_{4}+k_{12}k_{34}}{k_{1234}^{3}}+\frac{3\left(k_{1}k_{2}k_{34}+k_{3}k_{4}k_{12}\right)}{k_{1234}^{4}}+\frac{12k_{1}k_{2}k_{3}k_{4}}{k_{1234}^{5}}\right)~{}.$
(169)

We plot the total trispectrum and the signals therein in Fig. 10. Again, the
cut result agrees well with that from numerical results, since the vector
component of the EFT background $B$ is subdominant in the chosen kinematic
configurations. It becomes clear from the plot that the non-local type CC
signal (for a explicit classification of signals, see Appendix B) receive
enhancement since it is interpreted as the dynamical phase of the particles
produced by chemical potential. The local type signal, in contrast, comes from
vertex particle production, for which dS symmetry is respected. For that
reason, it is subjected to the original Boltzmann suppression, making its
amplitude extremely small for large masses.

Figure 10: The CC signals in the $\langle\zeta^{4}\rangle$ for different
combinations of mass $\mu$ and chemical potential $\tilde{\kappa}$ . Left
column: The non-local type CC signal (oscillations in
$k_{12}k_{34}/k_{I}^{2}$), with normalization
$k_{1}^{9}\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\zeta(\mathbf{k}_{4})\rangle^{\prime}_{s}/\frac{8H^{10}}{M^{2}\dot{\phi}_{0}^{4}}$
and momentum configuration $k_{1}=k_{2}=k_{3}=k_{4}$, $\psi_{I}=0$. Right
column: The local type CC signal (oscillations in $k_{12}/k_{34}$), with
normalization
$k_{1}^{6}k_{3}^{6}k_{I}^{-3}\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\zeta(\mathbf{k}_{4})\rangle^{\prime}_{s}/\frac{H^{10}}{M^{2}\dot{\phi}_{0}^{4}}$
and momentum configuration $k_{1}=k_{2}$, $k_{3}=k_{4}$,
$\frac{k_{12}k_{34}}{k_{I}^{2}}=2\times 10^{4}$, $\psi_{I}=0$. The upper
panels show the full result while the lower panels show the oscillatory CC
signal only, with the blue solid lines representing the analytical cut result
and the red dashed line representing that from numerical results (the signals
are filtered from the total numeric integration result using high-pass filters
with a Blackman window function).

The trispectrum size estimator $g_{NL}$ is defined in the regular tetrahedron
limit ($k_{1}=k_{2}=k_{3}=k_{4}=\frac{\sqrt{3}}{2}k_{I}$ with
$\phi=\frac{\pi}{2}$) as,

$g_{NL}\equiv\frac{25}{216}\frac{\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\zeta(\mathbf{k}_{4})\rangle^{\prime}}{\langle\zeta(\mathbf{k}_{1})^{2}\rangle^{\prime
3}}\simeq
230\times|\alpha_{-2}|^{2}\left(\frac{m}{5H}\right)^{-2}\left(\frac{M}{20H}\right)^{-2}~{}.$
(170)

In the collapsed limit, the trispectrum simplifies to the following
characteristic structure,

$\displaystyle\lim_{k_{I}\to
0}\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\zeta(\mathbf{k}_{4})\rangle^{\prime}_{s}$
$\displaystyle=$
$\displaystyle\frac{H^{10}}{48M^{2}\dot{\phi}^{4}}\frac{\pi\mathcal{D}(\mu)}{\left(k_{1}k_{3}\right){}^{9/2}}\Bigg{\\{}P_{2}^{2}(\cos\theta_{1I})P_{2}^{-2}(\cos\theta_{3I})\Bigg{[}e^{-2i\phi_{I}}|\alpha_{+2}^{*}\beta_{+2}|\sin\left(\mu\ln\frac{k_{12}k_{34}}{k_{I}^{2}}+\xi(\mu,-\tilde{\kappa})\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+e^{2i\phi_{I}}|\alpha_{-2}^{*}\beta_{-2}|\times\sin\left(\mu\ln\frac{k_{12}k_{34}}{k_{I}^{2}}+\xi(\mu,\tilde{\kappa})\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+2\cos(2\phi_{I})e^{-\pi\mu}\times\sin\left(\mu\ln\frac{k_{12}}{k_{34}}\right)\Bigg{]}$
$\displaystyle\qquad\qquad\qquad\qquad+P_{2}^{0}(\cos\theta_{1I})P_{2}^{0}(\cos\theta_{3I})\Bigg{[}|\alpha_{0}^{*}\beta_{0}|\sin\left(\mu\ln\frac{k_{12}k_{34}}{k_{I}^{2}}+\eta(\mu)\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad+e^{-\pi\mu}\sin\left(\mu\ln\frac{k_{12}}{k_{34}}\right)\Bigg{]}\Bigg{\\}}+\mathcal{O}\left(|\beta_{0,\pm
2}|^{2}\right)~{},$ (171)

where $\xi(\mu,\tilde{\kappa}),\eta(\mu)$ are parameter-dependent phases whose
detailed form is of no importance here. And the overall factor is

$\mathcal{D}(\mu)\equiv\frac{\left(\mu^{2}+\frac{9}{4}\right)\left(\mu^{2}+\frac{81}{4}\right)}{\mu}~{}.$
(172)

The various angular and shape dependence of the trispectrum is apparent from
(171). The tensor modes $\lambda=\pm 2$ are separated from the scalar mode
$\lambda=0$ in terms of their dependence on $\theta_{1I}\theta_{3I},\psi_{i}$,
and each of them has their own amplitudes for the two types of CC signals. In
the absence of the chemical potential, the first two rows in (171) combine
into a real number with a size comparable to the third row. However,
introducing a non-zero chemical potential breaks this degeneracy. For
$\tilde{\kappa}>0$, $|\beta_{-2}|>e^{-\pi\mu}>|\beta_{+2}|$, thus the
dominating contribution comes from the second row, namely the non-local type
CC signal contributed by the helicity $\lambda=-2$ mode. In addition, the
trispectrum grows an imaginary part because of the misalignment of the first
two rows in (171). This is a unique signature of parity violation on the
trispectrum, as suggested in Liu:2019fag . The imaginary part is an odd
function of the dihedral angle $\psi_{I}$ between two planes spanned by
$\mathbf{k}_{1},\mathbf{k}_{2}$ and $\mathbf{k}_{3},\mathbf{k}_{4}$.
Furthermore, due to a no-go theorem on parity violation in the single-field
EFT, the parity-violating imaginary part must be proportional to a factor
$e^{-\pi\mu}$, which is non-perturbative in $\mu^{-1}$ and is thence invisible
at EFT level. This is indeed confirmed in (171):

$\lim_{k_{I}\to
0}\imaginary\langle\zeta(\mathbf{k}_{1})\zeta(\mathbf{k}_{2})\zeta(\mathbf{k}_{3})\zeta(\mathbf{k}_{4})\rangle^{\prime}|_{s}=e^{-\pi\mu}\sinh(\pi\tilde{\kappa})\sin(2\psi_{I})\times(\text{real
\& even function in $\psi_{I}$})~{}.$ (173)

This fact persists away from the collapsed limit even after taking into
account the contributions from $t$-channel and $u$-channel. We verify this
dependence on the dihedral angle in Fig. 11, where we have summed all three
channels for different momentum configurations. The perfect agreement between
the numerical results and the analytical cut result confirms that the
imaginary part of the trispectrum comes purely from the non-perturbative CC
signal $S_{\pm 2}$ cut from the transverse tensor modes.

Figure 11: The imaginary part of the trispectrum
$\imaginary\langle\zeta^{4}\rangle/\frac{H^{10}}{M^{2}\dot{\phi}_{0}^{4}}$ as
a function of the dihedral angle $\psi_{I}$ for different momentum
configurations and parameter choices. Left panel:
$k_{1}=k_{3}=k_{I}/2,\theta_{1I}=\theta_{3I}=\pi/3$. Middle panel:
$k_{1}=k_{3}/\sqrt{2}=k_{I}/2,\theta_{1I}=\pi/3,\theta_{3I}=\pi/4$. Right
panel: $k_{1}=k_{3}/2=k_{I}/2,\theta_{1I}=\pi/3,\theta_{3I}=\pi/4$. The blue
solid lines representing the analytical cut result and the red dashed line
representing that from numerical results.

###  4.3 Parameter space

Having exploited the rich phenomenology of the massive spin-2 chemical
potential, we proceed to analyze its allowed parameter space and its
feasibility for future observations. First, consistency requires the scale of
the CC process to happen below the various cutoffs of the EFT, as illustrated
in Fig. 2. Therefore, we impose the constraint (88), with the cutoffs given by
(93) and(94). In addition, to keep our perturbative computations reliable, we
need several perturbativity constraints by requiring the self-energy
corrections to be subdominant. First, the inflaton propagator receive a 1-loop
correction from the enhanced $h$ field,

$\displaystyle\begin{gathered}\includegraphics[width=85.35826pt]{zetaProphloop}\end{gathered}\quad<\begin{gathered}\includegraphics[width=91.04872pt]{zetaProp}\end{gathered}$
(176)
$\displaystyle\Rightarrow\frac{|\alpha_{-2}|^{4}}{(4\pi)^{2}}\left(\frac{H}{2\Lambda_{c}}\right)^{2}\frac{H^{4}}{m^{4}}<1~{}.$
(177)

The graviton propagator mixes into the massive spin-2 particle at tree level
and receives contribution at 1-loop level,

$\displaystyle\max\Bigg{\\{}\begin{gathered}\includegraphics[width=85.35826pt]{gammaProphChain}\end{gathered},~{}\begin{gathered}\includegraphics[width=85.35826pt]{gammaProphzetaloop}\end{gathered}\Bigg{\\}}<\begin{gathered}\includegraphics[width=71.13188pt]{gammaProp}\end{gathered}$
(181)
$\displaystyle\Rightarrow\max\left\\{\frac{|\alpha_{-2}|^{2}\rho^{2}\dot{\phi}_{0}^{2}}{M_{p}^{2}m^{2}}~{},~{}\frac{|\alpha_{-2}|^{2}}{(4\pi)^{2}}\frac{\rho^{2}H^{4}}{M_{p}^{2}m^{2}}\right\\}<1~{}.$
(182)

The massive spin-2 propagator receives corrections from a tree-level mixing
diagram and multiple 1-loop diagrams:

$\displaystyle\max\Bigg{\\{}$
$\displaystyle\begin{gathered}\includegraphics[width=85.35826pt]{hPropGammaChain}\end{gathered}~{},$
(184)
$\displaystyle\begin{gathered}\includegraphics[width=85.35826pt]{hProphzetaloop}\end{gathered}~{},~{}\begin{gathered}\includegraphics[width=85.35826pt]{hPropzetaloop}\end{gathered}~{},~{}\begin{gathered}\includegraphics[width=85.35826pt]{hPropgammazetaloop}\end{gathered}\Bigg{\\}}<\begin{gathered}\includegraphics[width=71.13188pt]{hProp}\end{gathered}$
(189) $\displaystyle\Rightarrow\max$
$\displaystyle\left\\{\frac{|\alpha_{-2}|^{2}\rho^{2}\dot{\phi}_{0}^{2}}{M_{p}^{2}m^{2}}~{},~{}\frac{|\alpha_{-2}|^{4}}{(4\pi)^{2}}\left(\frac{H}{2\Lambda_{c}}\right)^{2}\frac{H^{4}}{m^{4}}~{},~{}\frac{|\alpha_{-2}|^{2}}{(4\pi)^{2}}\frac{H^{4}}{M^{2}m^{2}}~{},~{}\frac{|\alpha_{-2}|^{2}}{(4\pi)^{2}}\frac{\rho^{2}H^{4}}{M_{p}^{2}m^{2}}\right\\}<1~{}.$
(190)

One important aspect to notice here is that although (for simplicity) we have
imposed perturbativity for the linear mixing diagram, there is nothing
preventing it from taking a non-perturbatively large value. Unlike the cases
with loops, we can still cope with this strongly coupled system by solving the
linear EoM and performing diagonalization An:2017hlx ; Bordin:2018pca . This
amounts to resumming all the chain diagrams. For the purpose of this work,
however, we will take a simpler route and work with the weak-mixing case. At
last, the current observational bounds set an upper limit of non-
Gaussianities. The most stringent constraints come from Planck 2018, where the
curvature bispectrum and trispectrum estimators are found to be
$f_{NL}^{\text{equil}}=-26\pm 47$ and $g_{NL}^{\text{loc}}=(-5.8\pm 6.5)\times
10^{4}$ (68%CL). We need to make sure our signal strength do not get over-
amplified by the chemical potential. Considering the leading diagrams, we
impose

$\displaystyle\begin{gathered}\includegraphics[width=85.35826pt]{zetaBispechloop}\end{gathered}\qquad\qquad\begin{gathered}\includegraphics[width=85.35826pt]{zetaBispechhzetaloop}\end{gathered}\qquad\qquad\begin{gathered}\includegraphics[width=85.35826pt]{zetaBispechhgammaloop}\end{gathered}$
(194)
$\displaystyle\Rightarrow~{}f_{NL}\sim\frac{\dot{\phi}}{H^{2}}\times\frac{1}{(4\pi)^{2}}\max\left\\{|\alpha_{-2}|^{6}\frac{H^{9}}{\Lambda_{c}^{3}m^{6}},|\alpha_{-2}|^{4}\frac{H^{6}}{M^{2}m^{4}},|\alpha_{-2}|^{4}\frac{\rho^{2}H^{6}}{M_{p}^{2}m^{4}}\right\\}<47~{},$
(195)
$\displaystyle\begin{gathered}\includegraphics[width=85.35826pt]{trispectrum}\end{gathered}\Rightarrow~{}g_{NL}\sim
230\times|\alpha_{-2}|^{2}\left(\frac{m}{5H}\right)^{-2}\left(\frac{M}{20H}\right)^{-2}<6.5\times
10^{4}~{}.$ (197)

The total allowed parameter space is obtained by combining all these
constraints.

Although no significant evidence for non-Gaussianities is found in any
present-day observations, the next generation of CMB and Large Scale Structure
(LSS) measurements are expected to improve the accuracy by at least one order
of magnitude. For instance, LiteBIRD Matsumura:2013aja  is estimated to reduce
the error bar on the mixed non-Gaussianity from the WMAP result ($\Delta
f_{NL,\zeta\zeta\gamma}=\mathcal{O}(100)$) down to $\Delta
f_{NL,\zeta\zeta\gamma}=\mathcal{O}(1)$ Shiraishi:2019yux . Other next-
generation CMB observations such as CMB-S4 CMB-S4:2016ple  or CORE
CORE:2017oje  are expected to reach a comparable performance. Even more
precision can be achieved via the futuristic 21-cm observations, which provide
a tomographic map of the neutral Hydrogen atoms during the Dark Age
Cooray:2006km ; Pillepich:2006fj . Interferometry based on radio telescopes on
the far side of the moon Silk:2020bsr  potentially grants us access to all
$10^{12}$ modes in the Dark Age, which is six orders of magnitude more than
that observable with Planck, thereby greatly reducing the statistical errors.
The forecast sensitivity for 21-cm interferometers with a
$\mathcal{O}(10^{3})$ km baseline can reach $\Delta
g_{NL}^{\text{loc}}=\mathcal{O}(10)$ for the analytical background shape and
$\Delta g_{NL}^{\text{clock}}=\mathcal{O}(100)$ for the oscillatory CC signals
Floss:2022grj .

We plot in Fig. 12 the detectability of our model for future observations,
under the constraints from perturbativity, consistency and current non-
Gaussianity bounds. The colored parameter regions give non-Gaussianities large
enough to fall into the forecast sensitivities of LiteBIRD and 21-cm
interferometry, while the gray, dashed regions are excluded by the constraints
above. We see that our model enjoys considerably large feasible parameter
space for the mixed bispectrum and curvature trispectrum. In particular, the
introduction of the chemical potential makes it possible to detect the CC
signals of the spin-2 particle even when its mass is much greater than the
Hubble scale, $i.e.$, $\mu\gg 1$. It is also worthwhile to note that mixing
perturbativity gives strongest constraint, as can be seen from Fig. 12.
However, as stated above, the mixing vertex needs not to be perturbative. In
principle, we can give it a non-perturbative treatment by resumming the chain
diagrams. This would further expand the feasible parameter space, possibly
even to the extent of unlocking the $\langle\gamma\gamma\zeta\rangle$ channel
for detection. We leave the detailed investigation of this interesting
possibility to future studies.

Figure 12: The feasible parameter regions in our model for future observations
such as LiteBIRD and 21-cm interferometry. The various colored regions give
detectably large non-Gaussian backgrounds/CC signals for the corresponding
experiment. The gray dashed regions are excluded by the imposed constraints.
The horizontal axis is the dimensionless chemical potential
$\tilde{\kappa}=\frac{\dot{\phi}_{0}}{H\Lambda_{c}}$ and the vertical axis is
the dS-invariant mass of the spin-2 particle
$\mu=\sqrt{\frac{m^{2}}{H^{2}}-\frac{9}{4}}$. Here we have assumed a tensor-
to-scalar ratio $r=0.05$, a value consistent with the Planck bounds. The
mixing coupling and the coupling to the inflaton are given in the
parenthesizes as $(\rho,M/H)$. The LiteBIRD forecast sensitivity is taken as
$\Delta f_{NL,\zeta\zeta\gamma}=\mathcal{O}(1)$ for both the background and CC
signals Shiraishi:2019yux , while the 21-cm interferometer sensitivity is
taken to be $\Delta g_{NL}^{\text{loc}}=\mathcal{O}(10)$ for the background
shape and $\Delta g_{NL}^{\text{clock}}=\mathcal{O}(100)$ for CC signals
Floss:2022grj .

##  5 Conclusion and outlooks

From massive gravity to extra dimensions and to string theory, massive spin-2
particles play an important role in different aspects of modern physics. In
particular, if they exist during inflation, their interactions with the
inflaton and graviton leave distinctive signatures on primordial non-
Gaussianities, providing us invaluable insight into the particle spectrum of
the early universe. Yet such spin-2 particles may be equipped with a large
mass. Thus their pure dS-invariant gravitational production is exponentially
suppressed by Boltzmann factors, resulting in cosmological collider signals
potentially too small to detect.

In this work, we have explored one possibility to enhance the signal strength
by introducing a symmetry-breaking chemical potential. We first showed that
the chemical potential operator is uniquely fixed at dimension-5 by the
requirement of linearity, symmetry and consistency. Analysis of the linear
theory shows that the spin-2 field propagates two tensor DoFs and a scalar
DoF, with the vector modes suppressed by the chemical potential. The
assistance of a positive chemical potential leads to the exponential
enhancement in the production rate of the $\lambda=-2$ mode. After coupling
the enhanced spin-2 particle to the visible sector, we found large non-
Gaussianities of the type $\langle\zeta\zeta\gamma\rangle$,
$\langle\zeta^{4}\rangle$ and large CC signals therein. Under the constraints
from EFT consistency, perturbativity and current observations, our model
encompass wide parameter regions feasible for future CMB observations and
21-cm tomography.

Despite the interesting dynamics and rich phenomenology of our model, there
are several questions untouched and we conclude by mentioning a few of them as
prospects in the future.

  1. $\bullet$

First, the mixing between the massive spin-2 particle and graviton is taken to
be perturbative in this work. It is interesting and straightforward to
generalize the analysis to the non-perturbative case. This will broaden the
parameter space and allow for large helical primordial gravitational wave
power spectrum $\langle\gamma^{2}\rangle$, and possible observation of the
$\langle\gamma\gamma\zeta\rangle$ type non-Gaussianity in addition.

  2. $\bullet$

Second, the disappearance of the vector modes even in the small chemical
potential limit seems a bit mysterious to us, as it suggests a somewhat
unphysical discontinuity in the $\tilde{\kappa}\to 0$ limit. It may be
possible that this discontinuity is an artifact due to the linear
approximation and will vanish in the nonlinear theory. Something like a
Vainshtein mechanism may then be responsible for the suppressed vector modes
and it is worthwhile to explore in more detail.

  3. $\bullet$

Third, we have worked in a heavily bottom-up fashion in this paper, using
Stueckelberg trick and NDA counting methods to keep unknown UV physics under
control. It is certainly important to have a more UV complete understanding of
the chemical potential operator, especially in light of the several lines of
work mentioned in Sect. 3.4. A better understanding of the UV picture should
also shed light on the previous point.

  4. $\bullet$

Fourth, going beyond spin-2, chemical potential of higher-spin fields may aid
probing the spin-mass relation of the inflationary particle spectrum.
Experience from hadron resonances and string theory suggests that particles
with a higher spin tend to have a larger mass. Assuming a naive Regge behavior
for the massive spin-$S$ particles ($S\sim\alpha^{\prime}m_{S}^{2}$, where
$\alpha^{\prime}$ is the Regge slope), the CC signal strength then decays
quickly as $e^{-\pi\sqrt{S/\alpha^{\prime}}/H}$. This deficiency motivates a
generalization of chemical potential enhancement to higher spins. The
generalization seems straightforward for bosons. Following the logic of Sect.
3.1, we can write down the following operator,

$\int
d^{4}x\epsilon^{\mu\nu\rho\sigma}\frac{\phi}{2\Lambda_{c,S}}\nabla_{\mu}\Sigma_{\nu\lambda_{1}\cdots\lambda_{S-1}}\nabla_{\rho}\Sigma_{\sigma}^{~{}\lambda_{1}\cdots\lambda_{S-1}}~{},$
(198)

where $\Sigma_{\mu_{1}\cdots\mu_{S}}=\Sigma_{(\mu_{1}\cdots\mu_{S})}$ is a
totally symmetric rank-$S$ tensor. Then the massive spin-$S$ particle may also
benefit from a chemical potential enhancement, with a boosted CC signal
strength
$\sim\exp\left[{-\frac{\pi}{H}\left(\sqrt{\frac{S}{\alpha^{\prime}}}-\frac{S\dot{\phi}_{0}}{\Lambda_{c,S}}\right)}\right]$.
It is even conceivable to detect a uniform family of signals from the whole
high spin tower if $\Lambda_{c,S}\propto\sqrt{S}$. We leave the detailed study
of the corresponding dynamics and interesting phenomenology to future studies.

###### Acknowledgements.

XT would like to thank Yuhang Zhu and Yi Wang for helpful discussions. XT and
ZX are supported in part by the National Key R&D Program of China
(2021YFC2203100). ZX is supported in part by an Open Research Fund of the Key
Laboratory of Particle Astrophysics and Cosmology, Ministry of Education of
China, and a Tsinghua University Initiative Scientific Research Program.

##  Appendix A The Feynman rules

We begin by noting the mode functions of inflaton and graviton,

$\displaystyle\varphi(\tau,k)$
$\displaystyle=\frac{H}{\sqrt{2k^{3}}}(1+ik\tau)e^{-ik\tau}~{},$ (199)
$\displaystyle\gamma_{\lambda}(\tau,k)$
$\displaystyle=\frac{H}{2\sqrt{k^{3}}}(1+ik\tau)e^{-ik\tau}~{}.$ (200)

The mode functions for the massive spin-2 particle are given in Sect. 3.3. The
Schwinger-Keldysh (SK) propagators of $\varphi,\gamma_{ij},h_{ij}$ are labeled
by $F,G,H$. Diagrammatically, we denote them by

$\displaystyle~{}~{}~{}~{}~{}\begin{gathered}\includegraphics[width=91.04872pt]{phiPropRule}\end{gathered}=~{}F^{\eta_{1}\eta_{2}}(\tau_{1},\tau_{2},k)~{},$
(202)
$\displaystyle\begin{gathered}\includegraphics[width=113.81102pt]{gammaPropRule}\end{gathered}=~{}G_{ij,kl}^{\eta_{1}\eta_{2}}(\tau_{1},\tau_{2},\mathbf{k})~{},$
(204)
$\displaystyle\begin{gathered}\includegraphics[width=113.81102pt]{hPropRule}\end{gathered}=~{}H_{ij,kl}^{\eta_{1}\eta_{2}}(\tau_{1},\tau_{2},\mathbf{k})~{}.$
(206)

The four SK propagators are all built from the basic Wightman functions with
non-time ordering $\eta_{1}=-,\eta_{2}=+$,

$\displaystyle F^{-+}(\tau_{1},\tau_{2},k)$
$\displaystyle=\varphi(\tau_{1},k)\varphi^{*}(\tau_{2},k)~{},$ (208)
$\displaystyle G_{ij,kl}^{-+}(\tau_{1},\tau_{2},\mathbf{k})$
$\displaystyle=\frac{2}{M_{p}^{2}}\sum_{\lambda=\pm
2}\gamma_{\lambda}(\tau_{1},k)e_{ij}^{\lambda}(\mathbf{\hat{k}})\left[\gamma_{\lambda}(\tau_{2},k)e_{kl}^{\lambda}(\mathbf{\hat{k}})\right]^{*}~{},$
(209) $\displaystyle H_{ij,kl}^{-+}(\tau_{1},\tau_{2},\mathbf{k})$
$\displaystyle=\sum_{\lambda=\pm
2}h_{\lambda}(\tau_{1},k)e_{ij}^{\lambda}(\mathbf{\hat{k}})\left[h_{\lambda}(\tau_{2},k)e_{kl}^{\lambda}(\mathbf{\hat{k}})\right]^{*}$
$\displaystyle+\left(g_{0}(\tau_{1},k)e_{ij}^{0}(\hat{\mathbf{k}})+\frac{1}{3}h_{0}(\tau_{1},k)\delta_{ij}\right)\left(g_{0}(\tau_{2},k)e_{kl}^{0}(\hat{\mathbf{k}})+\frac{1}{3}h_{0}(\tau_{2},k)\delta_{kl}\right)^{*}~{}.$
(210)

More explicitly, denoting $P\in\\{F,G,H\\}$ as a general propagator, then the
other three SK propagators are

$\displaystyle P^{+-}(\tau_{1},\tau_{2},\mathbf{k})$
$\displaystyle=P^{-+}(\tau_{1},\tau_{2},-\mathbf{k})^{*}~{},$ (211)
$\displaystyle P^{++}(\tau_{1},\tau_{2},\mathbf{k})$
$\displaystyle=\Theta(\tau_{1}-\tau_{2})P^{-+}(\tau_{1},\tau_{2},\mathbf{k})+\Theta(\tau_{2}-\tau_{1})P^{+-}(\tau_{1},\tau_{2},\mathbf{k})~{},$
(212) $\displaystyle P^{--}(\tau_{1},\tau_{2},\mathbf{k})$
$\displaystyle=P^{--}(\tau_{1},\tau_{2},-\mathbf{k})^{*}~{}.$ (213)

It is often convenient to extract the kinematic factors from the spinning
propagators as

$P_{ij,kl}^{\eta_{1}\eta_{2}}(\tau_{1},\tau_{2},\mathbf{k})=\sum_{\lambda}e_{ij}^{\lambda}(\mathbf{\hat{k}})e_{kl}^{\lambda*}(\mathbf{\hat{k}})P_{\lambda}^{\eta_{1}\eta_{2}}(\tau_{1},\tau_{2},k)~{}.$
(214)

Note that although the longitudinal part of the $H$ propagator cannot be
simply factored as above, only the $g_{0}e_{ij}^{0}$ part dominate the IR
limit. Hence when computing the CC signals, we can neglect the
$h_{0}\delta_{ij}$ contribution and factor out the kinematic structure as
above.

The vertex rules of the EFT operators (85), (86) (87) are given by

$\displaystyle\begin{gathered}\includegraphics[width=142.26378pt]{FeynRulephih2}\end{gathered}$
$\displaystyle\qquad\begin{gathered}\frac{\eta}{2\Lambda_{c}}\epsilon_{mrs}(\mathbf{k}_{1}-\mathbf{k}_{2})_{r}\int
d\tau
a^{-2}(\tau)\partial_{\tau}F^{\eta^{\prime\prime\prime}\eta}(\tau^{\prime\prime\prime},\tau,k_{3})\\\
\qquad\qquad\qquad\qquad\qquad\times
H_{mn,ij}^{\eta\eta^{\prime}}(\tau,\tau^{\prime},\mathbf{k}_{1})H_{sn,kl}^{\eta\eta^{\prime\prime}}(\tau,\tau^{\prime\prime},\mathbf{k}_{2})~{},\end{gathered}$
(218)
$\displaystyle\begin{gathered}\includegraphics[width=142.26378pt]{FeynRulehgamma}\end{gathered}$
$\displaystyle\qquad\begin{gathered}i\eta\rho\dot{\phi}_{0}\int d\tau
a(\tau)H_{ij,mn}^{\eta^{\prime}\eta}(\tau^{\prime},\tau,\mathbf{k})\partial_{\tau}G_{mn,kl}^{\eta\eta^{\prime\prime}}(\tau,\tau^{\prime\prime},\mathbf{k})~{},\end{gathered}$
(221)
$\displaystyle\begin{gathered}\includegraphics[width=142.26378pt]{FeynRulegammahphi}\end{gathered}$
$\displaystyle\qquad\begin{gathered}i\eta\rho\int d\tau
H_{mn,ij}^{\eta\eta^{\prime}}(\tau,\tau^{\prime},\mathbf{k}_{1})\partial_{\tau}G_{mn,kl}^{\eta\eta^{\prime\prime}}(\tau,\tau^{\prime\prime},\mathbf{k}_{2})\partial_{\tau}F^{\eta\eta^{\prime\prime\prime}}(\tau,\tau^{\prime\prime\prime},k_{3})~{},\end{gathered}$
(224)
$\displaystyle\begin{gathered}\includegraphics[width=142.26378pt]{FeynRulehphi2}\end{gathered}$
$\displaystyle\qquad\begin{gathered}\frac{2i\eta}{M}(\mathbf{k}_{1})_{m}(\mathbf{k}_{2})_{n}\int
d\tau
F^{\eta^{\prime}\eta}(\tau^{\prime},\tau,k_{1})F^{\eta^{\prime\prime}\eta}(\tau^{\prime\prime},\tau,k_{2})H_{mn,ij}^{\eta\eta^{\prime\prime\prime}}(\tau,\tau^{\prime\prime\prime},\mathbf{k}_{3})~{}.\end{gathered}$
(227)

The dynamical factor of the diagram contributing to
$\langle\zeta\zeta\gamma\rangle$ reads

$\displaystyle\mathcal{I}_{\lambda}$ $\displaystyle=-\frac{16\rho
H^{2}}{MM_{p}^{2}\dot{\phi}_{0}}\sum_{\eta_{1},\eta_{2}=\pm
1}\eta_{1}\eta_{2}\int
d\tau_{1}d\tau_{2}a(\tau_{2})F^{\eta_{1}-}(\tau_{1},0,k_{1})F^{\eta_{1}-}(\tau_{1},0,k_{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times
H_{\lambda}^{\eta_{1}\eta_{2}}(\tau_{1},\tau_{2},k_{3})\partial_{\tau_{2}}G_{\lambda}^{\eta_{2}-}(\tau_{2},0,k_{3})~{}.$
(228)

The dynamical factors of the two diagrams contributing to
$\langle\zeta\zeta\gamma\rangle$ are given by

$\displaystyle\mathcal{I}^{(\text{i})}_{\lambda_{1}\lambda_{2}}\equiv$
$\displaystyle\frac{16\rho^{2}H}{M_{p}^{4}}\sum_{\eta_{1},\eta_{2}}\eta_{1}\eta_{2}\int
d\tau_{1}d\tau_{2}a(\tau_{1})\partial_{\tau_{2}}F^{\eta_{2}-}(\tau_{2},0,k_{3})\partial_{\tau_{2}}G^{\eta_{2}-}_{\lambda_{2}}(\tau_{2},0,k_{2})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
H^{\eta_{2}\eta_{1}}_{\lambda_{1}}(\tau_{2},\tau_{1},k_{1})\partial_{\tau_{1}}G^{\eta_{1}-}_{\lambda_{1}}(\tau_{1},0,k_{1})$
$\displaystyle+(k_{1}\leftrightarrow k_{2})~{},$ (229)
$\displaystyle\mathcal{I}^{(\text{ii})}_{\lambda_{1}\lambda_{2}}\equiv$
$\displaystyle-\frac{32\rho^{2}\dot{\phi}_{0}Hk_{3}}{M_{p}^{4}\Lambda_{c}}\sum_{\eta_{1},\eta_{2},\eta_{3}}\eta_{1}\eta_{2}\eta_{3}\int
d\tau_{1}d\tau_{2}d\tau_{3}a(\tau_{1})a(\tau_{2})a(\tau_{3})^{-2}\partial_{\tau_{3}}F^{\eta_{3}-}(\tau_{3},0,k_{3})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
H^{\eta_{3}\eta_{1}}_{\lambda_{1}}(\tau_{3},\tau_{1},k_{1})\partial_{\tau_{1}}G^{\eta_{1}-}_{\lambda_{1}}(\tau_{1},0,k_{1})$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\times
H^{\eta_{3}\eta_{2}}_{\lambda_{2}}(\tau_{3},\tau_{2},k_{2})\partial_{\tau_{2}}G^{\eta_{2}-}_{\lambda_{2}}(\tau_{2},0,k_{2})~{},$
(230)

respectively. The dynamical factor of the diagram contributing to
$\langle\zeta^{4}\rangle$ reads

$\displaystyle\mathcal{J}_{\lambda}=$
$\displaystyle-\frac{4H^{4}}{M^{2}\dot{\phi}_{0}^{4}}\sum_{\eta_{1},\eta_{2}=\pm
1}\eta_{1}\eta_{2}\int
d\tau_{1}d\tau_{2}F^{\eta_{1}-}(\tau_{1},0,k_{1})F^{\eta_{1}-}(\tau_{1},0,k_{2})$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times
F^{\eta_{2}-}(\tau_{2},0,k_{3})F^{\eta_{2}-}(\tau_{2},0,k_{4})H_{\lambda}^{\eta_{1}\eta_{2}}(\tau_{1},\tau_{2},k_{I})~{}.$
(231)

##  Appendix B The cutting rule for CC signals

In this appendix, we give a lighting review of the cutting rule for CC
signals. At tree-level, the analytical expression of the oscillatory CC
signals can be conveniently extracted via the recently proposed cutting rule.
The key insight is that the CC signals originate from two distinct physical
processes during inflation, and can be classified according to their different
oscillation patterns,

I. Non-local Type:
$\displaystyle\quad\sin\left(\mu\ln\frac{k_{L}k_{R}}{k_{I}^{2}}+\xi\right)~{},$
(232) II. Local Type:
$\displaystyle\quad\sin\left(\mu\ln\frac{k_{L}}{k_{R}}\right)~{}.$ (233)

Here $k_{L,R}$ are the corresponding injection frequencies at the left/right
blobs to which the massive propagator is attached.

Figure 13: A cartoon illustration of the cutting rule for CC signals.

The non-local type signal represents non-local bulk pair production and
resonance decay events, while the local type signal represents local vertex
production and resonance decay888Notice that in 3-point correlators, the two
types of CC signals take the same form, since $k_{I}=k_{R}$. However, they
still represent different contributions and can be distinguished from the
external line helicities.. Since all the events happen in the form of
resonances localized in time, one can utilize their time ordering to discard
certain part of the Feynman diagram that is free of any signals. In
particular, when $k_{L}>k_{R}$, the complicated commutator integral from
flipping one of the Heaviside theta function can be thrown away, leaving a
simple factorized integral that is often easy to compute. The diagram is then
cut into simple product of two Laplace transformations. This procedure is
illustrated in Fig. 13. Based on this observation, one can approximate any
tree diagrams by a combination of CC signals and EFT backgrounds. We briefly
summarize the general algorithm below, and refer interested readers to
Tong:2021wai  for more details.

Cutting algorithm for CC signals

  1. 1.

Consider only completely time-ordered or totally anti-time-ordered diagrams.

  2. 2.

For all massive propagators in the given diagram, work with one at a time and
mimic other massive propagators by the corresponding large-mass EFT operators.

  3. 3.

Work out the left/right blobs and reduce them to a sum of products of
exponentials and polynomials.

  4. 4.

Flip the time-ordering Heaviside step function and discard the commutator
integral, obtaining two factorized integrals as Laplace transformations.

  5. 5.

Perform a left $\leftrightarrow$ right symmetrization.

  6. 6.

Repeat Step 2-5 for each massive propagator, with their union giving the total
signal $S$.

  7. 7.

Finally, dress the total signal $S$ by an EFT background $B$ where all massive
fields are integrated out.

##  Appendix C Some helpful identities and checklists of dimensionality

The dS curvature tensors are made of metric tensors alone, since it is a
maximally symmetric spacetime:

$\displaystyle R_{\mu\nu\rho\sigma}$
$\displaystyle=H^{2}(g_{\mu\rho}g_{\nu\sigma}-g_{\mu\sigma}g_{\nu\rho})~{},$
(234a) $\displaystyle R_{\mu\nu}$ $\displaystyle=3H^{2}g_{\mu\nu}~{},$ (234b)
$\displaystyle R$ $\displaystyle=12H^{2}~{}.$ (234c)

Then using the commutator of covariant derivatives,

$[\nabla_{\mu},\nabla_{\nu}]T^{\rho_{1}\cdots\rho_{m}}_{~{}~{}~{}~{}~{}~{}~{}~{}\sigma_{1}\cdots\sigma_{n}}=R^{\rho_{1}}_{~{}~{}\lambda\mu\nu}T^{\lambda\cdots\rho_{m}}_{~{}~{}~{}~{}~{}~{}~{}~{}\sigma_{1}\cdots\sigma_{n}}+\cdots-R^{\lambda}_{~{}~{}\sigma_{1}\mu\nu}T^{\rho_{1}\cdots\rho_{m}}_{~{}~{}~{}~{}~{}~{}~{}~{}\lambda\cdots\sigma_{n}}-\cdots~{},$
(235)

one can prove the following quantity vanishes in dS:

$\displaystyle\mathcal{E}^{\mu\nu\rho\sigma}[\nabla_{\mu},\nabla_{\nu}]T_{\rho\alpha_{1}\cdots\alpha_{n}}$
$\displaystyle=-\mathcal{E}^{\mu\nu\rho\sigma}R^{\lambda}_{~{}\rho\mu\nu}T_{\lambda\alpha_{1}\cdots\alpha_{n}}-\mathcal{E}^{\mu\nu\rho\sigma}R^{\lambda}_{~{}\alpha_{1}\mu\nu}T_{\rho\lambda\cdots\alpha_{n}}-\cdots$
$\displaystyle=0-H^{2}\mathcal{E}^{\mu\nu\rho\sigma}(\delta^{\lambda}_{\mu}g_{\alpha_{1}\nu}-\delta^{\lambda}_{\nu}g_{\alpha_{1}\mu})T_{\rho\lambda\cdots\alpha_{n}}-\cdots$
$\displaystyle=0~{},$ (236)

for any totally symmetric tensor
$T_{\lambda\alpha_{1}\cdots\alpha_{n}}=T_{(\lambda\alpha_{1}\cdots\alpha_{n})}$.

In Table. 1 and Table. 2, we list the mass and conformal dimensions of various
quantities that frequently appear in this work. They are often convenient in
consistency checks, for instance, when counting powers of couplings and scale
factors in EFT operators.

Mass dimension | Quantities
---|---
$4$ | $\mathcal{L},~{}\Delta\mathcal{L}$
$3$ | $\delta^{3}(\mathbf{x})$
$2$ | $\dot{\phi}$
$1$ | $m,~{}\kappa,~{}H,~{}M_{p},~{}M,~{}\mathcal{M}_{*,\pm 2},~{}\Lambda_{c},~{}\partial_{\tau},~{}\partial_{i},~{}k,~{}\varphi(x),~{}h_{\mu\nu}(x)$
$0$ | $\mu,~{}\tilde{\kappa},~{}\rho,~{}a(\tau),~{}f_{NL},~{}g_{NL},~{}\gamma_{\mu\nu}(x),~{}\zeta(x)$
$-1$ | $\tau,~{}\mathbf{x},~{}\langle\varphi^{2}\rangle^{\prime}$
$-2$ | $h_{ij}(\mathbf{k}),~{}\varphi(\mathbf{k})$
$-3$ | $\gamma_{ij}(\mathbf{k}),~{}\zeta(\mathbf{k}),~{}\delta^{3}(\mathbf{k}),~{}\langle\zeta^{2}\rangle^{\prime},~{}\langle\gamma^{2}\rangle^{\prime}$
$-6$ | $\langle\zeta^{2}\gamma\rangle^{\prime},~{}\langle\gamma^{2}\zeta\rangle^{\prime}$
$-9$ | $\langle\zeta^{4}\rangle^{\prime}$
Table 1: A list of the mass dimensions of various quantities. Conformal dimension | Quantities
---|---
$4$ | $\mathcal{L},~{}\Delta\mathcal{L}$
$3$ | $\delta^{3}(\mathbf{x})$
$2$ | $h_{\mu\nu}(x)$
$1$ | $a(\tau),~{}\partial_{\tau},~{}\partial_{i},~{}k$
$0$ | $H,~{}M_{p},~{}m,~{}M,~{}\mathcal{M}_{*,\pm 2},~{}\Lambda_{c},~{}\kappa,~{}\rho,~{}\mu,~{}\tilde{\kappa},~{}\dot{\phi},~{}f_{NL},~{}g_{NL},~{}\gamma_{\mu\nu}(x),~{}\zeta(x)$
$-1$ | $\tau,~{}\mathbf{x},~{}h_{ij}(\mathbf{k})$
$-2$ |
$-3$ | $\varphi(\mathbf{k}),~{}\gamma_{ij}(\mathbf{k}),~{}\zeta(\mathbf{k}),~{}\delta^{3}(\mathbf{k}),~{}\langle\varphi^{2}\rangle^{\prime},~{}\langle\zeta^{2}\rangle^{\prime},~{}\langle\gamma^{2}\rangle^{\prime}$
$-6$ | $\langle\zeta^{2}\gamma\rangle^{\prime},~{}\langle\gamma^{2}\zeta\rangle^{\prime}$
$-9$ | $\langle\zeta^{4}\rangle^{\prime}$
Table 2: A list of the conformal dimensions of various quantities.

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