# The Kalb-Ramond field and Gravitational Parity Violation

Tucker Manton<EMAIL_ADDRESS>Department of Physics, Brown
University, Providence, RI 02912, USA  Stephon Alexander
<EMAIL_ADDRESS>Department of Physics, Brown University,
Providence, RI 02912, USA

###### Abstract

Gravitational wave physics can probe theories that extend beyond General
Relativity. Motivated by recent attention on the Kalb-Ramond field as a dark
matter candidate, in this work, we study a parity violating dimension four
operator which couples the dual Riemann curvature to the 2-form field. After
mapping the equations of motion for the right- and left-handed gravitational
wave amplitudes to the novel parameterization recently presented in
Jenks:2023pmk , we discuss various constraints on the model parameters in
light of the coincident electromagnetic/gravitational wave signal of GW170817
and the GWTC-3 dataset.

###### pacs:

Valid PACS appear here

##  I Introduction

The advent of gravitational wave (GW) astronomy has ushered in a new era of
modern physics, pioneered by the growing catalog of observations from the
LIGO-Virgo-Kagra collaboration LIGOScientific:2014pky ; VIRGO:2014yos ;
LIGOScientific:2020ibl ; KAGRA:2021vkt  with recent data supporting a
stochastic GW background from the pulsar timing array collaborations
NANOGrav:2023gor ; NANOGrav:2023hvm ; EPTA:2023fyk ; Xu:2023wog ;
Reardon:2023gzh . GW signals provide a rich compliment to electromagnetic
signals of astrophysical phenomena LIGOScientific:2017zic , as well as
offering a new arena for testing General Relativity (GR) in strong field
regimes LIGOScientific:2021sio . An immense outstanding question in the field
is not if GR will fail to describe a phenomenon, but when and how. In
particular, observing gravitational parity violation would be a smoking gun
for physics beyond GR, as the Einstein-Hilbert term alone produces no parity
violation. Parity violating gravitational theories have gathered interest in
recent decades for a variety of reasons. Observing parity violation in CMB
polarization measurements was considered as far back as 1998 Lue:1998mq .

Dynamical Chern-Simons theory (dCS) is likely the most studied parity
violating model for gravity. In addition to the two usual massless spin-2
degrees of freedom in GR, dCS includes a pseudoscalar field $\phi$ with a
canonical kinetic term coupled to the Pontryagin density
$\tilde{R}_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$, where
$\tilde{R}_{\mu\nu\rho\sigma}=\frac{1}{2}\varepsilon_{\mu\nu\alpha\beta}R^{\alpha\beta}_{\
\ \ \rho\sigma}$ is the dual Riemann tensor, which arises in many contexts
Yunes:2009hc  (see Alexander:2009tp  and references therein for a review).
Interesting physics follows if the dCS pseudoscalar acts as the inflaton. This
includes a preheating mechanism Alexander:2014bsa , leptogenesis
Alexander:2004us , as well as the potential for parity violation to be
detected in large-scale structure via galaxy correlations Creque-
Sarbinowski:2023wmb . Perturbations to rotating black holes were also studied
in dCS theory recently in Wagle:2023fwl , and in Nojiri:2019nar , the
propagation of gravitational waves in dCS were compared to standard $f(R)$
modified gravity theories.

There are numerous other parity violating theories on the market. One such
class is the ghost-free higher derivative models Crisostomi:2017ugk ;
Qiao:2021fwi , which are characterized by Lagrangians of the form
$\mathcal{L}(g,\partial g,\partial^{2}g)$, and individual terms are
contractions of the metric, Riemann, and the Levi-Civita tensors. Higher
derivative theories are generically plagued by Ostrogradsky instabilities
Woodard:2015zca , therefore such theories can at best be valid up to the mass
scale of the ghost degrees of freedom from an effective field theory
standpoint. Adjacent to these are the ghost-free scalar tensor models, such as
those discussed in Nishizawa:2018srh . Here, derivatives of a scalar field
couple to contractions of the Riemann and Levi-Civita tensors. As in the
higher order derivative theories, constraints on the Lagrangian coefficients
must be imposed in order to avoid Ostrogradsky ghosts. There is also a parity
violating addition to the Hor̂ava-Lifshitz approach to quantum gravity
Horava:2009uw ; Wang:2017brl , which was studied in the context of
gravitational wave physics in Zhu:2013fja . A different construction was
proposed in Takahashi:2022mew ; here, the authors employed conformal
transformations of the metric where the conformal factor is a function of the
Pontryagin density. This produces a parity violating effect in the Einstein
frame, which was recently used to study modifications of the Kerr metric
Tahara:2023pyg .

Alternatively, one can relax the standard assumptions of metricity (the so-
called teleparallel gravity theories) and vanishing torsion Conroy:2019ibo ;
Iosifidis:2020dck ; Freidel:2005sn ; Mavromatos:2023wkk ; Shapiro:2014kma ;
Szczachor:2012hh . In theories of this type, parity violating couplings
between the Levi-Civita and either non-metricity or torsion tensors are
expected on general grounds, which has been known since the late 1970’s
Hojman:1980kv . The authors of Jenks:2023pmk  analyzed many of these examples,
notably dynamical Chern-Simons, ghost-free scalar tensor gravity, (symmetric)
teleparallel, and Hor̂ava-Lifshitz.

A common feature of many of these models is that the parity violating
operators have dimension greater than four. Returning back to dCS, the
interaction is a dimension five operator, and therefore necessitates a mass
scale,

$\mathcal{L}_{\text{dCS}}=\frac{\phi}{m_{\text{CS}}}R\tilde{R},$ (1)

implying that its potential affect on gravitational parity violation is
suppressed. Moreover, dCS being quadratic in curvature has the consequence
that the field equations are higher than second order PhysRevD.86.124031 . It
is therefore well worth exploring the possibility of lower dimensional
operators which will not come with an (inverse) mass scale and,
idealistically, be manifestly safe from instabilities arising from higher
derivatives. Studying such an interaction is precisely the purpose of this
note.

The presence of the Levi-Civita symbol contracted with the Riemann tensor
results in an overall sign difference between the equations of motion for
gravitational wave amplitudes when written in the circular basis. That is, for
a gravitational wave propagating in the $z$-direction with polarizations in
the $xy$-plane, we write

$h_{\mu\nu}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&0&0&0\\\
0&-(h_{\text{L}}+h_{\text{R}})&i(h_{\text{L}}-h_{\text{R}})&0\\\
0&i(h_{\text{L}}-h_{\text{R}})&h_{\text{L}}+h_{\text{R}}&0\\\
0&0&0&0\end{pmatrix},$ (2)

where the right- and left-handed strains are related to the usual $+,\times$
amplitudes by $h_{\text{R,L}}=\frac{1}{\sqrt{2}}(h_{+}\pm ih_{\times}).$
Varying the Einstein-Hilbert Lagrangian with respect to each amplitude results
in simple wave equations for each. Parity violation will then follow provided
an additional interaction term sources the left- and right-handed amplitudes
with a different sign, say

$\begin{split}\Box h_{\text{L}}&=+\Xi,\\\ \Box h_{\text{R}}&=-\Xi,\end{split}$
(3)

for some $\Xi$ that is an expression involving additional matter fields and
perhaps their derivatives111The authors of Contaldi:2008yz  approached the
problem by considering the right- and left-handed waves as propagating with
different effective Newton constants. This affects the vacuum fluctuations
during inflation and the parity violation would translate into anomalous CMB
polarization.. As usual, we define the graviton as

$g_{\mu\nu}=\bar{g}_{\mu\nu}+h_{\mu\nu},$ (4)

where $\bar{g}_{\mu\nu}$ in this work will be the flat FLRW metric in
conformal coordinates,

$d\bar{s}^{2}=a(\eta)^{2}\big{(}-d\eta^{2}+dx^{2}+dy^{2}+dz^{2}\big{)}.$ (5)

Parity violating effects in the gravitational sector generically demand a non-
minimal coupling to matter, such as the pseudoscalar in dCS theory. It is
interesting to entertain the possibility that such matter belongs to a dark
sector. Very recently, the Kalb-Ramond (KR) two-form field $B_{\mu\nu}$ was
studied as a dark matter candidate Capanelli:2023uwv . The authors showed that
the KR field and Kalb-Ramond-like-particles (KRLPs) can be produced via
freeze-in and freeze-out mechanisms and can account for the relic density of
dark matter today. Others have considered the KR field as a portal between the
SM and dark sectors Dashko:2018dsw ; Dick:2020eop , and more generally in
early universe scenarios BeltranAlmeida:2018nin ; Almeida:2019xzt  including
leptogenesis deCesare:2014dga , as well as late time cosmic acceleration
Orjuela-Quintana:2022jrg ; BeltranAlmeida:2019fou .

The KR 2-form is one of three fields in the spectrum of bosonic string
theories PhysRevD.9.2273 ; Ogievetsky:1966eiu , also arising in a
$(1,0)\oplus(0,1)$ representation of the Lorentz group Ivanov:2016lha . The KR
field is additionally relevant in the context of axion physics; in the non-
interacting (massless) case, the KR field strength can be dualized to obtain a
pseudoscalar often referred to as the KR axion Svrcek:2006yi  (see also
Smailagic:2001ch  for an earlier discussion).

Inspired in particular by the recent attention to the KR field as a dark
matter candidate in models with broken gauge invariance Capanelli:2023uwv , we
consider parity violating effects due to a coupling between the dual-Riemann
tensor and the KR;

$\mathcal{L}_{\text{int}}=-\frac{\xi}{2}\tilde{R}_{\mu\nu\rho\sigma}B^{\mu\nu}B^{\rho\sigma}.$
(6)

The KR field carries mass dimension one, therefore in difference from the dCS
coupling, the constant $\xi$ is dimensionless. This term was written down in
Altschul:2009ae  in the context of Lorentz-violating vector and scalar
theories, which we will discuss further in the conclusion. However, to the
best of our knowledge, (6) has not been studied in the context of parity
violation222In Maity:2004he , the authors considered couplings between the KR
and Maxwell fields in the context of parity violation, while in
Majumdar:2001ff ; Majumdar:1999jd , a similar construction was explored using
the KR axion. .

This paper is organized as follows. In Sec. II, we discuss aspects of the free
KR Lagrangian, including a symmetry breaking process that leads to a
simplified scenario where the field components are constant. In Sec. III, we
discuss properties of the interaction (6) in more detail before introducing
the full theory in Sec. IV. We then focus on the perturbations in Sec. V,
where we make contact with the novel, parity violating gravitational wave
parameterization recently presented in Jenks:2023pmk . Here we find the
solutions to our model in an arbitrary FLRW cosmology. In Sec. VI, we discuss
constraints on the model parameters from GW170817 and the GWTC-3 data set,
before concluding in Sec. VII.

##  II Features of the KR Lagrangian

The KR field is described by the Lagrangian PhysRevD.9.2273

$\mathcal{L}_{\text{B}}=\frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho}-V(B),$ (7)

where

$H_{\mu\nu\rho}=\partial_{[\mu}B_{\nu\rho]}=\partial_{\mu}B_{\nu\rho}+\partial_{\nu}B_{\rho\mu}+\partial_{\rho}B_{\mu\nu}$
(8)

is the KR field strength and $V(B)$ is some potential. When $V=0$, the theory
enjoys a gauge symmetry with transformation333We use the anti-symmetrization
convention $A_{[\mu}B_{\nu]}=A_{\mu}B_{\nu}-A_{\nu}B_{\mu}$.

$B_{\mu\nu}\rightarrow B_{\mu\nu}+\partial_{[\mu}A_{\nu]}$ (9)

for an arbitrary vector $A_{\mu}(x)$, corresponding to a shift in $B$ by an
exact 2-form. There is an additional transformation involving a scalar field,

$A_{\mu}\rightarrow A_{\mu}+\partial_{\mu}\lambda.$ (10)

Thus although $A_{\mu}$ has four components, only three are independent. The
subsidiary gauge transformation (10) is a shift involving an exact 1-form.

The gauge invariance (9), (10) is broken explicitly by the potential $V$ and
the interaction (6). This is analogous to the Proca theory of a massive vector
boson, which is known to be dual to the massive KR field in certain regimes
Hell:2021wzm ; Capanelli:2023uwv . Regarding interactions with the
gravitational waves, we will see that a simple and interesting scenario occurs
when the KR field’s magnitude settles at a non-zero, constant (classical)
value, $B^{2}=B_{(0)}^{2}$. This can be modeled by a potential of the form

$V(B_{\mu\nu}B^{\mu\nu}-v^{2}).$ (11)

When the KR field relaxes to the minimum, we have

$v^{2}\rightarrow\bar{g}^{\mu\alpha}\bar{g}^{\nu\beta}B_{\mu\nu}^{(0)}B_{\alpha\beta}^{(0)},$
(12)

where $\bar{g}^{\mu\nu}$ is the unperturbed metric. The explicit form of the
potential (11) is not relevant for this work, beyond requiring that it
possesses a nontrivial stable minimum. In theories of this type, such
potentials have the consequence that the physics can display (spontaneous)
Lorentz violation, despite the Lagrange density being Lorentz invariant
Altschul:2009ae .

The KR field can be parameterized in a particularly simple form
Altschul:2009ae ; Maluf:2021eyu . In general, a single KR field in a constant
background value breaks the isotropy of the cosmology Maluf:2021eyu ,
analogous to the case of a single vector field Golovnev:2008cf . Global
isotropy can be recovered if multiple vector fields take on constant values of
equal magnitude orthogonal to each other, and we expect a similar story to
hold here. We will save a treatment of such a scenario for future work, and
will solely consider the effect of the interaction on gravitational waves
approximating an isotropic cosmology. We therefore parameterize the background
field as444Note in difference from Altschul:2009ae ; Maluf:2021eyu , we
include the factor of $\tfrac{1}{\sqrt{2}}$ for convenience.

$B^{(0)}_{\mu\nu}=\frac{1}{\sqrt{2}}\begin{pmatrix}0&-\mathcal{V}_{e}a^{2}&0&0\\\
\mathcal{V}_{e}a^{2}&0&0&0\\\ 0&0&0&\mathcal{V}_{m}a^{2}\\\
0&0&-\mathcal{V}_{m}a^{2}&0\end{pmatrix},$ (13)

where $a(\eta)$ is the scale factor and $\mathcal{V}_{e},\mathcal{V}_{m}$ are
taken to be real numbers, and can be thought of as the “electric” and
“magnetic” parts of the KR field, respectively. It then follows that the
minimum of the potential is at

$v=\sqrt{\mathcal{V}_{m}^{2}-\mathcal{V}_{e}^{2}},$ (14)

independent of the background evolution of the spacetime. It is also natural
to expect a mass term for the KR field in this scenario. For the field to be
frozen at the bottom of the minimum surrounding $v$, we require the mass to be
slightly smaller than but on the order of $v$. This will guarantee that the
symmetry breaking minimum exists while simultaneously ensuring the field is
heavy enough such that no radial excitations occur, allowing us to treat the
field as constant. However, as we look to make contact with observations in
the late universe with this construction, this is not a requirement in early
universe physics.

An alternative justification for considering a frozen KR background is by
simply examining the background evolution in a flat FLRW spacetime without the
interaction term. The field equation in this case is

$B^{\prime\prime}_{\mu\nu}+\delta_{\eta[\mu}B^{\prime\prime}_{\nu]\eta}+2\mathcal{H}\delta_{\eta[\mu}B^{\prime}_{\nu]\eta}+a^{2}\frac{\partial
V}{\partial B^{\mu\nu}}=0,$ (15)

where we’ve assumed $B_{\mu\nu}=B_{\mu\nu}(\eta)$ and
$\mathcal{H}=a^{\prime}/a$ is the Hubble parameter. As is standard in an
expanding cosmological background, the presence of Hubble friction ultimately
dampens the evolution of the unperturbed field until it rests at a constant
value. This, of course, will be at the minimum of the potential where
$\frac{\partial V}{\partial B^{\mu\nu}}=0.$

##  III Features of the interaction

The interaction (6) is a specific case of a more general coupling

$\mathcal{L}\sim R_{\mu\nu\rho\sigma}P^{\mu\nu\rho\sigma},$ (16)

since here, we can move the Levi-Civita symbol over to one of the $B$-fields
and equivalently write

$\mathcal{L}_{\text{int}}=-\frac{\xi}{2}R_{\mu\nu\rho\sigma}\tilde{B}^{\mu\nu}B^{\rho\sigma},$
(17)

where $\tilde{B}^{\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu}_{\ \
\alpha\beta}B^{\alpha\beta}$. From (16), we see that in considering equations
of motion, we have the derivative

$\frac{\partial\mathcal{L}}{\partial
R_{\mu\nu\rho\sigma}}=P^{\mu\nu\rho\sigma},$ (18)

which implies that $P$ must inherit all of the symmetries of the Riemann
tensor. That fact is not manifest as it is written in (17), therefore care
must be taken when we derive the equations of motion covariantly. In
particular, we have that

$\begin{split}P^{\mu\nu\rho\sigma}&=\frac{1}{3}\Big{(}\tilde{B}^{\mu\nu}B^{\rho\sigma}+B^{\mu\nu}\tilde{B}^{\rho\sigma}\Big{)}\\\
&+\frac{1}{6}\Big{(}\tilde{B}^{\mu\rho}B^{\nu\sigma}+B^{\mu\rho}\tilde{B}^{\nu\sigma}+\tilde{B}^{\rho\nu}B^{\mu\sigma}+B^{\rho\nu}\tilde{B}^{\mu\sigma}\Big{)}.\end{split}$
(19)

It is tedious but straightforward to show that this indeed possesses all
symmetries of the Riemann tensor,

$P_{\mu\nu\rho\sigma}=-P_{\nu\mu\rho\sigma}=-P_{\mu\nu\sigma\rho}=P_{\rho\sigma\mu\nu},$
(20)

along with the cyclic property

$P_{\mu\nu\rho\sigma}+P_{\mu\rho\sigma\nu}+P_{\mu\sigma\nu\rho}=0.$ (21)

However, when contracted with the Riemann tensor, all of the terms combine
such that

$R_{\mu\nu\rho\sigma}P^{\mu\nu\rho\sigma}=R_{\mu\nu\rho\sigma}\tilde{B}^{\mu\nu}B^{\rho\sigma}.$
(22)

From an effective field theory standpoint, there are additional dimension-four
operators coupling the KR field to curvature tensors involving the Levi-Civita
symbol that we could consider at linear order in the curvature;

$\mathcal{L}_{B\tilde{B}R}=\xi_{2}B^{\mu\lambda}\tilde{B}^{\nu}_{\ \
\lambda}R_{\mu\nu}+\xi_{3}\tilde{B}_{\mu\nu}B^{\mu\nu}R.$ (23)

Firstly, these terms are proportional due to the identity $B_{\mu}^{\
\lambda}\tilde{B}_{\lambda\nu}=\frac{1}{4}g_{\mu\nu}B_{\alpha\beta}\tilde{B}^{\alpha\beta},$
but more relevant for our purposes, such operators do not contribute to parity
violation. In this work we will therefore focus our attention on (6).

##  IV Theory

We consider the full theory given by

$\mathcal{L}=\mathcal{L}_{\text{EH}}+\mathcal{L}_{\text{B}}+\mathcal{L}_{\text{int}}+\mathcal{L}_{m},$
(24)

where $\mathcal{L}_{m}$ is any additional matter content and

$\mathcal{L}_{\text{EH}}=\frac{m_{p}^{2}}{2}R$ (25)

is the usual Einstein-Hilbert term, while $\mathcal{L}_{\text{B}}$ and
$\mathcal{L}_{\text{int}}$ are given by (7) and (6), respectively. For an
arbitrary background, the field equations for the $B$-field and the metric are
given by

$\begin{split}\nabla_{\alpha}H^{\alpha}_{\ \ \mu\nu}&=-\frac{\partial
V}{\partial B^{\mu\nu}}+4\xi\tilde{R}_{\mu\nu\rho\sigma}B^{\rho\sigma}\\\
m_{p}^{2}G_{\mu\nu}&=\xi\Theta_{\mu\nu}+T^{\text{B}}_{\mu\nu}+T_{\mu\nu}^{\text{m}},\end{split}$
(26)

where $T_{\mu\nu}^{\text{m}}$ is the contribution from the additional matter
sector,

$\begin{split}T_{\mu\nu}^{\text{B}}&=-\frac{1}{2}H_{\mu}^{\ \
\alpha\beta}H_{\nu\alpha\beta}+g_{\mu\nu}\Big{(}\frac{1}{12}H_{\alpha\beta\rho}H^{\alpha\beta\rho}-V(B)\Big{)}\end{split}$
(27)

is the stress tensor of the free KR field and

$\Theta_{\mu\nu}=\nabla^{\alpha}\nabla^{\beta}P_{\mu\alpha\nu\beta}-\frac{1}{4}g_{\mu\nu}\tilde{R}_{\alpha\beta\rho\sigma}B^{\alpha\beta}B^{\rho\sigma}$
(28)

stems from the new interaction term, where $P$ is given by (19).

##  V GW solution in FLRW background

We will look to make contact with the parameterization given in Jenks:2023pmk
, where the equations of motion (EOM) for the left/right helicity amplitudes
are written

$\begin{split}&h^{\prime\prime}_{\text{R,L}}+\Big{\\{}2\mathcal{H}+\lambda_{\text{R,L}}\sum_{n=1}^{\infty}k^{n}\Big{[}\frac{\alpha_{n}(\eta)}{(\Lambda_{\text{PV}}a)^{n}}\mathcal{H}+\frac{\beta_{n}(\eta)}{(\Lambda_{\text{PV}}a)^{n-1}}\Big{]}\Big{\\}}h^{\prime}_{\text{R,L}}\\\
&+k^{2}\Big{\\{}1+\lambda_{\text{R,L}}\sum_{m=0}^{\infty}k^{m-1}\Big{[}\frac{\gamma_{m}(\eta)}{(\Lambda_{\text{PV}}a)^{m}}\mathcal{H}+\frac{\delta_{m}(\eta)}{(\Lambda_{\text{PV}}a)^{m-1}}\Big{]}\Big{\\}}h_{\text{R,L}}=0,\end{split}$
(29)

where primes denote partial derivatives with respect to conformal time $\eta$,
$\mathcal{H}\equiv a^{\prime}/a$ is the Hubble parameter,
$\lambda_{\text{R,L}}=\pm 1,$ and $\Lambda_{\text{PV}}$ is the EFT cutoff. The
summation indices are such that $n$ runs over odd integers and $m$ runs over
even integers, and the $\\{\alpha_{n},\beta_{n},\gamma_{m},\delta_{m}\\}$
functions generically depend on the conformal time and are theory dependent.
For metric fluctuations on top of an FLRW cosmology in conformal coordinates,
the equations of motion for the gravitational wave amplitudes in our model
become

$\begin{split}h_{\text{L}}^{\prime\prime}+2\mathcal{H}h_{\text{L}}^{\prime}-\partial^{2}_{z}h_{\text{L}}-\frac{\xi}{m_{p}^{2}}\Big{\\{}2\mathcal{V}_{m}\mathcal{V}_{e}h_{\text{L}}(3\mathcal{H}^{2}-a^{\prime\prime}/a)-i\mathcal{H}\big{(}(5\mathcal{V}_{m}^{2}+3\mathcal{V}_{e}^{2})\partial_{z}h_{\text{L}}+2i\mathcal{V}_{m}\mathcal{V}_{e}h_{\text{L}}^{\prime}\big{)}+i(\mathcal{V}_{m}^{2}+\mathcal{V}_{e}^{2})\partial_{z}h_{\text{L}}^{\prime}\Big{\\}}=0,\end{split}$
(30)

$h_{\text{R}}^{\prime\prime}+2\mathcal{H}h_{\text{R}}^{\prime}-\partial^{2}_{z}h_{\text{R}}-\frac{\xi}{m_{p}^{2}}\Big{\\{}2\mathcal{V}_{m}\mathcal{V}_{e}h_{\text{R}}(3\mathcal{H}^{2}-a^{\prime\prime}/a)+i\mathcal{H}\big{(}(5\mathcal{V}_{m}^{2}+3\mathcal{V}_{e}^{2})\partial_{z}h_{\text{R}}-2i\mathcal{V}_{m}\mathcal{V}_{e}h_{\text{R}}^{\prime}\big{)}-i(\mathcal{V}_{m}^{2}+\mathcal{V}_{e}^{2})\partial_{z}h_{\text{R}}^{\prime}\Big{\\}}=0.$
(31)

Due to the presence of the Riemann tensor at linear order, note that
$a^{\prime\prime}$ appears. In order to cast the (momentum space) equations in
the form of (29), we need to remove the dependence on $a^{\prime\prime}$. We
can do this by rescaling the amplitudes as
$\tilde{h}_{\text{R,L}}=a^{p}h_{\text{R,L}},$ then solve for $p$. It is
tedious but straightforward to show that
$p=2\xi\mathcal{V}_{m}\mathcal{V}_{e}/m_{p}^{2}$, and now writing
$\tilde{h}_{\text{R,L}}(z,\eta)=e^{ikz}\bar{h}_{\text{R,L}}(\eta),$ we obtain

$\bar{h}_{\text{L}}^{\prime\prime}+\Big{\\{}2\mathcal{H}-\big{[}A\mathcal{H}+Bk\big{]}\Big{\\}}\bar{h}_{\text{L}}^{\prime}+\Big{\\{}k^{2}-\big{[}Ck\mathcal{H}+D\mathcal{H}^{2}\big{]}\Big{\\}}\bar{h}_{\text{L}}=0,$
(32)
$\bar{h}_{\text{R}}^{\prime\prime}+\Big{\\{}2\mathcal{H}+\big{[}\tilde{A}\mathcal{H}+\tilde{B}k\big{]}\Big{\\}}\bar{h}_{\text{R}}^{\prime}+\Big{\\{}k^{2}+\big{[}\tilde{C}k\mathcal{H}+\tilde{D}\mathcal{H}^{2}\big{]}\Big{\\}}\bar{h}_{\text{R}}=0,$
(33)

where the constants are respectively

$\begin{split}A&=-6\xi\tilde{\mathcal{V}}_{m}\tilde{\mathcal{V}}_{e},\\\
B&=\xi(\tilde{\mathcal{V}}_{m}^{2}+\tilde{\mathcal{V}}_{e}^{2}),\\\
C&=2\xi^{2}\tilde{\mathcal{V}}_{m}\tilde{\mathcal{V}}_{e}(\tilde{\mathcal{V}}_{m}^{2}+\tilde{\mathcal{V}}_{e}^{2})-\xi(5\tilde{\mathcal{V}}_{m}^{2}+3\tilde{\mathcal{V}}_{e}^{2}),\\\
D&=-8\xi\tilde{\mathcal{V}}_{m}\tilde{\mathcal{V}}_{e}(1+\tilde{\mathcal{V}}_{m}\tilde{\mathcal{V}}_{e}),\end{split}$
(34)

defining dimensionless quantities
$\tilde{\mathcal{V}}_{m}=\mathcal{V}_{m}/m_{p},\
\tilde{\mathcal{V}}_{e}=\mathcal{V}_{e}/m_{p}$, and

$\tilde{A}=-A,\ \ \ \tilde{B}=B,\ \ \ \tilde{C}=C,\ \ \ \tilde{D}=-D.$ (35)

From the parameterization (29), two terms can be straightforwardly identified:

$\begin{split}\beta_{1}=B&=\xi(\tilde{\mathcal{V}}_{m}^{2}+\tilde{\mathcal{V}}_{e}^{2}),\\\
\gamma_{0}=C&=2\xi^{2}\tilde{\mathcal{V}}_{m}\tilde{\mathcal{V}}_{e}(\tilde{\mathcal{V}}_{m}^{2}+\tilde{\mathcal{V}}_{e}^{2})-\xi(5\tilde{\mathcal{V}}_{m}^{2}+3\tilde{\mathcal{V}}_{e}^{2}),\end{split}$
(36)

however the other two terms, $A$ and $D$, (a) contribute the same sign to both
helicity amplitudes and (b) do not clearly fall into the form of (29). For an
arbitrary $\mathcal{V}_{m},\mathcal{V}_{e},$ we see that there are parity
preserving and parity violating contributions to the EOM.

As is discussed in Altschul:2009ae , it is consistent to set the “electric”
part of $B_{\mu\nu}$ equal to zero, which in our notation, is
$\mathcal{V}_{e}=0.$ In this simplified scenario, the $a^{\prime\prime}$ term
in in (30), (31) vanishes, as well as $A=\tilde{A}=D=\tilde{D}=0$. The
transformation to remove $a^{\prime\prime}$ is then trivial (i.e. $p=0$), and
the equations of motion become

$\bar{h}_{\text{R,L}}^{\prime\prime}+\Big{\\{}2\mathcal{H}+\lambda_{\text{R,L}}\bar{B}k\Big{\\}}\bar{h}_{\text{R,L}}^{\prime}+\Big{\\{}k^{2}+\lambda_{\text{R,L}}\bar{C}k\mathcal{H}\Big{\\}}\bar{h}_{\text{R,L}}=0,$
(37)

such that we can easily map our model parameters to those of Jenks:2023pmk :

$\bar{B}=\xi\tilde{\mathcal{V}}_{m}^{2}=\beta_{1},\ \ \ \
\bar{C}=-5\xi\tilde{\mathcal{V}}_{m}^{2}=\gamma_{0}.$ (38)

This scenario has the additional interesting feature that the interaction (6)
does not affect the background cosmology. This is apparent from the form of
the interaction’s contribution to the Einstein equations;

$\Theta_{\mu\nu}=\frac{\mathcal{V}_{e}\mathcal{V}_{m}(a^{\prime\prime}/a-4\mathcal{H}^{2})}{a^{6}}\begin{pmatrix}\frac{-4a^{2}\mathcal{H}^{2}}{a^{\prime\prime}/a-4\mathcal{H}^{2}}&&&\\\
&2&&\\\ &&1&\\\ &&&1\end{pmatrix},$ (39)

which clearly vanishes when $\mathcal{V}_{e}=0.$ The cosmological evolution is
thus driven by standard matter and the free energy of the KR field, with
stress tensor

$T_{\mu\nu}^{\text{B}}=-2\mathcal{V}_{m}^{2}\mathcal{H}^{2}\begin{pmatrix}1&&&\\\
&1&&\\\ &&-1&\\\ &&&-1\end{pmatrix}-a^{2}V(B)\begin{pmatrix}-1&&&\\\ &1&&\\\
&&1&\\\ &&&1\end{pmatrix},$ (40)

independent of the value of $\mathcal{V}_{e}$.

Now, the authors of Jenks:2023pmk  go on to write down the solutions for the
polarization modes as a function of redshift $z$ in terms of the parameters of
(29), which take the general form

$h_{\text{R,L}}=h^{\text{GR}}_{\text{R,L}}\text{exp}\Bigg{\\{}\mp\frac{[k(1+z)]^{n}}{2}\Bigg{(}\frac{\alpha_{n_{0}}}{\Lambda_{\text{PV}}^{n}}z_{n}+\frac{\beta_{n_{0}}}{\Lambda_{\text{PV}}^{n-1}}D_{n+1}\Bigg{)}\Bigg{\\}}\text{exp}\Bigg{\\{}\pm
i\frac{[k(1+z)]^{m}}{2}\Bigg{(}\frac{\gamma_{m_{0}}}{\Lambda_{\text{PV}}^{m}}z_{m}+\frac{\delta_{m_{0}}}{\Lambda_{\text{PV}}^{m-1}}D_{m+1}\Bigg{)}\Bigg{\\}}.$
(41)

Here, the summations over $m$ and $n$ are implicit, $h^{\text{GR}}$ is the
standard oscillating solution from GR, $D_{\alpha}$ and $z_{\alpha}$ are
defined as

$\begin{split}D_{\alpha}&=(1+z)^{1-\alpha}\int\frac{(1+z)^{\alpha-2}}{H(z)}dz,\\\
z_{\alpha}&=(1+z)^{-\alpha}\int\frac{dz}{(1+z)^{1-\alpha}}.\end{split}$ (42)

The $n_{0},m_{0}$ subscript on $\alpha,\beta,\gamma,\delta$ denote their late
time, slowly varying approximations. Our solution has
$\beta_{1}=-\tfrac{1}{5}\gamma_{0}=\xi\tilde{\mathcal{V}}_{m}^{2},$ thus only
$m=0,$ $n=1$ are nontrivial. In the exponent, we therefore have
$z_{0}=\text{ln}(1+z)$ and $D_{2}=(1+z)^{-1}D_{C}\equiv D_{A},$ where $D_{C}$
and $D_{A}$ are respectively the comoving and angular-diameter distances. Our
solution is then

$h_{\text{R,L}}=h^{\text{GR}}_{\text{R,L}}\text{exp}\Big{\\{}\mp\tfrac{\xi\tilde{\mathcal{V}}_{m}^{2}}{2}\Big{(}k(1+z)D_{2}+5iz_{0}\Big{)}\Big{\\}}.$
(43)

As we discussed in Sec. I, the interaction being a dimension-four operator
implies that the coupling $\xi$ is dimensionless, such that there is not a
mass scale setting a clear UV cutoff. This manifests in (43) by way of only
$n=1,m=0$ contributions from (37) along with $\alpha_{n}=\delta_{m}=0$; no
factors of $\Lambda_{\text{PV}}$ survive in the parameterization (29).

The dispersion relation for any parity violating theory which can be
classified by the parameterization (29) is given by equation (30) in
Jenks:2023pmk . In our case, this becomes

$\begin{split}\omega_{\text{R,L}}^{2}&=k^{2}\big{\\{}1+\lambda_{\text{R,L}}k^{-1}\gamma_{0}\mathcal{H}\big{\\}}\\\
&=k^{2}\\{1+\tilde{\lambda}_{\text{R,L}}k^{-1}\\},\end{split}$ (44)

defining

$\tilde{\lambda}_{\text{R,L}}=-\lambda_{\text{R,L}}(5\xi\tilde{\mathcal{V}}_{m}^{2}\mathcal{H}).$
(45)

From this we easily obtain the phase $v_{p}=\omega/k$ and group
$v_{d}=d\omega/dk$ velocities,

$\begin{split}v_{p}&=\sqrt{1+\tilde{\lambda}_{\text{R,L}}k^{-1}},\\\
v_{g}&=\frac{1+\tfrac{1}{2}\tilde{\lambda}_{\text{R,L}}k^{-1}}{\sqrt{1+\tilde{\lambda}_{\text{R,L}}k^{-1}}},\end{split}$
(46)

which are clearly different between $h_{\text{R}}$ and $h_{\text{L}}$. Taking
a closer look at the group velocity, we have

$\begin{split}v_{g}&=(1+\tfrac{1}{2}\tilde{\lambda}_{\text{R,L}}k^{-1})(1-\tfrac{1}{2}\tilde{\lambda}_{\text{R,L}}k^{-1}+\tfrac{3}{8}\tilde{\lambda}_{\text{R,L}}^{2}k^{-2}+\cdots)\\\
&\simeq
1+\tfrac{3}{8}\tilde{\lambda}_{\text{R,L}}^{2}k^{-2}+O\Big{(}\tilde{\lambda}_{\text{R,L}}^{3}k^{-3}\Big{)}\end{split}$
(47)

for $(\xi\tilde{\mathcal{V}}_{m}^{2}\mathcal{H})\ll 1.$ Since the correction
appears at $O(\tilde{\lambda}^{2})$, the sign difference between the left and
right handed amplitudes squares away such that they propagate with equal group
velocity to leading order. And because the correction is non-negative, the
model predicts a propagation speed slightly larger than one. Comparing to the
phase velocity, we see that

$v_{p}\simeq
1+\tfrac{1}{2}\tilde{\lambda}_{\text{R,L}}k^{-1}-\tfrac{1}{8}\tilde{\lambda}_{\text{R,L}}^{2}k^{-2}+O\Big{(}\tilde{\lambda}_{\text{R,L}}^{3}k^{-3}\Big{)}.$
(48)

The correction to the phase velocity due to the parity violating interaction
clearly appears at a lower order than the group velocity, and the sign
difference between $h_{\text{R}}$ and $h_{\text{L}}$ survives.

##  VI Comments on constraints

The possibility of parity violation to be observed in gravitational wave
detectors was suggested well over 10 years ago Alexander:2007kv ; Yunes:2010yf
. The data currently being gathered from merger processes
LIGOScientific:2014pky ; VIRGO:2014yos ; LIGOScientific:2020ibl ;
KAGRA:2021vkt  can in principle constrain parity violating parameters,
although much more progress on this front will be made from third generation
detectors due to their increased sensitivities Maggiore:2019uih ;
Evans:2021gyd . Additionally, very recently, an analysis to constrain parity
violation was carried out in Callister:2023tws  using data from pulsar timing
arrays NANOGrav:2023gor ; NANOGrav:2023hvm ; EPTA:2023fyk ; Xu:2023wog ;
Reardon:2023gzh . The two primary effects of interest are velocity
birefringence and amplitude birefringence. The former refers to right- and
left-handed waves propagating with different frequency dependent phase shifts,
the latter corresponding to an attenuation or amplification of one
polarization over another. It is also the case that modified theories of
gravity generally predict a propagation speed different from unity. In what
follows, we will focus on propagation speed and amplitude birefringence.

###  VI.1 Propagation speed

The observation of the coincident gamma ray burst (GRB)/gravitational wave
signal GW170817 has constrained the GW propagation speed to be
LIGOScientific:2017zic

$-3\times 10^{-15}<c_{\text{gw}}-1<7\times 10^{-16}.$ (49)

The measurement was used to constrain parity violating effects in ghost-free
scalar tensor models in Nishizawa:2018srh  and in symmetric teleparallel
gravity in Conroy:2019ibo . Using our result for the group velocity (47), this
translates to a bound on the parameter
$\tilde{\mathcal{V}}_{m}=\mathcal{V}_{m}/m_{p}$. If we take the dimensionless
coupling to be $\xi\sim O(1)$, and use the upper bound on (49), then

$\mathcal{V}_{m}^{4}\Bigg{(}\frac{\mathcal{H}^{2}}{k^{2}m_{p}^{4}}\Bigg{)}<7\times
10^{-17}.$ (50)

This can be simplified in terms of the Hubble constant and proper wavenumber
as

$\mathcal{V}_{m}\sqrt{\frac{H_{0}}{ka}}\lesssim 10^{-4}m_{p}.$ (51)

###  VI.2 Amplitude birefringence

In Okounkova:2021xjv , the authors studied the GWTC-2 data set to construct
constraints on amplitude birefringence from binary black hole merger. They
took the ansatz for the GW amplitudes
$h_{\text{R,L}}=h^{\text{GR}}_{\text{R,L}}e^{\lambda_{\text{R,L}}\kappa
D_{C}}$, where $\kappa$ is referred to as the opacity parameter and $D_{C}$ is
again the coming distance. They showed that $\kappa\lesssim 0.74$ Gpc-1, and
the authors of Jenks:2023pmk  utilized this bound to constrain the
dimensionless parameter $\beta_{1}$ to be less than $0.7\times 10^{-20}$ at
90% confidence level. More recently, in Ng:2023jjt , GWTC-3 data was used to
constrain the opacity parameter to be $\kappa<0.04$ Gpc-1, which is currently
the most stringent bound. Repeating the analysis in Jenks:2023pmk  with the
new bound on $\kappa$ results in $\beta_{1}<3.8\times 10^{-22}$. We can easily
translate this to a bound on the parameter $\mathcal{V}_{m}$ in terms of the
Planck mass. Again setting the coupling as $\xi\sim O(1)$, since
$\xi\tilde{\mathcal{V}}_{m}^{2}=\beta_{1},$ we find that
$\mathcal{V}_{m}<1.9\times 10^{-11}m_{p}$, or

$\mathcal{V}_{m}\lesssim 2.4\times 10^{8}\text{ GeV}.$ (52)

That the upper bound on the $\mathcal{V}_{m}$ parameter lies in the $O(100)$
PeV range is quite interesting. From a purely theoretical standpoint, it has
been known for some time Wells:2004di  that the PeV scale is an attractive
range for supersymmetry (SUSY), admitting unified gauge couplings, disallowing
flavor changing neutral currents, and predicting a Higgs mass between 125-155
GeV Bhattiprolu:2023lfh . It has also gained attention as a candidate energy
scale for dark matter Berlin:2016vnh , partially motivated by the detection of
near-PeV gamma rays by the Tibet $\text{AS}_{\gamma}$ collaboration
TibetASgamma:2021tpz . Stringent constraints on PeV scale dark matter were
derived in Maity:2021umk , and future data from Tibet $\text{AS}_{\gamma}$ is
expected to further shed light on the viability of PeV scale dark matter (see
also Cohen:2016uyg ; Bhattacharya:2019ucd ; Ishiwata:2019aet ; Blanco:2018esa
for further relevant discussion of the gamma ray background).

##  VII Discussion

Third generation gravitational wave detectors such as the Einstein Telescope
Maggiore:2019uih  and Cosmic Explorer Evans:2021gyd  are anticipated to have
sensitivities capable of probing beyond GR effects such as parity violation
Califano:2023aji . Parity violation may also be observed in large scale
structure Vanzan:2023cze ; Creque-Sarbinowski:2023wmb . Any observation
definitively signalling parity violation would be a smoking gun for
gravitational physics beyond General Relativity. It is therefore of great
interest to explore various models which predict gravitational parity
violation in anticipation of such data. In this work, inspired by recent
attention on the Kalb-Ramond field as a dark matter candidate, we considered
parity violation from a dimension-four operator coupling the dual Riemann
tensor to the KR fields,
$\mathcal{L}_{\text{int}}=-\tfrac{\xi}{2}\tilde{R}_{\mu\nu\rho\sigma}B^{\mu\nu}B^{\rho\sigma}$.
For simplicity, we constructed a scenario where the KR fields can be treated
as being constant. This will happen provided the KR field experiences a
potential that satisfies $\frac{\partial V}{\partial
B^{\mu\nu}}\big{|}_{B^{(0)}_{\mu\nu}}=0$ for $B^{(0)}_{\mu\nu}\neq 0.$ Two
parameters $\mathcal{V}_{e},\mathcal{V}_{m}$, characterize the degrees of
freedom of $B_{\mu\nu}.$ After writing down the field equations for the
gravitational waves in the circular basis for arbitrary
$\mathcal{V}_{m},\mathcal{V}_{e},$ we restricted to the case
$\mathcal{V}_{e}=0$ before mapping the solution to the general
parameterization presented in Jenks:2023pmk . Constraints from the
GRB/gravitational wave signal GW170817 LIGOScientific:2017zic  and the GWTC-3
KAGRA:2021vkt  data set were considered, and the latter translated to the
bound $\mathcal{V}_{m}\lesssim 10^{8}$ GeV. As we discussed in Sec. VI.2, this
upper bound for dark matter being near the PeV scale is interesting from
theoretical Wells:2004di ; Bhattiprolu:2023lfh  and phenomenological
TibetASgamma:2021tpz ; Maity:2021umk  standpoints.

We assumed a generic potential was responsible for the KR field settling into
its background value, which triggers spontaneous Lorentz violation. As a
result, (pseudo-) Nambu-Goldstone modes associated to the field fluctuations
along the broken generators emerge Bluhm:2004ep ; Bluhm:2007bd ;
Higashijima:2001sq ; Hernaski:2016dyk . Consequences of this spontaneous
symmetry breaking were studied extensively in Altschul:2009ae , where the
authors derived reasonable constraints on the Lorentz violating coefficients
present in certain non-minimal curvature couplings (see, e.g. Cheng:2006us ;
Carroll:2004ai ; Lambiase:2005kb ; Colladay:1998fq ; Kostelecky:1988zi ;
Kostelecky:2008ts ; Kostelecky:2003fs ). However, such modes do not play an
important role in our work here.

There are numerous next directions to pursue. On one hand, couplings to the
Riemann tensor contribute nontrivially to gravitational entropy Parikh:2009qs
; LopesCardoso:1999cv . Many parity violating theories, such as the
interaction we studied in this work (17) and better studied theories such as
dynamical Chern-Simons gravity, entail couplings to the Riemann tensor and
therefore will modify the Wald entropy Wald:1993nt . Entanglement entropy in
quantum field theory was studied in the context of parity violating theories
in Hughes:2015ora , where the authors made contact with gravitational
anomalies in flat space. It would be interesting to explore a connection
between gravitational entropy corrections and parity violation in black hole
physics. More generally, black hole solutions with KR hair modify the shadow
and lensing properties in comparison to GR Kumar:2020hgm ; Lessa:2019bgi .
Repeating such an analysis would offer a different approach to constraining
our model parameters, as we certainly expect additional modifications as a
result of the interaction (6).

It is of significant importance to consider the consequences of the
interaction (6) in early universe physics. The KR field and more general
$p$-forms in inflationary scenarios have been been studied in Starobinsky
inflation Elizalde:2018now , $F(R)$ theories Elizalde:2018rmz , anisotropic
inflation Almeida:2019xzt , and in general cosmological backgrounds
BeltranAlmeida:2018nin . Each of these works assumed even parity and gauge
invariance. Gauge invariance being preserved implies the KR field can be
described in terms of one (pseudo)scalar degree of freedom, which is not an
approach that we can take due to the non gauge invariance of (6). It is
therefore interesting to explore the consequences of (6) in inflationary
scenarios, in particular the deviations from GR for the primordial
gravitational radiation, as well as consistency with other cosmological
observables (for example, parity violation in the CMB was considered very
recently in Gaztaaga2024:2401.08288v1 ). Allowing the KR fields to be fully
dynamical opposed to frozen in a constant background value will be relevant in
such scenarios. One approach would be to study the interactions with gravity
in their polarization basis, as written in Capanelli:2023uwv . On the other
hand, as mentioned in Section II, genuinely preserving cosmological isotropy
when the KR field is in its background value should necessitate multiple
fields, which may result in modifications to our results in this paper. We
leave this and other explorations to future work.

ACKNOWLEDGEMENTS

The authors thank Heliudson Bernardo, Leah Jenks, Evan McDonough, and Andrew
Svesko for helpful comments and discussions. SA and TM are supported by the
Simons Foundation, Award 896696.

## References

  * (1) L. Jenks, L. Choi, M. Lagos, and N. Yunes, “Parametrized parity violation in gravitational wave propagation,” Phys. Rev. D 108 no. 4, (2023) 044023, arXiv:2305.10478 [gr-qc].
  * (2) LIGO Scientific Collaboration, J. Aasi et al., “Advanced LIGO,” Class. Quant. Grav. 32 (2015) 074001, arXiv:1411.4547 [gr-qc].
  * (3) VIRGO Collaboration, F. Acernese et al., “Advanced Virgo: a second-generation interferometric gravitational wave detector,” Class. Quant. Grav. 32 no. 2, (2015) 024001, arXiv:1408.3978 [gr-qc].
  * (4) LIGO Scientific, Virgo Collaboration, R. Abbott et al., “GWTC-2: Compact Binary Coalescences Observed by LIGO and Virgo During the First Half of the Third Observing Run,” Phys. Rev. X 11 (2021) 021053, arXiv:2010.14527 [gr-qc].
  * (5) KAGRA, VIRGO, LIGO Scientific Collaboration, R. Abbott et al., “GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo during the Second Part of the Third Observing Run,” Phys. Rev. X 13 no. 4, (2023) 041039, arXiv:2111.03606 [gr-qc].
  * (6) NANOGrav Collaboration, G. Agazie et al., “The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background,” Astrophys. J. Lett. 951 no. 1, (2023) L8, arXiv:2306.16213 [astro-ph.HE].
  * (7) NANOGrav Collaboration, A. Afzal et al., “The NANOGrav 15 yr Data Set: Search for Signals from New Physics,” Astrophys. J. Lett. 951 no. 1, (2023) L11, arXiv:2306.16219 [astro-ph.HE].
  * (8) EPTA, InPTA: Collaboration, J. Antoniadis et al., “The second data release from the European Pulsar Timing Array - III. Search for gravitational wave signals,” Astron. Astrophys. 678 (2023) A50, arXiv:2306.16214 [astro-ph.HE].
  * (9) H. Xu et al., “Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I,” Res. Astron. Astrophys. 23 no. 7, (2023) 075024, arXiv:2306.16216 [astro-ph.HE].
  * (10) D. J. Reardon et al., “Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array,” Astrophys. J. Lett. 951 no. 1, (2023) L6, arXiv:2306.16215 [astro-ph.HE].
  * (11) LIGO Scientific, Virgo, Fermi-GBM, INTEGRAL Collaboration, B. P. Abbott et al., “Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A,” Astrophys. J. Lett. 848 no. 2, (2017) L13, arXiv:1710.05834 [astro-ph.HE].
  * (12) LIGO Scientific, VIRGO, KAGRA Collaboration, R. Abbott et al., “Tests of General Relativity with GWTC-3,” arXiv:2112.06861 [gr-qc].
  * (13) A. Lue, L.-M. Wang, and M. Kamionkowski, “Cosmological signature of new parity violating interactions,” Phys. Rev. Lett. 83 (1999) 1506–1509, arXiv:astro-ph/9812088.
  * (14) N. Yunes and F. Pretorius, “Dynamical Chern-Simons Modified Gravity. I. Spinning Black Holes in the Slow-Rotation Approximation,” Phys. Rev. D 79 (2009) 084043, arXiv:0902.4669 [gr-qc].
  * (15) S. Alexander and N. Yunes, “Chern-Simons Modified General Relativity,” Phys. Rept. 480 (2009) 1–55, arXiv:0907.2562 [hep-th].
  * (16) S. Alexander, S. Cormack, A. Marcianò, and N. Yunes, “Gravitational-Wave Mediated Preheating,” Phys. Lett. B 743 (2015) 82–86, arXiv:1405.4288 [gr-qc].
  * (17) S. H.-S. Alexander, M. E. Peskin, and M. M. Sheikh-Jabbari, “Leptogenesis from gravity waves in models of inflation,” Phys. Rev. Lett. 96 (2006) 081301, arXiv:hep-th/0403069.
  * (18) C. Creque-Sarbinowski, S. Alexander, M. Kamionkowski, and O. Philcox, “Parity-violating trispectrum from Chern-Simons gravity,” JCAP 11 (2023) 029, arXiv:2303.04815 [astro-ph.CO].
  * (19) P. Wagle, D. Li, Y. Chen, and N. Yunes, “Perturbations of spinning black holes in dynamical Chern-Simons gravity I. Slow rotation equations,” arXiv:2311.07706 [gr-qc].
  * (20) S. Nojiri, S. D. Odintsov, V. K. Oikonomou, and A. A. Popov, “Propagation of Gravitational Waves in Chern-Simons Axion Einstein Gravity,” Phys. Rev. D 100 no. 8, (2019) 084009, arXiv:1909.01324 [gr-qc].
  * (21) M. Crisostomi, K. Noui, C. Charmousis, and D. Langlois, “Beyond Lovelock gravity: Higher derivative metric theories,” Phys. Rev. D 97 no. 4, (2018) 044034, arXiv:1710.04531 [hep-th].
  * (22) J. Qiao, T. Zhu, G. Li, and W. Zhao, “Post-Newtonian parameters of ghost-free parity-violating gravities,” JCAP 04 no. 04, (2022) 054, arXiv:2110.09033 [gr-qc].
  * (23) R. P. Woodard, “Ostrogradsky’s theorem on Hamiltonian instability,” Scholarpedia 10 no. 8, (2015) 32243, arXiv:1506.02210 [hep-th].
  * (24) A. Nishizawa and T. Kobayashi, “Parity-violating gravity and GW170817,” Phys. Rev. D 98 no. 12, (2018) 124018, arXiv:1809.00815 [gr-qc].
  * (25) P. Horava, “Quantum Gravity at a Lifshitz Point,” Phys. Rev. D 79 (2009) 084008, arXiv:0901.3775 [hep-th].
  * (26) A. Wang, “Hořava gravity at a Lifshitz point: A progress report,” Int. J. Mod. Phys. D 26 no. 07, (2017) 1730014, arXiv:1701.06087 [gr-qc].
  * (27) T. Zhu, W. Zhao, Y. Huang, A. Wang, and Q. Wu, “Effects of parity violation on non-gaussianity of primordial gravitational waves in Hořava-Lifshitz gravity,” Phys. Rev. D 88 (2013) 063508, arXiv:1305.0600 [hep-th].
  * (28) K. Takahashi, M. Minamitsuji, and H. Motohashi, “Generalized disformal Horndeski theories: Cosmological perturbations and consistent matter coupling,” PTEP 2023 no. 1, (2023) 013E01, arXiv:2209.02176 [gr-qc].
  * (29) H. W. H. Tahara, K. Takahashi, M. Minamitsuji, and H. Motohashi, “Exact solution for rotating black holes in parity-violating gravity,” arXiv:2312.11899 [gr-qc].
  * (30) A. Conroy and T. Koivisto, “Parity-Violating Gravity and GW170817 in Non-Riemannian Cosmology,” JCAP 12 (2019) 016, arXiv:1908.04313 [gr-qc].
  * (31) D. Iosifidis and L. Ravera, “Parity Violating Metric-Affine Gravity Theories,” Class. Quant. Grav. 38 no. 11, (2021) 115003, arXiv:2009.03328 [gr-qc].
  * (32) L. Freidel, D. Minic, and T. Takeuchi, “Quantum gravity, torsion, parity violation and all that,” Phys. Rev. D 72 (2005) 104002, arXiv:hep-th/0507253.
  * (33) N. E. Mavromatos, P. Pais, and A. Iorio, “Torsion at different scales: from materials to the Universe,” arXiv:2310.13150 [gr-qc].
  * (34) I. L. Shapiro and P. M. Teixeira, “Quantum Einstein-Cartan theory with the Holst term,” Class. Quant. Grav. 31 (2014) 185002, arXiv:1402.4854 [hep-th].
  * (35) M. Szczachor, “Supersymmetric Holst action with matter coupling and parity violation,” Int. J. Geom. Meth. Mod. Phys. 09 (2012) 1261015, arXiv:1202.2232 [gr-qc].
  * (36) R. Hojman, C. Mukku, and W. A. Sayed, “Parity violation in metric torsion theories of gravitation,” Phys. Rev. D 22 (1980) 1915–1921.
  * (37) S. Dyda, E. E. Flanagan, and M. Kamionkowski, “Vacuum instability in chern-simons gravity,” Phys. Rev. D 86 (Dec, 2012) 124031. https://link.aps.org/doi/10.1103/PhysRevD.86.124031.
  * (38) C. R. Contaldi, J. Magueijo, and L. Smolin, “Anomalous CMB polarization and gravitational chirality,” Phys. Rev. Lett. 101 (2008) 141101, arXiv:0806.3082 [astro-ph].
  * (39) C. Capanelli, L. Jenks, E. W. Kolb, and E. McDonough, “Cosmological Implications of Kalb-Ramond-Like-Particles,” arXiv:2309.02485 [hep-ph].
  * (40) A. Dashko and R. Dick, “The shadow of dark matter as a shadow of string theory,” Eur. Phys. J. C 79 no. 4, (2019) 312, arXiv:1809.01089 [hep-ph].
  * (41) R. Dick, “The shadow of dark matter as a shadow of string theory: String origin of the dipole term,” Eur. Phys. J. C 80 no. 6, (2020) 525, arXiv:2003.13843 [hep-th].
  * (42) J. P. Beltrán Almeida, A. Guarnizo, and C. A. Valenzuela-Toledo, “Arbitrarily coupled $p-$forms in cosmological backgrounds,” Class. Quant. Grav. 37 no. 3, (2020) 035001, arXiv:1810.05301 [astro-ph.CO].
  * (43) J. P. B. Almeida, A. Guarnizo, R. Kase, S. Tsujikawa, and C. A. Valenzuela-Toledo, “Anisotropic inflation with coupled p$-$forms,” JCAP 03 (2019) 025, arXiv:1901.06097 [gr-qc].
  * (44) M. de Cesare, N. E. Mavromatos, and S. Sarkar, “On the possibility of tree-level leptogenesis from Kalb–Ramond torsion background,” Eur. Phys. J. C 75 no. 10, (2015) 514, arXiv:1412.7077 [hep-ph].
  * (45) J. B. Orjuela-Quintana and C. A. Valenzuela-Toledo, “Anisotropic Dark Energy from a Coupled 2-form Field,” arXiv:2202.07546 [gr-qc].
  * (46) J. P. Beltrán Almeida, A. Guarnizo, R. Kase, S. Tsujikawa, and C. A. Valenzuela-Toledo, “Anisotropic $2$-form dark energy,” Phys. Lett. B 793 (2019) 396–404, arXiv:1902.05846 [hep-th].
  * (47) M. Kalb and P. Ramond, “Classical direct interstring action,” Phys. Rev. D 9 (Apr, 1974) 2273–2284. https://link.aps.org/doi/10.1103/PhysRevD.9.2273.
  * (48) V. I. Ogievetsky and I. V. Polubarinov, “The notoph and its possible interactions,” Yad. Fiz. 4 (1966) 216–223.
  * (49) E. A. Ivanov, “Gauge Fields, Nonlinear Realizations, Supersymmetry,” Phys. Part. Nucl. 47 no. 4, (2016) 508–539, arXiv:1604.01379 [hep-th].
  * (50) P. Svrcek and E. Witten, “Axions In String Theory,” JHEP 06 (2006) 051, arXiv:hep-th/0605206.
  * (51) A. Smailagic and E. Spallucci, “The Dual phases of massless / massive Kalb-Ramond fields: Letter to the editor,” J. Phys. A 34 (2001) L435–L440, arXiv:hep-th/0106173.
  * (52) B. Altschul, Q. G. Bailey, and V. A. Kostelecky, “Lorentz violation with an antisymmetric tensor,” Phys. Rev. D 81 (2010) 065028, arXiv:0912.4852 [gr-qc].
  * (53) D. Maity, P. Majumdar, and S. SenGupta, “Parity violating Kalb-Ramond-Maxwell interactions and CMB anisotropy in a brane world,” JCAP 06 (2004) 005, arXiv:hep-th/0401218.
  * (54) P. Majumdar, “New parity violating photon axion interaction,” Mod. Phys. Lett. A 19 (2004) 1319, arXiv:hep-th/0105122.
  * (55) P. Majumdar and S. SenGupta, “Parity violating gravitational coupling of electromagnetic fields,” Class. Quant. Grav. 16 (1999) L89–L94, arXiv:gr-qc/9906027.
  * (56) A. Hell, “On the duality of massive Kalb-Ramond and Proca fields,” JCAP 01 no. 01, (2022) 056, arXiv:2109.05030 [hep-th].
  * (57) R. V. Maluf and J. C. S. Neves, “Bianchi type I cosmology with a Kalb–Ramond background field,” Eur. Phys. J. C 82 no. 2, (2022) 135, arXiv:2111.13165 [gr-qc].
  * (58) A. Golovnev, V. Mukhanov, and V. Vanchurin, “Vector Inflation,” JCAP 06 (2008) 009, arXiv:0802.2068 [astro-ph].
  * (59) S. Alexander, L. S. Finn, and N. Yunes, “A Gravitational-wave probe of effective quantum gravity,” Phys. Rev. D 78 (2008) 066005, arXiv:0712.2542 [gr-qc].
  * (60) N. Yunes, R. O’Shaughnessy, B. J. Owen, and S. Alexander, “Testing gravitational parity violation with coincident gravitational waves and short gamma-ray bursts,” Phys. Rev. D 82 (2010) 064017, arXiv:1005.3310 [gr-qc].
  * (61) M. Maggiore et al., “Science Case for the Einstein Telescope,” JCAP 03 (2020) 050, arXiv:1912.02622 [astro-ph.CO].
  * (62) M. Evans et al., “A Horizon Study for Cosmic Explorer: Science, Observatories, and Community,” arXiv:2109.09882 [astro-ph.IM].
  * (63) T. Callister, L. Jenks, D. Holz, and N. Yunes, “A New Probe of Gravitational Parity Violation Through (Non-)Observation of the Stochastic Gravitational-Wave Background,” arXiv:2312.12532 [gr-qc].
  * (64) M. Okounkova, W. M. Farr, M. Isi, and L. C. Stein, “Constraining gravitational wave amplitude birefringence and Chern-Simons gravity with GWTC-2,” Phys. Rev. D 106 no. 4, (2022) 044067, arXiv:2101.11153 [gr-qc].
  * (65) T. C. K. Ng, M. Isi, K. W. K. Wong, and W. M. Farr, “Constraining gravitational wave amplitude birefringence with GWTC-3,” Phys. Rev. D 108 no. 8, (2023) 084068, arXiv:2305.05844 [gr-qc].
  * (66) J. D. Wells, “PeV-scale supersymmetry,” Phys. Rev. D 71 (2005) 015013, arXiv:hep-ph/0411041.
  * (67) P. N. Bhattiprolu and J. D. Wells, “Precision unification and the scale of supersymmetry,” arXiv:2309.12954 [hep-ph].
  * (68) A. Berlin, D. Hooper, and G. Krnjaic, “PeV-Scale Dark Matter as a Thermal Relic of a Decoupled Sector,” Phys. Lett. B 760 (2016) 106–111, arXiv:1602.08490 [hep-ph].
  * (69) Tibet ASgamma Collaboration, M. Amenomori et al., “First Detection of sub-PeV Diffuse Gamma Rays from the Galactic Disk: Evidence for Ubiquitous Galactic Cosmic Rays beyond PeV Energies,” Phys. Rev. Lett. 126 no. 14, (2021) 141101, arXiv:2104.05181 [astro-ph.HE].
  * (70) T. N. Maity, A. K. Saha, A. Dubey, and R. Laha, “Search for dark matter using sub-PeV $\gamma$-rays observed by Tibet AS$\gamma$,” arXiv:2105.05680 [hep-ph].
  * (71) T. Cohen, K. Murase, N. L. Rodd, B. R. Safdi, and Y. Soreq, “$\gamma$ -ray Constraints on Decaying Dark Matter and Implications for IceCube,” Phys. Rev. Lett. 119 no. 2, (2017) 021102, arXiv:1612.05638 [hep-ph].
  * (72) A. Bhattacharya, A. Esmaili, S. Palomares-Ruiz, and I. Sarcevic, “Update on decaying and annihilating heavy dark matter with the 6-year IceCube HESE data,” JCAP 05 (2019) 051, arXiv:1903.12623 [hep-ph].
  * (73) K. Ishiwata, O. Macias, S. Ando, and M. Arimoto, “Probing heavy dark matter decays with multi-messenger astrophysical data,” JCAP 01 (2020) 003, arXiv:1907.11671 [astro-ph.HE].
  * (74) C. Blanco and D. Hooper, “Constraints on Decaying Dark Matter from the Isotropic Gamma-Ray Background,” JCAP 03 (2019) 019, arXiv:1811.05988 [astro-ph.HE].
  * (75) M. Califano, R. D’Agostino, and D. Vernieri, “Parity violation in gravitational waves and observational bounds from third-generation detectors,” arXiv:2311.02161 [gr-qc].
  * (76) E. Vanzan, M. Kamionkowski, and S. C. Hotinli, “Phenomenology of a vector-field-induced (and possibly parity breaking) compensated isocurvature perturbation,” arXiv:2311.18121 [astro-ph.CO].
  * (77) R. Bluhm and V. A. Kostelecky, “Spontaneous Lorentz violation, Nambu-Goldstone modes, and gravity,” Phys. Rev. D 71 (2005) 065008, arXiv:hep-th/0412320.
  * (78) R. Bluhm, S.-H. Fung, and V. A. Kostelecky, “Spontaneous Lorentz and Diffeomorphism Violation, Massive Modes, and Gravity,” Phys. Rev. D 77 (2008) 065020, arXiv:0712.4119 [hep-th].
  * (79) K. Higashijima and N. Yokoi, “Spontaneous Lorentz symmetry breaking by antisymmetric tensor field,” Phys. Rev. D 64 (2001) 025004, arXiv:hep-th/0101222.
  * (80) C. A. Hernaski, “Spontaneous Breaking of Lorentz Symmetry with an antisymmetric tensor,” Phys. Rev. D 94 no. 10, (2016) 105004, arXiv:1608.00829 [hep-th].
  * (81) H.-C. Cheng, M. A. Luty, S. Mukohyama, and J. Thaler, “Spontaneous Lorentz breaking at high energies,” JHEP 05 (2006) 076, arXiv:hep-th/0603010.
  * (82) S. M. Carroll and E. A. Lim, “Lorentz-violating vector fields slow the universe down,” Phys. Rev. D 70 (2004) 123525, arXiv:hep-th/0407149.
  * (83) G. Lambiase, “Lorentz invariance breakdown and constraints from big-bang nucleosynthesis,” Phys. Rev. D 72 (2005) 087702, arXiv:astro-ph/0510386.
  * (84) D. Colladay and V. A. Kostelecky, “Lorentz violating extension of the standard model,” Phys. Rev. D 58 (1998) 116002, arXiv:hep-ph/9809521.
  * (85) V. A. Kostelecky and S. Samuel, “Spontaneous Breaking of Lorentz Symmetry in String Theory,” Phys. Rev. D 39 (1989) 683.
  * (86) V. A. Kostelecky and N. Russell, “Data Tables for Lorentz and CPT Violation,” Rev. Mod. Phys. 83 (2011) 11–31, arXiv:0801.0287 [hep-ph].
  * (87) V. A. Kostelecky, “Gravity, Lorentz violation, and the standard model,” Phys. Rev. D 69 (2004) 105009, arXiv:hep-th/0312310.
  * (88) M. K. Parikh and S. Sarkar, “Beyond the Einstein Equation of State: Wald Entropy and Thermodynamical Gravity,” Entropy 18 (2016) 119, arXiv:0903.1176 [hep-th].
  * (89) G. Lopes Cardoso, B. de Wit, and T. Mohaupt, “Deviations from the area law for supersymmetric black holes,” Fortsch. Phys. 48 (2000) 49–64, arXiv:hep-th/9904005.
  * (90) R. M. Wald, “Black hole entropy is the Noether charge,” Phys. Rev. D 48 no. 8, (1993) R3427–R3431, arXiv:gr-qc/9307038.
  * (91) T. L. Hughes, R. G. Leigh, O. Parrikar, and S. T. Ramamurthy, “Entanglement entropy and anomaly inflow,” Phys. Rev. D 93 no. 6, (2016) 065059, arXiv:1509.04969 [hep-th].
  * (92) R. Kumar, S. G. Ghosh, and A. Wang, “Gravitational deflection of light and shadow cast by rotating Kalb-Ramond black holes,” Phys. Rev. D 101 no. 10, (2020) 104001, arXiv:2001.00460 [gr-qc].
  * (93) L. A. Lessa, J. E. G. Silva, R. V. Maluf, and C. A. S. Almeida, “Modified black hole solution with a background Kalb–Ramond field,” Eur. Phys. J. C 80 no. 4, (2020) 335, arXiv:1911.10296 [gr-qc].
  * (94) E. Elizalde, S. D. Odintsov, V. K. Oikonomou, and T. Paul, “Logarithmic-corrected $R^{2}$ Gravity Inflation in the Presence of Kalb-Ramond Fields,” JCAP 02 (2019) 017, arXiv:1810.07711 [gr-qc].
  * (95) E. Elizalde, S. D. Odintsov, T. Paul, and D. Sáez-Chillón Gómez, “Inflationary universe in $F(R)$ gravity with antisymmetric tensor fields and their suppression during its evolution,” Phys. Rev. D 99 no. 6, (2019) 063506, arXiv:1811.02960 [gr-qc].
  * (96) E. Gaztaaga and K. S. Kumar, “On the origins of cmb anomalies and testing a new theory of inflationary quantum fluctuations,” Arxiv:2401.08288v1. https://arxiv.org/abs/2401.08288.


# Interacting Brownian particles exhibiting enhanced rectification in an
asymmetric channel

Narender Khatri and P. S. Burada  Department of Physics, Indian Institute of
Technology Kharagpur, Kharagpur - 721302, India<EMAIL_ADDRESS>
###### Abstract

Rectification of interacting Brownian particles is investigated in a two-
dimensional asymmetric channel in the presence of an external periodic driving
force. The periodic driving force can break the thermodynamic equilibrium and
induces rectification of particles (or finite average velocity). The spatial
variation in the shape of the channel leads to entropic barriers, which indeed
control the rectification of particles. We find that by simply tunning the
driving frequency, driving amplitude, and shape of the asymmetric channel, the
average velocity can be reversed. Moreover, a short range interaction force
between the particles further enhances the rectification of particles greatly.
This interaction force is modeled as the lubrication interaction.
Interestingly, it is observed that there exists a characteristic critical
frequency $\Omega_{c}$ below which the rectification of particles greatly
enhances in the positive direction with increasing the interaction strength;
whereas, for the frequency above this critical value, it greatly enhances in
the negative direction with increasing the interaction strength. Further,
there exists an optimal value of the asymmetric parameter of the channel for
which the rectification of interacting particles is maximum. These findings
are useful in sorting out the particles and understanding the diffusive
behavior of small particles or molecules in microfluidic channels, membrane
pores, etc.

  * July 2021

##  1 Introduction

In general, in biological systems, solid-state nanopores, and artificial
microstructures, it is well known that the Brownian particles move in
constrained geometries [1, 2]. The diffusive behavior of particles in
constrained geometries has not only important theoretical significance but
also has potential applications in many processes, especially catalysis,
osmosis, particles selectivity, and particles separation [1, 2]. Note that
when the Brownian particles diffuse in a constrained geometry, the spatial
confinement to the particles produces an entropic barrier, which significantly
impacts the transport properties of the particles [3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13]. In particular, the important examples of constrained geometries
where entropic effects are ubiquitous are, for instance, biological cells, ion
channels, nanoporous materials, zeolites, microfluidic devices, ratchets, and
artificial channels [14, 1, 2, 15, 16].

Rectification of Brownian particles in ratchet systems in the presence of an
unbiased external force has been an intense area of research in the last few
decades [17, 16, 14, 18, 19, 20, 21, 22, 23, 24, 25]. Here, the external
oscillatory (unbiased) force can induce directed motion of the diffusing
particles, i.e., known as rectification, in ratchet systems possessing
dynamical or spatial symmetry breaking at submicron scales [14]. Particularly,
the rectification of Brownian particles in ratchet systems has been inspired
by the working principles of Brownian motors [26], molecular motors [27],
nanoscale friction [28], surface smoothing [29], coupled Josephson junctions
[30], separation of particles [31], optical ratchets and directed motion of
laser-cooled atoms [32], and mass separation and trapping schemes at the
microscale [33]. Note that the rectification depends on the physical
properties of the diffusing particles such as mass, size, shape, etc. [14,
26]. In order to observe rectification of particles, the ratchet setup
requires two basic ingredients [34]; (i) asymmetry (spatial and/or temporal)
in the system itself to violate the left/right symmetry in the diffusive
behavior of the particles, and (ii) a zero-mean external oscillatory force to
break the thermodynamic equilibrium, which forbids the appearance of directed
motion due to the second law of thermodynamics. Depending on various
nonequilibrium situations, several models have been proposed to obtain
rectification of particles such as rocking ratchets [35], flashing ratchets
[36], diffusion ratchets [37], and correlation ratchets [38]. In all these
models, the unbiased driving force can break the thermodynamic equilibrium and
induces rectification of particles. The rectification of non-interacting
Brownian particles is a well studied problem both experimentally [16, 14] and
theoretically [18, 19, 20]. However, the interaction between the particles may
further influence the diffusive behavior of particles in confined geometries
[5]. Usually, the short range interaction force between the Brownian particles
can be considered as (i) the lubrication interaction [5, 39], (ii) the
harmonic interaction [40], (iii) the Lennard-Jones interaction [41], and (iv)
the Yukawa interaction [42].

In this article, we numerically study how the interaction between particles
and the channel shape influences the rectification of the Brownian particles
when they diffuse in an asymmetric periodic channel in the presence of a
periodic driving force. Since we are dealing with the diffusive behavior of a
large number of Brownian particles in a confined environment, out of the above
mentioned interactions, we consider the lubrication interaction in the current
study. The lubrication interaction, as the name implies, results from the thin
layer of viscous fluid that separates the surfaces of nearly touching Brownian
particles [43]. In other words, it provides the hydrodynamic effect,
originated by the action of the fluid stress, on a particle due to its
neighbors. Also, it plays a vital role, particularly when the particles are
confined. Rest of this article is organized as follows. In Sec. 2, we
introduce the model for the interacting Brownian particles in a two-
dimensional asymmetric channel in the presence of an external periodic driving
force. The influences of the particles interaction and the channel shape on
rectification are discussed in Sec. 3. Finally, the main conclusions are
presented in Sec. 4.

##  2 Model

Consider the motion of a Brownian particle, suspended in a two-dimensional
asymmetric channel, driven by an external oscillatory force, and the
interaction force $\vec{F}_{\mathrm{int}}$ due to its neighboring particles
(see Fig. 1). In the overdamped regime, the equation of motion of the Brownian
particle is given by the Langevin equation,

$\eta\frac{d\vec{r}}{dt}=F_{0}\sin(\omega
t)\,\hat{x}+\vec{F}_{\mathrm{int}}+\sqrt{\eta k_{B}T}\,\vec{\xi}(t),$ (1)

where $\vec{r}$ is the position of the particle in two dimensions, $F_{0}$ is
the amplitude of the periodic driving force acting along the $x$ direction
only, $\omega$ is the frequency of the driving force, $\eta$ is the friction
coefficient, $k_{B}$ is the Boltzmann constant, and $T$ is the temperature.
The thermal fluctuations due to the coupling of the particle with the
surrounding heat bath are modeled by a zero-mean Gaussian white noise
$\vec{\xi}(t)$, obeying the fluctuation-dissipation relation
$\langle\xi_{i}(t)\xi_{j}(t^{\prime})\rangle=2\delta_{ij}\delta(t-t^{\prime})$
for $i,j=x,y$.

Figure 1: Schematic illustration of a two-dimensional asymmetric channel, with
the periodicity $L$, confining the motion of Brownian particles which are
subjected to an external non-adiabatic oscillatory force
$\vec{F}(t)=F_{0}\sin(\omega t)\hat{x}$ along the $x$ direction and
considering the short range interaction force between the particles. The
reflecting channel boundaries are defined by equation (4) which assure the
confinement of particles inside the channel.

We have considered a few types of interaction between the particles, and out
of all, we find that the lubrication interaction plays a vital role,
particularly when the particles are confined in a channel which is the case in
the current study (details not shown). Therefore, we consider the lubrication
interaction to study the rectification of particles. The short range
interaction force on a particle $i$ due to its neighbors reads [5, 39],

$\displaystyle{\vec{F}_{\mathrm{int}}=\sum_{j=1,\leavevmode\nobreak\ j\neq
i}^{n}\frac{\sigma_{ij}}{d_{ij}}\left(\cos\theta_{ij}\,\hat{x}+\sin\theta_{ij}\,\hat{y}\right)},$
(2)

where the sum is taken over all its nearest neighbors, $\sigma_{ij}$ is the
interaction strength between the particle $i$ and $j$, $d_{ij}$ is the
corresponding distance between the particles, $\theta_{ij}$ is the angle that
$d_{ij}$ makes with the channel axis ($x$ axis), and $\hat{x}$ and $\hat{y}$
are the unit vectors along $x$ and $y$ directions, respectively. In the limit
of high density of identical particles whose size is much smaller than both
the periodicity and width of the channel, this interaction force can be
approximated as [5]

$\vec{F}_{\mathrm{int}}=K\,(\cos\theta\,\hat{x}+\sin\theta\,\hat{y}),$ (3)

where $K$ denotes the total strength of the interaction force on a particle
due to its neighbors and $\theta$ is a random variable that can have values
between $0$ to $2\pi$. In this limit, equation (3) is similar to the Vicsek
interaction [44, 45], which has been used extensively to study the collective
behavior of active particles. In particular, the same model for the
lubrication interaction has been previously used to study the biased transport
of interacting Brownian particles in symmetric channels [5].

The shape of two-dimensional asymmetric and spatially periodic channel is
described by its half-width (see Fig. 1)

$H(x)=a\left[\sin\left(\frac{2\pi
x}{L}\right)+\frac{\Delta}{4}\sin\left(\frac{4\pi x}{L}\right)\right]+b,$ (4)

where $L$ corresponds to the periodicity of the channel, $\Delta$ denotes the
asymmetric parameter of the channel, and parameters $a$ and $b$ control the
slope and channel width at the bottleneck, respectively. Note that the channel
is biased towards the positive $x$ direction for $\Delta>0$ and negative $x$
direction for $\Delta<0$.

In order to have a dimensionless description, we henceforth scale all lengths
by the periodicity of the channel $L$ and time by $\tau=\eta
L^{2}/(k_{B}T_{R})$, which is the characteristic diffusion time at an
arbitrary but irrelevant reference temperature $T_{R}$ [46]. In dimensionless
form, the Langevin equation reads

$\frac{d\vec{r}}{dt}=f_{0}\sin(\Omega
t)\hat{x}+\vec{f}_{\mathrm{int}}+\sqrt{D}\,\vec{\xi}(t).$ (5)

Here, the noise intensity (or the rescaled temperature) is given by
$D=T/T_{R}$. The dimensionless interaction force becomes
$\vec{f}_{\mathrm{int}}=\vec{F}_{\mathrm{int}}L/(k_{B}T_{R})$. The other
parameters read $f_{0}=F_{0}L/(k_{B}T_{R})$, $\Omega=\omega\tau$, and
$k=KL/(k_{B}T_{R})$.

While solving the Langevin equation $(\ref{Langevin2})$ numerically, we have
considered that the particle elastically reflects at the channel boundary to
ensure the confinement within the channel. Figure 1 shows the reflection of a
particle at the channel boundary. Let us say the initial position of the
particle was $A(x_{1},y_{1})$, and in the next instant of time, the position
of particle is at $C(x_{2},y_{2})$, which is outside of the channel boundary
[see Fig. 1]. The line joining points $A$ and $C$ intersects the channel
boundary at a point $B(p,q)$, i.e., the reflection point which can be
calculated numerically using the bisection method. The desired position
$D(x_{3},y_{3})$ of the particle after reflection at point $B$ is given by [5]

$\displaystyle x_{3}$ $\displaystyle=p\pm\,\frac{l}{\sqrt{1+m_{3}^{2}}},$ (6a)
$\displaystyle y_{3}$ $\displaystyle=q+m_{3}(x_{3}-p),$ (6b)

where $m_{3}$ is the slope of line $BD$ and
$l=\sqrt{(y_{2}-q)^{2}+(x_{2}-p)^{2}}$. Note that depending on the driving
force acting on the particle, the particle may reflect multiple times at the
channel boundaries to reach the final position.

Knowingly, the Fokker-Planck equation corresponding to the Langevin equation
(5) cannot be solved analytically for the considered system. Therefore, we
stick to the Brownian dynamics simulations performed within the stochastic
Euler algorithm by the integration of the Langevin equation (5) using
reflecting boundary conditions at the channel walls. As initial conditions, we
have assumed that at $t=0$ all the particles are randomly distributed in a
cell of the channel. In the long time limit, the average velocity of
particles, which is an observable of the rectification of particles, along the
$x$ direction is given by

$v=\lim_{t\to\infty}\frac{\langle x(t)\rangle}{t}.$ (7)

In particular, the positive and negative average velocity means that the
particles drift to the positive and negative $x$ directions, respectively.
Note that the effective diffusion coefficient of particles in the long time
limit, which is an observable of the dispersion of particles, along the $x$
direction is calculated as [3]

$D_{eff}=\lim_{t\to\infty}\frac{\langle x^{2}(t)\rangle-\langle
x(t)\rangle^{2}}{2t}.$ (8)

##  3 Impact of particle-particle interaction

###  3.1 Enhanced rectification

Figure 2: Average velocity $v$ versus driving frequency $\Omega$ for various
values of the interaction strength between the particles. The value of
characteristic critical frequency $\Omega_{c}$ is approximately 1.4. (a) For
$\Omega\leq\Omega_{c}\approx 1.4$ and (b) for $\Omega\geq\Omega_{c}\approx
1.4$. The corresponding scaled effective diffusion coefficient $D_{eff}/D$ is
depicted in (c). The set parameters are $a=1/(2\pi)$, $b=1.2/(2\pi)$,
$\Delta=1$, $L=2\pi$, $f_{0}=2.5$, and $D=0.3$.

Figures 2(a)-(b) depict the average velocity $v$ as a function of the driving
frequency $\Omega$ for various values of the interaction strength between the
particles. It is important to point out that there exists a characteristic
critical frequency $\Omega_{c}\approx 1.4$ for which the average velocity is
zero. For $\Omega<\Omega_{c}$, because of the chosen structure of the channel,
particles move towards the long end (or the slanted side) resulting in a
positive average velocity. Whereas, for $\Omega>\Omega_{c}$, due to the higher
frequency, the particles do not get enough time to cross the long end. Thus,
the particles drift to the short end (or the steeper side) resulting in a
negative average velocity. Note that the characteristic critical frequency
$\Omega_{c}$ mainly depends on the periodicity of the channel $L$, noise
intensity $D$, and amplitude of the oscillating force $f_{0}$. The same is
reported by Ai [47] in the case of non-interacting Brownian particles
exhibiting rectification in an asymmetric channel. Here, we observe that the
scaled effective diffusion coefficient $D_{eff}/D$ enhances linearly with
increasing the interaction strength $k$ (see Fig. 2(c)). This enhancement in
$D_{eff}/D$ helps to clarify that dispersion of particles, which facilitates
the ratchet transport, increases with increasing $k$. Therefore, the average
velocity greatly enhances in both the positive and negative directions with
increasing the interaction strength (see (a) and (b) in Fig. 2). In
particular, in the very high frequency limit, i.e., $\Omega\rightarrow\infty$,
the average velocity tends to zero irrespective of the interaction strength.
It is because, in this limit, the particles experience a time-averaged
constant force $f=\int_{0}^{\frac{2\pi}{\Omega}}f(t)dt=0$.

Figure 3: Average velocity $v$ versus driving amplitude $f_{0}$ for various
values of the interaction strength between the particles. (a) For
$\Omega=0.1$, (b) for $\Omega=0.5$, and (c) for $\Omega=2.0$. The set
parameters are $a=1/(2\pi$), $b=1.2/(2\pi)$, $\Delta=1$, $L=2\pi$, and
$D=0.3$.

Figure 3 depicts the average velocity $v$ as a function of the driving
amplitude $f_{0}$ for various values of the interaction strength $k$. For the
non-interacting particles, $v$ remains positive as a function of $f_{0}$ for
the lower values of $\Omega$ [19]. With increasing $\Omega$, $v$ reversal
occurs as a function of $f_{0}$. As mentioned earlier, the interaction between
the particles facilitates the rectification of particles. Thus, for
$\Omega<\Omega_{c}$, we observe that $v$ greatly enhances in the positive
direction with increasing $k$. Whereas, for $\Omega>\Omega_{c}$, $v$ greatly
enhances in both the positive and negative directions with increasing $k$.
Interestingly, $v$ exhibits fluctuations for the moderate $\Omega$ values,
e.g., $\Omega=0.5$. Note that these fluctuations start to disappear with
increasing $k$ (see Fig. 3(b)). The latter is due to an increase in the
dispersion of particles with $k$. In particular, for the small and large
$f_{0}$ values, the average velocity tends to zero for various interaction
strength values. It is because the driving force can be neglected for small
driving amplitude values and the effect of the asymmetry of the channel
disappears for the large driving amplitude values [48].

Figure 4: Average velocity $v$ versus noise intensity $D$ for various values
of the interaction strength between the particles. (a) For $\Omega=0$, (b) for
$\Omega=0.01$, and (c) for $\Omega=2.0$. The set parameters are $a=1/(2\pi)$,
$b=1.2/(2\pi)$, $\Delta=1$, $L=2\pi$, and $f_{0}=2.5$.

Figure 4 depicts the average velocity $v$ as a function of the noise intensity
$D$ for various values of the interaction strength $k$ and at different values
of the driving frequency $\Omega$. For $\Omega=0$, as one would expect, $v$ is
zero for any value of $D$ and $k$ because the driving force is absent. For the
non-interacting particles, $v$ remains positive as a function of $D$ for the
lower value of $\Omega$. On the other hand, for the higher value of $\Omega$,
$v$ remains negative as a function of $D$ (see Fig. 4(c)). When $D=0$, the
non-interacting particles cannot explore the available area in the channel as
they move only along the channel direction (see Fig. 5(a)), and the effect of
the asymmetry of the channel disappears, resulting in a zero average velocity.
However, due to the interaction, particles can explore the two-dimensional
area and would feel the asymmetry of the channel. As a result, they exhibit a
finite $v$, which increases with $k$. On increasing $D$, $v$ greatly enhances
in the positive direction with increasing $k$ for $\Omega<\Omega_{c}$.
Whereas, for $\Omega>\Omega_{c}$, the same is observed in the negative
direction. Note that there exists a maximum in $v$ at an optimal value of $D$,
and it shifts toward the lower values of $D$ with increasing $k$. For the
higher values of $D$, the interaction effects start disappearing, and $v$
decreases.

Figure 5: Steady state distribution of the particles mapped into a single cell
of the two-dimensional asymmetric channel for two different values of the
interaction strength $k$ is depicted in (a) and (b). The average velocity $v$
as a function of the interaction strength $k$ is depicted in (c). The set
parameters are $a=1/(2\pi)$, $b=1.2/(2\pi)$, $\Delta=1$, $L=2\pi$,
$f_{0}=2.5$, $\Omega=2.0$, and $D=0$.

To understand the significance of $k$ better, we look at the steady state
distribution of particles mapped into a single cell of the two-dimensional
asymmetric channel for different values of $k$ when $D=0$ (see Fig. 5). As
mentioned before, in the limit $k\to 0$, $v$ is zero. On further increasing
$k$, the particles start exploring the available area in the cell due to the
interaction force from their neighbors, and the asymmetry of the channel
begins to show its signature. As mentioned earlier, particles do not get
enough time to cross the long end for the higher frequency; hence, most of
them enter into the cell from the short end. As a result, most of the
particles are found towards the long end. Accordingly, $v$ becomes negative
and monotonically increases with increasing $k$ (see Fig. 5(c)).

###  3.2 Impact of asymmetric parameter $\Delta$

Figure 6: Average velocity $v$ versus driving amplitude $f_{0}$ for various
values of the asymmetric parameter $\Delta$. The inset depicts the dependence
of $v$ on $\Delta$ for $f_{0}=6$. The set parameters are $a=1/(2\pi)$,
$b=1.2/(2\pi)$, $L=2\pi$, $k=40$, $D=0.3$, and $\Omega=2.0$.

Here, we study the impact of the asymmetric parameter $\Delta$ on the
rectification of interacting particles. In particular, $\Delta$ can have
values between -1.5 to 1.5 because for $|\Delta|>1.5$, the channel is blocked,
i.e., the minimum width of the channel becomes zero (see Fig. 1). Figure 6
depicts the average velocity $v$ as a function of the driving amplitude
$f_{0}$ for different values of the asymmetric parameter $\Delta$. The
qualitative behavior of the average velocity as a function of driving
amplitude remains the same for $\Delta\neq 0$. The peculiar behavior of the
average velocity as a function of the asymmetric parameter is shown in the
inset of Figure 6. Note that $\Delta$ controls the direction of the average
velocity. The average velocity is positive for $\Delta>0$, zero at $\Delta=0$,
and negative for $\Delta<0$. When $\Delta=0$, the channel is symmetric, so the
average velocity is zero. Whereas, when $|\Delta|>1.5$, the channel is
blocked, so the average velocity becomes zero. Therefore, there exists an
optimal value of $|\Delta|$ for which the average velocity is maximum.

Note that the observed interaction effects can also be studied experimentally
by creating a corrugated asymmetric channel by microprinting on a substrate
and measuring the drift velocity of the interacting particles [49]. These
particles can be fabricated from luminescent polystyrene with well defined
identical diameters [16]. In order to have an estimate in real units, which is
very useful for the experimentalists, the characteristic values of the main
parameters for the overdamped interacting Brownian particles in water moving
in a corrugated asymmetric channel with asymmetric parameter $\Delta=1$ and
period length $L\sim 1\leavevmode\nobreak\ \mu\mathrm{m}$, and the temperature
set at room temperature ($T\sim 300\leavevmode\nobreak\ \mathrm{K}$) are given
by $\eta\sim 2\times 10^{-9}\leavevmode\nobreak\ \mathrm{kg/s}$ [50],
$\tau\sim 0.15\leavevmode\nobreak\ \mathrm{s}$, $k_{B}T_{R}/L\sim
10^{-14}\leavevmode\nobreak\ \mathrm{N}$, and $L/\tau\sim
6.67\leavevmode\nobreak\ \mu\mathrm{m/s}$. For the parameters $F_{0}\sim
2.5\times 10^{-14}\leavevmode\nobreak\ \mathrm{N}$, $\omega\sim
0.07\leavevmode\nobreak\ \mathrm{s^{-1}}$, and $K\sim 4\times
10^{-13}\leavevmode\nobreak\ \mathrm{N}$, the interacting particles drift in
the positive direction with an average velocity $V\sim
0.25\leavevmode\nobreak\ \mu\mathrm{m/s}$, which is approximately $3.1$ times
higher with respect to no interaction case. The characteristic critical
frequency in the real unit is given by $\omega_{c}\sim
9.33\leavevmode\nobreak\ \mathrm{s^{-1}}$ below which the interacting
particles greatly enhance the average velocity in the positive direction,
whereas above this critical value it is greatly enhanced in the negative
direction. It is expected that the enhanced rectification can be used for the
efficient and selective continuous separation of particles mixture based on
their physical properties like mass, size, shape, etc. [14, 31, 51]. Also, it
may have high relevance in describing transport in mesoscopic systems [17, 52]
and controlled drug release [53].

##  4 Conclusions

In this work, we have numerically studied the rectification of interacting
Brownian particles in a two-dimensional asymmetric channel. With respect to no
interaction case, which has been studied earlier both experimentally [16, 14]
and theoretically [18, 19, 20], we have shown that for the interacting
Brownian particles, it is possible to greatly enhance the rectification of
particles. We found that the interaction between particles greatly enhances
the average velocity in both the positive and negative directions for various
parameter values. It has been observed that there exists a characteristic
critical frequency $\Omega_{c}$ below which the average velocity greatly
enhances in the positive direction with increasing the interaction strength;
however, for the frequency above this critical value, the average velocity
greatly enhances in the negative direction with increasing the interaction
strength. Moreover, we have shown that there exists an optimal value of the
asymmetric parameter of the channel for which the rectification of interacting
particles is maximum. These results have a wide application in many processes,
such as separation or spatial sorting of particles mixture based on their
physical properties [14, 31, 51], diffusion of ions and macromolecular solutes
through the channels in biological membranes [2], controlled drug release
[53], and many more. In the future, the current study can be extended to non-
Brownian particles under the same situation. Particularly, this will be
relevant for biological systems where it is quite common to observe anomalous
diffusion [54].

##  5 Acknowledge

This work was supported by the Indian Institute of Technology Kharagpur under
the Grant No. IIT/SRIC/PHY/TAB/2015-16/114.

## References

## References

  * [1] Kärger J and Ruthven D M.  Diffusion in zeolites and other microporous solids.  (J. Wiley, New York, 1992).
  * [2] Hille B.  Ion channels of excitable membranes.  (Sinauer, Sunderland, 2001).
  * [3] Reguera D, Schmid G, Burada P S, Rubi J M, Riemann P, and Hänggi P.  Phys. Rev. Lett., 96:130603, 2006.
  * [4] Reguera D and Rubi J M.  Phys. Rev. E, 64:061106, 2001.
  * [5] Khatri N and Burada P S.  J. Chem. Phys., 151:094103, 2019.
  * [6] Khatri N and Burada P S.  Phys. Rev. E, 102:012137, 2020.
  * [7] Hu C T, Ou Y L, Wu J C, and Ai B Q.  J. Stat. Mech., 033207, 2016.
  * [8] Wu J C, Chen Q, and Ai B Q.  J. Stat. Mech., P07005, 2015.
  * [9] Hu C T, Ou Y L, Wu J C, Chen Q, and Ai B Q.  J. Stat. Mech., P05025, 2015.
  * [10] Li F G and Ai B Q.  J. Stat. Mech., P04027, 2014.
  * [11] Ao X, Ghosh P K, Li Y, Schmid G, Hänggi P, and Marchesoni F.  Eur. Phys. J. Special Topics, 223:3227, 2014.
  * [12] Ghosh P K, Misko V R, Marchesoni F, and Nori F.  Phys. Rev. Lett., 110:268301, 2013.
  * [13] Li Y, Ghosh P K, Marchesoni F, and Li B.  Phys. Rev. E, 90:062301, 2014.
  * [14] Hänggi P and Marchesoni F.  Rev. Mod. Phys., 81:387, 2009.
  * [15] Han J and Craighead H G.  Science, 288:1026, 2000.
  * [16] Matthias S and Müller F.  Nature, 424:53, 2003.
  * [17] Astumian R D.  Science, 276:917, 1997.
  * [18] Ai B Q and Liu L G.  Phys. Rev. E, 74:051114, 2006.
  * [19] Ai B Q.  Phys. Rev. E, 80:011113, 2009.
  * [20] Malgaretti P, Pagonabarraga I, and Rubi J M.  J. Chem. Phys., 138:194906, 2013.
  * [21] Ai B Q, Zhu W J, He Y F, and Zhong W R.  J. Stat. Mech., 023501, 2017.
  * [22] Ghosh P K and Ray D S.  J. Stat. Mech., P03003, 2007.
  * [23] Huang X Q, Liao J J, and Ai B Q.  J. Stat. Mech., 023209, 2018.
  * [24] Gao T F, Ai B Q, Zheng Z G, and Chen J C.  J. Stat. Mech., 093204, 2016.
  * [25] Ai B Q and He Y F.  J. Stat. Mech., P04010, 2010.
  * [26] Hänggi P, Marchesoni F, and Nori F.  Ann. Phys., 14:51, 2005.
  * [27] Astumian R D and Derenyi I.  Eur. Biophys. J., 27:474, 1998.
  * [28] Krim J, Solina D H, and Chiarello R.  Phys. Rev. Lett., 66:181, 1991.
  * [29] Derenyi I, Lee C, and Barabasi A L.  Phys. Rev. Lett., 80:1473, 1998.
  * [30] Zapata I, Bartussek R, Sols F, and Hänggi P.  Phys. Rev. Lett., 77:2292, 1996.
  * [31] Reguera D, Luque A, Burada P S, Schmid G, Rubi J M, and Hänggi P.  Phys. Rev. Lett., 108:020604, 2012.
  * [32] Faucheux L P, Bourdieu L S, Kaplan P D, and Libchaber A J.  Phys. Rev. Lett., 74:1504, 1995.
  * [33] Borromeo M and Marchesoni F.  Phys. Rev. Lett., 99:150605, 2007.
  * [34] Denisov S, Hänggi P, and Mateos J L.  Am. J. Phys., 77:602, 2009.
  * [35] Magnasco M O.  Phys. Rev. Lett., 71:1477, 1993.
  * [36] Hänggi P and Bartussek R.  Nonlinear physics of complex system: Current status and future trends.  (Springer, Berlin, 1996).
  * [37] Reimann P, Bartussek R, Haussler R, and Hänggi P.  Phys. Lett. A, 215:26, 1996.
  * [38] Doering C R, Horsthemke W, and Riordan J.  Phys. Rev. Lett., 72:2984, 1994.
  * [39] O’Neill M E and Stewartson K.  J. Fluid Mech., 27:705, 1967.
  * [40] Wu J C, Lv K, Zhao W W, and Ai B Q.  Chaos, 28:123102, 2018.
  * [41] Rein M and Speck T.  Eur. Phys. J. E, 39:84, 2016.
  * [42] Mazroui M and Boughaleb Y.  Physica A, 227:93, 1996.
  * [43] Li J, Jiang X, Singh A, Heinonen O G, Hernández-Ortiz J P, and de Pablo J J.  J. Chem. Phys., 152:204109, 2020.
  * [44] Vicsek T, Czirok A, Ben-Jacob E, Cohen I, and Shochet O.  Phys. Rev. Lett., 75:1226, 1995.
  * [45] Czirok A, Stanley H E, and Vicsek T.  J. Phys. A: Math. Gen., 30:1375, 1997.
  * [46] Burada P S, Schmid G, Reguera D, Vainstein M H, Rubi J M, and Hänggi P.  Phys. Rev. Lett., 101:130602, 2008.
  * [47] Ai B Q.  J. Chem. Phys., 131:054111, 2009.
  * [48] Schmid G, Burada P S, Talkner P, and Hänggi P.  Adv. Solid State Phys., 48:317, 2009.
  * [49] Mahmud G, Campbell C J, Bishop K J, Komarova Y A, Chaga O, Soh S, Huda S, Kandere-Grzybowska K, and Grzybowski B A.  Nat. Phys., 5:606, 2009.
  * [50] Cussler E L.  Diffusion: Mass transfer in fluid systems.  (Cambridge University Press, 1997).
  * [51] Mukhopadhyay A K, Liebchen B, and Schmelcher P.  Phys. Rev. Lett., 120:218002, 2018.
  * [52] Jülicher F, Ajdari A, and Prost J.  Rev. Mod. Phys., 69:1269, 1997.
  * [53] Siegel R A.  J. Controlled Release, 69:109, 2000.
  * [54] Metzler R, Jeon J H, Cherstvy A G, and Barkai E.  Phys. Chem. Chem. Phys., 16:24128, 2014.