# 50 Years of Horndeski Gravity: Past, Present and Future

Gregory W. Horndeski<EMAIL_ADDRESS>or<EMAIL_ADDRESS>Applied Math Department, University of Waterloo, 200 University Avenue,
Waterloo, N2L 3G1, Ontario, Canada  Alessandra Silvestri
<EMAIL_ADDRESS>Institute Lorentz, Leiden University, PO Box
9506, Leiden 2300 RA, The Netherlands

(August 28, 2024)

###### Abstract

An essay on Horndeski gravity, how it was formulated in the early 1970s and
how it was ’re-discovered’ and widely adopted by Cosmologists more than thirty
years later.

##  I Introduction

In September, 2023, Prof. Andreas Wipf, the Editor in Chief of the
International Journal of Theoretical Physics, contacted me (Gregory Horndeski)
concerning my paper entitled "Second-Order Scalar-Tensor Field Equations in a
Four-Dimensional Space." That paper originally appeared in his Journal back in
1974, and he thought it would be appropriate for me to write an essay, to
appear in the International Journal of Theoretical Physics in 2024, to
celebrate the 50th anniversary of that paper’s publication. At first I was a
little apprehensive about accepting his proposal because he wanted the essay
to cover how the theory came into being and how it has been used. Well I
certainly was familiar with the theory’s inception, but I really was not
qualified to discuss how it was being used. To allay my concerns Prof. Wipf
said that he would not mind if the essay had a co-author who could explain the
theories value to the Physics community. Well given that I would be permitted
to have a colleague to do the hard work, I agreed to participate in this essay
project. My next task was to find a co-author. To that end I contacted a
colleague of mine at the University of Waterloo, Prof. Ghazal Geshnizjani, who
has worked with my scalar-tensor equations, to see if she could provide me
with a list of names of physicists who might be able to assist me in my
endeavor. She provided me with the names of a handful of physicists, all of
whom were familiar to me from what little I knew about applications of my
equations. At the top of the list was Prof. Alessandra Silvestri, with whom I
had prior contact concerning my work as an artist. So I asked her if she would
be interested in co-authoring an essay with me and she said yes.

This joint project between Prof. Silvestri and me will consist basically of
two parts. In the next Section I shall discuss how I developed my scalar-
tensor equations. This will then be followed by a section, with several parts,
in which Prof. Silvestri elucidates upon the cosmological implications of what
has become known as "Horndeski Gravity, " or "Horndeski Scalar Theory," or
"Horndeski Theory." The essay will end with us commenting on the possible
future for scalar-tensor theories.

Since this is an essay, and not a survey article, we shall certainly not be
trying to discuss all possible ways in which my equations have been employed.
We just want to give the reader a feel for what has been going on during the
past 50 years. Consequently we shall not be able to mention everyone who has
made a valuable contribution to the theory’s development. Those whose work has
not been cited in this essay should not be chagrined, since you were not left
out because of malice, but because there are just so many of you.

##  II Horndeski Gravity: the formulation

I arrived at the University of Waterloo in June, 1970, when I was 22 years old
and had just completed my undergraduate work, receiving a B.Sc. in Engineering
Physics from Washington University of St.Louis. Prof. Lovelock was assigned to
be my supervisor, and at our first meeting it was clear that I needed to learn
tensor calculus. So he asked me to work through the first two chapters of
Eisenhart’s book "Riemannian Geometry" Eisenhart (1964). I did that in about
three weeks and afterwards visited Prof. Lovelock to find out what comes next.
He promptly went up to the blackboard in his office and wrote down the 5x5
generalised Kronecker delta, which is the determinant of a 5x5 matrix whose
entries are Kronecker deltas, and asked me what does that equal in a space of
four-dimensions. I had never seen that creature before in my life and really
had no idea of what it might equal. But then I quickly remembered something I
had learned in an algebra course I had taken about eight months earlier. That
course was taught by a post-doc from the University of Chicago, and he told
me, Greg, if someone ever writes something down on a blackboard and asks you
what it equals and you are not sure, you say 0 if you are in a vector space,
and the identity if you are in a group. How could he have been so prescient!
So I told Prof. Lovelock that the tensor in question equaled 0. He was
slightly taken aback by my reasonably quick response, and then went on to
explain why it equaled 0, as if he needed to convince me that he knew why it
vanished. He then went on to discuss how the generalised Kronecker delta could
be used to prove various things, such as the Cayley-Hamilton Theorem. Little
did I know at that time how important the generalised Kronecker delta would be
in my later studies.

The next thing Prof.Lovelock wanted me to learn was some general relativity.
To that end I was required to read and work through most of Adler, Bazin and
Schiffer’s (1965) book "An Introduction to General Relativity" Adler  _et al._
(1965). Once that was done I was ready to begin reading Prof. Rund’s papers on
tensorial concomitants Rund (1993, 1966), and Prof. Lovelock’s papers dealing
with the uniqueness of the Einstein field equations. While doing that I am
sure that Prof. Lovelock was looking for a topic that I could write my Masters
thesis on. That is when he came upon this paper of Bergmann Bergmann (1968)
dealing with the Brans-Dicke Theory Brans and Dicke (1961), and scalar-tensor
field equations. The equations of such theories were required to be derivable
from a variational principle in which locally the field variables were a
scalar function $\phi$ and the components of a Lorentzian metric tensor,
$g_{\alpha\beta}$. Bergmann claimed that in a four-dimensional space the most
general second-order scalar-tensor field equations derivable from a Lagrangian
of the form

$L=L(g_{\rm\alpha\beta};g_{\rm\alpha\beta,\gamma};g_{\rm\alpha\beta,\gamma\delta};\varphi;\varphi_{,\rm\gamma})$
(1)

can be obtained from the Lagrangian

$L_{\rm B}:=\sqrt{-g}\left(f_{1}R+f_{2}X+f_{3}\right)$ (2)

where $f_{1},f_{2},f_{3}$ are arbitrary scalar functions of $\varphi$, a comma
denotes a partial derivative with respect to local coordinates,
$g:=det(g_{\alpha\beta})$, $R:=g^{\alpha\beta}R_{\alpha\beta}$,
$R_{\alpha\beta}:=R_{\alpha\,\,\,\beta\gamma}^{\,\,\,\gamma}$,
$R_{\alpha\,\,\,\beta\gamma}^{\,\,\,\iota}:=\Gamma^{\iota}_{\alpha\beta,\gamma}-\Gamma^{\iota}_{\alpha\gamma,\beta}+\Gamma^{\mu}_{\alpha\beta}\Gamma^{\iota}_{\mu\gamma}-\Gamma^{\mu}_{\alpha\gamma}\Gamma^{\iota}_{\mu\beta}$,
$\Gamma^{\iota}_{\alpha\beta}:=1/2\,g^{\iota\mu}(g_{\alpha\mu,\beta}+g_{\mu\beta,\alpha}-g_{\alpha\beta,\mu}),g^{\iota\mu}$
is the matrix inverse of $g_{\iota\mu}$, and
$X:=-1/2\,g^{\alpha\beta}\phi_{,\alpha}\phi_{,\beta}$. Clearly this result is
erroneous since we can replace $f_{2}X+f_{3}$ in $L_{B}$ by a more general
arbitrary function $F=F(\varphi,X)$ and still get second-order field
equations. In my Master’s thesis (1971) I developed, using the aforementioned
work of Rund, the machinery necessary to formally study Lagrangians of the
form presented in Eq.1. However the bulk of my thesis dealt with the Brans-
Dicke theory, which is derivable from the Lagrangian

$L_{\rm BD}:=\sqrt{-g}\left(\varphi R+2\omega X/\varphi\right)$ (3)

with $\omega$ being a constant. After finishing that thesis in early 1971,
Prof. Lovelock and I turned our attention to proving that in a Lorentzian
space of four-dimensions the most general Lagrangian of the form given in Eq.1
that yields second-order scalar-tensor field equations is given by

$\displaystyle L=$
$\displaystyle\sqrt{-g}\left\\{\beta_{1}\left[R^{2}-4R_{\rm\kappa\lambda}R^{\rm\kappa\lambda}+R_{\rm\kappa\lambda\mu\nu}R^{\rm\kappa\lambda\mu\nu}\right]+\beta_{2}G^{\rm\mu\nu}\varphi_{\rm,\mu}\varphi_{\rm,\nu}+\beta_{3}R+\eta\right\\}$
(4)
$\displaystyle+\alpha\epsilon^{\rm\kappa\lambda\mu\nu}R_{\,\,\,\,\,\,\rm\kappa\lambda}^{\rm\theta\iota}R_{\rm\theta\iota\mu\nu}$

where $\alpha$ is a real constant, $\beta_{1},\ \beta_{2},\ \beta_{3}$ are
functions of $\phi$ and $\eta=\eta(\phi\ ,X)$.

Our proof of the validity of Eq.4 appeared in Horndeski and Lovelock Horndeski
and Lovelock (1972) (which was a special issue of the Journal of the Tensor
Society celebrating the 70th birthday of the renowned Japanese Mathematician
A. Kawaguchi) and was a non-trivial generalization of Lovelock’s papers
Lovelock (1969, 1970). There he established (among other things) that:

(i) In a space of four-dimensions the most general Lagrangian of the form
$L=L(g_{\alpha\beta};g_{\alpha\beta,\mu};g_{\alpha\beta,\mu\nu})$ which yields
second-order field equations is

$L_{\rm L}:=\sqrt{-g}\left\\{\lambda
R+2\Lambda+\beta\left[R^{2}-4R_{\rm\kappa\lambda}R^{\rm\kappa\lambda}+R_{\rm\kappa\lambda\mu\nu}R^{\rm\kappa\lambda\mu\nu}\right]\right\\}+\alpha\epsilon^{\rm\kappa\lambda\mu\nu}R_{\,\,\,\,\,\,\rm\kappa\lambda}^{\rm\theta\iota}R_{\rm\theta\iota\mu\nu}$
(5)

where $\alpha$, $\beta$, $\lambda$ and $\Lambda$ are constants; and

(ii) In a space of four-dimensions the most general symmetric, contravariant
tensor density
$A^{\alpha\beta}=A^{\alpha\beta}(g_{\gamma\theta};g_{\gamma\theta,\mu};g_{\gamma\theta,\mu\nu})$
which is divergence-free, is given by

$A^{\rm\alpha\beta}=\sqrt{-g}\left\\{-\lambda G^{\rm\alpha\beta}+\Lambda
g^{\rm\alpha\beta}\right\\}\,,$ (6)

In fact $A^{\alpha\beta}$ is the variational derivative of $L_{L}$.

In passing I would like to point out that in Eq.5 the $\beta$ Lagrangian is
proportional to the Gauss-Bonnet Lagrangian, and the $\alpha$ Lagrangian is
proportional to the Pontryjagin Lagrangian. Both of these Lagrangians have
identically vanishing Euler-Lagrange tensors.

In view of my work in my Master’s thesis and the paper I wrote with Prof.
Lovelock; Prof. Rund (who was the chairman of the Applied Math Department at
the University of Waterloo at the time), and Prof. Lovelock decided that I was
qualified to enter the Ph.D. program in 1971. It was also quite clear what the
task of my thesis would be; $viz.,$ to construct in a space of four-dimensions
the most general second-order scalar-tensor field equations that can be
obtained from a variational principle using a Lagrangian of arbitrary
differential order in the derivatives of $\phi$ and $g_{\alpha\beta}$. The
approach I took in solving this problem was modeled on Prof. Lovelock’s
technique to establish (ii) above, and goes as follows.

If $L$ is a Lagrangian of the form

$L=L(g_{\rm\alpha\beta};g_{\rm\alpha\beta,\gamma};g_{\rm\alpha\beta,\gamma\delta};\cdots;\varphi;\varphi_{,\rm\gamma};\cdots)$
(7)

(where the derivatives of $g_{\rm\alpha\beta}$ and $\varphi$ stop at some
point) then the Euler-Lagrange tensor densities

$E^{\rm\alpha\beta}(L)\equiv\frac{\delta L}{\delta
g_{\rm\alpha\beta}}:=\frac{\partial L}{\partial
g_{\rm\alpha\beta}}-\frac{d\,\,}{dx^{\rm\gamma}}\frac{\partial L}{\partial
g_{\rm\alpha\beta,\gamma}}+\cdots$ (8)

and

$E(L)\equiv\frac{\delta L}{\delta\varphi}:=\frac{\partial
L}{\partial\varphi}-\frac{d\,\,}{dx^{\rm\gamma}}\frac{\partial
L}{\partial\varphi_{,\rm\gamma}}+\cdots$ (9)

are related by

$E^{\rm\alpha\beta}(L)_{|_{\beta}}=\frac{1}{2}\varphi^{\rm\alpha}E(L)\ $ (10)

where the vertical bar, | denotes covariant differentiation. (The proof of Eq.10 is given in Horndeski (1973), but those who use infinitesimal variations can derive it from the invariance of the Lagrangian under the coordinate transformation that replaces $x^{\iota}$, by $x^{\iota}+\xi^{\iota}.$) Eq.10 shows us that if the field tensors are of second-order then the divergence of $E^{\alpha\beta}(L)$ must also be of second-order, and not of third order. This leads us to consider the following problem: In a space of four-dimensions construct all second-order symmetric (0,2) tensor densities

$A^{\rm\alpha\beta}=A^{\rm\alpha\beta}(g_{\rm\gamma\delta};g_{\rm\gamma\delta,\mu};g_{\rm\gamma\delta,\mu\nu};\varphi;\varphi_{,\rm\mu};\varphi_{,\rm\mu\nu})$
(11)

which are such that there exists a second-order scalar-tensor concomitant A
with the property that

$A^{\rm\alpha\beta}_{\,\,\,\,\,\,|\rm\beta}=\varphi^{\rm\alpha}A\,.$ (12)

So not only are we requiring the divergence of $A^{\alpha\beta}$ to be of
second-order, but it must be parallel to $\phi^{\alpha}$.

By generalizing the techniques used in Lovelock (1970) I was able to
construct, using numerous generalized Kronecker deltas, all of the tensorial
concomitants $A^{\alpha\beta}$ and A satisfying Eq.11 and Eq.12. At that point
we know that all of the field tensors for a second-order scalar-tensor theory
will be contained somewhere in $A^{\alpha\beta}$ and A, but we are not sure if
there exists a Lagrangian that will yield $A^{\alpha\beta}$ itself as its
Euler-Lagrange tensor. Well, when Lovelock was confronted with a similar
problem he just said, why not try the scalar density
$g_{\alpha\beta}A^{\alpha\beta}$ as a possible Lagrangian. So that is what I
did and with some effort it worked! (Why it works will be explained by me in a
future publication.) In this way I was able to come up with a Lagrangian and
an algorithm that would show how to go from any $A^{\alpha\beta}$ and A that
satisfies Eq.11 and Eq.12, to a Lagrangian L, which was such that
$E^{\alpha\beta}(L)=A^{\alpha\beta}$. This result first appeared in print in
Horndeski (1974). In terms of the notation currently used (with $\nabla$ now
denoting covariant differentiation), my original Lagrangian yielding the most
general second-order scalar-tensor field equations in a four-dimensional space
is expressible as

$\displaystyle\mathcal{L}_{H}$ $\displaystyle=$
$\displaystyle\sqrt{-g}\Big{\\{}\delta^{\alpha\beta\gamma}_{\mu\nu\sigma}\Big{[}\kappa_{1}\nabla^{\mu}\nabla_{\alpha}\phi
R_{\beta\gamma}^{\,\,\,\,\,\,\nu\sigma}+\frac{2}{3}\kappa_{1X}\nabla^{\mu}\nabla_{\alpha}\phi\nabla^{\nu}\nabla_{\beta}\phi\nabla^{\sigma}\nabla_{\gamma}\phi+\kappa_{3}\nabla_{\alpha}\phi\nabla^{\mu}\phi
R_{\beta\gamma}^{\,\,\,\,\,\,\nu\sigma}$ (13) $\displaystyle+$ $\displaystyle
2\kappa_{3X}\nabla_{\alpha}\phi\nabla^{\mu}\phi\nabla^{\nu}\nabla_{\beta}\phi\nabla^{\sigma}\nabla_{\gamma}\phi\Big{]}+\delta_{\mu\nu}^{\alpha\beta}\Big{[}(F+2W)R_{\alpha\beta}^{\,\,\,\,\,\,\mu\nu}+2F_{X}\nabla^{\mu}\nabla_{\alpha}\phi\nabla^{\nu}\nabla_{\beta}\phi$
$\displaystyle+$ $\displaystyle
2\kappa_{8}\nabla_{\alpha}\phi\nabla^{\mu}\phi\nabla^{\nu}\nabla_{\beta}\phi\Big{]}-6(F_{\phi}+2W_{\phi}-X\kappa_{8})\Box\phi+\kappa_{9}\Big{\\}},$

where
$\delta^{\alpha_{1}\alpha_{2}\dots\alpha_{n}}_{\mu_{1}\mu_{2}\dots\mu_{n}}=n!\delta^{[\alpha_{1}}_{\mu_{1}}\delta^{\alpha_{2}}_{\mu_{2}}\dots\delta^{\alpha_{n}]}_{\mu_{n}}$
and $\kappa_{1},\kappa_{3},\kappa_{8},\kappa_{9}$ are arbitrary functions of
$\phi$ and $X$. $F=F(\phi,X)$ is determined up to a quadrature by the equation
$F_{X}=2(\kappa_{3}+2X\kappa_{3X}-\kappa_{1\phi})$, with $W=W(\phi)$
essentially being an integration "constant" that arises when computing F.
$\phi$ and $X$ at the subscript of the functions indicate derivation of the
latter with respect to the scalar field and its kinetic term, respectively. In
Section 3 it will be explained how one goes from the above Lagrangian, to the
one currently favored.

Upon comparing the Lagrangian ${\mathcal{L}}_{H}$ with the one presented in
Eq.4, one can’t help but wonder why there isn’t a term in ${\mathcal{L}}_{H}$
involving a Lagrangian that is quadratic in the curvature tensor. The answer
is that the quadratic Gauss-Bonnet type Lagrangian can actually be built from
the Lagrangians in Eq.13 (see Appendix G in Horndeski (1973)), and the
Pontryagin Lagrangian has an identically vanishing Euler-Lagrange tensor, and
has been dismissed.

It is interesting to note that if one demands that the field equations
derivable from the Lagrangian presented in Eq.13 are quasi-linear in the
second derivatives of the field variables; _i.e._ , the coefficients of
$g_{\alpha\beta,\gamma\delta}$ and $\varphi_{,\gamma\delta}$ in the field
equations are at most functions of $g_{\alpha\beta}$ and $\phi$, then (as I
point out in Horndeski (1974)) we recover the Bergmann Lagrangian given in
Eq.2, up to a divergence. It is the quasi-linearity of the second-order
scalar-tensor field equations which uniquely characterizes Bergmann’s
Lagrangian, up to a divergence.

When Prof. Lovelock and I saw how complex the Lagrangian which yields the most
general second-order scalar-tensor field equations in a space of four-
dimensions were, we felt that clearly puts the kibosh on scalar-tensor field
theories. There were just too many of them, and they are way too complicated.
We wondered who would be crazy enough to work with such equations. Then crazy
showed up! And is still here today. It will be the task of my colleague, Prof.
Silvestri to explain why these equations became so useful, after languishing
as mere mathematical curiosities for over thirty years, before being
rediscovered by Charmousis, et al Charmousis  _et al._ (2012).

##  III Horndeski Gravity in Cosmology

The discovery of cosmic acceleration in 1998 Riess  _et al._ (1998);
Perlmutter  _et al._ (1999) provided a new boost for the exploration of
alternatives to general relativity (GR) in the description of gravity, this
time in the cosmological context. In fact, a phase of accelerated expansion
consistent with the observations, within a Universe governed by GR, requires a
mysterious source which permeates space in an everlasting manner. In other
words, a component with pressure equal to minus its energy density, i.e.
characterized by an equation of state $w\sim-1$111The equation of state for a
given species is defined as the ratio between the pressure and energy density
of that species, i.e. $w\equiv P/\rho$.. The constituents we are familiar
with, i.e. matter in the form of dust and radiation, are characterised by
$w=0$ and $w=1/3$, respectively and cannot source an accelerated expansion.
The vacuum energy of quantum fields is an obvious, and perhaps inescapable,
candidate which shall contribute a constant energy density to the budget of
the Universe, of the order $\Lambda_{\rm th}\sim 10^{-60}-1\,{\rm M}_{P}^{4}$
depending on the ultraviolet cutoff on the theory Weinberg (2000); Joyce  _et
al._ (2015). Measurements of the expansion rate of the Universe give
$\Lambda_{\rm obs}\sim 10^{-120}\,{\rm M}_{P}^{4}$ which is $15-30$ orders of
magnitude smaller, in mass scale. This would require an unnatural fine tuning
of the bare constant in the action, $\Lambda_{0}$, so to cancel almost
perfectly the huge vacuum energy of quantum fields, leaving a very small
contribution that matches $\Lambda_{\rm obs}$. This discrepancy, known as the
_cosmological constant problem_ , has prompted the exploration of dynamical
scenarios that would alleviate the fine tuning by employing a _dark energy_
(DE) field Copeland  _et al._ (2006), and of modifications of the laws of
gravity (MG) that would provide self-accelerating solutions in absence of any
matter (Clifton  _et al._ , 2012) and, ideally, degravitate the vacuum energy
Dvali  _et al._ (2007). This activity intensified through the years, leading
to an intricate landscape of candidate models of gravity and DE. We refer the
reader to (Clifton  _et al._ , 2012) for a comprehensive review of the latter.

While GR has been confirmed to great accuracy in the laboratory and in the
Solar System Will (2014), and, more recently, with the direct detection of
gravitational waves Abbott  _et al._ (2016, 2017a) and the imaging of black
holes Akiyama  _et al._ (2019), none of these tests probe gravity on
cosmological scales, where it is characterised by the Hubble expansion of the
Universe. This is a completely new regime, corresponding to vastly different
lengthscales and curvatures, and the diffuse presence of matter. We shall
approach it with the same level of scrutiny achieved in local tests of
gravity. From the observational point of view, this will become a concrete
possibility with the advent of Stage IV Large Scale Structure (LSS) surveys,
such as Euclid and the Vera C. Rubin Observatory’s Legacy Survey. In
anticipation of this endeavour, on the theoretical side it has become
gradually clear that we cannot proceed on a model basis, but we rather need
unifying frameworks for a fundamentally driven exploration of gravity. In
fact, a valuable lesson learned from local tests of gravity, is that
frameworks such as the Parametrized Post Newtonian Will (1971); Will and
Nordtvedt (1972) are crucial to carry out a comprehensive and conclusive
analysis. It is in this context, that Horndeski gravity became very popular in
Cosmology, but let us proceed gradually, starting with the intricate
gravitational landscape that emerged after the discovery of cosmic
acceleration.

###  III.1 Modifying Gravity

According to Weinberg-Deser and Lovelock theorems, GR is the unique, Lorentz
invariant, low-energy theory of an interacting, massless helicity-2 field in a
4-dimensional space-time. These theorems help us identify and, to a certain
extent, classify the different approaches to beyond GR. The simplest choice is
that of introducing an additional scalar field: inspired by inflationary
models and the theory of Brans-Dicke Brans and Dicke (1961), the first models
to be explored were extensions of the standard cosmological scenario with the
introduction of a dynamical scalar field, _quintessence_ , minimally or non-
minimally coupled to gravity Copeland  _et al._ (2006). There followed
k-essence models, in which the field had a non standard (non-linear) kinetic
term to aid tracker solutions that would avoid fine-tuning Armendariz-Picon
_et al._ (2000, 2001). On the MG side, models were built via the inclusion of
non-linear functions of _Lovelock scalar invariants_ 222These scalars are
special combinations and contractions of the Riemann tensor which, if present
in the Lagrangian, only introduce second-order derivative contributions to the
equations of motion. Lovelock (1971) in the action for gravity, the most
famous example being $f(R)$ gravity De Felice and Tsujikawa (2010). A
Lagrangian written in terms of linear and non-linear functions of these
scalars propagate only additional scalar degrees of freedom (DOFs), and does
not excite extra, ghost-like spin-2 fields. This property makes models with
Lovelock scalars interesting candidate models for inflation and cosmic
acceleration. The addition of a non-linear function of the Ricci scalar,
$f(R)$, frees up one scalar DOF, the addition of $f(R,\mathcal{G})$, with
$\mathcal{G}$ the Gauss-Bonnet term, frees up two additional scalar DOFs De
Felice and Tsujikawa (2011), and so on.

The theories that we mentioned thus far, are very representative of the models
explored in the context of cosmic acceleration, but they certainly do not
exhaust the ways in which one can go beyond the assumptions at the basis of
the Weinberg-Deser/Lovelock theorems. One can, for instance, work in higher-
dimensional space-times, give mass to the graviton or break Lorentz-
invariance, as thoroughly reviewed in (Clifton  _et al._ , 2012). Here we
shall mention one of the earliest braneworld theories considered, the five-
dimensional Dvali-Gabadadze-Porrati (DGP) model Dvali  _et al._ (2000), where
all matter belong to a four-dimensional brane embedded in a 5-dimensional bulk
where gravity lives. We shall return to this theory shortly.

While models were being formulated, the community started to explore their
predictions not only for the expansion history of the Universe, but also for
the dynamics of scalar cosmological perturbations that bring to the formation
of LSS. As a result, we gained key insights into the prospect of testing
gravity with LSS, identifying the optimal probe combinations to use towards
this goal Zhao  _et al._ (2009); Amendola  _et al._ (2013); Sapone  _et al._
(2009); Harnois-Déraps  _et al._ (2015); Alonso  _et al._ (2017); Leonard  _et
al._ (2015); Baker  _et al._ (2011). f(R) gravity and the DGP model were among
the first to be thoroughly explored Song  _et al._ (2007); Pogosian and
Silvestri (2008); Hu and Sawicki (2007); Charmousis  _et al._ (2006); Song
(2008); Cardoso  _et al._ (2008); Lombriser  _et al._ (2009); Seahra and Hu
(2010); Xu (2014); Schmidt (2009); Bag  _et al._ (2018); Schmidt  _et al._
(2010). It quickly emerged that the cosmological phenomenology of DE/MG models
is significantly richer than the one in the standard cosmological scenario,
$\Lambda$CDM (which relies on GR to describe gravity, and includes Cold Dark
Matter and the cosmological constant $\Lambda$ as ingredients, along with
ordinary matter and radiation). In the latter case, the expansion at late
times is driven by the cosmological constant, with equation of state $w=-1$;
once inside the cosmological horizon, inhomogeneities in the density of matter
grow at the same rate on all scales; correspondingly, structure grows with a
rate set uniquely by the Hubble expansion; weak gravitational lensing (i.e.
the bending of light from background galaxies by foreground LSS) proceeds in a
way consistent with clustering, with relativistic and non-relativistic
particles following the same geodesics. These relations are generally broken
in models of DE and MG: the dynamics of the background can be characterized by
a $w\neq-1$, and more generally an evolving $w(z)$; the growth of structure is
not uniquely set by the expansion rate, and can proceed in a scale-dependent
way; lensing and clustering are not necessarily equivalent. These are all
consequences of the fact that a new degree of freedom, typically a scalar
field, has been introduced (or activated) in the theory; this field generally
introduces a characteristic length scale and mediates an extra force, _fifth
force_ , between matter particles.

These realizations, gradually inspired the formulation of a phenomenological
framework for cosmological tests of gravity in terms of the following
functions: the already popular equation of state of DE, $w(z)$, to parametrize
any deviation at the level of background dynamics; along with $\mu(k,z)$ and
$\Sigma(k,z)$ 333Here $z$ is the redshift, and $k$ represents the wavenumber
associated to a given scale in Fourier space., functions of time and scale, to
capture deviations from $\Lambda$CDM in the dynamics of cosmological
perturbations. Specifically, $\mu$ encodes modifications in the clustering,
while $\Sigma$ encodes modifications in the lensing, and together with $w(z)$
they allow to define a complete set of equations to study the dynamics of
scalar perturbations on linear scales Amendola  _et al._ (2008); Bertschinger
and Zukin (2008); Pogosian  _et al._ (2010). Anything that observations can
tell us about the growth of structure can be stored into $\mu$ and $\Sigma$,
and indeed this framework has been widely adopted by different collaborations,
e.g. Planck Ade  _et al._ (2016) and the Dark Energy Survey Abbott  _et al._
(2023), to place constraints on departures from $\Lambda$CDM with data from
the cosmic microwave background and LSS, respectively. Yet, there are certain
shortcomings: it is very agnostic about the underlying theory (e.g. there is
no guarantee that any functional form of these functions will correspond to a
viable theory Peirone  _et al._ (2017)) and it is restricted to scalar
perturbations, thus overlooking the dynamics of tensor perturbations and
missing out on the complementary information contributed by gravitational
waves. These limitations can be overcome if we use Horndeski gravity to inform
theoretically $w(z),\mu(k,z)$ and $\Sigma(k,z)$, as we shall discuss later.
Alternative frameworks, such as the Parametrized Post-Friedmann one Baker  _et
al._ (2011, 2013), have also been explored.

Meanwhile, the exploration of theoretical scenarios continued. Studying the
decoupling limit of the braneworld DGP model, Nicolis et al. Nicolis  _et al._
(2009) realized that the scalar field corresponding to the bending mode of the
brane obeyed the Galilean shift symmetry, inherited from the invariance under
higher-dimensional Lorentz transformations:
$\partial_{\mu}\phi\rightarrow\partial_{\mu}\phi+b_{\mu}$, where $b_{\mu}$ are
constants. The corresponding Lagrangian contained derivative coupling terms of
cubic order in the scalar field, e.g.
$\Box\phi\,\partial\phi\cdot\partial\phi$, but still led to second-order
equations of motion. Models of massive gravity in four dimensions lead to
analogous terms in the decoupling limit, but at quartic and quintic order in
the scalar field; see the dRGT theory de Rham  _et al._ (2011) for a ghost-
free massive gravity model which produces such terms. Interestingly, requiring
a theory for a scalar field to be Galilean invariant and to have equations of
motions at most of 2nd order, identifies a finite number of terms that can
enter the corresponding Lagrangian. Very much like for Lovelock gravity! In
this case, we have $D+1$ Galileon terms for a Lagrangian in $D$ dimensions,
following the general structure
$\partial\phi\cdot\partial\phi(\partial^{2}\phi)^{n-2}$ with $n=1,\dots,D+1$.
Expressions of order $n>D+1$, while still being Galilean invariant, correspond
to surface terms that do not contribute to the equations of motion in $D$
dimensions. All this held in flat, Minkowski space. If we extend the analysis
to curved space-time, the Lagrangians need to be promoted to a covariant form.
In 2009, Deffayet et al. Deffayet  _et al._ (2009) derived the corresponding
covariant Lagrangians that keep the field equations of second-order, while
generally breaking the Galilean symmetry (albeit, in most cases, softly). The
result is the Covariant Galileon action:

$\displaystyle S_{CG}$ $\displaystyle=$
$\displaystyle\int{}d^{4}x\sqrt{-g}\left\\{\frac{M_{P}^{2}}{2}R-\frac{1}{2}c_{1}M^{3}\phi+c_{2}X+\frac{2c_{3}}{M^{3}}X\Box\phi+\frac{c_{4}}{M^{6}}X^{2}R+\frac{2c_{4}}{M^{6}}X\left[(\Box\phi)^{2}-\phi^{\mu\nu}\phi_{\mu\nu}\right]\right.$
(14)
$\displaystyle\left.-\frac{3c_{5}}{M^{9}}X^{2}G_{\mu\nu}\phi^{\mu\nu}+\frac{c_{5}}{M^{9}}X\left[(\Box\phi)^{3}-3\Box\phi\,\phi^{\mu\nu}\phi_{\mu\nu}+2\phi^{\mu\nu}\phi_{\mu\sigma}\phi_{\sigma}^{\nu}\right]\right\\}\,,$

where $M_{P}^{2}$ is the Planck mass, $X$ is the kinetic term of the scalar
field, $X\equiv-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$, greek
subscripts and superscripts indicate covariant derivatives, the mass scale $M$
is related to the horizon scale by $M^{3}=M_{P}H_{0}^{2}$ and
$c_{2},c_{3},c_{4},c_{5}$ are dimensionless constants. We can use the latter
to identify 5 blocks inside the Lagrangian, each one separately contributing
second-order equations of motion. In particular, the non-minimal couplings,
$X^{2}R$ and $X^{2}G_{\mu\nu}\phi^{\mu\nu}$, and the specific linear
combinations of operators in the $c_{4}$ and $c_{5}$ terms, respectively, are
necessary in order to avoid higher derivatives in the equations of motion.

With the breaking of the Galilean symmetry, more freedom is available in
setting the coefficients of each $n$-th order term in the Lagrangian, while
still maintaining second-order equations of motion. The most general scalar-
tensor field theory with an action that depends on derivatives of order two or
less of the fields, and results in second-order equations of motion in a
general curved space-time was identified in Deffayet  _et al._ (2011) and is
known as Generalized Galileons:

$S_{\rm GG}=\int d^{4}x\sqrt{-g}\sum_{i=2}^{5}{\cal L}_{i}\ ,$ (15)

with

$\displaystyle{\cal L}_{2}$ $\displaystyle=$ $\displaystyle G_{2}(\phi,X),$
$\displaystyle{\cal L}_{3}$ $\displaystyle=$ $\displaystyle-
G_{3}(\phi,X)\Box\phi,$ $\displaystyle{\cal L}_{4}$ $\displaystyle=$
$\displaystyle
G_{4}(\phi,X)\,R+G_{4X}\,[(\Box\phi)^{2}-(\nabla_{\mu}\nabla_{\nu}\phi)\,(\nabla^{\mu}\nabla^{\nu}\phi)]\,,$
$\displaystyle{\cal L}_{5}$ $\displaystyle=$ $\displaystyle
G_{5}(\phi,X)\,G_{\mu\nu}\,(\nabla^{\mu}\nabla^{\nu}\phi)-\frac{1}{6}\,G_{5X}\,[(\Box\phi)^{3}-3(\Box\phi)\,(\nabla_{\mu}\nabla_{\nu}\phi)\,(\nabla^{\mu}\nabla^{\nu}\phi)$
(16) $\displaystyle+$ $\displaystyle
2(\nabla^{\mu}\nabla_{\alpha}\phi)\,(\nabla^{\alpha}\nabla_{\beta}\phi)\,(\nabla^{\beta}\nabla_{\mu}\phi)]\
,$

where $G_{i}(\phi,X)$ are free functions of the scalar field and its kinetic
term, and $G_{iX}=\partial G_{i}/\partial X$. Theories are referred to as
cubic, quartic and quintic depending on whether $G_{3},G_{4},G_{5}$ are
included, respectively. Models for which $\partial G_{i}/\partial\phi=0$ are
commonly referred to as shift-symmetric, due to their invariance under
$\phi\rightarrow\phi+c$.

Galileon models gained significant interest in Cosmology because they allow
for self accelerating solutions that could describe both the inflationary
epoch and the late time accelerated expansion Kobayashi  _et al._ (2011); De
Felice and Tsujikawa (2011); Chow and Khoury (2009). In fact, action (III.1)
includes as subcases most of the models previously explored in the context of
inflation and cosmic acceleration, including those mentioned at the beginning
of this subsection. It reduces to the Covariant Galileons when
$G_{2}=-\frac{c_{1}}{2}M^{3}\phi+c_{2}X,G_{3}=-\frac{2c_{3}}{M^{3}}X,G_{4}=\frac{M_{P}^{2}}{2}+\frac{c_{4}}{M^{6}}X^{2},G_{5}=-\frac{3c_{5}}{M^{9}}X^{2}$.

Galileons have the interesting property that the $(n+1)$-th galileon
Lagrangian corresponds to $(\partial\phi)^{2}$ times the equation of motion of
the $n$-th galileon Lagrangian. This property is referred to as _Euler
hierarchy_ Fairlie and Govaerts (1992a); Fairlie  _et al._ (1992); Fairlie and
Govaerts (1992b). Another property of Generalized Galileons is non-
renormalization, i.e. the fact that they do not receive quantum corrections at
any order in perturbation theory Luty  _et al._ (2003); Hinterbichler  _et
al._ (2010).

###  III.2 The re-emergence of Horndeski Gravity

In 2011 Charmousis et al. Charmousis  _et al._ (2012) revisited Horndeski
gravity, with the goal of studying it on a FLRW background and looking for
subclasses of it that would incorporate a viable self-tuning mechanism as an
approach to the cosmological constant problem. Their paper brought Horndeski
gravity back into light, making it known to the cosmological community. It
also highlighted how most of the candidate scalar-tensor models of DE and MG
were special cases of it. It was at that point that Kobayashi and
collaborators, who had just studied the cosmology of Generalized Galileons in
(Kobayashi  _et al._ , 2011), noticed the equivalence between the latter, as
formulated in Deffayet  _et al._ (2011), and Horndeski gravity. They added a
proof of this equivalence in an Appendix to their original 2011 paper
(Kobayashi  _et al._ , 2011), where they showed that upon identifying:

$\displaystyle G_{2}$ $\displaystyle=$
$\displaystyle\kappa_{9}+4X\int^{X}dX^{\prime}(\kappa_{8\phi}-2\kappa_{3\phi\phi}),$
$\displaystyle G_{3}$ $\displaystyle=$ $\displaystyle
6F_{\phi}-2X\kappa_{8}-8X\kappa_{3\phi}+2\int^{X}dX^{\prime}(\kappa_{8}-2\kappa_{3\phi})$
$\displaystyle G_{4}$ $\displaystyle=$ $\displaystyle 2F--4X\kappa_{3}$
$\displaystyle G_{5}$ $\displaystyle=$ $\displaystyle-4\kappa_{1}$

and performing integration by parts, action (15) becomes exactly the original
Horndeski action of (13). A painting by Gregory Horndeski, depicting the
equations of Horndeski gravity is presented in Fig. 1. More will be said about
Fig. 1 below. Note that in this figure $\rho=-2X$.

Figure 1: "Horndeski Scalar Theory, Past, Present and Future." (2016),
painting by Gregory Horndeski depicting the equations of Horndeski gravity.

###  III.3 Screening mechanisms

Local tests of gravity have provided a spectacular confirmation of GR in the
Solar System and around compact objects. Any alternative theory that
introduces additional DOFs must incorporate a _screening mechanism_ to ’hide’
them in dense regions, effectively reducing to GR. There is only a handful of
screening mechanisms and they all rely on the scalar field (i.e. the extra DOF
in most DE/MG models, and certainly in those relevant for this review)
developing non-linearities in certain environments. We can organize them in
three broad classes: those which become active in regions of high Newtonian
potential, known as Chameleon and symmetron, typical of Generalized Brans-
Dicke models Khoury and Weltman (2004); mechanisms in which first derivatives
of the field become important in dense regions, typical of k-mouflage models
Babichev  _et al._ (2009); and the Vainshtein screening Vainshtein (1972),
where second derivatives of the field become dominant in dense regions,
hallmark of the pure Galileon models. Horndeski gravity contains all these
three types of mechanisms: non-minimal couplings like the one introduced by a
$G_{4}(\phi)$, source the Chameleon; non-standard kinetic terms that still
involve only first derivatives of the field, e.g. non-linear choices for
$G_{2}(X)$, source the kinetic screening; higher-derivative couplings that
enter the cubic, quartic and quintic Lagrangians are at the heart of the
Vainshtein mechanism. Which mechanism will be at play for a given Horndeski
model, depends on the corresponding $G_{i}$ functions, and potentially also on
the environment, as discussed in Gratia  _et al._ (2016).

Screening mechanisms produce a rich phenomenology, such as macroscopic
violations of the equivalence principle Hui  _et al._ (2009); Sakstein  _et
al._ (2017a); Bartlett  _et al._ (2021), modifications to stellar structure
Chang and Hui (2011) and pulsations Sakstein  _et al._ (2017b) and to the
galaxy morphology Vikram  _et al._ (2018); Desmond and Ferreira (2020), and
much more. For a complete review of the extensive literature on this, we refer
the reader to Jain  _et al._ (2013); Brax (2021); Sakstein (2020). Screening
is crucial to model when studying growth of structure on smaller, non-linear
scales, since it can modify the collapse mechanism and hence the halo
formation model. Interestingly, it can leave characteristic imprints on the
position of the splashback radius in clusters Adhikari  _et al._ (2018);
Contigiani  _et al._ (2019). Most studies to date have focused on screening
around spherically symmetric, static (or slowly rotating) sources, however it
is known that the efficiency of the mechanism depends on the symmetry of the
source. This feature has been exploited towards powerful laboratory tests of
Chameleon models Brax  _et al._ (2011); Burrage (2019). One shall also expect
to see some impact of this when upgrading the modeling of the collapse in
structure formation from spherical to ellipsoidal. Finally, screening is
expected to produce several interesting features around sources which have
some dynamics, e.g. scalar gravitational radiation Dar  _et al._ (2019);
Silvestri (2011)

###  III.4 Gravitational Waves propagation in a Horndeski Universe

When modifying the action for gravity in the cosmological context, one needs
also to take into account possible effects on the propagation of gravitational
waves (GWs). Even more so now, that we are witnessing the dawn of the GW
Astronomy era, started with the historical first direct detection of a GW from
a binary black hole merger in 2015 Abbott  _et al._ (2016). Generalized
Galileon models can impact the propagation of GWs in two ways: modifying their
speed and contributing an additional friction term in their propagation
equation. The latter further damps the amplitude of the GW, thus affecting the
luminosity distance inferred from the GW amplitude, as follows:

$\frac{d_{L}^{\rm GW}(z)}{d_{L}^{\rm
EM}(z)}=\frac{M_{*}(0)}{M_{*}(z)}\,,\,\,\,{\rm
where}\,\,\,\,\,M^{2}_{*}(z)\equiv
2\left[G_{4}-2XG_{4X}+XG_{5\phi}-\dot{\phi}HXG_{5X}\right]$ (18)

where $M_{*}^{2}$ corresponds to the running Planck mass of Generalized
Galileons, and is induced by the non-minimal coupling terms in the theory;
$d_{L}^{\rm EM}$ is the luminosity distance that would be associated to an
electromagnetic source at the same redshift of the GW. This opens the way to
interesting tests of gravity with standard sirens Amendola  _et al._ (2018);
Belgacem  _et al._ (2018, 2019).

The propagation speed instead receives the following contributions:

$c^{2}_{T}=1+\frac{X\left[2G_{4X}-2G_{5\phi}-\left(\ddot{\phi}-H\dot{\phi}\right)G_{5X}\right]}{G_{4}-2XG_{4X}+XG_{5\phi}-\dot{\phi}HXG_{5X}}\,.$
(19)

These originate from the non-linearities generated by higher order kinetic
energy terms, which modify the light-cone structure of GWs with respect to
that of photons. Simpler forms of non-minimal couplings, corresponding to
$G_{4}(\phi)$ and $G_{5}(\phi)$, do not affect the light-cone but contribute
to the damping term (18). Interestingly, all these components, which do not
exist for canonical scalar fields, generate a non-zero shear component of the
scalar field energy-momentum, which manifest as an effective anisotropic
stress in LSS, introducing a difference between lensing and clustering. For
these reasons, the latter is often considered a smoking gun signature of MG
since it is in close correspondence with a modified propagation of tensor
modes Sawicki  _et al._ (2017); Matos  _et al._ (2023).

2017 saw the first detection of a GW from a binary neutron star inspiral,
GW170817 Abbott  _et al._ (2017a) together with its electromagnetic
counterpart, GRB170817A Savchenko  _et al._ (2017); Abbott  _et al._ (2017b);
Goldstein  _et al._ (2017). This event allowed to place severe constraints on
the speed of tensors, setting it to be equal to that of light Abbott  _et al._
(2017c). As it was pointed out in a series of papers Baker  _et al._ (2017);
Creminelli and Vernizzi (2017); Ezquiaga and Zumalacárregui (2017); Langlois
_et al._ (2018), this constraint would effectively rule out a significant
portion of the Generalized Galileons, reducing the quartic lagrangian to
$G_{4}(\phi)R$ and eliminating the quintic one. However, it is important to
keep in mind that the source associated to this event is very close by in
cosmological terms, at a redshift of $z\sim 0.01$, while cosmological data
comes from higher redshifts; and, as pointed out in de Rham and Melville
(2018), its detection corresponds to energy scales that are close to the cut-
off scale at which low-energy actions like (15) become invalid. With all due
caveats, the first bright siren, GW170817, certainly delivered a strong
message on the importance of the synergy between LSS and GWs when exploring
gravity in the cosmological context.

###  III.5 Cosmological tests of gravity

Since its re-emergence in Cosmology, Horndeski gravity quickly became a
popular framework for cosmological tests of gravity and within a few years,
the word reached the original author, Gregory Horndeski, as he discusses in
the Conclusions of this essay. Shortly after, he created a painted depiction
of the equations of Horndeski gravity, Fig. 1, which was eventually purchased
by the Institute Lorentz for Theoretical Physics of Leiden University. As
shall be clear from the discussion in the previous subsections, Horndeski
gravity encompasses most of the candidate models of DE/MG explored in the
decades since the discovery of cosmic acceleration, providing a compact way of
organizing them within a unified action. This has the important advantage of
facilitating the theoretical and observational comparison of the different
scenarios.

What provided an additional boost, was the formulation of the effective field
theory of DE (EFTofDE) Bloomfield  _et al._ (2013); Gubitosi  _et al._ (2013);
Piazza and Vernizzi (2013); Frusciante and Perenon (2020), inspired by the EFT
of inflation Cheung  _et al._ (2008) and of quintessence Creminelli  _et al._
(2009). EFTofDE builds on fundamental symmetry arguments, treating
cosmological perturbations as the Nambu-Goldstone modes associated to a
cosmological state that spontaneously breaks time-translations, as is the case
with any model that sources cosmic acceleration in a dynamical way. The
formalism identifies a set of Lagrangian operators containing an increasing
number of cosmological perturbations and derivatives acting on them. In its
original formulation the EFTofDE identifies the most general Lagrangian,
quadratic in perturbations around the Friedmann-Lemaitre-Robertson-Walker
(FLRW) Universe, which propagates a massless tensor and a scalar field with
second-order equations of motion. In other words, the quadratic action of
Horndeski gravity! Thus providing Horndeski gravity with a new interpretation:
the low-energy effective action of gravity on the large scales that
characterize the expanding FLRW Universe, further confirming that it
represents an ideal framework for exploring gravity with LSS!

The EFTofDE action, in conformal time, is written in terms of a handful of
operators:

$\displaystyle\mathcal{S}_{\rm DE}=$ $\displaystyle\int
d^{4}x\sqrt{-g}\bigg{\\{}\frac{M_{P}^{2}}{2}(1+\Omega(\tau))R+\Lambda(\tau)-c(\tau)\,a^{2}\delta
g^{00}+\frac{M_{2}^{4}(\tau)}{2}\left(a^{2}\delta g^{00}\right)^{2}$
$\displaystyle-$ $\displaystyle\frac{\bar{M}_{1}^{3}(\tau)}{2}a^{2}\delta
g^{00}\,\delta{K}{{}^{\mu}_{\mu}}-\frac{\bar{M}_{2}^{2}(\tau)}{2}\left[\left(\delta{K}{{}^{\mu}_{\mu}}\right)^{2}-\delta{K}{{}^{\mu}_{\nu}}\,\delta{K}{{}^{\nu}_{\mu}}-\frac{a^{2}}{2}\delta
g^{00}\,\delta R^{(3)}\right]\bigg{\\}}\,,$ (20)

where $\Omega,\Lambda,c,M_{2}^{4},\bar{M}_{1}^{3},\bar{M}_{2}^{2}$ are free
functions of time; we have chosen a specific foliation of space-time by
identifying constant-time hypersurfaces with uniform field ones;
$K^{\mu}_{\nu}$ and $R^{(3)}$ are, respectively, the extrinsic curvature and
three dimensional spatial Ricci scalar of these hypersurfaces. As with all
previous actions, we are focusing on gravity and the scalar DOF, with the
understanding that matter fields, $\chi_{i}$ appear in an accompanying action
of the form $S_{m}=\int d^{4}x\sqrt{-g}\mathcal{L}_{m}[g_{\mu\nu},\chi_{i}]$.
Following the EFT prescription, one could include further operators in the
expansion in field and derivatives, extending to Beyond Horndeski. In fact,
there is a handful of operators that one can add to the action while
maintaining the resulting equations of motion of second-order for the
propagating DOFs, thanks to degeneracy properties of the kinetic matrix as it
was shown in (Gleyzes  _et al._ , 2015a, b) where the authors identified the
so-called GLPV theories. Healthy Beyond Horndeski theories can also be
identified by applying a disformal transformation Bekenstein (1993) to the
Horndeski Lagrangian Zumalacárregui and García-Bellido (2014). The broad class
of Degenerate Higher Order Scalar-Tensor theories, that encompasses also the
former cases, was introduced in Langlois and Noui (2016); Ben Achour  _et al._
(2016a); Motohashi  _et al._ (2016). See Traykova  _et al._ (2019); Hiramatsu
(2022); Sugiyama  _et al._ (2023) for some recent explorations of the
cosmological phenomenology of these models. For a more exhaustive overview of
Beyond Horndeski models, we refer the reader to Langlois (2019), and
references therein. Interestingly, some of the Beyond Horndeski terms in the
action can be constrained by considering the potential decay of GWs into DE, a
first discussed in Creminelli  _et al._ (2018, 2020).

Equation (20) is equivalent to the Horndeski action expanded to quadratic
order in perturbations, and in fact any theory belonging to the Horndeski
family, can be uniquely mapped into (20) via a simple procedure, as reviewed
in (Frusciante and Perenon, 2020). We refer the reader to the latter reference
for the complete mapping prescription, and report here only the expressions
for some of the EFT functions that will come in handy for the discussion
below:

$M_{P}^{2}\left(1+\Omega\right)=M_{*}^{2}-\bar{M}_{2}^{2}=2\left[G_{4}-XG_{5\phi}-X\ddot{\phi}G_{5X}\right]\,,\,\,\,\,\,\,\bar{M}_{2}^{2}=-2X\left[2G_{4X}-2G_{5\phi}-\left(\ddot{\phi}-H\dot{\phi}\right)G_{5X}\right]\,.$
(21)

Thanks to the mapping, Eq. (20) provides a unifying action to explore at once
the cosmological dynamics of a broad landscape of DE/MG models; one can work
both in a model specific way, resorting to the mapping procedure, or adopt a
more agnostic approach, exploring the phenomenology produced by different
choices of five free functions of time
$\Omega,\Lambda,M_{2}^{4},\bar{M}_{1}^{3},\bar{M}_{2}^{2}$ (the Friedmann
equation provides a constraint equation for the function $c$); see Raveri  _et
al._ (2014); Bellini  _et al._ (2016); Raveri (2020); Frusciante  _et al._
(2019); Melville and Noller (2020); Spurio Mancini  _et al._ (2019); Noller
and Nicola (2019); Kreisch and Komatsu (2018); Noller (2020) for some examples
of constraints with current data. In terms of (numerical) feasibility, the
latter represents a significant improvement with respect to using the generic
functions $G_{i}(\phi,X)$. These characteristics were exploited in the
creation of EFTCAMB Raveri  _et al._ (2014); Hu  _et al._ (2014), a patch that
implements EFTofDE into the public Einstein-Boltzmann solver CAMB Lewis  _et
al._ (2000). This can be used to create precise predictions for the
phenomenology of LSS and GWs on linear, cosmological scales for any Horndeski
gravity model which can then be translated into corresponding functional forms
of $w(z),\mu(k,z),\Sigma(k,z)$ Peirone  _et al._ (2018); Perenon  _et al._
(2017).

Furthermore, action (20) facilitates the simultaneous analysis of LSS and the
propagation of GWs on the cosmological background, which is a crucial
improvement given the recent advent of GW Cosmology. It allows us to factor
any constraints from the direct detection of GWs, like those discussed in the
previous subsection, into the exploration of LSS phenomenology. Using the
mapping prescription (21), the constraint on the speed of GWs, translates very
simply into $\bar{M}^{2}_{2}=0$, correspondingly simplifying the EFTofDE
action; $\Omega$ is connected both to the running Planck mass, $M_{*}^{2}$,
and to $\bar{M}_{2}^{2}$. In the case of luminally propagating tensors,
$\Omega$ reduces to $M_{*}^{2}$, corresponding to the additional friction term
that further damps the amplitude of GWs. As shown in Peirone  _et al._ (2018),
the $c_{T}^{2}=1$ constraint can have a significant impact on the LSS
phenomenology, limiting the possibility for a difference between $\mu$ and
$\Sigma$, i.e. between clustering and lensing, to the smaller scales only
Pogosian and Silvestri (2016). Another opportunity offered by (20), is that of
studying the joint power of galaxy surveys and future GW surveys, such as LISA
Amaro-Seoane  _et al._ (2017) and Einstein Telescope Maggiore  _et al._
(2020), in constraining gravity Balaudo  _et al._ (2023a, b).

Starting from a unifying action like (20), allows also to perform a very
efficient stability analysis, by further expanding the different operators to
quadratic order in the perturbations to the metric, and looking at the
properties of the resulting kinetic, gradient and mass matrices for the
propagating DOFs. This leads to a set of inequalities involving the free
functions in the action and their time derivatives, that represent general
criteria of viability Gleyzes  _et al._ (2013); Frusciante  _et al._ (2016).
The latter can provide additional constraining power when exploring gravity
with cosmological data Raveri  _et al._ (2014); Salvatelli  _et al._ (2016);
Melville and Noller (2020) and, importantly, physically inform the
parametrizations of the phenomenological functions, thus ensuring that we
sample only regions of the parameter space that correspond to viable theories
Peirone  _et al._ (2017). In fact, the combination of Horndeski gravity,
written in terms of EFTofDE, with the phenomenological framework
$(w,\mu,\Sigma)$, crucially addresses the above mentioned shortcomings of the
latter, and can provide insights into what can be learned about gravity from
cosmology. In Pogosian and Silvestri (2016) for instance, a _Horndeski
conjecture_ was identified for $\mu$ and $\Sigma$ through an analytical
exploration of the implications of Horndeski gravity for these two functions.
More interestingly, action (20), in combination with EFTCAMB, allows a Monte
Carlo sampling of the viable gravitational landscape, leading to the
formulation of powerful _theoretical priors_ for the phenomenology of LSS
Espejo  _et al._ (2019); Peirone  _et al._ (2018); Raveri  _et al._ (2017);
Traykova  _et al._ (2021); priors that can also take into account constraints
on the speed of tensors from the direct detection of gravitational waves. All
these aspects were used in a recent reconstruction of gravity from currently
available cosmological data Pogosian  _et al._ (2022); Raveri  _et al._
(2023), which besides showcasing the potentiality of current data, sets the
stage for future tests with LSS surveys.

Much more could be said, but it would elude the purpose of this short,
celebratory essay. We refer the reader to Kobayashi (2019) for a more
technical review of Horndeski and Beyond Horndeski. We will conclude this
part, noting how 50 years after its formulation, Horndeski theory has gained a
central role in cosmological tests of gravity. Exploring Horndeski models has
been insightful so far, but the true potential of this framework will be at
display with Stage IV LSS surveys that are seeing light in these years.

##  IV Concluding Remarks

I "retired" from Physics to pursue a career as an artist in 1981, maintaining
only a passing interest in the developments of General Relativity. Then in
July, 2014 I received a phone call from Prof. Jim Isenberg, who was a Post-Doc
at the University of Waterloo in the late 1970’s. Jim and I did some work on
my vector-tensor theory of gravity and electromagnetism Horndeski (1976). Jim
told me that he had just gotten back from a relativity conference in Italy,
and while walking down a hallway during the conference he had overheard some
physicists discussing Horndeski theory. So he interrupted their conversation
and asked if they were talking about Horndeski’s vector theory. They replied
no, they were interested in his scalar theory. Jim was bowled over by that,
since he never knew that I did anything on scalar-tensor theory. He suggested
that I Google Horndeski scalar theory to find out what was going on. I did and
was amazed to find out that there were 242 citations to my paper! Then about a
year later I heard from a young physicist, Dr. Alejandro Guarnizo-Trilleras,
who wanted to use the image of one of my paintings on the back cover of his
thesis. I, of course let him do that, and then in the course of reading his
thesis I discovered that some physicists were "going beyond Horndeski Theory"
Langlois and Noui (2016); Gleyzes  _et al._ (2015a); Ben Achour  _et al._
(2016b). Well, as long as I am alive no one is going beyond Horndeski Theory
without me. That is when I got back into doing research on scalar-tensor
theory. For an excellent review of Beyond Horndeski theory please see
Kobayashi (2019). Rather than remark on all the various things I have been
exploring involving scalar-tensor theories since 2015, I would like to mention
that I have been working on the bi-scalar tensor generalization of
"conventional" Horndeski theory. A great deal of progress on this problem has
been made by Ohashi, $et$ $al,$in Ohashi  _et al._ (2015). They have found the
basic form of $A^{\alpha\beta}$ for second-order bi-scalar tensor field
theories in a four-dimensional space, but were unable to find a Lagrangian
that could generate $A^{\alpha\beta}$. I feel fairly comfortable that I have
found that Lagrangian and it will appear in a future publication. Hopefully we
won’t have to wait another 30 years before physicists find a use for that
result.

The work presented in the previous section by Prof. Silvestri clearly
illustrates that the equations I developed back in the 1970s have proved to be
immensely valuable to various lines of research in Cosmology. One of the main
reasons for their usefulness has been the versatility they offer due to the
presence of the four coefficient functions of $\phi$ and X appearing in them.
So what I originally surmised to be the main drawback of my scalar-tensor
equations, has actually turned out to be their saving grace.

###### Acknowledgements.

We are grateful to Justin Khoury and Tsutomu Kobayashi for their precious
feedback and input. We would also like to thank Prof. A. Wipf, Editor in Chief
of the International Journal of Theoretical Physics, for providing us with the
opportunity to present this essay on Horndeski gravity, and also for the
patience he has evinced as we prepared it. AS acknowledges support from the
NWO and the Dutch Ministry of Education, Culture and Science (OCW) (through
NWO VIDI Grant No. 2019/ENW/00678104 and ENW-XL Grant OCENW.XL21.XL21.025
DUSC) and from the D-ITP consortium.

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# Target-centered Subject Transfer Framework for EEG Data Augmentation
111††thanks: This work was supported by Institute of Information &
Communications Technology Planning & Evaluation (IITP) grants funded by the
Korea government (No. 2015-0-00185, No. 2017-0-00451)

Kang Yin
Dept. Artificial Intelligence
Korea University
Seoul, Republic of Korea
<EMAIL_ADDRESS>Byeong-Hoo Lee
Byoung-Hee Kwon
Dept. Brain and Cognitive Engineering
Korea University
Seoul, Republic of Korea
<EMAIL_ADDRESS>Dept. Brain and Cognitive Engineering
Korea University
Seoul, Republic of Korea
<EMAIL_ADDRESS>Jeong-Hyun Cho  Dept. Brain and Cognitive Engineering
Korea University
Seoul, Republic of Korea
<EMAIL_ADDRESS>
###### Abstract

Data augmentation approaches are widely explored for the enhancement of
decoding electroencephalogram signals. In subject-independent brain-computer
interface system, domain adaption and generalization are utilized to shift
source subjects’ data distribution to match the target subject as an
augmentation. However, previous works either introduce noises (e.g., by noise
addition or generation with random noises) or modify target data, thus, cannot
well depict the target data distribution and hinder further analysis. In this
paper, we propose a target-centered subject transfer framework as a data
augmentation approach. A subset of source data is first constructed to
maximize the source-target relevance. Then, the generative model is applied to
transfer the data to target domain. The proposed framework enriches the
explainability of target domain by adding extra real data, instead of noises.
It shows superior performance compared with other data augmentation methods.
Extensive experiments are conducted to verify the effectiveness and robustness
of our approach as a prosperous tool for further research.

Keywords–Data augmentation, brain-computer interface, generative adversarial
network, electroencephalogram

##  I Introduction

Motor imagery (MI) based Brain-Computer Interfaces (BCI) systems have
experienced rapid developments since last century 90s [1]. The users are
instructed to perform mental tasks, during which the electroencephalogram
(EEG) signals are collected for the system to translate human’s intention and
makes corresponding response. It is generally believed that the successful
decoding of users’ intention heavily relies on their motivation and cognitive
arousal for skill acquisition [2, 3], while distraction and drowsiness are
harmful, thus results in poor signal qualities. Besides, individual uniqueness
and specificity makes it difficult to conduct generalized analysis among
subjects. Consequently, calibration process is usually inevitable before each
online experiment, to fine-tune the pre-trained model to adapt a specific
subject. This repeated process is tedious and consumes huge energy of subjects
thus impairs their following MI performance.

It should be worth noticing that, unlike images or texts, EEG data may contain
various noises originally, e.g. eye blinking, teeth clenching and flexing of
both arms, thus honestly it’s unwise to manually introduce more noises for
augmentation. On one hand, researches are conducted to eliminate external
perturbations [4, 5], while on the other hand, noises are added for
generalization ability [6, 7, 8], regardless of which, a reasonable balance
ought to be struck between the two, and arbitrarily introducing noises goes
against the aim of EEG data distillation and disentanglement, although
experimental performance might get improved somehow. Besides, given a large
amount of augmented data, it’s not an optimal to learn a shared latent space
of the augmented ones and the original ones [9, 10, 11], as the newly included
training data also transform the distribution of target subject in order for
generalization. Consequently, the proportion of augmented data and raw one is
usually very sensitive and required to be carefully searched.

Given the above observations, we argue that, theoretically, a practical DA
approach in BCI should meet the following two conditions: (a) improve
generalization ability without adding extra noises; (b) the data before and
after augmentation should match with each other. It is encouraging that the
basic SI BCI approach – directly training with the whole dataset, actually
doesn’t introduce any noise, thus, the only problem lies in the distribution
gap between a specific subject and the whole dataset. Inspired by remarkable
works [12, 13, 14] that gain huge successes on image style transformation and
regeneration in the field of computer vision, we assert that this kind of
style transfer shows strong potentials also for EEG signals in BCI domain,
with which the distribution difference is minimized between an individual and
the all.

Based on the above assumption, we propose a style transfer founded data
augmentation framework. Different from a similar work [15, 16, 17, 18] that
modifies target subject’s data distribution and create a co-domain for source
and target data, the proposed one is designed to fit and perfect the domain
for each target subject, i.e., the target domain’s distribution remains
relatively fixed and is gradually updated along with the arrival of
transferred data from source domains. Substantial experiments have been
conducted, and the proposed method shows superior performance on a very
challenging multi-task motor imagery datasets. The visualization analysis on
the generated data further proves the practicability and stability of our
proposed method.

The main contributions of this paper are summarized as follows:

  * •

The proposed framework is the first to emphasize the importance of using real
data for augmentation, and minimize the intersubject variability by
distribution transfer to target domain.

  * •

The source-target relevance maximization strategy optimizes the amount and
quality of source data to ensure effectiveness and stability of the transfer.

##  II Methods

###  II-A Dataset description and preprocessing

The OpenBMI dataset [19] is used in this work. It contains 200 trials for each
of 54 subjects and for each of left and right hand motor imagery (MI) task.
The MI task lasts for 4 s for each trial, and EEG signals were recorded with a
sampling rate of 1,000 Hz and collected with 62 Ag/AgCl electrodes.

The EEG data were band-pass filtered between 8 and 30 Hz with a 5th order
Butterworth digital filter, and is gone through z-score normalization.
Besides, we reduce the original 62 channels to 8 for simplicity, where only
[F3, C3, P3, Cz, Pz, F4, C4, P4] are reserved.

###  II-B Self-adjustable generative learning

####  II-B1 Problem formulation

Given a motor imagery dataset $X=[x^{1},x^{2},\dots,x^{m}]^{T}$, which
contains EEG signals from $m$ subjects, and each $x^{i}$, where
$i\in\\{1,2,\cdots,m\\}$, is a $t\times c\times p$ matrix, where $t$, $c$, $p$
denote the number of trials, channels and sample points, respectively. Trials
from different subjects are concatenated by row in matrix $X$. Each subject is
in turn taken as the target subject. Let $k$ be the current target subject
number, accordingly, $(x^{k})^{\complement}$ are taken as source subjects. For
simplicity (Table I), let $a$ and $b$ denote source subjects and target
subject. Data after subject transfer are marked by $\tilde{x}$.

####  II-B2 Source-target relevance maximization

Intuitively, it is claimed that not all data is suitable for transfer, where
outliers might exist, and it is of vital importance to select the most
relative data for a target subject. In this section, we propose a two-step
data filtering approach that optimize source data quality in order for better
transfer performance in Sect. II-B3.

To be specific, principal component analysis (PCA) is firstly applied to
reduce the dimension of original data, where data for each subject after
dimension reduction is denoted by $y^{i}$. Then, we rule out inner-subject
outliers, where the anomaly removal happened for each subject $y^{i}$.
Finally, the cluster head $y_{c}$ of target data is calculated and the intra-
subject optimal data selection is performed.

To perform PCA, we first calculate the mean $\overline{X}$ over channels:

$\overline{X}=\frac{\sum_{j=1}^{C}x_{:,j,:}}{C},$ (1)

where $C$ is the number of channels. By applying singular value decomposition
(SVD):

$Y=H\overline{X}V,$ (2)

where $H$ and $V$ are centering matrix and eigenvector matrix, respectively.
$Y$ represents the data after dimension reduction.

After that, for each subject $y^{i}$, the cluster center $y^{i}_{c}$ is
calculated:

$y_{c}^{i}=\frac{\sum_{j=1}^{t}y_{i,:}}{t},$ (3)

and the L1-distance $d_{j}$ is calculated to measure the relevance between a
certain trial $y_{j}^{i}$ and its mean.

For all subjects except the target one, a threshold $\beta_{1}$ is set to keep
a proportion of data in the order of distance to the center point. Similarly,
the distance between all remaining data and the target center point
$y_{c}^{target}$ is calculated, and a threshold $\beta_{2}$ here will filter
out those irrelevant trials which are unsuitable for transfer.

####  II-B3 Target-centered subject transfer

Following the idea of Cycle-GAN in [12], we aim to make source data $x_{a}$
and target data $x_{b}$ learn an approach to be capable of transferring to
each other’s style, by assuming them belong to two different domains. The
thinking behind this assumption is intuitive: subjects vary a lot in many
aspects, e.g., imagining ability, health condition and level of concentration,
which are unique in individual level, thus every single subject should be
accounted as a domain. We simply view source subjects as a whole domain in
order to reduce the complexity of our method. After the subject transfer, data
in source domain (various subjects) should obey the distribution in target
domain (a single target subject), where exterior differences caused by
subjects no longer exists. The subject transfer learning proceeds in a bi-
directional manner, where two GANs are trained simultaneously, in which
$(G_{A},D_{A})$ is used to transfer source data $x_{a}$ to target domain, and
$(G_{B},D_{B})$ transfers target data $x_{b}$ to source domain. Accordingly,
discriminators attempt to tell the real target (source) domain from source
(target) domain, and generators gradually learn to transfer between domain by
adversarially training with discriminators. As GAN is notorious for the
unstable training process, here we follow previous works of Wasserstein GAN
[20], and combine it with traditional Cycle-GAN for better training, thus, the
updated GAN loss is formulated as:

$\displaystyle\mathcal{L}_{\rm gan}(G,D,A,B)$ (4)
$\displaystyle={\mathbb{E}}_{x_{b}\sim{\mathbb{P}}_{B}}[\log
D_{A}(G_{B}(x_{b}))]-{\mathbb{E}}_{x_{a}\sim{\mathbb{P}}_{A}}[\log
D_{A}(x_{a})]$ $\displaystyle+{\mathbb{E}}_{x_{a}\sim{\mathbb{P}}_{A}}[\log
D_{B}(G_{A}(x_{a}))]-{\mathbb{E}}_{x_{b}\sim{\mathbb{P}}_{B}}[\log
D_{B}(x_{b})]$
$\displaystyle+\lambda{\mathbb{E}}_{\hat{x}\sim{\mathbb{P}}_{\hat{x}}}[(||\nabla_{\hat{x}}D_{A}(\hat{x})||_{2}-1)^{2}+(||\nabla_{\hat{x}}D_{B}(\hat{x})||_{2}-1)^{2}],$

where $\lambda$ is a hyperparameter, and the distribution
${\mathbb{P}}_{\hat{x}}$ obeys the uniform sampling rule along the line of
pair-wise points sampled from ${\mathbb{P}}_{A}$ and ${\mathbb{P}}_{B}$.

With the constraint of cycle consistence, the transferred data will be able to
recover back to its original style by passing through the generator in
corresponding domain, where the recovered data and the original one are asked
to be as close as possible, therefore, the cycle consistence loss is written
as follows:

$\displaystyle\mathcal{L}_{\rm cyc}(G,A,B)$
$\displaystyle={\mathbb{E}}_{x_{a}\sim{\mathbb{P}}_{A}}[||G_{B}(G_{A}(x_{a}))-x_{a}||_{1}]$
(5)
$\displaystyle+{\mathbb{E}}_{x_{b}\sim{\mathbb{P}}_{B}}[||G_{A}(G_{B}(x_{b}))-x_{b}||_{1}].$

##  III Experiments and discussion

The experiment consists generative learning part, where augmented data are
generated, and the downstream part for motor imagery classification. In the
generative learning part, we implement comparative method including both noise
addition (noise, multiple and flip) and augmentation approaches in previous
works ($\beta$-VAE [21], DCGAN [22] and EEG-GAN [23]). In the classification
part, EEGNet [24] is chosen as the backbone, and we following the
implementations and hyperparameter settings in the original paper. Three RTX
3080 Ti GPUs are used in this experiment.

Three typical noise addition approaches are applied here. Let $d$ denotes the
raw data, and the first approach is to add noise from a uniform distribution.
The artificial data is generated by:
$d_{noise}=d+\frac{u\cdot\text{std}(d)}{\gamma}$, where $u$ is sample from
U(-1,1), and std($\cdot$) calculates the variance. The hyperparameter $\gamma$
is for scaling. Similarly, by multiplying and flip the raw data, we generate
$d_{multi}=(1+\gamma)\times d$ and $d_{flip}=-d+\min(d)$.

###  III-A Classification

TABLE I: Comparison of average classification accuracy (%) for different
methodologies.

Methodology | Mean (SD) | Median | Range (Max–Min)
---|---|---|---
Baseline | 79.95 (9.91) | 79.30 | 38.93 (97.28-58.35)
Noise | 80.82 (9.77) | 80.45 | 38.36 (98.52-60.16)
Multiple | 80.07 (9.59) | 79.02 | 36.76 (97.61-60.85)
Flip | 80.11 (9.56) | 79.34 | 35.56 (97.78-62.22)
$\beta$-VAE [21] | 80.40 (9.03) | 79.75 | 36.74 (94.87-58.13)
DCGAN [22] | 81.85 (9.85) | 81.40 | 39.84 (98.52-58.68)
EEG-GAN [23] | 81.03 (9.97) | 81.85 | 41.73 (98.77-57.04)
Cycle-GAN (Ours) | 82.34 (8.98) | 81.52 | 34.05 (99.08-65.03)

Table I shows the classification results for different augmentation methods.
Performance improvement is observed for all. Particularly, in noise addition
approaches, multiple and flip operations could not effectively improve the
accuracy, while by adding uniform distribution noises, 0.87% classification
accuracy increase is achieved. For generative models, we witness a clear
superiority over than by simply adding noise. Also, the GAN based models
obtain obviously better performance than VAE, which might be due to the poor
reconstruction ability caused by its inherent architecture deficiency, where
more comparison studies are provided in Sect. III-B1. Most importantly, our
proposed method outperforms the other two GAN methods which are specially
designed for EEG data, which shows the significance of using real signal data
for generation, given that the other two both use random noises as the model
input.

(a) Raw

(b) x10

(c) x20

(d) x30

(e) x40

(f) x50

Figure 1: Topographic visualization of the power spectral density.

(a) Raw data

(b) $\beta$-VAE

(c) DCGAN

(d) EEG-GAN

(e) Ours

Figure 2: Visualization of cross-spectral density (CSD) matrix.

###  III-B Discussion

####  III-B1 The quality of generated signals

Figure 3: Comparison of generated signals by different methods.

In this section, we explore the relationship between augmentation size and
classification accuracy. As shown in Table II, we manually set 5 groups of
augmentation ratio, which are 1:10, 1:20, 1:30, 1:40, 1:50, respectively, and
the results indicates that, bigger augmentation size does help for the
generalization, therefore, improve performance. However, it saturates when
reaching a certain threshold. It might because the augmented data can no
longer introduce additional information to the target data distribution,
(which also proves that information redundancy exists in the dataset). The
topographic map (Fig. 1) proves our hypothesis in another way. By comparing
between raw data and generated ones, we first observe a growing resemblance
and after the data size increases to a certain degree, it almost imitates the
unique characteristics of the target distribution. If keeping training in this
phase, the uniqueness of augmented data will be compromised. Nonetheless, we
state that the raw data is still of critical role in augmentation settings.

####  III-B2 The impact of augmentation size

TABLE II: Influence of augmentation size.

Baseline | x10 | x20 | x30 | x40 | x50
---|---|---|---|---|---
79.95$\pm$9.91 | 81.78$\pm$9.34 | 82.34$\pm$8.98 | 82.03$\pm$9.53 | 81.09$\pm$9.85 | 81.33$\pm$9.33

To validate the proposed method’s effectiveness as a data augmentation
approach, in Fig. 3, we plot the averaged trials for different methods. It is
remarkable that, our proposed method brings more detailed information
especially when comparing with VAE, and it provides greater varieties compared
with simply adding noise. It depicts the target distribution’s flow and also
introduce differences, hence, improve the generalization ability. The
covariance matrix (Fig. 2) suggests similar conclusion that both
generalization and specificity are well considered in this approach.

##  IV Conclusion and future works

In this paper, we propose a novel data augmentation approach by transferring
source data into target domain. The proposed method jointly consider the
deficiencies of previous works on EEG data augmentation and the feasibility
for further analysis on the target domain. For future work, we will explore
the possibility of class-specific data transfer, which might provide target
domain better disentanglement once upon the complete of data transfer.

## Acknowledgment

The authors thank to H.-B. Shin for useful discussions of the methods and
experiments.

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