# Chameleon perfect scalar field as a geometric correction in $f(R)$ gravity

Laura L. Parrilla<EMAIL_ADDRESS>Instituto de
Ciencias Nucleares, Universidad Nacional Autónoma de México, Circuito Exterior
C.U., A.P. 70-543, México D.F. 04510, México.  Celia Escamilla-Rivera
0000-0002-8929-250X<EMAIL_ADDRESS>Instituto de Ciencias
Nucleares, Universidad Nacional Autónoma de México, Circuito Exterior C.U.,
A.P. 70-543, México D.F. 04510, México.

###### Abstract

In this work, we derive the analytical form for a $f(R)$ model that describes
a perfect scalar field $\phi$ by assuming the existence of a chameleon
mechanism. Based on four statements, at the background and perturbative level,
it is possible to relate the extra terms from this theory as a geometrical
perfect fluid term, whose has been expressed as possible candidates to explain
the nature of the dark sector, and possibly, in the case of a perfect scalar
chameleon during inflation, satisfy the big bang nucleosynthesis (BBN)
constraints until late times.

##  I Introduction

Since the discovery of the current accelerated cosmic expansion of the
Universe using observational data from astrophysical objects like Supernovae
type Ia independently by the high-redshift Supernovae Search Team
SupernovaSearchTeam:1998fmf  and the Supernovae Cosmology Project Team
SupernovaCosmologyProject:1998vns , has been attributed the source of this
phenomena to the so-called dark energy. However, even with the many efforts to
pursue the understanding of the nature of such a dark component, this has not
been observationally identified yet. One of the main characteristics of dark
energy is attributed to a negative pressure, which leads to an accelerated
expansion phenomenon by counteracting the force of gravity. The fact that the
negative pressure leads to the cosmic acceleration may look counter-
intuitive111In Newtonian gravity the pressure is related to a force associated
with a local potential that depends on the position in space.. The time-
dependent pressure $p(t)$ in homogeneous and isotropic spacetimes appears in
general relativity (GR) and mechanisms that generate this negative pressure
and cosmic acceleration are one of the main research topics in Cosmology.

The straightforward candidate proposed for dark energy is the cosmological
constant $\Lambda$, whose energy density remains constant and allows Einstein
field equations to preserve the conservation of energy. The $\Lambda$ term can
be interpreted as a perfect fluid by shifting to the right-hand side
(r.h.s)222There are several alternative models of dark energy also on the
r.h.s of the field equations, $\Lambda$ being the simplest of all of them. ,
and if dark energy modelled as $\Lambda$, for which equation-of-state
$w_{\text{fld}}=-1$, is interpreted as a perfect fluid with
$p_{\Lambda}\propto-\rho_{\Lambda}$. From the point of view of particle
physics, the Cosmological Constant can be related to a vacuum energy density,
where if we sum up zero-point energies of all normal modes of some field and
consider the cut-off of the momentum at Planck scale, the vacuum energy
density is around $10^{121}$ times larger than the observed $\Lambda$ density,
the so-called vacuum catastrophe. On the one hand, if the Cosmological
Constant is truly the consequence of the present current cosmic acceleration,
we need to find a mechanism to obtain the tiny value of $\Lambda$ which could
be consistent with observations.

On the other hand, if the origin of dark energy is not the cosmological
constant, we need to search for some alternative (or extended) gravity models
to explain the current cosmic acceleration. There are two approaches to
building dark energy models other than using $\Lambda$. The first approach is
to modify the r.h.s of Einstein field equations, ($G_{\mu\nu}=8\pi
GT_{\mu\nu}$), by considering specific forms of the energy-momentum tensor
$T_{\mu\nu}$ that includes the possibility of a fluid with negative pressure.
The most popular models that belong to this class are the so-called
quintessence Caldwell:2000wt , k-essence Rendall:2004kh  and perfect fluid
models Faraoni:2022gry . On this path, the methodology to follow is to
consider the existence of scalar fields333In quantum field theory, several
species of elementary particles corresponding to a field are produced. The
fields are classified as boson or fermionic depending on the spin of the
particle. In the case of scalar fields, this spin has a value of zero. At the
quantum level, each field corresponds to an operator, however, bosonic fields
can be considered classical in a suitable regime. with slowly/smooth varying
potentials, whereas in k-essence it is the scalar field kinetic energy that
drives the acceleration. The latter class is based on a perfect fluid of a
specific equation-of-state.

The second approach to constructing dark energy models is to modify the l.h.s
of Einstein field equations. We denote the representative models that belong
to this class as modified gravity Clifton:2011jh . While scalar field models
correspond to a modification of the energy-momentum tensor, the approach in
this path corresponds to the modified gravity in which the gravitational
theory is modified compared to general relativity. For example, the Lagrangian
density for general relativity is given by $f(R)=R-2\Lambda$, where $R$ is the
Ricci scalar and the constant allows us to have the acceleration phenomena
required. A possible modification of this scenario can be described by a non-
linear arbitrary function $f$ in terms of $R$, which is called $f(R)$ theories
of gravity. Dark energy models based on these kinds of theories have been
studied extensively including metric formalism Capozziello:2002rd ;
Capozziello:2003gx , observational test Amendola:2006we ; Bajardi:2022ocw ;
Jaime:2018ftn  and modifications to the spectra of galaxy clustering
Capozziello:2017zow ; DeMartino:2013zua .

Historically, Brans-Dicke theory, an important class of scalar-tensor
theories, gives rise to a constant coupling $Q$ between the scalar field
$\phi$ and the matter component. In this sense, modified gravity theories can
be regarded as a coupled quintessence scenario in Einstein’s frame. In the
absence of a scalar field potential $V(\phi)$, the Solar system tests
constraint the strength of this coupling Q to be smaller than the order of
$\approx 10^{-3}$. In this case, is not possible to satisfy the local gravity
constraints unless we have a $V(\phi)$ with a large mass that can be capable
to suppress the $Q$ coupling in the regions of high density. Furthermore, if
the same field $\phi$ is responsible for the current cosmic acceleration
phenomena, the potential $V(\phi)$ needs to be sufficiently flat in the
regions of low density. On this line of thought, these requirements are
possible to fulfill for large coupling models that satisfy the local gravity
constraints through the chameleon mechanism Khoury:2003rn . The existence of a
matter coupling gives rise to an extremum of the scalar field potential where
the field can be stabilised. In high-density regions, such as the interiors of
astrophysical objects, the field mass would be sufficiently large to avoid the
propagation of the fifth force. Meanwhile, the field would have a much lighter
mass in low-density environments, far away from compact objects, so it could
be responsible for the present cosmic acceleration. In this work, we are going
to consider a particular class of the chameleon field with an inverse power-
law potential of the form $V(\phi)=M^{4+n}\phi^{-n}$, with $n\geq 1$, so local
gravity constraints can be satisfied for M $\sim 10^{-2}$eV. Interestingly,
these constrictions correspond to the energy scale required for the current
cosmic acceleration observed.

Recently, the correspondence between the modified gravity geometrical terms
inspired by perfect fluid components has been pointed out as a natural way to
understand the nature of dark energy, and by extension, the full dark sector
Unnikrishnan:2010ag ; Capozziello:2022sbo . In resume, beyond the Ricci
curvature scalar $R$ or any other geometric invariant in a gravitational
action, we can modelled these extra terms as perfect fluids. As an extension
of these ideas, in this work, we propose to extend the conditions that allow
us to model the scalar field as perfect fluids using a chameleon scalar field.
Perfect fluids play a crucial role in general relativity being the natural
sources of Einstein field equations compatible with Bianchi identities. This
characteristic allows any source of field equations to be reformulated in
adequate perfect fluid form, in principle, to solve the dynamics related to
the Cauchy problem. In Cosmology, perfect fluids can represent, at least in a
coarse-grained image Hubble’s effective flow behaviour ranging from inflation
to dark energy epochs. For these reasons, the compatibility of perfect fluid
solutions with modified (or extended) theories of gravity is a crucial topic
to be investigated.

This paper is divided as follows: In Sec.II we discuss the perfect fluid form
scheme for a FLRW spacetime and the three conditions to fulfill to have the
perfect fluid description relationship. In Sec.III we present a fourth
condition at a perturbative level to describe a perfect scalar field. In
Sec.IV it is described the $f(R)$ theory at the background and perturbative
level. In Sec.V we introduce the chameleon scalar field in $f(R)$ theories and
the analytical form of $f(R)$ to obtain a perfect scalar field. Finally, our
discussions are given in Sec.VI.

##  II Background conditions: perfect scalar field in FLRW spacetimes

We can define a FLRW spacetime by considering a zero Weyl tensor,
$C_{jklm}=0$, with a time-like unit vector field in a covariant configuration
as $u^{k}u_{k}=-1$ Mantica:2016iun , as

$\displaystyle
ds^{2}=-dt^{2}+a^{2}(t)\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right],$ (1)

in spherical coordinates and $k=0,-1,+1$. Under such a description, we can
compute the following covariant derivative

$\nabla_{j}u_{k}=H(g_{ij}+u_{j}u_{k}),\quad\nabla_{j}H=-u_{j}u^{k}\nabla_{k}{H},$
(2)

where $H\equiv\dot{a}/a$. Here we are considering the notation $\mu,\nu=1,2,3$
and $i,j,k,l,m=0,1,2,3$. In a previous analysis Capozziello:2022sbo  was
considered the approach given by $h_{jk}=g_{jk}+u_{j}u_{k}$, as the projection
on the vector space locally orthogonal to $u_{k}$, therefore, we can compute
the Riemann tensor from
$R_{jklm}u^{m}=(\nabla_{j}\nabla_{k}-\nabla{k}\nabla_{j})u_{l}$, and its
contraction, the Ricci tensor as

$R_{kl}=\frac{1}{3}\left(R-4\zeta\right)u_{k}u_{l}+\frac{1}{3}\left(R-3\zeta\right)g_{kl},$
(3)

where $R=R^{k}R_{k}$ is the curvature scalar and $\zeta=3\ddot{a}/a$. Notice
that this form of the Ricci tensor has a perfect fluid form since the terms
related to the time-like vector and the shear/vorticity/acceleration-free term
are separated.

To reach a perfect fluid form like in Eq.(3), we need to consider the
following three geometrical background tests on each of the parameters
involved Capozziello:2022sbo  :

  1. 1.

If $\phi$ is perfect, therefore a function $f(\phi)$ is perfect as long as $f$
is smooth. This characteristic applies also to products and time-derivatives
of such functions, e.g. if $H$ is perfect, $H^{2}$ and $\dot{H}$ are perfect.

  2. 2.

Considering (1), the derivatives of a perfect scalar field $\phi$,
$\nabla_{i}\nabla_{j}{\phi}$, also has a perfect fluid form.

  3. 3.

Under (1) and (2), we conclude from Eq.(3) that $R$ is a perfect scalar.

##  III Scalar perturbation conditions: non-adiabatic pressure

While the three conditions described in the latter section restrict us to work
with perfect scalar forms, we need to be sure that such statements are also
conserved at the perturbative level. In this way, we need to consider the FLRW
(1) as a perturbed metric:

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle(1+2A)dt^{2}-2a(t)(\partial_{i}B)dtdx^{i}-a^{2}{t}\left[(1-2\psi)\delta_{ij}+2(\partial_{i}\partial_{j}E)dx^{i}dx^{j}\right],$
(4)

where A, B, $\psi$, and $E$ will be related to the Bardeen functions in the
standard scalar perturbations Bardeen:1980kt . From this point forward we are
going to consider linear order perturbations, therefore, we can derive the
standard conformal time coordinate potential equation as:

$\displaystyle\Psi^{\prime\prime}-3\mathcal{H}(1+c^{2}_{a})\Psi^{\prime}-c^{2}_{a}\nabla^{2}\Psi+[2\mathcal{H}^{\prime}+(1+3c^{2}_{a})\mathcal{H}^{2}]\Psi=(4\pi
Ga^{2})\delta p,$ (5)

where the prime denotes conformal time derivatives, $\mathcal{H}=aH$ and
$c^{2}_{a}=p^{\prime}/\rho^{\prime}$ denotes the adiabatic speed444This
definition is also preserved in physical time units., where $p$ and $\rho$ are
the pressure and density, respectively. $\delta p$ are the perturbations
related to the pressure of the matter field. If we consider Eq.(5) in the
standard perturbed Einstein field equations at first order we can recover

$\displaystyle\Psi^{\prime\prime}-3\mathcal{H}(1+c^{2}_{a})\Psi^{\prime}-c^{2}_{a}\nabla^{2}\Psi+[2\mathcal{H}^{\prime}+(1+3c^{2}_{a})\mathcal{H}^{2}]\Psi=\left(c^{2}_{s}-c^{2}_{a}\right)\nabla^{2}\Psi,$
(6)

where $c_{s}$ is referred to the speed of the perturbations. If we compare
Eq.(5) and Eq.(6) we can derive the expression that relates both speeds of the
perturbations as:

$\displaystyle\delta p=\left(\frac{c^{2}_{s}-c^{2}_{a}}{4\pi
Ga^{2}}\right)\nabla^{2}\Psi.$ (7)

By using the conservation equation $\rho^{\prime}+3\mathcal{H}(\rho+p)=0$,
derived from the energy momentum tensor
$T^{\mu}_{\nu}=(\partial\mathcal{L}/\partial
X)(\partial^{\mu}\phi\partial_{\nu}\phi)-\delta^{\mu}_{\nu}\mathcal{L}$, with
$X=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$, we can obtain

$\displaystyle c^{2}_{a}=\left(\frac{p^{\prime}}{\rho^{\prime}}\right)$
$\displaystyle=$
$\displaystyle\left(\frac{\dot{p}}{3H(\rho+p)}\right)=-\left(\frac{\mathcal{L}_{\phi}+\ddot{\phi}\mathcal{L}_{X}}{3H\dot{\phi}\mathcal{L}_{X}}\right).$
(8)

We performed a change rule to transform to physical time, and the subindex
denotes derivatives concerning $X$ and $\phi$. And

$\displaystyle c^{2}_{s}=\frac{\left(\frac{\partial\mathcal{L}}{\partial
X}\right)}{\frac{\partial\mathcal{L}}{\partial
X}+2X\left(\frac{\partial^{2}\mathcal{L}}{\partial X^{2}}\right)}.$ (9)

To have a perfect fluid described by an equation-of-state in the background,
we need to have a vanishing $\delta p$ in the perturbative scheme, therefore,
to achieve such a scenario we notice from Eq.(7) that $c^{2}_{s}=c^{2}_{a}$,
this is the so-called non-adiabatic condition.

The latter condition implies that Unnikrishnan:2010ag

$\displaystyle\frac{\partial}{\partial
X}\left[\frac{\mathcal{L}_{\phi}}{X\mathcal{L}_{X}}\right]=0,$ (10)

therefore, to satisfy the condition for a perfect fluid form like we need to
consider, at the perturbative level, the following geometrical perturbed test:

4\. If Eq.(10) is satisfied, therefore $\phi$ is a perfect scalar field.

##  IV Geometric perfect fluid from $f(R)$

Our next step is to follow the analogies from the statements 1-4, in the
context of $f(R)$ gravity. First, we need to verify if the high derivatives of
$R$ in this scheme are fulfilled according to 1-3, this will denote that the
extra terms rising from the geometrical part can be associated with a perfect
fluid form by comparing them with the standard matter terms. Second, at the
perturbative level, we need to derive the equations related to the non-
adiabatic condition in $f(R)$ and compute the possible constrictions on a
$f(R)$ smooth function to satisfy a modified version of Eq.(10), and therefore
fulfill the condition 4.

###  IV.1 Background conditions: perfect fluid in $f(R)$ gravity

We need to consider an effective fluid approach to add fluids in extended
theories from Einstein’s gravity, as $f(R)$ landscape. Many works have been
done in this direction Tsujikawa:2007gd ; Escamilla-Rivera:2016aca ;
Bahamonde:2017ize ; Nesseris:2022hhc , are references cited in there. However,
it is standard in all of them, to begin with, a fluid of the form
$\rho=\rho_{\text{fld}}+\rho{{}_{m}}$, where $\rho_{\text{fld}}$ is usually
associated with dark energy and $m$ already includes the dark matter rate in
the baryonic component. In this context, we specify the modified Einstein-
Hilbert action as:

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2\kappa}f\left(R\right)+\mathcal{L}_{\mathrm{m}}\right],$
(11)

where $\mathcal{L}_{\mathrm{m}}$ is the matter contribution Lagrangian and
$\kappa\equiv 8\pi G$, where $G$ is the Newton’s constant. Varying Eq. (11)
with respect to the metric $g_{\mu\nu}$, we can derive the following field
equations:

$\displaystyle
f_{R}G_{\mu\nu}-\frac{1}{2}\left[f-Rf_{R}\right]g_{\mu\nu}+\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)f_{R}=\kappa\,T_{\mu\nu}^{(m)},\quad$
(12)

where $G_{\mu\nu}$ is the standard Einstein tensor, $f_{R}=\partial f/\partial
R$, and $T_{\mu\nu}^{(m)}$ is the energy-momentum tensor of the matter fields,
which is one contribution of a general
$T_{\mu\nu}=T_{\mu\nu}^{(m)}+T_{\mu\nu}^{(\text{fld})}$. According to this
latter, we can consider that $T_{\mu\nu}^{(\text{fld})}$ can be a contribution
related to the geometrical part of Eq.(12) as

$\displaystyle\kappa
T_{\mu\nu}^{(\text{fld})}\equiv[1-f_{R}]G_{\mu\nu}+\frac{1}{2}[f-Rf_{R}]g_{\mu\nu}-\left(g_{\mu\nu}\Box-\nabla_{\mu}\nabla_{\nu}\right)f_{R},\quad$
(13)

which satisfy the conservation equation
$\nabla^{\mu}T_{\mu\nu}^{(\text{fld})}=0$. The evolution equations associated
to Eq.(12) considering Eq.(13) are given by

$\displaystyle 3\mathcal{H}^{2}$ $\displaystyle=$ $\displaystyle\kappa
a^{2}\left(\rho_{\mathrm{m}}+\rho_{\mathrm{fld}}\right),$ (14) $\displaystyle
6\dot{\mathcal{H}}$ $\displaystyle=$ $\displaystyle-\kappa
a^{2}\big{[}\left(\rho_{\mathrm{m}}+3p_{\mathrm{m}}\right)+\left(\rho_{\mathrm{fld}}+3p_{\mathrm{fld}}\right)\big{]},$
(15)

where

$\displaystyle\kappa\rho_{\mathrm{fld}}$ $\displaystyle=$
$\displaystyle-\frac{f}{2}+3\frac{\mathcal{H}^{2}}{a^{2}}-3\frac{\mathcal{H}\dot{f_{R}}}{a^{2}}+3\frac{f_{R}\dot{\mathcal{H}}}{a^{2}},$
(16) $\displaystyle\kappa p_{\mathrm{fld}}$ $\displaystyle=$
$\displaystyle\frac{f}{2}-\frac{\mathcal{H}^{2}}{a^{2}}-\frac{2f_{R}\mathcal{H}^{2}}{a^{2}}+\frac{\mathcal{H}\dot{f_{R}}}{a^{2}}-\frac{\dot{\mathcal{H}}}{a^{2}}-\frac{f_{R}\dot{\mathcal{H}}}{a^{2}}+\frac{\ddot{f_{R}}}{a^{2}},\quad\quad$
(17)

Using Eqs. (16) and (17) we can derive the equation-of-state (EoS) for the
fluid as:

$\displaystyle w_{\mathrm{fld}}=\frac{p_{\mathrm{fld}}}{\rho_{\mathrm{fld}}}$
$\displaystyle=\frac{2\left[(1+2f_{R})\mathcal{H}^{2}-\mathcal{H}\dot{f_{R}}+(2+f_{R})\dot{\mathcal{H}}-\ddot{f_{R}}\right]-a^{2}f}{a^{2}f-6(\mathcal{H}^{2}-\mathcal{H}\dot{f_{R}}+f_{R}\dot{\mathcal{H}})},\quad\quad$
(18)

which for $f(R)=R$, we obtain the standard EoS from Einstein’s gravity
Copeland:2006wr . Furthermore, if we consider $\dot{R}=0$ and a flat space
$k=0$, we obtain the Eos for a perfect fluid $p=w\rho$. Following this idea
and the conditions from 3 -1, Eq.(12) describe perfect fluids through the
contributions from the geometric part in $f(R)$.

###  IV.2 Scalar perturbation conditions: non-adiabatic pressure in $f(R)$

Since the contribution due to $T_{\mu\nu}^{(m)}$ is effectively associated
with a standard perfect fluid, in such case with the standard matter, and we
proved that $T_{\mu\nu}^{(\text{fld})}$ can describe a perfect fluid
concerning the extra geometric contributions, then we are ready to demonstrate
is this condition is fulfilled at the perturbative level. Full analyses on
$f(R)$ scalar perturbations have been presented in Clifton:2011jh ;
Duniya:2022vdi . In this work, we redo these calculations to obtain the
quantities to satisfy the condition 4.

We start with the gravitational field constraint equations given by

$\displaystyle\Psi^{\prime}+{\cal
H}\Phi=-\dfrac{1}{2}\kappa^{2}a^{2}\sum\left(\rho+p\right)V,$ (19)
$\displaystyle\nabla^{2}\Psi-3\mathcal{H}\left({\cal
H}\Phi+{\Psi}^{\prime}\right)=\dfrac{1}{2}\kappa^{2}a^{2}\sum{\delta\rho},$
(20)

where $\rho$ and $p$ denote the density and pressure of all the fluid/matter
components, in our case, matter (m) and the fluid (fld), and $V$ is the
effective velocity potential. Since we are associated our perfect fluid with
the geometrical contribution from a $f(R)$ gravity, we define in conformal
time

$\displaystyle\kappa^{2}(\rho_{\mathrm{fld}}+p_{\mathrm{fld}})V_{\mathrm{fld}}$
$\displaystyle{\equiv}$ $\displaystyle a^{-2}\left(\Phi f^{\prime}_{R}+{\cal
H}\delta{f}_{R}-\delta{f}^{\prime}_{R}\right)+\;2a^{-2}\left(\Psi^{\prime}+{\cal
H}\Phi\right)f_{R},$ (21)

where $\rho_{\mathrm{fld}}$ and $p_{\mathrm{fld}}$ are given by Eqs.(16)-(17).
The fluid density perturbation can be derived as

$\displaystyle\kappa^{2}\delta\rho_{\mathrm{fld}}$ $\displaystyle{\equiv}$
$\displaystyle 2a^{-2}\left[3{\cal H}(\Psi^{\prime}+{\cal
H}\Phi)-\nabla^{2}\Psi\right]f_{R}+\;a^{-2}\left(\nabla^{2}+3{\cal
H}^{\prime}\right)\delta{f}_{R}-3a^{-2}{\cal
H}\delta{f}^{\prime}_{R}+\;3a^{-2}\left(\Psi^{\prime}+2{\cal
H}\Phi\right)f^{\prime}_{R}.$ (22)

In order to arrive to an analogous expression as Eq.(7), we can combine
Eq.(19)-(20) to obtain

$\displaystyle\nabla^{2}\Psi\;=\;\dfrac{1}{2}\kappa^{2}a^{2}\sum\rho\xi,$ (23)

where $\xi=\Omega_{m}\xi_{m}+\Omega_{\mathrm{fld}}\xi_{\mathrm{fld}}$, denotes
the comoving overdensity with

$\displaystyle\kappa^{2}a^{2}\rho_{\mathrm{fld}}\xi_{\mathrm{fld}}$
$\displaystyle=$ $\displaystyle 3\left(\Psi^{\prime}+{\cal
H}\Phi\right)f^{\prime}_{R}-2f_{R}\nabla^{2}\Psi+\;\left[\nabla^{2}+3\left({\cal
H}^{\prime}-{\cal H}^{2}\right)\right]\delta{f}_{R},$ (24)

by using Eqs.(21)-(22). As in (5), we can establish the relation between
Bardeen potentials through
$\Psi-\Phi=\kappa^{2}a^{2}\sum\left(\rho+p\right)\Pi$, where $\Pi$ denotes the
anisotropic stress or anisotropic pressure dimensionless and define the
evolution equation for the fluid as

$\displaystyle\kappa^{2}\left[\delta{p}_{\mathrm{fld}}+\dfrac{2}{3}\left(\rho_{\mathrm{fld}}+p_{\mathrm{fld}}\right)\nabla^{2}\Pi_{\mathrm{fld}}\right]$
$\displaystyle{\equiv}$
$\displaystyle\;a^{-2}\Big{\\{}\delta{f}^{\prime\prime}_{R}+{\cal
H}\delta{f}^{\prime}_{R}-2\Phi f^{\prime\prime}_{R}-\left(4\nabla^{2}+{\cal
H}^{2}+a^{2}R\right)\dfrac{\delta{f}_{R}}{6}-2f_{R}\Big{[}\Psi^{\prime\prime}+2{\cal
H}\Psi^{\prime}+{\cal H}\Phi^{\prime}$ (25) $\displaystyle+\;\left({\cal
H}^{2}+2{\cal
H}^{\prime}\right)\Phi+\;\dfrac{1}{3}\nabla^{2}(\Phi-\Psi)\Big{]}-\left[\Phi^{\prime}+2(\Psi^{\prime}+{\cal
H}\Phi)\right]f_{R}^{\prime}\Big{\\}},$

and using (5) we obtain

$\displaystyle\Psi^{\prime\prime}+{\cal
H}\left(2+3c^{2}_{a}\right)\Psi^{\prime}+$ $\displaystyle\;{\cal
H}\Phi^{\prime}+\left[2{\cal H}^{\prime}+\left(1+3c^{2}_{a}\right){\cal
H}^{2}\right]\Phi=\;\dfrac{1}{3}\nabla^{2}\left(\Psi-\Phi\right)+\dfrac{3}{2}{\cal
H}^{2}c^{2}_{s}\xi,$ (26)

where

$\displaystyle\delta{p}=c^{2}_{a}\delta\rho+\left(c^{2}_{s}-c^{2}_{a}\right)\rho\xi.$
(27)

Under the non-adiabatic condition we recover from the latter that
$c^{2}_{s}=c^{2}_{a}$, henceforth, by assuming pressureless matter
($p_{m}=\delta{p}_{m}=0$), the fluid fld comes from a geometrical $f(R)$
contribution. Furthermore, such scenarios have been directly related to
barotropic perfect fluids, i.e with vanishing non-adiabatic pressure
perturbations. Interesting scenarios on this matter have been discussed in
Unnikrishnan:2010ag . However, this assumption is restricted to standard
gravity models. In this work, we extended this assumption by coupling
minimally a scalar field to the action (11). We will adopt this analysis from
a chameleon scalar field scheme in what follows.

##  V Chameleon perfect scalar field in $f(R)$

In the context of scalar field-driven expansion dynamics, e.g. inflation
epoch, a possible candidate so-called chameleon scalar field Khoury:2003aq ;
Khoury:2003rn ; Waterhouse:2006wv ; Khoury:2013yya ; Saba:2017xur  has been
suggested as to drive an inflationary expansion. One particular characteristic
of this scalar field is that its mass depends on the matter density effects,
i.e a scalar field with a varying mass in a dense scenario, where the scalar
field can acquire a large mass in a short range. Some observational tests have
been performed at CMB scales Sheikhahmadi:2019gzs  to associate this scalar
field to an early accelerated expansion phase. In the context of $f(R)$, the
chameleon scalar field has been studied in Burrage:2017qrf ; Brax:2008hh .
However, if the chameleon scalar field can have the property a perfect scalar
needs to be proved.

To develop the latter, we are going to consider the following action:

$\displaystyle S=\int
d^{4}x\sqrt{-g}\frac{1}{2}\left[M^{2}_{\text{pl}}f(R)-\partial_{\mu}\phi\partial^{\mu}\phi-2V(\phi)+\mathcal{L}_{\mathrm{m}}\right].\quad$
(28)

In this landscape, we are going to consider a conformal transformation to
relate this action with a standard one in a scalar-tensor theory through

$\exp\left(-\frac{2\beta\phi}{M_{\text{pl}}}\right)=f^{\prime}(R).$ (29)

where $\beta=\sqrt{\frac{1}{6}}$ Brax:2008hh . In Einstein frame we can write
the metric as $\bar{g}_{\mu\nu}$ by a conformal transformation defined as:

$\bar{g}_{\mu\nu}=e^{-\frac{2\beta\phi}{M_{\text{pl}}}}g_{\mu\nu},$ (30)

therefore, we can rewrite (28) as

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left(\frac{R}{2\kappa}-\frac{1}{2}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right)+S_{\mathrm{m}}[e^{\frac{2\beta\phi}{M_{\text{pl}}}}],\quad$
(31)

where

$V(\phi)=\frac{M_{\text{pl}}^{2}\left[Rf^{\prime}(R)-f(R)\right]}{2f^{\prime}(R)^{2}}.$
(32)

Notice that $f(R)$ theories are equivalent to scalar-tensor theories in this
chameleon scheme. Now, from the conditions described in Sec.IV, we can compute
the explicit form for the chameleon geometric scheme in $f(R)$. First, we need
to consider a Lagrangian density of the form

$\displaystyle\mathcal{L}(X,\phi)=f(X)-V(\phi).$ (33)

While a form of $f(X)\propto\log(X)$ satisfies directly the condition 4, in
this scheme we need to verify this condition on the chameleon potential. In a
$f(R)$ theory to have a chameleon mechanism it is required that the
derivatives of the potential behave as:
$V^{\prime}(\phi)<0,~{}V^{\prime\prime}(\phi)>0,~{}\text{and,}~{}V^{\prime\prime\prime}(\phi)<0.$
From (32) we can derive

$\displaystyle V^{\prime}(\phi)$ $\displaystyle=$ $\displaystyle\frac{\beta
M_{\text{pl}}}{f^{\prime\,2}}\left[Rf^{\prime}-2f\right],$ (34) $\displaystyle
V^{\prime\prime}(\phi)$ $\displaystyle=$
$\displaystyle\frac{1}{3}\left[\frac{R}{f^{\prime}}+\frac{1}{f^{\prime\prime}}-\frac{4f}{f^{\prime\,2}}\right],$
(35) $\displaystyle V^{\prime\prime\prime}(\phi)$ $\displaystyle=$
$\displaystyle\frac{2\beta}{3M_{\text{pl}}}\left[\frac{3}{f^{\prime\prime}}+\frac{f^{\prime}f^{\prime\prime\prime}}{f^{\prime\prime\,3}}+\frac{R}{f^{\prime}}-\frac{8f}{f^{\prime\,2}}\right].$
(36)

Generally, these functions give tight constraints on the form selected for
$f(R)$, therefore for a specific form of $V(\phi)$ we can find a specific
$f(R)$. To perform this calculation, a potential that follows the chameleon
mechanism given by Eqs.(34)-(35)-(36) is a power law potential of the form:

$\displaystyle V(\phi)=M^{4+n}\phi^{-n},$ (37)

and introduce this expression in Eq.(32) and solve the differential equation
to obtain an exact form for $f(R)$:

$\displaystyle
f(R)=\frac{A+2^{2/3}B-32M^{n+4}M_{\text{pl}}^{4}-32M_{\text{pl}}^{4}\phi^{n}+4R^{2}M_{\text{pl}}^{8}}{16M_{\text{pl}}^{6}},\quad$
(38)

where

$\displaystyle A$ $\displaystyle=$
$\displaystyle\frac{2\sqrt[3]{2}RM_{\text{pl}}^{10}\left(R^{3}M_{\text{pl}}^{6}+2\right)}{B},$
(39) $\displaystyle B$ $\displaystyle=$
$\displaystyle{(C-2R^{6}M_{\text{pl}}^{24}+10R^{3}M_{\text{pl}}^{18}+M_{\text{pl}}^{12})}^{1/3},$
(40) $\displaystyle C$ $\displaystyle=$
$\displaystyle\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}.$
(41)

Eq.(38) is the specific $f(R)$ form for a perfect chameleon scalar field.
Also, we can perform the derivation of this latter expression to obtain:

$\displaystyle
f_{R}=\frac{1}{8}M_{\text{pl}}^{2}(f_{1}+f_{2}-f_{3}+f_{4}+4R),$ (42)

where

$\displaystyle f_{1}$ $\displaystyle=$
$\displaystyle\frac{3\sqrt[3]{2}R^{3}M_{\text{pl}}^{8}}{\sqrt[3]{-2R^{6}M_{\text{pl}}^{24}+10R^{3}M_{\text{pl}}^{18}+\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}+M_{\text{pl}}^{12}}},$
(43) $\displaystyle f_{2}$ $\displaystyle=$
$\displaystyle\frac{\sqrt[3]{2}M_{\text{pl}}^{2}\left(R^{3}M_{\text{pl}}^{6}+2\right)}{\sqrt[3]{-2R^{6}M_{\text{pl}}^{24}+10R^{3}M_{\text{pl}}^{18}+\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}+M_{\text{pl}}^{12}}},$
(44) $\displaystyle f_{3}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt[3]{2}R^{3}M_{\text{pl}}^{20}\left(R^{3}M_{\text{pl}}^{6}+2\right)\left(-2R^{3}M_{\text{pl}}^{6}-\frac{3\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}}{M_{\text{pl}}^{12}-4R^{3}M_{\text{pl}}^{18}}+5\right)}{\left(-2R^{6}M_{\text{pl}}^{24}+10R^{3}M_{\text{pl}}^{18}+\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}+M_{\text{pl}}^{12}\right){}^{4/3}},$
(45) $\displaystyle f_{4}$ $\displaystyle=$
$\displaystyle\frac{2^{2/3}R^{2}M_{\text{pl}}^{10}\left(-2R^{3}M_{\text{pl}}^{6}-\frac{3\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}}{M_{\text{pl}}^{12}-4R^{3}M_{\text{pl}}^{18}}+5\right)}{\left(-2R^{6}M_{\text{pl}}^{24}+10R^{3}M_{\text{pl}}^{18}+\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}+M_{\text{pl}}^{12}\right){}^{2/3}}.$
(46)

Introducing Eqs.(38)-(42) in Eq.(13), we obtain for the time-time component:

$\displaystyle-6\kappa\left(g_{3}+g_{4}-g_{5}+g_{6}+4R\right)M_{\text{pl}}^{2}\left(3p_{\text{fld}}+\rho_{\text{fld}}\right)+32\kappa\rho_{\text{fld}}=\frac{3\left(g_{1}+2^{2/3}g_{2}-32M^{n+4}M_{\text{pl}}^{4}-32M_{\text{pl}}^{4}\phi^{n}+4R^{2}M_{\text{pl}}^{8}\right)}{M_{\text{pl}}^{6}},$
(47)

where

$\displaystyle g_{1}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt[3]{2}RM_{\text{pl}}^{10}\left(R^{3}M_{\text{pl}}^{6}+2\right)}{\sqrt[3]{-2R^{6}M_{\text{pl}}^{24}+10R^{3}M_{\text{pl}}^{18}+\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}+M_{\text{pl}}^{12}}},$
(48) $\displaystyle g_{2}$ $\displaystyle=$
$\displaystyle\sqrt[3]{-2R^{6}M_{\text{pl}}^{24}+10R^{3}M_{\text{pl}}^{18}+\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}+M_{\text{pl}}^{12}},$
(49) $\displaystyle g_{3}$ $\displaystyle=$
$\displaystyle\frac{3\sqrt[3]{2}R^{3}M_{\text{pl}}^{8}}{\sqrt[3]{-2R^{6}M_{\text{pl}}^{24}+10R^{3}M_{\text{pl}}^{18}+\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}+M_{\text{pl}}^{12}}},$
(50) $\displaystyle g_{4}$ $\displaystyle=$
$\displaystyle\frac{\sqrt[3]{2}M_{\text{pl}}^{2}\left(R^{3}M_{\text{pl}}^{6}+2\right)}{\sqrt[3]{-2R^{6}M_{\text{pl}}^{24}+10R^{3}M_{\text{pl}}^{18}+\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}+M_{\text{pl}}^{12}}},$
(51) $\displaystyle g_{5}$ $\displaystyle=$
$\displaystyle\frac{2\sqrt[3]{2}R^{3}M_{\text{pl}}^{20}\left(R^{3}M_{\text{pl}}^{6}+2\right)\left(-2R^{3}M_{\text{pl}}^{6}-\frac{3\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}}{M_{\text{pl}}^{12}-4R^{3}M_{\text{pl}}^{18}}+5\right)}{\left(-2R^{6}M_{\text{pl}}^{24}+10R^{3}M_{\text{pl}}^{18}+\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}+M_{\text{pl}}^{12}\right){}^{4/3}},$
(52) $\displaystyle g_{6}$ $\displaystyle=$
$\displaystyle\frac{2^{2/3}R^{2}M_{\text{pl}}^{10}\left(-2R^{3}M_{\text{pl}}^{6}-\frac{3\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}}{M_{\text{pl}}^{12}-4R^{3}M_{\text{pl}}^{18}}+5\right)}{\left(-2R^{6}M_{\text{pl}}^{24}+10R^{3}M_{\text{pl}}^{18}+\sqrt{-M_{\text{pl}}^{24}\left(4R^{3}M_{\text{pl}}^{6}-1\right){}^{3}}+M_{\text{pl}}^{12}\right){}^{2/3}}.$
(53)

Notice that the expression (47) and its second order derivative satisfy the
conditions from 1 -3, therefore the chameleon field $\phi$ with potential
Eq.(32) describe a perfect fluid through the contributions from the geometric
part in $f(R)$.

##  VI Discussion

In this work, we considered a particular class of chameleon field $\phi$ with
a power-law potential of the form $V(\phi)=M^{4+n}\phi^{-n}$. Under this
assumption, we found that the chameleon can be associated with a perfect
scalar field since its behaviour on the evolution and conservative background
equations satisfy the conditions described by 1 -3. Furthermore, at the
perturbative level, $\phi$ fulfill directly the condition 4 if we consider a
Lagrangian of the form (33). For $n\geq 1$, we notice from Eq.(47) that local
gravity constraints are satisfied for M $\sim 10^{-2}$eV. As we mentioned,
this is a restricted condition on the energy scale required for the current
cosmic acceleration observed. While the conditions at the background evolution
seem a natural candidate for dark energy for this kind of potential, it is
interesting to notice that at a perturbative level, a chameleon perfect scalar
$\phi$ could emerge at first during a phase transition during the inflation
era and, finally end with a random position within its potential. Following
this evolution, there has been a study on attractors conditions in which the
chameleon $\phi$ could satisfy the big bang nucleosynthesis (BBN) constraints
until today Mota:2011nh , which behind the idea of a perfect scalar field at
perturbative level can help to set up an ensemble of initial conditions to
study furthermore its behaviour at the early universe. This study will be
reported elsewhere.

###### Acknowledgements.

LLP is supported by CONACyT National Grant. CE-R acknowledges the Royal
Astronomical Society as FRAS 10147 and is supported by DGAPA-PAPIIT-UNAM
Project TA100122. This article is based upon work from COST Action CA21136
Addressing observational tensions in cosmology with systematics and
fundamental physics (CosmoVerse) supported by COST (European Cooperation in
Science and Technology).

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# Practical Bias Mitigation through Proxy Sensitive Attribute Label Generation

Bhushan Chaudhari, Anubha Pandey, Deepak Bhatt, Darshika Tiwari

###### Abstract

Addressing bias in the trained machine learning system often requires access
to sensitive attributes. In practice, these attributes are not available
either due to legal and policy regulations or data unavailability for a given
demographic. Existing bias mitigation algorithms are limited in their
applicability to real-world scenarios as they require access to sensitive
attributes to achieve fairness. In this research work, we aim to address this
bottleneck through our proposed unsupervised proxy-sensitive attribute label
generation technique. Towards this end, we propose a two-stage approach of
unsupervised embedding generation followed by clustering to obtain proxy-
sensitive labels. The efficacy of our work relies on the assumption that bias
propagates through non-sensitive attributes that are correlated to the
sensitive attributes and, when mapped to the high dimensional latent space,
produces clusters of different demographic groups that exist in the data.
Experimental results demonstrate that bias mitigation using existing
algorithms such as Fair Mixup and Adversarial Debiasing yields comparable
results on derived proxy labels when compared against using true sensitive
attributes.

## Introduction

Machine Learning has attained high success rates in practically every field,
including healthcare, finance, and education, based on the accuracy and
efficiency of the model’s outcome (Dastile, Çelik, and Potsane 2020; Bakator
and Radosav 2018). However, these models are biased and exhibit a propensity
to favor one demographic group over another in various applications, including
credit and loan approval, criminal justice, and resume-based candidate
shortlisting (Mehrabi et al. 2021; Gianfrancesco et al. 2018; Yapo and Weiss
2018). The idea of fairness has received a lot of attention recently to combat
the discrimination from the outcome of ML models (Dwork et al. 2012; Beutel et
al. 2017; Hardt, Price, and Srebro 2016).

The existing bias mitigation techniques (Feldman et al. 2015; Sattigeri et al.
2019; Zafar et al. 2017) can be classified into three categories: pre-
processing (Feldman et al. 2015; Kamiran and Calders 2011), post-processing
(Hardt, Price, and Srebro 2016; Pleiss et al. 2017) and in-processing (Zafar
et al. 2017; Dwork et al. 2012). While pre-processing bias mitigation
techniques attempt to transform the input before feeding it to the model for
training, post-processing strategies filter out the output through certain
transformations. In order to produce fair output, in-processing strategies
strive to learn bias-invariant models by imposing certain constraints during
training. Nevertheless, most state-of-the-art algorithms require information
about sensitive attributes to produce an unbiased model. However, in practice,
these sensitive attributes are inaccessible due to difficulties in data
collection, privacy, and legal constraints imposed by the government, like
General Data Protection Regulation(GDPR) introduced by the European Union in
May 2018 and Equal Credit Opportunity Act (Grari, Lamprier, and Detyniecki
2022; Lahoti et al. 2020).

Fairness is challenging to achieve in the absence of sensitive attributes due
to a lack of supervision. While sensitive attributes are inaccessible in the
real-world setting, it has been found that some non-sensitive attributes have
strong correlations with the sensitive features, which leads to bias
propagating through AI models(Angwin et al. 2016; Coston et al. 2019). For
instance, Hispanic and black populations have a higher proportion of younger
people, resulting in the correlation between age and race (Vogel and Porter
2016). Similarly, zip codes can be correlated with race. Hence, the bias gets
embedded in the non-sensitive attributes that are used in the model training.
Based on this hypothesis, a few initial efforts have been made to mitigate
bias in the absence of protected attributes (Grari, Lamprier, and Detyniecki
2022; Lahoti et al. 2020; Yan, Kao, and Ferrara 2020; Zhao et al. 2022). The
most recent approach (Zhao et al. 2022) identifies related features that are
correlated with the sensitive attributes and would further minimize the
correlation between the related features and the model’s prediction to learn a
fair classifier with respect to the sensitive attribute. However,
identification of related features require domain knowledge and access to
sensitive attributes to determine the correlation.

This research aims to provide proxy labels for sensitive attributes to make
the present bias mitigation approaches suitable for real-world applications
where access to protected attributes during model training is constrained.
Ideally the likelihood of a positive outcome should be the same regardless of
person’s protected group. However in real life this does not hold true. The
group which has more likelihood of getting positive outcome just because of
their protected attribute is referred as favourable group and group which has
more likelihood of getting negative outcome just because of their protected
attribute is referred as unfavourable group in this paper. We determine proxy
for favorable and unfavorable groups by leveraging the bias information
embedded in the non-sensitive features available in the given dataset. This
proxy sensitive labels can then be passed as an input into the existing bias
mitigation techniques. Thus we address the bottleneck in the applicability of
the existing bias mitigation to real-world applications. We have proposed a
novel pipeline that involves two stages: (1) Stage-1: Learn embeddings using
self-supervised learning that captures inter-feature relationships and,
consequently, latent bias information. (2) Stage-2: Generate proxy for
demographic groups by clustering the samples based on the embeddings obtained
from Stage-1. Further, experimental analysis reveals that identical results
can be observed by using the proxy labels in the current bias mitigation
technique as opposed to the genuine labels of sensitive qualities.

## Related Work

A substantial amount of work has been done to address and mitigate bias in
data sets and models(Mehrabi et al. 2021; Yapo and Weiss 2018; Gupta, Bhatt,
and Pandey 2021). Based on the point of intervention of the modeling stage,
bias mitigation techniques broadly fall into three categories: pre-processing,
in-processing, and post-processing. Pre-processing techniques underpin the
first stage of the modeling and transform the training data so that the
underlying discrimination is removed (Bellamy et al. 2018; Zemel et al. 2013;
Calmon et al. 2017; du Pin Calmon et al. 2018). These techniques reduce or
eliminate the correlation between sensitive attributes and other features,
including the target labels. Unfortunately, due to the blindness of these
techniques to model’s inference of the data, some level of bias still can
creep into the model predictions. In-processing techniques modify learning
algorithms to remove bias during the model training process. Most of the
algorithms in this category solve constraint optimization problem for
different fairness objectives. To ensure independence between predictions and
sensitive attributes, (Woodworth et al. 2017) regularizes the covariance
between them. (Zafar et al. 2017) minimizes the disparity between the
sensitive groups by regularizing the decision boundary of the classifier.
(Zhang, Lemoine, and Mitchell 2018) proposed a data augmentation strategy for
optimizing group fairness constraints such as equalized odds and demographic
parity. Another efficient algorithm (Chuang and Mroueh 2021), tries to
maximize the predictor’s ability to predict the ground truth while minimizing
the adversary’s ability to predict the sensitive attribute. Post-processing
techniques treat the learned model as a black-box model and try to mitigate
bias from the prediction (Kamiran, Karim, and Zhang 2012; Hardt, Price, and
Srebro 2016; Pleiss et al. 2017; Canetti et al. 2019). Typically, post-
processing algorithms select a subset of samples and adjust the predicted
labels accordingly. An intriguing finding is that any sample can be altered to
meet the requirements of group fairness because the metrics are expectations.
The papers (Hardt, Price, and Srebro 2016; Pleiss et al. 2017) choose samples
at random, whereas (Kamiran, Karim, and Zhang 2012) choose the samples with
the greatest degree of uncertainty, reflecting the human tendency to give
unprivileged groups the benefit of the doubt.

Most of the current algorithms have restrictions on their use in real-world
scenarios since they need access to protected attributes for bias mitigation.
Very recently efforts have been made towards bias mitigation in the absence of
sensitive attributes (Grari, Lamprier, and Detyniecki 2022; Lahoti et al.
2020; Yan, Kao, and Ferrara 2020; Zhao et al. 2022). (Grari, Lamprier, and
Detyniecki 2022) introduced a framework based on bayesian variational
autoencoders that relies on knowledge of causal graph to derive proxy. The
algorithm estimates proxy in a multi dimensional space and then uses this
generated proxy to remove bias from the model. But, since the proxy are
generated in a multi dimensional space, they cannot be generalised to other
bias mitigation algorithms. The paper (Du et al. 2021a) introduced a framework
wherein it only performs debiasing on the classification head. The algorithm
neutralizes the training samples that have the same ground truth label but
with different sensitive attribute annotations. Proxy generation for the
sensitive attributes is done by training a bias intensified model and then
annotating samples based on its confidence level. However, the algorithm makes
a strong assumption that bias-amplified model tends to assign the privileged
group more desired outcome whereas assigning the under-privileged group a
less-desired outcome based on the obtained prediction scores. The most recent
approach (Zhao et al. 2022) identifies related features that are correlated
with the sensitive attributes and would further minimize the correlation
between the related features and the model’s prediction to learn a fair
classifier with respect to the sensitive attribute. To identify the related
features, however, this method needs access to sensitive attributes to
determine the correlation.

Figure 1: Proposed pipeline for proxy sensitive attribute label generation.

## Methodology

It is widely established that bias propagates to the models even when
protected attributes are not used during training (Grari, Lamprier, and
Detyniecki 2022; Lahoti et al. 2020; Yan, Kao, and Ferrara 2020; Zhao et al.
2022). This is attributed to the frequent incorporation of protected attribute
data into other correlated non-protected attributes. Zip codes, for instance,
can be associated to the race attribute. Based on this hypothesis, we utilize
the non-protected attributes to obtain proxy-sensitive labels. Assuming the
availability of all variables except the protected attribute, our goal is to
recover all the latent information associated with the protected attribute
embedded into the available non-protected features.

This section outlines our suggested method for generating a proxy for a
sensitive protected attribute. We break the objective down into two stages. In
the first stage, we utilize self-supervised learning to produce the contextual
embedding of the input samples. Our goal is to learn an embedding with maximum
information about the protected attribute. In the second stage, we obtain
proxy labels for favorable and unfavorable groups using an unsupervised
clustering approach on the embedding obtained from the first stage. Finally,
we pass the generated proxy through existing state-of-the-art bias mitigation
algorithms to mitigate bias from any model. Figure 1 outlines the proxy-
generation pipeline.

Figure 2: Overall pipeline for bias mitigation and evaluation

### Proxy Generation for Sensitive Attribute

Stage-1: In the first stage, as shown the figure 1, we obtain contextual
embedding of the input samples. Towards this goal, we train neural network
architectures in a self-supervised fashion to efficiently encode inter-feature
relationships. In this paper, we have experimented with two neural network
architectures: (1) Auto-encoders and, (2) Transformers.

We train an auto-encoder on the reconstruction task to obtain embeddings
containing crucial input data details. An auto-encoder consists of encoder and
decoder modules. In the encoding operation, we pass the input feature vector
that gets mapped to a lower dimensional latent representation. In the decoding
operation, the original input data gets reconstructed back from the latent
representation. We trained the network on a reconstruction loss that minimizes
the mean square error between the input and output embeddings. Input data $X$
is passed through the encoder to get latent representation $h$ and then
reconstructed as $\hat{X}$ by the decoder as shown in the equations 1 and 2.
We train the network on reconstruction loss $Loss_{AE}$ as shown in equation 3
where $n$ represents the number of data points in a batch. Here, $f1$ and $f2$
are activation functions, $W$ is weight matrix and $b$ is bias.The latent
embeddings obtained from the encoder module contain information about the
protected attribute as it is generated from features that are correlated with
the protected attribute.

$h=f1(W_{i}*X+b_{i})$ (1)
$\hat{X}=f2(W_{j}*h+b_{j})$ (2)
$Loss_{AE}=\frac{1}{n}\Sigma_{i=1}^{n}|{X_{i}-\hat{X_{i}}}|$ (3)

We experimented with another neural network architecture called Transformer
with a similar goal. Transformers utilize a self-attention (Vaswani et al.
2017) mechanism to learn the embeddings. To compute self-attention, first,
three vectors, Query(Q), Key(K), and (V), are learned corresponding to each
feature in the input, and then the attention is computed as shown in the
equation 4. Finally, self-attained embeddings $h$ are obtained as shown in the
equation 6. We train the Transformer on a self-supervised learning task called
Masked Language Modelling (MLM). Towards this, 15$\%$ of the input data fields
are chosen randomly and replaced with a masked token. The Transformer then
processes samples to produce contextual row embeddings. The MLM head, made up
of MLP layers, reconstructs the original fields from these row embeddings. The
model is trained end-to-end by minimizing cross-entropy loss as shown in the
equation 8. The loss is calculated only on masked fields. The latent
embeddings ($h$) obtained from the transformer contain information about the
protected attribute due to its inherent property to learn the inter-feature
relationships.

$Attention(Q,K,V)=softmax(\frac{QK^{T}}{\sqrt{d_{k}}})V$ (4)

$head_{i}=Attention(QW^{Q}_{i},KW^{K}_{i},VW^{V}_{i})$ (5)

$h=Concat(head_{1},...,head_{h})W^{O}$ (6)

$p_{i}=Softmax(MLP(h))$ (7)
$Loss_{T}=-\sum_{c=1}^{M}y_{i}\log(p_{i})$ (8)

Further to ensure that the generated embeddings do not corresponds to the true
labels of the downstream classification task, we have trained the above
described neural network models on KL Divergence loss. KL divergence loss
historically has been used in classification tasks to ensure class separation
between two different labels. The KL divergence loss is based on the
information theoretic measure of the Kullback-Leibler (KL) Divergence, which
measures the difference between two probability distributions. By introducing
the KL divergence loss, the model is able to learn the distinction between the
two different labels better, thus leading to improved embedding generation
which contains information related to protected attribute and not downstream
task labels.

In order to implement the Kullback-Leibler (KL) divergence in the proposed
neural network architecture, a multi-layer perceptron (MLP) layer has been
applied on the generated embedding vectors. In the autoencoder, the MLP is
applied on top of the latent vectors, while in the transformer, the MLP is fed
with the contextual vector (h). The calculation of the KL loss on top of the
MLP depends on the input embedding vector. Specifically, the input embedding
vector is fed into the MLP, which will generate the probability distribution.
Then, the KL divergence between the probability distribution and the target
distribution is calculated. This KL loss is then used to optimize the MLP
weights and biases.

Stage-2: In the second stage, as shown the figure 1, we use an unsupervised
clustering algorithm to identify various groups in the embeddings obtained
from the previous stage. As we know, clustering is a subjective statistical
analysis, and there are many algorithms suitable for each data set and problem
type. In this paper, we have experimented with centroid-based and hierarchical
clustering algorithms. In particular, we have experimented with K means,
Hierarchical and BIRCH to obtain two clusters that serves as a proxy for
favourable and unfavourable groups. We further evaluate the performance of
generated proxy from each clustering algorithm on bias mitigation.

### Bias Mitigation Through Generated Proxy Sensitive Attribute

Once the proxy labels are obtained corresponding to the favourable and
unfavourable groups, we pass them as input to the existing bias mitigation
algorithms. In this paper, we have experimented with two widely used
benchmarks for bias mitigation: Adversarial Debiasing and Fair Mixup. Both
algorithms require labels corresponding to the protected attribute in the
input. We pass the proxy for the protected attribute obtained from the
proposed pipeline as input to de-bias the model. We have compared the
performance on bias mitigation with the true labels and proxy labels for the
protected attributes in the results section. However, for fairness evaluation,
we use true sensitive labels. Figure 2 shows the pipeline for bias mitigation
and fairness evaluation.

## Experimental Details

### Dataset Description

We have evaluated the proposed pipeline on the Adult Income Dataset, generated
from 1994 US Census. The objective of the dataset is to predict the income
level based on personal individual information. The target variable,Y takes a
binary value depicting salary $\leq$ 50K or salary $>$ 50k.The dataset consist
of 14 independent attributes and the field ’Gender’ is considered as a
sensitive attribute in our case. It takes up two values, namely ’Male’ and
’Female’.The dataset is imbalanced: only 24% of the samples belong to class 1,
out of which only 15.13% are females. The dataset consist of 48,842
independent rows. During the training of our model, we do not take into
account the information provided by the ’Gender’ attribute.

Bias Mitigation Algortihm | Average Precision |  SPD |  EOD
---|---|---|---
w/o Bias Mitigation | 0.8 | 0.2 | 0.11
Fair Mixup | 0.78 | 0.1 | 0.03
Adversarial Debiasing | 0.78 | 0.19 | 0.05
Table 1: Performance evaluation of existing bias mitigation algorithms.

### Implementation Details

We have implemented the proposed pipeline in the Pytorch framework. All the
experiments were performed on Ubuntu 16.04.7 with the Nvidia GeForce GTX
1080Ti GPU. 16GB of RAM was utilized while experimenting on the Adult Income
Dataset.

In Stage-1, we experimented with two embedding generator networks,
autoencoders and transformers. The autoencoders used in the algorithm consist
of one hidden layer. The hidden layer’s output receives ReLU activation while
its input receives Tanh activation. The model was trained for 200 epochs with
a batch size of 32 and a learning rate of 0.001 using Adam as the optimizer.
The Transformer architecture contains only the encoder module. Three encoder
blocks are used with six attention heads. Each encoder module is a feed-
forward network with 128 hidden units. We used the implementation of the
Transformer provided in the hugging face library. In Stage-2, we experimented
with K-Means, BIRCH, and Hierarchical clustering algorithms to generate proxy
labels for protected attributes. We utilize the implementation of these
clustering algorithms given in python’s sklearn library.

We utilize all the data samples to train our proposed pipeline to obtain the
proxy labels for sensitive attributes. Next, we randomly split the dataset
into 80-20 train and test split and train the classification model using bias
mitigation algorithms on the train set. We employ existing bias mitigation
algorithms like Adversarial debiasing(Zhang, Lemoine, and Mitchell 2018)
provided in the IBM AIF360 toolkit and fair-mixup(Chuang and Mroueh 2021) an
open-source solution that is accessible on GitHub. During training, we use the
generated proxy instead of the actual labels of the protected attribute and
assess performance on the protected attribute’s actual labels.

### Fairness Metrics

Fairness in machine learning measures the degree of disparate treatment for
different groups (e.g., female vs. male), or individual fairness, emphasizing
similar individuals should be treated similarly. There exists various metrics
in the literature to quantify fairness, each focusing on different aspects of
fairness. We have used two popularly used metrics: Statistical Parity
Difference (SPD) and Equalized Odds Difference (EOD).

Statistical Parity Difference (SPD)  : A classifier is considered fair if the
prediction Y on input features X is independent from the protected attribute
S. The underlying idea is that each demographic group has the same chance for
a positive outcome. (Mehrabi et al. 2019)

$SPD=|{P(\hat{Y}=1|S=0)}-{P(\hat{Y}=1|S=1)}|$ (9)

Equalized Odds Difference (EOD) : An algorithm is considered fair if across
both privileged and unprivileged groups, the predictor Y has equal false
positive rate(FPR) and false negative rate(FNR). This constraint enforces that
accuracy is equally high in all demographics since the rate of positive and
negative classification is equal across the groups.The notion of fairness here
is that chances of being correctly or incorrectly classified positive should
be equal for every group.

$\displaystyle\triangle{FPR}=|\\{P(\hat{Y}=1|S=1,Y=0)-$ (10) $\displaystyle
P(\hat{Y}=1|S=0,Y=0)\\}|$

$\displaystyle\triangle{FNR}=|\\{P(\hat{Y}=0|S=1,Y=1)-$ (11) $\displaystyle
P(\hat{Y}=0|S=0,Y=1)\\}|$

$EOD=\frac{\triangle{FPR}+\triangle{FNR}}{2}$ (12)

| FairMixup |  Adversarial Debiasing
---|---|---
Embedding | Clustering | Avg Precision | SPD | EOD | Avg Precision | SPD | EOD
| K-Means | 0.76 | 0.16 | 0.09 | 0.78 | 0.06 | 0.06
AutoEncoder | Hierarchical | 0.78 | 0.11 | 0.06 | 0.76 | 0.05 | 0.01
| BIRCH | 0.77 | 0.13 | 0.07 | 0.79 | 0.16 | 0.09
| K-Means | 0.75 | 0.09 | 0.07 | 0.79 | 0.15 | 0.04
Transformer | Hierarchical | 0.77 | 0.07 | 0.05 | 0.77 | 0.12 | 0.04
| BIRCH | 0.75 | 0.13 | 0.07 | 0.79 | 0.11 | 0.09
Table 2: Fairness and performance results on two open bias mitigation
algorithms

## Results

In this section, we empirically assess the effectiveness of the proxy-
sensitive label obtained through the proposed pipeline. Towards this end, we
pass the proxy-sensitive labels through state-of-the-art bias mitigation
methods like adversarial debiasing and fair mixup and evaluate the fairness
and classification performance on a public dataset called UCI Adult Income. We
have reported the classification performance on Average Precision and fairness
on Statistical Parity Difference (SPD) and Equalized Odds Difference (EOD).

Fair mixup and adversarial debiasing bias mitigation algorithms require
protected attribute information to de-bias the models. To form the baseline,
we have passed true labels of the protected attribute gender through the
mentioned bias mitigation algorithms. Fair mixup has a trade-off parameter
between fairness and accuracy, called lambda. We set this parameter as 0.5 for
SPD and 2.5 for EOD. Table 1 compares the classification and fairness
performance of the model trained using bias mitigation algorithms like fair
mixup and adversarial debiasing against a classifier trained without using any
bias mitigation algorithm. From table 1, we can observe that the model trained
without any bias mitigation algorithm produce an average precision of 0.8 and
SPD and EOD metrics as 0.2 and 0.11. However, with model debiasing, we can see
an improvement in SPD and EOD values proving the efficacy of the bias
mitigation algorithms in achieving fairness.

In this paper, we concentrate on a more practical experimental setup, where we
have assumed that the protected attributes are unavailable during model
training. Here, we have used a proxy generated by our pipeline as an input to
the existing bias mitigation techniques discussed above rather than the true
labels of the protected attribute to test the efficacy of the generated proxy
in model debiasing. With proxy-sensitive labels, we aim to achieve a similar
performance as the baselines as shown in Table 1.

Figure 3: Average precision and fairness metrics obtained by different
techniques of clustering on embeddings generated by Autoencoder architecture.
Results are shown on two state-of-the-art bias mitigation techniques, Fair
Mixup (FM) and Adversarial Debiasing (AD), with proxy-sensitive labels as
input instead of true sensitive labels. Figure 4: Average precision and
fairness metrics obtained by different techniques of clustering on embeddings
generated by Transformer architecture. Results are shown on two state-of-the-
art bias mitigation techniques, Fair Mixup (FM) and Adversarial Debiasing
(AD), with proxy-sensitive labels as input instead of true sensitive labels.

We have experimented with several algorithms in both stages to generate proxy-
sensitive labels. In Stage 1, we experimented with Autoencoder and Transformer
architectures to generate the embeddings. And in Stage-2, we experimented with
clustering algorithms like K-means, Hierarchical, and BIRCH. Figure 3 shows
the performance of all the configurations when Autoencoder is used for
embedding generation, and Figure 4 shows the performance when Transformer is
utilized. From figure 3 we can observe that the proxy generated by
hierarchical clustering produces the best results with the adversarial
debiasing algorithm. In this configuration, we can observe an absolute
improvement of 0.14% in SPD with comparable average precision and EOD
performance when proxy labels are used instead of true labels for the
sensitive attribute Gender. Figure 4 shows that with the Fair mixup algorithm,
the best-performing configuration with proxy-sensitive labels has achieved an
average precision of 0.77 with EOD and SPD values of 0.05 and 0.07. This
performance is comparable to model performance with the true protected
attribute. On the other hand, with the Adversarial debiasing algorithm, the
embeddings obtained from the Transformer have led to a $1\%$ absolute lift in
the average precision while improving the fairness metrics compared to the
baseline model trained on true sensitive labels.

Transformer architecture to learn embedding in the proxy generation phase
produces a significant lift in fairness. The inherent properties of the
transformer architecture to learn the inter-feature relationships enables it
to generate informative embeddings for the tabular dataset. This is supported
by the experimental results shown in figures 3 and 4 on the Adult Income
dataset. However, the choice of the modeling architectures to obtain the
embedding and the clustering algorithms are dataset-dependent.

### Learned Embedding Analysis

The performance evaluation discussed in the above section indicates that the
proxy-sensitive labels can be used as a substitute for the true labels of
protected attributes in the existing bias-mitigation algorithms. In this
section, we analyze the quality of embeddings learned in Stage-1 of the
proposed pipeline through an auxiliary prediction task similar to (Du et al.
2021b).

Towards this effect, we train three linear classifiers
$C\textsubscript{Proxy}$, $C\textsubscript{True}$ and
$C\textsubscript{Downstream}$ that take the embeddings as input and predict
proxy attribute, true protected attribute, and target class labels
respectively. Next, we compare the learned weight matrix of
$C\textsubscript{Proxy}$ with $C\textsubscript{True}$ and
$C\textsubscript{Downstream}$ separately using cosine similarity.

The cosine similarity between the weight vectors of $C_{Proxy}$ and $C_{True}$
is 0.25, and between $C_{Proxy}$ and $C_{Downstream}$ is 0.02. A high value of
cosine similarity between weight parameters of $C_{Proxy}$ and $C_{True}$
indicates that embedding contains a substantial amount of information about
the true protected attribute. In contrast, a low cosine similarity value
between weights of $C_{Proxy}$ and $C_{Downstream}$ indicates that the
clusters formed over the embeddings are not along the downstream prediction
task.

## Conclusion

Bias mitigation with no access to sensitive attributes is a challenging
problem and has received little attention in the literature. Numerous relevant
research studies exist on fairness in AI, but most of these studies assume
that protected attributes are accessible at the time of training. This
assumption limits their use in modeling scenarios where protected labels are
unavailable. In an effort to reduce this dependency, we propose a novel
pipeline that leverages the inherent bias information in the non-protected
attributes to obtain proxy labels of protected attributes. In the current
state-of-the-art bias mitigation algorithms, these proxies are passed as input
rather than the true labels of the sensitive attribute. Experimental results
demonstrate that model trained using generated proxy labels results in
satisfactory bias metrics such as SPD and EOD with little or no reduction in
detection rate. In the future, we will continue to advance our research by
investigating more effective methods to incorporate additional bias
information into the embedding to improve proxy labels. Additionally, we would
validate the compatibility of the proposed approach with additional bias
mitigation algorithms beyond the algorithms studied in this work.

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