# Palatini $f(R)$ gravity tests in the weak field limit: Solar System,
seismology and galaxies

Alejandro Hernandez-Arboleda<EMAIL_ADDRESS>Cosmo-Ufes &
PPGCosmo,
Universidade Federal do Espírito Santo, Vitória, ES, Brazil.
Davi C. Rodrigues<EMAIL_ADDRESS>Departamento de Física & Cosmo-Ufes
& PPGCosmo,
Universidade Federal do Espírito Santo, Vitória, ES, Brazil.
Institut für Theoretische Physik,
Universität Heidelberg, Heidelberg, Germany.
Júnior D. Toniato<EMAIL_ADDRESS>Departamento de Química e Física &
Cosmo-Ufes,
Universidade Federal do Espírito Santo, Alegre, ES, Brazil.
Aneta Wojnar<EMAIL_ADDRESS>Departamento de Física Teórica & IPARCOS,
Universidad Complutense de Madrid, E-28040, Madrid, Spain.

((Day Month Year); (Day Month Year))

###### Abstract

Palatini $f(R)$ gravity is probably the simplest extension of general
relativity (GR) and the simplest realization of a metric-affine theory. It has
the same number of degrees of freedom as GR and, in vacuum, it is
straightforwardly mapped into GR with a cosmological constant. The mapping
between GR and Palatini $f(R)$ inside matter is possible but at the expense of
reinterpreting the meaning of the matter fields. The physical meaning and
consequences of such mapping will depend on the physical context. Here we
consider three such cases within the weak field limit: Solar System dynamics,
planetary internal dynamics (seismology), and galaxies. After revising our
previous results on the Solar System and Earth’s seismology, we consider here
the possibility of $f(R)$ Palatini as a dark matter candidate. For any $f(R)$
that admits a polynomial approximation in the weak field limit, we show here,
using SPARC data and a recent method that we proposed, that the theory cannot
be used to replace dark matter in galaxies. We also show that the same result
applies to the Eddington-inspired Born-Infeld gravity. Differently from the
metric $f(R)$ case, the rotation curve data are sufficient for this
conclusion. This result does not exclude a combination of modified gravity and
dark matter.

###### keywords:

modified gravity; galaxies; seismology; Solar System

##  1 Introduction

The $f(R)$ gravity models are simple and natural extensions of GR. Three large
classes of $f(R)$ gravity are the metric, the Palatini, and the hybrid cases
[1, 2, 3, 4, 5]. The Palatini case starts with a bold generalization: the
affine connection is independent of the metric. This suggests an increase in
the number of degrees of freedom concerning GR. However, the field equations
demand a relation between the metric and the connection, leading to two
degrees of freedom and becoming identical to GR with cosmological constant in
vacuum [6, 7]. Inside matter, the correspondence to GR will in general depend
on the definition of the physical system (e.g., [8, 9, 10, 11, 12, 13, 14]).

Due to the equivalence between $f(R)$ Palatini and GR in vacuum, it is evident
that the parameterised post-Newtonian (PPN) formalism in the Eddington-
Robertson-Schiff approach [15] will pose no constraints; this since this
approach is based on a vacuum solution. On the other hand, the PPN test due to
Will-Nordtvedt (WN) [16] is not based on Schwarschild solution or an extension
of it, moreover it tests 10 PPN parameters in its standard form, instead of
two from the Eddington-Robertson-Schiff approach. Nonetheless, in [10] we have
shown in detail that the WN PPN formalism cannot put constraints on $f(R)$
Palatini. The latter result depends on the redefinition of the pressure and
internal energy inside matter, but such redefinitions cannot be tested in the
context of the Solar System PPN application.

The PPN result above should not be seen as a statement that $f(R)$ Palatini
will always yield the same result of GR, independently on the physical
context. Indeed, in [17] some of us used seismology data from Earth and find a
constraint on the theory, which is here revised.

It is well known that some specific metric $f(R)$ theories can yield good
results for galaxy rotation curves without dark matter (e.g., [18, 19, 20,
21]). We test here if it is possible for Palatini to do something similar.
This is done using 122 galaxies form the SPARC dataset [22]. To this end, we
use an approach that some of us developed recently [23].

In the next section, we review $f(R)$ Palatini gravity in the weak field
limit. Section 3 is devoted to the PPN review, Sec. 4 for the seismology
review, and in Sec. 5 we present our main results. In A we briefly explain the
connection between the Palatini weak field limit and Eddington-inspired-Born-
Infeld (EiBI) theory [24]. This is relevant since some of our results only
depend on the weak field limit of $f(R)$ Palatini, these results are therefore
automatically valid for EiBI as well.

##  2 Palatini $f(R)$ and its weak field limit: a brief review

Here we review key aspects of Palatini $f(R)$ that are relevant for this work.
Our notation is close to the one we used in [10]. Extended reviews on Palatini
$f(R)$ gravity can be found in Refs. [25, 26, 3].

The (torsionless) Palatini $f(R)$ gravity action, in the presence of a matter
action $S_{{\mbox{\scriptsize\rm m}}}$ that depends on the matter fields
$\Psi$, reads,

$S[g,\Gamma,\Psi]=\frac{1}{2\kappa}\int
f\big{(}R(\Gamma,g)\big{)}\sqrt{-g}\,d^{4}x+S_{{\mbox{\scriptsize\rm
m}}}[g,\Psi]\,.$ (1)

In the above $\kappa$ is a constant, $g_{\mu\nu}$ is the spacetime metric,
${\Gamma}^{\lambda}_{\mu\nu}$ is an independent and torsionless affine
connection. That is, in the action, $\Gamma_{\mu\nu}^{\lambda}$ enters as an
independent field from the spacetime metric and
${\Gamma}^{\lambda}_{\mu\nu}={\Gamma}^{\lambda}_{\nu\mu}$. The Riemann tensor
is such that, for any vector field $A^{\lambda}$,
$R^{\lambda}_{\sigma\mu\nu}A^{\sigma}=[\nabla_{\mu},\nabla_{\nu}]A^{\lambda}$,
where $\nabla_{\mu}$ is the covariant derivative with respect to the affine
connection ${\Gamma}^{\lambda}_{\mu\nu}$. The Ricci tensor and Ricci scalar
are respectively $R_{\mu\nu}(\Gamma)\equiv
R^{\lambda}_{\mu\lambda\nu}(\Gamma)$ and $R\equiv
g^{\mu\nu}R_{\mu\nu}(\Gamma)$. Let us notice that in the case of Palatini
gravity the matter fields couple only to the spacetime metric, not to the
connection $\Gamma$.

The field equations derived from $g_{\mu\nu}$ and $\Gamma^{\mu}_{\nu\lambda}$
variations are, respectively,

$\displaystyle
f^{\prime}(R)R_{\mu\nu}-\dfrac{1}{2}\,f(R)g_{\mu\nu}=\kappa\,T_{\mu\nu}\,,$
(2)
$\displaystyle\nabla_{\lambda}\left(\sqrt{-g}f^{\prime}(R)g^{\mu\nu}\right)=0\,.$
(3)

The prime on $f(R)$ means the derivative with respect to $R$. $T_{\mu\nu}$ is
the energy-momentum tensor, which is defined by [27]

$T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta S_{\mbox{\scriptsize\rm
m}}}{\delta g^{\mu\nu}}\,.$ (4)

Equation (2), together with the matter field equations, leads to
$\nabla^{\mu}T_{\mu\nu}\not=0$ (where we recall that $\nabla_{\mu}$ is the
covariant derivative built from the connection $\Gamma^{\mu}_{\nu\lambda}$).
This is expected since the action variation with respect to $\Psi$, together
with diffeomorphism invariance of $S_{\mbox{\scriptsize\rm m}}$, implies
that111The appendix of Ref. [28] presents further general considerations about
$T_{\mu\nu}$ conservation, in a modified gravity context.

$\nabla_{\\!C}^{\mu}T_{\mu\nu}=0\,,$ (5)

where $\nabla_{\\!C}^{\mu}$ is the covariant derivative with respect to the
connection

$C_{\mu\nu}^{\lambda}=\frac{1}{2}g^{\lambda\sigma}\left(g_{\mu\sigma,\nu}+g_{\sigma\nu,\mu}-g_{\mu\nu,\sigma}\right)\,.$
(6)

In the above, a comma is used to denote a partial derivative. In particular,
this implies that test particles will follow geodesics from the connection
$C_{\mu\nu}^{\lambda}$, not $\Gamma_{\mu\nu}^{\lambda}$.

From the trace of eq. (2),

$f^{\prime}(R)R-2f(R)=\kappa T\,,$ (7)

which shows that $R$ can be algebraically expressed as a function of $T$. This
is in contrast to the metric $f(R)$ case.

In order to proceed with the weak field limit, and similarly to [10], we
consider the following $f(R)$ case,

$f(R)=R+\alpha R^{2}+O(R^{3})\,,$ (8)

where $\alpha$ is a constant. With the above, we are assuming that $f(R)$ can
be approximated by an analytical function about $R=0$, with $f(0)=O(R^{3})$.
We stress that the condition $f(0)=O(R^{3})$ is more general than $f(0)=0$.
This condition is natural and, considering the PPN context, it is necessary.
It is necessary to guarantee asymptotic flatness, otherwise PPN in its
standard form cannot be applied. Moreover, we are only assuming that a
cosmological constant-like term, if present, is not relevant for the
considered weak field system. Even if the complete $f(R)$ function may be non-
analytical, the approximation above is valid for any $f(R)$ without
singularities about $R=0$. In particular, this approximation is not valid for
models of the type $f(R)=1/R+O(R)$, which are commonly prone to strong
physical constraints (e.g., [29]).

Hence, eq. (7) implies that

$R=-\kappa T+O(R^{3}).$ (9)

It is convenient to express the connection $\Gamma_{\mu\nu}^{\lambda}$ as a
function of a “metric”, and indeed this is well known to be possible. Equation
(3) points out that $\Gamma^{\lambda}_{\mu\nu}$ can be written as a function
of a tensor ${\overline{g}}_{\mu\nu}$ that is conformally related to the
spacetime metric $g_{\mu\nu}$; indeed, let

${\overline{g}}_{\mu\nu}=f^{\prime}(R)\,g_{\mu\nu}\,,$ (10)

and hence the independent connection results as

$\Gamma_{\mu\nu}^{\lambda}=\frac{1}{2}{\overline{g}}^{\lambda\sigma}\left({\overline{g}}_{\mu\sigma,\nu}+{\overline{g}}_{\sigma\nu,\mu}-{\overline{g}}_{\mu\nu,\sigma}\right)\,.$
(11)

As usual for the weak field limit, the background is taken as Minkowski
$\eta_{\mu\nu}$,

${\overline{g}}_{\mu\nu}=\eta_{\mu\nu}+\overline{h}_{\mu\nu}\,.$ (12)

Using eq. (9), up to the first nontrivial order, the field equation (2) does
not depend on $\alpha$ and it is equivalent to the Einstein field equation for
the metric ${\overline{g}}_{\mu\nu}$, coupling constant $\kappa$ and energy-
momentum tensor $T_{\mu\nu}$.

For the three systems considered in this work, the relevant energy-momentum
tensor is that of a perfect fluid,

$\displaystyle
T^{\mu\nu}=\left(\rho+\rho\Pi+p\right)u^{\mu}u^{\nu}+pg^{\mu\nu}\,,$ (13)

where $\rho$ is the mass density, $\Pi$ is the fluid’s internal energy per
unity of mass, $p$ is the pressure and $u^{\mu}=u^{0}(1,\boldsymbol{v})$ is
the four-velocity of the fluid. Since the fluid particles are not assumed to
move at relativistic velocities, we consider $\rho\gg p$ and thus, as a first
approximation, $T\approx-\rho$.

Therefore, using $\overline{h}_{00}=-2\overline{\Phi}$,

$\nabla^{2}\overline{\Phi}\approx\frac{\kappa}{2}\rho\,.$ (14)

The difference with respect to GR comes from the Euler equation, which is
derived from eq. (5) and depends on the connection $C_{\mu\nu}^{\lambda}(g)$,
instead of $\Gamma^{\lambda}_{\mu\nu}({\overline{g}})$. This implies that test
particles will accelerate due to a gravitational potential $\Phi$, with

$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu},$ (15)

and $h_{00}=-2\Phi$. The Poisson equation becomes

$\nabla^{2}\Phi\approx\frac{\kappa}{2}\left(\rho+2\alpha\nabla^{2}\rho\right)\,.$
(16)

The above is the modified Poisson equation implied by Palatini $f(R)$ gravity
in the weak field regime. This equation was derived in [10]. We also point out
that Eddington-inspired Born-Infeld (EiBI) gravity [24] has precisely the same
weak field limit. This is shown in A. We focus this work on $f(R)$ Palatini,
but clearly several of our results can be immediately extended to EiBI
gravity.

##  3 Post-Newtonian analysis for Solar System tests

The study of the post-Newtonian (PN) equations of motion of a gravitational
theory is mandatory to confront Solar System observational data and
theoretical predictions. For torsionless Palatini $f(R)$ models, whose
Lagrangian is analytic about $R=0$, PN analysis has been made in Ref. [10]
under the optic of the parametrized post-Newtonian (PPN) formalism [30]. In
this section, we review their main results.

It is not possible to apply the PPN formalism to a fully arbitrary $f(R)$
function. But, for the large and relevant class of functions that can be
approximated by a polynomial expansion in the weak field limit, it can be
applied. For the PN analysis, it is relevant to require the knowledge of the
$f(R)$ approximation up to the $R^{3}$ term, hence let

$f(R)=R+a_{2}R^{2}+a_{3}R^{3}+O(R^{4}),$ (17)

with the coefficients $a_{2}$ and $a_{3}$ being constants. This notation is
the same used in [10]. We stress that the $\alpha$ introduced in eq. (8) is
identical to $a_{2}$ ($\alpha=a_{2}$). In this PPN context, we use $a_{2}$,
once $\alpha_{i}$ symbols usually represent PPN parameters.

We work with the same assumptions as in the standard PPN formalism, therefore,
asymptotic flatness demands no constant term in the expansion of $f(R)$, while
the coefficient for the linear term can be made equal to 1 due to the
arbitrariness of the theory’s coupling constant. The Newtonian limit must also
be assumed, and this is possible with the expansion (17).

To have complete PN equations of motion, it is only necessary to know $g_{00}$
up to fourth-order terms, $g_{0i}$ up to third order and $g_{ij}$ to second
order. After solving field equations, the expanded metric components will be
given by

$\displaystyle g_{00}\approx$ $\displaystyle\
-1+2U+2\Psi-4{\tilde{a}_{2}}\Phi_{P}+3{\tilde{a}_{3}}{\rho^{*}}^{2}$
$\displaystyle\
+{\tilde{a}_{2}}{\rho^{*}}\\!\left(2-10U+v^{2}\\!-2\Pi+6p/\\!{\rho^{*}}\\!-2{\tilde{a}_{2}}\rho^{*}\right)\\!,$
(18) $\displaystyle g_{0i}\approx$ $\displaystyle\ -4V^{i},$ (19)
$\displaystyle g_{ij}\approx$ $\displaystyle\
\delta_{ij}+2\left(U-{\tilde{a}_{2}}{\rho^{*}}\right)\delta_{ij}.$ (20)

where we have worked with units where $G=c=1$, taking $\kappa=8\pi$ and we
defined ${\tilde{a}_{2}}=8\pi a_{2}$ and ${\tilde{a}_{3}}=64\pi^{2}a_{3}$. The
${\rho^{*}}\equiv{u^{0}}\sqrt{-g}\,\rho$ is the conserved fluid’s rest mass
density. Also, $U$ is the negative of the Newtonian potential, $\Psi$ groups
all the fourth-order standard PPN potentials and $\Phi_{P}$ is a particular
potential for Palatini $f(R)$ gravity. All these potentials are defined in
terms of the conserved density.

The first issue related to the expression (18) is that in the absence of
matter, it does not recover the equivalent expansion of GR. This is so because
the potential $\Phi_{P}$ does not vanish in vacuum regions (the remaining
terms in $g_{00}$ with ${\rho^{*}}=p=0$ coincide with the ones obtained in
GR). Hence, in order to obtain the equivalence between Palatini $f(R)$ and
general relativity in vacuum, one must construct a redefinition of matter
quantities, namely

$\rho^{*}\Pi_{\mbox{\scriptsize eff}}+3p_{\mbox{\scriptsize
eff}}=\rho^{*}\Pi+3p-2{\tilde{a}_{2}}{\rho^{*}}^{2}\,.$ (21)

We will further comment on the implications of such redefinition.

It would be tempting to obtain all the PPN parameters from metric components
(18)–(20). However, there are several terms in these expressions that are
outside the standard formalism parametrization. It is common to see in the
literature analysis exploring definitions of effective PPN parameters that
would encompass the PN metric modifications brought by an alternative theory
and use them to constraint-free parameters. Notwithstanding, we disagree with
this method of work since all the PPN parameters have their physical meaning
(i.e. the precise relation to the observational phenomena) constructed with
the assumption that they are constant in time and space. The correct approach
should be to reanalyse conservation laws and equations of motion in order to
verify if and how the new PN terms affect the original PPN parameters.

In the specific case of Palatini $f(R)$, the trajectory of electromagnetic
waves (or light in particular) can be understood directly. Since all the
experiments used to constrain the light motion do not depend on a given
medium, and only include second-order metric perturbations, such
electromagnetic experiments cannot infer any difference with respect to GR.
Therefore, it is possible to say that $\gamma=1$ within Palatini $f(R)$
gravity. This is in contrast with Ref. [31]: the reason for the difference
comes from non-asymptotically flat effects which should not be considered in
PPN, unless a proper explanation of the validity of the method is included.

From equation (5), and considering the PN metric components (18)-(20), one can
extract three conservation statements. For $\mu=0$ the leading terms recover
the continuity equation for ${\rho^{*}}$ and a conserved mass $m$ is found.
The total energy $E$ which is conserved in post-Newtonian Palatini expansion
is obtained from the next leading order and it is the same as the one derived
in GR with a source whose effective internal energy is given by
$\Pi_{\mbox{\scriptsize eff}}=\Pi-{\tilde{a}_{2}}{\rho^{*}}/2$. Moreover, with
the definition of the gravitational mass as $M=m+E$, both theories also agree
on this quantity, which is consistent with their equivalence in vacuum.

The previous redefinition of the fluid’s internal energy, together with
condition (21), fixes an expression for $p_{\mbox{\scriptsize eff}}$ also that
is justly the pressure term appearing at the Newtonian-order of Euler’s
equation. The latter is derived from the leading order terms of the case
$\mu=i$ in eq. (5), with the next order determining the post-Newtonian
conserved momentum in Palatini $f(R)$. Thus, one can note that
$p_{\mbox{\scriptsize eff}}$ is the relevant pressure in the sense that it is
the one giving a force per unity of area that is locally generated by the
fluid. This is the usual method to macroscopically measure pressure. Within
this approach, the effective pressure and internal energy are the important
information from the fluid. One could also interpret that a fluid’s pressure
or its internal energy may be determined from microphysics principles, which
would put into test the aforementioned redefinitions (see Ref. [10] for a more
detailed discussion).

Notwithstanding, Palatini $f(R)$ does not violate total conservation of energy
and momentum in the PN regime and, within the context of the PPN formalism,
this indicates that the parameters $\zeta_{1},\zeta_{2},\zeta_{3},\zeta_{4}$
and $\alpha_{3}$ did not have their physical meaning altered by the presence
of the $\Phi_{P}$ potential in the PN expansion of the metric. In conclusion,
all these five parameters are zero in Palatini $f(R)$ gravity, which is
consistent with the observational bounds.

The remaining PPN parameters to be determined are $\beta$, $\alpha_{1}$ and
$\alpha_{2}$. Using the previous results, this task is completed through the
analysis of the equation of motion of massive bodies. The procedure is to
split the fluid description into $N$ well-separated bodies and describe their
motion in terms of centre-of-mass positions. The details are described in Ref.
[10].

The final results for a body $A$ acceleration can be written as

$\displaystyle\boldsymbol{a}_{\scriptscriptstyle A}$ $\displaystyle=$
$\displaystyle\sum_{\scriptscriptstyle B\neq A}\dfrac{M_{\scriptscriptstyle
B}}{r_{\scriptscriptstyle AB}^{2}}\bigg{\\{}\\!\bigg{[}1+v_{\scriptscriptstyle
A}^{2}-4(\boldsymbol{v}_{\scriptscriptstyle
A}\\!\cdot\\!\boldsymbol{v}_{\scriptscriptstyle B})+2v_{\scriptscriptstyle
B}^{2}\,-\dfrac{3}{2}(\boldsymbol{n}_{\scriptscriptstyle
AB}\\!\cdot\\!\boldsymbol{v}_{\scriptscriptstyle
B})^{2}-\dfrac{5M_{\scriptscriptstyle A}}{r_{\scriptscriptstyle
AB}}-\dfrac{4M_{\scriptscriptstyle B}}{r_{\scriptscriptstyle
AB}}\bigg{]}\boldsymbol{n}_{\scriptscriptstyle AB}$ $\displaystyle\ \ -\
\Big{[}\boldsymbol{n}_{\scriptscriptstyle
AB}\cdot(4\boldsymbol{v}_{\scriptscriptstyle
A}-3\boldsymbol{v}_{\scriptscriptstyle
B})\Big{]}(\boldsymbol{v}_{\scriptscriptstyle
A}-\boldsymbol{v}_{\scriptscriptstyle B})\,+\dfrac{7}{2}\sum_{C\neq
A,B}\\!\\!M_{\scriptscriptstyle C}\dfrac{r_{\scriptscriptstyle
AB}}{r_{\scriptscriptstyle BC}^{2}}\,\boldsymbol{n}_{\scriptscriptstyle BC}\
+$ $\displaystyle\ \ -\sum_{\scriptscriptstyle C\neq A,B}M_{\scriptscriptstyle
C}\left[\dfrac{4}{r_{\scriptscriptstyle AC}}+\dfrac{1}{r_{\scriptscriptstyle
BC}}-\ \dfrac{r_{\scriptscriptstyle AB}}{2r_{\scriptscriptstyle
BC}^{2}}(\boldsymbol{n}_{\scriptscriptstyle
AB}\cdot\boldsymbol{n}_{\scriptscriptstyle
BC})\right]\boldsymbol{n}_{\scriptscriptstyle AB}\bigg{\\}}\,.$

In the above expressions, we use the definitions $M_{\scriptscriptstyle
A}=m_{\scriptscriptstyle A}+E_{\scriptscriptstyle A}$,
$\boldsymbol{r}_{\scriptscriptstyle AB}=\boldsymbol{r}_{\scriptscriptstyle
A}-\boldsymbol{r}_{\scriptscriptstyle B}$, $r_{\scriptscriptstyle
AB}=|\boldsymbol{r}_{\scriptscriptstyle AB}|$ and
$\boldsymbol{n}_{\scriptscriptstyle AB}=\boldsymbol{r}_{\scriptscriptstyle
AB}/r_{\scriptscriptstyle AB}$.

These expressions for the accelerations are identical to the corresponding GR
expressions [32] (this is evident also by the absence on either $a_{2}$ or
$a_{3}$). Thus, one can compare the term $\boldsymbol{a}_{\scriptscriptstyle
A}^{\mbox{\tiny PN}}$ with the general PPN formula for the post-Newtonian
acceleration and conclude that $\beta=1$ and $\alpha_{1}=\alpha_{2}=\xi=0$.
Therefore, the values of all the PPN parameters in Paltini $f(R)$ are the same
as GR. Moreover, the physical meaning of all parameters are not changed by the
presence of a non-standard PPN potential in the metric expansion. In
conclusion, all the bounds on the PPN parameters not only can be safely used
for Palatini $f(R)$ but they also are all in complete agreement with
observations. It should be noted that this result does not imply complete
equivalence between Palatini $f(R)$ and GR, just that they cannot be
differentiated within the PN test frameworks.

Although the last results guarantee that Palatini $f(R)$ is in agreement with
all Solar System tests, it also points out that the theory can only be
constrained in systems where further knowledge of the matter source is
possible. As we will see next, planetary seismology should allow us to move
forward to this task.

##  4 Planetary Physics

Recent advancements in seismographic tools and laboratory experiments have
significantly increased our knowledge about the interior of our planet [33,
34, 35]. This includes the study of iron’s properties and behaviour under
extreme temperatures and pressures in the Earth’s core [36]. Furthermore, new
neutrino telescopes are providing valuable information on the density,
composition, and abundance of light elements in the outer core, reducing
uncertainties related to the Earth’s core characteristics [37, 38, 39].

There is a small but still notable impact of Palatini gravity effects on
planetary physics. The correction term in the Poisson equation (16) (and in
the resulting hydrostatic equilibrium equation, see below) has not only an
effect on the early [40] and late [41] evolution of the giant (exo-)planets,
but also on the interiors of the solid, Earth-like ones [42, 43].

Due to that fact, since modified gravity affects the densities and thicknesses
of the layers [44] in solid planets, we have a chance to constrain
gravitational models, especially in the light of more accurate data about the
Solar System planets and the Earth in particular. However, in order to begin
constraining theories, we need an Earth model as a reference. The most
commonly used global seismological Earth model is the Preliminary Reference
Earth Model (PREM), which provides pressure, density, and elastic moduli
profiles as functions of depth [34], see the figure 1 in [17]. This model is
based on velocity-depth profiles given by the travel-time distance curves for
seismic waves and periods of free oscillations. The missing element to
determine these profiles is a hydrostatic equilibrium equation which in case
of PREM is the Newtonian one

$\frac{dP}{dr}=-\rho\frac{GM(r)}{r^{2}}),$ (23)

thus it does depend on a model of gravity. Since we do not have in our
disposal a gravity-independent model of Earth yet, one assumes that PREM is an
accurate model of our planet. Being equipped with the mass and moment of
inertia of our planet, we will briefly discuss how one can use the Earth’s
seismic data to constrain a given gravitational proposal [17]. To do so, we
further assume that the planet is spherical-symmetric and that we are dealing
with adiabatic compression - that is, there is no heat exchange between the
layers such that we can neglect the temperature variation term in the
hydrostatic equilibrium equation:

$\displaystyle\frac{d\phi}{dr}$ $\displaystyle=-\rho^{-1}\frac{dP}{dr},.$ (24)

It can be derived from (16) for Palatini $f(R)$:

$\frac{dP}{dr}=-\rho\left(\frac{GM(r)}{r^{2}}+\alpha\kappa\rho^{\prime}(r)\right)=:-\rho
g_{\text{eff}},$ (25)

where the pressure $P$ and density $\rho$ are functions of the radial
coordinate $r$ while mass $M(r)$ is given by

$M(r)=4\pi\int_{0}^{r}r^{2}\rho(r)dr.$ (26)

Moreover, another assumption is made on the planet’s layers: they are radially
symmetric shells with the density jump between the inner and outer core
$\Delta\rho=600$, central density $\rho_{c}=13050$ and density at the mantle’s
base $\rho_{m}=5563$ (in kg/m3). The outer layers (starting from the upper
mantle) are described by the Birch law $\rho=a+bv_{p}$, in which $a$ and $b$
are parameters depending on the mean atomic mass of the material in the upper
mantle [34]. The velocity $v_{p}$ is the longitudinal elastic wave and it
appears in the seismic parameter $\Phi_{s}$ [33]

$\Phi_{s}=v_{p}^{2}-\frac{4}{3}v_{s}^{2},$ (27)

together with the transverse elastic one $v_{s}$. Since the seismic parameter
is related to the bulk modulus (incompressibility of a given material)

$K=\frac{dP}{d\mathrm{ln}\rho}$ (28)

as $\Phi_{s}=K/{\rho}$, it carries the information on the properties of
matter, that is, the equation of state,

$\Phi_{s}=\frac{dP}{d\rho}.$ (29)

Therefore, we can rewrite (25) as

$\frac{d\rho}{dr}=-\rho g_{\text{eff}}\Phi_{s}^{-1}.$ (30)

Mass (26) and moment of inertia ($R$ is Earth’s radius)

$I=\frac{8}{3}\pi\int_{0}^{R}r^{4}\rho(r)dr,$ (31)

are our constraints, given by observations with a high accuracy [45, 46].

The solution of the hydrostatic equilibrium equation (30) with the mass
equation (26) and seismic data (the longitudinal and transverse elastic waves
$v_{p}$ and $v_{s}$ given by [34]) provide the density profile. Because of the
modifications in (30), one expects to obtain different density profiles [44,
17] than the ones given by the PREM model. Indeed, choosing a few values of
the parameter $\alpha$, one obtains slightly different profiles with respect
to the Newtonian one, see figures in [42].

The effects of gravity models on a planet’s mass and moment of inertia are
given by calculating errors for fixed parameter values
$(\rho_{c},\rho_{m},\Delta\rho)$, see figure 4 in [17]. The results show that
the deviations exceed the 2$\sigma$ level when the parameter $\alpha$ is
around $10^{9}$ m2, similar to the upper limits in other works [47, 48]. The
planet’s mass is slightly more sensitive to changes in $\beta$ than its moment
of inertia due to the outermost layers’ significant contribution to the moment
of inertia calculation (which is assumed to satisfy Birch law). However, even
small changes in the planet’s core structure and composition can significantly
affect its moment of inertia, as we could see. Using this property and simple
analysis summarised here, one can constrain Palatini $f(R)$ ($\alpha\lesssim
10^{9}\text{m}^{2}$) and Eddington-inspired Born-Infeld gravity222See A. with
the seismic data.

##  5 Galaxies

###  5.1 General considerations

Here we consider the possible effects of $f(R)$ Palatini on galaxy dynamics.
In particular, we test the possibility of replacing dark matter with
gravitational effects. To this end, instead of doing a hard-to-implement and,
in this context, not particularly informative analysis of galaxy-by-galaxy
fits, we use the recently proposed Normalized Additional Velocity (NAV) method
[23].

For an axisymmetric system in the weak field limit (16) with cylindrical
radial coordinate $r$ and density $\rho(r,z)$, the circular velocity in the
plane $z=0$ is

$\displaystyle\frac{V^{2}}{r}$ $\displaystyle=\partial_{r}\Phi$
$\displaystyle=\partial_{r}\overline{\Phi}+\kappa\alpha\partial_{r}\,\rho\,.$
(32)

As revised in Sec. 3, some of us [10], in a post-Newtonian context, found that
there are no differences between Palatini $f(R)$ and GR considering the Solar
System dynamics (apart from a map between fluid quantities that cannot be
tested in that context). There is a Newtonian order correction to gravity,
given by $\kappa\alpha\partial_{r}\,\rho$, but this correction is irrelevant
for the center of mass trajectory of a body. Thus Newtonian gravity fixes
$\kappa=8\pi G$. However, this $\rho$ dependent correction is, in general,
relevant for the internal dynamics of fluids, in particular for the stability
of planets or galaxies [10]. We consider here this development to be galaxies.

Before proceeding, there is still a relevant consideration that needs to be
fixed: the fluid description. We assume here that a galaxy is a fluid that can
be decomposed into stellar and gaseous parts. We will not consider here the
dark matter component, since we look at the possibility of replacing it with
Palatini $f(R)$ effects. These two considered parts are assumed to have
negligible pressure, in the sense $p\approx 0$. Therefore, eq. (14) is here
interpreted as the Newtonian Poisson equation that is usually applied to
galaxies, with $\kappa=8\pi G$, and $\alpha$ parameterises Newtonian gravity
deviations.

###  5.2 Thin disks and $f(R)$ Palatini

Since disk galaxies have thin disks, it is important to verify if this
structure of thin disks is compatible with Palatini gravity without dark
matter. In particular, if the approximation of razor-thin disks [49] (i.e.,
disks with negligible thickness, $\rho(r,z)=\Sigma(r)\delta(z)$) is sensible
within Palatini. This analysis is sufficient for finding that $\alpha<0$, as
it will be shown.

For an axisymmetric disk, with $z=0$ as the plane of the disk, the Newtonian
acceleration $\mathbf{\bar{a}}=-\boldsymbol{\nabla}\Phi$ has a non-null
component that is perpendicular to the disk at $z\not=0$.

From eq. (16), one sees that the $z$-component acceleration is given by

$a_{z}=\bar{a}_{z}+8\pi G\alpha\partial_{z}\rho\,,$ (33)

where $a_{z}=(\mathbf{\bar{a}})_{z}$ is the Newtonian $z$-component
acceleration. Hence, outside the disk the Palatini contribution is null, that
is $\mathbf{a}=\mathbf{\bar{a}}$. In the disk, the acceleration becomes
infinite if $\rho\propto\delta(z)$. If this force is towards the disk, no
matter could escape the disk and the disk is stable, this corresponds to
$\alpha<0$. Clearly, the disk is unstable for $\alpha>0$. Hence, there is no
qualitative issue with using razor-thin disks in Palatini gravity with
$\alpha<0$. In general, disks with finite thickness would be more stable in
Palatini than in Newtonian gravity.

Another issue is the disk thickness influence on the circular velocity within
Palatini $f(R)$ gravity. For Newtonian gravity, there is an influence, but for
the Palatini non-Newtonian contribution there is nothing more. Indeed, for
$\rho(r,z)=\Sigma(r)Z(z)$, which is how disk thickness is commonly modelled,
the non-Newtonian Palatini contribution to the rotation curve is the same,
independently of the details of the function $Z(z)$. Since it only depends
locally on the density, at $z=0$ it only depends on the constant $Z(0)$.

In order to understand if Palatini $f(R)$ gravity can somehow replace dark
matter in galaxies, the standard galaxy-by-galaxy fit approach requires the
knowledge of the mass distribution of several galaxies. If it is known, then
it will be necessary to solve the modified Poisson equation (16) for each one
of them, while taking into account the observational errors on the circular
velocity, mass-to-light ratios and ideally on the inclination and distance as
well. After that, all the results should be put together and analysed if the
systematics of the individual galaxies are reasonable in some sense. A very
simple and common way to proceed is to compute the reduced chi-squared of each
one of the galaxies. Nonetheless, this demands a significative amount of work
that, by itself, does not clearly elucidate if the rotation curve contribution
from $f(R)$ Palatini can, on average, mimic the expected non-Newtonian
contribution. It is also important to stress that many well-known galaxy
samples do not provide a detailed and complete profile of the baryonic mass
distribution. On the other hand, the circular velocity contributions from the
gas and stellar parts are always provided (which is sufficient information for
testing dark matter profiles, but not modified gravity).

One can criticize the replacement of dark matter by Palatini gravity by other
arguments outside the scope of galaxy rotation curves, for instance by
considering gravitational lensing or the CMB. Nonetheless, our purpose here is
to study the effects of galaxy rotation curves alone, both because
understanding the issues in this context may lead to new ideas, and to
illustrate the usage of this recent method within a different model.

###  5.3 Normalized additional velocity definition

In [23] we introduced a method (the normalized additional velocity, NAV) to
deal with dark matter or modified gravity effects on galaxy rotation curves.
This method focuses on evaluating the shape of the dark matter or non-
Newtonian contribution, instead of the amplitude. What is tested is already
relevant, and it is sufficient for showing that the non-Newtonian contribution
from Palatini gravity is far from sufficient to replace dark matter. Also, the
results are independent of whether $\alpha$ is constant or changes from galaxy
to galaxy. The issue of Palatini effects together with dark matter models is
not here considered. One would first need to address what would be the dark
matter halo profile in such a case.

The observational additional velocity is defined as [23]333See also [50] for a
different approach that uses essentially the same definition.

$\Delta V^{2}_{{\mbox{\scriptsize\rm obs}}}\equiv V^{2}_{{\mbox{\scriptsize\rm
obs}}}-V^{2}_{{\mbox{\scriptsize\rm bar}}}\,.$ (34)

and the model additional velocity by

$\Delta V^{2}_{{\mbox{\scriptsize\rm mod}}}\equiv V^{2}_{{\mbox{\scriptsize\rm
mod}}}-V^{2}_{{\mbox{\scriptsize\rm bar}}}\,.$ (35)

In the above, $V_{\mbox{\scriptsize\rm obs}}$ is the observational rotation
curve (RC), $V_{\mbox{\scriptsize\rm mod}}$ is the model RC and
$V_{\mbox{\scriptsize\rm bar}}$ is the expected baryonic RC (which is
essentially inferred from the stars and the hydrogen surface brightness). To
be more explicit,

$V^{2}_{{\mbox{\scriptsize\rm
bar}}}={V^{2}_{\mbox{\scriptsize\rm*}}}+{V^{2}_{\mbox{\scriptsize\rm
gas}}}\,,$ (36)

where ${V^{2}_{\mbox{\scriptsize\rm*}}}$ and ${V^{2}_{\mbox{\scriptsize\rm
gas}}}$ are the squared circular velocities of the stellar disk and the gas
(atomic hydrogen and helium). The mass-to-light ratios are included in these
quantities. In general, it is common to divide the stellar contribution into
disk and bulge. Here we will only consider galaxies that do not have a
relevant bulge.

These squared velocities need not be positive since they are not truly the
square of a given physical velocity, they are actually the radial acceleration
component multiplied by $r$ (this is a common convention).

${r_{\mbox{\scriptsize\rm n}}}$$\delta V^{2}$ Figure 1: The observational NAV
plane from SPARC data. It is inferred from eq. (37) and 122 SPARC galaxies (it
uses the same quality cuts used to derived the radial acceleration relation,
reducing the sample from 175 to 153, and neglects galaxies with a relevant
bulge). The two contours show the densest data regions: the densest region
with 68% of the data in dark grey (1$\sigma$ region), and the densest region
which includes about 95% of the data is in light grey (2$\sigma$ region).
These regions are not confidence regions, they display the distribution of
observational data. The two vertically greyed regions display the regions with
larger uncertainty which are not considered in any NAV analysis.

The method is based on the normalized additional velocity (NAV), denoted by
$\delta V^{2}$, as a function of the normalized radius
(${r_{\mbox{\scriptsize\rm n}}}$). It is defined by

$\delta V^{2}({r_{\mbox{\scriptsize\rm n}}})\equiv\frac{\Delta
V^{2}({r_{\mbox{\scriptsize\rm n}}}\,r_{\mbox{\scriptsize\rm max}})}{\Delta
V^{2}(r_{\mbox{\scriptsize\rm max}})}\,,$ (37)

where $r_{\mbox{\scriptsize\rm max}}$ is the largest radial value with RC data
and ${r_{\mbox{\scriptsize\rm n}}}\equiv r/r_{\mbox{\scriptsize\rm max}}$. It
is relevant to stress that $r_{\mbox{\scriptsize\rm max}}$ is not intended to
reflect a physical property of a galaxy, it is simply the radius where the RC
observation stops. This end depends on the quality of the observational data.
Nonetheless, this parameter is important for studying galaxies of a given
sample. In [23] we comment on the use of different radial parameters to
generate a normalised radius.

From the NAV definition (37), and using the SPARC data [22] for the galaxies
without a relevant bulge, we find the NAV observational distribution as given
by Fig. 1.

The essence of the method is to compare the observational NAV distribution
$\delta V^{2}_{\mbox{\scriptsize\rm obs}}$ with the model-inferred one, which
is here due to Palatini gravity and denoted by $\delta
V^{2}_{\mbox{\scriptsize\rm P}}$. This is significantly easier to do than
performing hundreds of fits of galaxies, and moreover, this approach does not
require precise knowledge of the baryonic matter distribution: approximations
that are not expected to be suitable for all the individual galaxies can be
used for the sample, without impact on the $\delta V^{2}$ sample distribution.
This will be done in detail in the next subsection.

###  5.4 Baryonic matter approximations

For the majority of modified gravity theories, one needs to know the three-
dimensional baryonic matter distribution $\rho(r,z)=\rho_{*}(r,z)+\rho_{\rm
gas}(r,z)$. The first difficulty is that these data are not known in detail or
provided, and the second is that, for several cases, technically it is
significantly harder to derive the rotation curve from modified gravity models
than from Newtonian gravity. Hence, it is common that, to test modified
gravity, one considers baryonic matter approximations (indeed, [52, 53, 54]
are some examples that use SPARC data approximations). We remark that the
three-dimensional baryonic matter profile is not needed to test dark matter
profiles, the stellar and gaseous contributions to the circular velocity
profiles are sufficient for the latter, and these are commonly provided in
galaxy rotation curve samples.

For the stellar component, we use an exponential thin disk as an approximation
[55, 56]. The surface density is given by

$\Sigma(r)=\Sigma_{0}e^{-r/h}\,,$ (38)

where the constants $h$ and $\Sigma_{0}$ are found for each one of the 122
galaxies such that their corresponding rotation curve best approximates the
stellar rotation curve provided by SPARC (a similar procedure was also done in
[21], but for a subset of galaxies). The values we derive are different from
references that use the exponential profile to extend the observed stellar
profile.

The circular velocity due to an axisymmetric exponential profile of negligible
thickness has a well-known analytical expression [55, 49], which reads

$V^{2}=4\pi G\Sigma_{0}hy^{2}[I_{0}(y)K_{0}(y)-I_{1}(y)K_{1}(y)]\,,$ (39)

where $y\equiv r/(2h)$, and $I_{n}$ and $K_{n}$ are modified Bessel functions
of the first and the second kind, respectively. Therefore, we find the stellar
exponential component parameters ($h$, $\Sigma_{0}$) that best approximate the
stellar contribution to the SPARC circular velocity by fitting (39) to the
latter data.

Although the stellar component has a well-established, simple and good
approximation [55, 56], the gas component is known to be significantly more
diverse. For instance, [54] considered elaborated morphology-dependent
approximations for the gas; but it also introduced systematical deviations
with respect to SPARC data. Reference [52] uses an exponential approximation
for the gas component, considering a subset of the SPARC data. Contrary to the
previously cited approaches, here we do not consider individual galaxy fits,
hence the issue of whether such approximations are, or are not, good for
specific galaxies is not relevant. This since our focus here is on the sample
data distribution, not on the individual particularities. Thus, similarly to
[52], we will consider exponential approximations for the gas in the same way
we considered the stellar part. Each of the 122 galaxies will be associated
with four numbers that describe the stellar and gaseous components (two pairs
of $\Sigma_{0}$ and $h$). We will detail further these approximations in a
future work more specific to this subject, but this description is sufficient
to reproduce our results.

###  5.5 Application

From eq. (32) with $\kappa=8\pi G$, the expressions of $\Delta
V^{2}_{{\mbox{\scriptsize\rm mod}}}$ and $\delta V^{2}_{{\mbox{\scriptsize\rm
mod}}}$ for Palatini $f(R)$ read

$\Delta V^{2}_{\mbox{\scriptsize\rm P}}(r)=8\pi G\alpha r\rho^{\prime}(r)\,,$
(40)

hence

$\delta V^{2}_{\mbox{\scriptsize\rm P}}({r_{\mbox{\scriptsize\rm
n}}})={r_{\mbox{\scriptsize\rm
n}}}\frac{\rho^{\prime}({r_{\mbox{\scriptsize\rm n}}})}{\rho^{\prime}(1)}\,.$
(41)

${r_{\mbox{\scriptsize\rm n}}}$$\delta V^{2}$ Figure 2: The NAV plane for
$f(R)$ Palatini gravity contrasted with the observational NAV plane. The grey
regions are explained in Fig. 1. Each blue curve corresponds to one of the 122
bulgeless SPARC galaxies. These curves are generated using the approximations
of Sec. 5.4 together with the $\delta V^{2}_{\mbox{\scriptsize\rm P}}$
expression (41). The limits of the 1$\sigma$ and 2$\sigma$ highest density
regions for Palatini are shown with the dashed purple contours (dark purple
and light purple for the 1$\sigma$ and 2$\sigma$ regions respectively). These
model regions are significantly different from the corresponding observational
ones. The entire $2\sigma$ Palatini region is not displayed as it extends far
beyond the observational region.

It is important to stress that $\delta V^{2}_{\mbox{\scriptsize\rm P}}$ is
independent of $\alpha$. The NAV tests the shape of the additional
contribution to the rotation curve, being independent of the amplitude.

Since $\delta V^{2}_{\mbox{\scriptsize\rm P}}$ directly depends on the
baryonic matter distribution, we use the approximation detailed in Sec. 5.4 to
generate the Palatini NAV plane and contrast it with the observational NAV, as
done in Fig. 2.

From Fig. 2, one sees that Palatini has difficulties on covering the bottom
part of the observational NAV region, while there is a large excess of data
outside the observational NAV region. It is important to stress that these
results are for any $\alpha$ values, which may or may not change from galaxy
to galaxy.

In [23] we introduce a metric to quantify the concordance between a model and
the observational NAV, the mean efficiency $E_{\mbox{\scriptsize\rm M}}$. It
is such that the maximum concordance is $E_{\mbox{\scriptsize\rm M}}=1$ and a
model whose half of its predicted data is inside the observational region and
half of it is outside yields $E_{\mbox{\scriptsize\rm M}}=0$. There is no
lower limit for $E_{\mbox{\scriptsize\rm M}}$: the larger the predicted model
region outside the observational NAV region, the lower will be
$E_{\mbox{\scriptsize\rm M}}$. Thus good or acceptable models should yield
$E_{\mbox{\scriptsize\rm M}}\gtrsim 0.5$, while the unsuited models would
yield $E_{\mbox{\scriptsize\rm M}}\lesssim 0$. For instance, the well-known
dark matter and modified gravity models called Burkert, NFW and MOND yield
respectively 0.80, 0.70 and 0.53 for the mean efficiency [23]. For $f(R)$
Palatini, we infer that

$E_{\mbox{\scriptsize\rm M}}<-2.0\,.$ (42)

We conclude that Palatini $f(R)$ gravity, or any other theory that satisfies
eq. (41) is not compatible with the observational data. That is, these
theories cannot replace dark matter in galaxies (even if $\alpha$ changes from
galaxy to galaxy).

##  6 Conclusions

Here we considered Palatini $f(R)$ gravity in different physical scenarios. We
show that, although deviations from general relativity (GR) do not appear in
the Will Nordtvedt parametrized post-Newtonian (PPN) formalism [10], it can be
constrained within other physical systems. In particular, we revised a recent
approach that constrains modified gravity with Earth seismology data [34]. Our
main result refers to testing Palatini gravity as a possible replacement of
dark matter in galaxies, using SPARC [22] data for rotationally supported
galaxies.

We have found that, although it is reasonable to use a thin disk approximation
for Palatini, replacing dark matter with Palatini gravity leads to a strong
tension between the model and the observational data. Indeed, most of its
predictions are outside and far from the observational data, as shown in Fig.
2 and computed in eq. (42). This result is valid for any value of $\alpha$
(16) (it could be either a common $\alpha$ value for all the galaxies or one
that changes from galaxy to galaxy). There are clearly other models that are
better for rotation curve data, either within the dark matter approach (e.g.,
NFW or Burkert halos) or modified gravity (e.g., MOND) [23].

From eq. (16) and simple approximations for the baryonic part contribution, it
is not hard to conclude that Palatini $f(R)$ will have difficulties in trying
to replace dark matter in galaxies. This is so since dark matter is commonly
expected to play a major role in the outskirts of galaxies. In these regions,
the baryonic density is small and (apart from individual peculiarities) should
become even smaller with increasing radius. On the other hand, the
contribution $\nabla^{2}\rho$ is expected to be more relevant close to the
galaxy centre than far from it. This is in qualitative agreement with our
finds here. Nonetheless, this type of qualitative argument, although useful,
cannot quantify the discrepancy between the model and data, nor it is useful
to compare results from different models.

At last, we stress that the normalised additional velocity (NAV) method [23]
is a recent one, and we understand that its application to $f(R)$ Palatini is
a relevant illustration of how it works. The main code of the method, with
several applications, is available at [57].

## Acknowledgments

We gratefully acknowledge the organizers of the Metric-Affine Frameworks for
Gravity 2022 workshop for their outstanding work in creating a friendly
atmosphere for fruitful discussions and collaborations. AHA thanks CAPES
(Brazil) for financial support. DCR thanks CNPq (Brazil) and FAPES (Brazil)
for partial financial support (TO 1020/2022, 976/2022, 1081/2022). AW
acknowledges financial support from MICINN (Spain) Ayuda Juan de la Cierva -
incorporacíon 2020 No. IJC2020-044751-I.

## Appendix A

Note that (16) is also a weak field regime for the Eddington-inspired Born-
Infeld (EiBI) gravity [24]

$\nabla^{2}\phi=\frac{\kappa}{2}\Big{(}\rho+\alpha\nabla^{2}\rho\Big{)},$ (43)

therefore our results, arriving from the planetary seismology and galaxies,
will also refer to this proposal. It is so because the Poisson equation
similarity in both theories of gravity is not accidental: let us note that in
the 1st order approximation, $|R_{\mu\nu}|\ll\alpha^{-1}$, EiBI gravity
becomes Palatini gravity with a quadratic term [58]

$\mathcal{S}[g,\Gamma,\psi_{m}]\approx\frac{1}{\kappa}\int
d^{4}x\sqrt{g}\left[R-2\Lambda+\frac{\alpha}{2}(R^{2}-2R_{\mu\nu}R^{\mu\nu})\right],$
(44)

where $\Lambda$ is a cosmological constant. As we already know from the PPN
analysis, only the quadratic terms are manifested in the non-relativistic
regime in Palatini gravity and hence we are dealing with the same modified
Poisson equation. Therefore, the upper bound resulting from the seismic data
for the EiBI proposal is $\alpha\lesssim 2\cdot 10^{9}\text{m}^{2}$ [17].

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