# On the evolution of inhomogeneous perturbations
in the $\Lambda$CDM model and $f\left(R\right)$ modified gravity theories

T. Schiavone∗ Department of Physics “E. Fermi”, University of Pisa, Polo
Fibonacci,
Largo B. Pontecorvo 3, I-56127, Pisa, Italy
and
INFN, Istituto Nazionale di Fisica Nucleare, Sezione di Pisa, Polo Fibonacci,
Largo B. Pontecorvo 3, I-56127, Pisa, Italy
∗E-mail<EMAIL_ADDRESS>
G. Montani  ENEA, Fusion and Nuclear Safety Department, C.R. Frascati,
Via E. Fermi 45, I-00044 Frascati (Rome), Italy
and
Physics Department, “Sapienza” University of Rome,
P.le Aldo Moro 5, I-00185 Rome, Italy

###### Abstract

We focus on weak inhomogeneous models of the Universe at low redshifts,
described by the Lemaître-Tolman-Bondi (LTB) metric. The principal aim of this
work is to compare the evolution of inhomogeneous perturbations in the
$\Lambda$CDM cosmological model and $f(R)$ modified gravity theories,
considering a flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric for the
background. More specifically, we adopt the equivalent scalar-tensor formalism
in the Jordan frame, in which the extra degree of freedom of the $f(R)$
function is converted into a non-minimally coupled scalar field. We
investigate the evolution of local inhomogeneities in time and space
separately, following a linear perturbation approach. Then, we obtain
spherically symmetric solutions in both cosmological models. Our results allow
us to distinguish between the presence of a cosmological constant and modified
gravity scenarios, since a peculiar Yukawa-like solution for radial
perturbations occurs in the Jordan frame. Furthermore, the radial profile of
perturbations does not depend on a particular choice of the $f(R)$ function,
hence our results are valid for any $f(R)$ model.

###### keywords:

Cosmology, Modified gravity, Inhomogeneous Universe, Late Universe, Large
Scale Structure

## Introduction

The well-known $\Lambda$CDM model [1, 2], which includes a cosmological
constant $\Lambda$ and a cold dark matter (CDM) component, is based on General
Relativity (GR), and it provides a robust and accurate description of our
Universe, supported by cosmological data. Moreover, the Universe appears
homogeneous and isotropic at very large scales [3] (cosmological principle),
and this kind of geometry is described by the FLRW line element. In this work,
based on a paper in preparation [4], we investigate possible deviations from
two pillars of the standard $\Lambda$CDM model, focusing on a local
inhomogeneous description of the Universe and modified gravity models.

Here, we adopt the LTB metric [2], a spherically symmetric solution of the
Einstein field equations, to account for local deviations of the Universe from
the homogeneity. Hence, we regard the weakly inhomogeneous LTB solution as the
standard FLRW geometry plus small spherically symmetric perturbations.

Furthermore, in recent years increasing interest raised to provide features
able to distinguish between GR and modified gravity approaches. Some open
problems in cosmology, such as the Hubble constant tension [5], may require
new physics [6]. The aim of the present work is to compare the evolution of
linear inhomogeneous perturbations in the $\Lambda$CDM and $f\left(R\right)$
modified gravity models [7, 8]. More in detail, we used the equivalent
formalism in the Jordan frame [7, 8, 9], and we adopt the $f\left(R\right)$
Hu-Sawicki model [10], which is a promising theoretical framework to reproduce
the cosmic acceleration in the late Universe via the dynamics of a non-
minimally coupled scalar field without a cosmological constant.

Then, we investigate the dynamics of inhomogeneous perturbations in the two
cosmological models abovementioned. Moreover, the separable variables method
is implemented in the analysis at the first-order of perturbation, and our
approach is the same used in Ref. 11. We obtain an analytic expression only
for the radial dependence of linear perturbations, while their time evolution
must be numerically evaluated. The most relevant issue, emerging from our
analysis, is the different morphology of the radial dependence of the LTB
solution in the $\Lambda$CDM formulation and Hu-Sawicki dynamics. Indeed, a
distinctive Yukawa-like behavior occurs in the Jordan frame gravity regardless
of the choice of a specific $f\left(R\right)$ model. Hence, we have found a
peculiar feature to distinguish a modified gravity model from a cosmological
constant scenario.

This work is organized as follows: in Sec. 1 we introduce the framework of the
$f\left(R\right)$ extended gravity models; in Sec. 2 we adopt the LTB metric;
then, in Sec. 3 we implement a perturbation approach to deal with local
inhomogeneities. Finally, we summarize our work in Sec. 4. We adopt the metric
signature $\left(-,+,+,+\right)$, and we set $c=1$.

##  1 $f\left(R\right)$ models in the Jordan frame

A generalization of the Einstein-Hilbert theory is provided by the so-called
$f\left(R\right)$ modified gravity models [7, 8]. In this context, the
gravitational Lagrangian density is expressed as a free function $f$ of the
Ricci scalar $R$, namely an extra degree of freedom compared to GR. The
extended field equations in $f\left(R\right)$ gravity are fourth-order partial
differential equations in the metric tensor components. In particular, if
$f\left(R\right)=R$, the gravitational field equations end up in the Einstein
equations in GR, trivially.

It is often convenient to reformulate $f\left(R\right)$ gravity in the scalar-
tensor representation in the Jordan frame [7, 8, 9]. In doing so, the extra
degree of freedom of the $f\left(R\right)$ function is converted into a non-
minimally coupled scalar field. It can be checked that the following action

$S_{J}=\frac{1}{2\,\chi}\,\int_{\Omega}d^{4}x\,\sqrt{-g}\,\left[\phi\,R-V\left(\phi\right)\right]+S_{M}\left(g_{\mu\nu},\psi\right)$
(1)

is dynamically equivalent to the action in the $f\left(R\right)$ metric
formalism, if $f^{\prime\prime}\left(R\right)\neq 0$, where we have defined a
scalar field $\phi=f^{\prime}\left(R\right)=df/dR,$ and a scalar field
potential
$V\left(\phi\right)=\phi\,R\left(\phi\right)-f\left(R\left(\phi\right)\right)$.
In Eq. (1), $\chi\equiv 8\,\pi\,G$ is the Einstein constant, $G$ is the Newton
constant, $g$ is the determinant of the metric tensor $g_{\mu\nu}$, while
$S_{M}$ is the matter action, and $\psi$ denotes matter fields. The advantage
of the Jordan frame is that the corresponding field equations [8] are now
second-order differential equations, but one has to deal with the non-
minimally coupling.

The increasing attention to these extended gravity models is motivated by the
consideration that the actual cosmic acceleration of the Universe might be
obtained by geometry rather than a cosmological constant. A geometrical
modification, indeed, could be regarded as an effective matter source.

Among several suggested $f\left(R\right)$ models [7, 8, 12, 13], we focus on
the Hu-Sawicki proposal [10], which is largely studied in the late Universe.
The deviation $F\left(R\right)\equiv f\left(R\right)-R$ with respect to the GR
scenario for the Hu-Sawicki (HS) model with $n=1$ assumes the following form

$F\left(R\right)=-m^{2}\frac{c_{1}\,R/m^{2}}{c_{2}\,R/m^{2}+1}\,,$ (2)

where $m^{2}\equiv\frac{\chi\,\rho_{m0}}{3}$ is related to the present matter
density $\rho_{m0}$, while $c_{1}$ and $c_{2}$ are dimensionless parameters.
It can be checked [10] that an effective cosmological constant is obtained for
$R\gg m^{2}$. Furthermore, $c_{1}$ and $c_{2}$ can be constrained [10]
specifying $F_{R0}\equiv dF\,/\,dR\,\left(z=0\right)$ at the present cosmic
time (redshift $z=0$).

Finally, we write the scalar field potential $V\left(\phi\right)$ in the
Jordan frame for the HS model with $n=1$:

$V\left(\phi\right)=\frac{m^{2}}{c_{2}}\,\left[c_{1}+1-\phi-2\sqrt{c_{1}\,\left(1-\phi\right)}\right]\,.$
(3)

It can be shown [14] that the potential (3) ends in an asymptotically stable
de Sitter Universe.

##  2 Inhomogeneous solutions in GR and in the Jordan frame

An inhomogeneous Universe can be described by using the LTB line element [2],
which is a spherically symmetric solution of the Einstein equations. The
Universe in the LTB geometry appears inhomogeneous, but isotropic, from a
single preferred point located at the center of the spherical symmetry. For
instance, a cosmological dust or a spherical overdensity mass are well
described by such a metric.

In the synchronous gauge, the LTB line element may be regarded as a
generalization of the FLRW one, and it is given by

$ds^{2}=-dt^{2}+e^{2\alpha}dr^{2}+e^{2\beta}\,\left(d\theta^{2}+sin^{2}\theta\,d\phi^{2}\right)\,,$
(4)

where $r$ is the radial distance from the preferred point, while
$\alpha=\alpha\left(t,\,r\right)$ and $\beta=\beta\left(t,\,r\right)$ are two
metric functions.

Considering a pressure-less dust ($p=0$) and a cosmological constant $\Lambda$
in GR, we can write the Einstein equations in the LTB metric. For instance,
the 01 component of field equations is written as

$\frac{\dot{\beta}^{\prime}}{\beta^{\prime}}-\dot{\alpha}+\dot{\beta}=0$ (5)

where $\dot{\left(\right)}$ and $\left(\right){}^{\prime}$ denote derivatives
with respect to time $t$ and radial coordinate $r$, respectively. The other
non-zero equations are the 00 and 11 components [4], while different
components vanish due to spherical symmetry.

It is possible to simplify the LTB metric within the framework of GR, using
Eq. (5), which allows us to find a relation between the metric functions
$\alpha$ and $\beta$. Following the approach in Ref. 2, the LTB line element
(4) becomes

$ds^{2}=-dt^{2}+\frac{\left[\left(a\,r\right){}^{\prime}\right]{}^{2}}{1-r^{2}\,K^{2}}dr^{2}+\left(a\,r\right){}^{2}\,\left(d\theta^{2}+sin^{2}\theta\,d\phi^{2}\right)\,,$
(6)

where $a=a\left(t,r\right)=e^{\beta}/r$ is a generalized scale factor in a
inhomogeneous geometry, and $K=K\left(r\right)$. Note that if
$a\left(t,r\right)$ and $K\left(r\right)$ are independent of $r$, the metric
(6) coincides with the FLRW line element, describing an isotropic and
homogeneous geometry. Moreover, the 00 and 11 components of the Einstein
equations in the LTB metric provides a generalization of the Friedmann
equations in the $\Lambda$CDM model.

Concerning the $f\left(R\right)$ modified gravity scenario in the equivalent
Jordan frame, the 01 component of the gravitational field equations in the LTB
metric is written as

$\frac{\dot{\beta}^{\prime}}{\beta^{\prime}}-\dot{\alpha}+\dot{\beta}=-\frac{1}{2\,\phi\,\beta^{\prime}}\,\left(\dot{\phi}^{\prime}-\dot{\alpha}\,\phi^{\prime}\right)\,.$
(7)

Furthermore, we can compute [4] also the 00 and 11 of field equations, as well
as the scalar field equation in the Jordan frame. Observe that $\phi$, $\rho$,
$\alpha$, and $\beta$ are all functions of $t$ and $r$ in a inhomogeneous
scenario.

Comparing Eq. (7) with the respective equation (5) obtained in GR, an extra
term occurs in the Jordan frame due to non-minimal coupling between $\phi$ and
the metric functions $\alpha$, $\beta$. As a consequence, we can not relate
$\alpha$ and $\beta$ to rewrite the LTB metric as in the GR scenario.
Moreover, the scalar field potential $V\left(\phi\right)$ does not affect Eq.
(7), but it is included in the remaining field equations. $V\left(\phi\right)$
represents an extra degree of freedom with respect to GR, and the dynamics in
the Jordan frame deviates from that of the Einstein theory.

##  3 Perturbation approach in GR and in the Jordan frame

In this Section, we investigate the different evolution of local
inhomogeneities within the framework of GR and modified gravity [4]. Here, we
stress one important difference: we can not adopt the LTB metric in the form
(6) in the Jordan frame, but we have to refer to the line element in Eq. (4),
as motivated in Sec. 2.

Then, we consider the LTB metric to describe local inhomogeneities as small
linear perturbations with respect to a flat FLRW metric as background. Hence,
we can write all the physical quantities as a background term, denoted with a
bar, plus a linear perturbation, which depends on $t$ and $r$, and it is
denoted with $\delta$. For instance, the energy density is written as
$\rho\left(t,r\right)=\bar{\rho}\left(t\right)+\delta\rho\left(t,r\right)\,.$
Similarly, we rewrite the scale factor $a\left(t,r\right)$ in GR, the scalar
field $\phi\left(t,r\right)$, the metric functions $\alpha\left(t,r\right)$,
and $\beta\left(t,r\right)$ in the Jordan frame. We also assume that
perturbation terms are small corrections if compared with respective
background quantities, i.e. $\delta\rho\ll\bar{\rho}$. Furthermore, we
consider $K^{2}\left(r\right)$ in the LTB metric (6) as a linear term, since
it is related to inhomogeneities. It should be noted that, comparing the LTB
line element (4) and a flat FLRW metric, we have to require two constraints:
$\bar{\alpha}\left(t\right)=\ln\left(\bar{a}\left(t\right)\right)$ and
$\bar{\beta}\left(t,r\right)=\ln\left(\bar{a}\left(t\right)\,r\right)$.

Concerning the scalar-tensor formalism in the Jordan frame, we expand the
scalar field potential $V\left(\phi\right)$ to the first order in
$\delta\phi$. It can be easily checked that the zeroth order term
$V\left(\bar{\phi}\left(t\right)\right)$ depends only on the background scalar
field [4].

Once we have split background and linear contributions, we focus on the
dynamics in the $\Lambda$CDM cosmological model and in the Jordan frame
gravity. Hence, we rewrite the field equations in GR and in the Jordan frame,
separating background and perturbation terms.

###  3.1 Comparing background solutions

To study the time evolution, we define a dimensionless variable
$\tau=t/t_{0}$, where $t_{0}$ is the actual time in the synchronous gauge.
Note that $t_{0}$ is approximately $t_{0}\approx 1/H_{0}$ in terms of the
Hubble constant.

In GR, the 00 component of the Einstein equation in the LTB geometry becomes
the Friedmann equation in the FLRW metric at background level. We set
$\bar{a}_{0}=\bar{a}\left(\tau=1\right)$ today, and we neglect relativistic
componenents in the late Universe. Hence, we obtain an analytical solution
[4]: the evolution of the background scale factor in terms $\tau$ is written
as

$\bar{a}\left(\tau\right)=\left(\frac{\Omega_{m0}}{\Omega_{\Lambda
0}}\right)^{1/3}\,\,\left\\{\sinh\left[\frac{3}{2}\,\sqrt{\Omega_{\Lambda
0}}\,\left(\tau-1\right)+\textrm{arcsinh}\left(\sqrt{\frac{\Omega_{\Lambda
0}}{\Omega_{m0}}}\right)\right]\right\\}^{2/3}\,.$ (8)

We used the cosmological density parameters $\Omega_{m0}=\rho_{m0}/\rho_{c0}$,
and $\Omega_{\Lambda 0}=\Lambda/\,\left(\chi\,\rho_{c0}\right)$ for matter and
cosmological constant components, respectively, where
$\rho_{c0}=3H_{0}^{2}/\chi$ is the actual critical energy density of the
Universe. We recall also that
$\bar{\rho}\left(\tau\right)\propto\bar{a}\left(\tau\right)^{-3}$ for a matter
component.

In Fig. 1 we plot the deceleration parameter
$\bar{q}\left(\tau\right)\equiv-\ddot{\bar{a}}/\,\left(\bar{a}\,\bar{H}^{2}\right)$,
fixing $\Omega_{m0}=0.3111$ and $\Omega_{\Lambda 0}=1-\Omega_{m0}$ from Ref.
3. Here, the dot denotes the derivative with respect to $\tau$. The scale
factor $\bar{a}\left(\tau\right)$ increases for growing values of $\tau$, and
note that $\bar{q}\rightarrow-1$ in the future.

We point out that the solution (8) in the $\Lambda$CDM cosmological model is
viable only at late times ($\tau\gg 10^{-6}$) in the Universe, otherwise we
have to include relativistic components, and solve numerically the Friedmann
equation.

Figure 1: Background deceleration parameter $\bar{q}(\tau)$ in the FLRW
geometry in terms of a dimensionless variable $\tau$. The background scenario
in the $\Lambda$CDM model are graphically the same comparing it with the one
in the $f(R)$ Hu-Sawicki model in the Jordan frame.

Now, we focus on the background field equations in the Jordan frame gravity.
Starting from the 01, 00, 11 components of the gravitational field equations
in the LTB metric and the scalar field equation, if we do not include local
inhomogeneities, we get the respective equation system in the Jordan frame for
the background FLRW metric (see the field equations in Ref. 8). We use the Hu-
Sawicki model in the Jordan frame, which is characterized by the scalar field
potential in Eq. (3).

To solve numerically the equation system, we fix $\Omega_{m0}$ at the same
value used in the previous study for the $\Lambda$CDM model, and we choose
[10] $\left|F_{R0}\right|=10^{-7}$ to obtain $c_{1}=2.0\cdot 10^{6}$ and
$c_{2}=1.5\cdot 10^{5}$. We set the free parameters in a way that the Hu-
Sawicki model in the Jordan frame is very close to the $\Lambda$CDM scenario
for the background level. In Fig. 2 note that the scalar field potential (3)
exhibits a slow-roll mimicking a cosmological constant.

Finally, we solve numerically the equation system in the Jordan frame [4],
imposing the conditions at $\tau=1:$
$\bar{\phi}\left(1\right)=1-\left|F_{R0}\right|$ and
$d\bar{\phi}/d\tau\left(1\right)=0$. The resulting plot of the deceleration
parameter $\bar{q}\left(\tau\right)$ are graphically undistinguishable from
the respective ones in the $\Lambda$CDM model (see Fig. 1). Concerning the
evolution of the scalar field, we plot in Fig. 2 the deviation
$\left|1-\bar{\phi}\left(\tau\right)\right|$ from the GR limit
($\bar{\phi}=1$) with a scale $10^{-7}$. Note that the dynamics of the Jordan
frame is very close to the one in $\Lambda$CDM model at the background order
for any $\tau$. Now, we can move on to the linearized solutions to highlight
the differences between the two cosmological models.

Figure 2: Plots referred to the Hu-Sawicki model in the Jordan frame, adopting
the FLRW metric (background solution). Left panel: scalar field potential of
the Hu-Sawicki model in the Jordan frame. Right panel: deviation
$|1-\bar{\phi}\left(\tau\right)|$ from the GR scenario ($\bar{\phi}=1$) using
a scale $10^{-7}$.

###  3.2 Comparing linear solutions

To examine separately the time and radial evolution of local inhomogeneities,
we use the separable variables method at the first-order perturbation theory.
So, we factorize all the linear perturbations into time and radial functions.
More specifically, we rewrite the perturbations of the scale factor $\delta
a\left(\tau,r\right)\equiv\delta\texttt{a}\left(\tau\right)\,\delta\mathfrak{a}\left(r\right)$,
the metric functions $\delta\alpha\left(\tau,r\right)\equiv\delta
A\left(\tau\right)\,\delta\mathcal{A}\left(r\right)$ and
$\delta\beta\left(\tau,r\right)\equiv\delta
B\left(\tau\right)\,\delta\mathcal{B}\left(r\right)$, the energy density
$\delta\rho\left(\tau,r\right)\equiv\delta
P\left(\tau\right)\,\delta\varrho\left(r\right)$, and the scalar field
$\delta\phi\left(\tau,r\right)\equiv\delta\Phi\left(\tau\right)\,\delta\varphi\left(r\right)$.
The factorization is assumed at the linear level to find analytical solutions,
but the separation of variables is not a general method, if we also include
non-linear terms.

Then, we investigate the effect of inhomogeneities in the dynamics, according
to the cosmological model considered. We focus on the linearized equations in
the LTB metric within the framework of the $\Lambda$CDM cosmological model.

Using the factorization abovementioned, it is easy to show that the 11
component of the linearized Einstein equations in the LTB metric can be
separated in two parts [4]. The radial behavior is
$\delta\mathfrak{a}\left(r\right)=K^{2}\left(r\right)\,,$while we have an
ordinary differential equation in terms of $\tau$ for the time evolution:

$\delta\ddot{\texttt{a}}\left(\tau\right)+\frac{\dot{\bar{a}}\left(\tau\right)}{\bar{a}\left(\tau\right)}\,\delta\dot{\texttt{a}}\left(\tau\right)-\delta\texttt{a}\left(\tau\right)\,\left[\frac{\ddot{\bar{a}}\left(\tau\right)}{\bar{a}\left(\tau\right)}+\left(\frac{\dot{\bar{a}}\left(\tau\right)}{\bar{a}\left(\tau\right)}\right)^{2}\right]=0\,.$
(9)

This equation is solved numerically, once you specify the background solution
(8). We impose the following conditions at $\tau=1$ today:
$\delta\texttt{a}\left(1\right)=10^{-5}$ and
$\delta\dot{\texttt{a}}\left(1\right)=0$. In Fig. 3 we plot the numerical
solution, after we have defined the ratio $\eta\left(\tau\right)$ between the
perturbed scale factor and the background respective quantity, namely
$\eta\left(\tau\right)\equiv\left|\delta\texttt{a}\left(\tau\right)\,/\,\bar{a}\left(\tau\right)\right|$.
Note that perturbations of the scale factor remain stable, since the
background scale factor $\bar{a}\left(\tau\right)$ dominates for any $\tau$ in
the late Universe, i.e. $\eta\ll 1$.

After that, we consider the 00 component of the linearized Einstein equations
in the LTB metric, and we combine it with the continuity equation. After long
but straightforward calculations [4], we obtain $\delta\mathfrak{a}\propto
1/r^{3}$ in GR. According to the cosmological principle, inhomogeneities decay
for increasing values of $r$. Furthermore, we also obtain that the
perturbation of the energy density for a matter component follows the same
dependence of the respective background quantity in the FLRW metric, namely
$\delta P\left(\tau\right)\propto 1/\bar{a}^{3}\left(\tau\right)$. However, we
can provide $\delta P\left(\tau\right)\ll\bar{\rho}\left(\tau\right)$ at any
times, setting the constant of integration.

Figure 3: Left panel: time evolution of the linear perturbation of the scale
factor $\delta\text{a}\left(\tau\right)$. Right panel: the ratio between the
linear deviation and background scale factor.

Now, we repeat the same approach in the modified gravity dynamics in the
Jordan frame, considering the corresponding linearized field equations in the
LTB metric. Expanding the scalar field potential and its derivative around the
background scalar field, as mentioned at the beginning of Sec. 3, it can be
proved that the potential $V\left(\phi\right)$ is involved only in the time
evolution of the linearized field equations [4]. As a result, we expect that
the radial evolution of local inhomogeneities is completely independent of the
choice of a particular $f\left(R\right)$ model, since $V\left(\phi\right)$ is
directly related to $f\left(R\right)$.

In what follows, we do not focus on the time evolution of perturbations, but
we just mention that we find a numerical solution using again the Hu-Sawicki
model (Sec. 1), and we have checked that inhomogeneous perturbations remain
smaller than respective background contribution for any time $\tau$.

To obtain an analytical solution for the radial part with the separable
variables method, we assume two simplifying conditions:
$\delta\dot{A}\left(\tau\right)=\delta\dot{B}\left(\tau\right)$ and
$\delta\varrho\left(r\right)\propto\delta\varphi\left(r\right)$. After long
but straightforward calculations [4], combining the linearized field
equations, we obtain a Yukawa-like solution for the radial part of the linear
perturbation of the scalar field

$\delta\varphi\left(r\right)=\frac{C}{r}\,\exp\left(-\frac{r}{r_{c}}\right)\,,$
(10)

where $C$ and $r_{c}$ are constants. Moreover, $\delta\varrho\left(r\right)$
shows the same evolution, since we recall that it is proportional to
$\delta\varphi\left(r\right)$. We also obtain a similar behavior for
$\delta\mathcal{A}\left(r\right)$ and $\delta\mathcal{B}\left(r\right)$. It
should be emphasized that modified gravity introduces a typical spatial scale,
$r_{c}$, such that for $r\gg r_{c}$ the inhomogeneities decay faster than ones
in GR. We stress again that this kind of solutions applies to any $f(R)$
model, and the radial evolution is different from the one in GR.

##  4 Summary and conclusions

We have investigated local inhomogeneities in the LTB metric regarded as small
deviations from a flat FLRW background metric. To date, there are no modified
gravity models that predict significant deviations from GR, reconciling all
possible cosmological data. Hence, to try to discriminate between several
cosmological models, it is crucial to test gravity in different regimes or by
using other techniques. Here, we have suggested one possible method, studying
the different dynamics of inhomogeneous perturbations. Our results have
pointed out a distinctive element in the evolution of local inhomogeneities of
the Universe from a theoretical point of view, allowing to distinguish between
the $\Lambda$CDM cosmological model and $f\left(R\right)$ modified gravity
theories. We have shown that the radial evolution of inhomogeneous
perturbations within the framework of the Jordan frame gravity is independent
of the scalar field potential $V(\phi)$: the distinctive radial solution is a
feature of any $f(R)$ model. Furthermore, we have obtained a Yukawa-like
solution for the radial perturbations in the Jordan frame, which is completely
different from the one in GR. This work may be an interesting arena to account
for the effects of local inhomogeneities in cosmological observables [15],
when forthcoming missions such as Euclid Deep Survey [16], will be able to
test the large-scale properties of the Universe.

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