# Swiss-cheese cosmologies with variable $G$ and $\Lambda$ from the
renormalization group

Fotios K. Anagnostopoulos<EMAIL_ADDRESS>Department of
Informatics & Telecommunications, University of the Peloponnese, Karaiskaki
70, Tripoli 221 00, Greece  Alfio Bonanno<EMAIL_ADDRESS>INAF,
Osservatorio Astrofisico di Catania, Via S.Sofia 78, 95123 Catania, Italy
INFN, Sezione di Catania, Via S. Sofia 72, 95123 Catania, Italy  Ayan Mitra
<EMAIL_ADDRESS>The Inter-University Centre for Astronomy and
Astrophysics (IUCAA), Post Bag 4, Ganeshkhind,
Pune 411007, India  Nazarbayev University, School of Engineering, Republic of
Kazakhstan, 010000  Kazakh-British Technical University, Tole bi 59, Almaty,
Republic of Kazakhstan, 050000  Vasilios Zarikas<EMAIL_ADDRESS>University
of Thessaly,Department of Mathematics, 35100 Lamia, Greece  Nazarbayev
University, School of Engineering, Republic of Kazakhstan, 010000

###### Abstract

A convincing explanation for the nature of the dark energy and dark matter is
still missing. In recent works a RG-improved swiss-cheese cosmology with an
evolving cosmological constant dependent on the Schücking radius has been
proven to be a promising model to explain the observed cosmic acceleration. In
this work we extend this model to consider the combined scaling of the Newton
constant $G$ and the cosmological constant $\Lambda$ according to the IR-fixed
point hypothesis. We shall show that our model easily generates the observed
recent passage from deceleration to acceleration without need of extra energy
scales, exotic fields or fine tuning. In order to check the generality of the
concept, two different scaling relations have been analysed and we proved that
both are in very good agreement with $\Lambda$CDM cosmology. We also show that
our model satisfies the observational local constraints on $\dot{G}/G$.

##  I Introduction

Although the $\Lambda$CDM scenario reproduces rather well a large amount of
observational data, it is important to stress that its two main ingredients,
Dark Matter (DM) Rubin and Ford (1970) and Dark Energy (DE) Spergel et al.
(2007), do not have a direct physical explanation. In particular, the observed
accelerated expansion of the Universe is caused by the DE component a
mechanism which introduces new difficulties in terms of fine-tuning or cosmic
coincidences Straumann (1999) for the presence of a significant vacuum energy
(cosmological constant).

For this reason the possibility that the cosmological constant is dynamically
generated as a low-energy quantum effect has been recently investigated by
various authors Wang et al. (2017); Carlip (2019). In particular in Bonanno
and Reuter (2002) a cosmology with a time dependent cosmological constant and
Newton constant whose dynamics arises from an underlying renormalization group
flow near an infrared attractive fixed point has been proposed for the first
time. Its phenomenological implications have been discussed in Bentivegna et
al. (2004) and further developed in Bonanno and Carloni (2012).

The IR-fixed point hypothesis originates from the asymptotically safe (AS)
scenario for quantum gravity Weinberg (1979); Reuter (1998). According to AS a
consistent quantum theory of gravity can be realized nonperturbatively by
defining the theory around a non-gaussian fixed point for the dimensionless
Newton’s constant Litim (2011); Nagy (2014); Bonanno and Saueressig (2017);
Percacci (2017); Eichhorn (2019); Pereira (2019); Reuter and Saueressig
(2019); Reichert (2020); Platania (2020). In fact this mechanism is not
qualitatively different from what occurs in the non-linear $\sigma$-model in
$d=3$: here too, although the theory is not perturbatively renormalizable, a
continuum limit at non-vanishing value of the coupling constant can be defined
beyond perturbation theory Polyakov (1993); Codello and Percacci (2009).

Assuming that the relevant scale of energy or momentum scales with the
distance as the coarse-graining, “resolution” scale of the renormalization
group, the spatial or temporal evolution of the Newton’s constant and
cosmological constant is dictated by the renomalization group flow Platania
(2020). The possibility of explaining DM in spiral galaxies as an infrared
effect of the running of $G$ has been discussed in Reuter and Weyer (2004a,
b), while complete cosmological histories based on RG-trajectories emanating
from the UV fixed point up to the IR regime have been discussed in Reuter and
Saueressig (2005); Bonanno and Reuter (2007). Non-singular cosmological models
have appeared in Kofinas and Zarikas (2016); Bonanno et al. (2018) and a
mathematical formalism to couple the RG evolution to the Einstein equations
have been discussed in Reuter and Weyer (2004) and Bonanno et al. (2021).

In Kofinas and Zarikas (2018) a further step in the direction of explaining
the DE content in the Universe with a running cosmological constant has been
proposed. The basic idea is to consider the contribution of an homogeneous
distribution of antigravity sources associated with the matter content at
galactic and cluster scale. A swiss-cheese cosmology represents an elegant
mathematical framework to implement this idea which has been elaborated also
in Mitra et al. (2021); Anagnostopoulos et al. (2019a, b). In this work we
would like to extend the original RG-improved swiss-cheese model to include
the running of the Newton’s constant according to the IR-fixed point
mechanism. In this case, the scaling behavior of $G$ and $\Lambda$ is
determined by the simple law

$G_{k}\equiv G(k)\sim
g_{\ast}/k^{2},\quad\quad\Lambda_{k}\equiv\Lambda(k)\sim\lambda_{\ast}k^{2}$
(1)

in the $k\rightarrow 0$ limit. This behavior could be the result of a “tree-
level” renormalization induced by a non-zero positive cosmological constant in
the IR, according to the singular behavior of the $\beta$-functions in the IR
Nagy et al. (2012); Christiansen et al. (2014); Biemans et al. (2017). The aim
of this study is to further elaborate the IR-fixed point scaling within the
swiss-cheese cosmology. In particular it will be shown that it is possible to
consistently describe a phase of accelerated expansion of the universe without
the introduction of a DE component.

The structure of this paper is the following: in section II the mathematical
the swiss-cheese idea and the basic equations will be discussed. In Section
III the IR fixed point hypothesis and the RG evolution will be presented as a
function of the Schücking Radius and of the matter density. In section IV the
numerical results are discussed and section V is devoted to the conclusions.

##  II Swiss cheese cosmological model

The original Einstein-Strauss model or otherwise called swiss-cheese model,
Einstein and Straus (1946), describes many homogeneously distributed black
holes smoothly matched within a cosmological metric. The key idea is to match
a Schwarzshild spherical vacuole in a homogeneous isotropic cosmological
spacetime. The analysis finally generates a dust Friedman-Robertson-Walker
(FRW) cosmology assuming many Schwarzschild black holes as the matter content.

It is also possible to show that respecting the Israel-Darmois matching
conditions, Eisenhart (2016), a homogeneous and isotropic cosmological metric
with a de Sitter - Schwarzschild like metric can be successfully matched
across a spherical 3-surface $\Sigma$ which is at fixed coordinate radius of
the cosmology metric frame (see also Balbinot et al. (1988)). However, the
radius of the surface in the Black Hole (BH) frame is time evolving. The
matching requires the first fundamental form, intrinsic metric, and second
fundamental form (extrinsic curvature), calculated in terms of the coordinates
on $\Sigma$, to be equal (opposite for the second fundamental form) on both
sides of the surface Darmois (1927), Dyer and Oliwa (2000).

A homogeneous isotropic cosmological metric can be written in spherical
coordinates as

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

where $a(t)$ is the scale factor and $\kappa=0,\pm 1$ is the spatial curvature
constant.

The junction conditions Eisenhart (2016) allow us to use different coordinate
systems on both sides of the hypersurface. Thus, the cosmology exterior metric
(2) can be joined smoothly to the following quantum-modified Schwarzschild
metric,Koch and Saueressig (2014),

$ds^{2}=-F(R)\,dT^{2}+R^{2}\left(d\theta^{2}+\sin^{2}\theta
d\phi^{2}\right)+F(R)^{-1}\,dR^{2}\,$ (3)

where F(R) is defined by

$F\left(R\right)=1-\frac{2\,G_{k}\,M}{R}-\frac{\Lambda_{k}\,R^{2}}{3}.$ (4)

Note, that now the Newton constant and the cosmological constants are not
constants but functions of one characteristic length scale $\sim 1/k$. Both
will be functions of $R$ as we shall explain later. The first fundamental form
is the metric on $\Sigma$ induced by the spacetime in which it is embedded.
This may be written as

$\gamma_{\alpha\beta}=g_{ij}\frac{\partial}{\partial
u^{\alpha}}x^{i}\frac{\partial}{\partial u^{\beta}}x^{i}\,,$ (5)

where $u^{\alpha}=(u^{1}\equiv u,\,u^{2}\equiv v,\,u^{3}\equiv w)$ is the
coordinate system on the hypersurface. Greek indices run over $1,\ldots,3,$
while Latin indices over $1,\ldots,4.$
The second fundamental form Eisenhart (2016) is defined by

$K_{\alpha\beta}=n_{i;j}\,\frac{\partial}{\partial
u^{\alpha}}x^{i}\frac{\partial}{\partial
u^{\beta}}x^{i}=(\Gamma^{p}{}_{ij}n_{p}-n_{i,j})\frac{\gamma_{\alpha\beta}}{g_{ij}}\,,$
(6)

where $n_{a}$ is a unit normal to $\Sigma$ and $\Gamma^{p}{}_{ij}$ are the
Christoffel symbols. We use subscripts $F$ and $S$ to denote quantities
associated with the FLRW and modified Schwarzschild metrics, respectively Dyer
and Oliwa (2000).

Let us consider a spherical hypersurface $\Sigma$ given by the function
$f_{F}(x_{F}^{i})=r-r_{\Sigma}=0,$ where $r_{\Sigma}$ is a constant. This
hypersurface in the FLRW frame can be parametrized by
$x_{F}^{i}\,=\,(t=u,\,\theta=v,\,\phi=w,\,r=r_{\Sigma})$, while the
parametrization in the BH frame is
$x_{S}^{i}=\left(T=T_{S}(u),\,\theta=v,\,\phi=w,\,R=R_{S}(u)\right)$. The fact
that we model $R=R_{S}(u)$ implies that the hypersurface $\Sigma$ may not
remain in constant BH radial distance as the universe expands. This radial
distance is also called Schücking radius. The successful matching will prove
that this choice of the matching surface was the appropriate thing to do for
successful modeling of the junction. _From now on wherever in this section we
write $T$ and $R$ we mean $T_{S}$ and $R_{S}$ respectively_. The first Darmois
condition $\gamma_{F\alpha\beta}=\gamma_{S\alpha\beta}$ gives

$-1=-F(R)\,\left(\frac{dT}{du}\right)^{2}+F(R)^{-1}\left(\frac{dR}{du}\right)^{2}\,,$
(7)

and

$a^{2}\,r_{\Sigma}^{2}=R^{2}\,.$ (8)

where as we have explained, $R=R_{S}$. The last equation 8 is an important one
and describes that the matching takes place in the hypersurface $\Sigma$ where
the BH radial distance is the time dependent Schücking radius $R_{S}$ and the
cosmological coordinate distance is the constant $r=r_{\Sigma}$.

The second Darmois condition requires to estimate the two second fundamental
forms. First the normal vector ,$n_{i}$, to the spherical hypersurface
$\Sigma$ has to be evaluated. If $\Sigma$ is given by the function
$f[x^{a}(u^{\alpha})]=0$, then $n_{i}$ can be calculated from

$n_{i}=-\frac{f_{,i}}{\sqrt{|g^{ab}f_{,a}f_{,b}|}}\,,$ (9)

where $,i$ denotes $\frac{\partial}{\partial x^{i}}$ .

The (outward pointing) unit normal in the FLRW frame can be calculated from
Eq. (9) and $f_{F}(x_{F}^{i})=r-r_{\Sigma}=0.$ The result is

$n_{Fi}=(0,0,0,-(\frac{a^{2}}{1-k\,r_{\Sigma}})^{1/2}).$ (10)

Note, that the unit normal is spacelike, i.e. $n_{F}^{i}n_{Fi}=+1$. Since the
equation of the matching surface $f_{F}(x_{S}^{i})$ is uknown in the
Schwarzschild frame the normal vector $n_{S}^{i}$ cannot be calculated
directly from Eq. (9). A condition for $n_{S}^{i}$ can be derived from the
partial differentiation of $x_{S}^{i}(u^{\alpha})$ with respect to
$u^{\alpha}$ which generates a tangential vector to the hypersurface $\Sigma$.

$n_{Si}\frac{\partial x_{S}^{i}}{\partial u^{\alpha}}=0\,,$ (11)

From (11) one obtains $n_{S2}=n_{S3}=0$ and

$n_{S1}\frac{dT}{du}+n_{S4}\frac{dR}{du}=0.$ (12)

Furthermore, $n_{S}^{i}$ must satisfy the identity

$n_{S}^{i}n_{Si}\equiv n_{F}^{i}n_{Fi}=1\;\;$ (13)

Then (13) gives

$-\left(F(R)\right)^{-1}{n_{S1}}^{2}+F(R){n_{S4}}^{2}=1.$ (14)

Now using equation (7) and equations (14), (12) it is possible to evaluate
$n_{Si}$ as a function of $u^{\alpha}:$

$n_{Si}=\left(\mp\frac{dR}{du},\;0,\;0,\;\pm\frac{dT}{du}\right)\,.$ (15)

The second fundamental form can be easily calculated in the FLRW frame due to
the simple form of $n_{Fi},$ Eq. (10). Thus from (6) we get

$\displaystyle K_{F\alpha\beta}$ $\displaystyle=$
$\displaystyle\Gamma^{4}_{Fij}n_{F4}\frac{\partial x^{i}_{F}}{\partial
u^{\alpha}}\frac{\partial x^{j}_{F}}{\partial
u^{\beta}}-n_{F4,j}\frac{\partial x^{4}_{F}}{\partial
u^{\alpha}}\frac{\partial x^{j}_{F}}{\partial u^{\beta}}$ (16)

However, $\frac{\partial x^{4}_{F}}{\partial u^{\alpha}}=\frac{\partial
r_{\Sigma}}{\partial u^{\alpha}}=0$, and

$\frac{\partial x^{i}_{F}}{\partial u^{\alpha}}\frac{\partial
x^{j}_{F}}{\partial u^{\beta}}=\frac{\partial x^{\mu}_{F}}{\partial
u^{\alpha}}\frac{\partial x^{\nu}_{F}}{\partial
u^{\beta}}=\delta^{\mu}_{\alpha}\delta^{\nu}_{\beta}$ (17)

so the second fundamental form becomes

$\displaystyle K_{F\alpha\beta}$ $\displaystyle=$ $\displaystyle
n_{F4}\Gamma^{4}_{F\alpha\beta}$ (18) $\displaystyle=$
$\displaystyle-\frac{1}{2}|g_{F44}|^{1/2}g_{F}^{4i}(g_{F\alpha
i,\beta}+g_{F\beta i,\alpha}-g_{F\alpha\beta,i})$ $\displaystyle=$
$\displaystyle\frac{1}{2}|g_{F44}|^{1/2}g_{F}^{44}g_{F\alpha\beta,4}\,,$

and finally

$K_{F\alpha\beta}=\frac{1}{2}\left(\frac{a^{2}}{1-\kappa\,r^{2}}\right)^{-{1/2}}g_{F\alpha\beta,4}\,.$
(19)

From equation (19) it is obvious that the only non zero components of
$K_{F\alpha\beta}$ are $K_{F22}$ and $K_{F33}$

The estimation of the second fundamental form of $\Sigma$ in the Schwarzschild
frame is more complicated. First we have to calculate $n_{Si}$ as a function
of $x_{S}^{i}.$ If we once more differentiate Eq. (11) with respect to
$u^{\alpha}$ it follows that

$n_{Si,j}\frac{\partial x_{S}^{i}}{\partial u^{\alpha}}\frac{\partial
x_{S}^{j}}{\partial u^{\beta}}=-n_{Si}\frac{\partial^{2}x_{S}^{i}}{\partial
u^{\alpha}\partial u^{\beta}}\,,$ (20)

which with the help of the definition equation 6, gives

$K_{S\alpha\beta}=\Gamma^{p}_{Sij}n_{Sp}\frac{\partial x_{S}^{i}}{\partial
u^{\alpha}}\frac{\partial x_{S}^{j}}{\partial
u^{\beta}}+n_{Si}\frac{\partial^{2}x_{S}^{i}}{\partial u^{\alpha}\partial
u^{\beta}}\,.$ (21)

Using equation (21), we find that
$K_{S\alpha\beta}=0,\,\forall\,\alpha\neq\beta.$ This is true since the second
derivative is zero for $\alpha\neq\beta$ and the first term of equation (21)
also is zero for $\alpha\neq\beta$ due to the specific Christoffel symbols.

The second Darmois matching condition $K_{S\alpha\beta}=K_{F\alpha\beta}$ on
$\Sigma,$ provides the following differential equations

$\displaystyle K_{S11}$ $\displaystyle\equiv$
$\displaystyle\Gamma^{4}_{S11}n_{S4}\left(\frac{dT}{du}\right)^{2}+\Gamma^{4}_{S44}n_{S4}\left(\frac{dR}{du}\right)^{2}+2\Gamma^{1}_{S14}n_{S1}\frac{dT}{du}\frac{dR}{du}$
(22) $\displaystyle\
+n_{S1}\frac{d^{2}T}{du^{2}}+n_{S4}\frac{d^{2}R}{du^{2}}=0\equiv K_{F11}\,,$

since $p$ can be either 1 or 4 for non zero $n_{Sp}$ and the only relevant non
zero Christofell symbols $\Gamma^{1}_{Sij}$, $\Gamma^{4}_{Sij}$ are only
$\Gamma^{1}_{S14},\,\Gamma^{1}_{S41}$ and all $\Gamma^{4}_{Sii}$. It is also

$K_{S22}=K_{F22}\Leftrightarrow\Gamma^{4}_{S22}n_{S4}=a\,r_{\Sigma}\sqrt{1-\kappa\,r_{\Sigma}^{2}}$
(23)

and

$K_{S33}=K_{F33}\Leftrightarrow\Gamma^{4}_{S33}n_{S4}=a\,r_{\Sigma}\sin^{2}\theta\,\sqrt{1-\kappa\,r_{\Sigma}^{2}}\,.$
(24)

Now the relevant Christoffel symbols are
$\Gamma^{1}_{S14}=\frac{F^{\prime}}{2F},\,\,\Gamma^{4}_{S44}=-\frac{F^{\prime}}{2F},\,\,\Gamma^{4}_{S11}=\frac{1}{2}F\left(F^{\prime}\right),\,\,\Gamma^{4}_{S22}=-R\,F,\,\,\Gamma^{4}_{S33}=-\sin^{2}\theta\,R\,F,\,$
where notation ”′” is understood as derivative with respect to $R$.

Equations (23) and (24) being equivalent, we can use either of them and Eq.
(8) to obtain

$\frac{dT}{du}=\mp\frac{\sqrt{1-\kappa\,r_{\Sigma}^{2}}}{1-\frac{2\,G_{k}\,M}{R}-\frac{\Lambda_{k}\,R^{2}}{3}}\,.$
(25)

It then follows from Eq. (7) that

$\left(\frac{dR}{du}\right)^{2}=\frac{2\,G_{k}\,M}{R}+\frac{\Lambda_{k}\,R^{2}}{3}-\kappa\,r_{\Sigma}^{2}\,.$
(26)

Now differentiating eqs. (25) and (26) w.r.t. to $u$, we get the following
expressions

$\frac{d^{2}T}{du^{2}}=\pm\sqrt{1-\kappa\,r_{\Sigma}^{2}}\,\frac{d\Big{(}+\frac{2G_{k}M}{R}+\frac{1}{3}\Lambda_{k}R^{2}\Big{)}}{dR}\,\Big{(}1-\frac{2G_{k}M}{R}-\frac{1}{3}\Lambda_{k}R^{2}\Big{)}^{-2}\,\frac{dR}{du}\,.$
(27)

and

$\frac{d^{2}R}{du^{2}}=\frac{1}{2}\,\frac{d\Big{(}\frac{2G_{k}M}{R}+\frac{1}{3}\Lambda_{k}R^{2}\Big{)}}{dR}$
(28)

With equations (25)-(28), Eq. (22) is now identically satisfied. Thus, both
the first and second fundamental forms are continuous on $\Sigma$.

Note that the cosmic evolution through the swiss-cheese is determined by the
cosmic evolution of the matching surface that is happening at spherical
radius, in the black hole frame, $R_{S}$ (and constant $r_{\Sigma}$ for cosmic
frame). This quantity enters the differential equations.

Finally, the matching requirements provide the following equations that are
relevant for the cosmological evolution. _From this point and to the rest of
the paper we use again separately the symbol $R$ for the BH radial distance
and the symbol $R_{S}$ for the Shucking radius_,

$\displaystyle R_{S}$ $\displaystyle=$ $\displaystyle ar_{\Sigma}$ (29)
$\displaystyle\Big{(}\frac{dR_{S}}{dt}\Big{)}^{2}$ $\displaystyle=$
$\displaystyle 1\\!-\\!\kappa
r_{\Sigma}^{2}-\Big{(}1-\frac{2G_{k}(R_{S})\,M}{R_{S}}-\frac{1}{3}\Lambda_{k}(R_{S})\,R_{S}^{2}\Big{)}$
(30) $\displaystyle 2\frac{d^{2}R_{S}}{dt^{2}}$ $\displaystyle=$
$\displaystyle\frac{d\Big{(}\frac{2G_{k}(R)\,M}{R}+\frac{1}{3}\Lambda_{k}(R)\,R^{2}\Big{)}}{dR}|_{R_{S}}$
(31)

The matching radius $r_{\Sigma}$ can be also understood as the boundary of the
volume of the interior solid with energy density equal to the cosmic matter
density $\rho$. Thus, the interior energy content should equal the mass $M$ of
the astrophysical object, i.e.

$r_{\Sigma}=\frac{1}{a_{0}}\Big{(}\frac{2G_{\\!N}M}{\Omega_{m0}H_{0}^{2}}\Big{)}^{\\!\frac{1}{3}}\,,$
(32)

with density parameter $\Omega_{m}=\frac{8\pi
G_{\\!N}\rho}{3H^{2}}=\frac{2G_{\\!N}M}{r_{\Sigma}^{3}a^{3}H^{2}}$ and $a_{0}$
set to be 1 for the rest of the analysis.

A rather important information in the context of the aforementioned approach
is the exact physical element of our model, galaxies, clusters of galaxies or
even super-clusters. In order to answer this, we should look at the physical
interpretation of the Schücking radius, $r_{\Sigma}$, and its relation with
measurable scales of the above astrophysical objects. For Milky Way
($M_{MW}\sim 4\cdot 10^{11}\ M_{\odot}$) with typical values of cosmological
parameters ($\Omega_{m0}\sim 0.3,\ H_{0}\sim 67$ km/Mpc/s, Planck
Collaboration et al. (2020)) using Eq.((32)) we have $r_{\Sigma}\sim 1.4$ Mpc
where the astrophysical radius is $R_{as}\sim 0.15$ Mpc. Similarly, for the
Local Group $M_{LG}\sim(7-20)\cdot 10^{11}$ we find $r_{\Sigma}\sim 2.34$Mpc
and the astrophysical radius is $R_{as}\sim 1.53$ Mpc. Lastly, for the case of
our local super-cluster, VIRGO ($M_{VSC}\sim 10^{15}\ M_{\odot}$) it is
$R_{as}\sim 30.67$ Mpc and $r_{\Sigma}\sim 4$ Mpc. We observe that only in the
scale of galaxy clusters it is $r_{\Sigma}/R_{as}$ of order $\mathcal{O}(1)$,
thus thereafter we will consider galaxy clusters as the elementary entities of
the model.

Eqs. (29, 30, 31) for constant $G_{k}$ and $\Lambda_{k}$ reduce to the
conventional Friedman-Lemaitre-Robertson-Walker (FLRW) expansion equations.
However, although these equations generate the standard cosmological equations
of the $\Lambda\text{CDM}$ model, even in this simple case the interpretation
is different since the $\Lambda$ term appearing in the black hole metric, Koch
and Saueressig (2014), is like an average of all anti-gravity sources inside
the Schücking radius of a galaxy or cluster of galaxies. These anti-gravity
sources one can claim that may arise inside astrophysical black holes in the
centers of which the presence of a quantum repulsive pressure could balance
the attraction of gravity to avoid the not desired singularity, or they arise
due to IR quantum corrections of a concrete quantum gravity theory.
Furthermore, even for constant $G_{k}$ and $\Lambda_{k}$ the coincidence
problem can be removed if the anti-gravity sources, due to the local effective
positive cosmological constant, are related with the large scale structure
that appears recently. In our analysis, we allow the running of $G_{k}$ and
$\Lambda_{k}$ which is the most natural thing if there are IR quantum gravity
corrections.

##  III IR fixed point and cosmic acceleration

In this section the basic idea of the IR fixed point RG scaling will be
introduced and the cosmic evolution of this model will be discussed.

###  III.1 The IR fixed point scaling

The presence of an IR instability in the RG flow for $G$ and $\Lambda$ is
deeply rooted in the structure of the fluctuations determinant associated to
the Einstein-Hilbert action

$S_{k}=\frac{1}{16\pi G_{k}}\int dx^{4}\sqrt{g}(-\mathcal{R}+2\Lambda_{k}).$
(33)

Let us consider a family of off-shell spherically symmetric contributions
labelled by the radius $\phi$ in order to distinguish the contributions from
the two operators $\sqrt{g}\propto\phi^{4}$ and
$\sqrt{g}\mathcal{R}\propto\phi^{2}$ to the Einstein-Hilbert flow Reuter and
Weyer (2004b). It is possible to decompose the fluctuating metric $h_{\mu\nu}$
to the sphere into its irreducible pieces in terms of the corresponding
spherical harmonics. In particular, in the transverse-traceless sector the
operator $S^{(2)}_{k}$ is given by

$-g^{\mu\nu}\nabla_{\mu}\nabla_{n}u+8\phi^{-2}+k^{2}-2\Lambda_{k}$ (34)

up to a positive constant. If the spectrum of the covariant laplacian
$-g^{\mu\nu}\nabla_{\mu}\nabla_{n}u$ is non-negative, a singularity develops
at a finite $k$ in the case of a positive cosmological constant.

In a scalar field theory an identical mechanism is present when the hessian of
the action vanishes in the broken phase. In this case perturbation theory
fails and the path-integral is dominated by non-homogeneous configurations
Ringwald and Wetterich (1990) that produces a “tree-level” instability
renormalization already at level of the bare action Alexandre et al. (1999).
In spite of the complexity of the vacuum structure, the approach based on
functional flow equation is able to sum an infinite tower of (irrelevant UV)
operator and it reproduces the correct flattening of the effective potential
$U(\phi)$ in the broken phase Bonanno and Lacagnina (2004). In this case,
below the coexistence line, the potential behaves as
$U(\phi)\propto-k^{2}\phi^{2}$ and approaches a flat bottom as $k\rightarrow
0$ Berges et al. (2002).

In gravity the situation is more difficult because of the technical
difficulties in solving the RG flow equation beyond the Einstein-Hilbert
truncation in the $k\rightarrow 0$ limit Benedetti and Caravelli (2012);
Demmel et al. (2014); Dietz and Morris (2013). Based on the analogy with the
scalar theory one can imagine that a RG trajectory which realizes a tree-level
renormalization exists Ohta and Yamada (2021). In particular one can write
that in the $k\rightarrow 0$ limit the dimensionless Newton constant and
cosmological constant are attracted toward an IR fixed point
$(g_{*},\lambda_{*}$) according to the following trajectory

$g\left(k\right)=g_{*}+h_{1}k^{\theta_{1}},\quad\quad\lambda\left(k\right)=\
\lambda_{*}+h_{2}k^{\theta_{2}},$ (35)

where $(\theta_{1},\theta_{2}\geq 0)$ are two unknown critical exponents. In
terms of dimensionful quantities, we thus have

${G(k)}_{{IR}}=\frac{g_{*}}{k^{2}}+h_{1}k^{\theta_{1}-2}\quad\quad\Lambda\left(k\right)_{{IR}}=\
\lambda_{*}k^{2}+h_{2}k^{\theta_{2}+2}$ (36)

Here, $k$ should be considered not as a momentum flowing into a loop, but as
an inverse of a characteristic distance, of the physical system under
consideration, over which the averaging of the field variables is performed.
It can then be associated to the radial proper distance or the matter density.
F

For the rest of the paper in order to study the late cosmic evolution, we
assume that the values of Newton constant and cosmological constant at the
astrophysical scales will be those suggested in Eq. (36) i.e.
$G_{k}={G(k)}_{{IR}}$ and $\Lambda_{k}=\Lambda\left(k\right)_{{IR}}$ .

In the first case we need to calculate the radial proper distance and in
particular the Schücking proper radial distance, $D_{p}(R_{S})$, which can be
found from

$D_{p}\left(R_{S}\right)=\ \int_{R_{min}}^{R_{S}}\frac{R}{\sqrt{F(R)}}$ (37)

where $R_{min}$ is zero in our case or a very small value if AS provides a non
singular black hole center, see Aliferis and Zarikas (2021). Then, the
derivative with respect to $R$ is

$D_{p}^{{}^{\prime}}\left(R_{S}\right)=\ \frac{1}{\sqrt{F(R_{S})}}$ (38)

Next, from matching equations equations (29,30,31) we get

${\left(\frac{dR_{S}}{dt}\right)}^{2}=1-\kappa\,r_{\Sigma}^{2}-F(R_{S})$ (39)

which gives

$r_{\Sigma}^{2}{\dot{a}}^{2}=1-\kappa\,r_{\Sigma}^{2}-1+\frac{2\,G_{k}(R_{S})\,M}{R_{S}}+\frac{1}{3}\Lambda_{k}(R_{S})\,R_{S}^{2}$
(40)

and

$\frac{{\dot{a}}^{2}}{a^{2}}=-\frac{\kappa}{a^{2}}+\frac{2G_{k}(R_{S})M}{a^{3}\,r_{\Sigma}^{3}}+\frac{1}{3}\Lambda(R_{S})$
(41)

###  III.2 Scaling using the Cluster Radial proper distance

Here the scaling we assume uses the proper distance

$k=\frac{\xi}{D_{p}(z)}$ (42)

Using Eq.(41) we get

$\frac{{\dot{a}}^{2}}{a^{2}}+k\frac{1}{a^{2}}=\chi+\psi$ (43)

where

$\chi=2\left[\frac{g_{*}}{\xi^{2}}D_{p}^{2}+h_{1}\xi^{\theta_{1}-2}D_{p}^{2-\theta_{1}}\right]M\frac{1}{r_{\Sigma}^{3}a^{3}}$
(44)

and

$\psi=\frac{1}{3}\left[\lambda_{*}\xi^{2}D_{p}^{-2}+h_{2}\xi^{\theta_{2}+2}D_{p}^{-2-\theta_{2}}\right].$
(45)

One useful quantity for evaluating the dark energy is the rate of change of
the proper Shucking distance $D_{p}$. We have
$\frac{dD_{p}}{\text{dR}}=\frac{dD_{p}}{\text{dt}}\frac{\text{dt}}{\text{dR}}=\frac{1}{\sqrt{F(R_{S})}}$
and thus,

$\frac{dD_{p}}{dz}=\frac{-r_{\Sigma}}{{(1+z)}^{2}}\frac{1}{\sqrt{F}}$ (46)

where $z$ is the redshift $z=-1+\frac{1}{a}$ while the time derivative is

${\dot{D}}_{p}=\frac{r_{\Sigma}\,a\,H}{\sqrt{1-\frac{\text{2M}}{a\,r_{\Sigma}}\left(\frac{g_{*}}{\xi^{2}}D_{p}^{2}+h_{1}\xi^{\theta_{1}-2}D_{p}^{2-\theta_{1}}\right)-\frac{1}{3}\left(a\,r_{\Sigma}\right)^{2}(\lambda_{*}\,\xi^{2}D_{p}^{-2}+h_{2}\xi^{2+\theta_{2}}D_{p}^{-2-\theta_{2}})}}.$
(47)

The dark energy and dark matter part into the expansion equations is

$H^{2}+\frac{k}{a^{2}}=\frac{8\,\pi\,G_{N}\,\rho_{DM}}{3}+\frac{8\,\pi\,G_{N}\,\rho_{DE}}{3},$
(48)

where the dark energy and dark matter density are given by

$\rho_{DE}=\frac{3}{8\pi\,G_{N}}\left[-\frac{2G_{N}\,M}{a^{3}\,r_{\Sigma}^{3}}+\psi+\chi\right],\,\,\,\,\rho_{DM}=\frac{3}{8\pi\,G_{N}}\frac{2G_{N}M}{a^{3}\,r_{\Sigma}^{3}}$
(49)

. The cosmological definition of the dust matter refers to a quality that
evolves as $\sim a^{-3}$. In order to use the standard cosmological parameter
$\Omega_{m0}$ so to easily compare our models with the concordance one, we
extract from the ”dark bulk” a portion that evolves as matter. Thus, the
remaining one is correctly labeled as Dark Energy. Following the common
parametrization the expansion can be written as

$E^{2}=\frac{H^{2}}{H_{0}^{2}}=\Omega_{DM}+\Omega_{DE}+\Omega_{\kappa}$ (50)

where

$\Omega_{DM}=\frac{\rho_{DM}}{\rho_{cr}}\,\,\ \text{and}\,\,\,\
\Omega_{DE}=\frac{\rho_{DE}}{\rho_{cr}}\,\,\text{and}\,\,\,\
\Omega_{k}=\frac{-\kappa}{a^{2}H_{0}^{2}}$ (51)

with $\rho_{cr}=\frac{3H_{0}^{2}}{8\pi\,GN}$.

The cosmic acceleration, using Eq.(31), is then expressed as follows

$2\frac{\ddot{a}}{a}+\frac{{\dot{a}}^{2}}{a^{2}}+\frac{\kappa}{a^{2}}=-8\pi\,G_{N}\,P$
(52)

where the total pressure is the sum of cosmic fluid pressure plus dark energy
pressure $P=p+p_{DE}$. In the case of dust Universe $p=0$, which is valid for
small redshifts and since $H^{2}+\kappa/a^{2}=\chi+\psi$ we have

$2\frac{\ddot{a}}{a}+\chi+\psi=-8\pi\,G_{N}\,p_{DE}$ (53)

Finally, the dark energy coefficient of equation or state is given by

$w_{DE}=\frac{p_{DE}}{\rho_{DE}}$ (54)

In order to calculate the expansion of the Universe and the time evolution of
the $\Omega_{DM},\,\Omega_{DE},$ the equation of state parameter $w_{DE}$ and
the deceleration parameter $q=-\frac{a\ddot{a}}{\dot{a}^{2}}$ we must solve
numerically ODEs for the scale factor and the proper distance. The relavant
quantities as functions of the redshift are give n below.

The derivative with respect to the redshift of the proper distance is

$\frac{D_{p}}{dz}=-\frac{r_{\Sigma}}{(z+1)^{2}\sqrt{-\frac{2M(z+1)\left(\frac{g_{*}D_{p}(z)^{2}}{\xi^{2}}+h_{1}\left(\frac{\xi}{D_{p}(z)}\right)^{\theta_{1}-2}\right)}{r_{\Sigma}}-\frac{r_{\Sigma}^{2}\left(h_{2}\left(\frac{\xi}{D_{p}(z)}\right)^{\theta_{2}+2}+\frac{\lambda_{*}\xi^{2}}{D_{p}(z)^{2}}\right)}{3(z+1)^{2}}+1}}$
(55)

This is one of the differential equations we have to solve together with the
expansion rate ODE for the scale factor Eq.(50)

$\begin{split}q(z)=&V^{-1}\biggl{[}6JMD_{p}(z)^{5}\left(h_{1}\left(\frac{\xi}{D_{p}(z)}\right)^{\theta_{1}}+g_{*}\right)+\frac{6\sqrt{3}Mr_{\Sigma}D_{p}(z)^{4}\left(h_{1}(\theta_{1}-2)\left(\frac{\xi}{D_{p}(z)}\right)^{\theta_{1}}-2g_{*}\right)}{z+1}\\\
&-\frac{2J\xi^{4}r_{\Sigma}^{3}D_{p}(z)\left(h_{2}\left(\frac{\xi}{D_{p}(z)}\right)^{\theta_{2}}+\lambda_{*}\right)}{(z+1)^{3}}+\frac{\sqrt{3}\xi^{4}r_{\Sigma}^{4}\left(h_{2}(\theta_{2}+2)\left(\frac{\xi}{D_{p}(z)}\right)^{\theta_{2}}+2\lambda_{*}\right)}{(z+1)^{4}}\biggr{]}\end{split}$

where

$V={2J\left(6MD_{p}(z)^{5}\left(h_{1}\left(\frac{\xi}{D_{p}(z)}\right)^{\theta_{1}}+g_{*}\right)+\frac{\xi^{4}r_{\Sigma}^{3}D_{p}(z)\left(h_{2}\left(\frac{\xi}{D_{p}(z)}\right)^{\theta_{2}}+\lambda_{*}\right)}{(z+1)^{3}}\right)}$
(56)

and

$J=\sqrt{-\frac{6M(z+1)D_{p}(z)^{2}\left(h_{1}\left(\frac{\xi}{D_{p}(z)}\right)^{\theta_{1}}+g_{*}\right)}{\xi^{2}r_{\Sigma}}-\frac{\xi^{2}r_{\Sigma}^{2}\left(h_{2}\left(\frac{\xi}{D_{p}(z)}\right)^{\theta_{2}}+\lambda_{*}\right)}{(z+1)^{2}D_{p}(z)^{2}}+3}$
(57)

where the Sucking radius for a cluster of galaxies with mass $M$ is

$r_{\Sigma}=\sqrt[3]{\frac{2G_{N}M}{H_{0}^{2}\Omega_{DM\,0}}}$ (58)

the dark energy and dark matter part evolution is :

$\Omega_{DE}(z)=\frac{\Omega_{m0}(z+1)^{3}\left(\frac{g_{*}{Dp}(z)^{2}}{\xi^{2}}+h1\left(\frac{\xi}{{D_{p}}(z)}\right)^{\theta_{1}-2}\right)}{G_{N}}+\frac{{h2}\left(\frac{\xi}{Dp(z)}\right)^{\theta_{2}+2}+\frac{\lambda_{*}\xi^{2}}{Dp(z)^{2}}}{3\,H_{0}^{2}}-\Omega_{m0}(z+1)^{3}$
(59)

$\Omega_{DM}(z)=\Omega_{m0}(z+1)^{3}$ (60)

The evolution of the dark energy coefficient is

$\begin{split}w_{DE}(z)=&\tilde{V}^{-1}\,r_{\Sigma}\biggl{[}\sqrt{3}\,r_{\Sigma}^{3}\,\xi^{4}\big{(}2\lambda_{*}+h_{2}(2+\theta_{2})(\frac{\xi}{D_{p}(z)})^{\theta_{2}}\big{)}-3\tilde{J}r_{\Sigma}^{2}(1+z)\xi^{4}\big{[}\lambda_{*}+h_{2}(\frac{\xi}{D_{p}(z)})^{\theta_{2}}\big{]}D_{p}(z)\\\
&-6\sqrt{3}M(1+z)^{3}\big{(}2g_{*}-h_{1}(-2+\theta_{1})\left(\frac{\xi}{D_{p}(z)}\big{)}^{\theta_{1}}\right)D_{p}(z)^{4}\biggr{]}\end{split}$

where

$\tilde{V}=3\tilde{J}(1+z)^{4}\biggr{[}r_{\Sigma}^{3}\xi^{4}\big{(}\lambda_{*}+h_{2}\big{(}\frac{\xi}{D_{p}(z)}\big{)}^{\theta_{2}}\big{)}D_{p}(z)/(1+z)^{3}-6G_{N}M\xi^{2}D_{p}(z)^{3}+6M\big{(}g_{*}+h_{1}\big{(}\frac{\xi}{D_{p}(z)}\big{)}^{\theta_{1}}\big{)}D_{p}(z)^{5}\biggl{]}$
(61)

and

$\tilde{J}=\left\\{3-\frac{r_{\Sigma}^{2}\xi^{2}[\lambda_{*}+h_{2}(\frac{\xi}{D_{p}(z)})^{\theta_{2}}]}{(1+z)^{2}D_{p}(z)^{2}}-\frac{6M(1+z)[g_{*}+h_{1}(\frac{\xi}{D_{p}(z)})^{\theta_{1}}]D_{p}(z)^{2}}{r_{\Sigma}\xi^{2}}\right\\}^{1/2}$
(62)

###  III.3 Scaling using the Cluster Density Profile

Here, in this subsection we use another scaling,

$k=\tilde{\xi}\,\rho_{cl}^{1/4}$ (63)

where the density of the cluster $\rho_{cl}$ is

$\rho_{cl}=\frac{\rho_{c}}{\left(\frac{\tilde{R}}{\alpha}\right)^{\beta}\left(\frac{\tilde{R}}{\alpha}+1\right)^{3-\beta}}$
(64)

where $\beta=1/2$ and $\alpha=226\,kpc$, and $\rho_{c}=5.37\times
10^{6}\,\,M_{\odot}\,kpc^{-3}$ .

The above density profile contains the standard Navarro - Frenk - White
profile explicitly at $\beta=1$ and asymptotically at
$\tilde{R}_{core}\rightarrow 0$. The parameter $\beta$ describes the central
cusp (i.e. $\rho_{cusp}\sim\tilde{R}^{-\beta}$) and it holds $0<\beta<3$, Zhao
(1996). Following Newman et al. (2013), where the aforementioned density
profile was fitted at observational data, we take their ensemble mean value of
$\beta$, that is $\beta=0.5$. The parameter $\alpha$, is the radius of the
central cusp and we take $\alpha=226\,kpc$ Lastly, we assume
$\rho_{c}=5.37\times 10^{6}\,\,M_{\odot}\,kpc^{-3}$. This profile was used as
a typical example.

To proceed further we set $\tilde{R}=D_{p}$ and using the matching equations
we find that the derivative of the proper distance is given now by

$D_{p}(z)/dz=-\frac{r_{\Sigma}}{(z+1)^{2}\sqrt{-\frac{2M(z+1)A}{r_{\Sigma}}-\frac{r_{\Sigma}^{2}B}{3(z+1)^{2}}+1}}$
(65)

The $\Omega_{DM}$ remains the same but the $\Omega_{DE}$ is given by

$\Omega_{DE}=\frac{2AM(z+1)^{3}}{H_{0}^{2}r_{\Sigma}^{3}}+\frac{B}{3H_{0}^{2}}-\Omega_{m0}(z+1)^{3}$
(66)

where

$A=\left(\frac{g_{*}}{\tilde{\xi}^{2}\sqrt{\frac{\rho_{c}}{\sqrt{\frac{D_{p}(z)}{\alpha}}\left(\frac{D_{p}(z)}{\alpha}+1\right)^{5/2}}}}+h_{1}\left(\tilde{\xi}\sqrt[4]{\frac{\rho_{c}}{\sqrt{\frac{D_{p}(z)}{\alpha}}\left(\frac{D_{p}(z)}{\alpha}+1\right)^{5/2}}}\right)^{\theta_{1}-2}\right)$
(67)

and

$B=h_{2}\left(\tilde{\xi}\sqrt[4]{\frac{\rho_{c}}{\sqrt{\frac{D_{p}(z)}{\alpha}}\left(\frac{D_{p}(z)}{\alpha}+1\right)^{5/2}}}\right)^{\theta_{2}+2}+\lambda_{*}\tilde{\xi}^{2}\sqrt{\frac{\rho_{c}}{\sqrt{\frac{D_{p}(z)}{\alpha}}\left(\frac{D_{p}(z)}{\alpha}+1\right)^{5/2}}}$
(68)

Unfortunately, expressions for $q(z)$ and a $w_{DE}(z)$ are quite long and
thus are not presented here for economy of space.

##  IV Numerical Results

In the following, we discuss the ability of our models to reproduce known
observable quantities. Specifically, we check if there exist sets of free
parameters that are able to reproduce the observed Hubble evolution within the
acceptable range variation of the gravitational constant over time and the
recent passage to cosmic acceleration. The existence of allowed range of
parameters at the parameter space does not guarantees the fitting quality of
our models, however the non-existence implies the non-viability of our models.
The full assessment of viability for our models ought to be performed using
likelihood analysis and model selection methods on cosmological data-sets,
such as Supernovae Ia Mitra and Linder (2021); Linder and Mitra (2019); Riess
et al. (1998); Perlmutter et al. (1999), Baryonic Accoustic Oscillations Alam
et al. (2021) and direct measurements of the Hubble rate, in the same fashion
with Anagnostopoulos et al. (2019b), will be resented in a forthcoming paper.
This analysis is beyond the scope of the current work.

In what follows, we explicitly assume $\kappa=0$, based upon observational
results, i.e CMB analysis Planck Collaboration et al. (2020). In any case,
even if the Universe is not exactly flat, the contribution of $\Omega_{k}$ to
the expansion is negligible for recent red-shifts like the ones that are
relevant in our context.

We have decided to explore the free parameters space working with some
simplified but reasonable assumptions. One should take into account that the
evolution of $G_{k}$, must be smooth enough to comply with observational
tests, i.e. solar system constraints, Fienga et al. (2014); Hofmann and Müller
(2018) and astroseismology, Bellinger and Christensen-Dalsgaard (2019). A
particular way to achieve this is to consider the first term in Eq.(36) to be
subdominant to the second, thus allowing $G(k)$ to remain almost constant and
have little dependence of $k$. In the same lines of reasoning, $h_{1}$ must be
very close to $G_{N}$. Therefore in our numerical assessments we set
$h_{1}=G_{N}$ and assume that $\theta_{1}$ must be close to $2$. On the other
hand, as it has already been proven that a term proportional to $k^{2}$,
generates successful phenomenology Mitra et al. (2021); Anagnostopoulos et al.
(2019a, b) we also assume that $\theta_{2}$ must be close to $0$, and since
both terms are practically proportional to $k^{2}$, we can choose
$\lambda_{*}\ll 1$. We further simplify our search for a viable cosmological
model assuming

$g_{*}\ll h_{1}k_{IR}^{\theta_{1}},\quad\lambda_{*}\ll
h_{2}k_{IR}^{\theta_{2}}$ (69)

where $k_{IR}$ is the astrophysical scale. There are still many combinations
of the remaining free parameters that can provide the correct phenomenology. A
particularly simple way to find a subset of the parameter space that gives
reasonable behavior from the phenomenological viewpoint, is to ensure that

$\chi(z\simeq 0)\sim\psi(z\simeq 0),$ (70)

which happens if we use as initial condition for the differential equation
that provides the evolution of the proper distance $D_{P}(z)$, the value from
the requirement to have as $H(z=0)$ the current measured value of the Hubble
rate. In general, the initial value that one gets for the proper distance in
order to achieve the current value for $H(0)$ or $\Omega_{DE}$ to be the same
as in the concordance model, could possibly give an unnatural value for the
proper distance. In contrast, we obtain reasonable values that are of the same
order and little bit larger than the Schücking radius of real astrophysical
objects (clusters of galaxies) which is the correct thing. Furthermore, in
order not to have new scales or fine tuning we always give values close to
order of one O(1) for $\xi$ and the dimensionless $\tilde{\gamma}$ which is
related to $h_{2}$ from the definition $\tilde{\gamma}=\gamma\,G_{N}^{1-b/2}$.
Note, that in general the values of $\xi$ an $\tilde{\xi}$ can be different as
Eq.(63) entails a different scaling length. However, in the case of the first
scaling using the proper distance and not the density, the natural length that
appears in the calculations is the cluster length scale._It worths emphasizing
at this point, that using this naturally emerging astrophysical cluster length
scale (which means also setting $\xi$ of O(1)), we get the correct amount of
dark energy today after a recent passage from deceleration to acceleration and
at the same time interestingly, this length scale makes the swiss-cheese model
an acceptable approximation, since the Schücking matching radius is close to
the cluster length._

In order to compare the Hubble rate in the context of AS models with the
observations, we have to normalize accordingly the Hubble rate, i.e to demand
$H(0)=100h$, where $h$ is dimensionless parameter and $h\in(0,1)$. From this
requirement, we can eliminate $\tilde{\gamma}$ from the Hubble rate demanding
the matching to happen in $2r_{\Sigma}$, or we can have an initial condition
for the $D_{p}(0)$ equal a value close to $r_{\Sigma}$ and fix
$\tilde{\gamma}$. Thus, we can further manage to lower the number of free
parameters for our models, which is beneficial from both conceptual and
numerical viewpoints. In fact, we can also check if a particular instance of
the model could describe the accelerated expansion of the universe, while also
satisfying the available observational constraints on $\dot{G}/G$. In Fig. (3)
we illustrate the Hubble rate for both scaling relations, with the total of 3
parameters, that are ($\xi,\theta_{1},\theta_{2}$) for the first scaling
relation and ($\tilde{\xi},\theta_{1},\theta_{2}$) for the second scaling
relation. An indicative set of parameters that can give the desired
phenomenology are listed in Table (1).

We aslo plot the most recent data compilation, that have obtained via the
Cosmic Chronometers (CC) method, thus being approximately model-independent,
compiled at Yu et al. (2018). In addition we depict the Hubble rate of the
concordance model (dashed black line) corresponding to the parameters set
$\\{\Omega_{m0},h\\}=\\{0.3,0.68\\}$, to allow for qualitative comparison
between the two models. The Hubble rate for each scaling relation is depicted
with the solid line. Examination of Fig. (3) shows that both scenarios can
give Hubble rate in close agreement with the observations.

Regarding the value of the normalized derivative of the coupling constant,
$\dot{G}/G$ today, we calculate $\dot{G}/G=-0.001\cdot 10^{-13}yrs^{-1}$ and
$\dot{G}/G=0.0083\cdot 10^{-13}yrs^{-1}$ for the first and the second scaling
relations, respectively. From Fig. (1), is easy to observe that both models
are within $1\sigma$, that is in full agreement with the observational values
from Fienga et al. (2014); Hofmann and Müller (2018). The same with the result
from Bellinger and Christensen-Dalsgaard (2019) which is one order of
magnitude larger from the other two and is not depicted at Fig. (1).

Figure 1: Values of the normalized derivative of the Newton constant G around
z = 0, for both scaling relations labeled ”scaling-I” which is the proper
distance scaling and ”scaling-II” the scaling using density. We include also
the most recent observational values along with their error bars ($1\sigma$).
These are: (a) from Fienga et al. (2014) via calculating possible impact of
varying G on planetary orbits (labelled ’solar system orbits’) (b) from
Hofmann and Müller (2018) using lunar laser ranging data (labelled ’lunar
laser’).

Fig. (1) shows the comparison of these results with corresponding
observations. As it is apparent from the latter figure, the evolution of G is
within 1$\sigma$ of the observed values, for both scaling relations.

Scaling Model | $\xi\,\text{or}\,\tilde{\xi}$ | $\tilde{\gamma}$ | $\theta_{1}$ | $\theta_{2}$
---|---|---|---|---
Cluster Radial Proper Distance ($\xi,\theta_{1},\theta_{2}$) | 3.0 | 3.0 | 2.001 | 0.06
$>>$ | 1.0 | 0.1 | 2.005 | 0.05
Cluster Density Profile ($\tilde{\xi},\theta_{1},\theta_{2}$) | 1e-28 | 10 | 2.001 | 0.06
$>>$ | 1e-28 | 15 | 2.005 | 0.05
Table 1: Indicative points on the parameter space for each of the two scaling
relations (§42,63) that correspond to efficient phenomenology.

As another viability test for our models, we use the transition redshift,
defined via the requirement $q(z_{tr})\equiv 0$. The latter corresponds to the
moment of the transition between matter era and Dark Energy era in the cosmic
history. In particular, using the plot of the deceleration parameter, q,
(second column of Fig.(3) ) for these particular instances of the two
different scaling expressions, we find the transition redshift. For the case
of scaling with proper distance, we have $z_{tr}^{I}\simeq 0.8$ which lies
within $\sim 3\sigma$ of both Jesus et al. (2020) results, obtained via a
model independent method. For the case of scaling using density, we obtain
$z_{tr}^{II}\simeq 1$ which again is statistically compatible with Jesus et
al. (2020) results. From the latter fact we deduce that both models seem to be
viable.

Figure 2: Plot of the Hubble rate along with a subset of the observational
dataset, containing only data points obtained via the Cosmic Chronometers (CC)
method, compiled at Yu et al. (2018). To allow direct comparison, we depict
the Hubble rate of the concordance model (dashed black line) corresponding to
the parameters set $\\{\Omega_{m0},h\\}=\\{0.3,0.68\\}$. The Hubble rate for
each scaling relation is depicted with the solid line. In all cases the
initial condition for the solution of proper distance was set to be $2r_{s}$,
for reasons explained in the text. Left: Scaling using proper distance, for
the parameter set
$\\{\Omega_{m0},\xi,h,\theta_{1},\theta_{2}\\}=\\{0.3,0.68,1,2.001,0.06\\}$
Right: Scaling using density, for the parameter set
$\\{\Omega_{m0},h,\tilde{\xi},\theta_{1},\theta_{2}\\}=\\{0.3,0.68,10^{-28},2.005,0.05\\}$

Figure 3: Plots of some representative cosmological parameters. From left to
right: equation of state parameter for the effective Dark Energy fluid,
$w_{DE}$, deceleration parameter, $q$ and normalized Dark Energy Density,
$\Omega_{DE}$ as a function of the redshift $z$, for the two different scaling
relations (top and bottom panel), see text. The parameters used for these
plots are (top panel) $\Omega_{m0}=0.3,\ \xi=1,\ \theta_{1}=2.001,\
\theta_{2}=0.04$, (bottom panel) $\Omega_{m0}=0.3,\ \tilde{\xi}=10^{-28},\
\theta_{1}=2.005,\ \theta_{2}=0.06$.

##  V Conclusions

In this work we have extended the swiss-cheese models presented in Kofinas and
Zarikas (2018); Anagnostopoulos et al. (2019b) to include an explicit running
of $G$ according to the IR fixed point hypothesis. The presented analysis
generalizes the original Einstein-Strauss model by considering an interior
vacuole spacetime described by a RG-improved Schwarzschild spacetime in the
presence of a cosmological constant and Newton’s constant that depend either
on the proper distance $D_{s}$ , or on the matter density according to a RG
trajectory for which both the dimensionless Newton constant and cosmological
constant approach a fixed point at large distances.

The Darmois matching conditions determine the evolution of the FLRW spacetime
outside the Schücking Radius. The RG trajectories emanating from the IR fixed
point have been parametrized by two critical exponents $\theta_{1}$ and
$\theta_{2}$ and by the scaling parameter $\xi$ which relates the RG cutoff
$k$ with a characteristic distance scale. It turns out that the present phase
of accelerated expansion is the result of the cumulative antigravity sources
contribution within the Schücking radius. In particular no fine-tuning emerges
at late times. Our formalism allows us also to take into account of the
constraints on $\dot{G}/G$. Albeit the parameter space is large and a full
numerical analysis is beyond the scope of the present work, it was possible to
find many set parameters which reproduce the current $\Lambda CDM$ standard
model. The results are very encouraging, as for both scaling expressions we do
obtain reasonable behavior for the cosmological quantities
$H(z),q,\,\,w_{DE},$ and $\Omega_{DE}$.

Interesting enough for the interesting cases both $\theta_{1}$ and
$\theta_{2}$ can be taken as positive and this is consistent with the idea
that the IR fixed point must be attractive in the $k\rightarrow 0$ limit. It
would therefore be interesting to further test this cosmological model against
the current observational data.

##  VI Acknowledgments

The authors acknowledge the support of the Faculty Development Competitive
Research Grant Program (FDCRGP) Grant No. 110119FD4534. We are grateful to
Alessia Platania for important remarks and comments on the manuscript.

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