∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail<EMAIL_ADDRESS>e4e-mail<EMAIL_ADDRESS>
11institutetext: Department of Mathematics and Applications “R. Caccioppoli”,
University of Naples Federico II, Via Cintia, 80126 Naples, Italy
22institutetext: Institute for Theoretical Physics, Karlsruhe Institute of
Technology (KIT), 76128 Karlsruhe, Germany 33institutetext: Institute for
Nuclear Physics, Karlsruhe Institute of Technology (KIT), Hermann-von-
Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen, Germany 44institutetext:
Università degli studi di Napoli “Federico II”, Dipartimento di Fisica “Ettore
Pancini”, Complesso Universitario di Monte S. Angelo, Via Cintia Edificio 6,
I-80126 Napoli, Italy 55institutetext: Scuola Superiore Meridionale, Largo S.
Marcellino 10, 80138 Napoli, Italy 66institutetext: Istituto Nazionale di
Fisica Nucleare, Sezione di Napoli, Complesso Universitario di Monte S.
Angelo, Via Cintia Edificio 6, 80126 Napoli, Italy 77institutetext: Tomsk
State Pedagogical University, ul. Kievskaya, 60, 634061 Tomsk, Russia
88institutetext: Lab.Theor.Cosmology,Tomsk State University of Control Systems
and Radioelectronics(TUSUR), 634050 Tomsk, Russia

# Reconstructing wormhole solutions in curvature based Extended Theories of
Gravity

Vittorio De Falcoe1,addr1  Emmanuele Battistae2,addr2,addr3  Salvatore
Capozzielloe3,addr4,addr5,addr6,addr7  Mariafelicia De
Laurentise4,addr4,addr6,addr8

(Received: / Accepted: )

###### Abstract

Static and spherically symmetric wormhole solutions can be reconstructed in
the framework of curvature based Extended Theories of Gravity. In particular,
extensions of the General Relativity, in metric and curvature formalism give
rise to modified gravitational potentials, constituted by the classical
Newtonian potential and Yukawa-like corrections, whose parameters can be, in
turn, gauged by the observations. Such an approach allows to reconstruct the
spacetime out of the wormhole throat considering the asymptotic flatness as a
physical property for the related gravitational field. Such an argument can be
applied for a large class of curvature theories characterising the wormholes
through the parameters of the potentials. According to this procedure,
possible wormhole solutions could be observationally constrained. On the other
hand, stable and traversable wormholes could be a direct probe for this class
of Extended Theories of Gravity.

###### Keywords:

Physics of black holes alternative gravity wormhole

††journal: Eur. Phys. J. C

##  1 Introduction

Wormholes (WHs) are exotic compact objects characterized by no horizon and
singularities, and endowed with a traversable bridge, called WH neck,
connecting two universes or two different regions of the same spacetime Visser
(1995). They have been extensively investigated in the literature, and the
papers on such a topic can be distinguished in two categories: (1)
finding/constructing solutions in General Relativity (GR) or in extended/
alternative theories of gravity Visser (1989); Barcelo and Visser (1999);
Böhmer  _et al._ (2012); Anchordoqui  _et al._ (2000); Bahamonde  _et al._
(2016); Capozziello  _et al._ (2020); (2) proposing methods to observationally
determine their existence considering reliable fluids or astrophysical probes
capable of making these objects stable and traversable structures Cardoso  _et
al._ (2016); Konoplya and Zhidenko (2016); Paul  _et al._ (2019); Dai and
Stojkovic (2019); Banerjee  _et al._ (2019); Hashimoto and Tanahashi (2017);
Dalui  _et al._ (2019); De Falco  _et al._ (2020). Regarding the last point,
the mentioned strategies are all based on analysis in full and strong
gravitational field regimes. However, at the best of our knowledge, there are
no methods to investigate these systems in the weak field limit and then
observe their features as realistic gravitational fields. Here we develop a
possible strategy to reconstruct WH solutions considering properties of
spacetime in the weak field limit of curvature based _Extended Theories of
Gravity_ Nojiri and Odintsov (2011); Capozziello and De Laurentis (2011);
Clifton  _et al._ (2012); Nojiri  _et al._ (2017).

In this class of theories, the Ricci curvature scalar $R$ of the Hilbert-
Einstein action is replaced by generic functions of curvature invariants and
auxiliary scalar fields. Quantum and cosmological motivations for adopting
these GR extensions are discussed in details in Ref. Capozziello and Faraoni
(2011). A straightforward realization of this approach is done by considering
$f(R)$ gravity theories, where GR is just a particular case of a wide class of
models (i.e., $f(R)=R$). In this context, also WH solutions can be
investigated considering the fact that further degrees of freedom, related to
curvature based Extended Theories of Gravity, can give rise to effective
perfect fluids acting as sources in their field equations Zubair  _et al._
(2016); Mazharimousavi and Halilsoy (2016); Capozziello  _et al._ (2018);
Elizalde and Khurshudyan (2018). Thanks to these additional geometrical
contributions, WHs could be, eventually, stable and traversable depending on
the parameters of the belonging theory.

However, WH solutions depend also on the external gravitational field and its
asymptotic behaviour, so an _inverse scattering technique_ could be employed
to reconstruct them from the weak field limit of curvature based Extended
Theories of Gravity. It has been shown that from the _Post-Newtonian (PN)
expansion_ of extended gravity field equations, _Yukawa-like corrections_ to
the standard Newtonian potential naturally emerge Capozziello and Stabile
(2010); Napolitano  _et al._ (2012); Liu  _et al._ (2017). It seems a general
feature of several classes of gravitational models, where the particular case
is just represented by GR, because such corrections are not present. The
parameters of the Yukawa-like corrections can be constrained by different sets
of data De Martino  _et al._ (2018); De Laurentis  _et al._ (2018);
Capozziello  _et al._ (2020, 2020), allowing thus to reconstruct reliable WH
models in agreement with the observations.

_An idea is to describe static and spherically symmetric WHs by Taylor-
expanding their metric components in the weak-field limit in order to control
how these solutions behave asymptotically. Adopting the weak field limit of
curvature based Extended Theories of Gravity, where the parameters are gauged
by the observations, and comparing them with the WH expansions, one is able to
determine the coefficients of the WH metric and then reconstruct such
solutions within different gravity frameworks with the aim to obtain stable
and traversable solutions_. This is the central argument on which this paper
is based.

An important remark is in order at this point. Theories we are going to take
into account are just a particular class of possible extensions of GR. Here we
are going to consider gravity models based on the Riemann tensor, and the
other curvature invariants, constructed by the Levi-Civita connection of
spacetime metric, in other words gravity formulated in the so-called metric
approach. However, Extended Theories of Gravity can involve also an affine
connection independent of the metric, the so-called metric-affine gravity (see
e.g. Olmo (2011)), or a purely affine formulation Poplawski (2014). Important
sub-classes of metric-affine theories are the Poincaré gauge gravity Obukhov
(2006), the modified teleparallel gravity based on the Weitzenböck connection
Cai  _et al._ (2016); Hohmann  _et al._ (2018), the modified symmetric
teleparallel gravity Conroy and Koivisto (2018). All these formulations point
out that the debate on the fundamental variables describing the gravitational
field is still open.

The article is organized as follows: in Sec. 2 we summarize the properties of
static and spherically symmetric WHs; in Sec. 3, the PN expansion in the
$f(R)$ gravity framework is discussed as a valid and general paradigm for all
curvature based Extended Theories of Gravity; in Sec. 4 we apply the above-
mentioned strategy to constrain the WH solutions through the entries of
extended gravity models; finally in Sec. 5 we draw the conclusions.

##  2 Static and spherically symmetric wormhole solutions

A static and spherically symmetric WH can be described, in spherical
coordinates $(t,r,\theta,\varphi)$ and geometrical units $G=c=1$, by the
metric Morris and Thorne (1988),

$\displaystyle
ds^{2}=-e^{2\chi(r)}dt^{2}+\frac{dr^{2}}{1-b(r)/r}+r^{2}d\Omega^{2},$ (1)

where $d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2}$, and $\chi(r)$ and
$b(r)$ represent the redshift and shape functions, respectively. Eq.(1)
describes a two-parameters family of metrics depending on $\chi(r),b(r)$. It
represent a class of solutions valid both in GR and in extended/alternative
theories of gravity, which can be built up to be traversable and stable.

Figure 1: Sketch of WH geometry.

In Fig. 1 a sketch of the WH geometry is displayed.

Referring to this picture, the significant geometrical proprieties of metric
(1) can be summarized as follows Morris and Thorne (1988) (see Fig. 1): (1)
the absence of horizons and singularities, which entails that $\chi(r)$ and
$b(r)$ are real smooth functions, and $\chi(r)$ is everywhere finite; (2) the
condition $(1-b(r)/r)\geq 0$ allows to define a finite proper radial distance
$l$ Morris and Thorne (1988); (3) the flaring outward condition Hochberg and
Visser (1998a, b); Capozziello  _et al._ (2015, 2014) requires that
$b^{\prime}(r)<b(r)/r$ near and on the throat. It defines the minimum radius
such that $r_{\rm min}=b_{0}$ and $b(r_{\rm min})=b_{0}$; (4) the asymptotic
flatness entails that $b(r)/r\to 0$ and $\chi(r)\to 0$ for $r\to+\infty$; (5)
the WH traversability depends on the underlying theory of gravity. It can be
achieved by considering some form of exotic matter Hochberg  _et al._ (1997);
Bronnikov  _et al._ (2013); Garattini (2019), topological defects Lobo and
Oliveira (2009); Harko  _et al._ (2013); Di Grezia  _et al._ (2017) or
alternative theories of gravity Capozziello  _et al._ (2012). It is not
satisfied in GR for standard perfect fluids; (6) the mass $M$ is defined
according to the Arnowitt, Deser, Misner (ADM) formalism. It represents the
total mass of the system contained in the whole spacetime Visser (1995). It is

$M\equiv\lim_{r\to+\infty}m(r)=\frac{c^{2}b_{0}}{2G}+4\pi
c^{2}\int^{\infty}_{b_{0}}\rho(x)x^{2}dx.$ (2)

By a rapid inspection of the above requirements, in particular properties (3)
and (4), it is clear that the external gravitational field and the asymptotic
conditions play a major role in defining self-consistent WH solutions. Here we
want to investigate a class of alternative theories of gravity (Extended
Gravity) which can potentially satisfy the above conditions. The paradigm is
that constraining the parameters of gravitational potentials of such models is
possible, in principle, to reconstruct WH solutions.

##  3 The Post-Newtonian expansion and the corrections to the gravitational
potential

We consider the external solution of an isolated compact object, where gravity
is described within Extended Theories of Gravity. As said above, we mean
extensions of GR, where the GR itself is a particular case of a large class of
theories. The aim is to reconstruct, in this framework, the features of WH
solutions. A straightforward extension of GR is given by the following action

$\mathcal{A}=\int f(R)\sqrt{-g}\ {\rm d}^{4}x,$ (3)

where the Lagrangian density $f(R)$ is a generic analytic function of the
Ricci curvature scalar $R$ (for $f(R)=R$, GR is restored). Here $g$ is the
determinant of the metric tensor $g_{\mu\nu}$, and $\sqrt{-g}\ {\rm d}^{4}x$
is the invariant volume.

Minimizing the action (i.e., $\delta\mathcal{A}=0$) and computing the integral
(3) in the local inertial frame (see Ref. Capozziello and Faraoni (2011), for
details), we obtain the well known fourth-order field equations within the
$f(R)$ gravity theories

$\displaystyle
f^{\prime}(R)R_{\mu\nu}-\frac{1}{2}f(R)g_{\mu\nu}-f^{\prime}(R)_{;\mu\nu}$ (4)
$\displaystyle\hskip 28.45274pt+g_{\mu\nu}\Box f^{\prime}(R)=\kappa
T_{\mu\nu},$

with the trace given by

$3\Box f^{\prime}(R)+f^{\prime}(R)R-2f(R)=\kappa T,$ (5)

where $\kappa=8\pi G/c^{4}$, $f^{\prime}(R)\equiv df(R)/dR$,
$(\cdot)_{;\mu}=\nabla_{\mu}(\cdot)$ is the covariant derivative, $\Box\equiv
g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ is the curved d’Alembert operator, and
$T_{\mu\nu}$ is the stress-energy tensor of a perfect fluid matter. Equations
(4) generate a fourth-order dynamics in the metric tensor $g_{\mu\nu}$ 111The
field equations (4) can be recast as  $\displaystyle G_{\mu\nu}$
$\displaystyle=\frac{1}{f^{\prime}(R)}\left[T^{\rm(curv)}_{\mu\nu}+T_{\mu\nu}\right]\equiv\frac{1}{f^{\prime}(R)}\left\\{f^{\prime}(R)_{;\mu\nu}-g_{\mu\nu}\Box
f^{\prime}(R)\right.$ (6) $\displaystyle\left.\hskip
42.67912pt+g_{\mu\nu}\frac{[f(R)-f^{\prime}(R)R]}{2}\right\\}+\frac{T_{\mu\nu}}{f^{\prime}(R)},$
where we recover the Einstein tensor $G_{\mu\nu}$ and the higher order terms
become part of an effective stress-energy tensor $T^{\rm(curv)}_{\mu\nu}$,
which behaves as a perfect fluid, see Ref. Capozziello  _et al._ (2018), for
more details..

Since $f(R)$ is an analytic function of $R$, we can expand it in Taylor series
around $R=0$ (corresponding to the Minkowski Ricci curvature) obtaining
Capozziello and Faraoni (2011)

$f(R)\approx f_{1}R+\frac{1}{2}f_{2}R^{2}+\dots,\quad
f_{n}=\frac{1}{n!}\frac{d^{n}f(R)}{dR^{n}}\Big{|}_{R=0}.$ (7)

Let us consider a class of static and spherically symmetric metrics described
in geometric units $G=c=1$ and spherical coordinates $(t,r,\theta,\varphi)$ as

$ds^{2}=-g_{tt}(r)dt^{2}+g_{rr}(r)dr^{2}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}),$ (8)

where the unknown functions are $g_{tt}(r),g_{rr}(r)$. Then, we take into
account the weak field limit of the metric tensor $g_{\mu\nu}$ with respect to
the Minkowski background $\eta_{\mu\nu}$, i.e.,
$g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $1=|\eta_{\mu\nu}|\gg|h_{\mu\nu}|$.
The PN expansion of the metric components $g_{\mu\nu}(r)$ up to the order
$\mathcal{O}(2)$ are given by the following expressions Capozziello and
Faraoni (2011)

$\displaystyle g_{tt}$ $\displaystyle=$
$\displaystyle-1+g_{tt}^{(2)}(r)+g_{tt}^{(4)}(r),$ (9) $\displaystyle g_{rr}$
$\displaystyle=$ $\displaystyle-1+g_{rr}^{(2)}(r),$ (10) $\displaystyle
g_{\theta\theta}$ $\displaystyle=$ $\displaystyle r^{2},\qquad
g_{\varphi\varphi}=r^{2}\sin^{2}\theta,$ (11)

and, at the same order, the Ricci curvature scalar is

$R\approx R^{(2)}+R^{(4)}+\dots\ .$ (12)

To find the asymptotic expressions of the functions $R^{(2)}$, $g_{tt}^{(2)}$,
$g_{rr}^{(2)}(r)$, we need three independent equations. Therefore, considering
the $tt$ and $rr$ components of Eqs. (4) together with the trace (5), we
respectively have Capozziello and Faraoni (2011)

$\displaystyle
g_{tt,rr}^{(2)}+\frac{2}{r}g_{tt,r}^{(2)}-\left[\frac{R^{(2)}(f_{1}+4f_{2})+8f_{2}R_{,r}^{(2)}}{f_{1}}\right]=0,$
(13) $\displaystyle
g_{rr,r}^{(2)}-\frac{f_{1}rR^{(2)}+8f_{2}R_{,r}^{(2)}-f_{1}rg_{tt,rr}^{(2)}}{2f_{1}}=0,$
(14) $\displaystyle
R_{,rr}^{(2)}+\frac{2}{r}R_{,r}^{(2)}+\frac{f_{1}}{6f_{2}}R^{(2)}=0.$ (15)

Here we are considering _solutions in vacuum_ (i.e., $T_{\mu\nu}=0$), but with
the presence of matter the extension of the above solutions can be
straightforwardly obtained.

Solving first Eq. (15) in terms of $R^{(2)}$, then Eq. (13) for
$g_{tt}^{(2)}$, and finally Eq. (14) for $g_{rr}^{(2)}$, we obtain Capozziello
and Faraoni (2011)

$\displaystyle R^{(2)}$ $\displaystyle=$
$\displaystyle\delta_{2}\frac{e^{-r/L}}{r}+\delta_{3}\frac{e^{r/L}}{2r},$ (16)
$\displaystyle g_{tt}^{(2)}$ $\displaystyle=$
$\displaystyle\delta_{0}-\frac{\delta_{1}}{f_{1}r}+\delta_{2}L^{2}\frac{e^{-r/L}}{3r}+\delta_{3}L^{3}\frac{e^{r/L}}{6r},$
(17) $\displaystyle g_{rr}^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{\delta_{1}}{f_{1}r}-\delta_{2}L^{2}\left(1+\frac{r}{L}\right)\frac{e^{-r/L}}{3r}$
(18)
$\displaystyle+\delta_{3}L^{3}\left(1-\frac{r}{L}\right)\frac{e^{r/L}}{6r},$

where $L=\sqrt{-6f_{2}/f_{1}}$, $\delta_{0},\delta_{1},\delta_{2},\delta_{3}$
are all constants.

In agreement with whether $L$ is real or complex, we can have different
solutions (see Ref. Capozziello and Faraoni (2011), for details). Assuming $L$
real, which strictly depends on the sign of $f_{1},f_{2}$, and imposing that
at infinity Eqs. (16) – (18) must reduce to the Minkowski metric, we have

$\displaystyle g_{tt}(r)$ $\displaystyle=$ $\displaystyle
1-\frac{GM}{f_{1}r}+\frac{\delta_{2}L^{2}e^{-r/L}}{3r},$ (19) $\displaystyle
g_{rr}(r)$ $\displaystyle=$ $\displaystyle
1+\frac{GM}{f_{1}r}+\frac{\delta_{2}L^{2}(1+r/L)e^{-r/L}}{3r},$ (20)
$\displaystyle R$ $\displaystyle=$
$\displaystyle\delta_{2}\frac{e^{-r/L}}{r}.$ (21)

We can combine the constants $f_{1}$ and $\delta_{2}$ in one single constant
$\delta$ and rewrite the metric components $g_{tt}$ and $g_{rr}$ as
gravitational potentials $\Phi$ and $\Psi$, i.e., $g_{tt}=-1+\Phi$ and
$g_{rr}=1+\Psi$, which respectively leads to Capozziello and Faraoni (2011)

$\displaystyle\Phi(r)$ $\displaystyle=$
$\displaystyle-\frac{2GM}{rc^{2}(\delta+1)}(1+\delta e^{-r/L}),$ (22)
$\displaystyle\Psi(r)$ $\displaystyle=$
$\displaystyle\Phi(r)-\frac{2GM}{rc^{2}(\delta+1)}\left(\delta\frac{r}{L}e^{-r/L}-2\right),$
(23)

where

$\delta_{2}=-\frac{6GM}{L^{2}}\left(\frac{\delta}{1+\delta}\right),\qquad\delta=-\frac{\delta_{2}f_{1}L^{2}}{6GM}.$
(24)

Readjusting the parameters in the following way

$G\to\frac{2G}{1+\delta},\qquad\alpha\to\delta,\qquad L\to\frac{1}{\lambda},$
(25)

the gravitational potential $\Phi$ becomes

$\Phi=-\frac{GM}{r}\left(1+\alpha e^{-\lambda r}\right),$ (26)

which is composed by the standard Newtonian potential $\Phi_{N}=-GM/r$ and a
Yukawa-like correction $\Phi_{Y}=\alpha\Phi_{N}e^{-r/\lambda}$, where $\alpha$
gives the strength of the correction and $\lambda$ is the length scale over
which such potential acts.

The potential $\Psi$ can be also written as

$\Psi(r)=\Phi(r)+\delta\Phi(r),$ (27)

where $\delta\Phi(r)$ is an extra contribution derived from the above
gravitational potential. Clearly $\Psi(r)\sim\Phi(r)$, as soon as the PN limit
of GR is recovered and also it has been showed that in the weak field limit
gives very small contributions, implying thus that $\Psi(r)\sim\Phi(r)$ De
Laurentis  _et al._ (2018).

Similar calculations, developed here for $f(R)$ gravity, can be easily
performed for any Extended Theory of Gravity involving higher-order curvature
invariants or scalar-tensor terms Capozziello and De Laurentis (2011);
Capozziello and Faraoni (2011). In Table 1, we report several examples of
Extended Theories with the proper corrections and parameterizations of
gravitational potentials.

##  4 Constraining wormhole solutions by Extended Theories of Gravity

The parameters of Yukawa-like corrections $\Phi_{Y}$ can be observationally
determined considering self-gravitating systems. In Capozziello  _et al._
(2020), the authors adopted _the Fundamental Plane of elliptical galaxies_ to
fulfil this aim. In the specific case of $f(R)$ theories, the range of
parameters $f_{1},f_{2}$ (i.e. $\alpha$ and $\lambda$) were fixed by
reconstructing models compatible with the Fundamental Plane of elliptical
galaxies. In that case, the goal was showing that gravitational corrections
could fit dynamics without resorting to the dark matter’s existence
hypothesis.

A similar procedure can be adopted here to “reconstruct” WH solutions and
their asymptotic behavior in the weak-field limit. Specifically, we can
develop the gravitational potential $\Phi$ in power series of $1/r$, together
with the Yukawa term, but avoiding singular expansions at infinity. This
procedure will allow thus to restore the prescriptions for physical WHs
summarized in Sec.2.

The approach consists in Taylor expanding the WH metric components at spatial
infinity, obtaining thus two power series of $1/r$ (one for $g_{tt}$ and
another one for $g_{rr}$), where the coefficients are functions of the
redshift and shape functions, respectively. Then we can match the coefficients
of the WH metric with those of the given extended theory of gravity to
reconstruct the WH solution. A summary of the models, the modified
gravitational potentials, and WH parameters is reported in Table 1.

###  4.1 Asymptotic expansion of the modified Newtonian potential

The expansion of the modified Newtonian potential $\Phi$ of Eq. (26), in power
series, means to expand the Yukawa-like potential. Let us define the variable
$x=1/r$, converting thus the limit from $r\to\infty$ to $x\to 0$. The function
$e^{-\lambda/x}$ has a singular point at $x=0$, which can be extended for
continuity, i.e., for $x\to 0$ the function $e^{-\lambda/x}\to 0$ for all
positive values of $\lambda$. If we consider the Taylor expansion of this
function at $x=0$, we obtain that the related power series has all
coefficients identically equal to zero. This means that, in such a point, the
function $e^{-\lambda/x}$ is approximated by the horizontal straight line
$y=0$. If we expand this function in terms of the Laurent series around $x=0$,
it gives $\sum_{k}\frac{(-1)^{k}}{k!}\lambda^{-k}x^{-k}$. This expansion
diverges close to $x=0$, being therefore physically not meaningful for our
tasks.

Let us develop a feasible approach combining a correct mathematical treatment
with an admissible physical interpretation. We Taylor-expand the Yukawa term
$e^{-\lambda/x}$ around $x=\varepsilon\ll 1$ up to $x^{2}$, having thus

$e^{-\frac{\lambda}{x}}=c_{1}(\varepsilon,\lambda)+c_{2}(\varepsilon,\lambda)x+\mathcal{O}\left[(x-\varepsilon)^{2}\right],$
(28)

where

$c_{1}(\varepsilon,\lambda)=\frac{e^{-\frac{\lambda}{\varepsilon}}(\varepsilon-\lambda)}{\varepsilon},\qquad
c_{2}(\varepsilon,\lambda)=\frac{\lambda
e^{-\frac{\lambda}{\varepsilon}}}{\varepsilon^{2}}.$ (29)

Equation (28) strictly depends on $\varepsilon$, which, in turn, depends on
the sensibility of the chosen observational instruments, $\Theta_{\rm
obs}^{\pm}$222We refer here to the sensibility of the instrument, $\Theta_{\rm
obs}^{\pm}$, as related to the error of the measurement process (e.g., $\delta
r=\pm 0.01$ Kpc). In Sec. 4.3.1 we show how to calculate it through an
example.. If the measurements are performed at $X=1/R_{\infty}$, with
$R_{\infty}$ very far from the WH throat, we can find the value of
$\varepsilon$ such that the remainder of the Taylor expansion is included
between $\Theta_{\rm obs}^{-},\Theta_{\rm obs}^{+}$, i.e.,

$\displaystyle\left|\mathcal{O}\left[(X-\varepsilon)^{2}\right]\right|\equiv\left(\frac{\lambda}{2\varepsilon^{4}}e^{-\frac{\lambda}{\varepsilon}}|2\varepsilon-\lambda|\right)(X-\varepsilon)^{2},$
(30) $\displaystyle\hskip 42.67912pt\Theta_{\rm
obs}^{-}<\left|\mathcal{O}\left[(X-\varepsilon)^{2}\right]\right|<\Theta_{\rm
obs}^{+}.$

The final Taylor expansion of the $\Phi$ potential is

$\Phi=\gamma_{1}x+\gamma_{2}x^{2}+\mathcal{O}(x^{3}),$ (31)

where

$\displaystyle\frac{\gamma_{1}}{GM}$ $\displaystyle=-\left[1+\alpha
c_{1}(\varepsilon,\lambda)\right],\quad\frac{\gamma_{2}}{GM}=-\alpha
c_{2}(\varepsilon,\lambda).$ (32)

With these considerations in mind, let us discuss the asymptotic expansion of
the WH metric.

###  4.2 Asymptotic expansion of the wormhole metric

Let us Taylor-expand the $g_{tt}$ and $g_{rr}$ components of the WH metric (1)
around $x=0$, which contain the unknown functions $\chi(x)$ and $b(x)$, having
thus333We follow the same approach used to determine the asymptotic expansion
of the Schwarzschild metric. For example, the radial component
$g_{rr}(r)=1/(1-2M/r)$ can be correctly expanded at infinity by using the
variable $x=1/r$. We have the function $g_{rr}(x)=1/(1-2Mx)$, and we can
expand it in Taylor series around $x=0$, having
$g_{rr}(x)=1+2Mx+4M^{2}x^{2}+\mathcal{O}(x^{3})$.

$\displaystyle g_{tt}$ $\displaystyle\equiv$ $\displaystyle
e^{-2\chi(x)}=-1+\alpha_{1}x+\alpha_{2}x^{2}+\mathcal{O}(x^{3}),$ (33)
$\displaystyle g_{rr}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{1-b(x)x}=1+\beta_{1}x+\beta_{2}x^{2}+\mathcal{O}\left(x^{3}\right).$
(34)

where

$\displaystyle\alpha_{1}=-2\chi^{\prime}(0),\qquad\alpha_{2}=-2\chi^{\prime}(0)^{2}-\chi^{\prime\prime}(0),$
(35) $\displaystyle\beta_{1}=b(0),\qquad\quad\ \
\beta_{2}=b(0)^{2}+b^{\prime}(0).$ (36)

From these equations, we can easily write the unknown functions and their
derivatives evaluated in $x=0$ in terms of the coefficients of their
expansions, i.e.,

$\displaystyle\chi^{\prime}(0)=-\frac{\alpha_{1}}{2},\qquad\chi^{\prime\prime}(0)=\alpha_{1}-\alpha_{2},$
(37) $\displaystyle b(0)=\beta_{1},\qquad\quad\ \
b^{\prime}(0)=\beta_{2}-\beta_{1}^{2},$ (38)

with

$\displaystyle\chi^{\prime}(0)=\left.\frac{d\chi(x)}{dx}\right|_{x=0},\quad\chi^{\prime\prime}(0)=\left.\frac{d^{2}\chi(x)}{dx^{2}}\right|_{x=0},$
(39) $\displaystyle b(0)=\left.b(x)\right|_{x=0},\qquad\ \
b^{\prime}(0)=\left.\frac{db(x)}{dx}\right|_{x=0}.$

The redshift and shape functions are therefore

$\displaystyle\chi(x)$ $\displaystyle=$
$\displaystyle\chi^{\prime}(0)x+\frac{\chi^{\prime\prime}(0)}{2}x^{2},$ (40)
$\displaystyle b(x)$ $\displaystyle=$ $\displaystyle b(0)+b^{\prime}(0)x.$
(41)

Now, we can explicitly determine the parameters (37) – (38) to reconstruct the
WH metric (1) by matching them with those of the Newtonian potential and the
Yukawa potential (32) within a precise gravity framework and, eventually,
benchmark them with the observational data.

###  4.3 Reconstructing wormhole solutions

Figure 2: WH metric time $g_{tt}(x)$ and radial $g_{rr}(x)$ components, and
redshift $\chi(x)$ and shape $b(x)$ functions plotted in terms of the inverse
of the radial distance $x$ for $f(R)$ (continuous lines) and
$R+\alpha_{0}R^{2}+\alpha_{1}R\Box R$ (dashed lines) theories. The WH solution
has been reconstructed by considering the globular cluster NGC 4649, having an
effective radius of $R_{\infty}=10.00$ Kpc, $\varepsilon=0.111828\
\mbox{Kpc}^{-1}$ for the expansion of the Yuakawa potential, and the best fit
values for the modified potential (26) are $\lambda=1\ \mbox{Kpc}^{-1}$, and
$\alpha=0.01$ for $f(R)$ theories, and $d_{0}=0.56$, $d_{1}=-0.12$,
$L_{0}=3.77$, $L_{1}=1.95$ for $R+\alpha_{0}R^{2}+\alpha_{1}R\Box R$ theories.

We consider the modified theories of gravity reported in Table I of Ref.
Capozziello  _et al._ (2020). In Table 1 we summarise these theories of
gravity together with related modified Newtonian potentials, and coefficients
$\gamma_{1},\gamma_{2}$ of their asymptotic expansions given in Eqs. (32). By
considering Eqs. (35) – (36), we can reconstruct the WH expansion by matching
the coefficients of the two power series, i.e.,

$\alpha_{1}=\gamma_{1},\quad\alpha_{2}=\gamma_{2},\quad\beta_{1}=\gamma_{1},\quad\beta_{2}=\gamma_{2}.$
(42)

Substituting them in Eqs. (37) – (38), we reconstruct the WH redshift and
shape functions in Eqs. (40) and (41).

####  4.3.1 Application of the method to specific examples

As an example, let us consider the globular cluster NGC 4649, also known as
M60, because it is located relatively close to the Earth ($\sim 60$ millions
of light years), it is the third-brightest giant elliptical galaxy of the
Virgo cluster of galaxies, and it has several other advantageous proprieties,
which permit to be easily detected and studied (see Ref. De Bruyne  _et al._
(2001), for more details). It has an effective radius $R_{\infty}=10.00$ Kpc
De Bruyne  _et al._ (2001), and it hosts at its center a supermassive black
hole of mass $(4.5\pm 1.0)\times 10^{9}\ M_{\odot}$ (see Ref. Shen and
Gebhardt (2010), for details), which we clearly assume to be a WH, in lack of
data on such exotic compact objects.

Following the procedure in Capozziello  _et al._ (2020) to constrain the
modified Newtonian potential (26), we use their best fit values, i.e.,
$R_{\infty}\cdot\lambda=10$ or $\lambda=1\ \mbox{Kpc}^{-1}$, and $\alpha=0.01$
(see Fig. 7 in Ref. Capozziello  _et al._ (2020), for details). To determine
the value of $\varepsilon$, we know that the error in the measurement of the
effective radius (or also distance from the compact object) is $\delta
R_{\infty}=\pm 0.01$ Kpc. Therefore the detection sensibility is $\Theta_{\rm
obs}^{\pm}\equiv e^{-\lambda(R_{\infty}\pm\delta R_{\infty})}$, where
$\Theta_{\rm obs}^{-}=4.5\times 10^{-5}$ and $\Theta_{\rm obs}^{+}=4.6\times
10^{-5}$. Adopting the inequality (30) with $X=1/R_{\infty}$, we find that
$0.111781\ \mbox{Kpc}^{-1}\lesssim\varepsilon\lesssim 0.111876\
\mbox{Kpc}^{-1}$, and we choose $\varepsilon=0.111828\ \mbox{Kpc}^{-1}$. Now,
we can calculate the coefficients $\gamma_{1},\gamma_{2}$ by selecting two
extended theories of gravity from Table 1. We consider for simplicity $f(R)$
and $R+\alpha_{0}R^{2}+\alpha_{1}R\Box R$, choosing $\alpha_{0}=-3$ and
$\alpha_{1}=-9$444Looking at Table 1, we see that in order to have $d_{0,1}$
and $L_{0,1}$ real, it is easy to impose the following inequalities on the
theory parameters $\alpha_{0},\alpha_{1}$, namely $\alpha_{1}\leq 0$ and
$\alpha_{0}\leq-\sqrt{-2a_{1}/3}$. . In such hypothesis we obtain the
following results:

  * •

$f(R)$ theory: the expansion coefficients (32) for $\alpha=0.01$, $\lambda=1\
\mbox{Kpc}^{-1}$ are $\gamma_{1}=-1$, $\gamma_{2}=-10^{-4}$;

  * •

$R+\alpha_{0}R^{2}+\alpha_{1}R\Box R$ theory: the expansion coefficients (32)
for $d_{0}=0.56$, $d_{1}=-0.12$, $L_{0}=3.77$, $L_{1}=1.95$ are
$\gamma_{1}=-0.95$, $\gamma_{2}=-0.85$.

In Fig. 2, we provide the WH reconstruction by showing the behaviours of
$g_{tt}(x),g_{rr}(x),\chi(x),b(x)$, which should be considered valid only for
values very close to $x=0$, although they have been all plotted up to $r=2M$.
This expansion can be easily extended to a generic order $n$, giving more
precise results. In addition, such a strategy can be complemented with other
techniques in full and strong gravitational field regimes to accurately infer
more information on the WH solutions.

Table 1:  Summary of different theories of gravity and related modified
Newtonian potentials (see Table I in Ref. Capozziello  _et al._ (2020)), and
the first two coefficients of the asymptotic expansion of the $\Phi$
potential, see Eq. (32). The lightgray cells refer to GR.

---
Theory of gravity | Modified Newtonian potential |  $\bm{\gamma_{1}}$ |  $\bm{\gamma_{2}}$
R | $\Phi_{N}=-GMx$ | $-GM$ | $0$
$f(R)$ | $\Phi=\Phi_{N}\left(1+\alpha e^{-\frac{\lambda}{x}}\right)$ | $-GM[1+\alpha c_{1}(\varepsilon,\lambda)]$ | $-GM\alpha c_{2}(\varepsilon,\lambda)$
$R+\alpha_{0}R^{2}+\alpha_{1}R\Box R$ | $\Phi=\Phi_{N}\left(1+d_{0}e^{\frac{-1}{L_{0}x}}+d_{1}e^{\frac{-1}{L_{1}x}}\right)$ | $-GM[1+d_{0}c_{1}(\varepsilon,1/L_{0})$ | $-GM[d_{0}c_{2}(\varepsilon,1/L_{0})$
| $d_{0,1}=\frac{1}{6}\mp\frac{a_{0}}{2\sqrt{9a_{0}^{2}+6a_{1}}},$ | $+d_{1}c_{1}(\varepsilon,1/L_{1})]$ | $+d_{1}c_{2}(\varepsilon,1/L_{1})]$
| $L_{0,1}=\sqrt{-3a_{0}\pm\sqrt{9a_{0}^{2}+6a_{1}}}$ |  |
$R+\sum_{k=0}^{p}\alpha_{k}R\Box^{k}R$ | $\Phi=\Phi_{N}\left(1+\Sigma_{k=0}^{p}d_{k}e^{\frac{-1}{L_{k}x}}\right)$ | $-GM\left[1+\sum_{k=0}^{p}d_{k}c_{1}(\varepsilon,1/L_{k})\right]$ | $-GM\sum_{k=0}^{p}d_{k}c_{2}(\varepsilon,1/L_{k})$
|  $d_{k}$ and $L_{K}$ are functions of $a_{k}$, |  |
| see Ref. Capozziello  _et al._ (2020), for more details |  |
$f(R,R_{\alpha\beta}R^{\alpha\beta})$ | $\Phi=\Phi_{N}\left[1+\frac{1}{3}\,e^{\frac{-m_{R}}{x}}-\frac{4}{3}\,e^{\frac{-m_{Y}}{x}}\right]$ | $-GM\left[1+\frac{c_{1}(\varepsilon,m_{R})}{3}-\frac{4c_{1}(\varepsilon,m_{Y})}{3}\right]$ | $-GM\left[\frac{c_{2}(\varepsilon,m_{R})}{3}-\frac{4c_{2}(\varepsilon,m_{Y})}{3}\right]$
| $Y=R_{\mu\nu}R^{\mu\nu},\quad m_{Y}^{2}=\frac{1}{f_{Y}(0)},$ |  |
| $m_{R}^{2}=-\frac{1}{3f_{RR}(0)+2f_{Y}(0)}$ |  |
$f(R,R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu})$ | $\Phi=\Phi_{N}\left[1+\frac{1}{3}\,e^{\frac{-m_{1}}{x}}-\frac{4}{3}\,e^{\frac{-m_{2}}{x}}\right]$ | $-GM\left[1+\frac{c_{1}(\varepsilon,m_{1})}{3}-\frac{4c_{1}(\varepsilon,m_{2})}{3}\right]$ | $-GM\left[\frac{c_{2}(\varepsilon,m_{1})}{3}-\frac{4c_{2}(\varepsilon,m_{2})}{3}\right]$
| $Z=R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu},\quad m_{2}^{2}=\frac{1}{f_{Y}(0)+4f_{Z}(0)},$ |  |
| $m_{1}^{2}=-\frac{1}{3f_{RR}(0)+2f_{YY}(0)+2f_{Z}(0)}$ |  |
$f(R,\phi)+\omega(\phi)\phi_{;\alpha}\phi^{;\alpha}$ | $\Phi=\Phi_{N}\left[1+g(\xi,\eta)\,e^{\frac{-m_{R}\tilde{k}_{R}}{x}}\right.$ | $-GM\left[1+g(\varepsilon,\eta)c_{1}(\varepsilon,m_{R}\tilde{k}_{R})\right.$ | $-GM\left[g(\varepsilon,\eta)c_{2}(\varepsilon,m_{R}\tilde{k}_{R})\right.$
| $\left.+\left(\frac{1}{3}-g(\xi,\eta)\right)\,e^{\frac{-m_{R}\tilde{k}_{\phi}}{x}}\right]$ | $\left.+\left(\frac{1}{3}-g(\varepsilon,\eta)\right)c_{1}(\varepsilon,m_{R}\tilde{k}_{\phi})\right]$ | $\left.+\left(\frac{1}{3}-g(\varepsilon,\eta)\right)c_{2}(\varepsilon,m_{R}\tilde{k}_{\phi})\right]$
| $\eta=\frac{m_{\phi}}{m_{R}},\quad m_{R}^{2}=-\frac{1}{3f_{RR}(0,\phi_{0})}$ |  |
| $m_{\phi}^{2}=-\frac{f_{\phi\phi}(0,\phi_{0})}{2\omega(\phi_{0})},\quad\xi=\frac{3{f_{R\phi}(0,\phi_{0})}^{2}}{2\omega(\phi_{0})},$ |  |
| $g(\xi,\eta)=\frac{1-\eta^{2}+\xi+\sqrt{\eta^{4}+(\xi-1)^{2}-2\eta^{2}(\xi+1)}}{6\sqrt{\eta^{4}+(\xi-1)^{2}-2\eta^{2}(\xi+1)}}$ |  |
| $\tilde{k}_{R,\phi}^{2}=\frac{1-\xi+\eta^{2}\pm\sqrt{(1-\xi+\eta^{2})^{2}-4\eta^{2}}}{2}$ |  |
$f(R,R_{\alpha\beta}R^{\alpha\beta},\phi)+\omega(\phi)\phi_{;\alpha}\phi^{;\alpha}$ | $\Phi=\Phi_{N}\left[1+g(\xi,\eta)\,e^{\frac{-m_{R}\tilde{k}_{R}}{x}}\right.$ | $-GM\left[1+g(\varepsilon,\eta)c_{1}(\varepsilon,m_{R}\tilde{k}_{R})\right.$ | $-GM\left[g(\varepsilon,\eta)c_{2}(\varepsilon,m_{R}\tilde{k}_{R})\right.$
| $\left.+\left(\frac{1}{3}-g(\xi,\eta)\right)\,e^{\frac{-m_{R}\tilde{k}_{\phi}}{x}}-\frac{4}{3}\,e^{\frac{-m_{Y}}{x}}\right]$ | $+\left(\frac{1}{3}-g(\varepsilon,\eta)\right)c_{1}(\varepsilon,m_{R}\tilde{k}_{\phi})$ | $+\left(\frac{1}{3}-g(\varepsilon,\eta)\right)c_{2}(\varepsilon,m_{R}\tilde{k}_{\phi})$
| $\eta=\frac{m_{\phi}}{m_{R}},\quad m_{R}^{2}=-\frac{1}{3f_{RR}(0,0,\phi_{0})+2f_{Y}(0,0,\phi_{0})},$ | $\left.-\frac{4}{3}c_{1}(\varepsilon,m_{Y})\right]$ | $\left.-\frac{4}{3}c_{2}(\varepsilon,m_{Y})\right]$
| $m_{Y}^{2}=\frac{1}{f_{Y}(0,0,\phi_{0})},\quad m_{\phi}^{2}=-\frac{f_{\phi\phi}(0,0,\phi_{0})}{2\omega(\phi_{0})}$ |  |
| $\xi=\frac{3{f_{R\phi}(0,0,\phi_{0})}^{2}}{2\omega(\phi_{0})},$ |  |
| $g(\xi,\eta)=\frac{1-\eta^{2}+\xi+\sqrt{\eta^{4}+(\xi-1)^{2}-2\eta^{2}(\xi+1)}}{6\sqrt{\eta^{4}+(\xi-1)^{2}-2\eta^{2}(\xi+1)}}$ |  |
| $\tilde{k}_{R,\phi}^{2}=\frac{1-\xi+\eta^{2}\pm\sqrt{(1-\xi+\eta^{2})^{2}-4\eta^{2}}}{2}$ |  |

##  5 Discussion and Conclusions

We have developed a strategy to reconstruct WH solutions through Extended
Theories of Gravity (tuned by fitting the observational data) in the weak
gravitational field limit. The method is based on a model independent
framework employing a family of static and spherically symmetric metrics (1),
depending upon the two unknown $\chi(r)$ (redshft) and $b(r)$ (shape)
functions, see Sec. 2.

In the weak field limit, GR simply reduces to the Newtonian theory, but the
observations on the rotational curves and on mass-to-light rations of several
galaxies showed a clear departure from the classical description. Therefore,
to solve such an issue, modified theories of gravity have been proposed, whose
true nature can be reconstructed only by the fit of the data. Such theories
solve this observational puzzle by adding to the Newtonian potential a Yukawa-
like correction (26), whose parameters can be gauged by the data, see Sec. 3

Our method consists in first Taylor-expanding the modified Newtonian potential
(26) around $x\equiv 1/r=0$ (or in weak field limit), where the most delicate
part is related to the expansion of the Yukawa-like correction, which behaves
singularly at $x=0$. However, we have developed a new procedure, based on
Taylor expanding such a function around $x=\varepsilon\ll 1$, and determining
the value of this parameter through the current sensibility of the
observational instruments, see Sec. 4.2. Then, we Taylor expand also the WH
$g_{tt},g_{rr}$ metric components of Eq. (1) around $x=0$, see Sec. 4.2.
Finally, we match the coefficients of the two expressions to reconstruct the
WH solution from the modified theories of gravity, see Sec. 4.3. In Table 1 we
show different theories of gravity with the related modified Newtonian
potentials and coefficients $\gamma_{1},\gamma_{2}$ of their asymptotic
expansion. As an example of our strategy, we show in Fig. 2 the asymptotic
behaviour of the WH metric components, redshift, and shape functions for two
different modified theories of gravity, see Sec. 4.3.1 for more details.

The considerations developed in this paper are useful, in general, to build up
a stable and traversable WH solution within Extended Gravity, because the
above procedure permits to impose conditions on the WH behaviour in the weak
field limit ruled by the given theory of gravity. Depending on the WH model,
this last feature can have also important consequences for selecting the class
of (geometric or matter) fluids used to make the WH stable and traversable and
of course for the related energy conditions. Taking into account these
constraints, it would be possible to rule out solutions not in agreement with
the aforementioned requests and restrict thus the class of admissible and
physical WH solutions.

In summary, we are developing, in this paper and in others (see e.g. De Falco
_et al._ (2020)) a _model-independent approach_ , which can be advantageous
because it allows: (1) to search for WH observational existence; (2) to reduce
the set of WH solutions capable of being compared to observational data; (3)
to impose constraints on extended or modified theories of gravity; (4) to
adapt such strategies for investigating also black holes and other compact
objects. In a future paper, we aim at extending this strategy for rotating and
axially symmetric WH metrics.

## Acknowledgements

V.D.F. thanks Gruppo Nazionale di Fisica Matematica of Istituto Nazionale di
Alta Matematica (INDAM) for support. S.C. and M.D.L. acknowledge the support
of Istituto Nazionale di Fisica Nucleare (INFN) sez. di Napoli, Iniziative
Specifiche MOONLIGHT2, QGSKY, and TEONGRAV. The authors thank the anonymous
referees for the useful comments.

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