# The accelerated expansion in $F(G,T_{\mu\nu}T^{\mu\nu})$ gravity

Mihai Marciu<EMAIL_ADDRESS>Faculty of Physics, University of
Bucharest, Bucharest-Magurele, Romania  Dana Maria Ioan<EMAIL_ADDRESS>Faculty of Physics, University of Bucharest, Bucharest-Magurele, Romania

###### Abstract

In the present manuscript the basic Einstein–Hilbert cosmological model is
extended, by adding a new functional $F(G,T_{\mu\nu}T^{\mu\nu})$ in the
fundamental action, encoding specific geometrical effects due to a nontrivial
coupling with the Gauss-Bonnet invariant ($G$), and the energy–momentum
squared term ($T_{\mu\nu}T^{\mu\nu}$). After obtaining the corresponding
gravitational field equations for the specific decomposition where
$F(G,T_{\mu\nu}T^{\mu\nu})=f(G)+g(T_{\mu\nu}T^{\mu\nu})$, we have explored the
physical features of the cosmological model by considering the linear
stability theory, an important analytical tool in the cosmological theory
which can reveal the dynamical characteristics of the phase space. The
analytical exploration of the corresponding phase space structure revealed
that the present model can represent a viable dark energy model, with various
stationary points where the effective equation of state corresponds to a
de–Sitter epoch, possible explaining the early and late time acceleration of
the Universe.

modified gravity and dark energy

##  I Introduction

In the present cosmological context the accelerated expansion [1] represents
an enigmatic phenomenon associated with the evolution of our Universe at the
level of background dynamics. This phenomenon has been discovered almost two
decades ago [2, 3], triggering various developments in science and technology.
The simplest dark energy model explaining the accelerated expansion of our
Universe is associated to the $\Lambda$CDM model [4, 5, 6, 7], a specific
cosmological theory which is based on a cosmological constant $\Lambda$ added
to the Einstein’s field equations. The $\Lambda$CDM model suffers from various
theoretical limitations [8, 9, 10] and cannot explain the dynamical evolution
of the dark energy equation of state, as probed through various astrophysical
observations [11, 12, 13, 14, 15]. In principle the $\Lambda$CDM model [10]
can be regarded as an effective approximate approach which is associated with
a constant equation of state, without addressing the $H_{0}$ tension [16, 17,
18] in a fundamental manner.

In the cosmological theories the modified gravity approaches [19, 20, 21, 22]
represent a novel paradigm which further extends the fundamental action,
embedding various invariant components, aiming for a more complete and
consistent theory of gravitation. The most natural extension of gravity is
represented by the $f(R)$ theory, a specific approach based on a functional
which depends on scalar curvature [23]. Since then, many alternative theories
have been proposed [4, 24, 25, 26], aiming for a more consistent theory [27,
28, 29] which can explain the accelerated expansion of our Universe, embedding
various dynamical effects associated to the dark sector. In these theories the
interplay between matter and geometry has been questioned in different
approaches [30, 31, 32, 33].

In the modified gravity theories a particular extension is related to the
energy–momentum squared gravity [34, 35, 36], a novel theory which can explain
various physical effects at cosmological scales [37, 38, 39]. The latter
theory is constructed by considering the interplay between matter and
geometry, taking into account an invariant which is based on a specific
self–contraction of the energy–momentum tensor [34]. The energy–momentum
squared gravity has attracted some attention in modern cosmological theories
[40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52], representing a viable
approach also from the astrophysical point of view [53, 54]. Specific
wormholes solutions have been considered in the energy–momentum squared
gravity [55], analyzing the physical implications. The inclusion of the
Gauss–Bonnet topological invariant in the energy–momentum squared gravity has
been considered recently [56, 57, 44, 58] for specific relativistic systems.

An important approach in modern cosmological theories is related to the
Gauss–Bonnet invariant, a special topological component in the
four–dimensional space–time [59, 60]. The inclusion of the Gauss–Bonnet
invariant has been considered in various modern theories of gravitation [61,
62, 63, 64, 64, 64, 65, 66, 67], representing a viable approach for specific
physical systems, possible explaining the dark energy phenomenon [68].

In this paper we shall consider a modified gravity model build in the
fundamental framework of the Einstein–Hilbert action, embedding the
geometrical interplay between the Gauss–Bonnet invariant and the
energy–momentum–squared component [56, 57, 44, 58]. The fundamental action in
our model contains a generic functional which depends on the Gauss–Bonnet
invariant [64] and the energy–momentum–squared term in a decomposed manner.
The physical characteristics are evaluated by considering the linear stability
theory for an exponential decomposition, analyzing the phase space structure
and the possibility of reaching the accelerated expansion. Such an approach
further extends the Einstein–Hilbert action by taking into account the effects
due to the geometrical characteristics of space–time, including also the
interplay with the matter sector, embedding the elementary properties of the
latter component.

The plan of our paper is the following. In Sec. II we propose the fundamental
action for our toy model, obtaining the corresponding modified Friedmann
equations. Then, in Sec. III we discuss the physical properties and the
emergence of the accelerated expansion in the current cosmological model by
considering the linear stability theory in the case of an exponential
behavior. Lastly, in Sec. IV we summarize the principal obtained results and
give the main concluding remarks.

##  II The action and the field equations

In what follows we shall propose a cosmological model described by the
following action [56, 57, 44]:

$S=\int d^{4}x\sqrt{-\tilde{g}}\Bigg{[}\frac{R}{2}+F(G,T^{2})\Bigg{]}+\int
d^{4}x\sqrt{-\tilde{g}}L_{m},$ (1)

where the generic function embedded into the Einstein–Hilbert action can be
decomposed in two specific terms, $F(G,T^{2})=f(G)+g(T^{2})$. In this case we
have assumed a non–linear dependence of the action by the Gauss–Bonnet
invariant $(G)$, and the energy–momentum squared invariant
$(T^{2}=T_{\mu\nu}T^{\mu\nu})$ [56, 57]. Before proceeding to the computations
of the modified Friedmann relations, we have to specify that the background
dynamics can be described by the Robertson–Walker metric:

$ds^{2}=-dt^{2}+a^{2}(t)(dx^{2}+dy^{2}+dz^{2}),$ (2)

where $a(t)$ is the cosmic scale factor which characterizes the expansion of
the Universe at the large scale structure. In general, the Gauss–Bonnet
invariant is defined in the following way,

$G=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\xi\sigma}R^{\mu\nu\xi\sigma}.$ (3)

For the above metric (2) the Gauss–Bonnet invariant reduces to the following
expression [64],

$G=24H^{2}(H^{2}+\dot{H}),$ (4)

where $H(t)$ represents the Hubble parameter defined as:
$H=\frac{\dot{a}}{a}$, where the dot represents the derivative with respect to
the cosmic time. In this case the scalar curvature acquires the following
expression,

$R=6(2H^{2}+\dot{H}).$ (5)

The second invariant in our action (1) is represented by the
energy–momentum–squared term [37], defined as:

$T^{2}=T_{\mu\nu}T^{\mu\nu},$ (6)

where $T_{\mu\nu}$ describes the energy–momentum of the matter sector,
$T_{\mu\nu}=diag[\rho,p,p,p]$, with $\rho$ the density and $p$ the pressure of
the matter component which behaves closely as a non–relativistic fluid with
zero pressure. If we further assume a barotropic equation of state for the
matter sector,

$p=w\rho,$ (7)

with $w$ the barotropic constant parameter, then the energy–momentum–squared
invariant takes the following expression:

$T^{2}=\rho^{2}(1+3w^{2}).$ (8)

The variation of the action (1) with respect to the inverse metric
$g^{\mu\nu}$ lead to the following modified Friedmann relations [44, 37]:

$3H^{2}=\rho+\Bigg{[}Gf^{\prime}(G)-f(G)-24H^{3}\dot{f^{\prime}(G)}\Bigg{]}+2\frac{\partial
g(T^{2})}{\partial T^{2}}(\rho^{2}+4\rho p+3p^{2})-g(T^{2}),$ (9)

$-3H^{2}-2\dot{H}=p+\Bigg{[}f(G)-Gf^{\prime}(G)+16(\dot{H}+H^{2})\dot{f^{\prime}(G)}\\\
+8H^{2}\ddot{f^{\prime}(G)}\Bigg{]}+g(T^{2}),$ (10)

where we have assumed the following definitions:

${}^{\prime}=\frac{d}{dG},$ (11) $\dot{}=\frac{d}{dt},$ (12)
$\ddot{}=\frac{d^{2}}{dt^{2}}.$ (13)

We can further define the energy density associated to the geometrical dark
energy component [37],

$\rho_{de}=Gf^{\prime}(G)-f(G)-24H^{3}\dot{f^{\prime}(G)}+2\frac{\partial
g(T^{2})}{\partial T^{2}}(\rho^{2}+4\rho p+3p^{2})-g(T^{2}),$ (14)

and the corresponding pressure:

$p_{de}=f(G)-Gf^{\prime}(G)+16(\dot{H}+H^{2})\dot{f^{\prime}(G)}\\\
+8H^{2}\ddot{f^{\prime}(G)}+g(T^{2}).$ (15)

Then, we can define the dark energy equation of state due to the geometrical
coupling of the invariant constituents,

$w_{de}=\frac{p_{de}}{\rho_{de}},$ (16)

and the total/effective equation of state for the background dynamics:

$w_{tot}=-1-\frac{2}{3}\frac{\dot{H}}{H^{2}}.$ (17)

Lastly, we define the matter density parameter in the usual manner,

$\Omega_{m}=\frac{\rho}{3H^{2}}$ (18)

and the geometrical dark energy density parameter,

$\Omega_{de}=\frac{\rho_{de}}{3H^{2}},$ (19)

satisfying the constraint equation: $\Omega_{m}+\Omega_{de}=1$.

##  III Dynamical properties for an exponential model

In order to study the dynamical properties for the exponential model where
$F(G,T_{\mu\nu}T^{\mu\nu})=f_{0}e^{\alpha G}+g_{0}e^{\beta T^{2}}$, with
$\alpha,\beta,f_{0},g_{0}$ constant parameters, we need to introduce the
following auxiliary variables:

$s=\Omega_{m}=\frac{\rho}{3H^{2}},$ (20)

$x=\frac{G}{3H^{2}}\frac{df(G)}{dG},$ (21)

$y=8H\dot{f^{\prime}(G)},$ (22)

$z=2\frac{dg(T^{2})}{d(T^{2})}\rho(1+4w+3w^{2}),$ (23)

$u=\frac{f(G)}{3H^{2}},$ (24)

$v=\frac{g(T^{2})}{3H^{2}}.$ (25)

Figure 1: The variation of the matter density parameter $s$ for the $A$
cosmological solution in a specific region of interest ($v=1,\beta=-1,x=1$).

In terms of these auxiliary variables, we can write the Friedmann constraint
equation (9) in the following way:

$s=\frac{1+u+v-x+y}{1+z},$ (26)

expressing the matter density parameter in terms of the remaining auxiliary
variables.

Next, we introduce the e–fold number $N=log(a)$ and write the corresponding
autonomous system of differential equations:

$\frac{dx}{dN}=\frac{1}{4\alpha uz(z+1)}\Big{[}9\beta u^{2}vw^{2}y+12\beta
u^{2}vwy+3\beta u^{2}vy+9\beta uv^{2}w^{2}y+12\beta uv^{2}wy+3\beta
uv^{2}y-18\beta uvw^{2}x^{2}+9\beta uvw^{2}y^{2}\\\ +9\beta uvw^{2}y-24\beta
uvwx^{2}+12\beta uvwy^{2}+12\beta uvwy-6\beta uvx^{2}+3\beta uvy^{2}+3\beta
uvy+8\alpha uxz^{2}+8\alpha uxz\\\ -18\beta v^{2}w^{2}x^{2}+9\beta
v^{2}w^{2}xy-24\beta v^{2}wx^{2}+12\beta v^{2}wxy-6\beta v^{2}x^{2}+3\beta
v^{2}xy+18\beta vw^{2}x^{3}-18\beta vw^{2}x^{2}-27\beta vw^{2}x^{2}y+9\beta
vw^{2}xy^{2}\\\ +9\beta vw^{2}xy+24\beta vwx^{3}-24\beta vwx^{2}-36\beta
vwx^{2}y+12\beta vwxy^{2}+12\beta vwxy+6\beta vx^{3}-6\beta vx^{2}-9\beta
vx^{2}y+3\beta vxy^{2}+3\beta vxy\Big{]},$ (27)

$\frac{dy}{dN}=\frac{1}{4\alpha uz(z+1)}\Big{[}-12\alpha u^{2}wz-12\alpha
u^{2}z^{2}-12\alpha u^{2}z-18\beta uvw^{2}x-9\beta uvw^{2}xy-24\beta
uvwx-12\beta uvwxy\\\ -12\alpha uvwz-6\beta uvx-3\beta uvxy-12\alpha
uvz^{2}-12\alpha uvz+12\alpha uwxz-12\alpha uwyz-12\alpha uwz+12\alpha
uxz^{2}+12\alpha uxz\\\ -4\alpha uyz^{2}-4\alpha uyz-4\alpha uz^{2}-4\alpha
uz-18\beta v^{2}w^{2}x-9\beta v^{2}w^{2}xy-24\beta v^{2}wx-12\beta
v^{2}wxy-6\beta v^{2}x-3\beta v^{2}xy\\\ +18\beta vw^{2}x^{2}+9\beta
vw^{2}x^{2}y-18\beta vw^{2}x-9\beta vw^{2}xy^{2}-27\beta vw^{2}xy+24\beta
vwx^{2}+12\beta vwx^{2}y-24\beta vwx-12\beta vwxy^{2}-36\beta vwxy\\\ +6\beta
vx^{2}+3\beta vx^{2}y-6\beta vx-3\beta vxy^{2}-9\beta vxy\Big{]},$ (28)

$\frac{dz}{dN}=\Big{[}-9uw^{3}z^{2}-9uw^{2}z^{3}-9uw^{2}z^{2}-3uwz^{2}-3uz^{3}-3uz^{2}-18vw^{3}z^{2}-9vw^{3}z-18vw^{2}z^{3}-39vw^{2}z^{2}\\\
-21vw^{2}z-12vwz^{3}-30vwz^{2}-15vwz-6vz^{3}-9vz^{2}-3vz+9w^{3}xz^{2}-9w^{3}yz^{2}-9w^{3}z^{2}\\\
+9w^{2}xz^{3}+9w^{2}xz^{2}-9w^{2}yz^{3}-9w^{2}yz^{2}-9w^{2}z^{3}-9w^{2}z^{2}+3wxz^{2}-3wyz^{2}-3wz^{2}+3xz^{3}\\\
+3xz^{2}-3yz^{3}-3yz^{2}-3z^{3}-3z^{2}\Big{]}\cdot\\\
\Big{[}(3w^{2}+1)z^{2}(u-x+y+1)+v(w^{2}(6z^{2}+6z+3)+4w(2z^{2}+3z+1)+2z^{2}+2z+1)\Big{]}^{-1},$
(29)

$\frac{du}{dN}=\frac{1}{4\alpha z(z+1)}\Big{[}-18\beta uvw^{2}x+9\beta
uvw^{2}y-24\beta uvwx+12\beta uvwy-6\beta uvx+3\beta uvy+8\alpha
uz^{2}+8\alpha uz\\\ -18\beta v^{2}w^{2}x+9\beta v^{2}w^{2}y-24\beta
v^{2}wx+12\beta v^{2}wy-6\beta v^{2}x+3\beta v^{2}y+18\beta
vw^{2}x^{2}-18\beta vw^{2}x-27\beta vw^{2}xy+9\beta vw^{2}y^{2}\\\ +9\beta
vw^{2}y+24\beta vwx^{2}-24\beta vwx-36\beta vwxy+12\beta vwy^{2}+12\beta
vwy+6\beta vx^{2}-6\beta vx-9\beta vxy+3\beta vy^{2}+3\beta vy\Big{]},$ (30)
$\frac{dv}{dN}=\Big{[}-v(u^{2}(3w^{2}+1)z^{2}(3\beta
v(w+1)(3w+1)x+2\alpha(z+1)(3w+z+3))+u(3\beta v^{2}(w+1)(3w+1)x((w(9w+8)\\\
+3)z^{2}+6w(w+2)z+w(3w+4)+2z+1)-2vz(3\beta(w+1)(3w+1)(3w^{2}+1)xz(x-y-1)\\\
+\alpha(z+1)(-9w^{3}z+3w^{2}(z^{2}+z+2)+w(z(16z+21)+8)+z^{2}+z+2))-2\alpha(3w^{2}+1)z^{2}(z+1)(3w+z+3)(x-y-1))\\\
+3\beta
v(w+1)(3w+1)x(v-x+y+1)(v(w^{2}(6z(z+1)+3)+4w(z+1)(2z+1)+2z(z+1)+1)-(3w^{2}+1)z^{2}(x-y-1)))\Big{]}\cdot\\\
\Big{[}2\alpha
uz(z+1)((3w^{2}+1)z^{2}(u-x+y+1)+v(w^{2}(6z(z+1)+3)+4w(z+1)(2z+1)+2z(z+1)+1))\Big{]}^{-1}.$
(31)

In this case, where we have an exponential decomposition, we have obtained
four critical points which are associated to a de–Sitter behavior
($w_{tot}=-1$). For these solutions the cosmological model acts as a
geometrical dark energy component, driving the accelerated expansion of the
Universe as a cosmological constant. In what follows, we shall describe each
cosmological solution in detail, analyzing the dynamical consequences.

The first cosmological solution found for the exponential case is located at
the following coordinates:

$A:\left(y=0,z=-w-1,u=-\frac{3\beta v(3w+1)x(v-x+1)}{9\beta vwx+3\beta
vx-4\alpha w}\right),$ (32)

describing a de–Sitter epoch where the matter density parameter is equal to:

$s=-\frac{4\alpha(v-x+1)}{4\alpha w-3\beta v(3w+1)x}.$ (33)

For this solution we note that the $x,z$ and $v$ variables are independent.
The $y$ component which is related to the time variation of the Gauss–Bonnet
geometrical invariant is set to zero. We can note that the location in the
phase space structure is influenced by the barotropic equation of state of the
matter sector, and $\alpha,\beta$ parameters which are describing the strength
of the coupling functions. In Fig. 1 we have presented a possible region of
interest for the matter density parameter where $s\in(0,1)$. In the general
case where all the parameters are not set, the final expressions for the
specific eigenvalues are too complex to be written in the manuscript. However,
by setting some of the parameters ($\alpha=1,v=1,x=1,\beta=-1$), we have
obtained some simple expressions of the resulting eigenvalues:

$\Big{[}0,0,-\frac{3(w+1)(w(51w+22)+7)}{w(w(51w+86)+19)+4},\\\
\frac{1}{2}\left(\frac{w(w+1)(3w+1)(397w+99)(w(w(51w+86)+19)+4)}{\sqrt{w^{2}(w+1)^{2}(3w+1)^{2}(13w+3)(397w+99)(w(w(51w+86)+19)+4)^{2}}}-3\right),\\\
\frac{1}{2}\left(-\frac{w(w+1)(3w+1)(397w+99)(w(w(51w+86)+19)+4)}{\sqrt{w^{2}(w+1)^{2}(3w+1)^{2}(13w+3)(397w+99)(w(w(51w+86)+19)+4)^{2}}}-3\right)\Big{]}.$
(34)

As can be observed, the solution is non–hyperbolic, due to the existence of
two zero eigenvalues. Hence, we can use the linear stability theory only to
study the specific cases where the dynamics corresponds to a saddle behavior.
A specific region where we have obtained a saddle behavior is presented in
Fig. 3. The evolution in the phase space structure towards the A cosmological
solution can be seen in Fig. 2. In this case the corresponding eigenvalues
have the following values:

$\big{[}0.,0.,-1.00194,-1.5+3.36584i,-1.5-3.36584i\big{]}.$ (35)

The second cosmological solution is found at the coordinates:

$B^{\pm}:\Big{(}x=\frac{3\beta\pm\sqrt{3}\sqrt{\beta(3w+1)(3\beta+16\alpha
w+9\beta w)}+9\beta w}{6\beta+18\beta w},y=0,z=-w-1,\\\
v=\frac{\pm\sqrt{3}\sqrt{\beta(3w+1)(3\beta+16\alpha w+9\beta
w)}-3\beta(3w+1)}{6(\beta+3\beta w)}\Big{)},$ (36)

with the corresponding matter density parameter equal to: $s=-\frac{u}{w}$.
Due to the specific form of the matter density parameter, we can observe that
the case of a pressure–less dark matter fluid cannot be considered, leading to
a divergence. Hence, we can only approximate the case of a pressure–less dark
matter fluid, $w\to 0$. The eigenvalues for the $B^{+}$ solutions are the
following:

$\Big{[}0,0,-\frac{3(w+1)(3w+1)\left(6\beta
u+9\beta(2u-1)w^{2}+\sqrt{3}w\sqrt{\beta(3w+1)(3\beta+16\alpha w+9\beta
w)}-3\beta w\right)}{6\beta
u(w+1)(3w+1)\left(3w^{2}+1\right)+(3w+5)w^{2}\left(\sqrt{3}\sqrt{\beta(3w+1)(3\beta+16\alpha
w+9\beta w)}-3\beta(3w+1)\right)},E_{4},E_{5}\Big{]},$ (37)

where $E_{4},E_{5}$ have complicated expressions and are not displayed in the
manuscript. In Fig. 4 we have displayed a saddle region for the $B^{+}$
solution, taking into account also the existence conditions which imply
$s\in(0,1)$.

Figure 2: The evolution towards the A cosmological solution in the phase space
structure ($w=0.0001,\alpha=-2,\beta=-1$). Figure 3: A specific region of
interest for the $A$ cosmological solution where the dynamics corresponds to a
saddle dynamical behavior. ($v=1,\beta=-1,x=1$). Figure 4: A specific region
of interest for the $B^{+}$ cosmological solution where the dynamics
corresponds to a saddle dynamical behavior. ($u=-0.1,\beta=1$).

The last de–Sitter cosmological solution found in the present analysis is
located in the phase space structure at the coordinates:

$C:\Big{(}x=\frac{4\left(\alpha+\alpha v+6\alpha vw^{2}+\alpha vw+3\alpha
w^{2}\right)}{4\alpha+3\beta v^{2}+27\beta v^{2}w^{2}+18\beta v^{2}w+12\alpha
w^{2}},y=0,z=-w-1,u=\\\ -\frac{3\beta
v^{2}(3w+1)^{2}\left(6vw^{2}+vw+v+3w^{2}+1\right)}{\left(3w^{2}+1\right)\left(4\alpha+3\beta
v^{2}+27\beta v^{2}w^{2}+18\beta v^{2}w+12\alpha w^{2}\right)}\Big{)},$ (38)

with the matter density parameter influenced by the dark matter equation of
state and specific coupling of the energy–momentum–squared function,

$s=\frac{v(3w+1)}{3w^{2}+1}.$ (39)

The variation of the corresponding matter density parameter for the $C$
cosmological solution is presented in Fig. 5. It can be seen that in the case
of a pressure–less dark matter component the matter density parameter
satisfies the observational constraints, being influenced also by the $v$
variable which encodes geometrical effects due to the specific form of the
energy–momentum–squared function.

For this critical line we have obtained the following eigenvalues
($v=1,\beta=1$):

$\Bigg{[}0,0,0,\frac{C_{1}\pm\frac{\sqrt{2}\sqrt{w^{2}(3w+1)^{4}\left(w^{2}-1\right)^{2}\left(3w^{2}+1\right)\left(9w^{2}+w+2\right)\left(50\alpha^{2}\left(3w^{2}+1\right)\left(9w^{2}+w+2\right)+12\alpha\left(3w^{2}+1\right)(3w+1)^{2}+9(3w+1)^{4}\right)(4\alpha+3w((4\alpha+9)w+6)+3)^{2}}}{\alpha(w-1)w(w+1)(3w+1)^{2}\left(3w^{2}+1\right)\left(9w^{2}+w+2\right)}}{4(4\alpha+3w((4\alpha+9)w+6)+3)}\Bigg{]},$
(40)

where we have defined:

$C_{1}=6(-4\alpha-3w((4\alpha+9)w+6)-3).$ (41)

In Fig. 6 we have presented a region where the dynamical corresponds to a
saddle behavior, possible explaining the late time acceleration of the
Universe in the background dynamics. Note that the last critical line is also
non–hyperbolic, having three zero eigenvalues. The transition from the
critical point $C$ towards the $A$ cosmological solution in the xOy plane can
be observed in Fig. 7 for specific initial conditions near the $C$ solution,
validating the obtained analytical solutions. Lastly, in Fig. 8 we have
represented the variation of the total (effective) equation of state for the
present cosmological scenario. We note that the evolution can pass from a
matter domination epoch towards a super–accelerated era, attaining the
cosmological constant boundary from below at late times, crossing the phantom
divide line in the early stages.

Figure 5: The variation of the matter density parameter $s$ for the $C$
cosmological solution. Figure 6: A specific region of interest for the $C$
cosmological solution where the dynamics corresponds to a saddle dynamical
behavior. ($v=1,\beta=1$). Figure 7: The evolution from the critical point C
towards the A cosmological solution in the xOy plane where
$w=0.0001,\alpha=-2,\beta=-1$. Figure 8: The variation of the total equation
of state in the case where $w=0.0001,\alpha=-2,\beta=-1$.

##  IV Summary and Conclusions

In this paper we have proposed a model in the theoretical framework of
modified gravity, where the fundamental Einstein–Hilbert action is extended,
by considering a more complete theory. The latter theory denoted as
$F(G,T_{\mu\nu}T^{\mu\nu})$ is based on two specific components. The first
component takes into account possible physical effects due to the
consideration of the Gauss–Bonnet invariant $(G)$, encoding geometrical
aspects in the generic theory. The second component in our action is based on
the energy–momentum–squared invariant $(T_{\mu\nu}T^{\mu\nu})$, embedding
geometrical effects from the specific form of the energy–momentum tensor. In
this cosmological model we have assumed that the generic function which
depends on the Gauss–Bonnet invariant and the energy–momentum–squared term can
be decomposed in an independent manner,
$F(G,T_{\mu\nu}T^{\mu\nu})=f(G)+g(T_{\mu\nu}T^{\mu\nu})$. After we have
proposed the generic action for our present model, we have obtained the
modified Friedmann relations by varying the action with respect to the inverse
metric, assuming that the background can be described by the Robertson–Walker
metric. Here we note that the continuity equation is not satisfied due to the
inclusion of the energy–momentum–squared term in the specific action, a
particular aspect for these theories. After obtaining the dynamical equations,
we have studied the physical aspects of our cosmological model by considering
the linear stability theory. In this study we have assumed an exponential
representation for the generic function in our action,
$F(G,T_{\mu\nu}T^{\mu\nu})=f_{0}e^{\alpha G}+g_{0}e^{\beta T^{2}}$, where
$\alpha,\beta,f_{0},g_{0}$ are constant parameters. In this particular case we
have introduced the auxiliary variables which are required in order to apply
the linear stability theory. After introducing the auxiliary variables
associated to the phase space structure, we have computed the critical points
of the present cosmological model for the exponential case.

In the phase space structure we have identified various critical points which
correspond to a de–Sitter epoch where the model behaves closely as a
cosmological constant, particular solutions which can explain the late time
stage of the Universe. As can be seen from the analysis, for these solutions
the effective matter density parameter is influenced by different coupling
terms, and the barotropic equation of state for the matter sector. From a
dynamical point of view we have identified possible regions of interest for
the coupling constants which correspond to a saddle dynamical behavior in the
late time stage of the Universe. These critical points are particular
solutions which correspond in principle to various epochs in the dynamical
trajectory of our Universe. In this case the phase space structure has four
critical points, associated to a de–Sitter epoch. For each of the
corresponding critical points we have established the dynamical behavior,
obtaining possible regions of interest for different parameters which can
describe the present model.

The analysis of the phase space structure showed that the present cosmological
model can describe the late–time accelerated expansion of the Universe and the
dynamical behavior of the effective equation of state. However, due to the
de–Sitter solutions found in the phase space structure, we have to further
assume that the matter and the radiation epochs appear by fine–tuning the
initial conditions of the current trajectory. The present paper can be
extended in various cosmological applications. For example, it would be
interesting to study the cosmological model by considering an observational
study with the recent cosmological data, obtaining constraints for various
parameters from an astrophysical point of view. Another possible aspect is
represented by the inflationary era, a particular stage in the evolution of
the Universe which can be further analyzed. These particular extensions can
provide support for a more complete theory of gravity and are left as future
projects.

###### Acknowledgements.

We would like to thank Prof. Dr. Virgil Baran for various discussions which
lead to the development of the present project. The computational part of this
work was performed using the computer stations provided by CNFIS through the
project CNFIS-FDI-2020-035.

## References

  * Frieman  _et al._ [2008] J. Frieman, M. Turner, and D. Huterer, Dark Energy and the Accelerating Universe, Ann. Rev. Astron. Astrophys. 46, 385 (2008), arXiv:0803.0982 [astro-ph] .
  * Li  _et al._ [2013] M. Li, X.-D. Li, S. Wang, and Y. Wang, Dark Energy: A Brief Review, Front. Phys. (Beijing) 8, 828 (2013), arXiv:1209.0922 [astro-ph.CO] .
  * Joyce  _et al._ [2016] A. Joyce, L. Lombriser, and F. Schmidt, Dark Energy Versus Modified Gravity, Ann. Rev. Nucl. Part. Sci. 66, 95 (2016), arXiv:1601.06133 [astro-ph.CO] .
  * Bamba  _et al._ [2012] K. Bamba, S. Capozziello, S. Nojiri, and S. D. Odintsov, Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests, Astrophys. Space Sci. 342, 155 (2012), arXiv:1205.3421 [gr-qc] .
  * Copeland  _et al._ [2006] E. J. Copeland, M. Sami, and S. Tsujikawa, Dynamics of dark energy, Int. J. Mod. Phys. D 15, 1753 (2006), arXiv:hep-th/0603057 .
  * Peebles and Ratra [2003] P. J. E. Peebles and B. Ratra, The Cosmological Constant and Dark Energy, Rev. Mod. Phys. 75, 559 (2003), arXiv:astro-ph/0207347 .
  * Padmanabhan [2003] T. Padmanabhan, Cosmological constant: The Weight of the vacuum, Phys. Rept. 380, 235 (2003), arXiv:hep-th/0212290 .
  * Moore [1994] B. Moore, Evidence against dissipationless dark matter from observations of galaxy haloes, Nature 370, 629 (1994).
  * Boylan-Kolchin  _et al._ [2011] M. Boylan-Kolchin, J. S. Bullock, and M. Kaplinghat, Too big to fail? The puzzling darkness of massive Milky Way subhaloes, Monthly Notices of the Royal Astronomical Society: Letters 415, L40 (2011), https://academic.oup.com/mnrasl/article-pdf/415/1/L40/2881682/415-1-L40.pdf .
  * Perivolaropoulos and Skara [2022] L. Perivolaropoulos and F. Skara, Challenges for $\Lambda$CDM: An update, New Astron. Rev. 95, 101659 (2022), arXiv:2105.05208 [astro-ph.CO] .
  * Aghanim  _et al._ [2020] N. Aghanim _et al._ (Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Spergel  _et al._ [2003] D. N. Spergel _et al._ (WMAP), First year Wilkinson Microwave Anisotropy Probe (WMAP) observations: Determination of cosmological parameters, Astrophys. J. Suppl. 148, 175 (2003), arXiv:astro-ph/0302209 .
  * Alam  _et al._ [2017] S. Alam _et al._ (BOSS), The clustering of galaxies in the completed SDSS-III Baryon Oscillation Spectroscopic Survey: cosmological analysis of the DR12 galaxy sample, Mon. Not. Roy. Astron. Soc. 470, 2617 (2017), arXiv:1607.03155 [astro-ph.CO] .
  * Tonry  _et al._ [2003] J. L. Tonry _et al._ (Supernova Search Team), Cosmological results from high-z supernovae, Astrophys. J. 594, 1 (2003), arXiv:astro-ph/0305008 .
  * Suzuki  _et al._ [2012] N. Suzuki _et al._ (Supernova Cosmology Project), The Hubble Space Telescope Cluster Supernova Survey: V. Improving the Dark Energy Constraints Above z$>$1 and Building an Early-Type-Hosted Supernova Sample, Astrophys. J. 746, 85 (2012), arXiv:1105.3470 [astro-ph.CO] .
  * Escamilla-Rivera and Torres Castillejos [2023] C. Escamilla-Rivera and R. Torres Castillejos, H0 Tension on the Light of Supermassive Black Hole Shadows Data, Universe 9, 14 (2023), arXiv:2301.00490 [astro-ph.CO] .
  * Poulin  _et al._ [2019] V. Poulin, T. L. Smith, T. Karwal, and M. Kamionkowski, Early Dark Energy Can Resolve The Hubble Tension, Phys. Rev. Lett. 122, 221301 (2019), arXiv:1811.04083 [astro-ph.CO] .
  * Efstathiou [2014] G. Efstathiou, H0 Revisited, Mon. Not. Roy. Astron. Soc. 440, 1138 (2014), arXiv:1311.3461 [astro-ph.CO] .
  * Nojiri  _et al._ [2017] S. Nojiri, S. D. Odintsov, and V. K. Oikonomou, Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution, Phys. Rept. 692, 1 (2017), arXiv:1705.11098 [gr-qc] .
  * Nojiri and Odintsov [2006] S. Nojiri and S. D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy, eConf C0602061, 06 (2006), arXiv:hep-th/0601213 .
  * Capozziello and De Laurentis [2011] S. Capozziello and M. De Laurentis, Extended Theories of Gravity, Phys. Rept. 509, 167 (2011), arXiv:1108.6266 [gr-qc] .
  * Tsujikawa [2010] S. Tsujikawa, Modified gravity models of dark energy, Lect. Notes Phys. 800, 99 (2010), arXiv:1101.0191 [gr-qc] .
  * De Felice and Tsujikawa [2010] A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13, 3 (2010), arXiv:1002.4928 [gr-qc] .
  * Cai  _et al._ [2016] Y.-F. Cai, S. Capozziello, M. De Laurentis, and E. N. Saridakis, f(T) teleparallel gravity and cosmology, Rept. Prog. Phys. 79, 106901 (2016), arXiv:1511.07586 [gr-qc] .
  * Bahamonde  _et al._ [2015] S. Bahamonde, C. G. Böhmer, and M. Wright, Modified teleparallel theories of gravity, Phys. Rev. D 92, 104042 (2015), arXiv:1508.05120 [gr-qc] .
  * Bahamonde  _et al._ [2023] S. Bahamonde, K. F. Dialektopoulos, C. Escamilla-Rivera, G. Farrugia, V. Gakis, M. Hendry, M. Hohmann, J. Levi Said, J. Mifsud, and E. Di Valentino, Teleparallel gravity: from theory to cosmology, Rept. Prog. Phys. 86, 026901 (2023), arXiv:2106.13793 [gr-qc] .
  * Koyama [2016] K. Koyama, Cosmological Tests of Modified Gravity, Rept. Prog. Phys. 79, 046902 (2016), arXiv:1504.04623 [astro-ph.CO] .
  * Bull  _et al._ [2016] P. Bull _et al._ , Beyond $\Lambda$CDM: Problems, solutions, and the road ahead, Phys. Dark Univ. 12, 56 (2016), arXiv:1512.05356 [astro-ph.CO] .
  * Bahamonde  _et al._ [2021] S. Bahamonde, M. Marciu, S. D. Odintsov, and P. Rudra, String-inspired Teleparallel cosmology, Nucl. Phys. B 962, 115238 (2021), arXiv:2003.13434 [gr-qc] .
  * Harko  _et al._ [2011] T. Harko, F. S. N. Lobo, S. Nojiri, and S. D. Odintsov, $f(R,T)$ gravity, Phys. Rev. D 84, 024020 (2011), arXiv:1104.2669 [gr-qc] .
  * Harko and Lobo [2010] T. Harko and F. S. N. Lobo, f(R,$L_{m}$) gravity, Eur. Phys. J. C 70, 373 (2010), arXiv:1008.4193 [gr-qc] .
  * Haghani  _et al._ [2013] Z. Haghani, T. Harko, F. S. N. Lobo, H. R. Sepangi, and S. Shahidi, Further matters in space-time geometry: f(R,T,R$\mu$$\nu$T$\mu$$\nu$) gravity, Phys. Rev. D 88, 044023 (2013), arXiv:1304.5957 [gr-qc] .
  * Jaybhaye  _et al._ [2023] L. V. Jaybhaye, S. Bhattacharjee, and P. K. Sahoo, Baryogenesis in f(R,Lm) gravity, Phys. Dark Univ. 40, 101223 (2023), arXiv:2304.02482 [gr-qc] .
  * Roshan and Shojai [2016] M. Roshan and F. Shojai, Energy-Momentum Squared Gravity, Phys. Rev. D 94, 044002 (2016), arXiv:1607.06049 [gr-qc] .
  * Katırcı and Kavuk [2014] N. Katırcı and M. Kavuk, $f(R,T_{\mu\nu}T^{\mu\nu})$ gravity and Cardassian-like expansion as one of its consequences, Eur. Phys. J. Plus 129, 163 (2014), arXiv:1302.4300 [gr-qc] .
  * Board and Barrow [2017] C. V. R. Board and J. D. Barrow, Cosmological Models in Energy-Momentum-Squared Gravity, Phys. Rev. D 96, 123517 (2017), [Erratum: Phys.Rev.D 98, 129902 (2018)], arXiv:1709.09501 [gr-qc] .
  * Bahamonde  _et al._ [2019] S. Bahamonde, M. Marciu, and P. Rudra, Dynamical system analysis of generalized energy-momentum-squared gravity, Phys. Rev. D 100, 083511 (2019), arXiv:1906.00027 [gr-qc] .
  * Kazemi  _et al._ [2020] A. Kazemi, M. Roshan, I. De Martino, and M. De Laurentis, Jeans analysis in energy–momentum-squared gravity, Eur. Phys. J. C 80, 150 (2020), arXiv:2001.04702 [gr-qc] .
  * Chen and Chen [2020] C.-Y. Chen and P. Chen, Eikonal black hole ringings in generalized energy-momentum squared gravity, Phys. Rev. D 101, 064021 (2020), arXiv:1910.12262 [gr-qc] .
  * Sharif and Gul [2023] M. Sharif and M. Z. Gul, Study of Charged Anisotropic Karmarkar Stars in f(R,T2)$f(\mathcal{R},\mathbb{T}^{2})$ Theory, Fortsch. Phys. 71, 2200184 (2023).
  * Sharif and Naz [2023] M. Sharif and S. Naz, Study of decoupled gravastars in energy–momentum squared gravity, Annals Phys. 451, 169240 (2023), arXiv:2304.06732 [gr-qc] .
  * Sharif and Zeeshan Gul [2023] M. Sharif and M. Zeeshan Gul, Stability Analysis of the Inhomogeneous Perturbed Einstein Universe in Energy–Momentum Squared Gravity, Universe 9, 145 (2023).
  * Fazlollahi [2023] H. R. Fazlollahi, Energy–momentum squared gravity and late-time Universe, Eur. Phys. J. Plus 138, 211 (2023).
  * Yousaf  _et al._ [2023a] Z. Yousaf, M. Z. Bhatti, H. Aman, and P. K. Sahoo, Non-singular bouncing model in energy momentum squared gravity, Phys. Scripta 98, 035002 (2023a), arXiv:2301.13610 [gr-qc] .
  * Akarsu and Uzun [2023] O. Akarsu and N. M. Uzun, Cosmological models in scale-independent energy-momentum squared gravity, Phys. Dark Univ. 40, 101194 (2023), arXiv:2301.11204 [gr-qc] .
  * Sharif and Anjum [2022] M. Sharif and A. Anjum, Complexity factor for static cylindrical system in energy-momentum squared gravity, Gen. Rel. Grav. 54, 111 (2022), arXiv:2210.02196 [gr-qc] .
  * Tangphati  _et al._ [2022] T. Tangphati, I. Karar, A. Banerjee, and A. Pradhan, The mass–radius relation for quark stars in energy–momentum squared gravity, Annals Phys. 447, 169149 (2022), arXiv:2206.10371 [gr-qc] .
  * Nari and Roshan [2018] N. Nari and M. Roshan, Compact stars in Energy-Momentum Squared Gravity, Phys. Rev. D 98, 024031 (2018), arXiv:1802.02399 [gr-qc] .
  * Sharif and Iltaf [2022] M. Sharif and S. Iltaf, Charged anisotropic solutions by decoupling approach in energy–momentum squared gravity, Chin. J. Phys. 79, 173 (2022).
  * Sharif and Naz [2022] M. Sharif and S. Naz, Gravastars in energy–momentum squared gravity, Mod. Phys. Lett. A 37, 2250125 (2022).
  * Sharif and Akram [2022] M. Sharif and H. M. S. Akram, Study of anisotropic compact stars with Krori–Barua spacetime in $f(R,T^{2})$ gravity, Mod. Phys. Lett. A 37, 2250115 (2022).
  * Sharif and Saeed [2022] M. Sharif and M. Saeed, Study of gravastars admitting conformal motion in $f(R,T^{2})$ gravity, Chin. J. Phys. 77, 583 (2022).
  * Akarsu  _et al._ [2018a] O. Akarsu, N. Katirci, S. Kumar, R. C. Nunes, and M. Sami, Cosmological implications of scale-independent energy-momentum squared gravity: Pseudo nonminimal interactions in dark matter and relativistic relics, Phys. Rev. D 98, 063522 (2018a), arXiv:1807.01588 [gr-qc] .
  * Akarsu  _et al._ [2018b] O. Akarsu, J. D. Barrow, S. Çıkıntoğlu, K. Y. Ekşi, and N. Katırcı, Constraint on energy-momentum squared gravity from neutron stars and its cosmological implications, Phys. Rev. D 97, 124017 (2018b), arXiv:1802.02093 [gr-qc] .
  * Sharif and Zeeshan Gul [2021] M. Sharif and M. Zeeshan Gul, Viable Wormhole Solutions in Energy-Momentum Squared Gravity, Eur. Phys. J. Plus 136, 503 (2021), arXiv:2105.04416 [gr-qc] .
  * Yousaf  _et al._ [2022a] Z. Yousaf, M. Z. Bhatti, S. Khan, and P. K. Sahoo, f(G,T$\alpha$$\beta$T$\alpha$$\beta$) theory and complex cosmological structures, Phys. Dark Univ. 36, 101015 (2022a), arXiv:2112.00575 [gr-qc] .
  * Yousaf  _et al._ [2022b] Z. Yousaf, M. Z. Bhatti, and S. Khan, Non-static charged complex structures in $f({\mathbb{G}},{\mathbf{T}}^{2})$ gravity, Eur. Phys. J. Plus 137, 322 (2022b).
  * Yousaf  _et al._ [2023b] Z. Yousaf, A. M. Galal, M. Z. Bhatti, and A. Farhat, Sources of irregular energy density in energy–momentum squared gravity with charge, New Astron. 98, 101930 (2023b).
  * Li  _et al._ [2007] B. Li, J. D. Barrow, and D. F. Mota, The Cosmology of Modified Gauss-Bonnet Gravity, Phys. Rev. D 76, 044027 (2007), arXiv:0705.3795 [gr-qc] .
  * Bamba  _et al._ [2010] K. Bamba, S. D. Odintsov, L. Sebastiani, and S. Zerbini, Finite-time future singularities in modified Gauss-Bonnet and F(R,G) gravity and singularity avoidance, Eur. Phys. J. C 67, 295 (2010), arXiv:0911.4390 [hep-th] .
  * Elizalde  _et al._ [2010] E. Elizalde, R. Myrzakulov, V. V. Obukhov, and D. Saez-Gomez, LambdaCDM epoch reconstruction from F(R,G) and modified Gauss-Bonnet gravities, Class. Quant. Grav. 27, 095007 (2010), arXiv:1001.3636 [gr-qc] .
  * Shah  _et al._ [2022] H. Shah, S. Khan, D. Lu, Z. Ahmad, S. M. Hussain, and H. H. Shah, Gravitational collapse of dissipative fluid in f(R,G) gravity, Int. J. Mod. Phys. A 37, 2250143 (2022).
  * De Felice and Tsujikawa [2009a] A. De Felice and S. Tsujikawa, Solar system constraints on f(G) gravity models, Phys. Rev. D 80, 063516 (2009a), arXiv:0907.1830 [hep-th] .
  * de la Cruz-Dombriz and Saez-Gomez [2012] A. de la Cruz-Dombriz and D. Saez-Gomez, On the stability of the cosmological solutions in $f(R,G)$ gravity, Class. Quant. Grav. 29, 245014 (2012), arXiv:1112.4481 [gr-qc] .
  * Ganiou  _et al._ [2022] M. G. Ganiou, M. J. S. Houndjo, C. Aïnamon, L. Ayivi, and A. Kanfon, Reconstruction method applied to bounce cosmology and inflationary scenarios in cosmological $f(G)$ gravity, Eur. Phys. J. Plus 137, 208 (2022).
  * Nojiri  _et al._ [2021] S. Nojiri, S. D. Odintsov, V. K. Oikonomou, and A. A. Popov, Ghost-free $F(R,\mathcal{G})$ gravity, Nucl. Phys. B 973, 115617 (2021), arXiv:2111.09457 [gr-qc] .
  * Marciu [2019] M. Marciu, Prospects of the cosmic scenery in a quintom dark energy model with generalized nonminimal gauss-bonnet couplings, Phys. Rev. D 99, 043508 (2019).
  * De Felice and Tsujikawa [2009b] A. De Felice and S. Tsujikawa, Construction of cosmologically viable f(G) dark energy models, Phys. Lett. B 675, 1 (2009b), arXiv:0810.5712 [hep-th] .