singularity, this can never occur when $-2<\beta<-1$, since the term
$\sim(t-t_{s})^{-\beta-2}$ is present in the variable $y$. On the contrary,
when $\beta<-2$, the two variables $x$, $y$ never become singular, so in the
case of a Type IV singularity, this can be developed by the physical system
for the values of $\beta$ in the range $\beta<-2$. Regarding the Type II case,
which recall that it occurs for $-1<\beta<0$, the dynamical system variable
$x$ is singular in this case since it depends on the term
$\sim(t-t_{s})^{-\beta-1}$, therefore a Type II singularity can never occur
for the model at hand. Regarding the Type III case, it cannot also occur due
to the term $\sim(t-t_{s})^{\frac{3\beta}{2}-\frac{3}{2}}$, which is singular
for $0<\beta<1$. Regarding, the Big Rip singularity, it can always occur,
because for $\beta>1$, the dynamical system variables $x$ and $y$ never
develop singular behavior. The results of our analysis are presented in TABLE
6.

Table 6: Type of Physical Singularity for $H(t)=H_{s}(t)+h_{s}(t)(t-t_{s})^{-\beta}$ Type of the Singularity | Occurrence/Non-occurrence
---|---
Type I | It occurs for $\beta>1$
Type II | Does not occur
Type III | Does not occur
Type IV | Does not occur for $-2<\beta<-1$, but occurs for $\beta<-2$

Therefore, in conclusion, if the dominant behavior of the Hubble rate solution
for the model at hand is given in Eq. (678), the Type III, and Type II never
occur in the physical system. However, the Type IV and Type I singularities
occur when $\beta<-2$ and for any $\beta>1$, respectively.

###  VIII.4 Dynamical System of Interacting Multifluids in LQC

In this section, we shall investigate the occurrence of finite-time
singularities in the context of a LQC interacting multifluid system. The full
analysis of the subject of this section was performed in Ref. Odintsov:2018awm
on which the presentation will be based. The three fluids that are considered
are a non-interacting baryonic fluid and the interacting DE-DM fluids. Also
the DE fluid will contain a bulk viscosity term. The Friedmann equation in the
context of LQC in the presence of the three fluids and for a FLRW spacetime,
has the following form,

$\displaystyle
H^{2}=\frac{\kappa^{2}\rho_{\mathrm{tot}}}{3}\left(1-\frac{\rho_{\mathrm{tot}}}{\rho_{c}}\right)\,,$
(681)

where, once again, $\rho_{c}$ denotes the critical density of LQC and
$\rho_{\mathrm{tot}}=\rho_{\rm DM}+\rho_{\rm DE}+\rho_{b}$ is the total energy
density of all the three fluids present in which $\rho_{\rm DM}$, $\rho_{\rm
DE}$ and $\rho_{b}$ denote the energy density of DM, DE and baryons,
respectively. Upon differentiation with respect to the cosmic time, Eq. (681)
yields,

$\displaystyle\dot{H}=-\frac{\kappa^{2}}{2}\left(\rho_{\rm
tot}+p_{\mathrm{tot}}\right)\left[1-2\left(\frac{\rho_{\rm
tot}+p_{\mathrm{tot}}}{\rho_{c}}\right)\right]\,,$ (682)

where $p_{\mathrm{tot}}$ denotes the total pressure of the three fluids, which
is basically equal to the pressure of the DE fluid, since DM and the baryonic
fluids are pressureless. For the DE fluid, we assume the following EoS
Nojiri:2005sr ,

$\displaystyle p_{\rm DE}=-\rho_{\rm DE}-A\kappa^{4}\rho_{\rm DE}^{2}\,,$
(683)

with $A$ being a real dimensionless parameter. The energy-momentum
conservation yields the following three continuity equations,

$\displaystyle\dot{\rho}_{b}+3H\rho_{b}=0\,,\quad\dot{\rho}_{\rm
DM}+3H\rho_{\rm DM}=Q\,,\quad\dot{\rho}_{\rm DE}+3H(\rho_{\rm DE}+p_{\rm
DE})=-Q\,,$ (684)

where $Q$ denotes the interaction between the dark fluids, and once again
$Q>0$ indicates that the DE fluid loses energy and $Q<0$ indicates that the DE
fluid gains energy at the expense of the DM fluid. We shall consider the
following interaction between the dark sector fluids,

$\displaystyle Q=3H(c_{1}\rho_{\rm DM}+c_{2}\rho_{\rm DE})\,,$ (685)

which is phenomenologically interesting Caldera-Cabral:2008yyo ; Pavon:2005yx
; Quartin:2008px ; Sadjadi:2006qp ; Zimdahl:2005bk  and here $c_{1}$, $c_{2}$
are the coupling parameters. For the three-fluid LQC system with equations
(681) and (682), we can construct an autonomous dynamical system by
appropriately introducing some dimensionless variables. After this, we shall
investigate whether the resulting polynomial dynamical system develops
singularities at some finite-time, using the dominant balance technique we
presented in previous sections of this chapter. We shall also discriminate the
dynamical system singularities from the actual physical finite-time
singularities. The major outcome of this section is the fact that we shall
analytically prove that the resulting LQC dynamical system has no finite-time
singularities, which verifies that LQC actually leads to non-singular
behavior. We now construct the autonomous dynamical system from Eqs. (681),
(682) and (684). We choose the dimensionless dynamical system variables as
follows,

$\displaystyle x_{1}=\frac{\kappa^{2}\rho_{\rm DE}}{3H^{2}}\,,\quad
x_{2}=\frac{\kappa^{2}\rho_{\rm DM}}{3H^{2}}\,,\quad
x_{3}=\frac{\kappa^{2}\rho_{b}}{3H^{2}}\,,\quad
z=\frac{H^{2}}{\kappa^{2}\rho_{c}}\,,$ (686)

which are constrained to satisfy the Friedmann constraint,

$\displaystyle x_{1}+x_{2}+x_{3}-z\left(x_{1}+x_{2}+x_{3}\right)^{2}=1\,,$
(687)

for all cosmic times. Furthermore, the total EoS parameter $w_{\mathrm{eff}}$
corresponding to the multifluid system, expressed in terms of the
dimensionless dynamical system variables (686), takes the following form,

$\displaystyle w_{\mathrm{eff}}=-x_{1}-3A\kappa^{4}\rho_{c}x_{1}^{2}z\,.$
(688)

Now we can form the polynomial autonomous dynamical system for the three
multi-fluids system, by combining Eqs. (681), (682), (684), and (686), so
after some algebra, we get,

$\displaystyle\frac{dx_{1}}{dN}=$
$\displaystyle\,-\frac{\kappa^{2}Q}{3H^{3}}+9Ax_{1}^{3}z-27Ax_{1}^{2}z+3w_{\rm
DE}x_{1}^{2}-3w_{\rm
DE}x_{1}-18x_{1}^{3}z+3x_{1}^{2}+3x_{1}x_{2}+3x_{1}x_{3}-3x_{1}$
$\displaystyle\,-18w_{\rm DE}x_{1}^{3}z-18w_{\rm
DE}x_{1}^{2}x_{2}z-36x_{1}^{2}x_{2}z-36x_{1}^{2}x_{3}z-18x_{1}x_{2}^{2}z$
$\displaystyle\,-54Ax_{1}^{4}z^{2}-54Ax_{1}^{3}x_{2}z^{2}-54Ax_{1}^{3}x_{3}z^{2}-18w_{\rm
DE}x_{1}^{2}x_{3}z-36x_{1}x_{2}x_{3}z-18x_{1}x_{3}^{2}z\,,$
$\displaystyle\frac{dx_{2}}{dN}=$
$\displaystyle\,\frac{\kappa^{2}Q}{3H^{3}}+9Ax_{1}^{2}x_{2}z+3w_{\rm
DE}x_{1}x_{2}-18x_{1}^{2}x_{2}z+3x_{1}x_{2}+3x_{2}^{2}+3x_{2}x_{3}-3x_{2}$
$\displaystyle\,-18w_{\rm DE}x_{1}^{2}x_{2}z-18w_{\rm
DE}x_{1}x_{2}^{2}z-36x_{1}x_{2}^{2}z-36x_{1}x_{2}x_{3}z-18x_{2}^{3}z$
$\displaystyle\,-54Ax_{1}^{3}x_{2}z^{2}-54Ax_{1}^{2}x_{2}^{2}z^{2}-54Ax_{1}^{2}x_{2}x_{3}z^{2}-18w_{\rm
DE}x_{1}x_{2}x_{3}z-36x_{2}^{2}x_{3}z-18x_{2}x_{3}^{2}z\,,$
$\displaystyle\frac{dx_{3}}{dN}=$ $\displaystyle\,9Ax_{1}^{2}x_{3}z-18w_{\rm
DE}x_{1}^{2}x_{3}z+3w_{\rm
DE}x_{1}x_{3}-18x_{1}^{2}x_{3}z+3x_{1}x_{3}+3x_{2}x_{3}+3x_{3}^{2}-3x_{3}$
$\displaystyle\,-18w_{\rm DE}x_{1}x_{2}x_{3}z-18w_{\rm
DE}x_{1}x_{3}^{2}z-36x_{1}x_{2}x_{3}z-36x_{1}x_{3}^{2}z-18x_{2}^{2}x_{3}z-36x_{2}x_{3}^{2}z$
$\displaystyle\,-54Ax_{1}^{3}x_{3}z^{2}-54Ax_{1}^{2}x_{2}x_{3}z^{2}-54Ax_{1}^{2}x_{3}^{2}z^{2}-18x_{3}^{3}z\,,$
$\displaystyle\frac{dz}{dN}=$ $\displaystyle\,-9Ax_{1}^{2}z^{2}+18w_{\rm
DE}x_{1}^{2}z^{2}-3w_{\rm DE}x_{1}z+18x_{1}^{2}z^{2}-3x_{1}z-3x_{2}z-3x_{3}z$
$\displaystyle\,+18w_{\rm DE}x_{1}x_{2}z^{2}+18w_{\rm
DE}x_{1}x_{3}z^{2}+36x_{1}x_{2}z^{2}+36x_{1}x_{3}z^{2}+18x_{2}^{2}z^{2}$
$\displaystyle\,+54Ax_{1}^{3}z^{3}+54Ax_{1}^{2}x_{2}z^{3}+54Ax_{1}^{2}x_{3}z^{3}+36x_{2}x_{3}z^{2}+18x_{3}^{2}z^{2}\,,$
(689)

and note that, as in the previous sections in this chapter, we used the
$e$-foldings number to quantify the dynamical evolution of the variables,
instead of the cosmic time. Furthermore, by choosing the interaction term $Q$
as in Eq. (685), the $Q$-containing terms in the dynamical system (VIII.4),
can be expressed in terms of the variables $x_{1}$ and $x_{2}$ in the
following way,

$\displaystyle\frac{\kappa^{2}Q}{3H^{3}}=3c_{1}x_{2}+3c_{2}x_{1}\,.$ (690)

Now having the dynamical system (VIII.4) at hand, we can apply the dominant
balance analysis in order to reveal the dynamical system singularities and we
shall also investigate whether physical singularities occur too. This is the
subject of the next subsection.

####  VIII.4.1 Dominant Balance Analysis of the three-fluid Cosmological
Dynamical System

We shall use the method of dominant balances in order to investigate whether
the dynamical system (VIII.4) develops finite-time singularities. As we shall
show, the dynamical system is singularity-free and in fact this result holds
true for a general set of initial conditions. One should have in mind of
course that there exists a limited set of initial conditions that may lead in
principle to finite-time dynamical system singularities, but this limited set
is not of interest. Near a singularity, the dimensionless variables
$x_{1}(N)$, $x_{2}(N)$, $x_{3}(N)$ and $z(N)$ at leading order behave as
follows,

$\displaystyle
x_{1}(N)=a_{1}(N-N_{c})^{p_{1}}\,,\quad\,x_{2}(N)=a_{2}(N-N_{c})^{p_{2}}\,,\quad
x_{3}(N)=a_{3}(N-N_{c})^{p_{3}}\,,\quad z(N)=a_{4}(N-N_{c})^{p_{4}}\,.$ (691)

The dynamical system (VIII.4) can be rewritten as follows
$\frac{d\vec{x}}{dN}=f(\vec{x})$, with $\vec{x}$ being
$\vec{x}=(x_{1},x_{2},x_{3},z)$, and also $f(x_{1},x_{2},x_{3},z)$ is defined
to be equal to,

$\displaystyle
f(x_{1},x_{2},x_{3},z)=\left(\begin{array}[]{c}f_{1}(x_{1},x_{2},x_{3},z)\\\
f_{2}(x_{1},x_{2},x_{3},z)\\\ f_{3}(x_{1},x_{2},x_{3},z)\\\
f_{4}(x_{1},x_{2},x_{3},z)\\\ \end{array}\right)\,,$ (696)

and the corresponding functions $f_{i}(x_{1},x_{2},x_{3})$, $i=1,2,3,4$ are,

$\displaystyle f_{1}(x_{1},x_{2},x_{3},z)=$
$\displaystyle\,3c_{1}x_{2}+3c_{2}x_{1}+9Ax_{1}^{3}z-27Ax_{1}^{2}z+3w_{\rm
DE}x_{1}^{2}-3w_{\rm
DE}x_{1}-18x_{1}^{3}z+3x_{1}^{2}+3x_{1}x_{2}+3x_{1}x_{3}-3x_{1}$
$\displaystyle\,-18w_{\rm DE}x_{1}^{3}z-18w_{\rm
DE}x_{1}^{2}x_{2}z-36x_{1}^{2}x_{2}z-36x_{1}^{2}x_{3}z-18x_{1}x_{2}^{2}z$
$\displaystyle\,-54Ax_{1}^{4}z^{2}-54Ax_{1}^{3}x_{2}z^{2}-54Ax_{1}^{3}x_{3}z^{2}-18w_{\rm
DE}x_{1}^{2}x_{3}z-36x_{1}x_{2}x_{3}z-18x_{1}x_{3}^{2}z\,,$ $\displaystyle
f_{2}(x_{1},x_{2},x_{3},z)=$
$\displaystyle\,3c_{1}x_{2}+3c_{2}x_{1}+9Ax_{1}^{2}x_{2}z+3w_{\rm
DE}x_{1}x_{2}-18x_{1}^{2}x_{2}z+3x_{1}x_{2}+3x_{2}^{2}+3x_{2}x_{3}-3x_{2}$
$\displaystyle\,-18w_{\rm DE}x_{1}^{2}x_{2}z-18w_{\rm
DE}x_{1}x_{2}^{2}z-36x_{1}x_{2}^{2}z-36x_{1}x_{2}x_{3}z-18x_{2}^{3}z$
$\displaystyle\,-54Ax_{1}^{3}x_{2}z^{2}-54Ax_{1}^{2}x_{2}^{2}z^{2}-54Ax_{1}^{2}x_{2}x_{3}z^{2}-18w_{\rm
DE}x_{1}x_{2}x_{3}z-36x_{2}^{2}x_{3}z-18x_{2}x_{3}^{2}z\,,$ $\displaystyle
f_{3}(x_{1},x_{2},x_{3},z)=$ $\displaystyle\,9Ax_{1}^{2}x_{3}z-18w_{\rm
DE}x_{1}^{2}x_{3}z+3w_{\rm
DE}x_{1}x_{3}-18x_{1}^{2}x_{3}z+3x_{1}x_{3}+3x_{2}x_{3}+3x_{3}^{2}-3x_{3}$
$\displaystyle\,-18w_{\rm DE}x_{1}x_{2}x_{3}z-18w_{\rm
DE}x_{1}x_{3}^{2}z-36x_{1}x_{2}x_{3}z-36x_{1}x_{3}^{2}z-18x_{2}^{2}x_{3}z-36x_{2}x_{3}^{2}z$
$\displaystyle\,-54Ax_{1}^{3}x_{3}z^{2}-54Ax_{1}^{2}x_{2}x_{3}z^{2}-54Ax_{1}^{2}x_{3}^{2}z^{2}-18x_{3}^{3}z\,,$
$\displaystyle f_{4}(x_{1},x_{2},x_{3},z)=$
$\displaystyle\,-9Ax_{1}^{2}z^{2}+18w_{\rm DE}x_{1}^{2}z^{2}-3w_{\rm
DE}x_{1}z+18x_{1}^{2}z^{2}-3x_{1}z-3x_{2}z-3x_{3}z$ $\displaystyle\,+18w_{\rm
DE}x_{1}x_{2}z^{2}+18w_{\rm
DE}x_{1}x_{3}z^{2}+36x_{1}x_{2}z^{2}+36x_{1}x_{3}z^{2}+18x_{2}^{2}z^{2}$
$\displaystyle\,+54Ax_{1}^{3}z^{3}+54Ax_{1}^{2}x_{2}z^{3}+54Ax_{1}^{2}x_{3}z^{3}+36x_{2}x_{3}z^{2}+18x_{3}^{2}z^{2}\,.$
(697)

Now we shall seek for consistent truncations of the vector-valued function
$f(x_{1},x_{2},x_{3},z)$ defined in Eq. (696), and investigate whether
singular solutions exist or not. Among many distinct truncations of the vector
function $f(x_{1},x_{2},x_{3},z)$, one is the following,

$\displaystyle\hat{f}(x_{1},x_{2},x_{3},z)=\left(\begin{array}[]{c}3x_{1}(N)x_{2}(N)\\\
3(w_{\rm DE}+1)x_{1}(N)x_{2}(N)\\\ -54Ax_{1}^{2}(N)x_{3}^{2}(N)z^{2}(N)\\\
54Ax_{1}^{2}(N)z^{3}(N)x_{2}(N)\\\ \end{array}\right)\,.$ (702)

The corresponding vector $\vec{p}$, which obviously corresponds to the
solution $\rho_{\mathrm{tot}}=\rho_{c}$, is easily found,

$\displaystyle\vec{p}=(-1,-1,-1,1)\,,$ (703)

and the corresponding vectors $\vec{a}$ are,

$\displaystyle\vec{a}_{1}=\left(-\frac{1}{3(w_{\rm
DE}+1)},-\frac{1}{3},-\frac{1}{3},\frac{w_{\rm
DE}+1}{\sqrt{-2A}}\right)\qquad\mbox{and}\qquad\vec{a}_{2}=\left(-\frac{1}{3(w_{\rm
DE}+1)},-\frac{1}{3},-\frac{1}{3},-\frac{w_{\rm DE}+1}{\sqrt{-2A}}\right).$
(704)

The corresponding Kovalevskaya matrix is given by

$\displaystyle K(\vec{a}_{1})=\left(\begin{array}[]{cccc}0&-\frac{1}{w_{\rm
DE}+1}&0&0\\\ -(w_{\rm DE}+1)&0&0&0\\\ -2(w_{\rm
DE}+1)&0&-1&\frac{2\sqrt{-2A}}{3(w_{\rm DE}+1)}\\\ -\frac{6(w_{\rm
DE}+1)^{2}}{\sqrt{-2A}}&-\frac{3(w_{\rm DE}+1)}{\sqrt{-2A}}&0&2\\\
\end{array}\right)\,,$ (709)

with eigenvalues

$\displaystyle(r_{1},r_{2},r_{3},r_{4})=(-1,2,1,-1)\,.$ (710)

In conclusion, the major outcome of this section is the fact that LQC utterly
erases any finite-time singularities that may have occurred in the classical
interacting three fluid system. We have to take into account that the
variables $x_{i}$ $(i=1,2,3)$ diverge when the Hubble rate vanishes, and this
happens when $\rho_{\mathrm{tot}}=\rho_{c}$ or $\rho_{\mathrm{tot}}=0$. That
means, the singularity, in the $x_{i}$ variables, exhibits that the total
energy density coincides with the critical one, and thus, this singularity in
the variables $x_{i}$ is not a physical singularity in the sense that neither
the Hubble rate diverges nor the corresponding energy densities diverge for
our interacting three fluid system in the context of LQC.

We shall conclude the analysis by finding analytically the fixed points of the
dynamical system. Also, we shall reveal the behavior of the total EoS
parameter of Eq. (688), in terms of the coefficients $c_{1}$ and $c_{2}$
appearing in the interaction term $Q$. We can find the fixed points of the
dynamical system which well-known linearization techniques for dynamical
systems. In the case that the fixed point is hyperbolic, in which case the
eigenvalues of the matrix that results after the linearization of the
dynamical system contain non-zero real part, the stability of the fixed point
can be determined. In the case of a hyperbolic fixed point, when the
eigenvalue of the linearization matrix has a negative real part, then the
fixed stable, and if the real part is negative, then the fixed point is
unstable. We shall denote the fixed points of the dynamical system (VIII.4) as
$\phi_{*}$, and the Jacobian of the linearized dynamical system near the fixed
points as, $\mathcal{J}(g)$, with the latter being equal,

$\displaystyle\mathcal{J}=\sum_{i}\sum_{j}\left[\frac{\partial f_{i}}{\partial
x_{j}}\right]\,.$ (711)

Now we shall solve the equation $f(x_{1},x_{2},x_{3},z)=0$, with
$f(x_{1},x_{2},x_{3},z)$ appearing in Eq. (696), and this process will reveal
the fixed points of the dynamical system (VIII.4), which are,

$\displaystyle\phi_{1}^{*}=$ $\displaystyle\,\left\\{x_{1}\to 0,x_{2}\to
0,x_{3}\to 0\right\\}\,,$ $\displaystyle\phi_{2}^{*}=$
$\displaystyle\,\left\\{x_{1}\to 0,x_{2}\to 0,x_{3}\to 0,z\to 0\right\\}\,,$
$\displaystyle\phi_{3}^{*}=$ $\displaystyle\,\left\\{x_{1}\to 0,x_{2}\to
0,x_{3}\to 1,z\to 0\right\\}\,,$ $\displaystyle\phi_{4}^{*}=$
$\displaystyle\,\left\\{x_{1}\to\frac{-\sqrt{(c_{1}-c_{2}-w_{\rm
DE})^{2}+4c_{1}w_{\rm DE}}-c_{1}+c_{2}+w_{\rm DE}}{2w_{\rm DE}},\right.$
$\displaystyle\qquad\left.x_{2}\to\frac{\frac{c_{1}^{2}}{w_{\rm
DE}}+\frac{c_{1}\sqrt{(c_{1}-c_{2}-w_{\rm DE})^{2}+4c_{1}w_{\rm DE}}}{w_{\rm
DE}}-\frac{c_{1}c_{2}}{w_{\rm DE}}+c_{1}}{2c_{1}},x_{3}\to 0,z\to
0\right\\}\,,$ $\displaystyle\phi_{5}^{*}=$
$\displaystyle\,\left\\{x_{1}\to\frac{\sqrt{(c_{1}-c_{2}-w_{\rm
DE})^{2}+4c_{1}w_{\rm DE}}-c_{1}+c_{2}+w_{\rm DE}}{2w_{\rm DE}},\right.$
$\displaystyle\qquad\left.x_{2}\to\frac{\frac{c_{1}^{2}}{w_{\rm
DE}}-\frac{c_{1}c_{2}}{w_{\rm DE}}-\frac{c_{1}\sqrt{(c_{1}-c_{2}-w_{\rm
DE})^{2}+4c_{1}w_{\rm DE}}}{w_{\rm DE}}+c_{1}}{2c_{1}},x_{3}\to 0,z\to
0\right\\}\,.$ (712)

Apparently, the fixed points $\phi_{1}^{*}$ and $\phi_{2}^{*}$ are not
hyperbolic, and the fixed points $\phi_{3}^{*}$, $\phi_{4}^{*}$ and
$\phi_{5}^{*}$ are hyperbolic. Omitting the exact form of the eigenvalues for
brevity, a thorough analysis of the phase space indicates that the fixed
points are unstable, regardless the values of the parameters $w_{\rm DE}$,
$c_{1}$, $c_{2}$. Also the total EoS in Eq. (688) must be studied for various
values of the free parameters. We shall consider the case that we fix the DE
EoS parameter $w_{\rm DE}$ to have various values, varying from quintessential
values ($w_{\rm DE}=-0.5$) to phantom values ($w_{\rm DE}=-1.5$). A numerical
analysis of the total EoS parameter (688) indicates that in the context of the
three fluid interacting cosmology, it is possible to realize various
cosmological evolutions, for example quintessential, phantom and also matter
dominated eras.

####  VIII.4.2 The Choice of the DE EoS and its Physical Implications

For the analysis performed in the previous subsections, the choice of the DE
EoS was chosen to be that of Eq. (683). However, a more general EoS can be
used, which may include higher powers of the Hubble rate and also further
effects of viscosity might be included. In this subsection we report the
implications of the chosen EoS (683) for DE phenomenology. Using the formalism
of Astashenok:2012iy  which we extend in the context of LQC, a comparison of
the classical theory with the LQC one can be done for the chosen EoS (683). We
note that when the LQC effects are not considered, the EoS of Eq. (683) drives
the cosmological system to finite-time singularities of the Type III form
Nojiri:2005sx . On the other hand, in the case of LQC, with one fluid, the
singularities are eliminated Sami:2006wj , and as we demonstrated previously,
the same applies in the three fluid system in which DM and DE are interacting
with each other. Thus, LQC eliminates the finite-time singularities. Our
findings and comparisons are presented in TABLE 7.

Table 7: Classical versus LQC Singularities for the DE EoS $p_{\rm DE}=-\rho_{\rm DE}-A\kappa^{4}\rho_{\rm DE}^{2}$ .  Case Study |  Cosmological Scenario | Occurrence of Singularities
---|---|---
Classical Case | Interacting Multi-fluids | Type III singularity occurs
LQC | One DE Fluid | No singularity occurs
LQC | Interacting Multi-fluids | No singularity occurs

##  IX The Avoidance of Finite-time Future Singularities

Near the singularity, especially in the case of Type I and Type III, the
Hubble rate becomes very large, which means that the curvature and the
temperature become very large and therefore the quantum effect and the thermal
effect become important. One may naturally wonder whether it is possible to
avoid the finite-time singularities from the cosmological picture so that we
realize a non-singular Universe. In this section, we shall examine the
possibility of avoiding the finite-time singularities in the cosmological
scenarios. More specifically, we shall discuss what could happen with the
future singular dark Universe when the effects of quantum or thermal radiation
are included. This section is based on Refs. Frampton:2011sp ; Brevik:2011mm ;
Nojiri:2020sti ; Carroll:2003st .

###  IX.1 Little Rip Scenario

The possibility of a phantom DE EoS (i.e., $w_{\mathrm{DE}}<-1$) is very hard
to exclude from the cosmological picture DiValentino:2020vnx . The phantom DE
can solve the Hubble constant tension DiValentino:2021izs  quite excellently.
However, one of the serious features that the phantom DE carries is the
following. For $w_{\mathrm{DE}}<-1$, the DE density increases with the
increasing scale factor and both of them can blow up at some finite-time in
the future leading to a Big Rip singularity Caldwell:2003vq . However, this
future singularity can be avoided even if $w_{\mathrm{DE}}<-1$ and this
heuristic scenario, dubbed as “Little Rip” was introduced in Ref.
Frampton:2011sp . With such fantastic proposal, the Little Rip scenario
received significant attention to the scientific community Frampton:2011sp ;
Brevik:2011mm ; Frampton:2011rh ; Nojiri:2011kd ; Granda:2011kx ; Ito:2011ae ;
Xi:2011uz ; Liu:2012iba ; Saez-Gomez:2012uwp ; Brevik:2012nt ; Brevik:2012ka ;
Balakin:2012ee ; Frampton:2013fxa ; Makarenko:2014nca ; Bouhmadi-Lopez:2014cca
; Albarran:2015cda ; Morais:2016bev . In this section, we shall describe how
the Big Rip singularity could be avoided even if $w_{\mathrm{DE}}$ is less
than $-1$.

In order to realize a non-singular scenario, one can proceed with a non-
singular scale factor $a(t)$ having the following form

$\displaystyle a=\mathrm{e}^{f(t)}\,,$ (713)

where $f(t)$ is any arbitrary non-singular function satisfying $\dot{f}=H$.
From the Friedmann equation $H^{2}=(\kappa^{2}/3)\rho$, one can derive that
$\dot{f}^{2}(t)=(\kappa^{2}/3)\rho$. And from the condition that $\rho$ is an
increasing function of $a$ which implies
$\frac{d\rho}{da}=\frac{6}{\kappa^{2}\dot{a}}\dot{f}\ddot{f}>0$, gives the
restriction

$\displaystyle\ddot{f}>0\,,$ (714)

because we are dealing with an expanding Universe, i.e., with $\dot{f}=H>0$.
Thus, one may note that all Little Rip scenarios which are described by the
evolution of the scale factor given by Eq. (713), with non-singular $f$ should
satisfy the Eq. (714).

To proceed further in this direction, let us now consider the well-known non-
linear EoS Nojiri:2005sr

$\displaystyle p=-\rho-f(\rho)\quad\Longrightarrow\quad
w=-1-\frac{f(\rho)}{\rho}\,,$ (715)

where $f(\rho)>0$ ensures that the $\rho$ increases with the scale factor,
because in that case $w<-1$. From the conservation equation,
$\dot{\rho}+3H(p+\rho)=0$, one can find the scale factor as

$\displaystyle
a=a_{0}\exp\left(\int_{\rho_{0}}^{\rho}\frac{d\rho}{3f(\rho)}\right)\,,$ (716)

where we set $a_{0}$, the present value of the scale factor to be unity and
$\rho_{0}$ denotes the present value of the energy density. Now, using the
conservation equation $\dot{\rho}=3Hf(\rho)$ and the Friedmann equation
$H^{2}=\frac{\kappa^{2}\rho}{3}$, one finds

$\displaystyle
t-t_{0}=\frac{1}{\sqrt{3}\kappa}\int_{\rho_{0}}^{\rho}\frac{d\rho}{\sqrt{\rho}f(\rho)}\,.$
(717)

We assume that $f(\rho)$ is given by the power law
$f(\rho)=\frac{A}{\sqrt{3}\kappa}\rho^{\nu+\frac{1}{2}}$, where $A>0$ and
$\nu$ are constants. We select $\nu=0$ in the above form (i.e., we consider
$f(\rho)=\frac{A}{\sqrt{3}\kappa}\sqrt{\rho}$) and for which one can find the
time from (717) as

$\displaystyle t-t_{0}=\frac{1}{A}\,\ln\frac{\rho}{\rho_{0}}\,,$ (718)

which clearly shows that $\rho\rightarrow\infty$ is not reached in some
finite-time. That means even if we have $w<-1$ but we can avoid the future
singularity and it is known as the Little Rip scenario.

Now, solving (716) for $f(\rho)=\frac{A}{\sqrt{3}\kappa}\sqrt{\rho}$, one can
express $\rho$ as a function of $a$ as Brevik:2011mm :

$\displaystyle\rho(a)=\rho_{0}\left(1+\frac{\sqrt{3}A}{2\kappa\sqrt{\rho_{0}}}\ln
a\right)^{2}\,,$ (719)

and consequently, using the first Friedmann equation, one can explicitly find
the scale factor in terms of the time as Brevik:2011mm :

$\displaystyle
a(t)=\exp\left[\frac{2\kappa\sqrt{\rho_{0}}}{\sqrt{3}A}\left\\{\exp\left(\frac{A}{2}(t-t_{0})\right)-1\right\\}\right]\,,$
(720)

which does not exhibit any finite-time singularity in the future.

####  IX.1.1 Little Rip in Viscous Universe

The Little Rip scenario can also be realized in the viscous Universe as well.
We refer to section III.1 for the basic dynamical equations for a viscous
Universe. In the viscous Universe, the effective pressure term as in Eq. (37)
or (38) which includes the bulk viscosity component plays the crucial role. We
generalize (715) in presence of the bulk viscosity as

$\displaystyle p_{\mathrm{eff}}=-\rho-f(\rho)-\eta(H)\,,$ (721)

where notice that we have generalized the last term in the right hand side of
Eq. (38) as $3\xi(t)H\rightarrow\eta(H)$ in which $\xi(t)$ refers to the
coefficient of the bulk viscosity. We investigate the cosmological scenario
with

$\displaystyle\eta(H)=\bar{\eta}=\mathrm{constant}\,.$ (722)

For the constant $\eta(H)$, using the conservation equation
$\dot{\rho}+3H(p_{\mathrm{eff}}+\rho)=0$, one can find the scale factor as

$\displaystyle
a=\exp\left(\frac{1}{3}\int_{\rho_{0}}^{\rho}\frac{d\rho}{\bar{\eta}+f(\rho)}\right)\,,$
(723)

where we have explicitly set $a_{0}=1$. Further, using the conservation and
Friedmann equations, one can express the time as

$\displaystyle
t-t_{0}=\frac{1}{\sqrt{3}\kappa}\int_{\rho_{0}}^{\rho}\frac{d\rho}{\sqrt{\rho}\,[\bar{\eta}+f(\rho)]}\,.$
(724)

Now, for the model $f(\rho)=\frac{A}{\sqrt{3}\kappa}\sqrt{\rho}$ (in this
case, the dimension of $A$, in natural units, is of the Planck’s mass), we can
solve

$\displaystyle
t-t_{0}=\frac{2}{A}\ln\left(\frac{\sqrt{3}\kappa\bar{\eta}+A\sqrt{\rho}}{\sqrt{3}\kappa\bar{\eta}+A\sqrt{\rho_{0}}}\right)\,.$
(725)

and consequently,

$\displaystyle\rho(t)=\frac{1}{A^{2}}\left[\left(\sqrt{3}\kappa{\bar{\eta}}+A\sqrt{\rho_{0}}\right)\exp(\frac{A}{2}(t-t_{0}))-\sqrt{3}\kappa{\bar{\eta}}\right]^{2}\,.$
(726)

Thus, we can see that in order to realize $\rho\rightarrow\infty$,
$t\rightarrow\infty$ is essential. This is the Little Rip scenario in the
viscous Universe. However, it is important to mention that the above Little
Rip scenario is dependent on the above EoS (721) with the constant $\eta(H)$.

Interestingly, one can also realize the Little Rip phenomenon in a more
generalized scenario where in addition $\eta(H)$ is not strictly a constant.
To illustrate this, we consider the following example:

$\displaystyle
f(\rho)=\frac{A}{\sqrt{3}\kappa}\,\rho^{\nu+1/2},\qquad\xi(t)=\frac{b}{3\kappa^{2}}\,\rho^{\gamma}\,,$
(727)

where $\nu$, $b$ and $\gamma$ are constants.777One can quickly note that for
$\nu=0$ and $\gamma=-1/2$, we realize the previous scenario
$f(\rho)=A\sqrt{\rho}$ and $\eta(H)=\mathrm{constant}$.  Thus, for the above
choice, one can solve for the cosmic time as

$\displaystyle
t-t_{0}=\int_{\rho_{0}}^{\rho}\frac{d\rho}{\rho\,(A\,\rho^{\nu}+b\,\rho^{\gamma})}\,.$
(728)

which can be expressed after integration in terms of the hypergeometric
function as

$\displaystyle t-t_{0}=\Bigg{[}\frac{\nu\,\rho^{-\gamma}}{\gamma
b(\gamma-\nu)}+\frac{\rho^{-\gamma}}{\gamma
b(\gamma-\nu)}\left\\{-\nu+(\nu-\gamma)\,{}_{2}F_{1}\left(1,\frac{\gamma}{\gamma-\nu},1+\frac{\gamma}{\gamma-\nu},-\frac{A}{\beta}\rho^{\nu-\gamma}\right)\right\\}\Bigg{]}\,.$
(729)

For constant bulk viscous coefficient, i.e., $\xi(t)=\mathrm{constant}$
(equivalently, $\gamma=0$ in Eq. (727)) together with $\nu=0$ in Eq. (727), we
obtain (note that in that case $\nu$ and $\gamma$ have units of the Planck
mass):

$\displaystyle t-t_{0}=\frac{1}{(A+b)}\ln\frac{\rho}{\rho_{0}}\,,$ (730)

and this presents the Little Rip phenomenon because $\rho\rightarrow\infty$
requires $t\rightarrow\infty$, that means the energy density does not diverge
at some finite-time Brevik:2011mm .

###  IX.2 Geometrical invariants to remove the finite-time future
singularities

In Section IV we have seen that different modified gravity theories may lead
to a variety of finite time future singularities. However, with the proper
choice of the geometrical invariants describing the underlying gravitational
theory, it is possible to remove such finite-time singularities. In this
section we shall describe briefly how one can avoid such finite time future
singularities from various modified gravity theories.

In the context of $F(R)$ gravity theory, one can reconstruct the $F(R)$
gravity models in such a way so that the finite-time future singularities may
disappear from the picture. However, such reconstructions should be consistent
with the existing theories of the Universe and the reconstructed $F(R)$ model
should pass the essential tests of gravity. This point was first raised in
Ref. Abdalla:2004sw  and subsequently discussed in Refs. Nojiri:2008fk ;
Bamba:2008ut . We recall that the general action of $F(R)$ gravity in the
Jordan frame that depends solely on the curvature [i.e. Eq. (174) without
matter] having the form

$\displaystyle S=\int d^{4}x\sqrt{-g}\;\frac{F(R)}{2\kappa^{2}}\,,$ (731)

Introducing the auxiliary fields $A$, $B$, one can rewrite the action (731) as
Eq. (221) and consequently as Eq. (222). Now, using the conformal
transformation $g_{\mu\nu}=e^{\sigma}g_{\mu\nu}$ where $\sigma=-\ln
F^{\prime}(A)$, one can obtain the action in the Einstein frame in Eq. (225)
and the corresponding potential $V(A)$ in Eq. (226) [see section IV.5.1 for
more details].

Dealing with the model

$\displaystyle F(R)=A-\gamma R^{-n}+\eta R^{2}.$ (732)

we consider the following case $-1<n<-\frac{1}{2}$, $\gamma<0$, and $\eta>0$
and with these choices, the field $\sigma$ turns out to be
$\sigma=-\ln\left(1+n\gamma A^{-n-1}+2\eta A\right)$ where $1+n\gamma
A^{-n-1}+2\eta A>0$, and the potential has the following form

$\displaystyle V(A)=\frac{(n+1)\gamma A^{-n}+\eta A^{2}}{\left(1+n\gamma
A^{-n-1}+2\eta A\right)^{2}}\ .$ (733)

Now, since

$\displaystyle\frac{d\sigma}{dA}=-\frac{F^{\prime\prime}(A)}{F^{\prime}(A)}=-\frac{-n(n+1)\gamma
A^{-n-2}+2\eta}{1+n\gamma A^{-n-1}+2\eta A},$ (734)

then there is a branch point, at
$A=A_{0}\equiv\left\\{\frac{n(n+1)\gamma}{2\eta}\right\\}^{\frac{1}{n+2}}$ or
$\sigma=\sigma_{0}\equiv-\ln\left(1+(n+2)\left(\frac{2\eta}{n+1}\right)^{\frac{n+1}{n+2}}(n\gamma)^{\frac{1}{n+2}}\right)$
where $\frac{d\sigma}{dA}=0$ or $F^{\prime\prime}(A)=0$. For small values of
$A$, the behavior of the potential $V(A)$ approaches to
$\frac{A^{n+2}}{(n+1)\gamma}$. However, when $A$ is very large, then $V(A)$
approaches to a constant, $V(A)\to\frac{1}{4\eta}$. Also, the potential $V(A)$
vanishes at
$A=A_{1}\equiv\left\\{-\frac{(n+1)\gamma}{\eta}\right\\}^{1/(n+2)}$ (since
$0<\frac{n\gamma}{2}<-\gamma$, hence, $A_{0}<A_{1}$). Further, looking at the
expression for $V^{\prime}(A)$ given by

$\displaystyle V^{\prime}(A)=\frac{\left\\{-n(n+1)\gamma
A^{-n-2}+2\eta\right\\}A\left\\{1-(n+2)\gamma
A^{-n-1}\right\\}}{\left(1+n\gamma A^{-n-1}+2\eta A\right)^{3}},$ (735)

one can notice that $V^{\prime}(A)$ has an extremum at $A=A_{0}$. Now, as
there is a branch point at $\sigma=\sigma_{0}$, then if we start from the
small curvature, the growth of the curvature stops at $R=A_{0}$, where
$\sigma=\sigma_{0}$. In fact, at the branch point, where
$h^{\prime\prime}(A)=0$, the mass
$m_{\sigma}~{}(\propto\frac{d^{2}V}{d\sigma^{2}})$ of $\sigma$ becomes
infinite since

$\displaystyle\frac{d^{2}V}{d\sigma^{2}}=\frac{F^{\prime}(A)}{F^{\prime\prime}(A)}\frac{d}{dA}\left(\frac{F^{\prime}(A)}{F^{\prime\prime}(A)}\frac{dV(A)}{dA}\right)=-\frac{3}{F^{\prime\prime}(A)}+\frac{A}{F^{\prime}(A)}+\frac{2F(A)}{F^{\prime}(A)^{2}}\to+\infty.$
(736)

Note also that $F^{\prime\prime}(0)<0$ when $A<A_{0}$. Then the growth of
$\sigma$ is finished at $\sigma=\sigma_{0}$. Hence, with the addition of the
$R^{2}$ term, cosmic doomsday does not occur but the Universe ends up in a de
Sitter phase.

The same situation arises for the Hořava–Lifshitz $F(R)$ gravity too. As
explored in Ref. Carloni:2010nx , the Hořava–Lifshitz $F(R)$ gravity has a
quite rich cosmological structure and the inclusion of $R^{2}$ in $F(R)$ could
avoid the finite time future singularities from the picture.

On the other hand, recalling the $F(T)$ gravitational theory, it is also
possible to construct some models of $F(T)$ gravity where the finite-time
future singularities can be avoided. We consider the correction of the power
law model $F(T)=AT^{\alpha}$ of Eq.(411) as follows

$\displaystyle F(T)=AT^{\alpha}+BT^{\delta},$ (737)

where $BT^{\delta}$ ($\delta\neq 0$, $B\neq 0$) is the correction term. This
model has some interesting features. For $\delta>1$, all four types finite
time future singularities can be avoided in this model Bamba:2012vg . We
emphasize that the model $F(T)=AT^{\alpha}+BT^{2}$ where $T^{2}$ is treated as
the correction term, which is obtained for $\delta=2$, the minimum integer
satisfying the condition $\delta>1$, therefore removes all four types of
finite time future singularities. This has similarity with the $F(R)$ gravity
theory where it has been observed that the inclusion of $R^{2}$ term could
cure the finite time future singularities Abdalla:2004sw ; Nojiri:2008fk ;
Bamba:2008ut . Apart from the above models, there are other type of models
which can avoid the finite-time future singularities:

  * •

Exponential model: In the exponential model of type

$\displaystyle F(T)=C\exp(\lambda T),$ (738)

where $C$ and $\lambda$ are nonzero constants, we do not realize any finite-
time future singularities Bamba:2012vg .

  * •

Logarithmic model: In the logarithmic model of the form

$\displaystyle F(T)=D\ln\left(\gamma T\right),$ (739)

where $D~{}(\neq 0)$ and $\lambda~{}(>0)$ are constants, we do not realize any
finite-time future singularities Bamba:2012vg .

In the context of non-local gravity (section IV.9), one can also avoid the
finite time future singularities. We consider the scenario where a correction
term of the form $uR^{2}/\left(2\kappa^{2}\right)$ where $u~{}(\neq 0)$ is any
real number, is added to the action (414) as follows Bamba:2012ky :

$\displaystyle S=\int
d^{4}x\sqrt{-g}\Bigg{[}\frac{1}{2\kappa^{2}}\bigg{\\{}R\left(1+f(\Box^{-1}R)\right)+uR^{2}-2\Lambda\bigg{\\}}+\mathcal{L}_{\mathrm{matter}}\left(Q;g\right)\Bigg{]}\,.$
(740)

Along with the two fields $\eta$, $\xi$ as in section IV.9, introducing a
scalar field $\zeta$. the action in Eq. (740) can be expressed as

$S=\int
d^{4}x\sqrt{-g}\Bigg{[}\frac{1}{2\kappa^{2}}\bigg{\\{}R\left(1+f(\eta)\right)-\partial_{\mu}\xi\partial^{\mu}\eta-\xi
R+u\left(2\zeta
R-\zeta^{2}\right)-2\Lambda\bigg{\\}}+\mathcal{L}_{\mathrm{matter}}\Bigg{]}\,.$
(741)

Note that by varying action (741) with respect to $\zeta$, one obtains
$\zeta=R$ and hence by substituting $\zeta=R$ in (741), one gets back (740).

Now, in the context of a flat FLRW universe, the gravitational field equations
for the action (740) can be written as Bamba:2012ky

$\displaystyle-3H^{2}\left(1+f(\eta)-\xi\right)+\frac{1}{2}\dot{\xi}\dot{\eta}-3H\left(f^{\prime}(\eta)\dot{\eta}-\dot{\xi}\right)+\Theta+\Lambda+\kappa^{2}\rho_{\mathrm{m}}=0\,,$
(742)
$\displaystyle\left(2\dot{H}+3H^{2}\right)\left(1+f(\eta)-\xi\right)+\frac{1}{2}\dot{\xi}\dot{\eta}+\left(\frac{d^{2}}{dt^{2}}+2H\frac{d}{dt}\right)\left(f(\eta)-\xi\right)+\Xi-\Lambda+\kappa^{2}p_{\mathrm{m}}=0\,,$
(743)

where $\rho_{\mathrm{m}}$, $p_{\mathrm{m}}$ are respectively the energy
density and pressure of the matter sector; $\Theta$ and $\Xi$ are respectively
the contributions from the additional term $uR^{2}/\left(2\kappa^{2}\right)$
given by

$\displaystyle\Theta\equiv
u\left(-6H^{2}R+\frac{1}{2}R^{2}-6H\dot{R}\right)=18u\left(-6H^{2}\dot{H}+\dot{H}^{2}-2H\ddot{H}\right)\,,$
(744) $\displaystyle\Xi\equiv
u\left[2\left(2\dot{H}+3H^{2}\right)R-\frac{1}{2}R^{2}+2\ddot{R}+4H\dot{R}\right]=6u\left(9\dot{H}^{2}+18H^{2}\dot{H}+2\dddot{H}+12H\ddot{H}\right)\,.$
(745)

Now, for the Hubble parameter as in (63) or (114), in the limit $t\to
t_{\mathrm{s}}$, $\Theta$ of Eq. (744) and $\Xi$ of Eq. (745) are
approximately given by

$\displaystyle\Theta\sim
18u\left[-6h_{\mathrm{s}}^{2}\beta\left(t_{\mathrm{s}}-t\right)^{-\left(3\beta+1\right)}+h_{\mathrm{s}}^{2}\beta^{2}\left(t_{\mathrm{s}}-t\right)^{-2\left(\beta+1\right)}-2h_{\mathrm{s}}^{2}\beta\left(\beta+1\right)\left(t_{\mathrm{s}}-t\right)^{-2\left(\beta+1\right)}\right]\,,$
(746) $\displaystyle\Xi\sim
6u\left[9h_{\mathrm{s}}^{2}\beta^{2}\left(t_{\mathrm{s}}-t\right)^{-2\left(\beta+1\right)}+18h_{\mathrm{s}}^{3}\beta\left(t_{\mathrm{s}}-t\right)^{-\left(3\beta+1\right)}+2h_{\mathrm{s}}\beta\left(\beta+1\right)\left(\beta+2\right)\left(t_{\mathrm{s}}-t\right)^{-\left(\beta+3\right)}+12h_{\mathrm{s}}^{2}\beta\left(\beta+1\right)\left(t_{\mathrm{s}}-t\right)^{-2\left(\beta+1\right)}\right]\,.$
(747)

We now examine the r.h.s. of Eq. (416) with $\Theta$ in Eq. (744). For
$\beta>1$, $\sigma<0$, the first term of Eq. (746), i.e.
$-108uh_{\mathrm{s}}^{2}\beta\left(t_{\mathrm{s}}-t\right)^{-\left(3\beta+1\right)}$,
becomes the leading term. As $u\neq 0$, $h_{\mathrm{s}}\neq 0$ and $\beta\neq
0$, therefore, this leading term does not vanish, which means that the
additional $R^{2}$ term can remove the finite-time future singularities. On
the other hand, for $-1<\beta<0\,,\,0<\beta<1$, the second and third terms of
Eq. (746) become the leading terms. Again since $u\neq 0$, $h_{\mathrm{s}}\neq
0$, $\beta\neq 0$ and $\beta\neq-2$, hence, these leading terms do not vanish.
This means that the additional $R^{2}$ term can remove the finite-time future
singularities.

We remark that similar to the $F(R)$ gravity, where the inclusion of a
correction term $R^{2}$ could cure the singularities Bamba:2012ky , in the
non-local gravity too, this situation happens.

###  IX.3 Scalar field models avoiding finite-time future singularities

According to Carroll:2003st , it is possible to construct scalar field phantom
DE models where the big rip singularity can be avoided. Considering a phantom
scalar field $\phi$ with potential $V(\phi)$ as in Carroll:2003st , in the
background of a FLRW universe, the energy density and pressure of the phantom
field, are respectively given by Copeland:2006wr

$\displaystyle\rho_{\phi}=-\dot{\phi}^{2}/2+V(\phi),\quad
p_{\phi}=-\dot{\phi}^{2}/2-V(\phi),$ (748)

and the equation of state of this phantom field, $w=p_{\phi}/\rho_{\phi}$
takes the form

$\displaystyle
w_{\phi}=\frac{\dot{\phi}^{2}/2+V(\phi)}{\dot{\phi}^{2}/2-V(\phi)},$ (749)

which satisfies $w_{\phi}<-1$ if $\dot{\phi}^{2}/2<V(\phi)$. The equation of
motion of the phantom field is given by

$\displaystyle\ddot{\phi}+3H\dot{\phi}-a^{-2}\nabla^{2}\phi-V^{\prime}(\phi)=0,$
(750)

where prime denotes the derivative with respect to $\phi$ and $\nabla^{2}$
denotes the spatial Laplacian, i.e.
$\nabla^{2}\phi=\partial_{x}^{2}\phi+\partial_{y}^{2}\phi+\partial_{x}^{2}\phi$.
In this model scenario, the dynamics of the universe heavily depends on the
potential function $V(\phi)$. As argued in Carroll:2003st , the following
gaussian potential has some interesting consequences in the context of finite
time singularities:

$\displaystyle V(\phi)=V_{0}e^{-(\phi^{2}/\sigma^{2})}\ ,$ (751)

where $V_{0}$ and $\sigma$ are constants. For this particular choice of the
potential, the evolution of the scalar field, the density parameter of phantom
scalar field and its equation of state have been numerically investigated in
Carroll:2003st . From the evolution of the equation of state $w_{\phi}$, one
finds that Carroll:2003st : during the initial stages of evolution, the
phantom scalar field is frozen by the expansion and it behaves like the
cosmological constant (i.e. $w_{\phi}\simeq-1$). After that the field starts
to evolve quite rapidly towards the maximum of its potential and the energy
density of the phantom field becomes dominant where $w_{\phi}$ crosses the
phantom divide line $w_{\phi}=-1$ and becomes more negative. Finally, in the
very late phase of the universe, the field comes to rest at the maximum of the
potential and the accelerating expansion begins with $w_{\phi}=-1$. In the
late phase, as $w_{\phi}$ does not go beyond $-1$, hence no future singularity
appears. In fact, the universe approaches to a de Sitter phase.

On the other hand, it is interesting to note that for $F(R)=R+\alpha R^{2}$
(i.e. the Starobinksy model Starobinsky:1980te ) where $\alpha$ is a
dimensionless constant, one can derive the corresponding quintessence scalar
field potential of exponential type Chaichian:2022apa . As due to the presence
of $R^{2}$ in $F(R)$, no finite-time future singularities appear
Abdalla:2004sw , therefore, we conclude that quintessence scalar field model
with exponential potential could avoid the finite-time future singularities.

###  IX.4 Inhomogeneous equation of state

In the context of various modified gravity theories, using the effective
energy density $\rho_{\rm eff}$ and pressure $p_{\rm eff}$, one can derive an
effective equation of state $p_{\rm eff}-w\rho_{\rm
eff}=G(H,\dot{H},\ddot{H})$, see Eq. (178) for the $F(R)$ gravity case [while
this is true for other gravity theories as well] in which
$G(H,\dot{H},\ddot{H})=-\frac{1}{\kappa^{2}}\left(2\dot{H}+3(1+w)H^{2}\right)$,
see Eq. (179). The explicit form of $G(H,\dot{H},\ddot{H})$ that involves the
geometric invariants and their time derivatives is given in Eq. (IV.5).
Therefore, the modified gravity theories can lead to various inhomogeneous
equation of state parameters depending on the choice of the underlying
gravitational theory. In this section we shall investigate the conditions on
$G(H,\dot{H},\ddot{H})$ that may prevent the appearance of finite time future
singularities. If Eq. (179) is found to be inconsistent for kind of finite
time future singularities, then one can conclude that no finite time future
singularities are realized in this case.

We consider the case when $H$ evolves as (115). In that case, the r.h.s. of
(179) behaves as Bamba:2008ut

$\displaystyle
G(H,\dot{H},\ddot{H})=-\frac{1}{\kappa^{2}}\left(2\dot{H}+3(1+w)H^{2}\right)\sim\left\\{\begin{array}[]{lcl}-\frac{3(1+w)h_{s}^{2}}{\kappa^{2}}\left(t_{0}-t\right)^{-2\beta}&\mbox{when}&\beta>1\\\
-\frac{2\beta
h_{s}+3(1+w)h_{s}^{2}}{\kappa^{2}}\left(t_{0}-t\right)^{-2}&&0<\beta<1\\\
-\frac{2\beta
h_{s}}{\kappa^{2}}\left(t_{0}-t\right)^{-\beta-1}&&0>\beta>-1\end{array}\right.\
.$ (755)

Now for $\beta>-1$, $\beta\neq 0$, which correspond to Type I, II and III
singularities, notice that the l.h.s. of (755), i.e.,
$G\left(H,\dot{H},\cdots\right)$ of (179) diverges. Thus, in order to avoid
the appearance of such finite time singularities, $G$ must be bounded and in
this case, Eq. (179) becomes inconsistent with the behavior of the r.h.s. of
(755).

An example in this direction is the following Bamba:2008ut

$\displaystyle
G\left(H,\dot{H},\cdots\right)=G_{0}\left(\frac{1+aH^{2}}{1+bH^{2}}\right)\,$
(756)

where $a$ and $b$ are positive real numbers and $G_{0}$ is a constant.

We further note that for $\beta>0$ corresponding to Type I or III singularity,
the r.h.s. of (755) becomes negative. Hence, if
$G\left(H,\dot{H},\cdots\right)$ is positive for large $H$, we do not realize
any finite time future singularity.

Another chance is that if $G\left(H,\dot{H},\cdots\right)$ contains the term
like $\sqrt{1-a^{2}H^{2}}$, which becomes imaginary for large $H$, then Eq.
(755) becomes inconsistent. Thus, singularities where the curvature blows up
(Type I, II, III) could be avoided.

We remark that such mechanism could be applied even if we have the phantom
EoS, i.e. $w<-1$. In that case one can add an extra term
$G_{1}(H)=G_{0}\left(\sqrt{1-H^{2}/H_{0}^{2}}-1\right)$ to
$G\left(H,\dot{H},\cdots\right)$. Here, $G_{0}$ and $H_{0}$ are real numbers.
If $H_{0}$ is considered to be large enough, $G_{1}(H)$ is not relevant for
the small curvature but relevant for large scale and hence the possibility of
the curvature singularity is avoided.

###  IX.5 Future Singularities with the Account of Quantum Effects

In this subsection we will extend our analysis of singularities performed when
we have studied semi-classical gravity. It is well known that, when the
Universe approaches the future singularity, its curvature and other
geometrical invariants grow up. As a result, the quantum effects may change
the behavior of the future spacetime singularity. For example, one can show
that quantum effects may change the structure of future singularity, see
Nojiri:2004ip ; Nojiri:2004pf  (see also Kamenshchik:2013naa ; Bates:2010nv ;
Tretyakov:2005en ; Calderon:2004bi ; Carlson:2016iuw ). We will use simple
qualitative arguments of Ref. Nojiri:2010wj  to show the role of quantum
effects in conformally-invariant theories to future singularity.

It is well-known that the generalized conformal anomaly $T_{A}$ has the
following form:

$\displaystyle T_{A}=b\left(\mathcal{F}+\frac{2}{3}\Box
R\right)+b^{\prime}{G}+b^{\prime\prime}\Box R\,,$ (757)

where ${G}$ is the Gauss-Bonnet invariant in (161) and $\mathcal{F}$ denotes
the square of the 4D Weyl tensor, given by

$\displaystyle\mathcal{F}=\frac{1}{3}R^{2}-2R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\,.$
(758)

In case when matter is conformally-invariant and there appear $N_{0}$ scalars,
$N_{1/2}$ spinors, $N_{1}$ vector fields, $N_{2}$ ($=1$) gravitons, and
$N_{\mathrm{HD}}$ higher-derivative conformal scalars, $b$ and $b^{\prime}$,
which are obtained using adiabatic regularization, take the following forms,

$\displaystyle
b=\frac{N_{0}+6N_{1/2}+12N_{1}+611N_{2}-8N_{\mathrm{HD}}}{120(4\pi)^{2}}\,,\quad
b^{\prime}=-\frac{N_{0}+11N_{1/2}+62N_{1}+1411N_{2}-28N_{\mathrm{HD}}}{360(4\pi)^{2}}\,.$
(759)

As given in (759), for the usual matter, $b$ is positive and $b^{\prime}$ is
negative. An exception is the higher-derivative conformal scalar. Note that
the value of $b^{\prime\prime}$ can always be shifted with the addition of
$R^{2}$ to the classical action.

If one writes the energy density $\rho_{A}$ and pressure $p_{A}$ corresponding
to the trace anomaly $T_{A}$, then one has $T_{A}=-\rho_{A}+3p_{A}$. Now with
the use of the energy conservation law in the FLRW Universe, one gets

$\displaystyle 0=\frac{d\rho_{A}}{dt}+3H\left(\rho_{A}+p_{A}\right)\,,$ (760)

which can be expressed by eliminating $p_{A}$ as

$\displaystyle T_{A}=-4\rho_{A}-\frac{1}{H}\frac{d\rho_{A}}{dt}\,,$ (761)

and with the help of integration, $\rho_{A}$ can be found as Nojiri:2005sx :

$\displaystyle\rho_{A}=-\frac{1}{a^{4}}\int dta^{4}HT_{A}\,.$ (762)

Now, using the above expression and identifying
$\rho_{\mathrm{eff}}=\rho_{A}$, we will consider the FLRW equation, later.
Before considering the FLRW equation, as in Nojiri:2010wj , however, we first
consider the trace of the Einstein equation, for simplicity, by including the
trace anomaly, as follows,

$\displaystyle
R=-\frac{\kappa^{2}}{2}\left(T_{\mathrm{matter}}+T_{A}\right)\,.$ (763)

Here, $T_{\mathrm{matter}}$ refers to the trace of the matter energy-momentum
tensor. Now, for the FLRW Universe, $\mathcal{F}$ and ${G}$ are as follows

$\displaystyle\mathcal{F}=0\,,\quad{G}=24\left(\dot{H}H^{2}+H^{4}\right)\,.$
(764)

What we would like to show is that, if there is a singularity, then the trace
equation (764) cannot be consistent. In particular, we will show that the
contribution from the conformal anomaly in the r.h.s. of Eq. (763) is more
singular than the scalar curvature in the l.h.s. of Eq. (763). Note that a
rigorous study on the quantum effects may be done following Carlson:2016iuw ,
however, this needs an extensive numerical investigations depending on the
particles content of the Universe as well as effective dark fluid.

Now we assume that $H$ behaves as in (114) or (115) and neglect the
contribution from matter and put $T_{\mathrm{matter}}=0$. In the case of Type
I singularity, the scalar curvature $R$, which is given by
$R=12H^{2}+6\dot{H}$, $R$ behaves as $R\sim\left(t_{s}-t\right)^{-2\beta}$ and
in the case of Type III singularity, $R$ behaves as
$R\sim\left(t_{s}-t\right)^{-\beta-1}$. On the other hand, in the case of Type
II singularity, $R$ behaves as $R\sim\left(t_{s}-t\right)^{-\beta-1}$. When
Type IV singularity appears, if $H_{s}(t)\neq 0$ in (115), $R$ is finite but
if $H_{s}(t)=0$, $R\sim\left(t_{s}-t\right)^{-\beta-1}$.

In the case of Type I singularity, near the singularity, $t\sim t_{s}$, as
seen from Eq. (764), the Gauss-Bonnet invariant ${G}$ behaves as ${G}\sim
24H^{4}\sim\left(t_{s}-t\right)^{-4\beta}$ and therefore ${G}$ becomes very
large and the contribution from the matter $T_{\mathrm{matter}}$ in (763) can
be neglected. On the other hand, one finds $\Box
R\sim\left(t_{s}-t\right)^{-2\beta-2}$. Then since
$R\sim\left(t_{s}-t\right)^{-2\beta}$, $T_{A}$ becomes much larger than $R$
and therefore Eq. (763) cannot be satisfied. This shows that the quantum
effects coming from the conformal anomaly remove Type I singularity.

In the case of Type II singularity, we find that ${G}$ behaves as ${G}\sim
24\dot{H}H^{2}\sim\left(t_{s}-t\right)^{-3\beta-1}$. Since
$R\sim\left(t_{s}-t\right)^{-\beta-1}$, the Gauss-Bonnet term in $T_{A}$ is
less singular and therefore negligible compared with $R$ and the contribution
from the matter. Therefore, the Gauss-Bonnet term in $T_{A}$ does not help to
prevent Type II singularity. Note, however, $\Box R$ behaves as $\Box
R\sim\left(t_{s}-t\right)^{-\beta-3}$, which is more singular than the scalar
curvature. Then if $2b/3+b^{\prime\prime}\neq 0$, the contribution from
$T_{A}$ becomes much larger than $R$ near the singularity $t\sim t_{s}$ and
Eq. (763) cannot be satisfied. Therefore, if $2b/3+b^{\prime\prime}\neq 0$,
even Type II singularity can also be prevented when the quantum effects due to
conformal anomaly are included.

In the case of Type III singularity, the Gauss-Bonnet invariant behaves as
${G}\sim 24\dot{H}H^{2}\sim\left(t_{s}-t\right)^{-3\beta-1}$ and $\Box R$
behaves as $\Box R\sim\left(t_{s}-t\right)^{-\beta-3}$. Because the scalar
curvature behaves as $R\sim\left(t_{s}-t\right)^{-\beta-1}$, both of the
terms, $\Box R$ and ${G}$, are more singular than the scalar curvature $R$ and
Type III singularity is also prevented. Thus, we demonstrated that quantum
effects may remove finite-time future singularities. Note that account of
quantum gravity effects in specific models is also known to remove the Big Rip
singularity Elizalde:2004mq .

###  IX.6 FLRW equation including Trace Anomaly

Here we consider the FLRW equation including the effective energy density
$\rho_{A}$ induced by the trace anomaly in (762). By using (757), (764), and
(762), we find the explicit form of $\rho_{A}$ as follows

$\displaystyle\rho_{A}=-\frac{1}{a^{4}}\int
dta^{4}H\left\\{24b^{\prime}\left(\dot{H}H^{2}+H^{4}\right)-6\left(b+b^{\prime\prime}\right)\left(\frac{d^{2}}{dt^{2}}+3H\frac{d}{dt}\right)\left(\dot{H}+2H^{2}\right)\right\\}\,.$
(765)

Then the FLRW equation is given by

$\displaystyle\frac{3}{\kappa^{2}}H^{2}=\rho+\rho_{A}\,.$ (766)

Here $\rho$ is the energy density corresponding to the phantom DE, which
behaves as $\rho\sim\rho_{0}a^{-3\left(1+w\right)}$ with $w<-1$. By combining
(765) and (766) and assuming $\rho=\rho_{0}a^{-3\left(1+w\right)}$ (we take
$a_{0}=1$), we obtain

$\displaystyle\frac{3}{\kappa^{2}}H^{2}a^{4}=\rho_{0}a^{1-3w}-\int
dta^{4}H\left\\{24b^{\prime}\left(\dot{H}H^{2}+H^{4}\right)-6\left(b+b^{\prime\prime}\right)\left(\frac{d^{2}}{dt^{2}}+3H\frac{d}{dt}\right)\left(\dot{H}+2H^{2}\right)\right\\}\,.$
(767)

Differentiating the above expression with respect to $t$, we obtain

$\displaystyle\frac{3}{\kappa^{2}}\left(\dot{H}+4H^{2}\right)=$
$\displaystyle\,\left(1-3w\right)\rho_{0}a^{-3\left(1+w\right)}-\left\\{24b^{\prime}\left(\dot{H}H^{2}+H^{4}\right)-6\left(b+b^{\prime\prime}\right)\left(\frac{d^{2}}{dt^{2}}+3H\frac{d}{dt}\right)\left(\dot{H}+2H^{2}\right)\right\\}\,.$
(768)

If we assume

$\displaystyle
H(t)\sim{\widetilde{h}}_{s}\left(t_{s}-t\right)^{\widetilde{\beta}}\,,$ (769)

with a positive constant ${\widetilde{h}}_{s}$, we find

$\displaystyle\dot{H}\sim\left(t_{s}-t\right)^{\widetilde{\beta}-1}\,,\quad
H^{2}\sim\left(t_{s}-t\right)^{2\widetilde{\beta}}\,,\quad
a^{-3\left(1+w\right)}\sim\left\\{\begin{array}[]{ll}\mathrm{e}^{\frac{3\left(1+w\right)}{\widetilde{\beta}+1}{\widetilde{h}}_{s}\left(t_{s}-t\right)^{\widetilde{\beta}+1}}&\mbox{if
$\widetilde{\beta}\neq-1$}\\\
\left(t_{s}-t\right)^{3\left(1+w\right){\widetilde{h}}_{s}}&\mbox{if
$\widetilde{\beta}=-1$}\end{array}\right.\,,$ (772)
$\displaystyle\dot{H}H^{2}\sim\left(t_{s}-t\right)^{3\widetilde{\beta}-1}\,,\quad
H^{4}\sim\left(t_{s}-t\right)^{4\widetilde{\beta}}\,,\quad\frac{d^{2}\dot{H}}{dt^{2}}\sim\left(t_{s}-t\right)^{\widetilde{\beta}-3}\,,\quad
H\frac{d\dot{H}}{dt}\sim\left(t_{s}-t\right)^{2\widetilde{\beta}-2}\,,$
$\displaystyle\frac{d^{2}H^{2}}{dt^{2}}\sim\left(t_{s}-t\right)^{2\widetilde{\beta}-2}\,,\quad
H\frac{dH^{2}}{dt}\sim\left(t_{s}-t\right)^{3\widetilde{\beta}-1}\,.$ (773)

By using the above behaviors, we investigate what kind of singularity can be
realized or prohibited.

First, we consider the case of Type I singularity, where
$\widetilde{\beta}\leq-1$. If $\widetilde{\beta}<-1$, the energy density
$\rho=\rho_{0}a^{-3\left(1+w\right)}$ of matter grows up very rapidly when
$t\to t_{s}$, other terms in (768) cannot cancel the term and therefore Eq.
(768) is not satisfied. If $\widetilde{\beta}=-1$, the l.h.s. of (768) behaves
as

$\displaystyle\frac{3}{\kappa^{2}}\left(\dot{H}+4H^{2}\right)\sim\frac{3}{\kappa^{2}}\left({\widetilde{h}}_{s}+4\widetilde{h}_{s}^{2}\right)\left(t_{s}-t\right)^{-2}>0\,.$
(774)

On the other hand, the first term in the r.h.s. of (768), which comes from the
matter, as found in (772),

$\displaystyle\left(1-3w\right)\rho_{0}a^{-3\left(1+w\right)}\sim\left(1-3w\right)\rho_{0}\left(t_{s}-t\right)^{3\left(1+w\right){\widetilde{h}}_{s}}>0\,,$
(775)

and the second term in the r.h.s. of (768), which comes from the trace
anomaly, behaves as

$\displaystyle-\left\\{24b^{\prime}\left(\dot{H}H^{2}+H^{4}\right)-6\left(b+b^{\prime\prime}\right)\left(\frac{d^{2}}{dt^{2}}+3H\frac{d}{dt}\right)\left(\dot{H}+2H^{2}\right)\right\\}$
$\displaystyle=$
$\displaystyle\,-24b^{\prime}\left(\widetilde{h}_{s}^{3}+\widetilde{h}_{s}^{4}\right)+6\left(b+b^{\prime\prime}\right)\left(6{\widetilde{h}_{s}}+18\widetilde{h}_{s}^{2}+12\widetilde{h}_{s}^{3}\right)\left(t_{s}-t\right)^{-4}\,.$
(776)

Because $b>0$ and $b^{\prime}<0$ as in (759) and $b^{\prime\prime}$ is
arbitrary, as long as $b+b^{\prime\prime}\geq 0$, the second term in the
r.h.s. of (768) is positive. Compared with (774) and (IX.6), we find that the
l.h.s. of (768) cannot balance with the second term in the r.h.s. of (768).
Furthermore, because both of the first and second terms of (768) are positive
as $b+b^{\prime\prime}\geq 0$, they cannot cancel each other. This tells that
the behavior with $\widetilde{\beta}=-1$ does not satisfy Eq. (768). By
combining the results with $\widetilde{\beta}<-1$ and $\widetilde{\beta}=-1$,
we find that the Type I singularity is prohibited.

We now consider Type III singularity, where $-1<\widetilde{\beta}<0$. We
should note that in Eq. (768), in the limit of $t\to t_{s}$, the first term in
the r.h.s. is finite and the l.h.s. and the second term in the r.h.s. diverge.
Therefore we may neglect the first term in the r.h.s. of (768). Because now
$2\widetilde{\beta}>\widetilde{\beta}-1>2\widetilde{\beta}-1$, the l.h.s.
cannot balance with the second term in the r.h.s. and therefore (768) cannot
be satisfied and the Type III singularity is prohibited.

In case of Type II, where $0<\widetilde{\beta}<-1$, n the limit of $t\to
t_{s}$, the first term in the r.h.s. is finite, again. On the other hand, in
the limit, the l.h.s. (768) diverges as
$\left(t_{s}-t\right)^{\widetilde{\beta}-1}$ and the second term in the r.h.s.
behaves as $\left(t_{s}-t\right)^{\widetilde{\beta}-3}$ if
$b+b^{\prime\prime}\geq 0$, or $\left(t_{s}-t\right)^{3\widetilde{\beta}-1}$
if $b+b^{\prime\prime}=0$, the l.h.s. cannot balance with the second term in
the r.h.s. and therefore (768) cannot be satisfied, again and the Type II
singularity is prohibited.

By combining the above results, we find the Type I, II, and III singularities
are prohibited by the quantum effects coming from the trace anomaly. The Type
IV singularity can be allowed in general when there is any matter.

When the matter can be neglected, to see what could happen, we solve (768) by
putting $\rho_{0}=0$ and by adjusting $b^{\prime\prime}$ so that
$b+b^{\prime\prime}=0$. Then we can rewrite (768) as follows,

$\displaystyle 1=$
$\displaystyle-\frac{\dot{H}\left(1+8\kappa^{2}b^{\prime}H^{2}\right)}{4H^{2}\left(1+2\kappa^{2}b^{\prime}H^{2}\right)}\,,$
(777)

which can be integrated to be

$\displaystyle
t-t_{s}=\frac{1}{4H}+\frac{3\kappa\sqrt{-2b^{\prime}}}{8}\ln\left|\frac{1+\kappa\sqrt{-2b^{\prime}}H}{1-\kappa\sqrt{-2b^{\prime}}H}\right|\,.$
(778)

The solution has two branches for positive $H$, that is, the region
$0<H<\frac{1}{\kappa\sqrt{-2b^{\prime}}}$ and
$\kappa\sqrt{-2b^{\prime}}<H<+\infty$. When $H\to 0$, $t$ goes to the plus
infinity and $H\to\sqrt{-2b^{\prime}}$, $t$ goes to the plus infinity, again.
That is, there is a local minimum in the region
$0<H<\frac{1}{\kappa\sqrt{-2b^{\prime}}}$. In the region,
$\kappa\sqrt{-2b^{\prime}}<H<+\infty$, when $H\to\sqrt{-2b^{\prime}}$, $t$
goes to the plus infinity and after that $t$ monotonically decreases as a
function of $H$ and when $H\to+\infty$, $t-t_{s}$ vanishes as
$t-t_{s}\sim\frac{1}{H}$. Then there could be three scenarios for the
cosmology. In one scenario, the Universe starts from the finite $t$
corresponding to the local minimum in the region
$0<H<\frac{1}{\kappa\sqrt{-2b^{\prime}}}$. After that, there are two
possibilities. In one possibility, when $t$ increases, $H\to 0$ and in another
possibility, when $t$ increases, $H\to\frac{1}{\kappa\sqrt{-2b^{\prime}}}$. In
another scenario, the Universe starts at $t\to t_{s}$ with $H\to+\infty$.
After that, $H$ monotonically decreases and goes to
$H\to\frac{1}{\kappa\sqrt{-2b^{\prime}}}$ when $t\to+\infty$.

###  IX.7 Future Singularities with the Account of Thermal Effects

We may remind that the large Hubble rate $H$ means the large temperature of
the Universe. The Hawking radiation effectively should be generated at the
apparent horizon of the FLRW Universe Gibbons:1977mu ; Cai:2008gw .
Eventually, it should give an important contribution to the energy density of
the late-time Universe, especially right before the Rip time. In other words,
at a large temperature that may even diverge at the Rip time, there should
appear thermal radiation.

####  IX.7.1 Type I Singularity with Thermal Effects: Transition to Type II
Singularity

Near the Type I (Big Rip) singularity, the temperature of the Universe becomes
large and we may expect the generation of thermal radiation as in the case of
the Hawking radiation. The Hawking temperature $T$ is proportional to the
inverse of the radius $r_{\mathrm{H}}$ of the apparent horizon and the radius
$r_{\mathrm{H}}$ is proportional to the inverse of the Hubble rate $H$.
Therefore the temperature $T$ is proportional to the Hubble rate $H$. As well-
known in statistical physics, the energy-density $\rho_{\mathrm{t\\_rad}}$ of
the thermal radiation is proportional to the fourth power of the temperature.
Then when $H$ is large enough, we may assume that the energy-density of the
thermal radiation is given by

$\displaystyle\rho_{\mathrm{t\\_rad}}=\alpha H^{4}\,,$ (779)

with a positive constant $\alpha$. At the late time, the FLRW equation should
be modified by the account of thermal radiation,

$\displaystyle\frac{3}{\kappa^{2}}H^{2}=\rho+\alpha H^{4}\,.$ (780)

Here $\rho$ is the energy density corresponding to the phantom DE, which
behaves as $\rho\sim\rho_{0}a^{-3\left(1+w\right)}$ with $w<-1$. At the late-
time but much before the Big Rip time, the first term in the equation (780)
dominates and therefore the Universe expands to the Big Rip singularity, where
the Hubble rate $H$ behaves as in (114) with $\beta=1$. Then near the Big Rip
time $t_{s}$, the second term in (780) should dominate and we obtain

$\displaystyle\frac{3}{\kappa^{2}}H^{2}\sim\alpha H^{4}\,,$ (781)

whose non-trivial solution is given by

$\displaystyle H^{2}=H_{\mathrm{crit}}^{2}\equiv\frac{3}{\kappa^{2}\alpha}\,.$
(782)

As $H$ goes to a constant, we might expect that the spacetime goes to the
asymptotically de Sitter spacetime but it is not true. Even in the de Sitter
spacetime, the scale factor $a$ becomes larger and larger as an exponential
function of $t$, then the first term in the equation (780) should dominate
finally. The Hubble rate $H$ is, however, already larger than
$H_{\mathrm{crit}}$, there is no solution of (780). Then the Universe should
end up at the finite-time with some kind of the singularity.

For more quantitative analysis, we solve (780), with respect to $H^{2}$ as
follows,

$\displaystyle
H^{2}=\frac{\frac{3}{\kappa^{2}}\pm\sqrt{\frac{9}{\kappa^{4}}-4\alpha\rho_{0}a^{-3\left(1+w\right)}}}{2\alpha}\,.$
(783)

As $H^{2}$ is a real number, we find that there is a maximum for the scale
factor $a$,

$\displaystyle a\leq
a_{\mathrm{max}}\equiv\left(\frac{9}{4\kappa^{4}\alpha\rho_{0}}\right)^{-\frac{1}{3\left(1+w\right)}}\,.$
(784)

Then we consider the behavior of $a$ or $H$ around the maximal
$a=a_{\mathrm{max}}$ by writing the scale factor $a$ as

$\displaystyle a=a_{\mathrm{max}}\mathrm{e}^{N}\,.$ (785)

Here, $N$ corresponds to the $e$-folding number but $N$ should be negative
because $a<a_{\mathrm{max}}$. Furthermore, as we are interested in the region
$a\sim a_{\mathrm{max}}$, we assume $\left|N\right|\ll 1$. Then by using
$H=\frac{dN}{dt}$, Eq. (783) can be rewritten as

$\displaystyle\left(1\mp\frac{1}{2}\sqrt{3\left(1+w\right)N}\right)dN\sim
dt\sqrt{\frac{3}{2\alpha\kappa^{2}}}\,,$ (786)

which can be integrated as

$\displaystyle
N\mp\frac{1}{3}\left(-N\right)^{\frac{3}{2}}\sqrt{-3\left(1+w\right)}\sim-\left(t_{\mathrm{max}}-t\right)\sqrt{\frac{3}{2\alpha\kappa^{2}}}\,.$
(787)

Here $a=a_{\mathrm{max}}$ when $t=t_{\mathrm{max}}$. Because we are assuming
$\left|N\right|\ll 1$, Eq. (787) can be rewritten as

$\displaystyle
N\sim-\left(t_{\mathrm{max}}-t\right)\sqrt{\frac{3}{2\alpha\kappa^{2}}}\mp\frac{\sqrt{-3\left(1+w\right)}}{3}\left(\left(t_{\mathrm{max}}-t\right)\sqrt{\frac{3}{2\alpha\kappa^{2}}}~{}\right)^{\frac{3}{2}}\,.$
(788)

Because $H=\frac{dN}{dt}$, we find

$\displaystyle H\sim$
$\displaystyle\sqrt{\frac{3}{2\alpha\kappa^{2}}}\mp\frac{\sqrt{-3\left(1+w\right)}}{2}\left(\sqrt{\frac{3}{2\alpha\kappa^{2}}}\right)^{\frac{3}{2}}\left(t_{\mathrm{max}}-t\right)^{\frac{1}{2}}\,,$
$\displaystyle\dot{H}\sim$
$\displaystyle\mp\frac{\sqrt{-3\left(1+w\right)}}{4}\left(\sqrt{\frac{3}{2\alpha\kappa^{2}}}\right)^{\frac{3}{2}}\left(t_{\mathrm{max}}-t\right)^{-\frac{1}{2}}\,.$
(789)

Then in the limit $t\to t_{\mathrm{max}}$, although $H$ is finite but
$\dot{H}$ diverges. Therefore, the Universe ends up with Type II singularity
at $t=t_{\mathrm{max}}$. Thus, we demonstrated that the account of thermal
effects near the Big Rip singularity changes the Universe evolution to the
finite-time Type II singularity.

####  IX.7.2 Type III Singularity with the Account of Thermal Effects:
Transition to Type II Singularity

The scale factor which generates Type III singularity can be expressed as

$\displaystyle
a(t)=a_{s}\mathrm{e}^{-\frac{h_{s}}{1-\beta}\left(t_{s}-t\right)^{1-\beta}}\,,$
(790)

with $a_{s}$, $t_{s}$, $\beta$, and $h_{s}$ some constants. In order to
generate Type III singularity we restrict the value of $\beta$ as

$\displaystyle 0<\beta<1\,.$ (791)

Then the Hubble rate $H$ is given by

$\displaystyle H=h_{s}\left(t_{s}-t\right)^{-\beta}\,.$ (792)

Hence, in the limit $t\to t_{s}$, $H$ diverges but the scale factor $a$ is
finite. From Eq. (112) it follows

$\displaystyle\rho_{\mathrm{eff}}=\frac{3h_{s}^{2}}{\kappa^{2}}\left(t_{s}-t\right)^{-2\beta}\,,\quad
p_{\mathrm{eff}}=-\frac{1}{\kappa^{2}}\left(-2\beta
h_{s}\left(t_{s}-t\right)^{-\beta-1}+3h_{s}^{2}\left(t_{s}-t\right)^{-2\beta}\right)\,.$
(793)

By deleting $\left(t_{s}-t\right)$, we find the following EoS,

$\displaystyle
p_{\mathrm{eff}}=-\rho_{\mathrm{eff}}-\frac{2h_{s}\beta}{\kappa^{2}}\left(\frac{\kappa^{2}\rho_{\mathrm{eff}}}{3h_{s}^{2}}\right)^{\frac{\beta+1}{2\beta}}\,.$
(794)

Using (790) and (793), one gets

$\displaystyle\rho_{\mathrm{eff}}=\frac{3}{\kappa^{2}}h_{s}^{2}\Bigg{(}\frac{1-\beta}{h_{s}}\ln\left(\frac{a_{s}}{a(t)}\right)\Bigg{)}^{-\frac{2\beta}{1-\beta}}\,.$
(795)

With the account of the thermal radiation, instead of (780), we have

$\displaystyle\frac{3}{\kappa^{2}}H^{2}=A\left(\ln\left(\frac{a_{s}}{a(t)}\right)\right)^{-B}+\alpha
H^{4}\,,\quad
A\equiv\frac{3}{\kappa^{2}}\left(\frac{h_{s}^{2}}{(1-\beta)^{\beta}}\right)^{\frac{1}{1-\beta}}\,,\quad
B\equiv\frac{2\beta}{1-\beta}>0\,.$ (796)

Then instead of (783), we obtain

$\displaystyle
H^{2}=\frac{\frac{3}{\kappa^{2}}\pm\sqrt{\frac{9}{\kappa^{4}}-4\alpha
A\left(\ln\left(\frac{a_{s}}{a(t)}\right)\right)^{-B}}}{2\alpha}\,.$ (797)

Then in order that $H^{2}$ to be real, we find that there is a maximum
$a_{\mathrm{max}}$ for $a(t)$,

$\displaystyle a(t)\leq a_{\mathrm{max}}\equiv
a_{s}\mathrm{e}^{-\left(\frac{9}{4A\alpha\kappa^{2}}\right)^{-\frac{1}{B}}}<a_{s}\,.$
(798)

Because $a_{\mathrm{max}}$ is smaller than $a_{s}$, we find that dark Universe
with the future Type III singularity is transited to the one with Type II
singularity due to the account of thermal effects.

####  IX.7.3 Thermal Radiation for Type II and Type IV Singularities

When we consider Type II and Type IV singularities, $H_{s}(t)$ in (115) is
finite and positive, $0<H=H_{s}(t_{s})<\infty$ When one considers general
matter, the first FLRW equation where usual matter and the thermal radiation
as in (780) are included, is given by

$\displaystyle\frac{3}{\kappa^{2}}H^{2}=\rho+\alpha H^{4}\,.$ (799)

Here, $\rho$ is the matter energy-density. In case of Type II or Type IV
singularity, if $H_{s}(t_{s})\neq 0$, near the singularity, the l.h.s. goes to
a finite value
$\frac{3}{\kappa^{2}}H^{2}\to\frac{3}{\kappa^{2}}H_{s}(t_{s})^{2}$ and the
contribution from the thermal radiation in the r.h.s. also becomes finite,
$\alpha H^{4}\to\alpha H_{s}(t_{s})^{4}$. Therefore, the thermal radiation
does not change the structure of the singularity. Even if $H_{s}(t_{s})=0$,
the r.h.s. behaves as $\left(t_{s}-t\right)^{-4\beta}$ and the contribution
from the thermal radiation behaves as $\left(t_{s}-t\right)^{-2\beta}$.
Because $\beta<0$, the contribution from the thermal radiation is less
dominant and therefore the thermal radiation does not change the structure of
the singularity.

###  IX.8 Combination of Quantum Effect and Thermal Effect

We now combine the quantum effect and the thermal effect. In Subsection IX.5
the trace part of the Einstein equation is used. As the radiation is usually
conformal, the trace part of the energy-momentum tensor of the radiation
should vanish and the thermal radiation does not contribute to the trace
equation. We should be, however, more careful in the present situation. The
energy density of the thermal radiation is only determined by the temperature.
Therefore, the Universe expands and its volume with the thermal radiation
increases, the total energy should also be increased if the temperature is not
changed or increases as in the case of Type I (Big Rip) or Type III
singularity. In other words, say, in the phantom Universe, there should exist
effectively negative pressure. The energy of the thermal radiation is not
conserved because the expansion produces the new thermal radiation. We should
note, however, in order that the effective pressure, which includes the effect
of the expansion, is consistent with the FLRW equations, the energy-density of
the thermal radiation and the effective pressure must satisfy the conservation
law

$\displaystyle
0=\frac{d\rho_{\mathrm{t\\_rad}}}{dt}+3H\left(\rho_{\mathrm{t\\_rad}}+p_{\mathrm{t\\_rad}}\right)\,.$
(800)

To show the conservation law, we may start from the first FLRW equation where
usual matter and the thermal radiation as in (780) are included,

$\displaystyle\frac{3}{\kappa^{2}}H^{2}=\rho+\rho_{\mathrm{t\\_rad}}\,,\quad\rho_{\mathrm{t\\_rad}}=\alpha
H^{4}\,.$ (801)

Here $\rho$ is matter energy-density. By considering the derivative of Eq.
(801) with respect to time $t$, we obtain,

$\displaystyle\frac{6}{\kappa^{2}}H\dot{H}=\dot{\rho}+4\alpha H^{3}\dot{H}\,.$
(802)

Then by using the standard conservation law for matter,

$\displaystyle 0=\dot{\rho}+3H\left(\rho+p\right)\,,$ (803)

with the matter pressure, and combining (801) and (802), we obtain

$\displaystyle-\frac{1}{\kappa^{2}}\left(2\dot{H}+3H^{2}\right)=p-\alpha\left(H^{4}+\frac{4}{3}H^{2}\dot{H}\right)\,,$
(804)

which is nothing but the second FLRW equation and we can identify the
effective pressure of the thermal radiation as follows,

$\displaystyle
p_{\mathrm{t\\_rad}}=-\alpha\left(H^{4}+\frac{4}{3}H^{2}\dot{H}\right)\,.$
(805)

Thus, effectively, the energy density and the effective pressure of the
thermal radiation satisfy the conservation law (800) or we can find the exact
and unique form of the effective pressure in (805) directly by using the
conservation law (800) and assuming the form of the energy density of the
radiation in (779).

Then the trace part
$T_{\mathrm{t\\_rad}}=-\rho_{\mathrm{t\\_rad}}+3p_{\mathrm{t\\_rad}}$ of the
energy-momentum tensor for the radiation including the effect of the expansion
of the Universe is given by

$\displaystyle
T_{\mathrm{t\\_rad}}=-4\alpha\left(H^{4}+H^{2}\dot{H}\right)\,.$ (806)

Let us assume the behavior of the Hubble rate $H$ as in (114). Then in the
case of Type I (Big Rip) case ($\beta\geq 1$), near the singularity, we find
$T_{\mathrm{t\\_rad}}\sim\left(t_{s}-t\right)^{-4\beta}$, whose behavior is
not so changed from that of $T_{A}$ although we need to compare $b^{\prime}$
with $\alpha$ to see which term is the dominant one.

In case of Type II singularity ($-1<\beta<0$), we find
$T_{\mathrm{t\\_rad}}\sim\left(t_{s}-t\right)^{-3\beta-1}$. As
$R\sim\left(t_{s}-t\right)^{-\beta-1}$, the contribution from
$T_{\mathrm{t\\_rad}}$ is negligible. In case $b^{\prime\prime}\neq 0$, which
is arbitrary and can be put to vanish if we do not add $R^{2}$ term, the
contribution from $\Box R$ in $T_{A}$ dominates and the Type II singularity
does not occur.

In case of Type III singularity ($0<\beta<1$), we find
$T_{\mathrm{t\\_rad}}\sim\left(t_{s}-t\right)^{-3\beta-1}$, whose behavior is
not changed from that of the Gauss-Bonnet invariant ${G}$ in $T_{A}$ but
weaker than the behavior of $\Box R$. Then if $b^{\prime\prime}\neq 0$, the
contribution from the thermal radiation is less dominant than that of the
conformal anomaly $T_{A}$. If $b^{\prime\prime}=0$, the contribution from
$T_{\mathrm{t\\_rad}}$ is not changed from that from $T_{A}$ and we need to
compare $b^{\prime}$ with $\alpha$ to see which could be dominant, again.
Thus, we demonstrated that when quantum effects dominate over thermal effects
then future singularities are removed. However, in some cases which depend on
the specific features of the theory under consideration, the dominant
contribution is due to the thermal effects. In this case, the most possible
future of the Universe is the Type II singularity.

###  IX.9 Quantum Effects May Change the Occurrence of a Finite-time Future
Singularity

In this section, for the scalar-tensor theory, which often generates the Big
Rip singularity, we consider the quantum correction found in Ref.
Barvinsky:1993zg . The action of the scalar-tensor theory with a single scalar
is given by,

$\displaystyle
L=\frac{1}{2\kappa^{2}}\left(R+\frac{\tilde{\gamma}}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\right)\,,$
(807)

where $\tilde{\gamma}=\pm 1$. When $\tilde{\gamma}=1$, the scalar field is the
phantom universe and therefore often generates the Big Rip singularity It
would be interesting to investigate the quantum properties of such scalar-
tensor gravity.

The calculation of the one-loop effective action in the model (807) has been
performed as follows,

$\displaystyle W_{\mathrm{1-loop}}=$
$\displaystyle\,-\frac{1}{2}\ln\frac{L^{2}}{\mu^{2}}\int
d^{4}x\sqrt{-g}\left\\{\frac{5}{2}V^{2}-\tilde{\gamma}\left(V^{\prime}\right)^{2}+\frac{1}{2}\left(V^{\prime\prime}\right)^{2}+\left[\frac{\tilde{\gamma}}{2}V-2V^{\prime\prime}\right]\phi_{,\mu}\phi^{,\mu}-\left[\frac{13}{3}V+\frac{\tilde{\gamma}}{12}V^{\prime\prime}\right]R\right.$
$\displaystyle\left.+\frac{43}{60}R_{\alpha\beta}^{2}+\frac{1}{40}R^{2}-\frac{\tilde{\gamma}}{6}R\phi_{,\mu}\phi^{,\mu}+\frac{5}{4}\left(\phi_{,\mu}\phi^{,\mu}\right)^{2}\right\\}\,.$
(808)

The above one-loop action is found in Ref. Barvinsky:1993zg . We may regard
this effective action (IX.9) as a finite quantum correction to the classical
one. In (IX.9), the cut-off $L$ should be identified with the corresponding
physical quantity like the GUT scale or the Planck scale. When the universe is
approximated by the de Sitter spacetime, the natural choice is
$L^{2}=\left|R\right|$ because the curvature is large enough and constant
Buchbinder:1984giy ; Buchbinder:1992rb . On the other hand, in the case that
$\left|V\right|\gg\left|R\right|$, $L^{2}$ should be identified with $|V|$.

The phantom terms may be induced even if $V=0$ and $\tilde{\gamma}=-1$, which
corresponds to the canonical scalar. This happens if the universe goes through
a region with negative curvature. By the fine-tuning of $V$, there may appear
the quantum gravity-induced phantom theory, which subsequently may change the
universe’s evolution.

Here, we consider the action where $L^{2}$ is replaced with $\left|R\right|$
as a simple example,

$\displaystyle W_{\mathrm{1-loop}}=$ $\displaystyle\,-\frac{1}{2}\int
d^{4}x\sqrt{-g}\ln\frac{\left|R\right|}{\mu^{2}}\left\\{\frac{5}{2}V^{2}-\tilde{\gamma}\left(V^{\prime}\right)^{2}+\frac{1}{2}\left(V^{\prime\prime}\right)^{2}+\left[\frac{\tilde{\gamma}}{2}V-2V^{\prime\prime}\right]\phi_{,\mu}\phi^{,\mu}\right.$
$\displaystyle\,\left.-\left[\frac{13}{3}V+\frac{\tilde{\gamma}}{12}V^{\prime\prime}\right]R+\frac{43}{60}R_{\alpha\beta}^{2}+\frac{1}{40}R^{2}-\frac{\tilde{\gamma}}{6}R\phi_{,\mu}\phi^{,\mu}+\frac{5}{4}\left(\phi_{,\mu}\phi^{,\mu}\right)^{2}\right\\}\,.$
(809)

The variations of this action with respect the scalar field $\phi$ and the
metric $g_{\mu\nu}$ are given by

$\displaystyle\frac{1}{\sqrt{-g}}\frac{\delta
W_{\mathrm{1-loop}}}{\delta\phi}=$
$\displaystyle\,-\frac{1}{2}\ln\frac{\left|R\right|}{\mu^{2}}\left\\{\left\\{\frac{5}{2}V^{2}-\tilde{\gamma}\left(V^{\prime}\right)^{2}+\frac{1}{2}\left(V^{\prime\prime}\right)^{2}\right\\}^{\prime}+\left[\frac{\tilde{\gamma}}{2}V-2V^{\prime\prime}\right]^{\prime}\phi_{,\mu}\phi^{,\mu}\right.$
$\displaystyle\,\left.-2\nabla_{\mu}\left\\{\left[\frac{\tilde{\gamma}}{2}V-2V^{\prime\prime}\right]\phi^{,\mu}\right\\}-\left[\frac{13}{3}V+\frac{\tilde{\gamma}}{12}V^{\prime\prime}\right]^{\prime}R+\frac{\tilde{\gamma}}{3}\nabla_{\mu}\left\\{R\phi^{,\mu}\right\\}-5\nabla_{\mu}\left\\{\left(\phi_{,\rho}\phi^{,\rho}\right)\phi^{,\mu}\right\\}\right\\}\,,$
(810) $\displaystyle\frac{1}{\sqrt{-g}}\frac{\delta
W_{\mathrm{1-loop}}}{\delta g_{\mu\nu}}=$
$\displaystyle\,-\frac{1}{2}\ln\frac{\left|R\right|}{\mu^{2}}\left[\frac{1}{2}g^{\mu\nu}\left\\{\frac{5}{2}V^{2}-\tilde{\gamma}\left(V^{\prime}\right)^{2}+\frac{1}{2}\left(V^{\prime\prime}\right)^{2}+\left[\frac{\tilde{\gamma}}{2}V-2V^{\prime\prime}\right]\phi_{,\mu}\phi^{,\mu}\right.\right.$
$\displaystyle\,\left.-\left[\frac{13}{3}V+\frac{\tilde{\gamma}}{12}V^{\prime\prime}\right]R+\frac{43}{60}R_{\alpha\beta}^{2}+\frac{1}{40}R^{2}-\frac{\tilde{\gamma}}{6}R\phi_{,\mu}\phi^{,\mu}+\frac{5}{4}\left(\phi_{,\mu}\phi^{,\mu}\right)^{2}\right\\}$
$\displaystyle\,-\left[\frac{\tilde{\gamma}}{2}V-2V^{\prime\prime}\right]\phi^{,\mu}\phi^{,\nu}+\left[\frac{13}{3}V+\frac{\tilde{\gamma}}{12}V^{\prime\prime}\right]R^{\mu\nu}-\left(\nabla^{\mu}\nabla^{\nu}-g^{\mu\nu}\nabla^{2}\right)\left[\frac{13}{3}V+\frac{\tilde{\gamma}}{12}V^{\prime\prime}\right]$
$\displaystyle\,-\frac{43}{30}R^{\mu}_{\
\rho}R^{\nu\rho}+\frac{43}{60}\left\\{\left(\nabla_{\alpha}\nabla^{\nu}R^{\alpha\mu}+\nabla_{\alpha}\nabla^{\mu}R^{\alpha\nu}\right)-\nabla^{2}R^{\mu\nu}-g^{\mu\nu}\nabla_{\rho}\nabla_{\sigma}R^{\rho\sigma}\right\\}+\frac{1}{20}RR^{\mu\nu}$
$\displaystyle\,+\frac{1}{20}\left(\nabla^{\mu}\nabla^{\nu}-g^{\mu\nu}\nabla^{2}\right)R+\frac{\tilde{\gamma}}{6}R^{\mu\nu}\phi_{,\rho}\phi^{,\rho}-\frac{\tilde{\gamma}}{6}\left(\nabla^{\mu}\nabla^{\nu}-g^{\mu\nu}\nabla^{2}\right)\left(\phi_{,\rho}\phi^{,\rho}\right)$
$\displaystyle\,+\frac{\tilde{\gamma}}{6}R\partial^{\mu}\phi\partial^{\nu}\phi-\frac{5}{2}\phi_{,\rho}\phi^{,\rho}\phi^{,\mu}\phi^{,\nu}+\left(-R^{\mu\nu}+\nabla^{\mu}\nabla^{\nu}-g^{\mu\nu}\nabla^{2}\right)\left[-\frac{1}{2R}\left\\{\frac{5}{2}V^{2}-\tilde{\gamma}\left(V^{\prime}\right)^{2}+\frac{1}{2}\left(V^{\prime\prime}\right)^{2}\right.\right.$
$\displaystyle\,\left.\left.+\left[\frac{\tilde{\gamma}}{2}V-2V^{\prime\prime}\right]\phi_{,\rho}\phi^{,\rho}-\left[\frac{13}{3}V+\frac{\tilde{\gamma}}{12}V^{\prime\prime}\right]R+\frac{43}{60}R_{\alpha\beta}^{2}+\frac{1}{40}R^{2}-\frac{\tilde{\gamma}}{6}R\phi_{,\rho}\phi^{,\rho}+\frac{5}{4}\left(\phi_{,\rho}\phi^{,\rho}\right)^{2}\right\\}\right]\,.$
(811)

In the case when the Big Rip singularity occurs, the curvature becomes quickly
very large. This means that quantum effects (e.g., quantum gravity effects)
become important not only for the early universe but also for the future
universe. These quantum effects may even become dominant when the universe
approaches the Big Rip singularity. In fact, the quantum correction becomes
dominant because $W_{\mathrm{1-loop}}$ contains higher derivative terms, when
we may neglect the classical terms. In order to simplify the situation more,
we assume that the curvature and the scalar field $\phi$ are constant
$R_{\mu\nu}=\frac{3}{l^{2}}g_{\mu\nu}$, $R=\frac{12}{l^{2}}$, and $\phi=c$. We
also choose the potential $V(\phi)$as the exponential function of $\phi$,
$V(\phi)=V_{0}\mathrm{e}^{-2\frac{\phi}{\phi_{0}}}$. Then by using (IX.9) and
(IX.9), we obtain

$\displaystyle 0=$ $\displaystyle\,\frac{1}{\sqrt{-g}}\frac{\delta
W_{\mathrm{1-loop}}}{\delta\phi}$ $\displaystyle=$
$\displaystyle\,-\frac{1}{2}\ln\frac{\left|R\right|}{\mu^{2}}\left[-\frac{4}{\phi_{0}}\left(\frac{5}{2}-\frac{4\tilde{\gamma}}{\phi_{0}^{2}}+\frac{8}{\phi_{0}^{4}}\right)V_{0}^{2}\mathrm{e}^{-\frac{4c}{\phi_{0}}}+\frac{2}{\phi_{0}}\left(\frac{13}{3}+\frac{\tilde{\gamma}}{3\phi_{0}^{2}}\right)V_{0}\mathrm{e}^{-\frac{2c}{\phi_{0}}}\frac{12}{l^{2}}\right]\,.$
(812) $\displaystyle 0=$ $\displaystyle\,\frac{1}{\sqrt{-g}}\frac{\delta
W_{\mathrm{1-loop}}}{\delta g_{\mu\nu}}$ $\displaystyle=$
$\displaystyle\,g^{\mu\nu}\left[-\frac{1}{4}\ln\left(\frac{12}{l^{2}\mu^{2}}\right)\left\\{\left(\frac{5}{2}-\frac{4\tilde{\gamma}}{\phi_{0}^{2}}+\frac{8}{\phi_{0}^{4}}\right)V_{0}^{2}\mathrm{e}^{-\frac{4c}{\phi_{0}}}-\left(\frac{13}{3}+\frac{\tilde{\gamma}}{3\phi_{0}^{2}}\right)V_{0}\mathrm{e}^{-\frac{2c}{\phi_{0}}}\frac{12}{l^{2}}+\frac{147}{5l^{4}}\right\\}\right.$
$\displaystyle\,\left.+\frac{1}{8}\left(\frac{5}{2}-\frac{4\tilde{\gamma}}{\phi_{0}^{2}}+\frac{8}{\phi_{0}^{4}}\right)V_{0}^{2}\mathrm{e}^{-\frac{4c}{\phi_{0}}}+\frac{3}{2l^{2}}\left(\frac{13}{3}+\frac{\tilde{\gamma}}{3\phi_{0}^{2}}\right)V_{0}\mathrm{e}^{-\frac{2c}{\phi_{0}}}-\frac{441}{40l^{4}}\right]\,.$
(813)

Eq. (IX.9) can be solved with respect to $l^{2}$ as follows,

$\displaystyle
R=\frac{12}{l^{2}}=2\left(\frac{5}{2}-\frac{4\tilde{\gamma}}{\phi_{0}^{2}}+\frac{8}{\phi_{0}^{4}}\right)\left(\frac{13}{3}+\frac{\tilde{\gamma}}{3\phi_{0}^{2}}\right)^{-1}V_{0}\mathrm{e}^{-\frac{2c}{\phi_{0}}}\,.$
(814)

We should note, however, that Eq. (IX.9) is not consistent with the expression
in Eq. (814) in general. Then, Eq. (IX.9) might be regarded as an equation
determining $\mu$. We should also note that the r.h.s. in (814) is not always
positive. In the case $\tilde{\gamma}>0$, when $\tilde{\gamma}^{2}<5$, the
r.h.s. in Eq. (814) is positive, but when $\tilde{\gamma}^{2}>5$, it is
positive if
$\phi_{0}^{2}>\frac{4}{5}\left(\tilde{\gamma}+\sqrt{\tilde{\gamma}^{2}-5}\right)$
or
$\phi_{0}^{2}<\frac{4}{5}\left(\tilde{\gamma}-\sqrt{\tilde{\gamma}^{2}-5}\right)$.
On the other hand, in a phantom case $\tilde{\gamma}<0$, the r.h.s. in (814)
is positive if $\phi_{0}^{2}>-\frac{\tilde{\gamma}}{13}$. Anyway, there may
occur an (asymptotically) de Sitter solution. Thus, the universe becomes a
quantum de Sitter space before entering the Big Rip singularity. This
qualitative discussion indicates that the finite time future singularity may
never occur (or, at least may become milder) under the conjecture that quantum
effects become dominant just before the Big Rip. Due to the sharp increase of
the curvature invariants near the Big Rip, such a conjecture looks quite
natural.

##  X Summary and Conclusions

In this review we aimed to present an overview of how finite-time cosmological
singularities may arise in various cosmological contexts and to show the
nature of singularities. There are various types of singularities that may be
developed from cosmological theories, and we focused to demonstrate how soft
and crushing types singularities may be developed. In all the cases, it is
vital to understand that these cosmic singularities cannot be developed from
standard GR approaches, unless a phantom ingredient is added in the theory. On
the other hand, standard and non-standard modifications of GR give rise
naturally to cosmic singularities without the need of a phantom ingredient.
Thus, we presented all the distinct cases that may give rise to finite-time
cosmological singularities. Our aim was to provide the theoretical frameworks
that allow this singularity occurrence to happen. But our inherent reasoning
for this work is to further motivate the study of finite-time cosmological
singularities. The point is that cosmic singularities have their own physical
significance, apart from the mathematical structures they imply. The fabric of
spacetime that can accommodate such spacelike future singularities may have
its own mysteries to reveal. A singularity, especially if it is a crushing
type singularity, indicates our inability to describe physics adequately. The
reason might be unknown for the moment, but this should be the aim of future
scientists. The fact that GR cannot produce such singularities, while some
modifications of it can, indicates the fact that GR may be an effective theory
active in less strong gravity regimes, while the modifications of it might be
the more fundamental theories that correctly describe to some extent nature in
strong gravity limits. The fact that the modifications of GR predict cosmic
singularities on the other hand, might be an indication that these theories
link the classical physics to the unknown quantum nature of the Universe.
Indeed, as gravity effects become stronger, the modifications of GR come into
play, so they might act as a direct link between the classical GR physics and
the unknown quantum nature of our Universe. The links are the singularities
developed by the modifications of GR. The singularities are always mysterious,
but our experience from electrodynamics indicates that they are the links
between the classical and quantum physics. Their appearance indicates the need
for a fundamental quantum theory. GR itself does not lead to finite-time
singularities, unless a phantom ingredient is added to the description. On the
other hand, GR modifications lead to singularities, that in some sense touch
the quantum theory, they are in between the classical GR theory and the
quantum theory of the Universe. The singularities themselves might be a direct
indication that these theories are the correct description of nature, and they
are closer to the physical reality. These theories are a step closer to the
underlying fundamental quantum theory that governs our Universe at high energy
and at strong gravity regimes. There are three things that are remarkable to
point out, firstly spacelike singularities in GR occur only at the center of
the Schwarzschild black holes, the nature of which is unknown, regarding the
interior of the black hole, thus the singularity itself points to an
underlying fundamental theory of quantum nature beyond GR. Secondly, string
corrections to the GR Lagrangian always introduce higher curvature terms in
the Lagrangian, and these corrections may lead to finite-time singularities.
Thirdly the Big Bang singularity, present in some theories, indeed points out
a quantum era for our Universe. Thus singularities always point out the urge
for an underlying fundamental theory governing the Universe at strong gravity
and high energy regimes, that is yet to be found.

There are many fundamental questions that future scientists should address at
some point. Was spacetime itself created along with matter? What is the
relation of spacetime itself with matter, from a fundamental point of view? Is
spacetime evolving with matter or it evolves independently? What is spacetime
in the end, is it another manifestation of matter or are these independent?
Obviously, the Big Bang singularity relates all these questions, and of course
a future crushing singularity brings all these questions to the mainstream of
theoretical physics. A singularity in spacetime means geodesics incompleteness
and the latter indicates our inability to describe physics, to reach that era
physically. This might indicate the change in the topology of the Universe
occurring at the time instance of the finite-time future singularity. These
questions are not easy to address. Thus, with this review we provided a
comprehensive overview of all the distinct theories associated with the
finite-time singularities, which may potentially yield indications of an
underlying fundamental quantum theory.

##  XI Acknowledgments

The work of JdH has been supported by grant PID2021-123903NB-I00 funded by
MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. This work
was supported by MINECO (Spain), project PID2019-104397GB-I00 and also
partially supported by the program Unidad de Excelencia Maria de Maeztu
CEX2020-001058-M, Spain (S.D.O). SP acknowledges the financial support from
the Department of Science and Technology (DST), Govt. of India under the
Scheme “Fund for Improvement of S&T Infrastructure (FIST)” [File No.
SR/FST/MS-I/2019/41].

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