#### Generalized second law of thermodynamic in modified teleparallel theory

Eur. Phys. J. C
Generalized second law of thermodynamic in modified teleparallel theory
M. Zubair 2
Sebastian Bahamonde 1
Mubasher Jamil 0
0 Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST) , H-12, Islamabad , Pakistan
1 Department of Mathematics, University College London , Gower Street, London WC1E 6BT , UK
2 Department of Mathematics, COMSATS Institute of Information Technology Lahore , Lahore , Pakistan
This study is conducted to examine the validity of the generalized second law of thermodynamics (GSLT) in flat FRW for modified teleparallel gravity involving coupling between a scalar field with the torsion scalar T and the boundary term B = 2∇μT μ. This theory is very useful, since it can reproduce other important well-known scalar field theories in suitable limits. The validity of the first and second law of thermodynamics at the apparent horizon is discussed for any coupling. As examples, we have also explored the validity of those thermodynamics laws in some new cosmological solutions under the theory. Additionally, we have also considered the logarithmic entropy corrected relation and discuss the GSLT at the apparent horizon.
1 Introduction
The rapid growth of observational measurements on
expansion history reveals the expanding paradigm of the universe.
This fact is based on accumulative observational evidence
mainly from Type Ia supernova and other renowned sources
[
1–3
]. The expanding phase implicates the presence of a
repulsive force which compensates the attractiveness
property of gravity on cosmological scales. This phenomenon
may be translated as the existence of exotic matter
components and most acceptable understanding for such enigma is
termed dark energy (DE); it has a large negative pressure.
Various DE models and modified theories of gravity have been
proposed to incorporate the role of DE in cosmic expansion
history (for reviews see [4–6]).
In contrast to Einstein’s relativity and its proposed
modifications where the source of gravity is determined by
curvature scalar terms, another formulation has been presented
which comprises a torsional formulation as gravity source
[
7–11
]. This theory is labeled TEGR (teleparallel equivalent
of general relativity) and it is determined by a Lagrangian
density involving a zero curvature Weitzenböck connection
instead of a zero torsion Levi-Civita connection with the
vierbein as a fundamental tool. The Weitzenböck connection
is a specific connection which characterizes a globally flat
space-time endowed with a non-zero torsion tensor. Using
that connection, one can construct an alternative and
equivalent theory of GR. The latter appears, since the scalar torsion
2
only differs by a boundary term B = e ∂μ(eT μ) with the
scalar curvature by the relationship R = −T + B, making
the two variations of the Einstein–Hilbert and TEGR actions
the same. Thus, these two theories have the same field
equations. However, these two theories have different
geometrical interpretations, since in TEGR the torsion acts as a force;
meanwhile in GR, the gravitational effects are understood
due to the curved space-time. TEGR is then further extended
to a generalized form by the inclusion of a f (T ) function in
the Lagrangian density (as f (R) is the extension of GR) and
it has been tested cosmologically by numerous researchers
[12–14]. It is important to mention that f (T ) and f (R) are
no longer equivalent theories, and in order to consider the
equivalent teleparallel theory of f (R), one needs to
consider a more generalized function f (T , B), incorporating the
boundary term in the action [15]. In [16], the authors studied
some cosmological features (reconstruction method,
thermodynamics and stability) within f (T , B) gravity and, in [17],
some cosmological solutions were found using the Noether
symmetry approach. Additionally, it has been proved that
when one considers Gauss–Bonnet higher order terms, an
additional boundary term BG (related to the contorsion
tensor) needs to be introduced to find the equivalent teleparallel
modified Gauss–Bonnet f (R, G) theory [18].
Later, Harko et al. [19] proposed a comprehensive form
of this theory by involving a non-minimal torsion matter
interaction in the Lagrangian density. In Ref. [20], Zubair
and Waheed recently have investigated the validity of energy
constraints for some specific f (T ) models and discussed the
feasible bounds of involved arbitrary parameters. They also
discussed the validity of the generalized second law of
thermodynamic in the cosmological constant regime [21].
Another very much studied approach in modified
theories of gravity is to change the matter content of the universe
by adding a scalar field in the matter sector. These
models have been considered several times in cosmology, using
different kinds of scalar fields such as quintessence,
quintom, k-essence, etc. (see [22–24]). Moreover, we can also
extend that idea by adding a coupling between the scalar field
and the gravitational sector (see [
25–27
]) where,
cosmologically speaking, we can have new interesting results such as
the possibility of crossing the phantom barrier. Motivated by
these theories, other interesting modified theories of gravity
have also been discussed in the literature [28–30] on the
cosmological landscape. Recently, Bahamonde and Wright [31]
presented a new model of teleparallel gravity by introducing
a scalar field non-minimally coupled to both the torsion T
and the boundary term B = 2∇μT μ. It is shown that such
a theory can describe the non-minimal coupling to torsion
and also the non-minimal coupling to scalar curvature under
certain limits.
Black hole thermodynamics suggests that there is a
fundamental connection between gravitation and
thermodynamics [32]. Hawking radiation [33], the proportionality relation
between the temperature and surface gravity, and also the
connection between horizon entropy and the area of a black
hole [34] support this idea. Jacobson [35] was the first to
deduce the Einstein field equations from the Clausius
relation, Th d Sˆh = δ Qˆ , together with the fact that the entropy
is proportional to the horizon area. In the case of a general
spherically symmetric space-time, it was shown that the field
equations can be constituted as the first law of
thermodynamics (FLT) [36]. The relation between the FRW equations
and the FLT was shown in [37] for Th = 1/2π RA, Sh =
π R2 /G. The investigation of the validity of
thermodynami
A
cal laws in GR as well as modified theories has been carried
out by numerous researchers in the literature [38–53].
In this work we focus on the validity of the
thermodynamical laws in a modified teleparallel gravity involving a
non-minimal coupling between both torsion scalar and the
boundary term with a scalar field.
In Sect. 1, we give a brief introduction of this theory and
then we derive the respective field equation for flat FRW
geometry with a perfect fluid as the matter contents. In Sects.
2 and 3, we formulate the first and second law of
thermodynamics at the apparent horizon for any coupling.
Additionally, as examples, we study the validity of the
thermodynamics laws using new cosmological solutions based on
powerlaw and exponential-law cosmology (see Appendix A). In
Sect. 4, we discuss the validity of GSLT for the entropy
functional with quantum corrections. In each case, suitable limits
of the parameters are chosen in order to visualize the validity
of GSLT in quintessence, scalar field non-minimally
coupled to torsion (known as teleparallel dark energy) and
nonminimally coupled to the scalar curvature theories. Finally,
in Sect. 5, a discussion of the work is presented.
In the following paper, the notation used is the same as in
[31], where the tetrad and the inverse of the tetrad fields are
denoted by a lower letter eμa and a capital letter Eaμ,
respectively, with the (+, −, −, −) metric signature.
2 Teleparallel quintessence with a non-minimal coupling to a boundary term
In this study we consider the modified teleparallel model
which involves a scalar field non-minimally coupled to the
torsion T and a boundary term defined in terms of a
divergence of the torsion vector, B = 2e ∂μ(eT μ), where e =
det(eμa). The action of this theory is given by
S =
1
2κ2 ( f (φ)T + g(φ)B)
1
+ 2 ∂μφ∂μφ − V (φ) + Lm e d4x ,
(1)
where Lm determines the matter contents, κ2 = 8π G, V (φ)
is the energy potential and f (φ) and g(φ) are coupling
functions. For simplicity, we will use the notation
NMC(B+T) to refer to this theory ( f (φ) = g(φ) = 0). In Ref.
[31], the authors consider a special case of this action where
f (φ) = 1 + κ2ξ φ2 and g(φ) = κ2χ φ2. A non-minimally
coupled scalar field with the boundary term B (NMC-B) is
recovered if ξ = 0. Using dynamical system techniques, the
cosmology in NMC-B was studied in [31]. If χ = 0, we get
the same action as in [
25
], which is known as “teleparallel
dark-energy theory” (TDE). By choosing χ = −ξ we obtain
scalar field models non-minimally coupled to the Ricci scalar
that hereafter, for simplicity we will label as NMC-R [26].
In addition, the so-called “Minimally coupled quintessence”
theories arise when we take χ = ξ = 0 [27].
Variation of the action (1) with respect to the tetrad field
yields the following field equations:
κ2 f (φ) e−1∂μ(eSa μν ) − EaλT ρ μλ Sρ νμ − 41 Eaν T
2
− E ν
a
1
2 ∂μφ∂μφ − V (φ)
1
+ Eaμ∂ν φ∂μφ + κ2 2(∂μ f (φ) + ∂μg(φ))Eaρ Sρ μν
+ Eaν g(φ) − Eaμ∇ν ∇μg(φ) = Taν ,
where = ∇α∇α; ∇α is the covariant derivative linked
with the Levi-Civita connection symbol and Taν is the
matterenergy momentum tensor.
If we vary the action (1) with respect to the scalar field, that
yields the following modified Klein–Gordon equation:
1
φ + V (φ) = 2κ2 f (φ)T + g (φ)B .
Throughout this paper, a prime denotes differentiation with
respect to φ. We will assume the homogeneous and isotropic
flat FRW metric in Euclidean coordinates defined by
ds2 = dt 2 − a2(t )(dx 2 + dy2 + dz2),
where a(t ) represents the scale factor and the corresponding
tetrad components are eμi = (1, a(t ), a(t ), a(t )). It is
important to mention that despite f (T ) gravity not being
invariant under local Lorentz transformations, the latter tetrad is a
“good tetrad” to consider, since it does not constrain its field
equations [54]. Hence, this tetrad can be used safely within
this scalar field theory too.
The energy-momentum tensor of matter is defined as
Tμν = (ρm + pm )uμuν − pm gμν ,
where uμ is the four velocity of the fluid and ρm and pm
define the matter energy density and pressure, respectively.
Using the tetrad components for the flat FRW metric, the
field Eq. (2) leads to
3H 2 f (φ) = κ2 ρm + V (φ) + 21 φ˙ 2
+ 3H φ˙ g (φ),
3H 2 f (φ) + 2H˙ f (φ) = −κ2 pm − V (φ) + 21 φ˙ 2
− 2H φ˙ f (φ) + (g (φ)φ˙ 2 + g (φ)φ¨).
Here, H = a˙ (t )/a(t ) is the Hubble parameter and dots and
primes denote differentiation with respect to the time
coordinate and the argument of the function, respectively. We can
also rewrite these equations in a fluid representation,
3H 2 = κe2ffρeff,
2H˙ = −κe2ff(ρeff + peff),
κ2
where κe2ff = f (φ) , ρeff = ρm + ρφ is the total energy density
and peff = pm + pφ is the total pressure. The energy density
and the pressure of the scalar field ρφ and pφ are, respectively,
defined as follows:
ρφ = 21 φ˙ 2 + V (φ) + κ32 H φ˙ g (φ),
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
1 2
pφ = 2 φ˙ − V (φ)
1
+ κ2 2H φ˙ f (φ) − (g (φ)φ˙ 2 + g (φ)φ¨) .
In this theory the standard continuity equation reads
ρ˙eff + 3H (ρeff + peff) = 0 ,
ρ˙m + 3H (ρm + pm ) = 0 .
Hereafter, we will assume a standard equation of state for the
matter given by a barotropic equation pm = wρm , with w
being the state parameter. If we use the above equation, we
can directly find that the energy density becomes
ρm = ρ0a(t )−3(1+w),
where ρ0 is an integration constant. It is proper to mention
that, for a flat FRW metric, the torsion scalar and the boundary
term are
(10)
(11)
(12)
(13)
(14)
(15)
T = −6H 2 ,
B = −18H 2 − 6H˙ ,
and hence the Ricci scalar is recovered via R = −T + B =
−12H 2 − 6H˙ .
Finally, the equation for the scalar field, the so-called Klein–
Gordon equation, takes the form
1
φ¨ + 3H φ˙ − 2κ2 f (φ)T + g (φ)B
+ V (φ) = 0.
(16)
Note that the Klein–Gordon equation can also be obtained
directly from the field equations (6) and (7), so that it is not
an extra equation.
3 Thermodynamics in modified teleparallel theory
In this section we are interested to explore the general
thermodynamic laws in the framework of the theory studied in
[31], where the authors considered a quintessence theory
non-minimally coupled between both a torsion scalar T and
the boundary term B with the scalar field (NMC-B+T). The
main aim of the next sections will be to formulate the first and
second laws of thermodynamics in this theory. The complete
and general thermodynamics law will be derived. After that,
we will use the cosmological solutions found in Appendix A
to study some interesting examples to visualize if they do or
do not satisfy the thermodynamic laws.
3.1 First law of thermodynamics
This section is devoted to an investigation of the validity
of the first law of thermodynamics in NMC-(B+T) at the
apparent horizon for a flat FRW universe.
The condition hαβ ∂α RA∂β RA = 0 gives the radius RA of
apparent horizon for flat FRW metric by
The associated temperature is Th = κsg/2π , where the
surface gravity κsg is given by [55]
1
κsg = 2√−h ∂α(
√
1
−hhαβ ∂β RA) = − RA
R˙ A
1 − 2H RA
= − R2A (2H 2 + H˙ ).
By using Eq. (9) and (III A) we easily get
f (φ)
G
d RA = 4π R3A H (ρeff + peff)dt.
Now, multiplying both sides of this equation by a factor
−2π RA Th = 1 − R˙ A/(2H RA), we can rewrite the above
equation as follows:
= −(4π R3A H dt − 2π R2Ad RA)
π
×(ρeff + peff ) + G R2A Th d f (φ),
where we have used A = 4π R2A. Thus, we can identify the
entropy as the quantity which is multiplied by Th , namely
Now, we define energy of the universe within the
apparent horizon. The Misner–Sharp energy is defined as E =
RA/(2Geff ), which can be written as
In terms of the volume V = 4π R3A/3, we find that the energy
density is given by
E¯ =
3H 2 f (φ)
8π G
V ≡ ρeffV .
Taking the differential of the energy relation, we get
RA
d E¯ = 2G d f (φ)+4πρeff RA2d RA−4π H RA3(ρeff+ peff)dt.
Combining Eqs. (19) and (23), it results in
Th dSh = d E¯ − 2π RA2(ρeff − peff)d RA
RA (2π RA Th − 1)d f (φ).
+ 2G
A f (φ)
4G
Sh =
By defining the work density, we get
Here T¯ (de)αβ hαβ is the energy density of the dark compo
nents. Using the above definition of the work density in Eq.
(24), we obtain
RA (2π RA Th − 1)d f (φ),
Th dSh = d E¯ − W¯ dV + 2G
which can be rewritten as
Th dSh + Th dSp = d E¯ − W¯ dV ,
(25)
(26)
(27)
where dSp = −(RA/(2G Th ))(2π RA Th − 1)d f (φ), which
is the first law of thermodynamics in this teleparallel
theory. The extra term dSp defined in Eq. (27) can be treated
as an entropy production term in non-equilibrium
thermodynamics. In gravitational theories such as Einstein, Gauss–
Bonnet and Lovelock gravities [56–58], the usual first law
of thermodynamics is satisfied by the respective field
equations. In fact, these theories do not involve any surplus term
in universal form of the first law of thermodynamics, i.e.,
T dS = d E − W dV .
Initially Akbar and Cai used this approach to discuss
thermodynamic laws in f (R) gravity [59]. It is shown that an
additional entropy term is produced, to be compared to other
modified theories. Later Bamba et al. [41,42,60,61]
developed the first law of thermodynamics in Palatini f (R), f (T ),
f (R, φ, X ) (where X = −1/2gμν ∇μφ∇ν φ is the kinetic
term of a scalar field φ) and f (R, φ, X, G) (where G = R2 −
4Rμν Rμν + Rμνρσ Rμνρσ is the Gauss–Bonnet invariant) the
ories and formulated an additional entropy production term.
A similar approach is applied to discuss the thermodynamic
laws in f (R, T ), f (R, Lm ) and f (R, T , Rμν T μν ) theories,
and one can see that the presence of non-equilibrium entropy
production terms is necessary in such theories [46,47,62].
Bamba et al. [41,42,60,61] have shown that one can
manipulate the FRW equations in order to redefine the entropy
relation, which results in an equilibrium description of
thermodynamics so that the first law of thermodynamics takes
the form T dS = d E − W dV . Moreover, in all these theories
it has been the case that the usual form of the first law of
thermodynamics, i.e., T dSeff = d E − W dV , can be obtained by
defining the general entropy relation as a sum of an horizon
entropy and an entropy production term.
Here, we may define the effective entropy term being the
sum of horizon entropy and entropy production term as Seff =
Sh + Sp so that Eq. (27) can be rewritten as
Th dSeff = d E¯ − W¯ dV ,
(28)
where Seff is the effective entropy related to the contributions
involving a scalar field non-minimally coupled to the torsion
T and a boundary term at the apparent horizon of FRW
spacetime.
3.2 Generalized second law of thermodynamics
According to the generalized second law of thermodynamic
(GSLT), the entropy of matter and energy sources inside the
horizon plus the entropy associated with a boundary of the
horizon must be non-decreasing. In the previous section we
have shown that the usual first law of thermodynamics does
not hold in this theory. Therefore, to study GSLT, we would
use the modified first law of thermodynamics. In fact the
generalized entropy relation satisfies the condition
S˙tot = S˙h + S˙ p + S˙in ≥ 0,
where S˙h represents the entropy associated with the horizon,
S˙ p represents the entropy production term and S˙in is the sum
of all entropy components inside the horizon.
Let us proceed with the modified first law of
thermodynamics,
Ti dSi = d Ei + pi dV − Ti dSp,
which can be represented as
4
Tin S˙in = (ρi + pi )4π Rh 2(R˙h − H Rh ) + 3 π Rh3 Qi − Ti S˙ p,
where Rh represents the radius of the horizon, Tin denotes
the temperature for all the components inside the horizon and
Qi is an interaction term for the i th component. Summing
up the total entropy inside the horizon, we find
Tin S˙in = (ρeff + peff)4π Rh 2(R˙h − H Rh ) − Tin S˙ p.
(32)
Here,
i
Qi = 0,
(ρi + pi ) = ρeff + peff.
i
Now, let us study the validity of GSLT at the apparent
horizon. In de Sitter space-time RA = 1/H and the future event
horizon becomes the same as the Hubble horizon. The time
derivative of (20) results in
π 2H˙
S˙h = H 2G {φ˙ f (φ) − H
f (φ)}.
Here, we set the thermal equilibrium with Tin = Th , so that
Eq. (32) implies
(29)
(30)
(31)
(33)
1
S˙in + S˙ p = Th (ρeff + peff)4π RA2(R˙ A − H RA).
After simplification, we get
S˙in =
4π
H˙ (H˙ + H 2)
G (H˙ + 2H 2)H 3
Hence, Eq. (29) implies the relation of GSLT of the form
π
S˙tot = G H 2
4H˙ (H˙ + H 2) 2H˙
H (H˙ + 2H 2) + φ˙ f (φ) − H
f (φ)
In the following sections we will see if the GSLT is
satisfied for some interesting cosmological solutions that can be
constructed from our solutions found before (see Sects. A.1–
A.3).
3.2.1 Specific model: power-law solutions for
f (φ) = 1 + ξ κ2φ2 and g(φ) = χ κ2φ2
In this section we will analyze if the GSLT is valid for
different power-law models. In Sect. A.1, we found some specific
solutions to the specific coupling f (φ) = 1 + ξ κ2φ2 and
g(φ) = χ κ2φ2. Here, we will focus our study for quartic
(m = −1) and inverse potentials (m = 2/3). Additionally,
by setting the coupling constants ξ and χ , we will also focus
our study on different non-minimally coupled scalar–tensor
theories.
1
(a) Power-law potential with χ = 4
For this case, the energy potential reads
V (φ) = 21 φ2−2/m φ02/m
m2 + 12mnχ − 6n2ξ ,
χ = 41 , m = 2n(2ξ+3χ)+2χ±√42(2(4nχξ+−(13)n+1)χ)2+8nξ(1−4χ) .
In Fig. 1, we present the evolution of energy density ρ
and potential V (φ) for the power-law case. It can be seen
that V (φ) is a decreasing function of time, where we set
ξ ≤ 0.
Here, we set the validity condition for GSLT in terms of the
parameters m, ξ and χ . We choose m to show the particular
representation of potential.
(b) Case m = −1
This choice of m corresponds to a quartic potential, V (φ) ∝
1−6χ
φ4. Here, we find the relation for n of the form n = 6ξ+χ
and the constraint n > 1 results in the following condition:
χ <
1
|| χ > 6
(36)
(37)
Here, we discuss the specific cases NMC-B (ξ = 0) and
TDE (χ = 0). The validity of GSLT for a quartic potential
is shown in Fig. 2. For NMC-B theory, the GSLT is valid
in the range 0 < χ < 112 and in the case of TDE we need
0 < ξ < 16 . These constraints are set in accordance with the
condition of power-law solutions, n > 1.
We also the evolution of GSLT in Fig. 3, where we set
n = 2 and choose the particular values for χ and ξ . The
black curve corresponds to NMC-B with χ = 118 and the
blue curve represents the plot of GSLT for TDE with ξ = 118 .
(c) Case m = 2/3
In this case one can recover the inverse potential i.e., V (φ) ∝
φ−1. Here, we show the validity of GSLT in Figs. 4 and 5.
In Fig. 4, the left plot corresponds to NMC-B, which shows
the validity in the range − 81 < χ < 0 and the right plot
corresponds to TDE with − 23 < ξ < 0. It can be seen that
Fig. 3 Validity of GSLT for power-law potential with n = 2
GSLT is violated in case of TDE, whereas it is valid in both
NMC-B and NMC-R.
1
(d) Power-law potential with χ = 4
For this case the potential takes the following form:
−(3n+1) . Finally, we have GSLT dependence on n, and
valid12n
ity is shown in Fig. 6.
n2ξ(8ξ + 3) 3n2(8ξ + 3) − 1 φ− 32nn+ξ1 −2φ 32nn+ξ1 +4
0
(n(8ξ + 3) + 1)2
χ = 41 , m = 8ξ n +4n3ξn + 1 , n = − 8ξ 1+ 3 .
,
For this model, we have a fixed value of χ , so we cannot
discuss the TDE and NMC-B theories. In the case of
NMCR (χ = −ξ ), we find that this representation does not show
realistic results as we need to fix n < 0 for inverse and
quadratic potentials.
Moreover, in the case of NMC-(B+R), we explore the
validity for quartic potential, where we need to set ξ =
3.2.2 Specific model: exponential solutions for
f (φ) = 1 + κ2αemφ and g(φ) = κ2βekφ
This section is devoted to an analysis of the question if the
GSLT is satisfied for some exponential solutions found in
Sect. A.3. Since mostly all the solutions are similar, we will
only analyze the case where w = 2 −3nφ0 − 1 (see Sect. A.3.2).
In this case, the energy potential contains three exponentials
that can represent different kind of potentials. For example,
√3κφ0 , we find
by taking m = √2(φ0−1)
which is known as a double exponential potential. This kind
of potential has been widely studied in the literature. From
Fig. 7, it is observed that the scalar potential increases as the
field increases. Thus the cosmic acceleration will be driven
by the potential energy of the scalar field.
Another interesting example that can be constructed is by
taking φ0 = 4 and α = 24κ482κ−4m2 , which gives us the following
potential:
,
(39)
which is an interesting kind of potential studied in the
literature (see [
63, 64
]). The exponential function models generally
lead to an accelerating expansion behavior of the universe.
Let us now study the GSLT for this kind of potential. In
this model, Eq. (36) becomes
where for simplicity we took κ = G = 1. Clearly, the above
inequality will hold depending on the values of the
parameters φ0, n and α. Since n > 1, if φ0 ≥ −2 and α > 0,
GSLT will be valid at any time. Additionally, if φ0 ≤ −2
and α < 0, the GSLT will be always true. For all the other
cases, the validity of GSLT will depend on time. Figures 8
and 9 show the behavior of the GSLT inequality for different
values of the parameters. For φ0 > 0 and α < 0, the
inequality will be more constrained for bigger |φ0|. Additionally, it
can be seen that GSLT will be always true at very late times
for −2 < φ0 < 0.
4 Generalized second law of thermodynamics with logarithmic entropy corrections
The entropy–area relation involving quantum corrections
leads to curvature corrections in the Einstein–Hilbert action
[65,66]. The logarithmic corrected entropy is defined
through the relation [67–70]
S =
where α, β and γ are dimensionless constants; however, the
exact values of these constants are yet to be determined.
These corrections arise in black hole entropy in loop quantum
gravity due to thermal equilibrium fluctuations and quantum
fluctuations [71,72]. In [70] Sadjadi and Jamil investigated
the validity of GSLT for FRW space-time with logarithmic
correction. They found that in a (super-) accelerated universe
the GSL is valid whenever α(<) > 0, leading to a
(negative) positive contribution from logarithmic correction to the
entropy. In the following section, we will present the
validity of the GSLT with modified entropy relations involving
logarithmic corrections.
First note that the time derivative of Eq. (41) yields
S˙ =
A˙ f
A + f φ˙
A f
4G
1 + α
4G
A f
− β
4G
A f
2
. (42)
(41)
In the case of the apparent horizon, Eq. (41) implies
S˙ A =
Here, we used the fact that the Hubble horizon is RA = 1/H .
Now, by using Eqs. (35) and (43), we get
,
+
1 + α
which can be represented in terms of effective components
as follows:
f 3(φρe(ftf)) + ff φ˙
Gκ2ρeff
3π f (φ (t ))2
− β
3π f (φ (t ))2
Gκ2ρeff
Gκ2ρeff
3π f (φ (t ))2
2
6π(weff + 1)(3weff + 1)
Gκ(1 − 3weff)
3 f (φ (t ))
ρeff
The GSL with quantum corrections have natural
implications in the studies of the very early universe since quantum
corrections are directly linked with high energy and short
distance scales.
5 Conclusions
The non-minimally coupling models in cosmology are
frequently used to study guaranteed late time acceleration,
phantom crossing and the existence of finite time future
singularity. In this paper, the thermodynamic study is executed
in modified teleparallel theory which involves scalar field
non-minimally coupled to both the torsion and a
boundary term [31]. One interesting factor in this theory is that
in a suitable limit, one can recover very well-known
theories of gravity such as quintessence, teleparallel dark energy
and non-minimally coupled scalar field with the Ricci scalar
R.
In this paper, we investigated the GSLT for an
expanding universe with apparent horizon. The laws of
thermodynamics are universally valid and hence they must apply
in a modified form to the whole universe. The existence of
an apparent horizon in a space-time provides the
opportunity to formulate the first law and the generalized second
law of thermodynamics. According to the laws of
thermodynamics, the first law must always be true for a physical
system (provided it is non-dissipative); however, the GSLT
holds exactly using the apparent horizon. We obtained these
laws in the non-minimal coupled teleparallel theory with
a boundary term and a scalar field. Our results generalize
many previous GSLT studies, while extending them to the
logarithmic corrected entropy–area law. Moreover, GSLT
with quantum corrections have natural implications in the
studies of the very early universe, since quantum
corrections are directly linked with high energy and short distance
scales.
We explored the existence of power-law solutions for
two different choices of Lagrangian coefficients, which are
functions of the scalar field, f (φ) and g(φ). In the first
case we set f (φ) = 1 + κ2ξ φ2 and g(φ) = κ2χ φ2
and choose the power-law forms of scale factor and scalar
field to discuss possible forms of the power-law potential
V (φ) (see [28]). Here, one can recover the quadratic,
quartic, inverse and Ratra–Peebles potentials. We also discuss
the Brans–Dicke theory as a particular case in power-law
cosmology. Moreover, we also discuss the power-law
solutions for the choice of f (φ) = 1 + κ2αemφ and g(φ) =
κ2βekφ with φ (t ) = φm0 log(t ). Here, we explored different
forms of V (φ) depending on the equation of state
parameter.
We showed the behavior of matter density ρ and
potential V (φ) for the power-law potential; it can be seen that
both are decreasing functions of time as shown in Fig. 1. In
our discussion of the validity of GSLT, we considered the
quartic and inverse potentials in all the viable special cases
like TDE, NMC-B, NMC-R and NMC-(B+R) and presented
results in Figs. 2, 3, 4, 5 and 6. In Figs. 2 and 3, we found that
GSLT is true for NMC-B (with 0 < ξ < 16 ) and TDE (with
0 < χ < 112 ). For the inverse potential, we find that GSLT
is valid for NMC-B (with −81 < χ < 0) and TDE (with
− 23 < ξ < 0), whereas it is violated for TDE, as shown in
Figs. 4 and 5. For the choice of χ = 41 , we found the validity
region only in the case of NMC-(B+R) with n > 21 as shown
in Fig. 6. For the exponential forms of the Lagrangian
coefficients, we found that GSLT is always true for (φ0 ≤ −2 &
α < 0) and (φ0 ≥ −2 & α > 0), whereas for other choices
of the parameters the validity of GSLT is time dependent.
Finally, we also formulated the GSLT for logarithmic entropy
corrections.
Acknowledgements S.B. is supported by the Comisión Nacional de
Investigación Científica y Tecnológica (Becas Chile Grant No. 72150066).
The authors would like to thank the anonymous referee for the
interesting feedback and useful comments on the paper.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
A New cosmological solutions
A.1 Power-law solutions for f (φ) = 1 + κ2ξ φ2 and
g(φ) = κ2χ φ2
In this section we are going to find analytical power-law
solutions for the NMC-(B+T) case where the coupling functions
are
f (φ) = 1 + κ2ξ φ2,
g(φ) = κ2χ φ2,
where ξ and χ are coupling constants. As we discussed
before, for this specific choice we can recover NMC-B
(ξ = 0), TDE (χ = 0), NMC-R (ξ = −χ ) and quintessence
theories (χ = ξ = 0). Let us now study power-law
cosmology where the scale function and the scalar field take a
power-law form
(n(8ξ + 3) + 1)2
⎧⎪⎪⎨ 21 φ2−2/m φ02/m m2
n2ξ(8ξ + 3) 3n2(8ξ + 3) − 1 φ− 32nn+ξ1 −2φ 32nn+ξ1 +4
0
+ 12mnχ − 6n2ξ , χ = 41 , m = 2n(2ξ+3χ)+2χ±√42(2(4nχξ+−(13)n+1)χ)2+8nξ(1−4χ) ,
1 4nξ 1
, χ = 4 , m = 8ξ n + 3n + 1 , n = − 8ξ + 3 .
Since n = 0, the exponents of the scalar field of the first and
second terms on the r.h.s. must match otherwise the above
equation will not hold. Therefore, the state parameter needs
to be
2
w = 3n − 1.
Using this condition, Eq. (A6) becomes
2φ02/m φ−2/m κ2ρ0a0−2/n − 3n2
This condition will be true for different choices of the
parameters. For simplicity, we will assume the case that a0 = 1,
which gives
Thus, the energy potential (A5) and the energy density are
simplified to
For the first potential (χ = 1/4), we can recover a
selfinteracting scalar field case V (φ) ∝ φ4 if m = −1,
which is the case where the coupling parameters satisfied
ξ = −6nχ6−n6χ+1 . It corresponds to the inverse potential if
m = 2/3 with the additional constraint ξ = − 2(1−χ3n+9nχ)
and the Ratra–Peebles potential is recovered by choosing
0 < m < 1.
A.2 Power-law solutions for Brans–Dicke: f (φ) = κ2ξ φ2
and g(φ) = κ2χ φ2
It is easy to see that if we choose
f (φ) = κ2ξ φ2,
g(φ) = κ2χ φ2,
3n2
ρ0 = κ2 ,
4(2nξ + (3n + 1)χ )2 + 8nξ(1 − 4χ )
2(4χ − 1)
4nξ 1 1
⎪⎩⎪ 8ξ n + 3n + 1 , χ = 4 , n = − 8ξ + 3 .
1
, χ = 4 ,
where a0, φ0, m and n are constants. Under these ansatzes,
from (6) we directly find that the energy potential becomes
V (φ) = −ρ0a0−3(w+1)φ 2−3nm(w+1) − m2 φ03n(wm+1)
1 φ2− m2 φ02/m
− 2
+
3n2φ−2/m φ2/m
0
κ2
.
m2 + 12mnχ − 6n2ξ
Now, by replacing the above expression in (7) we find that
the parameters need to satisfy
κ2φ2/m φ2−2/m m2(4χ − 1) − 2m(2nξ + 3nχ + χ ) + 2nξ
0
= κ2ρ0(w + 1)a0−3(w+1)φ03n(wm+1) φ− 3n(wm+1) + 2nφ02/m φ−2/m .
our action becomes a Brans–Dicke one in a canonical form
with the Brans–Dicke parameter being wB D = 1. The latter
theory has been widely studied in the literature, so that it is
an important coupling to consider.
To find solutions, we will follow the same approach as
the previous section. Moreover, since the equations are so
similar, we will not state all the steps. It is easy to see that if
w = −1 + 2(13−nm) = −1 we obtain the following solutions:
needs to hold. For the specific case where ρ0 = 0, we have
three possibles cases for the state parameter,
3nφ02a0n2 − 2nm −4m2χ+m2+4mnξ+6mnχ+2mχ−2nξ ,
2(m−1)
,
, .
A.3 Exponential solutions for f (φ) = 1 + κ2αemφ and
g(φ) = κ2βekφ
In this section, we will assume that the coupling functions
are exponential:
2ακ2m2n(φ0 − 1)a03w+3e mφ(3n(w+1)+φ0)
φ0
+a03w+3 κ φ0 − 2m2n e
2 2
3mn(w+1)φ
φ0
= βκ2kφ0a03w+3(kφ0 − m(3n + 1))e
2mφ
−κ2m2ρ0(w + 1)e φ0
(A21)
(A22)
(A23)
.
(A25)
(A15)
(A16)
(A17)
(A18)
(A19)
(A20)
Now, we will explore all the possible solutions under these
state parameters. In general, all the solutions are very similar
but they are not exactly the same.
A.3.1 Case 1: w = 2m3−mknφ0 − 1
For this special case, from (A20) the other parameters must
satisfy (k = 2m)
ρ0 = −
3βκ2k 3κ2 − 2km + 2m2 a0
,
Note that for the case k = 2m we find that n = 1/3, which
is not allowed. Hence, if k = 2m we find that the energy
density and energy potential for the first state parameter, w1,
are given by
f (φ) = 1 + κ2αemφ ,
g(φ) = κ2βekφ ,
a(t ) = a0t n,
φ0 log(t ),
φ (t ) = m
n > 1,
where α, β, m and k are parameters. The cosmology for the
specific case where β = −α and k = m was studied in [73].
Additionally, we assume that the scale factor and the scalar
field behave as
where n, m, φ0 and a0 are constants. By using (6), we find
that the energy potential becomes
V (φ) = −ρ0a0−3(w+1)e−
3mn(w+1)φ
φ0
2mφ
− e− φ0
×
3βknφ0ekφ
m
Now, if we replace this equation in (7), we find that
3κ2+2km−4m2 t mk −2
4m4
.
(A24)
2−φ0
A.3.2 Case 2: w = 3n − 1
For this case, we need to have φ0 = 2 and also from (A20)
the parameters must obey
−
+
4m4
.
ρ0 =
3ακ4(φ0 − 1)φ04a0κ2φ02 − κ2φ0
2m4(φ0 − 2)
,
k =
3mn + m
φ0
,
and hence the energy density and the potential take the
following form:
A.3.3 Case 3: w = −1 (dark-energy fluid)
For a dark-energy fluid with w = −1, from (A20) the
parameters need to be
φ0 = 1,
k = 3mn + m,
κ2
n = 2m2 ,
and therefore the energy density is a constant ρ0 and the
potential becomes
3ακ4e−mφ
4m4
+
−
4m4
4m4
− ρ0.
(A30)
φ 32κm2 +m −2mφ
(A26)
(A27)
(A28)
(A29)
3κ2 − 2m2 e−2mφ
3β 3κ4 + 2κ2m2 e 1 . S. Perlmutter et al., Astrophys . J. 483 , 565 ( 1997 ) 2 . S. Perlmutter et al., Nature 391 , 51 ( 1998 ) 3 . A.G. Riess et al., Astrophys . J. 607 , 665 ( 2004 ) 4 . V. Sahni, Lect. Notes Phys . 653 , 141 ( 2004 ) 5 . M. Sharif , M. Zubair , Int. J. Mod. Phys. D 19 , 1957 ( 2010 ) 6 . K. Bamba , S. Capozziello , S. Nojiri , S.D. Odintsov , Astrophys.
Space Sci . 342 , 155 ( 2012 ) 7 . Einstein , A. , Sitzungsber. Preuss. Akad. Wiss. Phys. Math. KI. 217
( 1928 ) 8 . Einstein , A. , Sitzungsber. Preuss. Akad. Wiss. Phys. Math. KI. 224
( 1928 ) 9 . K. Hayashi , Phys. Rev. D 19 , 3524 ( 1979 ) 10 . H.I. Arcos , J.G. Pereira , Int. J. Mod. Phys. D 13 , 2193 ( 2004 ) 11 . J.W. Maluf, Ann. Phys. 525 , 339 ( 2013 ) 12 . R. Ferraro , F. Fiorini , Phys. Rev. D 75 , 084031 ( 2007 ) 13. G.R. Bengochea , R. Ferraro , Phys. Rev. D 79 , 124019 ( 2009 ) 14 . E.V. Linder , Phys. Rev. D 81 , 127301 ( 2010 ) 15 . S. Bahamonde, C.G. Böhmer , M. Wright , Phys. Rev. D 92 ( 10 ),
104042 ( 2015 ) 16 . S. Bahamonde, M. Zubair , G. Abbas. arXiv: 1609 .08373 [gr-qc] 17 . S. Bahamonde , S. Capozziello . Eur. Phys. J. C 77 , 107 ( 2017 ).
doi:10 .1140/epjc/s10052-017-4677-0 18. S. Bahamonde, C.G. Böhmer , Eur. Phys. J. C 76 ( 10 ), 578 ( 2016 ) 19. T. Harko , F.S.N. Lobo , G. Otalora , E.N. Saridakis , Phys. Rev. D
89 , 124036 ( 2014 ) 20. M. Zubair , S. Waheed , Astrophys. Space Sci . 355 , 361 ( 2015 ) 21. M. Zubair , S. Waheed , Astrophys. Space Sci . 360 , 68 ( 2015 ) 22 . E.J. Copeland , M. Sami , S. Tsujikawa, Dynamics of dark energy.
Int. J. Mod . Phys. D 15 , 1753 ( 2006 ). arXiv:hep-th/0603057 23. C. Armendariz-Picon , V.F. Mukhanov , P.J. Steinhardt , Phys. Rev.
D 63 , 103510 ( 2001 ) 24. Y.F. Cai , E.N. Saridakis , M.R. Setare , J.Q. Xia , Phys. Rept . 493 , 1
( 2010 ) 25. C.Q. Geng , C.C. Lee , E.N. Saridakis , Y.P. Wu , Phys. Lett. B . 704 ,
384 ( 2011 ) 26 . F. Perrotta , C. Baccigalupi , S. Matarrese, Phys. Rev. D 61 , 023507
( 1999 ) 27 . O. Bertolami , P.J. Martins , Phys. Rev. D 61 , 064007 ( 2000 ) 28 . O. Chandia , J. Zanelli , Phys. Rev. D 55 , 7580 ( 1997 ) 29. G. Kofinas, E.N. Saridakis , Phys. Rev. D 90 , 084044 ( 2014 ) 30. G. Kofinas, E.N. Saridakis , Phys. Rev. D 90 , 084045 ( 2014 ) 31 . S. Bahamonde, M. Wright , Phys. Rev. D 92 , 084034 ( 2015 ) 32 . J.M. Bardeen , B. Carter , S.W. Hawking , Commun. Math. Phys. 31 ,
161 ( 1973 ) 33 . S.W. Hawking, Commun. Math. Phys. 43 , 199 ( 1975 ) 34 . J.D. Bekenstein , Phys. Rev. D 7 , 2333 ( 1973 ) 35. T. Jacobson, Phys. Rev. Lett . 75 , 1260 ( 1995 ) 36. T. Padmanabhan, Phys. Rept . 406 , 49 ( 2005 ) 37 . R.G. Cai, S.P. Kim , JHEP 02 , 050 ( 2005 ) 38. C. Eling , R. Guedens , T. Jacobson, Phys. Rev. Lett . 96 , 121301
( 2006 ) 39. M. Akbar , R.G. Cai, Phys. Lett. B 648 , 243 ( 2007 ) 40. Y. Gong , A. Wang , Phys. Rev. Lett . 99 , 211301 ( 2007 ) 41. K. Bamba , C.Q. Geng , JCAP 06 , 014 ( 2010 ) 42. K. Bamba , C.Q. Geng , JCAP 11 , 008 ( 2011 ) 43 . S.-F. Wu , B. Wang , G.-H. Yang , P.-M. Zhang , Class. Quant. Gravit.
25 , 235018 ( 2008 ) 44. M.R. Setare , Phys. Lett. B 641 , 130 ( 2006 ) 45. M.R. Setare , JCAP 01 , 123 ( 2007 ) 46. M. Sharif , M. Zubair , J. Cosmol . Astropart. Phys. 11 , 042 ( 2013 ) 47. M. Sharif , M. Zubair , Adv. High Energy Phys . 2013 , 947898 ( 2013 ) 48 . Zubair , M. , Jawad , A. , 360 , 11 ( 2015 ) 49. M. Zubair , F. Kousar , S. Bahamonde, Phys. Dark Univ. 14 , 116
( 2016 ) 50 . H. Mohseni Sadjadi, Phys. Rev. D 76 , 104024 ( 2007 ) 51. M. Jamil , E.N. Saridakis , M.R. Setare , JCAP 1011 , 032 ( 2010 ) 52. K. Bamba , M. Jamil , D. Momeni , R. Myrzakulov , Astrophys. Space
Sci. 344 , 259 ( 2013 ) 53. U. Debnath, S. Chattopadhyay , I. Hussain, M. Jamil , R. Myrza-
kulov , Eur. Phys. J. C 72 , 1875 ( 2012 ) 54 . N. Tamanini , C.G. Boehmer , Phys. Rev. D 86 , 044009 ( 2012 ) 55 . R.G. Cai, S.P. Kim , JHEP 02 , 050 ( 2005 ) 56. M. Akbar , R.G. Cai, Phys. Rev. D 75 , 084003 ( 2007 ) 57 . R.G. Cai, L.M. Cao , Nucl. Phys. B 785 , 135 ( 2007 ) 58 . R.G. Cai, L.M. Cao , Y.P. Hu , JHEP 08 , 090 ( 2008 ) 59. M. Akbar , R.G. Cai, Phys. Lett. B 635 , 7 ( 2006 ) 60. K. Bamba , C.Q. Geng , S. Tsujikawa, Phys. Lett. B 688 , 101109
( 2010 ) 61. K. Bamba , C.Q. Geng , S. Nojiri , S.D. Odintsov , EPL 89 , 50003
( 2010 ) 62. M. Sharif , M. Zubair , J. Cosmol . Astropart. Phys. 03 , 028 ( 2012 ) 63 . V. Sahni , A.A. Starobinsky , Int. J. Mod. Phys. D 9 , 373 ( 2000 ) 64. T. Matos , L.A. Urena-Lopez , Phys. Rev. D 63 , 063506 ( 2001 ) 65. T. Zhu, J.-R. Ren , Eur. Phys. J. C 62 , 413 ( 2009 ) 66 . R.-G. Cai et al., Class. Quant. Gravit . 26 , 155018 ( 2009 ) 67 . S.K. Modak, Phys. Lett. B 671 , 167 ( 2009 ) 68. M. Jamil , M.U. Farooq, JCAP 03 , 00 ( 2010 ) 69 . S. Das , U. Debnath , A.A. Mamon . arXiv: 1510 . 02573v1 70. H.M. Sadjadi , M. Jamil , Europhys. Lett. 92 , 69001 ( 2010 ) 71. C. Rovelli, Phys. Rev. Lett . 77 , 3288 ( 1996 ) 72. A. Ashtekar , J. Baez , A. Corichi , K. Krasnov , Phys. Rev. Lett . 80 ,
904 ( 1998 ) 73 . V. Pettorino , C. Baccigalupi , G. Mangano, JCAP 0501 , 014 ( 2005 ).
doi:10 .1088/ 1475 - 7516 / 2005 /01/014. arXiv: astro-ph/0412334