#### Generalised nonminimally gravity-matter coupled theory

Eur. Phys. J. C
Generalised nonminimally gravity-matter coupled theory
Sebastian Bahamonde 0
0 Department of Mathematics, University College London , Gower Street, London, WC1E 6BT , UK
In this paper, a new generalised gravity-matter coupled theory of gravity is presented. This theory is constructed by assuming an action with an arbitrary function f (T , B, Lm ) which depends on the scalar torsion T , the boundary term B = ∇μT μ and the matter Lagrangian Lm . Since the function depends on B which appears in R = −T +B, it is possible to also reproduce curvature-matter coupled models such as f (R, Lm ) gravity. Additionally, the full theory also contains some interesting new teleparallel gravity-matter coupled theories of gravities such as f (T , Lm ) or C1T + f (B, Lm ). The complete dynamical system for flat FLRW cosmology is presented and for some specific cases of the function, the corresponding cosmological model is studied. When it is necessary, the connection of our theory and the dynamical system of other well-known theories is discussed.
1 Introduction
Nowadays, one of the most important challenges in physics
is try to understand the current acceleration of the Universe.
In 1998, using observations from Supernovae type Ia, it was
shown that the Universe is facing an accelerating expansion,
changing the way that we understand how our Universe is
evolving [
1
]. Later, other cosmological observations such as
CMB observations [
2–5
], baryon acoustic oscillations [6] or
galaxy clustering [
7
] also confirmed this behaviour of the
Universe. The responsible of this late-time acceleration of
the Universe is still not well understood and for that
reason it was labelled as the dark energy problem. In general,
there are two different approaches which try to deal with this
issue. First, one can assume that general relativity (GR) is
always valid at all scales and introduce a new kind of matter
which mimics this acceleration. This kind of matter known
as “exotic matter” needs to violate the standard energy
conditions to describe the evolution of the Universe. Up to now,
this kind of matter has not been discovered in the laboratory.
One can say that this approach lies on the idea of changing
the right hand side of the Einstein field equations. An
alternative approach to understand and study the dark energy is to
assume that GR is only valid at certain scales and therefore it
needs to be modified. In this approach, the left hand side of
the Einstein field equations is modified and there is no need to
introduce exotic matter. Different kind of modified theories
of gravity have been proposed in the literature to understand
the dark energy problem (see the reviews [
8–10
]).
One very interesting and alternative theory of gravity is
the teleparallel equivalent of general relativity (TEGR) or
“teleparallel gravity”. In this theory, the manifold is endorsed
with torsion but assumes a zero curvature. The
connection which satisfies this kind of geometry is the so-called
“Weitzenböck” connection, which was first introduced in
1922 [
11
]. It was then showed that this theory is equivalent to
GR in the field equations but the geometrical interpretation of
gravity is different. In TEGR, there is not geodesic equation
as in GR. Instead, forces equations describe the movement
of particles under the influence of gravity. Additionally, the
dynamical variable is the tetrad instead of the metric as in
GR. For more details about TEGR, see [
12–17
] and also
the book [18]. Similarly as in GR, there are also modified
theories starting from the teleparallel approach. The most
famous teleparallel modified theory is f (T ) gravity (where
T is the scalar torsion) which can describe very well the
current acceleration of the Universe and also other
cosmological observations (see [
19–33
] and also the review [
34
]).
The TEGR action contains the term T so f (T ) gravity is a
straightforward generalisation of it. This theory is analogous
to the well-known f (R) gravity, where instead of having the
scalar curvature R in the action, a more general theory with an
arbitrary function which depends on R is introduced. These
two theories are analogous but mathematically they are very
different. As we pointed out before, the TEGR field
equations are equivalent to the Einstein field equations. However,
their generalisations f (R) and f (T ) gravity have different
field equations. Further, f (R) gravity is a 4th order theory
and f (T ) gravity is a 2nd order theory. This characteristic
can be understood using the fact that R = −T + B, where
B is a boundary term. Hence, a linear combination of R or
T in the action will produce the same field equations since
B will not contribute to it. However, when one modifies the
action as an arbitrary function f (T ) or f (R), there will be
a difference in their field equations due to the fact that now
the boundary term B contributes. This was fully studied in
[
35
] where the authors introduced a new theory, the so-called
f (T , B) gravity, which can recover either f (T ) gravity or
f (T , B) = f (−T + B) = f (R) as special cases. Flat FLRW
cosmology of this theory was studied in [
36,37
].
Other kinds of modified theories of gravity have been
considered in the literature. Some interesting ones are theories
with non-minimally coupling between matter and gravity.
In standard metric approach, some alternatives models have
been proposed such as f (R, T ) [
38
], where T is the trace of
the energy-momentum tensor or non-minimally coupled
theories between the curvature scalar and the matter Lagrangian
f1(R)+ f2(R)Lm [
39
]. Further, another more general theory
is the so-called f (R, Lm ) where now an arbitrary function
of R and Lm is considered in the action [
38
]. Along the lines
of those theories, modified teleparallel theories of gravity
where couplings between matter and the torsion scalar have
been also considered. Some important theories are for
example: f (T , T ) gravity [
40
] and also non-minimally couplings
between the torsion scalar and the matter Lagrangian theory
f1(T ) + f2(T )Lm [
41
]. Along this line, in this paper, we
present a new modified teleparallel theory of gravity based
on an arbitrary function f (T , B, Lm ) where Lm is the
matter Lagrangian. In this theory, we have the possibility of for
example recover f (−T + B, Lm ) = f (R, Lm ) or a new
generalisation of [
41
] in the teleparallel framework with a
function f (T , Lm ) depending on T and Lm . The later new
theory is the analogous theory as f (R, Lm ) gravity. We will
explicitly discuss about how those models are related, with
B being the main ingredient which connects both the metric
and tetrad approaches.
After formulating the new f (T , B, Lm ) theory, the
conservation equation is obtained and exactly as in f (R, Lm ),
we will see that the conservation equation in f (T , B, Lm )
theory is not always valid. It will be proved that for the
flat FLRW case and assuming Lm = −2ρ, the
conservation equation is conserved exactly as happens in f (R, Lm )
or in f1(R) + f2(R)Lm (see [
42,43
]). The main aim of this
paper is to also formulate the dynamical system of this new
generalise theory, which is in general a ten-dimensional one.
This dynamical system is a generalisation of different
models such as the ones studied in [
43–45
]. After formulating
the full dynamical system, different special cases are
recovered. Some of them have been studied in the past, hence we
only mention how our dimensionless variables are related to
them and then we show that our dynamical system becomes
them for the special case studied. Then, using dynamical
system techniques, we will study new cases that can be
constructed from our action. Similarly as in f (R, Lm ) (see [43]),
a power-law and a exponential kind of coupling between Lm
and T is studied. Additionally, another new kind of
couplings between the boundary term B and Lm are studied.
For this theory, we study different power-law models with
f (T , B, Lm ) = C1T +C5 Bs +(C4 +C4 Bq )Lm . This model
depends highly on the power-law parameters s and q. The
critical points and their stability are then studied for different
models. For the readers interested on dynamical systems in
cosmology, see the review [
46
] and also see [
47,48
] for
further applications to dynamical systems in modified
teleparallel models with the boundary term B.
The notation of this paper is the following: the natural
units are used so that κ = 1 and the signature of the metric
is ηab = (+ 1, − 1, − 1, − 1). The tetrad and the inverse of
the tetrad are labelled as eμa and Eaμ respectively where Latin
and Greek indices represent tangent space and space-time
coordinates respectively.
This paper is organized as follows: Sect. 2 is devoted to
present a very brief review of teleparallel theories of
gravity and some interesting modified theories than can be
constructed from this approach. In Sect. 3 is presented the new
generalised gravity-matter coupled theory of gravity known
as f (T , B, Lm ) where T , B and Lm are the scalar torsion,
the boundary term and the matter Lagrangian respectively.
The corresponding field equations of the theory and the flat
FLRW cosmological equations are also derived in this
section. In Sect. 4 is presented the dynamical system of the
full model and for some specific theories, the
corresponding dynamical analysis of them is performed. Finally, Sect. 5
concludes the main results of this paper.
2 Teleparallel gravity and its modifications
Let us briefly introduce the teleparallel equivalent of general
relativity (TEGR) and some important modifications under
this theory. Basically, this theory is based on the idea of
having a globally flat manifold (zero curvature) but with a
nontrivial geometry for having a non-zero torsion tensor. Hence,
the concept of paralellism is globally defined in TEGR. The
dynamical variable of this theory is the tetrad which defines
orthonormal vectors at each point of the manifold and they
are directly related with the metric as follows
gμν = ηabeμaeνb,
where ηab is the Minkowski metric. The connection which
defines a globally flat curvature with a non-vanishing
torsion is the so-called Weitzenböck connection Wμa ν , which
(1)
defines the torsion tensor as taking its anti-symmetric part,
namely
Let us clarify here that the above definition is not the most
general form of the torsion tensor. The most general definition
also contains the spin-connection which needs to be pure
gauge in order to fulfil the condition of teleparallelism (zero
curvature). In this paper is assumed that the spin-connection
is identically zero.
The TEGR action is defined with the so-called torsion
scalar T as follows
where e = det(eμa) and Sm is the matter action. The torsion
scalar is defined as the contraction of the super-potential
= 41 (T abc − T bac − T cab) + 21 (ηac T b − ηab T c) (4)
with the torsion tensor as T = Tabc Sabc. Here, Tμ = T λλμ
is the so-called torsion vector. The definition of T comes
directly from the condition of zero-curvature where one
arrives that the Ricci scalar is directly linked with it via
2
R = −Sabc Tabc + e ∂μ(eT μ) = −T + B,
where B refers to the boundary term which connects the
Ricci scalar with the torsion scalar. From (3) and the above
relationship, one can directly notice that the TEGR is
equivalent to the Einstein–Hilbert action up to a boundary term.
Hence, TEGR is an alternative formulation of gravity which
reproduces the same field equations as GR. Although, the
geometrical interpretation of these theories are different. GR
lies in a manifold with a non-zero curvature (in general) with
a zero torsion tensor whereas TEGR is the opposite.
Moreover, geodesic equations are replaced by forces equations in
TEGR (see [
18
] for more details about this theory).
A straightforward generalisation of the action (3) is to
replace T by an arbitrary function of f which depends on T ,
namely
(2)
(3)
(5)
(6)
S f (T ) =
e f (T ) d4x + Sm.
The former theory is the most popular modification of TEGR
and it was firstly introduced in [
20
] with the aim to study
inflation in cosmology. In some sense, this generalisation is
analogous as the famous modification of GR, the so-called
f (R) gravity, where instead of having R in the Einstein–
Hilbert action, an arbitrary function of R is introduced in
the action. The formulation described here for f (T ) gravity
where the spin-connection is identically zero is not invariant
under Lorentz transformations. This is due to the fact that T
itself is not invariant under local Lorentz transformations so
f (T ) gravity will also have this property [
49,50
]. In
standard TEGR where T is in the action, this problem is not
important since the action only differs by a boundary term
with respect to the Einstein–Hilbert action so one can say
that this theory is quasi-invariant under local Lorentz
transformation. The problem of the loose of the Lorentz invariant
produces that two different tetrads could give rise different
field equations so it depends on the frame used. For example,
the flat FLRW in spherical coordinates give rise to different
field equations as in Cartesian coordinates. At the level of the
field equations, this problem can be alleviated by choosing
“good tetrads” as it was introduced in [51]. In this approach,
one needs to rotate the tetrad fields and fix it accordingly
depending on the geometry studied. In [
52
], it was proposed
a new approach of teleparallel theories of gravity where a
non-zero spin-connection is assumed giving rise to a
covariant version of f (T ) gravity. Both approaches should arrive
at the same field equations and since almost all the works
based on f (T ) gravity used the approach presented above,
we will continue using this approach.
It is also possible to create other kind of modifications of
teleparallel theories of gravity. A very interesting
modification theory is given by the following action [
35
]
S f (T,B) =
e f (T , B) d4x + Sm,
(7)
where now the function also depends on the boundary term
B. Under this theory, it is possible to recover either f (−T +
B) = f (R) gravity or f (T ) gravity. Moreover, the theory
f (T , B) = C1T + f1(B) can also be obtained from this
action. From this theory one can directly see how f (R) and
f (T ) are connected by this boundary term. Since R = −T +
B, only if a linear combination of R and T is assumed in the
action (TEGR or GR), we will have equivalent theories at
the level of the field equations. It is known that f (R) gravity
is a 4th order theory whereas f (T ) gravity is a 2nd order
theory. Hence, f (T , B) gravity is also a 4th order theory.
f (T ) and f (R) gravity have different field equation orders
since the difference comes from integrating by parts twice
the boundary term B.
3 f (T, B, Lm) gravity
3.1 General equations
Inspired by the theories described in [
53
] in the curvature
approach and also from f (T , B) gravity, let us now consider
the following gravity model
S f (T,B,Lm ) =
β 1 δ(eLm ) .
Ta = − 2e δeβa
e f (T , B, Lm ) d4x ,
where the function f depends on the scalar curvature T , the
boundary term B and the matter Lagrangian Lm . The
energymomentum tensor of matter Taβ is defined as
Now, we will assume that the matter Lagrangian depends
only on the components of the tetrad (or metric) and not on
its derivatives, giving us
∂ Lm
2Taβ = −Lm Eaβ − ∂eβa .
Now, by a variation of action (8) with respect to the tetrad,
we obtain
δS f (T,B,Lm ) =
e fT δT +e f B δ B+e fL δδLeβam δeβa + f δe d4x,
=
e[ fT δT + f B δ B − fL (2Taβ +Lm Eaβ )δeβa
+ f Eaβ δeβa ]d4x,
where we have used Eq. (10) and fT = ∂ f /∂ T , f B = ∂ f /∂ B
and fL = ∂ f /∂ Lm . Variations with respect to the torsion
scalar and the boundary term are given by [
35
]
1
e fT δT = −4e e ∂μ(eSa μβ ) fT − fT T σ μa Sσ βμ
e f B δ B = e[2Eaσ ∇β ∇σ f B − 2Eaβ
f B − B f B Eaβ
+ (∂μ fT )Sa μβ δeβa ,
− 4(∂μ f B )Sa μβ ]δeβa ,
so that by imposing δ S f (T,B,Lm ) =
f (T , B, Lm ) field equations given by
2Eaσ ∇β ∇σ f B − 2Eaβ
f B − B f B Eaβ
− 4[(∂μ fT ) + (∂μ f B )]Sa μβ
− 4 fT (e−1∂μ(eSa μβ ) − T σ μa Sσ βμ)
+ f Eaβ − fL Lm Eaβ = 2 fL Taβ .
The above field equations can be also written only in
spacetime indices by contracting it by eλa giving us
2∇β ∇λ f B − 2δλβ
β
f B − B f B δλ
− 4[(∂μ fT ) + (∂μ f B )]Sλ μβ
− 4 fT eλa(e−1∂μ(eSa μβ ) − T σ μa Sσ βμ)
+ f δλβ − fL Lm δλβ = 2 fL Tλβ .
From these field equations, one can directly recover
teleparallel gravity by choosing f (T , Lm ) = T + Lm which
gives us the same action as (3). Moreover if we choose
f (T , Lm ) = T + f1(T ) + (1 + λ f2(T ))Lm we recover
the non-minimal torsion-matter coupling extension of f (T )
gravity presented in [
41
]. Note that in our case, we have
assumed that the matter Lagrangian does not depend on the
derivatives of the tetrads, which according to [
41
] is
equivalent as having
(17)
(18)
(20)
∂ Lm
∂(∂μeρa ) = 0.
Let us now study the conservation equation for this theory.
First, we will use that Rλβ = Gλβ + 21 (B − T )δλβ , where Gλβ
is the Einstein tensor. Using this relationship, we can rewrite
the field equation (16) as follows
Hλβ := fT Gλβ + ∇λ∇β f B − gλβ f B
1
− 2 (T fT + B f B + Lm fL − f )gλβ − 2Xν Sλν β
= fL Tλβ ,
where for simplicity we have also introduced the quantity
Xν = ( f BT + f B B + f BT )∇ν B + ( fT T
+ fT B + fT L )∇ν T + ( fT L + f B L + fL L )∇ν Lm .
(19)
By taking covariant derivative of Hλβ and after some
simplifications, we find that
1
∇λ Hλβ = 2Sσρ λ Kβσρ X λ − 2 gλβ ∇λ(Lm fL )
1
= − 2 gλβ ∇λ(Lm fL ),
where we have used the fact that the energy-momentum
tensor is symmetric and hence Sσρ λ Kβσρ X λ = 0. The latter
comes from the fact that field equations are symmetric, and
hence the energy-momentum tensor is also symmetric. Now,
we will find the condition that f needs to satisfy in order
to have the standard conservation equation for the energy
momentum tensor, i.e., ∇μT μν = 0. By taking covariant
derivative in (18) and assuming ∇μT μν = 0, one gets that the
standard conservation equation for the energy-momentum
tensor is satisfied if the function f satisfy the following form
∂ Lm
(2Tμν + gμν Lm )∇μ fL = −eμagβν ∂eβa ∇μ fL = 0,
(21)
which matches with the conservation equation presented in
[
53
]. Note that in our case, we have defined the
energymomentum tensor in a different way so that there is a minus
sign of difference between Eq. (13) presented in [
53
] and the
above equation. Thus, in general, f (T , B, Lm ) is not
covariantly conserved and depending on the metric, the model and
the energy-momentum tensor, this theory may or may not
be conserved. Hereafter, we will consider that the matter is
described by a perfect fluid whose energy-momentum tensor
is given by
Tμν = (ρ + p)uμuν − pgμν .
(22)
Here, ρ and p are the energy density and the pressure of
the fluid respectively and uμ is the 4-velocity measured by
a co-moving observer with the expansion so that it
satisfies uμuμ = 1. For a perfect fluid, if one assumes that
in the proper frame where the particle is static, the
matter Lagrangian is invariant under arbitrary rescaling of time
coordinate [
54
]. Therefore, from (10), one gets T00 = ρ =
−(1/2)Lm which is equivalent as having Lm = −2ρ. This is
a “natural choice” for a perfect fluid (see [
41,42,54
] for more
details). Hence, from Eq. (21) we can directly conclude that
the conservation law will be always satisfied when flat FLRW
and a perfect fluid are chosen without depending on the model
for the function f (T , B, Lm ). This statement was also
mentioned in [
53
], which is a special case of our theory, explicitly
when f (T , B, Lm ) = f (−T + B, Lm ) = f (R, Lm ).
3.2 Flat FLRW cosmology
In this section we will briefly find the corresponding modified
flat FLRW cosmology of our theory. Consider a spatially flat
FLRW cosmology whose metric is represented by
ds2 = dt 2 − a(t )2(d x 2 + d y2 + d z2),
where a(t ) is the scale factor of the universe. The tetrad
corresponding to this space-time in Cartesian coordinates
reads
eβa = diag(1, a(t ), a(t ), a(t )).
For the space-time given by (23), the modified FLRW
equations become
1
3H 2(3 f B + 2 fT ) − 3H f˙B + 3 f B H˙ + 2 f = 0,
(3 f B + 2 fT )(3H 2 + H˙ ) + 2H f˙T − f¨B
1
+ 2 f = − fL ( p + ρ),
where H = a˙ /a is the Hubble parameter and dots
represent derivation with respect to the cosmic time. Note that
the terms f˙B = f B B B˙ + f BT T˙ + f B L L˙ m and f˙T =
f BT B˙ + fT T T˙ + fT L L˙ m . It is clear that when f (T , B, Lm ) =
T + Lm = T − 2ρ, one recovers standard TEGR (or GR)
plus matter. The energy density of matter does not appear in
(25) explicitly since it is implicitly considered in the term
f /2. When f (T , B, Lm ) = f (−T + B, Lm ) = f (R, Lm ),
the above equations are the same as the ones reported in [
43
].
Note that in the latter paper, the authors used another
signature notation ηab = (− + ++), so that one needs to change
R → −R to match those equations.
(23)
(24)
(25)
(26)
As a consequence of the conservation law holds when
considering a perfect fluid as a matter content of the universe,
we also know that the standard continuity equation is valid
in our case. Hence, we have that the fluid satisfies
ρ˙ + 3H (ρ + p) = 0.
Let us now assume a barotropic equation of state p = wρ, so
that we can directly find that the energy density of the fluid
behaves as
ρ(t ) = ρ0a(t )−3(1+w),
where ρ0 is an integration constant. It is also useful to note
that the scalar torsion and the boundary term in this
spacetime satisfy the relationship (5), namely
T = − 6H 2 ,
B = − 6(H˙ + 3H 2) ,
→ R = − T + B = − 6(H˙ + 2H 2) .
4 Dynamical systems
4.1 Dynamical system for the full theory
(27)
(28)
(29)
In this section we will explore the dynamical system of
different theories of gravity coupled with matter. To do this, we
will first study the dynamical system of the general modified
FLRW by using the conservation equation given by (27) and
also the first modified FLRW equation (25). By replacing the
boundary term given by Eq. (29) in (25) and expanding the
derivatives of f we get
6H 2 fT − 3H ( f B B B˙ + f BT T˙ + 6H (1 + w)ρ f B L )
− 21 B f B + 21 f = 0, (30)
where we have used the conservation equation (27) to replace
L˙ m = −2ρ˙ = 6Hρ(1 + w). Let us now introduce the
following dimensionless variables
T˙ fT B B˙ f B B B˙ f BT ,
x1 = 2H fT , x2 = 2H fT , x3 = 2H fT
x4 = 2T˙HfTfTT , y1 = 12BHf2BfT , y2 = 12THf2BfT = − 2ffBT ,
(31)
f
z = − 12 fT H 2 , φ =
α =
3(w + 1)ρ fT L
fT
, θ =
3(w + 1)ρ f B L ,
fT
(w + 1)ρ fL
2 fT H 2
.
(32)
These dimensionless variables were chosen with the aim
of having a similar variables as the ones presented in [
43
].
Further, using these variables will help us to compare both
theories in the limit case where f (T , B, Lm ) = f (−T +
B, Lm ) = f (R, Lm ). Using these variables, the Friedmann
constraint given by (30) becomes
x1 + x2 + y1 + z + φ = 1.
Moreover, using the dimensionless variables, we can find the
following useful relations
f˙T
2H fT
f˙B
2H fT
f¨B
2H 2 fT
= x3 + x4 + α,
= x1 + x2 + φ,
y1
= (x1 + x2 + φ) y2
d x1 d x2 dφ ,
+ d N + d N + d N
H˙ y1
H 2 = y2 − 3,
where we have defined N = ln a as the number of e-folding
so that d/dt = H d/d N . The effective state matter and
the deceleration parameter can be written in terms of these
dimensionless parameters as follows
ptotal
weff = ρtotal = −
2H˙
3H 2 + 1
For acceleration universes, one needs that q˜ < 0 or
equivalently weff < −1/3.
By replacing the identities (34)–(37) and the
dimensionless variables defined as (32) into the second modified
Friedmann equation (26), we get
d x1 d x2 dφ
d N + d N + d N
y1
= −(x1 + x2 + φ) 2(α + x3 + x4) + y2 − 3
y1
+ 2(α + x3 + x4) + y2 + 3y1 − 3z + θ .
The conservation equation (27) can be also written in terms
of the dimensionless variables, which yields
dφ
d N = φ[3(w + 1)(2βB L L − 1) + 2x3(βB B L − 1)
fT f B B B
αB B B = f B B fT B , αT T B =
fT fT T B ,
fT2B
αT B B =
fT fT B B
fT2B
fT fT B L .
, αT B L = f B L fT B
The other quantities defined above will be useful in the full
dynamical system equations. Using the dimensionless
variables, the identities mentioned before and the above
definitions, one can find the dynamical system. This procedure is
very involved since the dynamical system is a ten
dimensional one. After all of those computations, the system can
be summarized with the following equations:
d x1
d N = 2(αT T B x12 + αT B B x1x3 + x3) + y1
− 4αT B (α + x3 + x4 + 6) +
+ 2βT BL x4φ,
4αT B 2x1
y2 − y2
12αT B y2(α + x3 + x4 + 3)
y1
d x2
d N = θ + φ 3(w + 2) − 6(w + 1)βBL L − 2βB BL x3
y1
− 4βT BL x4 − y2
− 2αT T B x12 + α(4αT B − 2x1 − 2x2 + 2)
x1 x2 4αT B 1
+ y1 y2 − y2 − y2 + y2 − 3
+ x1(−2(αT B B + 1)x3 − 2x4 + 3) + x2(−2x3 − 2x4 + 3)
+ 4αT B (x3 + x4 + 6)− 12αT B y2(α +y1x3 + x4 + 3) +2x4 − 3z,
βT L L = fTffB2TLL L , βT T L = ffTB LfTfTT BL ,
B fT B
2 fT
, αT T T =
fT fT T T ,
fT2T
αT B =
(44)
(45)
(46)
(47)
(49)
d x3 x3
d N = − x2
+ 2αT T B x12 −
− 2α − θ + 6(w + 1)βBL L φ − 3(w + 2)φ
2αT B B x12 y1 y1 αxT1B − 2 + y2 6 − αT B
x3
y2(y1 − 3y2)
+ 4βB BL x3φ − 2x3 + 4βT BL x4φ − 2x4
+ y1(3y2 y+2 φ − 1) + 3z
x1x3(x1 y12 − x3 y1 y2 − 3αT B (y1 − 3y2)2)
+ αT B x2 y2(y1 − 3y2)
2αB B B x3(−x1 y1 + x3 y2 + 2αT B (y1 − 3y2))
+ 2αT T B x1x3 − y2
− 2αx3 + 2(αT B B − 1)x32
x3 y1
+ 2αT BL x3φ − 2x3x4 − y2
d x4
d N = 2βT T L x1φ + 2αT T B x1x3
+ 3x3,
+ x4
y1(x3 y2 − x1 y1) 2y1
αT B y2(y1 − 3y2) − 2x3 + y2 − 6
− 2αx4 + 2(αT T T − 1)x42,
ddyN1 = − y1(x1 + x2y2+ 2y1+φ) − 2y1(α + x3+x4 − 3) + xα3TyB1 ,
(48)
d y2 x1 y1
d N = αT B − x1 − x2 − 2(y2(α + x3 + x4 − 3) + y1) − φ,
dz x1 y1 2y1z
d N = −θ + 2αT B y2 − 2z(α + x3 + x4 − 3) − y2
Additionally, one can use the Friedmann constraint (33) to
reduce the above system to a nine-dimensional one. In the
following sections we will explore the dynamical system of
different kind of matter coupled theories of gravity which
can be obtained from our approach.
4.2 Specific model: f (T , B, Lm ) = f˜(−T + B, Lm )
= f˜(R, Lm ) gravity
From our action it is possible to recover a very interesting
model which comes from the curvature approach. As we
discussed before, this is possible due to the fact that the function
also depends on the boundary term B so that it is always
possible to reconstruct theories which contains the scalar
curvature R. In this sense, one can construct a non-minimally
coupled theory between the Lagrangian matter and the scalar
curvature, explicitly by taking the function being
f (T , B, Lm ) = f˜(−T + B, Lm ) = f˜(R, Lm ).
(55)
This kind of models was first proposed in [
53
] where the
authors suggested that in general, this theory has some extra
terms in the geodesic equation. The complete analysis of the
dynamical system for flat FLRW for this model was
studied in [
43
]. From our full dynamical system, it is possible
to recover the same dynamical system equations reported in
the former paper. Let us recall here again that in general, the
signature of the metric for curvature theories as f (R) gravity
is usually taken as the opposite as in teleparallel theories of
gravity and hence the paper [
43
] is written in another
signature notation compared to our notation. The important of this
notation issue is that the scalar curvature, scalar torsion and
also the boundary term will have a minus sign of difference
with respect to our case. Hence, in their notation Eqs. (45)–
(54) will have a minus of difference in all those quantities.
Therefore, to recover the same dynamical systems found in
[
43
] we need to change R → −R (and of course T → −T
and B → −B) which makes that the important derivatives
appearing in the dimensionless variables become
f B = − f R , fT = f R , f B B = fT T = − fT B = f R R .
In this case, some dimensionless variables can be reduced.
It is possible to connect our dimensionless variables to the
dimensionless variables used in the mentioned paper by
working with the variables
1
y = 2y1 − 1 , y2 = 2 , x = 2(x1 + x2) = −2(x3 + x4) ,
z˜ = 2z , φ˜ = 2φ = −2α , θ˜ = 2θ ,
where tildes represent the variables chosen in [
43
]. By
replacing (56) and (32) in our dynamical system we directly find
that the corresponding dynamical system becomes a
fivedimensional one explicitly given by
d x
d N = x [x − y + φ˜ (1 + βB B L )] − 1 − y − 3z˜ + θ˜
+ φ˜ [3(1 + w)(2βB L L + 1) − y] ,
d y x x
d N = − 2αT B − y 2αT B + 2y − 4 ,
d z˜ x
d N = z˜(x + φ˜ + 2(2 − y)) − θ˜ − 2αT B (1 + y) ,
dφ˜
d N = φ˜ [x (1 − βB B L ) − 3(1 + w)(2βB L L + 1) + φ˜ ],
dθ˜
d N = θ˜ (6(w + 1)βL L + x − 2y − 3w + 1) + φ˜
(56)
(57)
(58)
(59)
(60)
(61)
(62)
+
x (y + 1)φ˜
2αT B
.
The above equations are the same reported in [
43
] for
f (R, Lm ) gravity if one changes the variables βB B L , βB L L ,
αT B and βL L accordingly. In this theory, it is possible to
reconstruct different interesting gravity-matter coupled
models as for example standard non-minimally curvature-matter
coupled models where f (R, Lm ) = f1(R)+ f2(R)Lm which
has been studied in the literature (see [
39
]). In [
44
], it was
studied the dynamical system for some of those models.
To find all the most important details regarding the above
dynamical system for the former model and also for other
more general models in f (R, Lm ) gravity, see [
43
].
4.3 Specific model: f (T , B, Lm ) = f˜(T , Lm ) gravity
Let us now introduce a new theory of gravity based on an
arbitrary function f which depends on T and Lm only. As
f (T ) was motivated by f (R) gravity, f (T , Lm ) gravity is
somehow, the teleparallel version of f (R, Lm ) discussed in
the previous section. Different particular cases of this
theory have been studied in the past. Let us first derive the full
dynamical system for the f (T , Lm ) gravity and then study
some particular theories. The Friedmann equation (33) for
this model reads
(63)
z = 1.
In this case, y1 = y2 = x1 = x2 = x3 = φ ≡ 0, so one needs
to be very careful with the general dynamical system (45)–
(54) since some of these equation will be also identically
zero. Let us clarify here the way that one needs to proceed
to find the correct dynamical system. There are two ways to
find out the correct dynamical system for an specific model.
Let us discuss how to proceed with the model that we are
interested here, i.e., where f = f (T , Lm ). The first way
to proceed is using the full dynamical system described by
(45)–(54). If one directly replaces f = f (T , Lm ) in the full
dynamical system, there will be some expressions that are
indeterminate or directly zero, for example terms like y1/y2
or terms divided by x2. Hence, one first needs to replace back
all the original definitions of the dimensionless variables and
after doing that, one can restrict f = f (T , Lm ). By doing
that, several equations are directly satisfied. Indeed, one can
verify that Eqs. (45), (47), (49), (50) and (52) are identically
zero, as expected. Then, for all the remaining equations, one
needs to introduce again the dimensionless variables needed
(in this case x4, α and θ ). A second approach is to directly
assume f = f (T , Lm ) in the Friedmann equations (30)–
(26) and then introduce the same dimensionless variables that
we defined. By doing that, we arrive of course at the same
dynamical system as the first approach. We will implement
the first procedure in this work. Eq. (46) gives us a constraint
for the variables, namely
y1
y2
= −2α − θ − 2x4 + 3z.
Let us here clarify again that even though y1 = y2 ≡ 0,
the quotient y = y1/y2 = B/ T = 3 + H˙ /H 2 is clearly
non-zero. If we replace the above equation and also use the
Friedmann constraint (63), the remaining three Eqs. (48),
(53) and (54) gives us the following set of equations,
d x4 1
d N = − 2α + θ [2αβ˜T T L (2α + θ )2
+ x4(4α2(6β˜T T L − β˜T L L + 3)
dα
d N = α[2α(β˜T L L − 2β˜T T L − 1) − 2β˜T T L θ
− 2(2β˜T T L + 1)x4 − 3w − 3],
dθ
d N = 4αθ + 4α2 + θ (2θ + 6(w + 1)γL L − 3w − 3)
+ 2x4(2α + θ ),
− 4(αT T T − 3)x43],
+ 6α(2β˜T T L θ +θ + w + 1)−3(w + 1)(2γL L − 1)θ )
2
+ x4 (−4α(αT T T − 4β˜T T L − 6) − 2(αT T T − 4)θ )
where
P = −
(64)
(65)
(66)
(67)
(70)
(72)
(73)
(74)
(75)
where for convenience we have introduced the following
quantities
β˜T L L = ffTB22LL βT L L = fTffT2TLL L , β˜T T L = T ffTT LT L .
In the following section we will study some interesting cases
that can be constructed from this theory.
(68)
4.3.1 Nonminimal torsion-matter coupling
f (T , Lm ) = f1(T ) + f2(T )Lm
In this section we assume that the function takes the following
form,
f (T , Lm ) = f1(T ) + f2(T )Lm ,
(69)
where f1(T ) and f2(T ) are arbitrary functions of the torsion
scalar. This model is an extension of f (T ) gravity, where
an additional nonminimal coupling between the torsion and
the matter Lagrangian is considered [
41
]. In this model we
have that γL L = β˜T L L ≡ 0. In [
45
], the dynamical system
of this model was carefully studied. The authors used other
dimensionless variables but one can verify that our dynamical
system (65)–(67) give rise to the the same dynamics. In that
paper, the authors used a different energy density which is
related to our as ρnew = −2ρ. The dimensionless variables
used in [
45
] are given by
f2(T )
Y = 12H 2 f2(T ) ,
f1(T )
X = 12H 2 f1(T ) ,
ρ
= − 3H 2 .
X = 2θα−+33((ww++11)) ,
It is possible to show that our variables are related to them
as follows
Y = 2θα , (71)
which gives us the following set of equations,
d X
d N = −
3(w + 1)Q(X + 1)(Y + 1)(2(W + 1)X + 1) ,
P(X + 1) − Q(2W (Y + 1) − X + Y )
dY
d N = −
3(w + 1)(X + 1)(Y + 1)( PY + 2QY + Q) ,
P(X + 1) − Q(2W (Y + 1) − X + Y )
X = − 1 − 2 Q(Y + 1),
T 2 f2 (T )
f1(T )
It can be shown that the Eqs. (72)–(74) are equivalent to
our equations (65)–(67) if the corresponding (71) is used
properly. This for sure is a good consistency check that our
equations are correct. The full study of the dynamical system
(72)–(74) was carried out in [
45
] where 6 different kind of
functions f1(T ) and f2(T ) were assumed. For some of those
models, they found some critical points representing
accelerating or decelerating solutions and also scaling solutions.
For more details about all of this models and their dynamical
analysis, see [
45
].
4.3.2 Exponential couplings for f (T , Lm ) gravity
Now, let us study a new model where the function takes the
following form
f (T , Lm ) = −
exp
−
(T + Lm ) ,
1
(76)
(77)
(78)
(79)
where is a positive cosmological constant. Let us take a
look at this model further. In the limit where the argument is
much less than one ( 1 (T + Lm ) 1), if one expands up to
first order in the argument, the function becomes
f (T , Lm ) ≈ −
+ T + Lm + · · · ,
hence in that limit, one recovers the TEGR plus matter case
with a cosmological constant. Therefore, the function (76)
is an interesting model to take into account. An analogous
model was proposed in [
53
], where instead of having T , the
authors considered the scalar curvature R. The dynamical
system of the former model was investigated in full detail in
[
43
].
Under this theory, we directly find that β˜T L L = αT T T = 1
and by manipulating the definitions of the other quantities,
βL L and β˜T T L can be written as
α α
βL L = 3(1 + w) , β˜T T L = − θ .
From the Friedmann constraint (63), Lm = −2ρ and by
using Eqs. (28) and (29) we directly find that for this model,
the universe is always expanding as a De-Sitter one with a
scalar factor being equal to
t
a(t ) ∝ e± 2 3 .
It is interesting to see that actually this model is very
different to its analogous in f (R, Lm ). In the model f (R, Lm ) =
− exp[−(R + Lm )/ ], it is not possible to directly find an
unique scale factor which rules out the whole dynamic for the
model. Hence, in that case, the dynamical system technique
is very useful to check how the dynamics evolves on time. In
our case, since the dynamics is always the same (described
by the above equation) it is not important to study its
dynamical properties since the solution is the standard De-Sitter
universe. Therefore, the model described by an exponential
coupling between Lm and T as it is given by (76) mimics a
De-Sitter universe.
4.3.3 Power-law couplings f (T , Lm ) gravity
Let us consider another interesting new model that one can
consider from our approach where the function takes the
following form
f (T , Lm ) = M − (T + Lm )1+ ,
where is a constant and M is another constant which
represents a mass characteristic scale. In this case, up to first order
in , the expansion of the above function becomes
f (T , Lm ) ≈ T + Lm + (T + Lm ) log
so that since is assumed to be very small (comparable with T
and Lm ), the above model could represents a small deviation
of the standard TEGR plus matter case. For this model we
find that
T + Lm
M
,
(81)
β˜T L L = αT T T = 1 −
α
βL L = 3(1 + w) .
−1, β˜T T L = −
( − 1)α
θ
,
Similarly as we did in the previous section, from the
Friedmann constraint (63), by replacing Lm = −2ρ and by using
Eqs. (28) and (29) we find the following equation for the
scale factor,
(ρ0 + 3a3w+1a˙ 2) (ρ0 − 3(2 + 1)a3w+1a˙ 2) = 0,
(83)
which gives us two different types of scale factors. One can
directly check that if = 0, the above equation is reduced
to the standard TEGR plus matter case, namely 3H 2 = ρ.
For the specific case where = −1/2, we must need ρ =
0, so that this special case is not a reliable model. There
is no point on going further with the dynamical system of
this model since the equation can be directly solved for the
scale factor. The above equation depends on the power-law
parameter . For negatives values of , the only possibility is
that the second bracket is zero whereas for positives values of
, there will be two kind of possible scale factor. This is again
different as the case f (R) = M − (R + Lm )− +1 studied in
[
43
]. Our model seems to be simpler than the former one due
to the fact that T only contains derivatives of a(t ) and not
second derivatives as R.
Let us know explore what kind of solutions we have from
our power-law model. The first type can be obtained by
assuming that the first bracket is zero, which is only valid
for > 0 giving us the following scale factor,
a±(t ) =
3 3w1+3
4
2
±i √ρ0t (w + 1) 3(w+1) , where
(80)
(82)
> 0,
(84)
where for simplicity we have chosen that the integration
constant is zero. Let us clarify here that this scale factor will rule
kind of theories have not been considered in the past, but
there are some studies for the specific case f˜(B, Lm ) =
f (B) + Lm , which is known as f (B) gravity [
36,37,55
].
The full dynamical system (45)–(54) is simplified since x1 =
x3 = x4 = α ≡ 0 which implies that Eqs. (45), (47), (48)
and (53) are also automatically zero. Hence, in our variables,
this theory is a five-dimensional dynamical system given by
d x2
d N = −6(w + 1)βB L L φ + θ + 3(w + 3)φ
+x2(6 − 2β˜B B L φ) − 3 −
y2
x2(y1 + βB B y2) + y1(2y1 − 6y2 + φ)
y1(x2 + φ − 1)
,
,
y2
d y1
d N = −
d y2
d N = −(x2 + φ),
dφ
d N = 2(3βB L L − 1)φ (w + 1) + 2β˜B B L x2φ,
dθ
d N = 3θ (2(w + 1)βL L − w + 1) +
βB B x2φ − 2θ y1
y2
out the dynamic only for > 0. The scale factor must be real
and positive so we must ensure that the imaginary term
disappears. This is possible for some values of w. If one assumes
that w > −1, for the solution a+(t ), the state parameter
must satisfy w+ = 61k − 1 for any positive integer number
k whereas for the solution a−, the state parameter must be
w− = 61k1 − 1 to ensure a positive real value of a±(t ).
Moreover, for these two solutions, a˙± > 0 and a¨± > 0 for both
w±, so this solution could describe an accelerating
expanding universe for those specific values of w±. However, only
the solution a− with w− = 11/6 ≈ 1.8 represents
powerlaw expanding accelerating universes without evoking exotic
matter.
Additionally, Eq. (83) can be solved by letting the
second bracket equal to zero, which is valid for all = −1/2,
yielding
(85)
(86)
a(t ) =
3ρ0
4(2 + 1)
1
3(w+1)
where again for simplicity we have assumed that the
integration constant is zero. This solution is very similar to (84)
but now the parameter plays a role in the dynamics of the
universe. Let us again consider the case where w > −1 for
studying this solution. For > −1/2, the scale factor and
its derivatives are always positive so that the scale factor will
mimic a power-law accelerating universe. For < −1/2, we
need to impose that
1
w = −1 + 6k , where k ∈ Z+,
otherwise the scale factor would be negative. Moreover, all
the derivatives of the scale factor would be also positive if w
satisfy the above condition .Hence, only special cases of w
will give rise to viable models when < −1/2. Further, all
those models are in the regime −1 < w < 0 which represents
exotic kind of matter. Thus, cases with < −1/2 needs
exotic matter to represent accelerating expanding universes.
Additionally, we can conclude that for > 1/2, the
powerlaw f (T , Lm ) will mimic power-law accelerating universes
without evoking exotic matter.
4.4 Specific model: C1T + f (B, Lm ) gravity
In this section we will study the case where the function takes
the following form
f (T , B, Lm ) = C1T + f˜(B, Lm ),
where C1 is a constant and the function f˜(B, Lm ) depends
on both the boundary term and the matter Lagrangian. The
first term represents the possibility of having TEGR (or GR)
in the background when we set C1 = 1. If this term does
not appear in the function, it is not possible to recover GR
since one cannot construct GR from f˜(B, Lm ) gravity. This
(87)
where for simplicity we have introduced the following
quantities
β˜B B L = βB B L
fT B fT f B B L , βB B = T ffBB B .
f B B = f B L f B B
Let us now concentrate on a specific model based on the
boundary term non-minimally coupled with the the matter
Lagrangian where the function takes the following form
f (T , B, Lm ) = C1T + f1(B) + f2(B)Lm ,
where C1 is a constant and f1(B) and f2(B) are functions
which depends on the boundary term B. This case is
analogous to the one studied in Sect. 4.3.1, but the dynamical
system is more complicated to deal since it is a five
dimensional one. The aim of this section is to study some specific
cases that can be constructed from the above model.
Let us further study the case where the functions are a
power-law type given by
f1(B) = C5 Bs , f2(B) = (C4 + C3 Bq )Lm ,
(95)
where C3, C4, C5, q and s are constants. Since we are
interested on studying non-trivial couplings between B and Lm
we will assume that C3 = 0. We directly find that βL L =
βB B L = 0. It can be proved that for this model, the
dynamical system can be reduced from five dimensional to a four
dimensional one. For this case, the dynamical system is
difficult to study. However, if one assumes that the exponents
are related as
(88)
(89)
(90)
(91)
,
(92)
(93)
(94)
q = 1 − s,
the system becomes easier to work since it becomes a three
dimensional dynamical system. Then, we will split the study
depending on different cases which depends on the constants.
4.4.1 f (T , B, Lm ) = C1T + C5 Bs + (C4 + C3 B)Lm
Let us first study a very special case where q = 1 in (95)
giving us a linear coupling between the boundary term B and the
matter Lagrangian Lm . This model will depend on the
powerlaw parameter s and also on the constants C3, C4 and C5. In
this model, one can relate two dynamical dimensionless
variables with the other ones making it a three dimensional one.
In our case, we will replace θ and y1 as follows
θ = −
y1 =
d x2
d N =
dφ
d N = −3(w + 1)(φ + x2),
dθ
d N =
3(w + 1)
φ
P : (x2, y2, φ) =
0,
which depends on the power-law parameter s. The case s = 1
can be discarded since a linear combination of the boundary
term does not affect the field equations. It is easily to see that
the effective state parameter for this critical point is always
−1, hence this critical point always represents acceleration.
To find out about the stability of this point, one needs to
check the eigenvalues evaluated at P. There are three
different eigenvalues given by
3
− 3(1 + w), − 2 ±
3√(8 − 7s)s
2s
.
(100)
One can directly see that when 1 < s ≤ 8/7 and w > −1,
the critical point P is stable.
4.4.2 f (T , B, Lm ) = C1T + (C4 + C3 Bq )Lm
Let us now consider the case where C5 = 0 in (95). This
model represents the case where f1(B) = 0. Let us also
assume that q = 1 to do not have the same model as the
previous case. For this case, it is possible to relate the terms
β˜B B L and βB B with the dynamical variables as follows
β˜B B L = −
βB B = −
Moreover, the dynamical system can be reduced from 5D to
3D since some of the variables are directly related, namely
(101)
(99)
This dynamical system has only one critical point given by
(x2∗, φ∗, θ∗) =
q(w + 1) q(w + 1) 3w + 3
, ,
− 2q + w − 1 2q + w − 1 2 − 2q
where we have assumed that q = (1 − w)/2. This point of
course depends on the parameters q and w. For this point,
there is acceleration when
q1 +− wq < − 13 ⇒ q > 21 (−3w − 1), (107)
where we have assumed that w > −1. Further, for the dust
case w = 0 we can see that this point requires q > −1/2
to represent and accelerating universe. It is also possible to
check that there are three Eigenvalues associated with this
C3(2φ + 2x2 − 1) C1φ
1 s
C12s(φ−3(w+1)y2) s−1 − C5s(w + 1) 2C1(φ−3(w+1)y2) s−1
C5s(w+1) C5s(w+1)
2C3C5(s − 1)(w + 1) 2C1(Cφ5−s(3w(w++1)1)y2) s −s1 + 4C1C43 s−11 φ
.
3C1(w + 1)φ (2φ + 2x2 − 1) C3
,
Thus, it is possible to replace the above equations into
Eqs. (90), (91) in order to reduce the dimensionality of the
dynamical system for this model. By doing that, we find that
the model only has one critical point given by
By replacing (101) and (102) in the dynamical system
(88)(92) we find that this system is reduced as follows
θ (4φ − 2) + 3(4(w + 2)φ2 − (3w + 5)φ + w + 1) + x2(2θ + 6(3w + 5)φ − 9(w + 1)) + 6(w + 1)x22
2φ
,
(97)
(98)
(103)
(104)
(105)
(106)
Fig. 1 Region plot for the state parameter w and the power-law
parameter q for the model described by (95) with C5 = 0. The figure represents
the regions where the point (106) is stable. The blank regions represent
the regions where the point is a saddle one
point. Those Eigenvalues are very long to present here but
Fig. 1 represents a region plot where the point is stable. Note
that besides of the values of q and w, this point is never
unstable.
4.4.3 f (T , B, Lm ) = C1T + C5 Bs + (C4 + C3 B1−s )Lm
Let us now assume the case where q = 1 − s and
the constant C5 = 0 which is a more generic model
which has an additional boundary power-law
contribution. As we have studied in the previous section, we can
again reduce the dynamical system as a three-dimensional
one. However, the models is much more complicated than
the previous two models. The dynamics of the model
highly depends on the parameter s. We can relate the
terms β˜B B L and βB B with the dimensionless variables
but now those quantities are very long for a generic
s. Moreover, those terms make the dynamical system
very long and difficult to treat for any s. One can also
relate two dimensionless variables with the other ones.
In this case, we will choose to work with the variables
(y2, θ , φ) since the dynamical system is slightly easier to
work with them. The variables x2 and y1 are then given
by
1
x2 = 6(w + 1)φ(2C1C4φ − 6C1C4(w + 1)y2 + 3C3C5s(w + 1))
[φ(−2θ (2C1C4φ + 3C3C5(2s − 1)(w + 1))
−3(w + 1)(2φ − 1)(2C1C4φ + 3C3C5s(w + 1)))
−6(w + 1)y2(θ 3C3C5(s − 1)2(w + 1) − 2C1C4φ
−3C1C4(w + 1)φ(2φ − 1))],
(108)
y1 = −
It is possible to write down the dynamical system for any
generic s but it is very long a cumbersome to present it here.
Moreover, the critical points highly depend on the parameter
s and it is not possible to obtain all the possible critical points
for any arbitrary s. Hence, we will only study some particular
models. We will concentrate only on models with integer
values of s. Table 1 represents various models with their
critical points, effective state parameter and their acceleration
regime. In general, for all the critical points for those models,
it is possible to have acceleration for the dust case w =
0.
It is also important to mention that for s ≥ 2, all the models
have only one critical point with the possibility of
describing acceleration depending on the state parameter w. It can
be proved that the critical points in the models s = 3, 4, 5
(there is only one critical point for each model) are always
saddle points. The model s = 2 can be either a saddle point
or an unstable point. Hence, for all of positive models of s,
the critical points cannot be stable. When negatives values
of s are considered, the system becomes more complicated.
For s ≤ −3, the dynamical system becomes highly
complicated to analyse. For the case s = −1, the critical point
is either a saddle or an unstable point so it cannot be
stable. Moreover, for the dust case (w = 0), the critical point
for the model s = −1 is always unstable spiral. For the
case s = −2, there are two critical points P1 and P2 (see
Table 1). The critical point P2 is either a saddle point or
stable whereas the point P1 is always a saddle point.
Figure 2 represents the regions where the point P2 is stable.
It is important to mention that only the term C = C3C5
appears in the Eigenvalues so that it is possible to make
2D region plots for the model. In this figure, it was
considered the case C1 = C4 = 1 which is equal to consider the
standard General Relativity plus matter model in the
background.
5 Conclusions
In this work we have presented a new modified theory of
gravity based on an arbitrary function f which depends on
the scalar torsion T , the boundary term B and the
matter Lagrangian of matter Lm . Different kind of modified
theories of gravity can be recovered from this theory. The
incorporation of B in this function is with the aim to have
the possibility to recover and connect standard metric
theories based on the curvature scalar. This is possible since
R = −T + B, so that it is possible to recover the generalised
curvature-matter Lagrangian coupled theory f (R, Lm ).
Figure 3 shows the most important theories that can be
conm
L
)
s
−
1
B
3
C
3 9
/ /
1 5 1
3
/
5
−
d 1 2 3
n f
e f − 2 − 3 − 4 (
p e
e w w w w
d
1 02
)
7
− 2 9
0
w + 1
3
) 5C 30
w +
( 8
− 9
tsn 604
iac 3)
i
po 9 , 1
l + )
1
it 4
r
c + (w
3
+ 4
e
h w
t 3 , 2 −
o 2( )φ w w
f
r 3 , 3 (
tee C5 ,θ 1 − 3
2 9 −
m C y
a 0 (
w
,
)
5 w
(
3
)
1 5
)
+ − 1 7
)
1 9
)
+ − 1 3
(w w + +
1 w
+ −
−
w 3 , 3
4
(w w + − −
2 , 2
(w w ) ,
1 )
1
(w w
− + 0
, 1 +
+ 2 ) w
1 3( (w
+ 8 , 3 2
) −
9 ,
)
4 − 3
,
w
1 −
7 −
w
( 2 +
3
w
(
3 1
P
e 3
tiffecev 53w+ 2B5+ L4m 3B5+ L4m 4B5+ L4m 5B5+ L4m C5 +B LC)4m C5 +2B 1L)m
d 7( C C C C C C C C + + +
an 2) l + + + + + + + + 2 3
ino C3 edo T1 C3 B T1 C3 2B T1 C3 3B T1 C3 4B T1 B3 + B3
t 5 C C
ra C M C C C C C ( T (
e 5(
l
e 2
1 2
3 C C 3 4 0
< C 5 5 C + 7 0
5 C C 5 3 4 37
w C 0 0 C C √
8 8 5 2
& ≤ 9 3 ≤ C
0 3 4
49 438 − 49 922 8035 4
9
< 5
−
9
3 5
5 3
7
>
−
)
1
+
5
(w5 C3
C C
3 0
3
< > C −
5
w w C ∨
−
w
−
∨
∨
,
)
3
+
w
3
7
(
w
−
)
3
, 5
+
)
3
4
+
8
9
) 5 0
+ 1 C 4
3 2
w + C 2
3 5
2( 5w 4
5 (
+
C 6
3 3 4
C 3 5
1
5 (
1 )
+
1
(
− 3
)
+
)
w
7
3
1
+
8
9
(
)
w
+
1
(
− )
1 1
1
( +
5
C w
3 5
(
C 2
5 1
1 1
−
−
4
5
2
2
9
4
+
−
3
C
5
≤
8C 3
<
43 C5 3 8C 3
C 43 C5
− 5C 5
≤ √ 3 C − 5C 9 5
3 9
9 < √ 3 3 3
w √ 7 3 9 5 7
9
<
>
w
&
9
8
9
2
9 5 560 61 √60 7 3+ 3+ 0 ≤− 39 5 74− 300
53 73 + − 5 C5 C5 < C35 5− 73+ √48 7
9
3 +
C 3
5 C
C 5
0 C
8 0
9 8
3 3
4
−
9
8
9
2
<
w
&
−
≤
3
C
5
C
∨
+
)
1
+
5
(w5 C3
C C
3 0
5C 30
9
0
1
8
9
+
,
)
3
+
)
3
5
+
)
w
3
7
( +
5
,
+
8
9 0 8
C 4 9
− ) 3 2
) 1 C 2 − )
3 5 ) 1
4 + 4 1
1 +
+ w +
w ( 4 − 5w
5
32 63 51 w 2(
3 3 () 7 1
13 1
C
5
C
5
1
2 −
P
w (
3
+
C
1 5
(
3 C
5
− 1
1
e
l
b
a
T s
Fig. 2 Region plot for the critical point P2 for the model s = −2 for
the constants C1 = C4 = 1 and C = C5C3. The figure is representing
the regions where the critical point for that model (see Table 1) is stable.
The point is never unstable. All the blank regions represents the regions
where the point is a saddle point
structed from our action. The graph is divided into three
main parts. The left part of the figure represents the
scalarcurvature or standard metric theories coupled with the
matter Lagrangian. Different interesting cases can be recovered
from this branch, such as a generalised f (R, Lm ) theory
or a non-minimally scalar curvature-matter coupled
gravity f1(B) + f2(B)Lm or just standard f (R) gravity. The
entries at the middle of the figure represent all the theories
based on the boundary term B and the matter Lagrangian
Lm . In this branch, new kind of theories are presented based
on a general new theory C1T + f (B, Lm ), where the term
C1T is added in the model to have TEGR (or GR) in the
background. The right part of the figure is related to
teleparallel theories constructed by the torsion scalar and the
matter Lagrangian. Under these models, a new general theory
f = f (T , Lm ) is highlighted in box, allowing to have new
kind of theories with new possible couplings between T
and Lm . As example, in this paper we have considered
theories with exponential or power-law couplings between T
and Lm . Under special limits, these theories can represent
a small deviation of standard TEGR with matter with or
without a cosmological constant. As special case, this
theory can also become a non-minimally torsion-matter coupled
gravity theory f = f1(T ) + f2(T )Lm , presented previously
in [
53
]. Thus, different gravity curvature-matter or
torsionmatter coupled theories can be constructed. Some of them
have been considered and studied in the past but others are
new. The relationship between all of those well-known
theories have not been established yet. From the figure, one
can directly see the connection between modified
teleparallel theories and standard modified theories. The quantity
B connects the right and left part of the figure. Hence, the
connection between the teleparallel and standard theories is
directly related to this boundary term B. Therefore, one can
directly see that the mother of all of those gravity theories
coupled with the matter Lagrangian is the one presented in
this work, the so-called f (T , B, Lm ).
In this work, we have also studied flat FLRW cosmology
for the general f (T , B, Lm ) theory of gravity. Explicitly, we
have focused our study on the dynamical systems of the full
theory. In general, the theory is very complicated to work
since it becomes a ten dimensional dynamical system. This
is somehow expected since the theory is very general and
complicated. Using the full dynamical system found for the
full theory, we then study different special interesting theories
of gravity. For the case f = f (−T + B, Lm ) = f (R, Lm ),
it was proved that our full dynamical system becomes a
fivedimensional one. Moreover, we have proved how one can
relate our dimensionless variables with the ones used in [
43
]
giving us a possibility of checking our calculations. We have
found that the dynamics of this model is the same as it was
described in [
43
].
The case f = f (T , Lm ) is also studied, where in
general the dynamical system can be reduced to be a
threedimensional one. This theory is analogous to f (R, Lm ) but
mathematically speaking, it is different. It is easier to solve
analytically the flat modified FLRW for a specific model for
the f (T , Lm ) than f (R, Lm ). Further, for the later theory,
for the exponential/power-law curvature-matter couplings
one needs to study the dynamical system to understand the
dynamics. For the f (T , Lm ) case, the
exponential/powerlaw torsion-matter couplings are directly integrated, giving
us a scale factor of the universe directly from the
modified FLRW equations. Hence, one does not need
dynamical system technique to analyse the dynamics of those
two examples. Another special interesting case studied was
f (T , Lm ) = f1(T ) + f2(T )Lm . The dynamical system for
this case is reduced as a two-dimensional one. We have
proved that our dimensionless variables can be directly
connected to the ones introduced in [
45
]. This also gives us a
good consistency check that our full ten-dimensional
dynamical system is correct mathematically, at least for those
special cases. Thus, the dynamics of those models are consistent
with the study made in [
45
].
Finally, we have also studied the dynamics of modified
FLRW for C1T + f (B, Lm ) gravity using dynamical
system. The dynamical system for this case becomes a
fivedimensional one, exactly as the f (R, Lm ) case. The
dynamics for this model is more complicated than f (T , Lm ). This
is somehow expected since B contains second derivatives
of the scale factor and T only contains first derivatives of
the scale factor (see Eq. 5). Further, R also contains
secf = f( −T + B, Lm)
f (T, B, Lm)
f = f(T, Lm)
f = C1T + f (B, Lm)
f (R, Lm)
C1T + f (B, Lm)
f (T, Lm)
f = f1(R) + f2(R)Lm
f = f1(B) + f2(B)Lm
f = f1(T ) + f2(T )Lm
f1(R) + f2(R)Lm
C1T + f1(B) + f2(B)Lm
f1(T ) + f2(T )Lm
ond derivatives of the scale factor, exactly as B, so it is
not so strange to see that the dimensionality of the
dynamical system of f (R, Lm ) is the same as C1T + f (B, Lm ).
Under the boundary-matter coupled model, we have
studied a specific case where the matter Lagrangian is
nonminimally coupled with B as f1(B) + f2(B)Lm . By
assuming some power-law boundary functions f1(B) = C5 Bs and
f2(B) = (C4 + C3 Bq ), we analysed the dynamics using
dynamical system techniques. In general, the dynamical
system for this power-law couplings are four dimensional but for
the specific case where q = 1 − s, becomes a three
dimensional one. Thus, we have analysed this model depending on
three different limit cases: (i) q = 1, (ii) C5 = 0 and lastly
the case (iii) C5 = 0, q = 1 − s. In general, the dynamics of
all of these models are similar. As we have seen, mainly only
one critical point is obtained for mainly all of them. The
stability of those points were also studied, showing the regions
where the critical points become stable.
As a future work, it might be interesting to study further
other models that can be constructed from the full theory.
In principle, one can use the same ten dynamical system
that we constructed here, and then simplify it by assuming
other new kind of couplings between T , B or Lm . In
addition, one can also use the reconstruction technique to find
out which model could represent better current cosmological
observations. Further, we can also incorporate the teleparallel
Gauss–Bonnet terms TG and BG to have a more general
theory f (T , B, Lm , TG , BG ) (see [
56
]) or even a more general
new classes of theories based on the squares of the
decomposition of torsion Tax, Tvec and Tten (see [
57
]). Then, one
can study the dynamics of the modified FLRW for this
general theory. By doing all of this, it will give a powerful tool
to determine which models are better describing the current
acceleration of the Universe, or other cosmological important
questions.
Acknowledgements The author would like to thank Christian Böhmer
for his invaluable feedback and for helping to improve the manuscript.
The author is supported by the Comisión Nacional de Investigación
Científica y Tecnológica (Becas Chile Grant no. 72150066).
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.
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