Reconstruction, Thermodynamics and Stability of $\Lambda$CDM Model in $f(T,\mathcal{T})$ Gravity

0.33 MB·English

by  Ednaldo L. B. Junior

Reconstruction, Thermodynamics and Stability of ΛCDM Model in f(T, ) Gravity
T
Ednaldo L. B. Junior(a,e),∗ Manuel E. Rodrigues(a,b),† Ines G. Salako(c),‡ and Mahouton J. S. Houndjo(c,d)§
(a) Faculdade de F´ısica, PPGF, Universidade Federal do Par´a, 66075-110, Bel´em, Par´a, Brazil
(b) Faculdade de Ciˆencias Exatas e Tecnologia, Universidade Federal do Par´a
Campus Universita´rio de Abaetetuba, CEP 68440-000, Abaetetuba, Par´a, Brazil
(c) Institut de Math´ematiques et de Sciences Physiques (IMSP), 01 BP 613, Porto-Novo, B´enin
(d) Facult´e des Sciences et Techniques de Natitingou - Universit´e de Parakou - B´enin and
(e) Faculdade de Engenharia da Computac¸˜ao, Universidade Federal do Par´a,
Campus Universitrio de Tucuru´ı, CEP: 68464-000, Tucuru´ı, Par´a, Brazil
We reconstruct the ΛCDM model for f(T, ) Theory, where T is the torsion scalar and
T T
the trace of the energy-momentum tensor. The result shows that the action of ΛCDM is a
6 combination of a linear term, a constant ( 2Λ) and a non-linear term given by the product
1 −
√ TF (T1/3/16πG)(16πG +T +8Λ) ,withF beingagenericfunction. Weshowthattomain-
0 − gh T i g
2 tainconservationofenergy-momentumtensorshouldimposethatFg[y]mustbelinearonthetrace
. Thisreconstructiondecaysinthef(T)TheoryforF Q,withQaconstant. Ourreconstruction
y T g ≡
describesthecosmological erastothepresenttime. Themodelpresentstabilitywithinthegeomet-
a
M ric and matter perturbations for the choice Fg = y, where y = (T1/3/16πG)(16πG +T +8Λ),
T
exceptforgeometricparttodeSittermodel. Weimposethefirstandsecondlawsofthermodynam-
ics to the ΛCDM and find the condition where they are satisfied, that is, T ,G > 0, however
8 A eff
wherethisisnotpossibleforcaseswherewechoose,leadingtoabreakdownofpositiveentropyand
1
Misner-Sharp energy.

]
c PACSnumbers: 98.80.-k,95.36.+x,04.50.Kd
q
-
r
g
[
I. INTRODUCTION

3
v
1 The description of the gravitational interaction can be done in different ways, the main and best known is that
2 using Riemannian geometry as a tool for the formulationof this description. This is knownas the Theory of General
6 Relativity (GR) of Einstein [1].
0 The Riemannian geometry is based on the differential geometry which deals with the so-called space-time as a
0 differentiable manifold of dimension four, where we have only the effect of the curvature. There are other geometries
.
1 that generalize this idea. A space-time may have curvature, but also torsion in their structure. There are two
0 fundamental concepts in differential geometry. The formulation of this type of geometry has been undertaken and
5 gravitation is known as Einstein-Cartan geometry [2]. In this theory the gravitational interaction is described by
1
both curvature and torsion of space-time, where the torsion is commonly attributed to the inclusion of spin, through
:
v fractional spin fields. A very particular case of this theory is when we take the identically zero curvature, and then
i only have a space-time with torsion. This type of geometry is known as the Weitzenbock geometry [3, 4], where the
X
torsion describes the gravitational interaction. Various analysis can be performed in this type of geometry, which
r
a is proven to be dynamically equivalent to GR [5]. In this context, recently a new formulation is started up, and
generalizesthe callTeleparallelTheory(TT), ofthe space-time wherethe gravitationalinteractionis describedsolely
by the torsion.
Through the standard Big Bang [6] theory, the ΛCDM models describe very well the evolution of our universe, in
a Riemannian geometry within the GR. There are several open issues, but the main one today is the so-called dark
energy. In order to make the cosmologicalWMAP data fitting with the theory, it is necessary to introduce an exotic
component in the equations of GR, i.e, the dark energy. This can be modelled as a perfect fluid with the equation
of state p =ω ρ , where the values of ω must be very close to the 1 today. An alternative to that data,
DE DE DE DE
−
being consistent with the theory, is the modification of the geometry. A good review can be seen in [7]. One of the
first general possibilities is the well known f(R) Theory [8], where R is the scalar curvature, obtained through the
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected]
§Electronicaddress: [email protected]
2
double contractionof the Riemann tensor indices. This theory places the analytic function f(R) in the action, where
the GR can be reobtained in a certain limit, such as from f(R)=a R+a R2, with a 0 one gets again GR. This
1 2 2
→
theory has proved effective in simulating the evolution of our universe, in various epochs. Other possibilities have
arisen through the generalisationof the GR, by changing the action. One such change is the case of f(R, ) Theory,
T
where is the trace of the energy-momentum tensor. In this case, the matter content should be taken into account
T
as having a kind of interaction with the geometry.
Adirectanalogycouldbe made betweentheorieswithonly the curvature andthe torsiononly. As the f(R)theory
is a generalizationof the GR, it is logicto alsothink of a generalizationof the TT,where the analogousobjectto the
riemannian scalar curvature, is the torsion scalar T, obtained from contractions between the torsion and contorsion
tensors. A change in the action of TT is made considering an analytic function f(T) which depends on the torsion
scalar. That was first thought of a theory arising from the Born-Infeld action [9]. Then, several studies have shown
the great accordance of this theory with the most varied approaches in Gravitation and Cosmology [10].
Another recent proposal is to consider not only the torsion scalar in the action, but also the trace of the energy-
momentum tensor, as an analogy to the f(R, ) Theory. This theory, called f(T, ) has been formulated recently
T T
[11], still requiring verification of compatibility with the cosmological data and the physical requirements for a good
cosmologicaltheory. That is the reason to check how should be the functional form of the action of this theory, such
that the ΛCDM model is valid. For this we use the scheme the reconstruction method for a modified gravity [12]. In
addition, we make a stability analysis for the ΛCDM model.
It also exists a wide interest in studying the thermodynamics of our universe. Various calculus have been done for
guaranteeing the system to obey the first and second laws of thermodynamics. In the GR, it has been shown that
the first law of thermodynamics can be shown as dE = TdS +WdV [13]. This law can also be represented in the
modified versions of gravity, but with an additive content of entropy production, for a non-equilibrium description
[14]. We have done here an analysis which yields the conditions for satisfying the classicalthermodynamics laws. We
adopt the units as k =c=~=1 and the Newton constant G−1/2 =M =1.2 1019GeV [15].
B Plank
×
The paper is organized as follows. In section II we make a brief description of f(T) and f(T, ) theories, with
T
the main elements and definitions necessary for their formulation. In section III we have analysed the conservation
of energy-momentum tensor, which results in strong constraints to the functional form of the action of
the theory, leading to function f(T, ) with a linear dependence on the trace . In the IV section, we
T T
use the reconstruction method for the actions to obtain the ΛCDM model for f(T, ) theory. The result shows
T
that the action of ΛCDM is a combination of a linear term, a constant ( 2Λ) and a non-linear term
given by the product √ TF (T1/3/16πG)(16πG +T +8Λ) , with F being−a generic function. Insection
g g
− T
V we make the stability analysis of the studied model. The model present stability within the geometric
and matter perturbations fo(cid:2)r the choice F = y, where y(cid:3)= (T1/3/16πG)(16πG +T +8Λ), except for
g
T
geometric part to de Sitter model. We do a thermodynamic analysis for f(T, ) theory in section VI. We
T
impose the first and second laws of thermodynamics to the ΛCDM and find the condition where
they are satisfied, that is, T ,G > 0, however where this is not possible for cases where we choose,
A eff
leading to a breakdown of positive entropy and Misner-Sharp energy. In section VII we have established
the conservation of energy-momentum tensor to previous results, showing an inconsistency for this approach.
We make our final considerations in section VIII.
II. f(T) AND f(T, ) GRAVITIES
T
Inthissectionwewillseethebasicpreliminaryconceptsforthereconstructionoftheoriesf(T)andf(T, )theories
T
of gravity.
In theory f(T), the geometry is determined solely by the matrices that transform the metric of space-time into
the Minkowski metric. To begin, we define the space-time as a differentiable manifold in which only the torsion is
non-zero, that is, the curvature is identically zero, then all Riemann tensor components are zero.
Now we define the line element as
dS2 =g dxµdxν. (1)
µν
Taking into account that we can define 1-forms in the co-tangent space of the manifold, and introduce Lorentz
symmetry in the line element, we can re-write the line element as
dS2 =g dxµdxν =η θaθb, (2)
µν ab
where θa = ea dxµ are 1-forms, index a = 0,...,3 and [η ] = diag[1, 1, 1, 1]is the Minkowski metric. Here the
µ ab − − −
Latin indices are related to the co-tangent space and the Greeks ones to the space-time. By the way of writting the
3
line element(2), we canestablishthe following relationsη =e µe νg , g =ea eb η , ea e ν =δν, e νeb =δb,
ab a b µν µν µ ν ab µ a µ a ν a
where e µ is the inverse of matrix called tetrad ea .
a µ
The connection is chosen such that all the Riemann tensor components are identically zero, and one has the
Weitzenbock connection [16]
Γα =e α∂ ea. (3)
νµ a ν µ
We can now define a tensor which gives sense of torsion to the space-time
Tα =Γα Γα =e α ∂ ea ∂ ea . (4)
µν νµ− µν a µ ν − ν µ
Through the components of the torsion tensor, we can define(cid:0)the contortion(cid:1)tensor components and the tensor S µν
α
1
Kµν = (Tµν Tνµ T µν) , (5)
α −2 α− α− α
1
S µν = (Kµν +δµTσν δνTσµ ) . (6)
α 2 α α σ− α σ
We can also define the analogous object to the scalar curvature in GR, the torsion scalar
1 1
T =Tα S µν = TµνσT + TµνσT T σµTν . (7)
µν α 4 µνσ 2 σνµ− σ νµ
It is this object that plays the curvature scalar role in GR, and that should also form the action of f(T) theory. The
action of f(T)theory is constructedin a way that we have a linear term in the torsionscalar,another containing the
correction term to the TT and another term related to the material content. Then we write the action as
1
S = d4xe[T +f(T)+ ],, (8)
m
16πG L
Z
with e = det[ea ] = √ g = det[g ], G the Newton’s constant, and setting c to unity. In f(T) theory, tetrads
µ − µν
are dynamic fields, then doing the functional variation of the action (8) in relation to them, one gets the following
p
equations of motion
1
(1+f ) e−1∂ (ee σS νµ) e λTσ S µν +e σS νµ∂ Tf + e ν[T +f]=4πGeσΘ ν, (9)
T µ a σ − a µλ σ a σ µ TT 4 a a σ
(cid:2) (cid:3)
where we use the nomenclature f = ∂f/∂T and f = ∂2f/∂T2, Θ ν represents the components of the matter
T TT σ
energy-momentum tensor.
Let’s take the example of the Friedmann-Lemaˆıtre-Robertson-Walker(FLRW) universe with flat spatial section
ds2 =dt2 a2(t) dx2+dy2+dz2 , (10)
−
where a(t) is the scale factor. The Hubble parameter is(cid:0)given by H(t) =(cid:1)a˙(t)/a(t). Now we specify our choice of
tetrads as
[ea ]=diag[1,a(t),a(t),a(t)]. (11)
µ
With this choice, we can represent the line element (2) through a set of 1-forms θ0 = dt, θ1 = a(t)dx,θ2 =
a(t)dy,θ3 = a(t)dz . Thus, the equations of motion (9) for f(T) theory, taking the material content as a perfect
(cid:2)
fluid Θ ν =diag[ρ , p , p , p ], are given as follows
µ m(cid:3)t − mt − mt − mt
8πG f
H2 = ρ 2H2f (12)
mt T
3 − 6 −
4πG(ρ +p )
H˙ = mt mt , (13)
−1+f 12H2f
T TT
−
Thesubscribedρ andp meansthe densityandpressureofthetotalmatterintheuniverse,weconsiderhereonly
mt mt
the components of the baryonic matter as ρ ,p and radiation as ρ ,p .
m m r r
{ } { }
Here, the torsion scalar is obtained by the definitions (4)-(7) with (11), resulting in
T = 6H2, (14)
−
4
We now see clearly that the equations of motion of f(T) theory are identical to that of the GR, and the equations of
Friedmann (flat spatial section), when the nonlinear term are zero, i.e making f(T)=f =f =0 in (12)-(13).
T TT
Now we can present the most recent generalizationof the f(T) theory. Following the analogy of the generalization
of the f(R) theory to f(R, ), where is the trace of the energy-momentum tensor. Here, we can also introduce
T T
in the action an analytic function that depends not only on the torsion scalar, but also on the trace . For such a
T
theory, the action is then given by
1
S = d4xe [T +f(T, )+16πG ] , (15)
m
16πG T L
Z
where f(T, ) is an arbitrary analytical function of the torsion scalar T and of the trace of the matter energy-
momentumTtensor Θ ν, and is the matter Lagrangiandensity. T
µ Lm
Here we must considerthat the Lagrangiandensity depends only on tetradsand notonits derivatives. For the
m
L
energy-momentum tensor of a perfect fluid we have the following trace
=Θ µ =ρ 3p . (16)
T µ mt− mt
We can then make the functional variation of the action (15) with respect to the tetrads, resulting in the following
equations of motion [11]
(1+f ) e−1∂ (ee αS σµ) e αTµ S νσ +(f ∂ T +f ∂ )e αS σµ
T µ a α − a να µ TT µ TT µT a α
f +T e αΘ σ +p e σ
+e σ (cid:2) f a α m a =(cid:3) 4πGe αΘ σ, (17)
a 4 − T 2 a α
(cid:18) (cid:19) (cid:18) (cid:19)
where f = ∂f/∂ and f = ∂2f/∂T∂ . Here it is evident that the particular case where the function f depend
T TT
T T
only on the torsionscalar T, i.e, f f(T)[39], the equation (17) reduces to the equation of motion of f(T) theory in
≡
(9).
Takingagainthemattercontentasaperfectfluidandchoosingthediagonaltetradsin(11),theequationsofmotion
of f(T, ) theory for a flat FLRW universe, are given by
T
1
3H2 =8πGρ f +12H2f +f (ρ +p ) , (18)
mt T T mt mt
− 2
(cid:0) (cid:1) ρ +p
H˙ = 4πG(ρ +p ) H˙ f 12H2f H(ρ˙ 3p˙ )f f mt mt . (19)
mt mt T TT mt mt TT T
− − − − − − 2
(cid:18) (cid:19)
(cid:0) (cid:1)
Inthis paper,inthe nextsection,weproposeto reconstructthe ΛCDMmodelandstudy the stabilityofthe de Sitter
and power law solutions.
III. CONSERVATION LAWS TO f(T, ) THEORY
T
This section is devoted to establishing a conservation law for the f(T, ) theory. For this, we must describe the
T
equationsofmotioninacovariantform. Letusmultiplytheequationofmotion(17)byg ea (tohavetheindexesµ
µσ ν
andν the free inthe end)andusingthe identity g ea [e−1∂ (ee αS σλ) e αTλ S γσ]=(1/2)[G (1/2)g T],
µσ ν λ a α − a γα λ µν − µν
where G is the Einstein tensor, we can rewrite our equation of motion as
µν
1 1 1 1
(1+f ) G g T +S λ(f ∂ T +f ∂ )+ g (f +T) f (Θ +g p )=4πGΘ . (20)
2 T µν − 2 µν νµ TT λ TT λT 4 µν − 2 T νµ µν mt νµ
(cid:18) (cid:19)
Now we take the divergence µ all two sides and isolate Θ µ, taking into account that G µ 0, what gives us
∇ ∇µ ν ∇µ ν ≡
1 1
Θ µ = (f ∂ T +f ∂ )ea [e−1∂ (ee αS σλ) e αTλ S γσ] (1+f )∂ T
∇µ ν (4πG+(1/2)fT)( TT σ TT σT ν λ a α − a γα λ − 4 T ν
+ S µλ(f ∂ T +f ∂ )+S µλ f ∂ T∂ T +f ∂ ∂ T +f ∂ T +f ∂ T∂
∇µ ν TT λ TT λT ν TTT µ λ TTT µT λ TT∇µ λ TTT µ λT
1 (cid:16) 1
+f ∂ ∂ +f ∂ + (f ∂ T +f ∂ +∂ T) (f ∂ T +f ∂ )(Θ µ+δµp )
TTT µT λT TT∇µ λT 4 T ν T νT ν − 2 TT µ TT µT ν ν mt
(cid:17)
1
f ∂ p . (21)
T ν mt
−2 )
5
This equation gives us two constraints, one for ν =0 and other for ν =1,2,3, as follows
1 dT d d dp
mt
ρ˙ +3H(ρ +p ) = 2(ρ +p ) f +f T f T +2f , (22)
mt mt mt mt mt TT TT T T
4( (cid:18) dt dt (cid:19)− dt dt )
p dT d
mt
0 = f +f T . (23)
TT TT
2 dt dt
(cid:20) (cid:21)
The second constraint results in
d
[f ]=0, (24)
T
dt
fixingf(T, )asalinearfunctionofthetrace oraconstant,whichisinagreementwiththeresultsobtainedrecently
T T
in [32] for the analogousf(R, ) Theory. Already throughthe first constraint,if we areto maintain the conservation
T
of energy-momentum tensor, we must impose ρ˙ +3H(ρ +p ) 0, as usually obtained in the ΛCDM model.
mt mt mt
≡
This provides us
dT d d dp
mt
2(ρ +p ) f +f T f T +2f =0. (25)
mt mt TT TT T T
dt dt − dt dt
(cid:18) (cid:19)
When we do the same analysis in f(R, ) Gravity, we also have a restriction on the functional form of the action
T
[31].
We will guard these results and reconstruct the ΛCDM model taking into account these two constraint.
IV. RECONSTRCTION OF ΛCDM MODEL IN f(T, ) THEORY
T
In this section we use the reconstruction method, through a particular model, for obtaining what should be the
functional form of the function f(T, ). This method basically consist in choosing a model consistent with the
T
cosmological data, and use the imposition of the characteristic equation for this specific model, must be satisfied at
anytimeoftheevolutionofouruniverse. Sowithanimposition,weintegratetheequationofmotioninordertomake
the modelbeing valid. This resultsin a functionalformoffixed functionthat makesup the actionofthe theory. The
use of this method in some cases of modified gravity can be seen in [12, 17].
Now, for reconstructing the ΛCDM model, it is necessary to impose an equation of motion of this model
3H2 =8πGρ +Λ, (26)
mt
where Λ is the cosmological constant. In this model, the matter is described through a perfect fluid formulation,
where the pressure satisfies to the following equation of state p = ω ρ +ω ρ = (1/3)ρ , with ρ and ρ being
mt m m r r r m r
the matter and the radiation densities. The trace of the energy-momentum tensor (16) is
=ρ 3p =(1 3ω )ρ +(1 3ω )ρ =ρ . (27)
mt mt m m r r m
T − − −
By inverting the relation (27) for the energy matter density in terms of the trace and using (14), imposing the
ΛCDM model from the equation (26)[40], we can substitute in the equation of motion (18) of the f(T, ) theory,
T
yielding
1 1 1 T
Λ= (f 2Tf ) f + +Λ . (28)
T T
−2 − − 3 T 2πG 2
(cid:20) (cid:18) (cid:19)(cid:21)
Integrating this differential equation we obtain the following action function
T1/3
f(T, )= 2Λ+√ TF (16πG +T +8Λ) , (29)
g
T − − 16πG T
(cid:20) (cid:21)
where F [x] is a generic arbitrary function of its argument x. Taking into account the constraint (24), we see that
g
F [x] x, linear function in x. The function F [x] can also take a constant value. We see that the reconstruction of
g g
≡
the action (15) is given by
1
S = d4xe[T +f(T, )+16πG ]
m
16πG T L
Z
1 T1/3
= d4xe T 2Λ+√ T (16πG +T +8Λ) +16πG . (30)
m
16πG − − 16πG T L
Z (cid:26) (cid:20) (cid:21) (cid:27)
6
We can now see that this action generalizes the f(T) theory, because being able to have terms representing the
interaction between the torsion and matter, generated by the function F in (29). We also see clearly that this
g
generalization is not for any functional form between T and , but only by the product given in the third term of
T
the above action. This restricts a lot the functional form of the theory, when it is imposes the validity of the ΛCDM
model, as we have here. Another important observation is that this action lies in the f(T) when we properly choose
F (T1/3/16πG)(16πG +T +8Λ) Q, where Q is a constant given in [18]. In the next section, we will use a
g
T ≡
model for this generic function, where F [y]=y, with y =(T1/3/16πG)(16πG +T +8Λ).
(cid:2) (cid:3) g T
Here it is still some important observations. Our model is able to reproduce well the eras that the evolution of
our universe goes. For exemple, with the imposition (26) we see daily that the era of radiation, where we write now
3H2 8πGρ +Λ, and of the matter, where 3H2 8πGρ +Λ, are now well reproduced. In radiation era, as
r m
≈ ≈
the term of the radiation density dominates over the matter term, the generic function is being approximated by
F (T1/3/16πG)(T +8Λ) . Already in the matter era, as now the dominant term is the matter density, we have the
g
same functional form of the action (30). In the dark energy era, it’s approach3H2 Λ, or equivalently (T/2) Λ,
(cid:2) (cid:3) ≈ − ≈
Which brings us to the same functional form of radiation era for F , but now T 2Λ. Here inflation can be
g
≈ −
described similarly, whereas the component that dominates is the vacuum energy density.
As seen, our reconstruction is compatible with the most varied eras of cosmological evolution of
our universe, leading us to believe that this model should also be compatible with the cosmological
experimental data. We know at least that the acceleration of the universe at this stage is consistent
with our model, for the very reconstruction is done to satisfy this condition. And we still have
compatibility with the measures of type IA SN, where ω 1 [36]. We can at least see that there
DE
∼ −
is a compatibility with ultra-stiff fluid where ω =1, made in a recent analysis [35].
m
One last important observation is that for the reconstruction satisfy the first constraint (25), we have to fix the
trace in terms of torsion scalar as follows
1
= 11+4608G2π2 T 8Λ . (31)
T 16πG −
(cid:2)(cid:0) (cid:1) (cid:3)
This shows us that the validity of the reconstruction of the ΛCDM model is subject to a fixation of the energy-
momentum tensor trace, given in (31). This show us that the f(T, ) theory should always be considered a theory
T
in which the trace of energy-momentumtensor depends linearly on the torsionscalar. However,this does not permit
us to conclude that the f(T, ) Gravity must alwaysfall back onf(T) Gravity,because for what f(T, ) fall back in
T T
f(T), it is necessary that F Q in (29), and does not have a linear dependence on the trace, as we have here.
g
≡
It may be possible to escape from the result that the action should have linear dependence on the
trace , which is a direct implication of the conservation of the energy-momentum tensor given in
T
(24). We can think about the possibilityof non-conservation of energy-momentum tensor, arising from
the non-minimal coupling between matter and torsion, similar to the case of theories modified only
curvature[33,34]. Thisshouldbeinterpretedasapossibilitythatthemovementofthematerialcontent
is not more geodesic having the appearance of an extra force term. This can also come to modify the
bending of light and add a term in acceleration exerted by gravity, astrophysical implications may
have to help in models of dark matter. This approach goes beyond our analysis here and should be
done in future work.
In the next section, we will show the stability of the ΛCDM model reconstructed here.
V. STABILITY OF DE SITTER AND POWER-LAW SOLUTIONS
In order to establish the validity of the ΛCDM model, let us make a simple stability analysis. To do so, we will do
a small perturbation in the geometry and the matter
H(t)=h(t)[1+δ(t)], ρ (t)=ρ (t)[1+δ (t)], (32)
mt mth m
where h(t),ρ (t) is an exact solution of the equations of motion for call background (obey to (17)-(18)) and
mth
{ }
δ(t),δ (t)<<1. With this, the torsion scalar is given by
m
T = 6H2 = 6h2(1+δ)2 =T (t)[1+2δ(t)], (33)
0
− −
where we have T (t)= 6h2(t). The trace of the energy-momentum tensor is given by
0
−
(t)=ρ (t) 3p (t)=(1 3ω )ρ (t)+(1 3ω )ρ (t)=ρ (t)= (t)[1+δ (t)], (34)
mt mt m m r r m 0 m
T − − − T
7
where (t)=ρ (t). The perturbations of the torsion scalar and the trace of the energy-momentum tensor are
0 mh
T
δ¯T(t)=T T =2T (t)δ(t), δ¯ (t)= = (t)δ (t) . (35)
0 0 0 0 m
− T T −T T
Let us express the function f(T, ) in the Taylor series until the first order around the point [T (t), (t)], by
0 0
T T
∂f ∂f
f(T, ) f(T , )+ δ¯T + δ¯ . (36)
0 0
T ≈ T ∂T ∂ T
(cid:20) (cid:21)T=T0 (cid:20) T (cid:21)T=T0
its derivatives, that are obtained deriving (36), can be expressed as
∂2f ∂2f
f (T, ) f (T , )+ δ¯T + δ¯ , (37)
T T ≈ T 0 T0 ∂T2 ∂T∂ T
(cid:20) (cid:21)T=T0 (cid:20) T (cid:21)T=T0,T=T0
∂2f ∂2f
f (T, ) f (T , )+ δ¯T + δ¯ . (38)
T T ≈ T 0 T0 ∂T∂ ∂ 2 T
(cid:20) T (cid:21)T=T0,T=T0 (cid:20) T (cid:21)T=T0
By substituting these expressions into (17), collecting the term until the first order, we get the following result
1
3h2(1+2δ)=8πGρ (1+δ ) f +f δ¯T +f δ¯ +12h2(1+2δ)
mth m − 2( 0 T0 T0 T ×
(cid:2) (cid:3)
4
f +f δ¯T +f δ¯ + + f +f δ¯T +f δ¯ ρ + ρ (1+δ )
× T0 T0T0 T0T0 T ) T0 T0T0 T0T0 T (cid:18) mh 3 rh(cid:19) m
(cid:2) (cid:3) (cid:2) (cid:3)
(39)
that replacing ρ = , h2 = T /6 and ρ = [(1/8πG)((T /2)+Λ)+ ] becomes
mh T0 0 − 0 rh − 0 T0
2 1 T T
0 0
T 1+f +2T f + +Λ f δ(t)= +Λ
− 0(cid:26) T0 0 T0T0 − 3(cid:20)T0 2πG(cid:18) 2 (cid:19)(cid:21) T0T0(cid:27) ( 2
1 1 T 1 1 T
0 0
5 + +Λ f +T f + +Λ f δ (t). (40)
−6(cid:20) T0 πG(cid:18) 2 (cid:19)(cid:21) T0 0T0 T0T0 − 3T0(cid:20)T0 2πG(cid:18) 2 (cid:19)(cid:21) T0T0) m
We maintain for the moment this result. Now we have to express the perturbation δ(t) in terms of δ (t). To do so,
m
let us take the perturbation of the conservation equation of the energy-momentum tensor
ρ˙ +3H(ρ +p )=0 (41)
mt mt mt
4
ρ˙ (1+δ )+ρ δ˙ +3h(1+δ) ρ + ρ (1+δ )=0
mth m mth m mh rh m
3
(cid:18) (cid:19)
ρ √6 [(T /2)+Λ]
δ(t)= mth δ˙ (t)= 0 δ˙ (t). (42)
m m
−3h(ρ +p ) −8πG√ T (1/8πG)[(T /2)+Λ]+
mth mth − 0{ 0 T0}
Here, the “dot” denotes the derivative with respect to time t. Now, by inserting (42) in (40), one gets
F (t)
δ =δ(0)exp 1 dt , (43)
m m F (t)
(cid:20)Z 2 (cid:21)
T 1 1 T 1 1 T
0 0 0
F (t)= +Λ 5 + +Λ f +T f + +Λ f , (44)
1 2 − 6 T0 πG 2 T0 0T0 T0T0 − 3T0 T0 2πG 2 T0T0
(cid:20) (cid:18) (cid:19)(cid:21) (cid:20) (cid:18) (cid:19)(cid:21)
F (t)= T √6T +2Λ 6πG[1+f +2T f ]+(4πG +T +2Λ)f 24πG
2 − 0 0 {− T0 0 T0T0 T0 0 T0T0} ×
p (cid:16) (cid:17) .h
(4πG +T +2Λ) . (45)
0 0
× T
i
This equation is valid for specific models, the ones we have chosen, the de Sitter and power-law models. We will
substitute the characteristic of these models in the above equation.
Let us first analyse the stability of de Sitter solution, where the characteristic is given by h=h , yielding through
0
the equation of continuity
ρ (t)=ρ exp[ 3h t]. (46)
mh 0 0
−
8
The torsion scalar is constant and is given by T = 6h2. The trace of the energy-momentum tensor is given by
0 − 0
=ρ exp[ 3h t]. (47)
0 0 0
T −
Now we have to use the reconstruction of f(T, ) for ΛCDM in (29), specifying the case of de Sitter solution. We
T
first consider the time depending torsionscalar,i.e, variable torsionscalar,and then fix for the de Sitter solution, we
can then correctly calculate the derivatives of the function f(T, ).
T
For the de Sitter solution, we take one case for action (30). We make the case where F [y] = y, with y =
g
(T1/3/16πG)(16πG +T +8Λ).
0 T0 0
For this case, we have the following integral in (43)
4 61/3(h5/3+61/6πG)(3h2 Λ) (3h2 Λ)t+ 2πGρ0e−3h0t
δ (t)=δ(0)exp × 0 0− 0− 3h0 (48)
m m (− h2/3(65/6h2 6πGh1/h3)(3√6h2 Λ) i)
0 0− 0 0−
and from (42)
δ(t)= δ (t)4×61/3(h50/3+61/6πG)(3h20−Λ) 3h20−Λ−2πGρ0e−3h0t . (49)
− m h50/3 (65/6h20−6πGh10(cid:2)/3)(3√6h20−Λ) (cid:3)
We do a numerical graph representing the temporal evolution of these perturbations in Figure 1. We can see
that the perturbations rapidly decrease to zero, showing a possible stability for the solution of de Sitter. There is a
serious problem here. The graphical representation to δ(t) in figure 1 shows that despite this function quickly drop
to zero, the initial values are of the order to 1041, thus contradicting the initial assumption for perturbation which is
δ(t)<<1. This showsus an impossibility for stability to the modelof de Sitter, due preciselyto the initial values for
perturbation of the geometric part.
1.0
7´1041
0.8 6´1041
5´1041
∆HL@DmtdeSitter 00..46 ∆HL@DtdeSitter 34´´11004411
2´1041
0.2
1´1041
0.0 0
0 1 2 3 4 0 1 2 3 4
t@1(cid:144)GeVD t@1(cid:144)GeVD
Figure 1: Theleftgraphshowsthetemporalevolutionofδm(t)(inred)andtheoftherightthetemporalevolutionofδm(t)(inblue)bothfor
thecaseFg[y]=y,withy=(T01/3/16πG)(16πGT0+T0+8Λ),thetwofortheparticularcaseofdeSittersolution. Theparametersarechosen
as{N =20,h0=2.1×0.7×10−42,δm(0)=1,ρ0=0.1×10−121,G=(1.2)2×10−38,Λ=10−42}.
Doing the same procedure for the case of powerlaw,where now h (t)=α/t, the equation(43), for the case F =y
0 g
with y =(T1/3/16πG)(16πG +T +8Λ), becomes
0 T0 0
F
δ (t)=δ(0)exp pl1dt (50)
m m F
(Z pl2 )
t −3α α α 1/3 t 3α
F = 4 61/3 α2+61/6πGt2 (Λt2 3α2) (Λt2 3α2) +2πGρ t2 , (51)
pl1 0
− × (cid:18)t0(cid:19) t (cid:20) (cid:16)t(cid:17) (cid:21) − " − (cid:18)t0(cid:19) #
α 1/3
F =α2(3√6α2 Λt2) 65/6α2 6πGt2 . (52)
pl2
− − t
(cid:20) (cid:16) (cid:17) (cid:21)
From (42), with (50), we find
F Λt2 3α2
pl1
δ(t)=δm(t)F  − −3α  (53)
pl2 αt Λ 3α2 +8πGρ t
 (cid:18) − t2 0(cid:16)t0(cid:17) (cid:19)
9
We numerically represent the evolution of perturbations δ (t) and δ(t) in the figure 2. Again we can see that the
m
perturbations quickly decay to zero, showing the stability of this solution.
0.01850 0.0370
∆HL@Dmtpowerlaw 00..0011884405 ∆HL@Dtpowerlaw 00..00336689
0.01835 0.0367
0 1 2 3 4 0 1 2 3 4
t@1(cid:144)GeVD t@1(cid:144)GeVD
Figure 2: Theleftgraphshowsthetemporalevolutionofδm(t)(inred)andtheoftherightfortemporalevolutionofδ(t)(inblue)bothfor
thecaseFg[y]=y,withy=(T01/3/16πG)(16πGT0+T0+8Λ),thetwofortheparticularcaseofpowerlawsolution. Theparametersarechosen
as{α=2,t0=3,h0=2.1×0.7×10−42,δm(0)=1,ρ0=0.1×10−121,G=(1.2)2×10−38,Λ=10−42}.
In the next section, we will study the two laws of thermodynamics for the ΛCDM model.
VI. THERMODYNAMIC LAWS ACCORDING TO ΛCDM MODEL
By analogyto the thermodynamics of black holes,the thermodynamics of the cosmologicalmodels canbe realised,
verifying if these models satisfy the classical thermodynamics. This has been done in the context of GR [20]. It has
also been of wide interest the study of these laws for the caseof the theories of modified gravities[21]. Inthe context
ofmodifiedgravityitiscommontouseadescriptionofnon-equilibriumthermodynamicsforrepresentingthephysical
system [22], we will use this representationhere.
In this section we will investigate the conditions of satisfying the first and second laws of thermodynamics.
A. First law
We will redefine now the action of the theory f(T, ) as
T
S = d4xe[f(T, )+16πG ]. (54)
m
T L
Z
This is necessary to get the possibility for defining an effective Newton constant G . Now the equations of motion
eff
are rewritten as
3H2 =8πG (ρ +ρ ) , (55)
eff mt DE
H˙ = 4πG (ρ +p +ρ +p ) , (56)
eff mt mt DE DE
−
with
G f
T
G = 1+ , (57)
eff
f 16πG
T (cid:18) (cid:19)
1 1
ρ = f p f , (58)
DE T mt
16πG f − 2
eff T (cid:18) (cid:19)
1
p =ρ +p ρ 12H2H˙f f H(ρ˙ 3p˙ ) . (59)
DE mt mt DE TT TT mt mt
− − 4πG f − −
eff T
h i
We impose the conservationof the matter sector ρ˙ +3H(ρ +p )=0, as discussed in the sections III and IV.
mt mt mt
But as we are doing a non-equilibrium thermodynamics description, the dark energy sector is does not conserve,
3H2 d 1
ρ˙ +3H(ρ +p )= , (60)
DE DE DE
8π dt G
(cid:18) eff(cid:19)
10
where we used (55)-(59). This concords with the theory analogue to f(T, ), in [23], and also for the particular case
T
where f f(T)in [14]. We still should compareour results with the ones of f(R)theory, in [24]. It is also clear here
≡
that, when G = G (in the particular case f(T, ) = T) the dark energy sector is conserved, the TT is recovered,
eff
T
or the GR analogously,within an equilibrium description of thermodynamics.
Now, we establish the basic tools for the first law of thermodynamics. The line element (10) can be rewritten as
dS2 =h dxµdxν +r2 dθ2+sin2θdφ2 , (61)
µν
where we have the metric of 2-dimensional space [h ] =(cid:0)[1, a2(t)], for(cid:1)µ,ν = 0,1 and x1 = t,x2 = r, a new
µν b −
radial coordinate r = ra(t), with r being the usual radial coordinate r2 = x2+y2+z2. Here the space-time can be
decomposed in a 2-dimensional space with the metric h and other 2-dimensionalspace with is a 2-sphere, with the
µν
line element dΩ2 =bdθ2+sin2θdφ2.
Through this line element (61), we can calculate the apparent horizon[41] and the associated temperature to this.
The apparent horizon is obtained from the expression hµν∂ r ∂ r =0, which results in
µ A ν A
1
r = b. b (62)
A
H
The temperature is calculated by the expression b
1
T = ∂ √ hhµν∂ r (63)
A µ ν A
4π√ h −
− (cid:12) h i(cid:12)
(cid:12) (cid:12)
where h=det[h ]. The temperature is given by (cid:12) b (cid:12)
µν
r
T = A H˙ +2H2 , (64)
A
4π
(cid:16) (cid:17)
b
where, from now, we impose H˙ +2H2 >0 for getting a definite positive temperature.
Using (62) and (56), we can calculate the derivative of r
A
dr
A =4πG r2b(ρ +p ) , (65)
dt eff A tot tot
b
withρtot =ρmt+ρDE andptot =pmt+pDE. Now,wewbill takethe entropyrelatedto apparenthorizon,for modified
gravitytheories[25],thatisS =A/(4G ),withA=4πr2. Here,appearsanewrestrictionforthethermodynamic
A eff A
system, G > 0 for getting a definite positive entropy S 0. Considering this entropy, and using G in (57)
eff A eff
≥
and (65), we get the following differential
b
1 1
dS =2πr 4πr2 (ρ +p )dt+ r d . (66)
A A A tot tot 2 A G
(cid:20) (cid:18) eff(cid:19)(cid:21)
NowwetaketheMisner-SharpenergyfobrmodibfiedgravitiesE =rb /(2G )=Vρ [42][27],withV =(4/3)πr3,
MS A eff tot A
where, using (65), the differential gives rise to
b b
1 1
dE =2πr2 (ρ +p )dt+ r d . (67)
MS A tot tot 2 A G
(cid:18) eff(cid:19)
Using (62) in (64), considering (65), we can rebwrite the temperaturbe as
2
r d 1 1 1 1dr
A A
T = +2 = 1 . (68)
A
4π "dt(cid:18)rA(cid:19) (cid:18)rA(cid:19) # 2πrA (cid:18) − 2 dt (cid:19)
b b
Making the product (68) for (66) we have b b b
T dS = dE +2πr2 (ρ p )dr +6πr2 (ρ +p )dt 4πr2ρ dr
A A − MS A tot− tot A A tot tot − A t A
1 1
+ r (1+2πr T )d . (69)
2 A Ab A G b b b b
(cid:18) eff(cid:19)
b b