A complete cosmological scenario from $f(R,T^\varphi)$ gravity theory

0.22 MB·

by  P. H. R. S. Moraes

Eur. Phys. J. C manuscript No.
(will be inserted by the editor)
A complete cosmological scenario from f(R,Tφ) gravity theory
P.H.R.S. Moraesa,1, J.R.L. Santosb,2
1ITA- InstitutoTecnolo´gico deAerona´utica - Departamento deF´ısica, 12228-900, Sa˜o Jos´e dos Campos, Sa˜o Paulo, Brazil
2UFCG - UniversidadeFederal deCampina Grande- Departamento deF´ısica, 58109-970, Campina Grande, Para´ıba, Brazil
6
1
0
Received: date /Accepted: date
2

n
a Abstract Recent elaborated by T. Harko and collab- second after the Big-Bang, named “inflationary era”.
J
orators,the f(R,T)theories of gravitycontemplate an It is common to describe this phenomenon via scalar
2
optimistic alternative to dark energy, for which R and fields [1]-[7], although another reputed form to do so
1
T standfortheRicciscalarandthetraceoftheenergy- comes from the f(R) gravity [8]-[11].

] momentumtensor,respectively.Althoughtheliterature Recently, a more general theory of gravity, named
c
q has shown that the T dependence on the gravitational f(R,T) gravity,was proposed by T. Harko and collab-
- part of the action - which is due to the considera- orators[12].A priori itcanalsocontributetoinflation-
r
g tion of quantum effects - may induce some novel fea- aryerastudiesthroughitsscalarfieldapproach,named
[ tures in the scope of late-time cosmological dynamics, f(R,Tφ) gravity [12].

1 in the radiation-dominated universe, when T = 0, no Although the late time acceleration of the universe
v contributions seem to rise from such theories. Appar- expansion[13]-[15]hasbeenbroadlyinvestigatedinsuch
1 ently, f(R,T) contributions to a radiation-dominated a theory of gravity [16]-[25], the inflationary era still
1
universe may rise only from the f(R,Tφ) approach, presents a lack of examination. The same happens for
8
2 which is nothing but the f(R,T) gravity in the case of the radiation-dominatedera of the universe.
0 a self-interactingscalarfieldwhose traceofthe energy- For the radiation era, the lack of f(R,T) applica-
1. momentum tensor is Tφ. We intend, in this article, to tions is quite predictable for the following reason. The
0 show how f(R,Tφ) theories of gravity can contribute equation of state (EoS) of the universe is p = ρ/3
6 to the study of the primordial stages of the universe. at this stage, with p and ρ representing the pressure
1
Our results predict a graceful exit from inflationary and density of the universe, respectively. Such an EoS
:
v stagetoaradiation-dominatedera.Theyalsopredicta yields a null trace for the energy-momentum tensor of
i
X late-time cosmic accelerationafter a matter-dominated a perfect fluid. A null trace for the energy-momentum
phase, making the f(R,Tφ) theories able to describe, tensor yields the f(R) formalism for the f(R,T) func-
r
a in a self-consistent way, all the different stages of the tionalformsfoundintheliterature(check,forinstance,
universe dynamics. [12],[16]-[25]). Hence the study of the radiation era of
theuniverseinf(R,T)gravityseemstobequitetricky.
Keywords f(R,T) gravity self-interacting scalar
· Apriori,itseemsreasonabletoaffirmthatthef(R,T)
field primordial stages of the universe
· gravitydoesnotcontributetothestudyoftheradiation-
dominateduniverse,sincethecontributioncomingfrom
the trace of the energy-momentum tensor in f(R,T)
1 Introduction
vanishes at this stage. Specifically, such a shortcoming
orincompletenesshasattractedattentionrecently[23]-
Plentyofeffortshavebeenmadeinthetheoreticalframe-
[26].
work with the purpose of explaining the accelerated
In [23], the authors argued that the high redshift
regime our universe has passed through a fraction of
f(R,T)cosmologicalsolutionstendtorecoverthestan-
ae-mail: [email protected] dardmodelofcosmologyifthef(R,T)functionalform
be-mail: [email protected] is linear in R precisely because when z >>1, the radi-
2
ationwithEoSp=ρ/3dominatestheuniversedynam- throughout this article we will assume units such that
ics, making the trace of the energy-momentum tensor 4πG = c = 1. Moreover, we are going to deal with
tovanish.Therefore,nonovelcontributionswouldcome a standard Lagrangian density for a real scalar field,
fromthef(R,T)theoriesofgravitationattheradiation whose form is
era. 1
= ∂ φ∂µφ V(φ), (2)
In[24],inordertomakef(R,T)gravitytheoryable L 2 µ −
to contribute also to a radiation-dominateduniverse,a for which it has been considered a self-interacting po-
compactified space-like extra dimension had to be in- tential V(φ).
voked. Furthermore,since the energy-momentumtensor of
In [25], with the purpose of obtaining well behaved the scalar field is given by
solutions at the radiation stage of the universe, the
∂
f(R,T) gravity was generalized, by allowing the speed Tφ =2 L g , (3)
µν ∂gµν − µνL
of light c to vary. Moreover, a scenario alternative to
inflation was obtained. its trace is
Bydevelopingthef(R,Tφ)approach,whichissub-
Tφ = ∂ φ∂µφ+4V(φ). (4)
missivetof(R,T)gravity,weintend,here,tomakethe µ
−
f(R,T) theories able to contribute also to inflationary The f(R,Tφ) = R/4 + f(Tφ) gravity model is
and radiation-dominated eras, in a self-consistent way. −
defined in the following form:
Recall that once T = 0 at the radiation stage, it is
natural to expect that f(R,T) theories retrieve f(R) S = d4x√ g R +f(Tφ)+ (φ,∂ φ) , (5)
µ
gravityorevengeneralrelativity(GR)whenf(R) R, − −4 L
∼ Z (cid:20) (cid:21)
in such a way one would not expect any new informa-
forwhichwehaveinsertedthe gravitationalpartofthe
tionsderivedfromf(R,T)gravity.Tomakethef(R,T)
action. By minimizing the above action, we obtain
gravity able to describe a radiation universe by itself,
G =2[T g f(Tφ) 2f′(Tφ)∂ φ∂ φ], (6)
withoutnecessarilyrecoveringthe f(R)formalismout- µν µν µν µ ν
− −
comes,willmakesuchatheoryabletocontributetoall whereG istheEinsteintensorandf′isthederivative
µν
the different stages of the universe dynamics. In order of f(Tφ) with respect to Tφ.
to do so, we will implement the first-order formalism Fortheaction(5),aflatFriedmann-Robertson-Walker
[27,28] to the f(R,Tφ) gravity. universe has the following Friedmann equations:
We will not omit the recent dynamical features our
3 1
universehaspresented.Inotherwords,alate-time cos- H2 = 2f′(Tφ) φ˙2 f +V , (7)
2 2 − −
mic acceleration,which is corroboratedby Type Ia Su- (cid:20) (cid:21)
pernovae observations [13,14] and described in ΛCDM 3 φ˙2
cosmologybythepresenceofthecosmologicalconstant H˙ + H2 = +f(Tφ) +V , (8)
2 −" 2 #
in standard Einstein’s field equations (FEs), shall also
be described in our model. in which φ = φ(t), H = a˙/a is the Hubble parameter,
a is the scale factor and dots denote derivation with
respect to time.
2 The f(R,Tφ) Gravity
We are also able to derive the equation of motion
for such a system, which is given by
Earlyandlatetimeaccelerationsoftheuniverseexpan-
sion can be explained in the framework of scalar field [1 2f′(Tφ)](φ¨+3Hφ˙) (9)
−
models [1]-[7], [29]-[35]. The consideration of f(R,T) 2f˙′(Tφ)φ˙ +[1 4f′(Tφ)]V =0,
φ
gravity in the case of a self-interacting scalar field φ − −
yields the f(R,Tφ) gravity [12]. where it was used the notation Vφ dV(φ)/dφ.
≡
Herewewillworkwiththecasef(R,Tφ)= R/4+
−
f(Tφ), for which R stands for the Ricci scalar, while
3 First-Order Formalism
Tφ is the trace of the energy-momentum tensor of the
scalar field, so that
Letusimplementthefirst-orderformalismtothemodel
above,byfollowingtherecipepresentedin[27,28].Firstly
Sφ = d4x√ g (φ,∂ φ) (1)
− L µ it is straightforward to observe from (7) and (8) that
Z
the Hubble parameter obeys the differential equation
is the action of such a field, with g being the deter-
minant of the metric with signature (+, , , ) and H˙ = (1 2f′) φ˙2. (10)
− − − − −
3
Then, the initial ingredient that we may consider in Here,foramatterofsimplicity,wefocusonthecase
order to establish the first-order formalism is based on n=1, which leads to
the definition
φ˙ W , (11) ′
φ f =α. (19)
≡−
with W dW(φ)/dφ and W is an arbitrary function
φ
≡
of the field φ. Such a procedure is commonly used in Then, by substituting f′ in Eq.(14), we find
cosmologicalscenarioscoupled with scalarfields, as we
mayseein[29]andreferencestherein.Thisassumption
implies that (10) is rewritten as h(φ)=(1 2α)a1 sin φ, (20)
−
H˙ = (1 2f′) W2. (12)
− − φ
which means that
If we establish that
H h(φ)+c, (13)
≡ H =(1 2α)a sin φ+c. (21)
1
where c is a real constant, then, Eq.(10) requires the −
following constraint to the function h:
Moreover, it is straightforward to obtain that the po-
′
h =(1 2f )W . (14)
φ φ tential V is
−
Moreover,theFriedmannequationsimposethatthepo-
tential V(φ) has to obey the relation 3 [(1 2α)a sin φ+c]2 1 α a2cos2 φ
V = 2 − 1 − 2 − 1 .(22)
3 1 1 4α
V = (h+c)2+f 2f′ W2. (15) − (cid:0) (cid:1)
2 − 2 − φ
(cid:18) (cid:19)
FromEq.(18)one cansee thatthe first-orderdiffer-
3.1 Examples
ential equation for this scenario has the form
We can apply the first-order formalism by consider-
ing a specific form for the function f(Tφ). As in [28], φ˙ = a cos φ, (23)
1
let us deal with f(Tφ) = α(Tφ)n, with α as a con- −
stant and n as an integer. Note that such a functional
form for f(Tφ) straightforwardly yields the GR case whose analytical solution is
when α = 0. Moreover, it is the analogous of the form
f(T)=αTn in the f(R,T) gravity with no scalar field φ(t)=2tan−1 tanh 1(b a t) , (24)
case, firstly suggested in [12] for deriving an acceler- 2 1− 1
(cid:26) (cid:20) (cid:21)(cid:27)
ated cosmologicalscenario and then applied to a num-
berofotherf(R,T)well-behavedcosmologicalmodels,
such as [18,19], [24,25], among many others. Such an with b being a real constant. The previous solution
1
assumption yields allows us to rewrite Eq.(21) as
f(Tφ)=α φ˙2+4V n =α W2+4V n , (16)
− − φ
f′ =αn (cid:16)φ˙2+4V n(cid:17)−1 =αn(cid:0) W2+4V(cid:1) n−1.(17) H =(1−2α)a1 tanh(b1−a1t)+c, (25)
− − φ
Inthis(cid:16)study,wea(cid:17)regoingtoc(cid:0)onsiderthat(cid:1)thefunc- whichispresentedindetailsintheupperpanelofFig.1
below, where we can see that H exhibits a kink-like
tion W is given by
profile as a direct consequence of the model defined in
W(φ)=a1 sin φ, (18) Eq.(18).
with a1 being a real constant. Such a form for W(φ) is We can use the last result to determine the expan-
a sine-Gordon type of model, and it has been broadly sion parameter (or scale factor) a(t), whose explicit
studied in the literature, specially in investigations re- form is given by
lated with classical field theory, as one can see in [36]
and references therein. Very recently, this model was
a=ect [sech(b a t)](1−2α) . (26)
applied in the context of braneworld scenarios for the 1 1
−
f(R,T) gravity [28].
4
Moreover,the analytical EoS parameter is
[12(α 2)α+7] a2 3 (1 2α)2a2+c2 cosh[2(b a t)]+6(2α 1)a c sinh[2(b a t)] 3c2
ω = − 1− − 1 1− 1 − 1 1− 1 − , (27)
[4(2 3α)α 3]a2+3[(1 2α)2a2+c2] cosh[2(b a t)]+6(1 2α)a c sinh[2(b a t)]+3c2
− − 1 (cid:2) − 1 (cid:3) 1− 1 − 1 1− 1
and its features can be observed in the central panel The last equation results in the following relation for
of Fig.1. We can verify from the analytical cosmolog- the Hubble parameter
ical parameters that small deviations on the values of
theconstantsproducecosmologicalscenarioswhichare a1
H(t)=c (1 2α) 3sinh(a t+b ) (32)
1 1
similar to those presentedin Fig.1. Therefore,the time − 6 −
(cid:26)
evolution of our parameters are not so strongly depen-
+sinh[3(a t+b )] sech3(a t+b ),
dent on the initial conditions of the model. 1 1 1 1
(cid:27)
Another interesting model is
whichispresentedindetailsintheupperpanelofFig.2
φ3 below. As in the previous example, it is possible to
W(φ)=a φ , (28)
1 3 − choosethearbitraryconstantsinordertoobtainakink-
(cid:18) (cid:19)
like profile for H.
whose first-order differential equation is such that
We can also show that the potential V for this an-
φ˙ =a 1 φ2 , (29) alytical model is
1
−
which is(cid:0)satisfie(cid:1)d by V = 1 1 (1 2α)a φ φ2 3 +3c 2 (33)
1
1 4α 6 − −
φ=tanh(a1t+b1), (30) −1 α(cid:26) a(cid:2)2 φ2 1 2 .(cid:0) (cid:1) (cid:3)
whereb isarealconstant.Thislastfirst-orderdifferen- − 2 − 1 −
1 (cid:18) (cid:19) (cid:27)
tial equation representsthe so-calledφ4 type of model, (cid:0) (cid:1)
Withal, the expansion parameter for such a case is
which is also broadly investigated in classical field the-
given by the following
ory, as one can see in [37] and references therein.
In the present case, we are dealing with f(Tφ) =
1
αTφ again.Therefore,wecancombinethepreviousin- a=exp ct+ (1 2α)sech2(a t+b ) (34)
1 1
6 −
gredients to determine the following form for the func- (cid:20) (cid:21)
tion h(φ): [cosh(a t+b )]32(2α−1) ,
1 1
×
φ2 while the EoS parameter reads
h=(1 2α)a φ 1 . (31)
1
− 3 −
(cid:18) (cid:19)
ω = 1+192(1 4α)a2cosh2(a t+b ) 8[4α(5α 2)+5]a2+3 4(4(α 3)α+1)a2+45c2 cosh[2(a t+b )]
− − 1 1 1 − − 1 − 1 1 1
(cid:26)
+ 6 4(1 2α)2a2+9c2 cosh[4(a t+b )]+16α2a2 cosh[6(a t+(cid:2)b )] 16αa2 cosh[6(a t+(cid:3)b )]
− 1 1 1 1 1 1 − 1 1 1
+ 4a2 cosh[6(a t+b )]+9c2 cosh[6(a t+b )]+216αa c sinh[2(a t+b )]+144αa c sinh[4(a t+b )]
(cid:2)1 1 1 (cid:3) 1 1 1 1 1 1 1 1
−1
+ 12(2α 1)a c sinh[6(a t+b )] 108a c sinh[2(a t+b )] 72a c sinh[4(a t+b )]+90c2 . (35)
1 1 1 1 1 1 1 1 1
− − −
(cid:27)
5
ThelatterisplottedinthecentralpanelofFig.2.Again, it is worth checking if the cosmological parameters de-
if we take small deviations of these parameters we de- rived previously indeed resemble, for small values of
termine similar cosmological scenarios. time, what is expected for the primordial universe dy-
namics.
Intheprevioussectionwehavealsoplottedtheevo-
2 lution of the trace of the energy-momentum tensor of
the scalar field, which shall be interpreted below.
1.5 The inflationary model states that early in the his-
toryoftheuniverse,itsexpansionhasaccelerated.From
H 1 standardFriedmannequations,this happensifthe uni-
verseisdominatedbyacomponentwithEoSω < 1/3
0.5 −
[38,39].Accordingto standardcosmology,duringinfla-
tion, the energy density of the universe was dominated
0
by a constant, say, Λ , in such a way the Friedmann
0 5 10 15 20 25 ι
equations read H2 Λ . The Hubble parameter was,
t ι ∼ ι
0.4
0.2
0 2.5
-0.2
Ω 2
-0.4
-0.6 1.5
H
-0.8
1
-1
0 5 10 15 20 25 30 0.5
t
0
0 5 10 15 20 25
4 t
0.4
3 0.01
0.2
Φ 0.005 0
T 2
-0.2
0 Ω
1 15 20 25 30 -0.4
-0.6
0 -0.8
0 5 10 15 20 25 30
-1
t
0 5 10 15 20 25 30 35
t
Fig. 1 The upper panel shows the evolution of H(t), the
central panel shows ω, and the lower panel shows Tφ, for
2
α = −1, a1 = 0.3, b1 = 3, and c = 0.988, in the W(φ) = 0.01
a1 sinφ model. 1.5
0.005
Φ
T 1 0
15 20 25 30 35
0.5
4 Cosmological Interpretations
0
In Section 3, we were able to depict the time evolution 0 5 10 15 20 25 30 35
of H and ω derived from the first-order formalism ap- t
plied to the f(R,Tφ) theory. Remind that our aim in
this work is to make f(R,T) gravity able to induce a
Fig. 2 The upper panel shows the evolution of H(t), the
complete cosmologicalscenario,whichincludesthe pri- central panel shows ω, and the lower panel shows Tφ, for
mordial eras of the universe, since the theory seems to α = −4, a1 = 0.2, b1 = −2.5, and c = 1.31, in the W(φ) =
present some shortcomings at these stages. Therefore, a1(φ3/3−φ)model.
6
then,constantduringinflationaryera.Indeed,onemay Firstly, one can note that the values assumed by
write, for inflation, aι eHιt [38,39]. sucha quantity are always 0,as they, indeed, should
∼ ≥
be.
From the upper panels of Figs.1-2, the Hubble pa-
The high values of density and temperature which
rametervaluespredictedfromourmodelindeedremain
are known to characterize the Big-Bang,together with
approximately constant during a small period of time
the inflationary EoS mentioned above, make us to ex-
after the Big-Bang,in agreementwith inflationary sce-
nario. In fact, H˙ 0.003 and 0.0003 at t = 0.01 pect to have the maximum values of the trace of the
∼ − ∼ − energy-momentum tensor of the scalar field during in-
for Figs.1 and 2, respectively.
flation, as corroboratedby Figs.1-2.
Afterinflation,H mustevolveas t−1,witht be-
∝ H H At the radiation era, Tφ must vanish. We zoomed
ingthe Hubble time [38,39],inaccordancewithFigs.1-
in on Figs.1-2 lower panels in order to explicit such a
2.
predictedfeature.Notethatafteraperiodassumingits
StillintheupperpanelsofFigs.1-2,oneshouldnote maximum values, the trace of the energy-momentum
that for high values of time, H behaves once again as tensor Tφ 0 at the same time scale ω 1/3 in
→ →
a constant. From standard cosmology, it is well known Figs.1-2 central panels. Such a property reinforces the
that for high values of time, H is proportional to ρΛ goodbehaviour of our solutions, specifically at the pri-
(the densityofthe cosmologicalconstant),whichis,in- mordial stages of the universe evolution.
deed, a constant. Therefore, the recent cosmic acceler- At matter-dominated era, Tφ must be = 0. The
6
ation our universe is undergoing can be described by a samehappensforthe secondperiodofacceleration.In-
scalefactorwhichevolvesaseH0t,withtheconstantbe- deed,the plots ofTφ show that after the radiationera,
haviourofH beingpredictedinFigs.1-2forhighvalues Tφ increasesits values.Still, suchvaluesare small,due
of time. to the low density of the universe at these stages, spe-
In the central panels of Figs.1-2, we have depicted cially when compared to the ρ values at inflation.
the evolution of the EoS parameter ω with time. By Remarkably,Tφ remainsconstantforhighvalues of
analysing it, one can see that for small values of time, time.SinceTφ ρ,thischaracterizesthelate-timeuni-
∝
ω 1. This is in agreement with some constraints verse dynamics to be dominated by a constant. Recall
tha∼t h−ave been put to inflationary EoS recently [40]- thatinstandardcosmology,forhighvaluesoftime,the
[42]. cosmologicalconstantΛdominatestheuniversedynam-
ics. As argued above, ρ remains, indeed, constant as
Λ
After representing inflation, ω smoothly evolves to
time passes by. Such an important feature of standard
1/3. This is the maximum value the EoS should
∼ cosmologyispredictedfromthef(R,Tφ)approachhere
assume during the universe evolution. Moreover, 1/3
presented.
is precisely the value of the EoS which describes the
radiation-dominatederaoftheuniverse[38,39],i.e.,the
model predicts a smooth transition from inflationary 5 Discussion and perspectives
stage to radiation-dominated era.
In this article we established a first-order formalism to
As time passes by, the central panels of Figs.1-2
determinetwoanalyticalmodelsrelatedtothef(R,Tφ)
show that ω 1. According to recent observations
→ − theory.Suchaformalismwasconstructedfromthedef-
ofanisotropiesinthe temperatureofcosmicmicrowave
inition observed in Eq.(11).
background radiation [15], this is the present value of
ThementionedexpressiontogetherwithEq.(7)and
the universe EoS, and is responsible for the recent cos-
(13) gave us a constraint for the potential V, which
mic acceleration.
canbeviewedin(15).Moreover,thearbitraryfunction
Thefactthatnowadaystheuniverseispassingthrough
h(φ) was introduced and its form is directly related to
its second phase of acceleratedexpansion justifies such ′
f .
an evolution for ω. Since an accelerated universe is de-
The firstanalyticalmodelwas generatedbyconsid-
scribedbyanegativeEoS,itisexpectedthatduringthe ering W = a sin φ and the specific form for f(Tφ)
1
whole universe evolution, the value of ω, coming from
showedin(16).Suchdefinitionsleadedusto determine
1,whichisrelatedtotheinflationarystage,firstin-
h(φ), besides the potential V, the Hubble parameter
∼−
creases,tendingtoadeceleratingphase,thendecreases,
H, as well as other cosmologicalquantities like the ex-
returning to an accelerated phase.
pansion and EoS parameters. In the second case, we
Furthermore, the lower panels of Figs.1-2 show the adopted W = a (φ3/3 φ), and one more time we
1
−
time evolution of the trace of the energy-momentum determinedthefunctionhandthecosmologicalparam-
tensor of the scalar field for the models. eters for the model.
7
We can directly see that the field φ(t) satisfies the cosmicaccelerationtheuniverseisundergoing[13]-[15],
equation of motion (9) for both models. Furthermore, from an EoS parameter whose late-time values agree
this first-order formalism may help us to obtain other with WMAP cosmic microwave background observa-
one-field analytical scenarios and also may lead us to tions [15].
hybridanalyticalmodels,whereitisconsideredthecou- Therefore, departing from a cosmological scenario
pling betweenanf(R,Tφ,χ) function with a two scalar derivedfromGR,wehaveobtainedanEoSwhichvaries
field Lagrangian. Such a scenario shall be derived and withtime, allowing,throughits evolution,the universe
presented in near future. dynamics to be dominated by different components in
Another further work may rise from an alternative a natural form. Moreover, in order to obtain a recent
formforf(R,Tφ),suchasf(R,Tφ)=g(R)h(Tφ),with cosmicacceleration,we did nothave to inserta cosmo-
g(R) and h(Tφ) being functions of R and Tφ, respec- logicalconstant in the model FEs, in this way, evading
tively. Such a functional form suggests a high coupling the cosmologicalconstant problem [49,50,51].
betweenmatterandgeometry,andthereforemayimply The f(R,Tφ) cosmological scenario derived in this
some novelties in gravitational and cosmological per- work is somehow similar to a decaying vacuum Λ(t)
spectives.Wecanconjecturethatbycouplingsuchkind model, such as those presented in [45,46], for instance.
oftheorieswithascalarfieldlagrangianmayleadusto Firstly, it should be stressed that recently in [26], the
anewsetofFriedmannequationsandtodifferentequa- f(R,T) theories have been interpreted as a generaliza-
tionsofmotion.Oncewehavetheseingredientswemay tion of the Brans-Dicke model [52]-[56]. As argued by
try to apply the first-order formalism and then com- the authors in [26], the extra terms in the f(R,T)FEs
pute the constraints that the superpotential will obey. may play the role of additional terms of effective grav-
In fact, the analogous of that model for the f(R,T) itational and cosmological“constants” G and Λ .
eff eff
gravitywithnoscalarfieldcasewasinvestigatedin[16]. ThoseextratermsinΛ aretheresponsibleforavac-
eff
Wehaveobtainedfromthefirst-orderformalismap- uum decay in Λ(t)-like models.
plied to the f(R,Tφ) gravity, two models able to de- Second,inbothdecayingvacuumandf(R,T)mod-
scribeallthedifferentdynamicalstagesoftheuniverse, els, the covariant derivative of the energy-momentum
from inflation, to radiation, matter and dark-energy tensor µT doesnotvanish.Indecayingvacuummod-
µν
∇
dominated eras. els, this requires necessarily some energy exchange be-
For the first time, an f(R,T) cosmological model tween matter and vacuum, throughvacuum decay into
was able to describe all the dynamical stages of the matter or vice versa.
universe continuously, including the primordial ones. The vacuum-decay models presented in [45,46], for
There was no need of invoking different EoS or func- instance,arecapableofdescribingthecompletehistory
tionalformsfor f(Tφ) indifferentstagesin orderto be ofthe universeinaneffective unifiedframework,inthe
abletoaccountforallthe transienterasoftheuniverse same way the present f(R,Tφ) model does. In those
dynamics, as made in [26], [43] and [44], for instance. models,the vacuumalwaysdecaysinthe dominantdy-
In fact, there was no need of invoking any EoS at all. namicalcomponent of eachstage.However,in order to
Such a quantity was obtained simply by dividing the be able to describe the various phases of the universe
solutions p by ρ. dynamics, the functional form taken for Λ(t) is pro-
Itisremarkablethefactthattheinflationaryscenar- posed on phenomenological grounds, by extrapolating
iosofourmodelshavesmoothlydecayedtoaradiation- backwards in time the present available cosmological
dominated era. This phenomenon is sometimes called data. On the other hand, the f(R,Tφ) scenario is free
“gracefulexit”anditisnottrivialtobeobtained(check of ansatz or phenomenology.
[45,46]forthegracefulexitachievementthroughdecay- Note also that the Λ(t) model presentedin [45] was
ing vacuum models). In fact, an ideal inflationary sce- mimic through a scalar field model (check Section 4 of
nario must naturally decay to a radiation-dominated sucha reference).This corroboratesthe argumentthat
era [47,48], as in the present case. itmighthavesomefundamentalanalogybetweenscalar
Alsoremarkableisthefactthatthemodelsarewell- field and decaying vacuum models.
behaved at radiation stage. The match between the In this article, we have also formally brought to
time scales for ω = 1/3 e Tφ =0 in Figs.1-2 reinforces thescientificcommunityattentiontheshortcomingsur-
that. rounding the f(R,T) theories of gravity in the regime
Not just the presentmodels were able to describe a T = 0. It is, indeed, straightforward to realize that in
smooth transition between inflationary and radiation- such a regime of f(R,T) gravity one automatically re-
dominated eras, but they were also able to predict the coversthe f(R) theories, or GR when f(R) is linear in
matter dominated phase, with ω = 0, and the recent R [23]. We have shown, as an unprecedented outcome,
8
thatitispossibletodescriberadiationinf(R,T)grav- tensor T is then set to zero in order to obtain the
µν
itynotnecessarilybytherecoveringofGR,butthrough wave equation for vacuum [65]-[67]. In the same way
extradynamicaltermscomingfromascalarfieldinthe the EoS ω = 1/3 yields a null trace for the energy-
f(R,Tφ) approach. momentum tensor, this will happen for vacuum, with
If one focus specifically on the radiation era pre- T = 0. Therefore it is expected that the study of
µν
dicted by the f(R,Tφ) model here presented, one is GWs, as the derivation of their primordial spectrum,
able to obtain a scale factor which evolves as the one for instance, in the f(R,T) formalism yields the same
of standard cosmology purely from the f(Tφ) contri- predictions as those obtained via f(R) gravity [68].
bution, which is a complementary novel feature of the
ApowerfultooltostudythepropertiesofGWswas
f(R,T) gravity that we will show below. developed in [69]. It consists in characterizing the po-
BytakingTφ =0in(4)yieldsφ˙2 =4V(φ).Sincein larizationstates ofGWs ina giventheory1.Inorderto
this case f(Tφ)=α(Tφ)n =0, one obtains, from (10), doso,oneevaluatestheNewman-Penrose(NP)quanti-
ties[70,71].In[63],fromtheNPquantitiescalculation,
H˙ = 4V(φ). (36)
− theauthorshavefoundextrapolarizationstatesforthe
FromEqs.(11),(15)and(36),onecanwriteH˙ = 2H2, f(R) gravity. Those extra states may be corroborated
−
which integrated for the scale factor a yields a t1/2. by experiment once the advanced GW detectors, such
∼
From standard cosmology [38,39], this is exactly the as LIGO [57] and VIRGO [58], start they next run.
time proportionality of the scale factor of a radiation- Since the NP formalism is applied to the vacuum FEs
dominateduniverse.Thereforetheformalismpresented of a given theory, such an application to the f(R,T)
in this article surpasses the f(R,T = 0) issue since theory is expected to yield the same extra polarization
it makes f(R,T) gravity able to describe a radiation- states as those of the f(R) formalism [63].
dominated universe purely from its dependence on Tφ. TheKerrmetricdescribesthegeometryofanempty
Besides some questions exposed in the Introduc- space(T =0)aroundarotatingunchargedblackhole
µν
tion,veryrecently,the drawbackinf(R,T)theorywas (BH)withaxialsymmetry.KerrBHsstudieshavebeen
raisedin[26],inwhichtheauthorshavesearchedfordif- constantly seen in the literature (check, for instance,
ferentformsforf(T)indifferentstagesoftheuniverse. [72]-[75]). Since BHs are characterizedby strong gravi-
Intheradiation-dominatedstage,theyfoundthatf(T) tational fields in quantum length scales, it is expected
should be a non-null constant. As the authors have that their physics bring up some new insights about
considered the f(R,T) theories as a generalization of quantum theories of gravity [76,77]. Once again, one
Brans-Dicke gravity,the f(T) cte case leads to Geff wouldnotobtainnew contributionsto the study ofthe
∼
and Λeff to be, indeed, constants, as it may be seen space-timearounda KerrBHfromthe f(R,T)formal-
in [26]. Therefore no contributions to a T =0 universe ism, in such a way the f(R) theories outcomes should
would come from the f(R,T) formalism on this ap- be expected [78]-[81].
proach,departing from the results of the presentwork.
Hence, one can see that the lack of contributions
One could argue that the f(R) formalism retrieval
fromf(R,T)gravitywillappearnotonlyinaradiation-
in f(R,T) theories when T = 0 is not a shortcoming,
dominateduniverse,butalsoinstudiesmadeinthevac-
being just an intrinsic property of such theories. How-
uumregime,sinceT =0 T =0.Inotherwords,in
µν
ever,belowwebringtothereader’sattentionsomefun- the absence of the f(R,Tφ→) approach presented here,
damental areas of knowledge which will have to be ne-
onewouldexpectthe f(R)formalismfeatures retrieval
glected if the f(R,T) theories, indeed, simply recovers
in the study, not only of a radiation-dominated uni-
the f(R) formalism when T =0.
verse, but also in those of GWs and space-times sur-
Many contributions to Physics, Astrophysics and
rounding BHs.
Cosmology will come from the imminent detection of
Forinstance,inthesearchfornewpolarizationstates
gravitationalwaves(GWs)[57,58].Theirdetectionwill
of GWs mentioned above, one works with the vacuum
clarify some persistent and important issues, such as
FEs of a theory. It is expected that if one considers
the absolute ground state of matter [59], and also con-
the vacuum on the rhs of standard f(R,T) FEs (check
templates a powerful tool in estimating parameters of
Eq.(11)of[12]),onegets,aftertheNPformalismappli-
compactbinarysystems[60],constrainingthe equation
cation,the same polarizationstates of the f(R) theory
of state of neutron stars [61] and brane cosmological
[63]. On the other hand, by taking T = 0 (and con-
parameters values in braneworld models [62], and dis- µν
tinguishingGRfromalternativetheoriesofgravity[63].
The study of the propagationof GWs is made once 1In general relativity, there are two polarization states of
they have been generated [64]. The energy-momentum gravitational waves, knownas plusand cross polarizations.
9
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34. T. Harko etal., Eur. Phys.J. C74(2014) 2784.
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40. K. Bamba et al., Class.Quant. Grav. 30(2013) 015008.
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