Diego Saez-Gomez

tudy bations of the are kept quasi-static [13, 15]. approximation Let us point out for II. that this Θ f(R,
F(R,T)
µν
kind this = T −2T ) approximation of GRAVITY models.
gravity
µν − pg µν
InTHEORIES
Section . may VI we apply our results to two particu (7
tion odelsof and thenumerical first-order results perturbed are obtained fields [16, and23] compared and therefore with the requires ΛCDM model. Finally in Section VII we conclu
us iththe Let theequations us main start conclusions bymotion writingbecome
of the this general investigation. action for f(R, T ) gravities [4],
on II, we briefly review the state-of-the-art of f(R, T
S =
II.
S G +
f(R,
S m T
= 1 ) gravity.
) GRAVITY 2κ 2 d 4 x √ Section
osmological equations f R R for µν f(R, − 1 2 fg T )=f µν − (g 1 (R) µν +f −∇ 2 (T µ ∇) ν models ) f R = − as
THEORIES
−g κwell 2
−(f(R, fas
T Tµν T )+L + f Tm pg ) µν , . (8
omentum conservation for such models. Then, Section IV addressed I. FORMULAE
ons ere we
where,
for have f(R,
Let us Action, κstart 2 dropped
=8πG,
T )=f 1
by writing R
(R)+f the explicit
is the 2 (T
general Ricci
) models dependences
scalar
while of
actionand forT Section f in R
f(R, represents
Vand deals T .
T ) gravities the
with
trace
the
[4], of the energy-momentum tensor, i.e., T =
tis is
while
kind straightforward
L
of models.
m is the matter
Into Section see that
Lagrangian.
VI we theapply usual
As usual
our continuity
the
results
energy-momentum
toequation two particular is not satisfied for the field equations (8), an
S = S c 1 (R/R HS ) n
f(R) =R − R HS
c 2 (R/R HS ) n +1 , f(R) =R + Rn (aR n G + S m = 1
tensor is defined as,
dsequently comparedthe with covariant the ΛCDM derivative model. ofFinally the energy-momentum in Section VII we
− b)
2κ 2 d 4 x √ tensor conclude is not null in general ∇ µ T µν = 0. In order t
n. ain the modified continuity equation, let us take the covariant−g (f(R, T )+L m ) ,
T 1+cR n .
µν = 2 δS derivative m
√ −g δg µν .
of the equation (8),
here, κ
Field 2 =8πG, R∇ is µ the f Ricci scalar and T represents the trace of the energy-momentum tensor, i.e., T = T
equations, R R µν − 1 (R,
S EH = d 4 x √ −g
hile Then, T ) GRAVITY
L m
byis varying the matter the THEORIES action Lagrangian. with2respect fg µν − (g µν −∇ µ ∇ ν ) f R = −
κ 2
+ f T Tµν − f T Θ µν
As usual to the energy-momentum metric field g µν , the tensor
R +2κ 2 field is equations defined as, are obtained,
L m → S = d 4 x √ −g f(R)+2κ 2
L m
or f(R, T ) fgravities T µν = √ 2
R (R, T )R[4],
µν − 1 δS m
2 f(R, T )g µν − (g µν −∇ µ ∇ ν ) f R (R, T )=−
−g δg µν . κ 2 + f T (R, T ) T µν − f T (R, T )Θ µν ,
where the energy-momentum tensor is defined as: T µν (m) = −√ 2 δ ( √ where = 1 →
the subscripts on the function f(R, T ) mean differentiation −gL m )
hen, by varying the action with respect to the metric field g µν with respect to R or T , and the tensor Θ
2κ , the field equations −g δg µν are obtained,
defined 2 d 4 x √ f R ∇ µ R µν + R µν ∇ µ f R − 1 2 g µν(f R ∇ µ R + f T ∇ µ T ) − (g µν ∇ µ −∇ µ ∇ µ ∇ ν ) f R =
−g (f(R, T )+L m ) , (1)
as,
f R (R, T )R µν − 1 ∇ µ − κ 2
+ f
T represents the trace of
Divergence of the field 2 f(R, the energy-momentum T )g µν − (g µν −∇ µ ∇ ν ) f R (R, )=− κ 2 + f T (R, T ) T µν − f T (R, T )Θ µν ,
equations Θ µν ≡ g αβ δT tensor, T Tµν − f
i.e., T T Θ
= µν . (9
T µ µ,
αβ
δg µν = −2T µν g µν L m +2g αβ δL m
the energy-momentum tensor is defined as,
us, using the identities ∇ µ R
.
µν − 1 2 Rg µν
= 0, and (∇ν − ∇ ν ) f R (R, T )=R δg µν g µν αβ
∇ µ f R , the covariant derivativ
the here energy-momentum the subscripts on the function f(R, T ) mean differentiation with respect to R or T , and the tensor Θ µν
efined Note T µν that as, = √for 2 δS m
tensor needs to satisfy,
a regular . f(R, T ) function, in absence of any kind of matter, (2) the corresponding f(R) gravity equa
−g δg
µν
are recovered, and consequently
∇ µ T
the corresponding properties and the well-known solutions for f(R) gravity are
µν =
f
T 1
he satisfied metric by fieldf(R, g µν T , the ) theories field equations inκclassical Θ 2 + f
µν ≡ gare αβ δT T 2vacuum g µν∇ µ T − (T µν + Θ µν )∇ µ ln f T −∇ µ Θ
obtained, αβ
δg µν = −2T (for a review on f(R)
µν − g µν L m +2g αβ theories, δL µν . (10
m see [1]). Moreover, here w
.
interested to study the behavior of this kind of theories for spatially flatδg FLRW µν g αβ spacetimes, which are express
nce, comoving for coordinates by the line element,
−∇
In
a
ote that µ ∇
general,
perfect
ν ) f
for R (R,
the
fluid
T )=−
divergence
with an
κ
a regular f(R, 2 +
of
equation
f
the
T ) function, T (R,
equations
T ) of state
T
in µν −
is not
p
f
absence T (R,
null,
= wρ,
T
which
beingw
)Θ
of any kind µν
of matter, the corresponding f(R) gravity equati
re recovered, and consequently the corresponding ds 2 properties = a 2 (η) ,
may give
a constant,
(3)
rise to an
the
anomalous
0−component
behavior
of
of the
the covarian
ivative matter (10) turns content. out to become,
dη and 2 −the dxwell-known 2 solutions for f(R) gravity are a
atisfied ) mean by differentiation f(R, T ) theories with respect in classical to Rvacuum or T , and (for the a review tensoron Θ µν f(R) is
κ 2 + w − 3
f theories, see [1]). Moreover, here we
terested where a(η) to study is the the scale behavior factor T − (1 + w)Tf
2 in of conformal TT T ˙ + 3(1 + w)
H(κ 2 − f
this kind of time theories η. Then, for spatially the main T ) − 2f
flatissue TR (4HḢ FLRW arises spacetimes, on + the Ḧ) T =0. (11
T. Harko, F. S. N. Lobo, S. Nojiri and S. D. Odintsov, content #Phys. which Rev. D84, are of 024020 the expressed
(2011) Unive