Diego Saez-Gomez

effective repulsive force 12h
R ∼ reproduced 0
+6h 0 by the F (R) terms . and compare (19) with the binding forces of
for example the Solar ⎪⎩ System). However, to ensure that a little rip is reproduced, one has to stu
acceleration but free of future singularities, which may have a stronger growth in time than thos
big rip singularities. Let us use the reconstruction technique using Eqs. (12-21). As an exam
iods okingof at super-accelerating some of the models expansion reconstructed and it does in not the contain previous any sections, future
(t s
singularity. we can see To for
that find β
the
=1
−t)
out
2 Hubble parameter (25)
a
ay
periodic
lead to
function
a little rip,
that
one
reproduces
has to determine
a cyclic
the
Universe.
duration
From
of each
the
cycle6βh effective
and 0
EoS
the strength
parameter
of the
(t s
for β < 1
−t) (41), it is clear that this
force
odel exhibits
reproduced
periods
by the
of
F
super-accelerating
(R) terms and compare
expansion
with the
and
binding
it does
forces β+1
not contain
of coupled
any
systems
future
(as
singularity. To find out
lar System). However, to ensure that a little rip is reproduced,
the model (25) may lead to a little rip, one hasfunction,
one has to f( study
to determine the duration ˜R) ∝ ˜R models n n = 2 of eternal
(20)
of each cycle and the strength of the
ee of future singularities, which may have a stronger
The
growth
Little
in time
than
ective repulsive force reproduced by the F (R) terms and compare d
2 those models containing
with the binding forces of coupled systems (as
s. Let us use the reconstruction technique using Eqs. (12-21). −3n As an example, we consider the U(φ) =e −βeαφ ,
r example the Solar System). However, to ensure that a little rip is dt 2 reproduced, +3H d Rip
R n−1 ∼ (2 − n)R n (21)
dt one has to study models of eternal
celeration but free of future singularities, which may have a stronger n = 2growth free of singularities time than those models containing (22)
g rip singularities. Let us useU(φ) the =e reconstruction technique using Eqs. (12-21). As an example, we consider the
nction, Little Rip in f(R) −βeαφ , (45)
n =
gravity
1 2 + 3µ
(23)
2(1 − 3λ +3µ)
constants. Then, by the expression (21), the Hubble parameter and the scale factor yield,
U(φ) =e −βeαφ , R 2 ˜f(R) (24) (45)
H(t) =h 0 e αt + h 1 , → a(t) =a 0 e 4βeαt +6αt . (46)
here α and
Hubble
β are constants.
parameter
φ = t, H= g (φ) =g (t) (25)
Then, by the expression (21), the Hubble parameter and the scale factor yield,
d h 1 =6α. It is straightforward to see that the function (46) describes aP Universe, (φ) =P 0 (cos where ωφ) 4 for small
(26)
H(t) =h 0 e αt be easily reconstructed by
+ h 1 , → a(t) =a 0 e 4βeαt +6αt the expressions (15) and (20),
ubbleparametercanbeapproximatedasaconstant,reproducingadeSittersolution,asinthe
H(t) =H 0 (cos ω t) 2 +2ω(2 . sin ω t − tan ω t) (27) (46)
del. For large times, the Universe ends in an eternal phantom phase, where the EoS parameter
here ithout h 0 big =4αβ rip singularity. and h 1 =6α. Nevertheless, It is straightforward a little rip to (dissolution see that the of bound function S = structures) dtd (46) 3 x describes g (3) NF( might ˜R) occur a Universe, where(28)
8for small
ilarly es t to α, the Action theHubbleparametercanbeapproximatedasaconstant,reproducingadeSittersolution,asinthe
model presented in Ref. [5], as it is pointed out below. The functions P (φ) and Q(φ)
se structed of ΛCDM by themodel. expressions For large (15) and times, (20), the Universe R = ends 6(2H 2 in+ Ḣ) an and eternal by using phantom expression
8
phase, (13), where which yields,
Here the couplings κ i are constants depending on C 1,2 .Hence,forsmallvaluesoftheRicciscalartheactionreduces EoS parameter
F (R) <
to−1, the
but
action
without
for General
big rip
Relativity
singularity.
plus power-law H(t) =H
Nevertheless,
curvature 0 [cos(ωt)]
a little
corrections, 2 2ωµ
+
rip (dissolution of bound structures)
might occur
a finite time, P (φ) =e similarly 4βeαφ , toQ(φ) the model =−6αpresented 2 (3 + 4βe αφ in)(3 Ref. + 8βe [5], αφ as )e 4βeαφ it is pointed . out below. F (R) = The
C (47) 1 functions + C 2 4 R
are constants depending on C 1,2 .Hence,forsmallvaluesoftheRicciscalartheactionreduces
1 − 3λ as +3µ R 2 , tan(ωt) 2µ
which is1known −
1 − 3λ recipe +3µ to [sin(ωt)]2 cure the
singularities and which has a viable behavior ([13]). R
12R
+
P + (φ) 75 and e Q(φ)
25
0
ral Relativity plus power-law curvature corrections, as R 2 action (48) represents a viable model where GR can (29)
be recovered while the curvature scalar corrections remain , which small. is It known is important recipe to that cure such theadditional Rcorrections
0
hn be easily reconstructed by the expressions (15) and (20),
hat hasreproduces abecome viablerelevant behavior the solution close ([13]). to(46) the Hence, little can be rip thecalculated evolution. action (48) Inby order represents inverting to estimate athe viable expression themodel time for where ofthe thelittle GR Ricci rip can scalar induced dissolution of
e
nd
curvature
by using
scalar
the expression
corrections
P
(13),
remain
(φ) =e
which
small.
yields,
It is important that such
P (φ)
additional
=P
bound structures in a naive way, one might compare the energy-density 4βeαφ , Q(φ) =−6α 2 (3 + 4βe αφ of a 0 (cos(ωφ))
corrections
− 2(1−3λ)
1−3λ+3µ (30)
bound
)(3 + 8βe αφ system as the Solar System with
to the the little density rip evolution. ρ )e 4βeαφ F (R) .Forthemodel(46),suchdensitycanbeapproximatedforlargetimesby
In order to estimate time little rip induced dissolution . of
(47)
S = d
he F (R) action that F (R) reproduces =
C 4 x √ −gβ(φ)L
naive way, one might compare the energy-density
the solution (46) can be calculated by inverting the expression of the Ricci scalar
1 + C 2 4 R of
m (31)
a
exponential
R
bound system as the Solar System with
12R
+
+ 75 e
25
0 16 functions are expanded power series,
themodel(46),suchdensitycanbeapproximatedforlargetimesby
terms adjusted with the dark energy density ρ F (R) = nowadays,
ρ. 0 e 2αt , (48)
(50)
= 6(2H 2 + Ḣ) and by using the expression R
F iner >F 0 (32)
0 (13), which yields,
where ρ R 2 R 3
0 is a constant that
ρ
F (R) =
C 1 + C 2 4 R
1 = −24e −39/12 R 0 ,andC 2 =2 √ can be set F (R) ∼ κ 1 R + κ 2 + κ 3
R 0 R
3R 0 . Hence, we have reconstructed F (R) R model
12R
+
+ 75 e
25
0
which is able to
2 + ··· .
F (R) = ρ 0 e 2αt by imposing that the current value of the energy-density is ρ
, (50) F (R) (t 0 )= 3 κ
H 2 0 2 ∼
10 −47 GeV 4 , where the age of the Universe is takenwhere
to be t 0 ∼ 13.73
ρ F Gyrs, (R) (t 0
accordingtoRef.[14].
)= 3 κ 2 H2 0 ∼ 10 −47 GeV
Onecansetthetime
4 (33)
of the little rip dissolution occurrence when the gravitational coupling of the 0 Sun-Earth 16
accelerated that can be (eternal) set by imposing phase, that whichthe does current not lead valueto ofa the future energy-density . (48)
R
singularity. isNote ρ F (R) that (t 0 )= the 3 system
action (48)
0 κ
H 2 is broken due to the
Einstein-Hilbert action plus some corrections for small
here R 0 = α 2 , C 1 = −24e −39/12 R 0 ,andC 2 =2 √ values of the Ricci curvature R, 2 0 ∼
cosmological expansion. By assuming a mean density of the Sun-Earth system given by ρ ⊙−⊕ =0.594×10
age of the Universe is taken to be t 0 ∼ 13.73 Gyrs, accordingtoRef.[14]. Onecansetthetime where the
−3 kg/m 3 ∼
10 −21 GeV 4 ,thetimeforthelittleripdissolutionofboundstructuresis,
tion ns are occurrence expanded
Time for
when in power
Little
the gravitational series,
Rip
coupling of the
3R
Sun-Earth
0 . Hence,
system
we have
is
reconstructed
broken due to the
F (R) model which is able to
n. By assuming a mean density of the Sun-Earth
produce a super-accelerated (eternal) Rphase, 2 tsystem which R 3 given by ρ
does not lead ⊙−⊕ =0.594×10
to a future singularity. −3 kg/m 3 LR = 13.73 Gyrs + 29.93
∼
Note that the action (48)
forthelittleripdissolutionofboundstructuresis,
α . (51)
rns out to be the Einstein-Hilbert
F (R) ∼ κ 1 R +
action
κ 2 +
R 0
plus
κ 3
Rsome 2 + ···
corrections
.
for small values of the Ricci
(49)
curvature R, where the
Hence, depending on the parameter α, the 0
ponential functions are expanded in power series,
t LR = 13.73 Gyrs + 29.93 appearance of the little rip may last shorter or longer. For example, when
α = 10 −1 Gyrs −1 ,thelittleripoccursattheUniverseageoft
α . LR = 313.03 Gyrs, while for α (51) ≥ 1Gyrs −1 ,thetimefor
the decoupling will be much shorter.
RS. 2 Nojiri, S. ROdintsov AlotofmodelscanbereconstructedinframesofF 3 and DSG, AIP Conf. Proc. 1458, 207 (2011), arXiv:1108.0767.
he parameter α, the appearance of the little F (R) rip∼ may κ 1 R last + κ(R) shorter 2
gravity + κ 3
which + are ··· able . to reproduce a super-accelerating (49)
R
or longer. For example, when
0
where α and β are constants. Then, by the expression (21), the Hubble parameter and the scale
H(t) =h 0 e αt + h 1 , → a(t) =a 0 e 4βeαt +6αt .
where h 0 =4αβ and h 1 =6α. It is straightforward to see that the function (46) describes a Univ
times t α, theHubbleparametercanbeapproximatedasaconstant,reproducingadeSitte
case of ΛCDM model. For large times, the Universe ends in an eternal phantom phase, where
w F (R) < −1, but without big rip singularity. Nevertheless, a little rip (dissolution of bound stru
in a finite time, similarly to the model presented in Ref. [5], as it is pointed out below. The funct
P (φ) =e 4βeαφ , Q(φ) =−6α 2 (3 + 4βe αφ )(3 + 8βe αφ )e 4βeαφ .
The F (R) action that reproduces the solution (46) can be calculated by inverting the expressio
where R 0 = α 2 , C 1 = −24e −39/12 R 0 ,andC 2 =2 √ 3R 0 . Hence, we have reconstructed F (R) mo
reproduce a super-accelerated (eternal) phase, which does not lead to a future singularity. Note
turns out to be the Einstein-Hilbert action plus some corrections for small values of the Ricci cur
One can set the time of the little rip dissolution occurrence when the gravitational coupling of the
Sun-Earth system is broken due to the cosmological expansion by assuming a density for the f(R)
A. López-Revelles,
R0
2 R. Myrzakulov and DSG, Phys. Rev. D 85, 103521 (2012), arXiv:1201.5647.
−1
phase free of singularities. Let us consider, for example, a Hubble parameter that reproduces a phantom Universe
16
.