Cyclic cosmology from Lagrange-multiplier modified gravity

0.3 MB·English

by  Yi-Fu Cai

Cyclic cosmology from Lagrange-multiplier modified gravity
Yi-Fu Cai1,2,∗ and Emmanuel N. Saridakis3,†
1 Department of Physics, Arizona State University, Tempe, AZ 85287, USA
2 Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918-4, Beijing 100049, China
3College of Mathematics and Physics,
Chongqing University of Posts and Telecommunications, Chongqing 400065, China
Weinvestigatecyclicandsingularity-freeevolutionsinauniversegovernedbyLagrange-multiplier
modifiedgravity,eitherinscalar-fieldcosmology,aswellasinf(R)one. Inthescalarcase,cyclicity
canbeinducedbyasuitablyreconstructedsimplepotential,andthemattercontentoftheuniverse
canbesuccessfully incorporated. Inthecaseoff(R)-gravity,cyclicity canbeinducedbyasuitable
1
reconstructedsecondfunctionf2(R)ofaverysimpleform,howeverthematterevolutioncannotbe
1
analyticallyhandled. Furthermore,westudytheevolutionofcosmologicalperturbationsforthetwo
0
scenarios. Forthescalarcasethesystempossessesnowavelikemodesduetoadust-likesoundspeed,
2
whileforthef(R)casethereexistanoscillation modeofperturbationswhichindicatesadynamical

n degree of freedom. Both scenarios allow for stable parameter spaces of cosmological perturbations
a through thebouncing point.
J

2 PACSnumbers: 98.80.-k,04.50.Kd
1

] I. INTRODUCTION [3] with bouncing solution studied in [20], was recently
O
gained new interest, since its extended version may re-
C duce irrelevant degrees of freedom in modified gravity
Inflation is now considered to be a crucial part of the
. models[21–23]. Inparticular,usingtwoscalarfields,one
h cosmological history of the universe [1], however the so
p called “standard model of the universe” still faces the ofwhichis the Lagrangemultiplier,leads to aconstraint
- ofspecial formon the secondscalarfield, andas a result
problem of the initial singularity. Such a singularity
o
the whole system contains a single dynamical degree of
r is unavoidable if inflation is realized using a scalar field
t freedom.
s whilethebackgroundspacetimeisdescribedbythestan-
a dard Einstein action [2]. As a consequence, there has In the present work we are interested in constructing
[ been a lot of effort in resolving this problem through a scenario of cyclic cosmology in universe governed by
2 quantum gravity effects or effective field theory tech- modified gravity with Lagrange multipliers. We do that
v niques. both in the case of scalar cosmology, as well as in the
4 case of f(R)-gravity, both in their Lagrange-multiplier
A potential solution to the cosmological singularity
0 modifiedversions. Asweshow,therealizationofcyclicity
problem may be provided by non-singular bouncing cos-
2 and the avoidance of singularities is straightforward.
mologies [3]. Such scenarios have been constructed
3
. through various approaches to modified gravity, such as This paper is organized as follows. In section II we
7 present modified gravity with Lagrange multipliers, sep-
the Pre-Big-Bang[4]andthe Ekpyrotic[5]models,grav-
0 ity actions with higher order corrections [6], braneworld arately for scalar cosmology (subsection IIA) and for
0
f(R)-gravity (subsection IIB). In section III we con-
1 scenarios [7], non-relativistic gravity [8], loop quantum
: cosmology [9] or in the frame of a closed universe [10]. structthescenarioofcyclicityrealization,reconstructing
v the requiredpotential (subsection IIIA) and the form of
Non-singular bounces may be alternatively investigated
Xi using effective field theory techniques, introducing mat- f(R)modification(subsectionIIIB). InsectionIVwein-
vestigate the stability of the cosmological perturbations
r ter fields violating the null energy condition [11] lead-
a ing to observable predictions too [12], or introduce non- of the two scenarios. Finally, section V is devoted to the
summary of our results.
conventional mixing terms [13]. The extension of all the
above bouncing scenarios is the (old) paradigm of cyclic
cosmology[14],inwhichtheuniverseexperiencesthepe-
riodicsequenceofcontractionsandexpansions,whichhas
II. MODIFIED GRAVITY WITH LAGRANGE
been rewaked the last years [15] since it brings different
MULTIPLIERS
insights for the origin of the observable universe [16–18]
(see [19] for a review).
Let us present the cosmological scenarios with La-
On the other hand, the interesting idea of modifying
grange multipliers. In order to be general and complete,
gravity using Lagrange multipliers, first introduced in
we present both the scalar case, as well as the f(R)-
gravity one. Throughout the work we consider a flat
Friedmann-Robertson-Walkergeometry with metric
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected] ds2 = dt2+a2(t)dx2, (1)
−
2
where a is the scale factor, although we could straight- equation(3) can be integratedfor positive U(φ), leading
forwardly generalize our results to the non-flat case too. to
φ dφ
t= . (9)
A. Scalar cosmology with Lagrange multiplier ± 2U(φ)
Z
Inverting this relation withprespect to φ, one can find
In this subsection, we consider a scenario with two
the explicit t-dependence of φ, that is the corresponding
scalars, namely φ and λ, where the second scalar is a
φ(t). Thus, substituting the expression of φ(t) into (5),
Lagrange multiplier which constrains the field equation
we obtain a differential equation for H(t):
of the first one. The action reads [21, 22]:
2H˙(t)+3[H(t)]2 = κ2 ω(φ(t))U(φ(t))
R ω(φ) − {
S = d4x√−g 2κ2 − 2 ∂µφ∂µφ−V(φ) −V(φ(t))+wmρm(t)}, (10)
Z (cid:26)
1 the solution of which determines completely the cosmo-
λ ∂ φ∂µφ+U(φ) + ,(2)
− 2 µ Lm logicalevolution. Finally,substitutingφ(t)andH(t)into
(cid:20) (cid:21) (cid:27)
(4), we canfind the t-dependence ofthe Lagrangemulti-
where λ is the Lagrange multiplier field. Furthermore, plier field λ:
V(φ) and U(φ) are potentials of φ, while the function
ω(φ) and especially its sign determines the nature of the λ(t)= 1 3 H(t)2 V (φ(t)) ρ (t)
scalar field φ, that is if it is canonical or phantom. Fi- 2U(φ(t)) κ2 − − m
(cid:26) (cid:27)
nally,theterm m accountsforthemattercontentofthe ω(φ(t))
universe. L . (11)
− 2
Variation over the λ-field leads to
However,onecouldalternativelyfollowtheinversepro-
φ˙2 cedure, that is to first determine the specific behavior of
0= U(φ). (3)
2 − H(t) and try to reconstruct the corresponding potential
V(φ), which is responsible for the φ-evolution that leads
As we observe, this equation acts as a constraint for the
to such a H(t). In particular, he first chooses a suit-
other scalar field φ, and it is the cause of the significant
able U(φ) which will lead to an easy integration of (9),
cosmological implications of such a construction. Now,
allowing for an interchangeable use of φ and t through
the Friedmann equations straightforwardlywrite
φ(t) t(φ). Therefore, the second Friedmann equation
↔
3 ω(φ)+λ (5) gives straightforwardly
H2 = φ˙2+V(φ)+λU(φ)
κ2 2 1
(cid:20) (cid:21) V(φ)= 2H˙ (t(φ))+3H(t(φ))2
= [ω(φ)+2λ]U(φ)+V(φ)+ρm, (4) κ2
h +ω(φ)U(φ)+wi ρ (t(φ)), (12)
m m
1 ω(φ)+λ
2H˙ +3H2 = φ˙2 V(φ) λU(φ) where ω(φ) can still be arbitrary. In summary, such a
− κ2 2 − − potential V(φ) produces a cosmological evolution with
(cid:16) (cid:17) (cid:20) (cid:21)
= ω(φ)U(φ) V(φ)+p , (5) the chosen H(t).
m
− Finally,letusmakeacommenthereconcerningthena-
where we have defined the Hubble parameter as H a˙, ture of models with a second, Lagrange-multiplier field,
≡ a
and we have also made use of the constraint (3). Ad- since it is obvious that the constraint (3) changes com-
ditionally, in these expressions ρm and pm stand respec- pletely the dynamics, comparing to the conventional
tivelyfortheenergydensityandpressureofmatter,with models. In principle, one expects to have a propagation
equation-of-state parameter wm pm/ρm. Observing mode for each new field. However, due to the constraint
≡
the above equations,we canstraightforwardlydefine the (3) and the form in which the λ-field appears in the ac-
energydensityandpressureforthedark-energysectoras tion, the propagating modes of φ and λ do not appear,
and the system is driven by a system of two first-order
ρ [ω(φ)+2λ]U(φ)+V(φ), (6)
de ordinary differential equations, one for each field. As a
≡
pde ω(φ)U(φ) V(φ), (7) consequence, there are no propagating wave-like degrees
≡ −
of freedom and the sound speed for perturbations is ex-
andthusthedarkenergyequation-of-stateparameterwill
actly zero irrespective of the background solution [21].
be
ω(φ)U(φ) V(φ)
wde = − . (8) B. f(R)-gravity with Lagrange multiplier
[ω(φ)+2λ]U(φ)+V(φ)
Let us now explore some general features of the sce- In this subsection we present f(R)-gravity with La-
nario at hand following [22]. First of all the constraint grange multipliers, following [22]. In this case, we start
3
byaconventionalf(R)-gravity,andweaddascalarfield nically very complicated. These subtleties are caused by
λ which is a Lagrangemultiplier leading to a constraint. the fact that the Lagrange multiplier field propagates in
In particular, the action reads: thiscase,whichisadisadvantageofthepresentscenario,
contrary to the scalar cosmology of the previous subsec-
S = d4x√ g f (R) λ 1∂ R∂µR+f (R) , tion.
1 µ 2
− − 2 Variation over λ leads to the constraint
Z (cid:26) (cid:20) (cid:21)(cid:27)
(13)
where f (R) andf (R)are two independent functions of 1
1 2 0= R˙2+f (R). (14)
theRicciscalarR. Notethatintheaboveactionwehave −2 2
notincludedthemattercontentoftheuniverse,sincethis
would significantly modify the multiplying terms of λ, Additionally,varyingoverthemetricg andkeepingthe
µν
making the subsequent reconstruction procedures tech- (0,0)-component we obtain
1 d d dR
0= f (R)+18λ(H¨ +4HH˙)2+ 3 H˙ +H2 3H f λf + +3H λ , (15)
−2 1 − dt 1,R− 2,R dt dt
(cid:26) (cid:16) (cid:17) (cid:27)(cid:26) (cid:18) (cid:19)(cid:18) (cid:19)(cid:27)
where denotes the derivativewith respectto the Ricci III. COSMOLOGICAL BOUNCE AND CYCLIC
,R
scalar. COSMOLOGY
For f (R)>0, the constraint (14) can be solved as
2
R dR Having presented the cosmological models with La-
t= , (16) grange multipliers, both in the scalar, as well as in the
2f (R)
Z 2 f(R)-gravitycase, in this sectionwe are interestedin in-
and inverting this relationpwith respect to R one can vestigating the bounce and cyclic solutions.
obtain the explicit t-dependence of R, that is the corre- Inprinciple,inordertoacquireacosmologicalbounce,
sponding R(t). On the other hand, the Ricci scalar is one has to have a contracting phase (H < 0), followed
given from R=6H˙ +12H2. Thus, inserting R(t) in this by an expanding phase (H > 0), while at the bounce
relation one obtains a differential equation in terms of point we have H =0. In this whole procedure the time-
H(t), namely derivativeoftheHubbleparametermustbepositive,that
is H˙ >0. Onthe other hand,in orderfor a cosmological
6H˙(t)+12[H(t)]2 =R(t), (17) turnaroundto be realized,one has to have anexpanding
phase (H >0) followed by a contracting phase (H <0),
the solution of which determines completely the cosmo-
while at the turnaround point we have H = 0, and in
logical behavior. Finally, using the obtained H(t) and
this whole procedure the time-derivative of the Hubble
R(t), equation (15) becomes a differential equation for
parameter must be negative, that is H˙ < 0. Observing
the Lagrange multiplier field λ(t) which can be solved.
the form of Friedmann equations (4), (5), as well as of
However,onecouldalternativelyfollowtheinversepro-
(15), we deduce that such a behavior can be easily ob-
cedure, that is to first determine the specific behavior
tained in principle. In the following two subsections we
of H(t) and try to reconstruct the corresponding f (R),
2
proceedtothe detailedinvestigationinthe caseofscalar
whichisresponsiblefortheR-evolutionthatleadstosuch
and f(R)-modified cosmology.
an H(t). In particular,with a known H(t) (17) provides
immediatelyR(t),whichcanbeinverted,givingt=t(R).
Thus,usingtheconstraint(14),theexplicitformoff (R)
2
is found to be
A. Scalar cosmology with Lagrange multiplier
2
1 dR
f (R)= . (18)
2
2 dt (cid:12)
(cid:18) (cid:19) (cid:12)t=t(R) In order to provide a simple realization of cyclicity in
(cid:12)
Itisinterestingtomentionthati(cid:12)ntheabovediscussion this scenario, we start by imposing a desirable form of
(cid:12)
H(t) that corresponds to a cyclic behavior. We consider
f (R) is arbitrary. That is, while in conventional f(R)-
1
a specific, simple, but quite general example, namely we
gravity the cosmological behavior is determined com-
assume a cyclic universe with an oscillatory scale factor
pletely by f (R), in Lagrange-multiplier modified f(R)-
1
of the form
gravitythedynamicsisdeterminedcompletelybyf (R).
2
In this case f (R) would become relevant only in the
1
presence of matter, and its effect on Newton’s law [22]. a(t)=Asin(ωt)+a , (19)
c
4
where we have shifted t in order to eliminate a possible also the λ(t) evolution, which according to (11) reads
additional parameter standing for the phase1. Further-
more, the non-zero constant ac is inserted in order to 2 Aω2[A+acsin(ωt)]
eliminate any possible singularities from the model. In λ(t)= 1
sucha scenariot varies between and+ ,andt=0 − − κ2m4 (− [Asin(ωt)+ac]2 )
is just a specific moment witho−ut∞any par∞ticular phys- 1 ρ Asin(ωt)+a −3. (25)
ical meaning. Finally, note that the bounce occurs at −m4 m0{ c}
a (t)=a A. Straightforwardlywe find:
B c
− In order to provide a more transparent picture of the
obtained cosmologicalbehavior, in Fig. 1 we present the
Aωcos(ωt)
H(t)= (20) evolution of the oscillatory scale factor (19) and of the
Asin(ωt)+a
c Hubble parameter(20), with A=1, ω =1 and a =1.3,
c
H˙(t)= Aω2[A+acsin(ωt)]. (21) whereallquantitiesaremeasuredinunitswith8πG=1.
− [Asin(ωt)+a ]2
c
Concerningthemattercontentoftheuniverseweassume )
t 2
it to be dust, namely w 0, which inserted in the (
m a
≈
evolution-equation ρ˙ +3H(1+w )ρ = 0 gives the
m m m
usual dust-evolution ρ =ρ /a3, with ρ its value at 1
m m0 m0
present.
Now, we first consider φ to be a canonical field, that 0
-20 -10 0 10 20
is we choose ω(φ) = 1. Concerning the potential U(φ)
)
thatwillgiveasthesolutionforφ(t)accordingto(9),we t
( 1
choose a simple and easily-handled form, namely H
0
m4
U(φ)= , (22)
2 -1
wheremisaconstantwithmass-dimension. Inthiscase, -20 -10 0 10 20
(9) leads to
t
φ=m2t, (23) Figure 1: The evolution of the scale factor a(t) and of the
Hubble parameter H(t) of the ansatz (19), with A=1, ω=1
and ac = 1.3. All quantities are measured in units where
having chosen the + sign in (9). As we have said in
8πG=1.
subsection IIA, such a simple relation for φ(t) allows to
replace t by φ/m2 in all the aforementioned relations.
Therefore, substitution of (20),(21) and (22) into (12),
and using φ instead of t, provides the corresponding ex-
pression for V(φ) that generates such an H(t)-solution: )
8
(
V
m4 1 Aω2 A+acsin ωmφ2
V(φ)= 2 + κ2 2− Ahsin ωmφ2 (cid:16)+ac 2(cid:17)i 6
h (cid:16) 2(cid:17) i
Aωcos ω φ  4
m2
+3 . (24)
Asin ωm(cid:16)φ2 +(cid:17)ac 
2
(cid:16) (cid:17)
Note that in the case of dust matter, thereconstructed
potential does not depend on the matter energy density
0
and its evolution. Finally, for completeness, we present
-20 -10 0 10 20
Figure 2: V(φ) from (24) that reproduces the cosmological
1 Note that if the average of an oscillatory scale factor keeps in- evolutionofFig.1,withω(φ)=1,U(φ)=m4/2andm=1.2.
creasingthroughouttheevolution,itcouldyieldarecurrentuni-
All quantities are measured in units where 8πG=1.
verse which unifies the early time inflation and late time accel-
eration[24,25].
5
NotethatH(t)byconstructionsatisfiestherequirements
)
for cyclicity, described in the beginning of this section. (t 4
Furthermore, in Fig. 2 we depict the corresponding po- a
tentialV(φ)givenby(24). Fromthesefiguresweobserve 3
thatanoscillatingandsingularity-freescalefactorcanbe
generated by an oscillatory form of the scalar potential 2
V(φ) (although of not a simple function as that of a(t),
-20 -10 0 10 20
ascanbeseenbytheslightlydifferentformofV(φ)inits
)
minima and its maxima). This V(φ)-form was more or (t 1
H
less theoretically expected, since a non-oscillatory V(φ)
would be physically impossible to generate an infinitely 0
oscillatingscalefactorandauniversewithaformoftime-
symmetry. Note also that, having presented the basic -1
mechanism, we can suitably choose the oscillation fre-
quency and amplitude in order to get a realistic oscilla- -20 -10 0 10 20
tion period and scale factor for the universe. Finally, we
t
stress that although we have presented the above spe-
cific simple example, we can straightforwardly perform Figure 3: The evolution of the scale factor a(t) and of the
the described procedure imposing an arbitrary oscillat- Hubble parameter H(t), for a scalar potential of the ansatz
ing ansatz for the scale factor. (26), with V0 = 3, ωV = 1 and Vc = 3, α(0) = 3.2 and
H(0) = −0.7. All quantities are measured in units where
The aforementioned bottom to top approach was en-
8πG=1.
lightening about the form of the scalar potential that
leads to a cyclic cosmological behavior. Therefore, one
can perform the above procedure the other way around,
starting from a specific oscillatory V(φ) and resulting to which is oscillatory and always non-zero. Inserting (27)
an oscillatory a(t), following the steps described in the and its derivative into (17) we obtain R(t) as
first part of subsection IIA. As a specific example we
consider the simple case R(t)=6AHωHcos(ωHt)+12A2Hsin2(ωHt), (29)
V(φ)=V0sin(ωV φ)+Vc, (26) a relation that can be easily inverted giving
withU(φ)chosenasin(22)andthus(23)holdstoo. De-
1 3ω + 3(48A2 4R+3ω2 )
spitethesimplicityofV(φ),thedifferentialequation(10) t(R)= arccos H H − H ,
cannotbesolvedanalytically,butitcanbeeasilyhandled ωH " p 12AH #
numerically. In Fig. 3 we depict the corresponding solu-
tion for H(t) (and thus for a(t)) under the ansatz (26), where we have kept the plus sign in the square root. Fi-
with V0 = 3, ωV = 1 and Vc = 3, with α(0) = 3.2 and nally, f2(R) can be reconstructed using (18), leading to
H(0)= 0.7(in units where 8πG=1). As expected, we
do obtai−n an oscillatory universe. f (R)= 1ω2 48A2 4R+3ω2
2 −4 H H − H
(cid:16) (cid:17)
2R+3ω2 +ω 3(48A2 4R+3ω2 ) (.30)
B. f(R)-gravity with Lagrange multiplier × − H H H − H
(cid:20) q (cid:21)
Insummary,suchanansatzforf (R)producesthecyclic
In order to provide a simple realization of cyclicity in 2
universewithscalefactor(28). Notethatf (R)hasare-
this scenario, we start by imposing a desirable form of 2
markably simple form, and this is an advantage of the
H(t) that corresponds to a cyclic behavior. Due to the
scenario at hand, since in conventional f(R)-gravity one
complicated structure of the reconstruction procedure,
needsveryrefinedandcomplicatedformsoff(R)inorder
we will choose an easier ansatz comparing to the previ-
to reconstruct a given cosmologicalevolution [26]. How-
oussubsection,whichallowsfortheextractionofanalyt-
ever, as we stated in the beginning of subsection IIB, in
ical results. In particular, having described the general
the f(R)-version of Lagrange-modified gravity one can-
requirementsforanH(t)thatgivesrisetocycliccosmol-
not incorporate the presence of matter in a convenient
ogy in the beginning of the present section, we choose
way that will allow for an analytical treatment. The ab-
H(t)(andnota(t))to bestraightawayasinusoidalfunc-
sence of matter evolution in a cyclic scenario is a disad-
tion, that is
vantage,since we cannotreproduce the evolutionepochs
H(t)=A sin(ω t), (27) of the universe. Therefore it would be necessary to con-
H H
structaformalismthatwouldallowforsuchamatterin-
which gives rise to a scale factor of the form
clusion,similarlytothecaseofLagrange-multipliermod-
AHcos(ωHt) ifiedscalar-fieldcosmology. Sucha projectis inprepara-
a(t)=A exp , (28)
H0 − ω tion.
(cid:20) H (cid:21)
6
IV. STABILITY ANALYSIS which is applicable in all scales. Thus, in this solution
there aretwo modes D(x) andS(x), whichare arbitrary
Acentralissueinallcyclicmodelsisthestabilityanal- functions of spatial coordinates, and their explicit forms
ysis of the cosmological perturbations along with back- can be determined by certain boundary conditions.
ground evolution. In Newtonian gauge the linear metric
Observing the second term of the right-hand-side of
perturbation is given by
(33), one may concern that the Newtonian potential
ds2 = (1+2Φ)dt2+a2(t)(1 2Ψ)dx2 , (31) might be divergent on the bouncing point, when H =0.
− − Fortunately, this does not happen in a generic bounce
model. As it was shown in phenomenological studies of
wherethevariableΦistheso-calledNewtonianpotential
genericbouncescenarios[27],onecanimposeanapproxi-
whichdescribesthescalarpartofmetricperturbation. In
mateparametrizationoftheHubbleparameterasalinear
the following we will study the evolution of the Newto-
function of the cosmic time, that is H = ξt, nearby the
nianpotentialnearthebouncingpoint,forthetwoabove
bouncing point t =0, with ξ being a positive constant.
scenarios respectively. B
Doing so, in the neighborhood of a bounce in a specific
cycle the Newtonian potential can be solved as
A. Scalar cosmology with Lagrange multiplier
sceAnnareioxpolifcistcaanlaarlycsoissmonoltohgeyNweiwthtonLiaagnrapnogteentmiaulltoifpltiheer ΦB ≃D 1− π2ξte−ξ2t2(eξ2t2 −1)21 +Saξt e−ξ2t2,
(cid:20) r (cid:21) B
was performed in [21]. A very remarkable feature of this (34)
modelisthatthesystempossessesnowavelikemodesand where a is the value of the scale factor at the bouncing
B
thus the sound speed of perturbations is identically zero point.
inallbackgrounds. Byvirtueofthisparticularproperty,
Relation (34) presents convergent behavior, and thus
it could be possible to control the dangerous growth of
the perturbations are able to pass through the bounce
unstablemodesofcosmologicalperturbationsinthecon-
smoothly and without any pathology. Therefore, the
tracting phases of the cyclic universe.
scenario at hand indeed provides a satisfactory way to
Assuming that there is no anisotropic stress in our
realize a cyclic picture without instability on its pertur-
model, then Ψ=Φ. Consequently, we only have one de-
bations.
greeoffreedomofcosmologicalperturbations. According
to the analysis of [21], one deduces that when the back-
ground evolution is dominated by the scalar field φ, the
perturbationequationoftheNewtonianpotentialcanbe
expressed as
H′′ H′ H′′
Φ′′+ 1 Φ′+ Φ 0, (32) B. f(R)-gravity with Lagrange multiplier
− H′ H − H′ ≃
(cid:18) (cid:19) (cid:18) (cid:19)
where the prime denotes the derivative with respect to
Whenwearedealingwithf(R)-gravitywithLagrange
lna. Note that the dust-like sound speed c = 0 has
s multiplier, we cannot use the simplifying relation Ψ =
been applied to eliminate the gradient term. Equation
Φ, even under the assumption of zero anisotropic stress.
(32) yields the generic solution [21]
Therefore, in order to study the number of degrees of
H a da H freedominthiscaseitisconvenienttoexpandtheaction
Φ D(x) 1 +S(x) , (33) into quadratic order,whichcanbe formallyexpressedas
≃ − a H a
(cid:18) Z (cid:19)
S(2) = d4x√ g f1,RRδ R2+f gµνδgµνδ R+δ R + gµνδgµν λC , (35)
1 1 1 2 2
− 2 − 2 2
Z (cid:20) (cid:16) (cid:17) (cid:16) (cid:17) (cid:21)
where δ R and δ R denote the perturbed Ricci scalar at is given by
1 2
first and second order respectively. Moreover, C is the
2
perturbed constraint equation at first order, which form
1
δgµν∂ R∂ R+∂ R∂µδ R+f δ R=0. (36)
2 µ ν µ 1 2,R 1
7
In principle, one could worry since the above action leads to a rich cosmological behavior. In particular, one
involves two scalar modes and higher derivative terms, canobtainanarbitrarycyclicevolutionfor the scalefac-
which could imply instabilities. However, this is not tor, by reconstructing suitably the scalar potential. As
the case since the higher derivative terms can be fixed expected, an oscillatory scale factor is induced by an os-
by the perturbed constraint equations. In addition, the cillatory potential, and we were able to perform such a
vanishing of the (i,j) component of the perturbed Ein- procedure starting either from the scale factor or from
steinequationallowsto eliminate one degreeoffreedom. thepotential. Additionally,themattersectorcanbealso
Therefore, there is still only one mode of metric pertur- incorporated easily, and this is an advantage of the sce-
bation which is able to propagate freely. nario, since it allows for the successful reproduction of
In the following, we would like to focus on the prop- the thermal history of the universe.
agation of the metric perturbation around the bouncing In the case of f(R)-gravity, we considered a Lagrange
point,whichiscrucialtothe stabilityanalysisofacyclic multiplier field and a second form f (R). This sce-
2
scenario. For calculation convenience we assume that nario leads also to a rich behavior, and one can acquire
the bounce takes place slowly, and then the universe ap- cycliccosmologybysuitablyreconstructingf (R),which
2
proaches a static phase around the bounce asymptoti- proves to be of a very simple form, contrary to the con-
cally. Under this assumption we can obtain the kinetic ventional f(R)-gravity. However, the complexity of the
terms of the perturbation equation, which take the fol- model does not allow for the easy incorporation of the
lowing approximate form: mattersector,sinceonecannotextractanalyticalresults,
and this is a disadvantage of the construction.
2 f
Q¨ ∇ Q+ 1 Q+...=0, (37) In summary, we saw that Lagrange-multiplier modi-
− a2 f
1,R fied gravity may lead to cyclic behavior very easily, and
the scenario is much more realistic in the case of scalar
in which Q Φ+Ψ. Although this equation is far from
≡ cosmology. Since a necessary test of every cosmologi-
a complete form, one can still extract a few important
cal scenario is to examine the evolution of perturbations
issues. Namely, the sound speed of the perturbation is
through the bounce [28], we extended our analysis be-
unity under the assumption of slow bounce, as it can
yond the background level, in both scenarios we consid-
be read from the coefficient before the gradient term.
ered. For the case of scalar cosmology the perturbation
Furthermore, we confirm that there exist only a single
behaves as a frozen mode without oscillations, since its
degree of freedom in the f(R) cosmology with Lagrange
sound speed is vanishing. For the case of f(R)-gravity
multiplier.
and under the assumption of slow bounce, we obtain a
Wementionherethatweinvestigatedthestabilityonly
dynamicaldegreeoffreedomofwhichthe soundspeedis
under the assumption of slow bounce. And the result in
almost unity. Thus, in conclusion, it is possible for the
this case illustrates that it is possible for the cosmolog-
cosmologicalperturbationtoevolvethroughthebounces.
ical perturbation to evolve through the bounces. How-
However, a more generic perturbation analysis, beyond
ever, it is necessary to point out that a more complete
the slow bounce assumption, is needed, but since it lies
andgenericanalysisshouldbeperformedinordertocon-
beyondthe scope of this workit is left for future investi-
straintheparameterspaceofthescenario. Suchageneral
gation.
analysis is left for future investigation.
V. CONCLUSIONS
Inthis workwe investigatedcyclic evolutionsin auni- Acknowledgments
verse governed by Lagrange-multiplier modified gravity.
In order to be more general we considered the scenario
ofmodifiedgravitythroughLagrangemultipliersinboth It is a pleasure to thank the anonymous referees for
scalar-field cosmology,as well as in f(R) one. valuable comments. The work of YFC is supported in
In the case of scalar cosmology, the use of a second part by the Arizona State University Cosmology Initia-
field that is the Lagrange multiplier of the usual scalar, tive.
[1] A. H. Guth, Phys. Rev. D 23, 347 (1981); A. D. Linde, Lett. 68, 1969 (1992).
Phys. Lett. B 108, 389 (1982); A. J. Albrecht and [4] G. Veneziano, Phys. Lett. B 265, 287 (1991);
P.J. Steinhardt,Phys.Rev. Lett.48, 1220 (1982). M. Gasperini andG.Veneziano,Astropart.Phys.1,317
[2] A. Borde and A. Vilenkin, Phys. Rev. Lett. 72, 3305 (1993) [arXiv:hep-th/9211021].
(1994) [arXiv:gr-qc/9312022]. [5] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok,
[3] V. F. Mukhanov and R. H. Brandenberger, Phys. Rev. Phys.Rev.D64,123522 (2001) [arXiv:hep-th/0103239];
8
J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Stein- Phys. Lett. B 666, 212 (2008) [arXiv:0805.0413 [astro-
hardt and N. Turok, Phys. Rev. D 65, 086007 (2002) ph]].
[arXiv:hep-th/0108187]. [17] J.E.Lidsey,D.J.Mulryne,N.J.NunesandR.Tavakol,
[6] R. Brustein and R. Madden, Phys. Rev. D 57, Phys. Rev. D 70, 063521 (2004) [arXiv:gr-qc/0406042];
712 (1998) [arXiv:hep-th/9708046]. T. Biswas, Y. S. Piao and Y. Z. Zhang, Nucl. Phys. B 725, 265
A. Mazumdar and W. Siegel, JCAP 0603, 009 (2006) (2005) [arXiv:gr-qc/0407027]; Y. S. Piao, Phys. Rev. D
[arXiv:hep-th/0508194]; T. Biswas, R. Brandenberger, 70, 101302 (2004) [arXiv:hep-th/0407258]; H. H. Xiong,
A. Mazumdar and W. Siegel, JCAP 0712, 011 (2007) T. Qiu, Y. F. Cai and X. Zhang, Mod. Phys. Lett. A
[arXiv:hep-th/0610274]. 24, 1237 (2009) [arXiv:0711.4469 [hep-th]]; X. Zhang,
[7] Y. Shtanov and V. Sahni, Phys. Lett. B 557, 1 (2003) Eur.Phys.J.C60,661(2009) [arXiv:0708.1408 [gr-qc]];
[arXiv:gr-qc/0208047]; E. N. Saridakis, Nucl. Phys. B T. Biswas and S. Alexander, Phys. Rev. D 80, 043511
808, 224 (2009) [arXiv:0710.5269 [hep-th]]. (2009)[arXiv:0812.3182[hep-th]];Y.S.Piao,Phys.Lett.
[8] Y. F. Cai and E. N. Saridakis, JCAP 0910, 020 (2009) B 677, 1 (2009) [arXiv:0901.2644 [gr-qc]].
[arXiv:0906.1789 [hep-th]];E.N.Saridakis,Eur.Phys.J. [18] S. Mukherji and M. Peloso, Phys. Lett. B 547,
C 67, 229 (2010) [arXiv:0905.3532 [hep-th]]. 297 (2002) [arXiv:hep-th/0205180]; J. Khoury,
[9] M. Bojowald, Phys. Rev. Lett. 86, 5227 (2001) P. J. Steinhardt and N. Turok, Phys. Rev. Lett.
[arXiv:gr-qc/0102069]. 92, 031302 (2004) [arXiv:hep-th/0307132];L.Baumand
[10] J. Martin and P.Peter, Phys.Rev.D 68, 103517 (2003) P. H. Frampton, Phys. Rev. Lett. 98, 071301 (2007)
[arXiv:hep-th/0307077]. [arXiv:hep-th/0610213].
[11] Y.F.Cai,T.Qiu,Y.S.Piao,M.LiandX.Zhang,JHEP [19] M.NovelloandS.E.P.Bergliaffa, Phys.Rept.463,127
0710, 071 (2007) [arXiv:0704.1090 [gr-qc]]; Y. F. Cai, (2008) [arXiv:0802.1634 [astro-ph]].
T. Qiu, R. Brandenberger, Y. S. Piao and X. Zhang, [20] R. H. Brandenberger, V. F. Mukhanov and
JCAP 0803, 013 (2008) [arXiv:0711.2187 [hep-th]]; A. Sornborger, Phys. Rev. D 48, 1629 (1993)
Y. F. Cai, E. N. Saridakis, M. R. Setare and J. Q. Xia, [arXiv:gr-qc/9303001].
arXiv:0909.2776 [hep-th]. [21] E. A. Lim, I. Sawicki and A. Vikman, JCAP 1005, 012
[12] Y. F. Cai and X. Zhang, JCAP 0906, 003 (2009) (2010) [arXiv:1003.5751 [astro-ph.CO]].
[arXiv:0808.2551 [astro-ph]]; Y. F. Cai, T. t. Qiu, [22] S. Capozziello, J. Matsumoto, S. Nojiri and
J. Q. Xia and X. Zhang, Phys. Rev. D 79, 021303 S. D. Odintsov, Phys. Lett. B 693, 198 (2010)
(2009) [arXiv:0808.0819 [astro-ph]]; Y. F. Cai, W. Xue, [arXiv:1004.3691 [hep-th]].
R. Brandenberger and X. Zhang, JCAP 0905, 011 [23] C.Gao,Y.Gong,X.WangandX.Chen,arXiv:1003.6056
(2009) [arXiv:0903.0631 [astro-ph.CO]]; Y. F. Cai, [astro-ph.CO]; S. Nojiri and S. D. Odintsov, Phys. Lett.
W. Xue, R. Brandenberger and X. m. Zhang, JCAP B 691, 60 (2010) [arXiv:1004.3613 [hep-th]].
0906, 037 (2009) [arXiv:0903.4938 [hep-th]]; Y. F. Cai [24] B. Feng, M. Li, Y. S. Piao and X. Zhang, Phys. Lett. B
and X. Zhang, Phys. Rev. D 80, 043520 (2009) 634, 101 (2006) [arXiv:astro-ph/0407432].
[arXiv:0906.3341 [astro-ph.CO]]. [25] Y.F.Cai,J.LiuandH.Li,Phys.Lett.B690,213(2010)
[13] E. N. Saridakis and J. Ward, Phys. Rev. D 80, [arXiv:1003.4526 [astro-ph.CO]].
083003 (2009) [arXiv:0906.5135 [hep-th]]; E. N. Sari- [26] S. Nojiri and S. D. Odintsov, J. Phys. Conf. Ser. 66,
dakis and S. V. Sushkov, Phys. Rev. D 81, 083510 012005 (2007) [arXiv:hep-th/0611071].
(2010) [arXiv:1002.3478 [gr-qc]]; E. N. Saridakis and [27] Y.F.Cai,T.t.Qiu,R.BrandenbergerandX.m.Zhang,
J. M. Weller, Phys. Rev. D 81, 123523 (2010) Phys. Rev. D 80, 023511 (2009) [arXiv:0810.4677 [hep-
[arXiv:0912.5304 [hep-th]]. th]].
[14] R.C.Tolman, Relativity, Thermodynamics and Cosmol- [28] R. H. Brandenberger, Phys. Rev. D 80, 023535 (2009)
ogy, Oxford U.Press (1934). [arXiv:0905.1514 [hep-th]]; J. Zhang, Z. G. Liu and
[15] P.J. SteinhardtandN.Turok,Phys.Rev.D65, 126003 Y. S. Piao, arXiv:1007.2498 [hep-th]; T. Qiu and
(2002) [arXiv:hep-th/0111098]; P. J. Steinhardt and K. C. Yang, arXiv:1007.2571 [astro-ph.CO].
N.Turok, Science296, 1436 (2002).
[16] H.H.Xiong,Y.F.Cai,T.Qiu,Y.S.PiaoandX.Zhang,