yi-fu cai and emmanuel n. saridakis- cyclic cosmology from lagrange-multiplier modified gravity

8/3/2019 Yi-Fu Cai and Emmanuel N. Saridakis- Cyclic cosmology from Lagrange-multiplier modified gravity

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arXiv:1007.3204v1

[astro-ph.CO

]19Jul2010

Cyclic cosmology from Lagrange-multiplier modified gravity

Yi-Fu Cai1,2, and Emmanuel N. Saridakis3,

1 Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918-4, Beijing 100049, P.R. China2 Department of Physics & School of Earth and Space Exploration & Beyond Center,

Arizona State University, Tempe, AZ 85287, USA3College of Mathematics and Physics,

Chongqing University of Posts and Telecommunications Chongqing 400065, P.R. China

We investigate cyclic and singularity-free evolutions in a universe governed by Lagrange-multipliermodified gravity, either in scalar-field cosmology, as well as in f(R) one. In the scalar case, cyclicitycan be induced by a suitably reconstructed simple potential, and the matter content of the universecan be successfully incorporated. In the case off(R)-gravity, cyclicity can be induced by a suitablereconstructed second function f2(R) of a very simple form, however the matter evolution cannot beanalytically handled.

PACS numbers: 98.80.-k, 95.36.+x

I. INTRODUCTION

Inflation is now considered to be a crucial part of thecosmological history of the universe [1], however the so

called standard model of the universe still faces theproblem of the initial singularity. Such a singularityis unavoidable if inflation is realized using a scalar fieldwhile the background spacetime is described by the stan-dard Einstein action [2]. As a consequence, there hasbeen a lot of effort in resolving this problem throughquantum gravity effects or effective field theory tech-niques.

A potential solution to the cosmological singularityproblem may be provided by non-singular bouncing cos-mologies [3]. Such scenarios have been constructedthrough various approaches to modified gravity, such asthe Pre-Big-Bang [4] and the Ekpyrotic [5] models, grav-

ity actions with higher order corrections [6], braneworldscenarios [7], non-relativistic gravity [8], loop quantumcosmology [9] or in the frame of a closed universe [10].Non-singular bounces may be alternatively investigatedusing effective field theory techniques, introducing mat-ter fields violating the null energy condition [11] lead-ing to observable predictions too [12], or introduce non-conventional mixing terms [13]. The extension of all theabove bouncing scenarios is the (old) paradigm of cycliccosmology [14], in which the universe experiences the pe-riodic sequence of contractions and expansions, which hasbeen rewaked the last years [15] since it brings differentinsights for the origin of the observable universe [1618]

(see [19] for a review).On the other hand, the interesting idea of modifyinggravity using Lagrange multipliers, first introduced in[3] with bouncing solution studied in [20], was recentlygained new interest, since its extended version may re-duce irrelevant degrees of freedom in modified gravity

Electronic address: [email protected] address: [email protected]

models [2123]. In particular, using two scalar fields, oneof which is the Lagrange multiplier, leads to a constraintof special form on the second scalar field, and as a resultthe whole system contains a single dynamical degree of

freedom.In the present work we are interested in constructing

a scenario of cyclic cosmology in universe governed bymodified gravity with Lagrange multipliers. We do thatboth in the case of scalar cosmology, as well as in thecase of f(R)-gravity, both in their Lagrange-multipliermodified versions. As we show, the realization of cyclicityand the avoidance of singularities is straightforward.

This paper is organized as follows. In section II wepresent modified gravity with Lagrange multipliers, sep-arately for scalar cosmology (subsection II A) and forf(R)-gravity (subsection II B). In section III we con-struct the scenario of cyclicity realization, reconstructing

the required potential (subsection IIIA) and the form off(R) modification (subsection IIIB). Finally, section IVis devoted to the summary of our results.

II. MODIFIED GRAVITY WITH LAGRANGE

MULTIPLIERS

Let us present the cosmological scenarios with La-grange multipliers. In order to be general and complete,we present both the scalar case, as well as the f(R)-gravity one. Throughout the work we consider a flatFriedmann-Robertson-Walker geometry with metric

ds2 = dt2 a2(t)dx2, (1)

where a is the scale factor, although we could straight-forwardly generalize our results to the non-flat case too.

A. Scalar cosmology with Lagrange multiplier

In this subsection, we consider a scenario with twoscalars, namely and , where the second scalar is a
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Lagrange multiplier which constrains the field equationof the first one. The action reads [21]:

S =

d4x

g

R

22 ()

2

V()

1

2

+ U()

+ Lm

, (2)

where is the Lagrange multiplier field. Furthermore,V() and U() are potentials of , while the function() and especially its sign determines the nature of thescalar field , that is if it is canonical or phantom. Fi-nally, the term Lm accounts for the matter content of theuniverse.

Variation over the -field leads to

0 =2

2 U(). (3)

As we observe, this equation acts as a constraint for theother scalar field , and it is the cause of the significantcosmological implications of such a construction. Now,

the Friedmann equations straightforwardly write

3

2H2 =

() +

2

2 + V() + U()

= [() + 2] U() + V() + m (4)

12

2H+ 3H2

=

() +

2

2 V() U()

= ()U() V() +pm, (5)where we have defined the Hubble parameter as H a

a,

and we have also made use of the constraint (3). Ad-

ditionally, in these expressions m and pm stand respec-tively for the energy density and pressure of matter, withequation-of-state parameter wm pm/m. Observingthe above equations, we can straightforwardly define theenergy density and pressure for the dark-energy sector as

de [() + 2] U() + V() (6)pde ()U() V(), (7)

and thus the dark energy equation-of-state parameter willbe

wde =()U() V()

[() + 2] U() + V(). (8)

Let us now explore some general features of the sce-nario at hand following [21]. First of all the constraintequation (3) can be integrated for positive U(), leadingto

t = d

2U(). (9)

Inverting this relation with respect to , one can findthe explicit t-dependence of , that is the corresponding

(t). Thus, substituting the expression of (t) into (5),we obtain a differential equation for H(t):

2H(t) + 3[H(t)]2 = 2 {((t))U((t))V((t)) + wmm(t)} , (10)

the solution of which determines completely the cosmo-logical evolution. Finally, substituting (t) and H(t) into

(4), we can find the t-dependence of the Lagrange multi-plier field :

(t) =1

2U( (t))

3

2H(t)2 V ( (t)) m(t)

( (t))2

. (11)

However, one could alternatively follow the inverse pro-cedure, that is to first determine the specific behavior ofH(t) and try to reconstruct the corresponding potentialV(), which is responsible for the -evolution that leadsto such a H(t). In particular, he first chooses a suit-able U() which will lead to an easy integration of (9),allowing for an interchangeable use of and t through(t) t(). Therefore, the second Friedmann equation(5) gives straightforwardly

V() =1

2

2H(t ()) + 3H(t ())

2

+()U() + wmm(t()), (12)

where () can still be arbitrary. In summary, such apotential V() produces a cosmological evolution withthe chosen H(t).

Finally, let us make a comment here concerning the na-ture of models with a second, Lagrange-multiplier field,

since it is obvious that the constraint (3) changes com-pletely the dynamics, comparing to the conventionalmodels. In principle, one expects to have a propagationmode for each new field. However, due to the constraint(3) and the form in which the -field appears in the ac-tion, the propagating modes of and do not appear,and the system is driven by a system of two first-orderordinary differential equations, one for each field. As aconsequence, there are no propagating wave-like degreesof freedom and the sound speed for perturbations is ex-actly zero irrespective of the background solution [21, 23].

B. f(R)-gravity with Lagrange multiplier

In this subsection we present f(R)-gravity with La-grange multipliers, following [21]. In this case, we startby a conventional f(R)-gravity, and we add a scalar field which is a Lagrange multiplier leading to a constraint.In particular, the action reads:

S =

d4x

g

f1(R)

1

2R

R + f2(R)

,

(13)


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where f1(R) and f2(R) are two independent functions ofthe Ricci scalar R. Note that in the above action we havenot included the matter content of the universe, since thiswould significantly modify the multiplying terms of ,making the subsequent reconstruction procedures tech-nically very complicated. These subtleties are caused bythe fact that the Lagrange multiplier field propagates inthis case, which is a disadvantage of the present scenario,

contrary to the scalar cosmology of the previous subsec-

tion.

Variation over leads to the constraint

0 = 12

R2 + f2(R). (14)

Additionally, varying over the metric g and keeping the

(0, 0)-component we obtain

0 = 12

f1(R) +

3

H+ H2 3H d

dt

f1(R) f2(R) +

d

dt+ 3H

dR

dt

. (15)

For f2(R) > 0, the constraint (14) can be solved as

t = R dR

2f2(R)

, (16)

and inverting this relation with respect to R one canobtain the explicit t-dependence of R, that is the corre-sponding R(t). On the other hand, the Ricci scalar is

given from R = 6H+ 12H2. Thus, inserting R(t) in thisrelation one obtains a differential equation in terms ofH(t), namely

6H(t) + 12[H(t)]2 = R(t), (17)

the solution of which determines completely the cosmo-

logical behavior. Finally, using the obtained H(t) andR(t), equation (15) becomes a differential equation forthe Lagrange multiplier field (t) which can be solved.

However, one could alternatively follow the inverse pro-cedure, that is to first determine the specific behaviorof H(t) and try to reconstruct the corresponding f2(R),which is responsible for the R-evolution that leads to suchan H(t). In particular, with a known H(t) (17) providesimmediately R(t), which can be inverted, giving t = t(R).Thus, using the constraint (14), the explicit form off2(R)is found to be

f2(R) =

1

2dR

dt2

t=t(R). (18)

It is interesting to mention that in the above discussionf1(R) is arbitrary. That is, while in conventional f(R)-gravity the cosmological behavior is determined com-pletely by f1(R), in Lagrange-multiplier modified f(R)-gravity the dynamics is determined completely by f2(R).In this case f1(R) would become relevant only in thepresence of matter, and its effect on Newtons law [21].

III. COSMOLOGICAL BOUNCE AND CYCLIC

COSMOLOGY

Having presented the cosmological models with La-grange multipliers, both in the scalar, as well as in thef(R)-gravity case, in this section we are interested in in-vestigating the bounce and cyclic solutions.

In principle, in order to acquire a cosmological bounce,one has to have a contracting phase (H < 0), followedby an expanding phase (H > 0), while at the bouncepoint we have H = 0. In this whole procedure the time-derivative of the Hubble parameter must be positive, thatis H > 0. On the other hand, in order for a cosmologicalturnaround to be realized, one has to have an expandingphase (H > 0) followed by a contracting phase (H < 0),while at the turnaround point we have H = 0, and in

this whole procedure the time-derivative of the Hubbleparameter must be negative, that is H < 0. Observingthe form of Friedmann equations (4), (5), as well as of(15), we deduce that such a behavior can be easily ob-tained in principle. In the following two subsections weproceed to the detailed investigation in the case of scalarand f(R)-modified cosmology.

A. Scalar cosmology with Lagrange multiplier

In order to provide a simple realization of cyclicity inthis scenario, we start by imposing a desirable form ofH(t) that corresponds to a cyclic behavior. We considera specific, simple, but quite general example, namely weassume a cyclic universe with an oscillatory scale factorof the form

a(t) = A sin(t) + ac, (19)


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where we have shifted t in order to eliminate a possibleadditional parameter standing for the phase1. Further-more, the non-zero constant ac is inserted in order toeliminate any possible singularities from the model. Insuch a scenario t varies between and +, and t = 0is just a specific moment without any particular phys-ical meaning. Finally, note that the bounce occurs ataB(t) = ac A. Straightforwardly we find:

H(t) =A cos(t)

A sin(t) + ac(20)

H(t) = A2 [A + ac sin(t)]

[A sin(t) + ac]2 . (21)

Concerning the matter content of the universe we assumeit to be dust, namely wm 0, which inserted in theevolution-equation m + 3H(1 + wm)m = 0 gives theusual dust-evolution m = m0/a

3, with m0 its value atpresent.

Now, we first consider to be a canonical field, thatis we choose () = 1. Concerning the potential U()that will give as the solution for (t) according to (9), wechoose a simple and easily-handled form, namely

U() =m4

2, (22)

where m is a constant with mass-dimension. In this case,(9) leads to

= m2t, (23)

having chosen the + sign in (9). As we have said insubsection IIA, such a simple relation for (t) allows toreplace t by /m2 in all the aforementioned relations.Therefore, substitution of (20),(21) and (22) into (12),and using instead of t, provides the corresponding ex-pression for V() that generates such an H(t)-solution:

V() =m4

2+

1

2

2

A2

A + ac sin

m2

A sin

m2

+ ac

2

+3

A cos

m2

A sin

m2

+ ac

2

. (24)

Note that in the case of dust matter, the reconstructedpotential does not depend on the matter energy densityand its evolution. Finally, for completeness, we present

1 Note that if the average of an oscillatory scale factor keeps in-creasing throughout the evolution, it could yield a recurrent uni-verse which unifies the early time inflation and late time accel-eration [24, 25].

also the (t) evolution, which according to (11) reads

(t) = 1 22m4

A

2 [A + ac sin(t)]

[A sin(t) + ac]2

1m4

m0 {A sin(t) + ac}3 . (25)

In order to provide a more transparent picture of theobtained cosmological behavior, in Fig. 1 we present theevolution of the oscillatory scale factor (19) and of theHubble parameter (20), with A = 1, = 1 and ac = 1.3,where all quantities are measured in units with 8G = 1.

- 2 0 - 1 0 0 1 0 2 0

H

(

t

)

- 2 0 - 1 0 0 1 0 2 0

a

(

t

)

Figure 1: The evolution of the scale factor a(t) and of theHubble parameterH(t) of the ansatz (19), withA = 1, = 1and ac = 1.3. All quantities are measured in units where8G = 1.

- 2 0 - 1 0 0 1 0 2 0

V

(

)

Figure 2: V() from (24) that reproduces the cosmologicalevolution of Fig. 1, with() = 1, U() = m4/2 andm = 1.2.All quantities are measured in units where 8G = 1.


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Note that H(t) by construction satisfies the requirementsfor cyclicity, described in the beginning of this section.Furthermore, in Fig. 2 we depict the corresponding po-tential V() given by (24). From these figures we observethat an oscillating and singularity-free scale factor can begenerated by an oscillatory form of the scalar potentialV() (although of not a simple function as that of a(t),as can be seen by the slightly different form of V() in its

minima and its maxima). This V()-form was more orless theoretically expected, since a non-oscillatory V()would be physically impossible to generate an infinitelyoscillating scale factor and a universe with a form of time-symmetry. Note also that, having presented the basicmechanism, we can suitably choose the oscillation fre-quency and amplitude in order to get a realistic oscilla-tion period and scale factor for the universe. Finally, westress that although we have presented the above spe-cific simple example, we can straightforwardly performthe described procedure imposing an arbitrary oscillat-ing ansatz for the scale factor.

The aforementioned bottom to top approach was en-lightening about the form of the scalar potential thatleads to a cyclic cosmological behavior. Therefore, onecan perform the above procedure the other way around,starting from a specific oscillatory V() and resulting toan oscillatory a(t), following the steps described in thefirst part of subsection IIA. As a specific example weconsider the simple case

V() = V0 sin(V ) + Vc, (26)

with U() chosen as in (22) and thus (23) holds too. De-spite the simplicity ofV(), the differential equation (10)cannot be solved analytically, but it can be easily handlednumerically. In Fig. 3 we depict the corresponding solu-tion for H(t) (and thus for a(t)) under the ansatz (26),

with V0 = 3, V = 1 and Vc = 3, with (0) = 3.2 andH(0) = 0.7 (in units where 8G = 1). As expected, wedo obtain an oscillatory universe.

B. f(R)-gravity with Lagrange multiplier

In order to provide a simple realization of cyclicity inthis scenario, we start by imposing a desirable form ofH(t) that corresponds to a cyclic behavior. Due to thecomplicated structure of the reconstruction procedure,we will choose an easier ansatz comparing to the previ-ous subsection, which allows for the extraction of analyt-

ical results. In particular, having described the generalrequirements for an H(t) that gives rise to cyclic cosmol-ogy in the beginning of the present section, we chooseH(t) (and not a(t)) to be straightaway a sinusoidal func-tion, that is

H(t) = AH sin(Ht), (27)

which gives rise to a scale factor of the form

a(t) = AH0 exp

AHcos(Ht)

H

, (28)

- 2 0 - 1 0 0 1 0 2 0

H

(

t

)

- 2 0 - 1 0 0 1 0 2 0

a

(

t

)

Figure 3: The evolution of the scale factor a(t) and of theHubble parameter H(t), for a scalar potential of the ansatz(26), with V0 = 3, V = 1 and Vc = 3, (0) = 3.2 andH(0) = 0.7. All quantities are measured in units where8G = 1.

which is oscillatory and always non-zero. Inserting (27)and its derivative into (17) we obtain R(t) as

R(t) = 6AHHcos(Ht) + 12A2Hsin

2(Ht), (29)

a relation that can be easily inverted giving

t(R) =1

Harccos

3H +

3(48A2H 4R + 32H)

12AH

,

where we have kept the plus sign in the square root. Fi-

nally, f2(R) can be reconstructed using (18), leading to

f2(R) = 14

2H

48A2H 4R + 32H

2R + 32H + H

3(48A2H 4R + 32H)

. (30)

In summary, such an ansatz for f2(R) produces the cyclicuniverse with scale factor (28). Note that f2(R) has a re-markably simple form, and this is an advantage of thescenario at hand, since in conventional f(R)-gravity oneneeds very refined and complicated forms off(R) in orderto reconstruct a given cosmological evolution [26]. How-ever, as we stated in the beginning of subsection II B, inthe f(R)-version of Lagrange-modified gravity one can-not incorporate the presence of matter in a convenientway that will allow for an analytical treatment. The ab-sence of matter evolution in a cyclic scenario is a disad-vantage, since we cannot reproduce the evolution epochsof the universe. Therefore it would be necessary to con-struct a formalism that would allow for such a matter in-clusion, similarly to the case of Lagrange-multiplier mod-ified scalar-field cosmology. Such a project is in prepara-tion.


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IV. CONCLUSIONS

In this work we investigated cyclic evolutions in a uni-verse governed by Lagrange-multiplier modified gravity.In order to be more general we considered the scenarioof modified gravity through Lagrange multipliers in bothscalar-field cosmology, as well as in f(R) one.

In the case of scalar cosmology, the use of a second

field that is the Lagrange multiplier of the usual scalar,leads to a rich cosmological behavior. In particular, onecan obtain an arbitrary cyclic evolution for the scale fac-tor, by reconstructing suitably the scalar potential. Asexpected, an oscillatory scale factor is induced by an os-cillatory potential, and we were able to perform such aprocedure starting either from the scale factor or fromthe potential. Additionally, the matter sector can be alsoincorporated easily, and this is an advantage of the sce-nario, since it allows for the successful reproduction ofthe thermal history of the universe.

In the case of f(R)-gravity, we considered a Lagrangemultiplier field and a second form f2(R). This sce-nario leads also to a rich behavior, and one can acquirecyclic cosmology by suitably reconstructing f2(R), whichproves to be of a very simple form, contrary to the con-ventional f(R)-gravity. However, the complexity of themodel does not allow for the easy incorporation of thematter sector, since one cannot extract analytical results,

and this is a disadvantage of the construction.

In conclusion, we saw that Lagrange-multiplier modi-fied gravity may lead to cyclic behavior very easily, andthe scenario is much more realistic in the case of scalarcosmology. A good test of the scenario would be to ex-amine the evolution of perturbations through the bounce[27] and the possibility to obtain the formation of a scale-invariant primordial spectrum. Such an analysis, neces-sary for every cyclic cosmology, lies beyond the scope ofthis work and is left for future investigation.

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yi-fu cai and emmanuel n. saridakis- cyclic cosmology from lagrange-multiplier modified gravity

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