Classical and quantum big brake cosmology for scalar field and tachyonic models

Alexander Y. Kamenshchik

Dipartimento di Fisica and INFN, Via Irnerio 46, 40126 Bologna, Italyand L.D. Landau Institute for Theoretical Physics of the Russian Academy of Sciences, Kosygin str. 2, 119334 Moscow, Russia

Serena Manti

Dipartimento di Fisica,Via Irnerio 46, 40126 Bologna, Italyand Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

(Received 1 February 2012; revised manuscript received 19 April 2012; published 11 June 2012)

We study a relation between the cosmological singularities in classical and quantum theory, comparing

the classical and quantum dynamics in some models possessing the big brake singularity—the model

based on a scalar field and two models based on a tachyon-pseudo-tachyon field. It is shown that the effect

of quantum avoidance is absent for the soft singularities of the big brake type while it is present for the big

bang and big crunch singularities. Thus, there is some kind of a classical-quantum correspondence,

because soft singularities are traversable in classical cosmology, while the strong big bang and big crunch

singularities are not traversable.

DOI: 10.1103/PhysRevD.85.123518 PACS numbers: 98.80.Qc, 04.60.Ds, 98.80.Jk

I. INTRODUCTION

The problem of cosmological singularities has beenattracting the attention of theoreticians since the early1950s [1–3]. In the 1960s the general theorems about theconditions for the appearance of singularities wereproven [4,5] and the oscillatory regime of the approachingto the singularity [6] also called the mixmaster universe [7]was discovered. The introduction of the notion of thequantum state of the Universe, satisfying the Wheeler-DeWitt equation (see [8]) has stimulated the diffusionof the hypothesis that in the framework of quantum cos-mology the singularities can disappear in some sense.Namely, the probability that the Universe acquires suchvalues of the parameters, describing its evolution, whichcorrespond to a classical cosmological singularity, can beequal to zero (for a recent review see [9]).

Basically, until the end of 1990s almost all the discus-sions about classical and quantum cosmology of singu-larities were devoted to the big bang and big crunchsingularities, which are characterized by the vanishingvalue of the cosmological radius. The situation waschanged after the discovery of the phenomenon of thecosmic acceleration [10,11]. Such a discovery was thestarting point for the formulation of cosmological modelscontaining a special type of substance, the so-called darkenergy, which, for its specific properties, was consideredresponsible for the accelerated expansion of the Universe[12–15], and consequently stimulated the study of thevarious possible candidates for the role of this substance.The fundamental feature of the dark energy, whichproduces an accelerated expansion, is that it possessesa pressure p such that the strong energy condition�þ 3p > 0 is violated (here � is the energy density).The construction of different cosmological models,describing dark energy, has attracted the attention of

researchers to the fact that other types of cosmologicalsingularities do exist. First of all, one should mentionthe big rip singularity [16,17] arising in the models wherethe phantom dark energy [18] is present. Under phantomdark energy one understands the substance whose pres-sure is negative and has an absolute value bigger than itsenergy density. Such a singularity is characterized byinfinite values of the cosmological radius (scale factor),of its time derivative, of the Hubble parameter and itstime derivative and, hence, of its energy density andpressure.Another class of singularities includes the so-called soft

or sudden singularities [19–22]. They occur at a finite valueof the scale factor and of its time derivative, and, hence, ofthe Hubble parameter and of the energy density, while thesecond derivative of the scale factor, the first derivative ofthe Hubble parameter, and the pressure are divergent. Theparticularity of the big brake cosmological singularity,belonging to this class, consists in the fact that the timederivative of the scale factor is equal exactly to zero. Thatmakes this singularity especially convenient for study. Thebig brake singularity was first described in paper [22],where it has arisen in the context of a particular cosmo-logical model with a tachyon field, whose potential de-pended on the trigonometrical functions. In the same paperit was noticed that a very simple cosmological model,based on the anti-Chaplygin gas, leads unavoidably to thebig brake singularity. The anti-Chaplygin gas, with theequation of state p ¼ A

� , where A is a positive constant,

arises in the theory of wiggly strings [23], which obtainedthis name [22] in analogy with the Chaplygin gas, whichsatisfies an equation of state of the type p ¼ � A

� and has

acquired a certain importance in cosmology as a candidatefor the unification between dark energy and dark matter[24,25].

PHYSICAL REVIEW D 85, 123518 (2012)

1550-7998=2012=85(12)=123518(11) 123518-1 � 2012 American Physical Society

Starting from the anti-Chaplygin gas cosmologicalmodel and using the technique of reconstruction of thepotentials for the scalar field models, one can constructthe scalar field model reproducing the cosmological evo-lution occurring in the anti-Chaplygin gas cosmologicalmodel. Such a potential was constructed and studied in[26]. In the same paper the quantum cosmology of thecorresponding model was studied, and it was shown thatthe requirement of the normalizability of the quantum stateof the Universe, satisfying the Wheeler-DeWitt equation,implies the disappearance of this quantum state at the bigbrake singularity. Thus, this result appears as confirmingthe hypothesis that in the framework of quantum cosmol-ogy the singularities can disappear. Similar researches,devoted to the properties of the solutions of the Wheeler-DeWitt equations for different cosmological models, con-nected in some way with dark energy hypothesis, havegiven analogous results [27]. However, two question arise:First, how general is this phenomenon? In other words,should the wave function of the Universe satisfying theWheeler-DeWitt equation disappear at the values of itsarguments, which classically correspond to the soft singu-larities? The second question is more subtle: Does thedisappearance of the wave function of the Universe atsome values of its arguments mean that the relevant proba-bility distribution disappears too? The point is that thewave function of the Universe satisfying the Wheeler-DeWitt equation does not have a direct probabilistic inter-pretation [28]. To provide such an interpretation one has tochoose a time-dependent gauge-fixing condition, and afterthat one should undertake the reduction of the set ofvariables to the smaller set of the physical degrees offreedom [28]. The explicit realization of this procedure israther complicated, but study of its general features cangive some interesting results.

To try to answer the questions formulated above we shallgive a comparative analysis of three cosmological models,encountering the big brake singularity: the scalar fieldmodel [26] and two tachyon models [22]. The classicaldynamics of the approach to the big brake singularity in thetachyon model with trigonometric potential was consid-ered in detail in Ref. [29], while the mechanism of thecrossing of this singularity was suggested in [30], where itwas shown that there was a large class of cosmologicalevolutions crossing the big brake singularity. Thus, wehave some kind of complementarity—the classical dynam-ics of the tachyon model with trigonometric potential [22]encountering the big brake singularity was studied in detailin [22,29,30], and it was shown that this singularity istraversable. On the other hand, the quantum dynamics ofthe model with the scalar field was studied in [26], andthere it was shown that the wave function of the Universedisappears encountering such a singularity. In the presentpaper we shall try to fill the ‘‘holes’’ of the precedingconsiderations. Namely, we shall study in detail the

classical dynamics of the model with the scalar field, andwe shall show that the cosmological trajectory arriving tothe big brake is unique. However, all other trajectoriescross the soft singularities of a more general kind, namely,the singularities where the deceleration is infinite, but theHubble parameter is finite and differs from zero. The wavefunction of the Universe disappears at such a singularity,but we shall argue that the corresponding probability isdifferent from zero, due to the Faddeev-Popov factor aris-ing in the process of reducing to the physical degrees offreedom. The situation with a quantum dynamics of thesimple pseudotachyon model with a constant potential isquite similar. The analysis of the rather complicatedtachyon model with trigonometric potential shows that inthis case the wave function of the Universe is not obliged todisappear at the values of its arguments, corresponding tothe classical big brake singularity.On the other hand, in the analysis of the big bang and the

big crunch singularities, which in these models are nottraversable classically, shows that the effect of quantumavoidance is present. Thus, we have some kind of thequantum-classical correspondence here. For the singular-ities which are not traversable in classical cosmology theeffect of quantumavoidance of singularities is present,whilefor the soft traversable singularities such an effect is absent.In concluding the Introduction, we would like to reiter-

ate that while the soft (sudden) singularities were knownlong before the discovery of the cosmic acceleration [19],the development of the dark energy models has stimulatedthe study of this kind of singularities. It was shown that thepresence of such singularities in some models of darkenergy does not contradict observational data onSupernovae of type Ia [29,31]. However, in this paper weare not concerned with the dark energy problem; rather, westudy some general questions arising in the classical andquantum cosmology in the presence of singularities.The structure of the paper is as follows. In Sec. II we

briefly illustrate the big brake cosmology, using the modelwith an anti-Chaplygin gas, in addition, we introduce therelated scalar field cosmological model. In Sec. III, westudy in detail the classical dynamics of this model. InSection IV, we review the results of [26] concerning thequantum cosmology of the model with the scalar field andcompare the classical and quantum cosmology of thismodel. In Sec. V we briefly recapitulate the basic infor-mation about the tachyon cosmological model [22] andabout its classical dynamics [29,30], and in Sec. VI wediscuss its quantum dynamics. Section VII is devoted toconcluding remarks.

II. THE COSMOLOGICAL MODELWITH THEANTI-CHAPLYGIN GAS AND THE RELATED

SCALAR FIELD POTENTIAL

Let us consider a flat Friedmann-Lemaıtre-Robertson-Walker universe with the metric

ALEXANDER Y. KAMENSHCHIK AND SERENA MANTI PHYSICAL REVIEW D 85, 123518 (2012)

123518-2

ds2 ¼ dt2 � a2ðtÞdl2; (1)

filled with an anti-Chaplygin gas [22,23] with the equationof state

p ¼ A�; (2)

where A is a positive constant. The Friedmann equation is

H2 ¼ �; (3)

where the Hubble parameter H is as usual

H � _aa; (4)

where the ‘‘dot’’ means the derivative with respect to thecosmic time t. The energy conservation condition is

_� ¼ �3Hð�þ pÞ: (5)

Equations (2) and (5) give immediately the dependenceof the energy density � on the cosmological radius a,

�ðaÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB

a6� A

s; (6)

where B> 0 is an integration constant.Substituting the expression (6) into Friedmann Eq. (3)

one can explicitly find the dependence of the cosmictime t on a (see [26]) but we shall not need it. Let us noticeinstead that at the beginning of the cosmological evolution,when the cosmological radius is very small, we have�� 1=a3 and the fluid behaves like dust. Then, when the

cosmological radius tends to the critical value a� ¼ðB=AÞð1=6Þ the energy density disappears and the pressure,according to Eq. (2), grows indefinitely. Thus, we encoun-ter some kind of cosmological singularity. Let us study it insome detail.

In the vicinity of the singularity the cosmological radiuscan be represented as

aðtÞ ¼ a� � �aðtÞ; (7)

where �aðtÞ is a small magnitude. Substituting (7) intoFriedmann Eq. (3) with the expression (6) in the right-handside and integrating it in the vicinity of the time moment tBsuch that aðtBÞ ¼ a�, we obtain

�aðtÞ ¼ CðtB � tÞ4=3; (8)

where C ¼ 3ð5=3Þ2�ð7=3ÞðABÞð1=6Þ. Then,aðtÞ ¼ a� � CðtB � tÞ4=3; _aðtÞ ¼ 4C

3 ðtB � tÞ1=3;€aðtÞ ¼ �4C

9 ðtB � tÞ�2=3:(9)

From these expressions we see that when the time tends totB, the cosmological radius tends to the finite value a�, itstime derivative disappears, i.e. the expansion of theUniverse stops, but the second time derivative instead tendsto �1, i.e. the deceleration is infinite. Thus, the stop of

expansion occurs in a singular way that justifies the nameof this singularity—big brake [22].Let us notice that the big brake singularity just like other

soft singularities possesses the important property that theChristoffel symbols at the singularity are finite (or evenzero) [32]. Thus, thematter can pass through this singularityand then the geometry of the spacetime can reappear [30].Different aspects of the soft singularities crossing were alsoconsidered in [33]. In the case of theUniverse filledwith theanti-Chaplygin gas the things look particularly simple. Onecan easily see that the expression (9) is well-defined att > tB. Namely, after arriving at the point of the big brakeaðtBÞ ¼ a�, which is at the same time the point of themaximal expansion of the Universe, the Universe beginscontracting and this contraction culminates in the encounterwith a big crunch singularity at a ¼ 0. Thus, themodel withthe anti-Chaplygin gas describes the evolution of theUniverse from the big bang to the big crunch passingthrough the soft big brake singularity at the moment ofthe maximal expansion of the Universe.However, the described evolution is the only evolution

present in thismodel. If wewant to have a little bit more richmodel for the analysis of its classical and quantum dynam-ics, we can use a standard procedure of the reconstruction ofpotentials of minimally coupled scalar fields [22,34–38],reproducing a given cosmological evolution. This proce-dure is based on the use of the following two equations:

_’ 2 ¼ �þ p; (10)

V ¼ 12ð�� pÞ: (11)

Herewe shall give the result of the reconstruction procedure(for details see [26]),

Vð’Þ ¼ �ffiffiffiffiA

p2

�sinh3’� 1

sinh3’

�: (12)

As a matter of fact we have two possible potentials, whichdiffers by the general sign. We choose the ‘‘plus’’ sign.Then, let us remember that the big brake occurs when theenergy density is equal to zero (the disappearance of theHubble parameter) and the pressure is positive and infinite(an infinite deceleration). To achieve this condition in thescalar fieldmodel it is necessary to require that the potentialis negative and infinite. It is easy to see from Eq. (12) thatthis occurs when ’ ! 0 being positive. Thus, to have themodel with the big brake singularity we can consider thescalar field with a potential which is a little bit simpler thanthat from Eq. (12) but still possesses rather rich dynamics.Namely, we shall study the scalar field with the potential

V ¼ �V0

’; (13)

where V0 is a positive constant. Section III will be devotedto the analysis of the classical cosmology of the model withthis potential.

CLASSICAL AND QUANTUM BIG BRAKE COSMOLOGY FOR . . . PHYSICAL REVIEW D 85, 123518 (2012)

123518-3

III. CLASSICAL DYNAMICS OF THECOSMOLOGICAL MODELWITH A SCALARFIELD WHOSE POTENTIAL IS INVERSELY

PROPORTIONAL TO THE FIELD

The Klein-Gordon equation for the scalar field with thepotential (13) is

€’þ 3H _’þ V0

’2¼ 0; (14)

while the first Friedmann equation is

H2 ¼ _’2

2� V0

’: (15)

We shall also need the expression for the time derivative ofthe Hubble parameter, which can be easily obtained fromEqs. (14) and (15),

_H ¼ �32 _’

2: (16)

Now we shall construct the complete classification of thecosmological evolutions (trajectories) of our model, usingEqs. (14)–(16).

First of all, let us announce briefly the main results of ouranalysis.

(1) The transitions between the positive and negativevalues of the scalar field are impossible.

(2) All the trajectories (cosmological evolutions) withpositive values of the scalar field begin in the bigbang singularity, then achieve a point of maximalexpansion, then contract and end their evolution inthe big crunch singularity.

(3) All the trajectories with positive values of the scalarfield pass through the point where the value of thescalar field is equal to zero. After that the value ofthe scalar field begins growing. The point ’ ¼ 0corresponds to a crossing of the soft singularity.

(4) If the moment when the Universe achieves the pointof the maximal expansion coincides with the mo-ment of the crossing of the soft singularity then thesingularity is the big brake.

(5) The evolutions with the negative values of the scalarfield belong to two classes—the first being an infi-nite expansion beginning from the big bang, and thesecond being the evolutions obtained by the timereversion of those of the first class, which are con-tracting and end in the big crunch singularity.

To prove these results, we begin with the considerationof the Universe in the vicinity of the point ’ ¼ 0. We shalllook for the leading term of the field ’ approaching thispoint in the form

’ðtÞ ¼ ’1ðtS � tÞ�; (17)

where’1 and� are positive constants and tS is the momentof the soft singularity crossing. The time derivative of thescalar field is now

_’ðtÞ ¼ �’1ðtS � tÞ��1: (18)

Because of the negativity of the potential (13) at positivevalues of ’, the kinetic term should be stronger than thepotential one to satisfy Friedmann Eq. (15). That impliesthat � � 2=3. However, if �< 2=3 we can neglect thepotential term and remain with the massless scalar field. Itis easy to show considering in Friedmann (15) and Klein-Gordon (14) equations the scalar field behaves like ’�lnðtS � tÞ, which is incompatible with the hypothesis of itssmallness (17). Thus, one remains with the only choice

� ¼ 23: (19)

Then, if the coefficient at the leading term in the kineticenergy is greater than that in the potential, it follows fromFriedmann Eq. (15) that the Hubble parameter behaves as

ðtS � tÞ�ð1=3Þ which is incompatible with Eq. (16). Thus,the leading terms of the potential and kinetic energy shouldcancel each other,

1

2�2’2

1ðtS � tÞ2��2 ¼ V0

’1

ðtS � tÞ�� (20)

that for � ¼ 2=3 gives

’1 ¼�9V0

2

�ð1=3Þ: (21)

Hence, the leading term for the scalar field in the presenceof the soft singularity is

’ðtÞ ¼�9V0

2

�ð1=3ÞðtS � tÞð2=3Þ: (22)

Now, integrating Eq. (16) we obtain

HðtÞ ¼ 2

�9V0

2

�ð2=3ÞðtS � tÞð1=3Þ þHS; (23)

where HS is an integration constant giving the value of theHubble parameter at the moment of the soft singularitycrossing. If this constant is equal to zero, HS ¼ 0, themoment of the maximal expansion of the Universe coin-cides with that of the soft singularity crossing and theUniverse encounters the big brake singularity. If HS � 0,we have a more general type of the soft cosmologicalsingularity where the energy density of the matter in theUniverse is different from zero. The sign ofHS can be bothpositive or negative, hence, the Universe can pass throughthis singularity in the phase of its expansion or of itscontraction.The form of the leading term for the scalar field in the

vicinity of the moment when ’ ¼ 0 (22) shows that, afterpassing the zero value, the scalar field begin growing beingpositive. Thus, it proves the first result from the list pre-sented above about impossibility of the change of the signof the scalar field in our model.We have already noted that the time derivative of the

scalar field had changed the sign crossing the soft singu-larity. It cannot change the sign in a nonsingular waybecause the conditions _’ðt0Þ ¼ 0, ’ðt0Þ � 0 are incompat-ible with Friedmann Eq. (15). It is seen from Eq. (22) that

ALEXANDER Y. KAMENSHCHIK AND SERENA MANTI PHYSICAL REVIEW D 85, 123518 (2012)

123518-4

before the crossing of the soft singularity the time deriva-tive of the scalar field is negative and after its crossing it ispositive. The impossibility of the changing the sign of thetime derivative of the scalar field without the soft singu-larity crossing implies the inevitability of the approachingof the Universe to this soft singularity. Thus, the third resultfrom the list above is proven.

It is easy to see from Eq. (16) that the value of theHubble parameter is decreasing during all the evolution.At the same time, the absolute value of its time derivative(proportional to the time derivative squared of the scalarfield) is growing after the soft singularity crossing. Thatmeans that at some moment the Hubble parameter shouldchange its sign, becoming negative. The change of the signof the Hubble parameter is nothing but the passing throughthe point of the maximal expansion of the Universe, afterwhich it begins contraction culminating in the encounterwith the big crunch singularity. Thus, the second resultfrom the list presented above is proven.

In summation, we can say that all the cosmologicalevolutions where the scalar field has positive values havethe following structure: They begin in the big bang singu-larity with an infinite positive value of the scalar field andan infinite negative value of its time derivative, then theypass through the soft singularity where the value of thescalar field is equal to zero and where the derivative of thescalar field changes its sign. All the trajectories also passthrough the point of the maximal expansion, and thispassage through the point of the maximal expansion canprecede or follow the passage through the soft singularity.In the case when these two moments coincide (HS ¼ 0),we have the big brake singularity (see fourth result fromthe list above). Thus, all the evolutions pass through thesoft singularity, but for only one of them this singularityhas a character of the big brake singularity. The family ofthe trajectories can be parametrized by the value of theHubble parameter HS at the moment of the crossing of thesoft singularity. There is also another natural parametriza-tion of this family—we can characterize a trajectory by thevalue of the scalar field ’ at the moment of the maximalexpansion of the Universe and by the sign of its timederivative at this moment (if the time derivative of thescalar field is negative that means that the passing throughthe point of maximal expansion precedes the passingthrough the soft singularity, and if the sign of this timederivative is positive, then passage trough the point ofmaximal expansion follows the passage through the softsingularity). If at the moment when the Universe achievesthe point of maximal expansion the value of the scalar fieldis equal to zero, then it is the exceptional trajectory cross-ing the big brake singularity.

For completeness, we shall say some words about thefifth result, concerning the trajectories with the negativevalues of the scalar field. Now, both the terms in the right-hand side of Friedmann Eq. (15), potential and kinetic, are

positive and, hence, the Hubble parameter cannot disap-pear or change its sign. It can only tend to zero asymptoti-cally while both these terms tend asymptotically to zero.Thus, in this case there are two possible regimes: Aninfinite expansion which begins with the big bang singu-larity and an infinite contraction which culminates in theencounter with the big crunch singularity. The secondregime can be obtained by the time reversal of the firstone and vice versa. Let us consider the expansion regime. Itis easy to check that the scalar field being negative cannotachieve the zero value because the suggestion ’ðtÞ ¼�’1ðt0 � tÞ�, where ’1 < 0, �> 0 is incompatible withEqs. (15) and (16). Hence, the potential term is alwaysnonsingular, and at the birth of the Universe from the bigbang singularity the kinetic term dominates and the dy-namics is that of the theory with the massless scalar field.Namely,

’ðtÞ ¼ ’0 þffiffi29

qlnt; HðtÞ ¼ 1

3t; (24)

where ’0 is a constant. At the end of the evolution theHubble parameter tends to zero, while the time growsindefinitely. That means that both the kinetic and potentialterms in the right-hand side of Eq. (15) should tend to zero.It is possible if the scalar field tends to infinity while itstime derivative tends to zero. The joint analysis of Eqs.(15) and (16) gives the following results for the asymptoticbehavior of the scalar field and the Hubble parameter:

’ðtÞ ¼ ~’0 � ð56Þð2=5ÞVð1=5Þ0 tð2=5Þ;

HðtÞ ¼ ð65Þð1=5ÞVð2=5Þ0 t�ð1=5Þ;

(25)

where ~’0 is a constant.

IV. THE QUANTUM DYNAMICS OF THECOSMOLOGICAL MODELWITH A SCALARFIELD WHOSE POTENTIAL IS INVERSELY

PROPORTIONAL TO THE FIELD

In this section we recapitulate briefly the results of [26],where the quantum cosmology of the model with a scalarfield, whose potential is inversely proportional to the field,was studied. We shall try to reinterpret some of the resultsobtained in [26], putting them in a wider context.As usual, we shall use the canonical formalism and the

Wheeler-DeWitt equation [8]. For this purpose, instead ofthe Friedmann metric (1), we shall consider the moregeneral metric

ds2 ¼ N2ðtÞdt2 � a2ðtÞdl2; (26)

where N is the so-called lapse function. The action of theFriedmann flat model with the minimally coupled scalarfield now looks as

S ¼Z

dt

�a3 _’2

2N� a3Vð’Þ � a _a2

N

�: (27)

CLASSICAL AND QUANTUM BIG BRAKE COSMOLOGY FOR . . . PHYSICAL REVIEW D 85, 123518 (2012)

123518-5

Variating the action (27) with respect to N and putting thenN ¼ 1 we come to the standard Friedmann equation. Now,introducing the canonical formalism, we define the canoni-cally conjugated momenta as

p’ ¼ a3 _’

N(28)

and

pa ¼ �a _a

N: (29)

The Hamiltonian is

H ¼ N

��p2

a

4aþ p2

’

2a3þ Va3

�(30)

and is proportional to the lapse function. The variation ofthe action with respect to N gives the constraint

� p2a

4aþ p2

’

2a3þ Va3 ¼ 0; (31)

and the implementation of the Dirac quantization proce-dure gives the Wheeler-DeWitt equation

�� p2

a

4aþ p2

’

2a3þ Va3

�c ða; ’Þ ¼ 0: (32)

Here c ða;�Þ is the wave function of the Universe and thehats over the momenta mean that the functions are sub-stituted by operators. Introducing the differential operatorsrepresenting the momenta as

p a � @

i@a; p’ � @

i@’(33)

and multiplying Eq. (32) by a3 we obtain the followingpartial differential equation:

�a2

4

@2

@a2� 1

2

@2

@’2þ a6V

�c ða; ’Þ ¼ 0: (34)

Finally, for our potential inversely proportional to thescalar field we have

�a2

4

@2

@a2� 1

2

@2

@’2� a6V0

’

�c ða; ’Þ ¼ 0: (35)

Note that in Eq. (32) and in the subsequent equations wehave ignored rather a complicated problem of the choiceof the ordering of noncommuting operators because thespecification of such a choice is not essential for ouranalysis. Moreover, the interpretation of the wave functionof the Universe is rather an involved question [28,39,40].The point is that to choose the measure in the space of thecorresponding Hilbert space we should fix a particulargauge condition, eliminating in such a way the redundantgauge degrees of freedom and introducing a temporaldynamics into the model [28]. We shall not dwell here onthis procedure, assuming generally that the cosmological

radius a is in some way connected with the chosen timeparameter, and that the unique physical variable is thescalar field ’. Then, it is convenient to represent thesolution of Eq. (35) in the form

c ða; ’Þ ¼ X1n¼0

CnðaÞ�nða;’Þ; (36)

where the functions �n satisfy the equation

�� 1

2

@2

@’2� a6V0

’

��ða; ’Þ ¼ �EnðaÞ�nða;’Þ; (37)

while the functions CnðaÞ satisfy the equation

a2

4

@2CnðaÞ@a2

¼ EnðaÞCnðaÞ; (38)

where n ¼ 0; 1; . . . . Requiring the normalizability of thefunctions �n on the interval 0 � ’<1, which, in turn,implies their nonsingular behavior at ’ ¼ 0 and ’ ! 1,and using the considerations similar to those used in theanalysis of the Schrodinger equation for the hydrogen-likeatoms, one can show that the acceptable values of thefunctions En are

En ¼ V0a12

2ðnþ 1Þ2 ; (39)

while the corresponding eigenfunctions are

�nða;’Þ ¼ ’ exp

��V0a

6’

nþ 1

�L1n

�2V0a

6’

nþ 1

�; (40)

where L1n are the associated Laguerre polynomials.

Rather often the fact that the wave function of theUniverse disappears at the values of the cosmologicalparameters corresponding to some classical singularity isinterpreted as an avoidance of such singularity. However,in the case of the soft singularity considered in the model athand, such an interpretation does not look too convincing.Indeed, one can be tempted to think that the probability offinding the Universe in the soft singularity state character-ized by the vanishing value of the scalar field is vanishingbecause the expression for functions (40) entering into theexpression for the wave function of the Universe (36) isproportional to ’. However, the wave function (36) canhardly have a direct probabilistic interpretation. Instead,one should choose some reasonable time-dependent gauge,identifying some combination of variables with an effec-tive time parameter, and interpreting other variables asphysical degrees of freedom [28]. The definition of thewave function of the Universe in terms of these physicaldegrees of freedom is rather an involved question; how-ever, we are in a position to make some semiqualitativeconsiderations. The reduction of the initial set of variablesto the smaller set of physical degrees of freedom impliesthe appearance of the Faddeev-Popov determinant whichas usual is equal to the Poisson bracket of the gauge-fixing

ALEXANDER Y. KAMENSHCHIK AND SERENA MANTI PHYSICAL REVIEW D 85, 123518 (2012)

123518-6

condition and the constraint. Let us, for example, choose asa gauge-fixing condition the identification of the new‘‘physical’’ time parameter with the Hubble parameter Htaken with the negative sign. Such an identification isreasonable because as it follows from Eq. (16) the variableHðtÞ is monotonously decreasing. The volume a3 is thevariable canonically conjugated to the Hubble variable.Thus, the Poisson bracket between the gauge-fixing con-dition � ¼ H � Tphys and the constraint (31) includes the

term proportional to the potential of the scalar field, whichis inversely proportional to this field itself. Thus, thesingularity in ’ arising in the Faddeev-Popov determinantcan cancel zero, arising in [36].

Let us confront this situation with that of the big bangand big crunch singularities. As it was seen in Sec. III suchsingularities classically arise at infinite values of the scalarfield. To provide the normalizability of the wave functionone should have the integral on the values of the scalar field’ convergent, when j’j ! 1. That means that, indepen-dently of details connected with the gauge choice, not onlythe wave function of the Universe but also the probabilitydensity of scalar field values should decrease rather rapidlywhen the absolute value of the scalar field is increasing.Thus, in this case, the effect of the quantum avoidance ofthe classical singularity is present.

V. THE TACHYON COSMOLOGICAL MODELWITH THE TRIGONOMETRIC POTENTIAL

The tachyon field, born in the context of the string theory[41], provides an example of matter having a large enoughnegative pressure to produce an acceleration of the expan-sion rate of the Universe. Such a field is today consideredas one of the possible candidates for the role of dark energyand, also for this reason, in recent years it has been in-tensively studied. The tachyon models represent a subclassof the models with nonstandard kinetic terms [42], whichdescend from the Born-Infeld model, invented already inthe 1930s [43]. Before considering the model with thetrigonometric potential [22], possessing the big brakesingularity, we write down the general formulas of thetachyon cosmology.

The Lagrangian of the tachyon field T is

L ¼ � ffiffiffiffiffiffiffi�gp

VðTÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� g��T;�T;�

q(41)

or, for the spatially homogeneous tachyon field,

L ¼ � ffiffiffiffiffiffiffi�gp

VðTÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� _T2

p; (42)

where g is the determinant of the metric. The energydensity and the pressure of this field are, respectively,

� ¼ VðTÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� _T2

p (43)

and

p ¼ �VðTÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� _T2

p; (44)

while the field equation is

€T

1� _T2þ 3H _T þ V;T

VðTÞ ¼ 0: (45)

We also shall introduce the pseudotachyon field with theLagrangian [22]

L ¼ ffiffiffiffiffiffiffi�gp

WðTÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_T2 � 1

p(46)

and with the energy density

� ¼ WðTÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_T2 � 1

p (47)

and the pressure

p ¼ WðTÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_T2 � 1

p: (48)

The Klein-Gordon equation for the pseudotachyon field is

€T

1� _T2þ 3H _T þ W;T

WðTÞ ¼ 0: (49)

We also shall write down the equations for the timederivative of the Hubble parameter in the tachyon andpseudotachyons models as

_H ¼ � 3

2

VðTÞ _T2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� _T2

p ; (50)

_H ¼ � 3

2

WðTÞ _T2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_T2 � 1

p : (51)

We see that the Hubble parameter in both these models isdecreasing just like in the scalar field model [see Eq. (16)].Note that for the case when the potential of the tachyon

field VðTÞ is a constant, the cosmological model with thistachyon coincides with the cosmological model with theChaplygin gas [44]. Analogously, the pseudotachyonmodel with the constant potential coincides with the modelwith the anti-Chaplygin gas. Hence, its classical dynamicsis that described in Sec. II. However, it is of interest tous to integrate explicitly the Klein-Gordon Eq. (49) forthe pseudotachyon field with the constant potential. Theresult is

_T 2 ¼ 1

1� a6

a6�

; (52)

where the initial conditions for the evolution are fixed bythe choice of the radius a� at which the Universe encoun-ters the big brake. Note that, when the Universe tends to thebig brake, the time derivative of the pseudotachyon fieldtends to �1, and while the Universe encounters the bigbang and big crunch singularities of the time derivativeof T tends to �1. In both cases the behavior of thetime derivative of the pseudotachyon field going to the

CLASSICAL AND QUANTUM BIG BRAKE COSMOLOGY FOR . . . PHYSICAL REVIEW D 85, 123518 (2012)

123518-7

singularities does not depend on the particular trajectory,parametrized by the value a�.

Now we shall study a very particular tachyon potentialdepending on the trigonometric functions which were sug-gested in [22]. Its form is

VðTÞ¼ �

sin2 32ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ð1þkÞp

T�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�ð1þkÞcos23

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�ð1þkÞ

pT

s;

(53)

where � is a positive constant and k is a parameter, whichis chosen in the interval �1< k< 1. The case of thepositive values of the parameter k is especially interesting.In this case one subset of the cosmological evolutions isinfinite and tends to a de Sitter regime of the exponentialexpansion with the asymptotic value of the tachyon fieldT¼T0¼ �

3ffiffiffiffiffiffiffiffiffi�ð1þkÞ

p . Other trajectories go to the points of the two-

dimensional phase space ðT; _TÞ, where the field acquiresthe values T¼T3¼ 2

3ffiffiffiffiffiffiffiffiffi�ð1þkÞ

p arccos 1ffiffiffiffiffi1þk

p or T¼T4¼ 2

3ffiffiffiffiffiffiffiffiffi�ð1þkÞ

p ð��arccos 1ffiffiffiffiffi1þk

p Þand where the expression under the sign of the square rootof the potential (53) vanishes. At the same moment, thetime derivative of the tachyon field becomes equal to �1and, hence, the other square root in the Lagrangian (41)vanishes too. It was shown in [22] that after that, thetransformation of the tachyon into the pseudotachyon be-comes unavoidable. The potential WðTÞ for the pseudo-tachyon is obtained from the potential VðTÞ for the tachyon(53) by the change of the sign of the expression under thesquare root. What happens after that? Let us suppose thatthe trajectory of the tachyon field crosses the point T ¼ T3,_T ¼ �1. Then, after crossing this point the Universe tendsto the big brake singularity in the regime described by thefollowing formulas [30]:

T ¼ TBB þ�

4

3WðTBBÞ�ð1=3Þðt� tBBÞð1=3Þ; (54)

H ¼�9W2ðTBBÞ

2

�ð1=3Þðt� tBBÞð1=3Þ; (55)

where tBB is the moment when the Universe encounters thebig brake singularity and TBB is the value of the tachyonfield at this moment. It was shown that TBB can accept thevalues in the interval 0< TBB � T3 [22] and, hence, thecosmological trajectories encountering the big brake sin-gularity constitute an infinite one-parameter set, whoseelements can be parametrized by the value of the tachyonfield TBB.

Then after the big brake crossing, the cosmologicalexpansion is followed by a contraction, which culminatesat the encounter with the big crunch singularity, whichoccurs at T ¼ 0 and _T¼�

ffiffiffiffiffi1þkk

p[30].

VI. THE QUANTUM COSMOLOGY OF THETACHYON AND THE PSEUDOTACHYON FIELD

Now, we would like to construct the Hamiltonian for-malism for the tachyon and pseudotachyon fields. Usingthe metric (26), one can see that the contribution of thetachyon field into the action is

S ¼ �Z

dtNa3VðTÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� _T2

N2

s: (56)

The conjugate momentum for T is

pT ¼ a3V _T

Nffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� _T2

N2

q ; (57)

and so the velocity can be expressed as

_T ¼ NpTffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2T þ a6V2

q : (58)

The Hamiltonian of the tachyon field is now

H ¼ Nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2T þ a6V2

q: (59)

Analogously, for the pseudotachyon field, we have

pT ¼ a3W _T

Nffiffiffiffiffiffiffiffiffiffiffiffiffiffi_T2

N2 � 1q ; (60)

_T ¼ NpTffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2T � a6W2

q (61)

and

H ¼ Nffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2T � a6W2

q: (62)

In what follows it will be convenient for us to fix the lapsefunction as N ¼ 1.Now, adding the gravitational part of the Hamiltonians

and quantizing the corresponding observables, we obtainthe following Wheeler-DeWitt equations for the tachyons:

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2T þ a6V2

q� a2p2

a

4

�c ða; TÞ ¼ 0 (63)

and for the pseudotachyons

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2T � a6W2

q� a2p2

a

4

�c ða; TÞ ¼ 0: (64)

The study of the Wheeler-DeWitt equation for theUniverse filled with a tachyon or a pseudotachyon field israther a difficult task because the Hamiltonian dependsnonpolynomially on the conjugate momentum of suchfields. However, one can come to interesting conclusions,considering some particular models.

ALEXANDER Y. KAMENSHCHIK AND SERENA MANTI PHYSICAL REVIEW D 85, 123518 (2012)

123518-8

First of all, let us consider a model with the pseudo-tachyon field having a constant potential. In this case theHamiltonian in Eq. (64) does not depend on the field T.Thus, it is more convenient to use the representation of thequantum state of the Universe where it depends on thecoordinate a and the momentum pT . Then the Wheeler-DeWitt equation will have the following form:

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2T � a6W2

qþ a2

4

@2

@a2

�c ða; pTÞ ¼ 0: (65)

It becomes algebraic in the variable pT . Now, we see thatthe Hamiltonian is well-defined at p2

T � a6W2. Looking atthe limiting value p2

T ¼ a6W2 and comparing it with therelation (61), we see that it corresponds to _T2 ! 1, which,in turn, corresponds to the encounter with the big brakesingularity as was explained in Sec. V. The only way to‘‘neutralize’’ the values of pT , which imply the negativityof the expression under the square root in the left-hand sideof Eq. (65) is to require that the wave function of theUniverse is such that

c ða; pTÞ ¼ 0 at p2T � a6W2: (66)

The last condition could be considered as a hint on thequantum avoidance of the big brake singularity. However,as it was explained in Sec. IVon the example of the scalarfield model, to speak about the probabilities in the neigh-borhood of the point where the wave function of theUniverse vanishes, it is necessary to realize the procedureof the reduction of the set of variables to a smaller set ofphysical degrees of freedom. Now, let us suppose that thegauge-fixing condition is chosen in such a way that the roleof time is played by a Hubble parameter. In this case theFaddeev-Popov determinant, equal to the Poisson bracketbetween the gauge-fixing condition and the constraint, willbe inversely proportional to the expression

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2T�a6W2

p[see

Eq. (62)], which tends to zero at the moment of theencounter with the big brake singularity. Thus, in thecase of a pseudotachyon model, just like in the case ofthe cosmological model based on the scalar field, theFaddeev-Popov determinant introduces the singular factor,which compensates the vanishing of the wave function ofthe Universe.

What can we say about the big bang and the big crunchsingularities in this model? It was noticed in the precedingsection that at these singularities _T2 ¼ 1. From the relation(61) it follows that such values of _T correspond tojpTj ! 1. A general requirement of the normalizabilityof the wave function of the Universe implies the vanishingof c ða; pTÞ at pT ! �1 which signifies the quantumavoidance of the big bang and the big crunch singularities.It is quite natural because these singularity are not travers-able in classical cosmology.

Now we consider the tachyon cosmological model withthe trigonometric potential, whose classical dynamics wasbriefly sketched in the preceding section. In this case the

Hamiltonian depends on both the tachyon field T and itsmomentum pT . The dependence of the expression underthe square root on T is more complicated than that on pT .Hence, it does not make sense to use the representationc ða; pTÞ instead of c ða; TÞ. Now, we have under the

square root the second-order differential operator � @2

@T2 ,

which is positively defined, and the function �a6W2ðTÞ,which is negatively defined. The complete expressionshould not be negative, but what does it mean inour case? It means that we should choose such wavefunctions for which the quantum average of the operatorp2T � a6W2ðTÞ is non-negative,

hc jp2T�a6W2ðTÞjc i

¼ZDTc �ða;TÞ

�� @2

@T2�a6WðTÞ2

�c ða;TÞ�0: (67)

Here the symbol DT signifies the integration on thetachyon field T with some measure. It is easy to guessthat the requirement (67) does not imply the disappearanceof the wave function c ða; TÞ at some range or at someparticular values of the tachyon field, and one can alwaysconstruct a wave function which is different from zeroevery where and thus does not show the phenomenon ofthe quantum avoidance of singularity. However, the formsof the potential VðTÞ given by Eq. (53) and of the corre-sponding potential WðTÞ for the pseudotachyon field aris-ing in the same model [22] are too cumbersome toconstruct such functions explicitly. Thus, to illustrate ourstatement, we shall consider a simpler toy model.Let us consider the Hamiltonian

H ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2 � V0x

2q

; (68)

where p is the conjugate momentum of the coordinate xand V0 is some positive constant. Let us choose as a wavefunction a Gaussian function

c ðxÞ ¼ expð��x2Þ; (69)

where � is a positive number and we have omitted thenormalization factor, which is not essential in the presentcontext. Then the condition (67) will look like

Zdx expð��x2Þ

�� d2

dx2� V0x

2

�expð��x2Þ

¼ffiffiffiffi�

2

r �3

4

ffiffiffiffi�

p � V0

2�ð3=2Þ

�� 0; (70)

which can be easily satisfied if

� �ffiffiffiffiffiffiffiffiffi2

3V0

s: (71)

Thus, we have seen that for this very simple model onecan always choose such a quantum state, which does notdisappear at any value of the coordinate x and whichguarantees the positivity of the quantum average of the

CLASSICAL AND QUANTUM BIG BRAKE COSMOLOGY FOR . . . PHYSICAL REVIEW D 85, 123518 (2012)

123518-9

operator, which is not generally positively defined. Comingback to our cosmological model we can say that therequirement of the well-definiteness of the pseudotachyonpart of the Hamiltonian operator in the Wheeler-DeWittequation does not imply the disappearance of the wavefunction of the Universe at some values of the variables andthus does not reveal the effect of the quantum avoidance ofthe cosmological singularity.

At the end of this section we would like also to analyzethe big bang and big crunch singularities in the tachyonmodel with the trigonometric potential. As was shown in[22] the big bang singularity can occur in two occasions(the same is true also for the big crunch singularity [30])—either atWðTÞ ! 1 (for example, for T ! 0) or at _T2 ¼ 1,WðTÞ � 0. One can see from Eqs. (43) and (60) that whenthe Universe approaches these singularities the momentumpT tends to infinity. As was explained before, the wavefunction of the Universe in the momentum representationshould vanish at jpTj ! 1 and, hence, we have the effectof the quantum avoidance.

VII. CONCLUDING REMARKS

We have studied a relation between the cosmologicalsingularities in classical and quantum theory, comparingthe classical and quantum dynamics in some models pos-sessing the big brake singularity—the model based on ascalar field and two models based on a tachyon (pseudo-tachyon) field. It was shown that in the tachyon model withthe trigonometric potential [22], the wave function of theUniverse is not obliged to vanish in the range of thevariables corresponding to the appearance of the classicalbig brake singularity. In a more simple pseudotachyoncosmological model the wave function, satisfying theWheeler-DeWitt equation and depending on the cosmo-logical radius and the pseudotachyon field, disappears atthe big brake singularity. However, the transition to thewave function depending only on the reduced set of physi-cal degrees of freedom implies the appearance of theFaddeev-Popov factor, which is singular and which singu-larity compensates the terms, responsible for the vanishingof the wave function of the Universe. Thus, in both thesecases, the effect of the quantum avoidance of the big brakesingularity is absent.

In the case of the scalar field model with the potentialinversely proportional to this field, all the classical tra-jectories pass through a soft singularity (which for oneparticular trajectory is exactly the big brake). Thewave function of the Universe disappears at the vanishingvalue of the scalar field which classically correspondsto the soft singularity. However, also in this case the

Faddeev-Popov factor arising at the reduction to thephysical degrees of freedom provides nonzero value ofthe probability of finding the Universe at the softsingularity.In spite of the fact that we have considered some

particular scalar field and tachyon-pseudo-tachyon mod-els, our main conclusions were based on rather generalproperties of these models. Indeed, in the case of thescalar field we have used the fact that its potential atthe soft singularity should be negative and divergent, toprovide an infinite positive value of pressure. In the caseof the pseudotachyon field both the possible vanishing ofthe wave function of the Universe and its ‘‘reemergence’’in the process of reduction were connected with thegeneral structure of the contribution of such a fieldinto the super-Hamiltonian constraint (62). Note that inthe case of the tachyon model with the trigonometricpotential, the wave function does not disappear at all.On the other hand, we have seen that for the big bang

and big crunch singularities not only the wave functions ofthe Universe but also the corresponding probabilities dis-appear when the Universe is approaching to the corre-sponding values of the fields under consideration, andthis fact is also connected with rather general propertiesof the structure of the Lagrangians of the theories. Thus, inthese cases the effect of quantum avoidance of singularitiestakes place.One can say that there is some kind of a classical—

quantum correspondence here. The soft singularities aretraversable at the classical level (at least for simplehomogeneous and isotropic Friedmann models) andthe effect of quantum avoidance of singularities isabsent. The strong big bang and big crunch singularitiescannot be passed by the Universe at the classical level,and the study of the Wheeler-DeWitt equation indicatesthe presence of the quantum singularity avoidanceeffect.It would be interesting also to find examples of the

absence of the effect of the quantum avoidance of singu-larities, for the singularities of the big bang-big crunchtype. Note that interest in studying the possibility of thecrossing of such singularities is growing and some modelstreating this phenomenon have been elaborated on duringthe last few years [45].

ACKNOWLEDGMENTS

We are grateful to A.O. Barvinsky and C. Kiefer forfruitful discussions and to P.V. Moniz and M. Bouhmadi-Lopez for useful correspondence. This work was partiallysupported by the RFBR Grant No. 11-02-00643.

ALEXANDER Y. KAMENSHCHIK AND SERENA MANTI PHYSICAL REVIEW D 85, 123518 (2012)

123518-10

[1] L. D. Landau and E.M. Lifshitz, The Classical Theory ofFields (Pergamon Press, Oxford, 1975), 4thed..

[2] C.W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation(W.H. Freeman, San Francisco, 1973).

[3] S.W. Hawking and G. F. R. Ellis, The Large ScaleStructure of Space-Time (Cambridge University Press,Cambridge, 1973).

[4] S.W. Hawking and R. Penrose, Proc. R. Soc. A 314, 529(1970).

[5] R. Penrose, Structure of Space-Time (W.A. Benjamin,New York, 1970).

[6] I.M. Khalatnikov and E.M. Lifshitz, Phys. Rev. Lett. 24,76 (1970); V.A. Belinsky, I.M. Khalatnikov, and E.M.Lifshitz, Adv. Phys. 19, 525 (1970).

[7] C.W. Misner, Phys. Rev. Lett. 22, 1071 (1969).[8] B. S. DeWitt, Phys. Rev. 160, 1113 (1967).[9] C. Kiefer, Quantum Gravity (Oxford University Press,

Oxford, 2007), 2nd ed.[10] A. Riess et al., Astron. J. 116, 1009 (1998).[11] S. J. Perlmutter et al., Astrophys. J. 517, 565 (1999).[12] V. Sahni and A.A. Starobinsky, Int. J. Mod. Phys. D 9, 373

(2000); 15, 2105 (2006).[13] T. Padmanabhan, Phys. Rep. 380, 235 (2003).[14] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. 75, 559

(2003).[15] E. J. Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod.

Phys. D 15, 1753 (2006).[16] A. A. Starobinsky, Gravitation Cosmol. 6, 157 (2000).[17] R. R. Caldwell, M. Kamionkowski, and N.N. Weinberg,

Phys. Rev. Lett. 91, 071301 (2003).[18] R. R. Caldwell, Phys. Lett. B 545, 23 (2002).[19] J. D. Barrow, G. J. Galloway, and F. J. Tipler, Mon. Not. R.

Astron. Soc. 223, 835 (1986).[20] Yu. Shtanov and V. Sahni, Classical Quantum Gravity 19,

L101 (2002).[21] J. D. Barrow, Classical Quantum Gravity 21, L79 (2004);

21, 5619 (2004); J. D. Barrow and C.G. Tsagas, ClassicalQuantum Gravity 22, 1563 (2005).

[22] V. Gorini, A. Yu. Kamenshchik, U. Moschella, and V.Pasquier, Phys. Rev. D 69, 123512 (2004).

[23] B. Carter, Phys. Lett. B 224, 61 (1989); A. Vilenkin, Phys.Rev. D 41, 3038 (1990).

[24] A. Yu. Kamenshchik, U. Moschella, and V. Pasquier, Phys.Lett. B 511, 265 (2001).

[25] J. C. Fabris, S. V. B. Goncalves, and P. E. de Souza, Gen.Relativ. Gravit. 34, 53 (2002); N. Bilic, G. B. Tupper, andR.D. Viollier, Phys. Lett. B 535, 17 (2002); M.C. Bento,O. Bertolami, and A.A. Sen, Phys. Rev. D 66, 043507

(2002); V. Gorini, A. Yu. Kamenshchik, and U. Moschella,Phys. Rev. D 67, 063509 (2003).

[26] A. Y. Kamenshchik, C. Kiefer, and B. Sandhofer, Phys.Rev. D 76, 064032 (2007).

[27] M. Bouhmadi-Lopez and P. V. Moniz, Phys. Rev. D 71,063521 (2005); M. P. Dabrowski, C. Kiefer, and B.Sandhofer, Phys. Rev. D 74, 044022 (2006); M.Bouhmadi-Lopez, C. Kiefer, B. Sandhofer, and P. V.Moniz, Phys. Rev. D 79, 124035 (2009).

[28] A. O. Barvinsky, Phys. Lett. B 241, 201 (1990); Phys. Rep.230, 237 (1993).

[29] Z. Keresztes, L. A. Gergely, V. Gorini, U. Moschella, andA.Y. Kamenshchik, Phys. Rev. D 79, 083504 (2009).

[30] Z. Keresztes, L. A. Gergely, A.Y. Kamenshchik, V. Gorini,and D. Polarski, Phys. Rev. D 82, 123534 (2010).

[31] M. P. Dabrowski, T. Denkiewicz, and M.A. Hendry, Phys.Rev. D 75, 123524 (2007).

[32] L. Fernandez-Jambrina and R. Lazkoz, Phys. Rev. D 70,121503 (2004).

[33] A. Balcerzak and M. P. Dabrowski, Phys. Rev. D 73,101301 (2006); H. Maeda, arXiv:1112.5178 [Phys. Rev.D. (to be published)]; Z. Keresztes, L. A. Gergely, andA.Y. Kamenshchik, arXiv:1204.1199.

[34] A. A. Starobinsky, Pis’ma Zh. Eksp. Teor. Fiz. 68, 721(1998); [JETP Lett. 68, 757 (1998)].

[35] A. B. Burd and J. D. Barrow, Nucl. Phys. B308, 929(1988).

[36] A. A. Andrianov, F. Cannata, A.Y. Kamenshchik, and D.Regoli, J. Cosmol. Astropart. Phys. 02 (2008) 015.

[37] A. Y. Kamenshchik, A. Tronconi, and G. Venturi, Phys.Lett. B 702, 191 (2011).

[38] A. Y. Kamenshchik and S. Manti, arXiv:1111.5183.[39] R. Brout and G. Venturi, Phys. Rev. D 39, 2436 (1989); G.

Venturi, Classical Quantum Gravity 7, 1075 (1990); C.Bertoni, F. Finelli, and G. Venturi, Classical QuantumGravity 13, 2375 (1996).

[40] A. O. Barvinsky and A.Y. Kamenshchik, ClassicalQuantum Gravity 7, L181 (1990).

[41] A. Sen, J. High Energy Phys. 04 (2002) 048.[42] C. Armendariz-Picon, V. F. Mukhanov, and P. J.

Steinhardt, Phys. Rev. D 63, 103510 (2001).[43] M. Born and L. Infeld, Proc. R. Soc. London, Ser. A 144,

425 (1934).[44] A. Frolov, L. Kofman, and A. Starobinsky, Phys. Lett. B

545, 8 (2002).[45] I. Bars, S. H. Chen, and N. Turok, Phys. Rev. D 84, 083513

(2011); I. Bars, S. H. Chen, P. J. Steinhardt, and N. Turok,arXiv:1112.2470.

CLASSICAL AND QUANTUM BIG BRAKE COSMOLOGY FOR . . . PHYSICAL REVIEW D 85, 123518 (2012)

123518-11