tachyons, scalar ﬁelds, and cosmology · gorini et al. physical review d 69, 123512 ~2004!...

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PHYSICAL REVIEW D 69, 123512 ~2004!

Tachyons, scalar fields, and cosmology

Vittorio GoriniDipartimento di Scienze Fisiche e Matematiche, Universita` dell’Insubria, Via Valleggio 11, 22100 Como, Italy

and INFN, sez. di Milano, Via Caloria 16, 20133 Milano, Italy

Alexander KamenshchikDipartimento di Scienze Fisiche e Matematiche, Universita` dell’Insubria, Via Valleggio 11, 22100 Como, Italy

and L.D. Landau Institute for Theoretical Physics of the Russian Academy of Sciences, Kosygin str. 2, 119334 Moscow, Ru

Ugo MoschellaDipartimento di Scienze Fisiche e Matematiche, Universita` dell’Insubria, Via Valleggio 11, 22100 Como, Italy

and INFN, sez. di Milano, Via Caloria 16, 20133 Milano, Italy

Vincent PasquierService de Physique The´orique, C.E. Saclay, 91191 Gif-sur-Yvette, France

~Received 7 November 2003; published 11 June 2004!

We study the role that tachyon fields may play in cosmology as compared to the well-established use ofminimally coupled scalar fields. We first elaborate on a kind of correspondence existing between tachyons andminimally coupled scalar fields; corresponding theories give rise to the same cosmological evolution for aparticular choice of the initial conditions but not for any other. This leads us to study a specific one-parameterfamily of tachyonic models based on a perfect fluid mixed with a positive cosmological constant. For positivevalues of the parameter, one needs to modify Sen’s action and use thes process of resolution of singularities.The physics described by this model is dramatically different and much richer than that of the correspondingscalar field. For particular choices of the initial conditions, the universe, which does mimic for a long time a deSitter–like expansion, ends up in a finite time in a special type of singularity that we call abig brake. Thissingularity is characterized by an infinite deceleration.

DOI: 10.1103/PhysRevD.69.123512 PACS number~s!: 98.80.Cq, 98.80.Jk

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I. INTRODUCTION

The recently discovered cosmic acceleration@1–3# hasput forward the problem of unraveling the nature of thecalled dark energy responsible for such a phenomenon~for areview, see@4,5#!. The crucial feature of the dark energwhich ensures an accelerated expansion of the universthat it breaks the strong energy condition.

The tachyon field arising in the context of string theo@6–9# provides one example of matter which does the jThe tachyon has been intensively studied during the pastyears also in application to cosmology@10–30#; in this caseone usually takes Sen’s effective action@6# for granted andstudies its cosmological consequences without worryabout the string-theoretical origin of the action itself. Wtake this attitude in the present paper.

However, it would be reasonable, before considering ccrete tachyon cosmological models, to answer a simple qtion: is the tachyon field of real interest for cosmology?deed, it is well known that for isotropic cosmologicmodels, for a given dependence of the cosmological radon time it is always possible to construct a potential fominimally coupled scalar field~in brief, scalar field! model,which would reproduce this cosmological evolution~see,e.g.,@31#!, provided rather general reasonable conditionssatisfied. Since a similar statement holds also for cosmolcal models with tachyons, one can find some kind of corspondence between minimally coupled scalar field mod

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and tachyon ones describing the same cosmological evtion @14#. Therefore, a natural question arises: does it msense at all to study tachyon cosmological models in placthe traditional scalar field models?

In our opinion, the point is that the correspondencetween tachyon and scalar field cosmological models irather limited one and amounts to the existence of ‘‘corsponding’’ solutions of the models, obtained by imposicertain special initial conditions. If one moves away frothese conditions, the dynamics of the tachyon modelbecome more complicated and very different from that ofscalar field cousin.

In this paper, we consider some examples of scalartachyon field isotropic cosmological models having coindent exact solutions. All the exact solutions considered hactually arise as solutions of some isotropic perfect fluid cmological models.

Some of the tachyon and scalar field potentials thatconsider here are well known in the literature and are widused as quintessential models. We will introduce also a n~at least to our knowledge! tachyon model that is based othe cosmic evolution driven by the mixture of a cosmologicconstant and of a fluid with equation of statep5k«, with21,k

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GORINI et al. PHYSICAL REVIEW D 69, 123512 ~2004!

zero for finite values of the tachyon fieldT. Thus, it mightseem that these features would kill the model; on the ctrary, using the methods of the qualitative theory of ordinadifferential equations, it is possible to extract from the moa dynamics which is perfectly meaningful also for positivek.We find that this dynamics is quite rich. In particular, whthe value of the parameterk is positive, the model possesstwo types of trajectories: trajectories describing an eternexpanding universe and those hitting a cosmological sinlarity of a special type that we have chosen to call abigbrake. Thus, starting with a simple perfect fluid model atrying to reproduce its cosmological evolution in scalar fieand tachyon models, we arrive at a relatively simple scafield potential with a correspondingly simple dynamics aat a complicated tachyon potential. The latter provideswith a model having a very interesting dynamics giving rto very different cosmological evolutions and opening in tuopportunities for some nontrivial speculations about theture of the universe.

The structure of the paper is as follows. In the secosection we study the problem of the correspondence betwtachyon and scalar field cosmological models, and we gsome examples in the third section. In the fourth sectionsuggest a way to go beyond the limitations of tachyomodels, and we make use of this idea in the fifth and sisections, where we study the dynamics and the cosmologa particular~toy! tachyon model introduced in Sec. III. In thlast section we conclude our paper with some speculaton the future evolution of the universe arising from tanalysis of our model.

II. THE CORRESPONDENCE

We consider a flat Friedmann cosmological modelds2

5dt22a2(t)dl2 of a universe filled with some perfect fluiand suppose that the cosmological evolutiona5a(t) isgiven; the Friedmann equation

ȧ2

a25« ~1!

provides the dependence«5«(t) of the energy density of thefluid on the cosmic time~we have set 8pG/351 for conve-nience!. Then, the equation for energy conservation

«̇523ȧ

a~«1p! ~2!

fixes the pressurep5p(t); therefore, an equation of statp5p(«) can in principle be written describing the uniqufluid model compatible with the given cosmic evolutio~provided that«̇Þ0).

Now, let us suppose that the matter content of the univeis modeled by a homogeneous tachyon fieldT(t) describedby Sen’s Lagrangian density@6,7#,

L52V~T!A12Ṫ2. ~3!

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The energy momentum tensor is diagonal and the cosponding field-dependent energy density and pressuregiven by

«5V~T!

A12Ṫ2, ~4!

p52V~T!A12Ṫ2, ~5!

while the field equation for the tachyon is written as follow

T̈

12Ṫ21

3ȧṪ

a1

V,TV

50. ~6!

One can try and find a potentialV(T) so that, for certainsuitably chosen initial conditions on the tachyon field, tscale factor of the universe is precisely the givena(t). Asimilar construction can be attempted for a minimacoupled scalar field, described by the Lagrangian density

L51

2ẇ22U~w!. ~7!

A given cosmological evolutiona5a(t) can therefore beused to establish a sort of correspondence between thepotentialsV(T) andU(w) ~see also@14#!.

Let us see in some detail how this works and whatsought correspondence really means. It is often more pracal to use the scale factora to parametrize the cosmic timethis can be done providedȧÞ0. From Eqs.~4! and ~5!, itfollows that

Ṫ25p1«

«. ~8!

By using Eqs.~1! and~2!, the latter equation can be rewritteas follows:

T851

aA~«1p!

«25

1

aA2«8a

3«2, ~9!

where a prime denotes the derivative with respect toa. Oncea(t) and therefore«(a) is specified@see Eq.~1!#, Eq.~9! canbe integrated to give

T5F~a!5Eadxx A2«8~x!x3«2~x! ~10!~an arbitrary additive integration constant is hidden in tunspecified integration limit!. By inverting Eq. ~10!, a5F21(T), and making use of the relation

V5A2«p5A~«2a6!86a5

, ~11!

the shape of the required tachyon potentialV5V(a)5V@F21(T)# can be found.

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TACHYONS, SCALAR FIELDS, AND COSMOLOGY PHYSICAL REVIEW D69, 123512 ~2004!

For the minimally coupled scalar field, we have similformulas. The analogs of Eqs.~4! and~5! for the scalar fieldare

«5 12 ẇ21U~w!, ~12!

p5 12 ẇ22U~w!. ~13!

From these we get

w851

aA~«1p!

«5

1

aA2«8a

3«~14!

and

U51

2~«2p!5

~«a6!8

6a5. ~15!

Integration of Eq.~14! gives

w5F~a!5Eadxx A2«8~x!x3«~x! ~16!and, as before,U5U(a)5U@F21(w)#. The formulas aboveestablish a kind of correspondence between the potentiaUandV in the sense that the cosmologies resulting from spotentials admit the given cosmological evolutiona5a(t)for suitably chosen initial conditions on the fields. Howevit has to be stressed that for arbitrary initial conditions onfields, the cosmological evolutions~within the correspondingmodels! may be drastically different. Furthermore, changithe initial conditions, say, for a minimally coupled scalfield theory, one gets different cosmological evolutiowhich, in turn, can be reproduced in entirely differetachyon theories and vice versa. Thus, any scalar field potial has a whole~one-parameter! family of correspondingtachyon potentials, and the same is true the other waround.

We can learn something more by expressing the fieldTand w and the potentialsV(T) and U(w) in terms of theHubble variableh5ȧ/a. Since

ḣ52 32 ~«1p! ~17!

one has easily that

Ṫ25«1p

«52

2ḣ

3h2, V~T!5Ah2~ 23 ḣ1h2!, ~18!

and that

ẇ25«1p52 23 ḣ, U~w!5h21 13 ḣ. ~19!

Equations~18! and ~19! call for the condition«1p>0, orequivalentlyḣ2 32 h2, ~20!

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This tachyonic model has been studied in@13,14#, where alsothe correspondence with a minimally coupled scalar fitheory with exponential potential was noticed. There exisvast literature on the latter~see, e.g.,@32–41#!.

~ii ! Now we add a positive cosmological constant to tprevious model, i.e., we consider the mixture

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with the following exact solutions:

T~ t !56Ag2l21

l S 32 g~2l21!t D2l/(122l)

1T0 .

~39!

The scalar field model with the potential~36! was studied in@37# while the corresponding tachyon model with the potetial ~38! was discussed in@13,14#.

~iv! In our last example, we consider the Chaplygin gdescribed by the following equation of state:

p52A

«, A.0. ~40!

In this example, the evolutionh(t) is given only implicitly@42# by the formula

t51

6A1/4S ln h1A1/4

h2A1/422 arctan

h

A1/41p D . ~41!

However, as shown in@42#, the scalar potential can be recostructed using the known explicit dependence of« on a, andone gets

U~w!51

2AAS cosh 3~w2w0!1 1cosh 3~w2w0! D . ~42!

The corresponding field configuration is also given implitly,

w~ t !571

3arccoshS S h2~ t !1 ḣ~ t !3 D

AA

1AS h2~ t !1 ḣ~ t0!3 D 2

A21D 1w0 . ~43!

Similarly, using the dependence of« on a, one can recon-struct the tachyon potential,

V~T!5AA5const. ~44!

It is easy to see@15# that the tachyon model with a constapotential is exactly equivalent to the Chaplygin gas modIndeed, in the case of a constant tachyon potential, the rtion between the tachyon energy density~4! and the pressure~5! is just that of the Chaplygin gas~40!, where p«52V2(T)52A. The Chaplygin gas cosmological modwas introduced in@42# and further developed in@43–46# andmany other papers. Comparison with observational dataalso been extensively performed@47#.

IV. TRANSGRESSING THE BOUNDARIES

We now take a step back and consider the problemfinding a tachyonic field theory admitting the same cosmevolution as the one produced by perfect fluid with equat

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of statep5k«, where nowk.0. As we have stated, it isimpossible to reproduce this dynamics using Sen’s tachyoaction ~3!. One way out is to introduce a new field theobased on a Born-Infeld type action with Lagrangian

L5W~T!AṪ221. ~45!

In this new field theory, the energy and pressure are given

«5W~T!

AṪ221~46!

and

p5W~T!AṪ221. ~47!

The pressure~if well defined! is now positive. On the otheside, the equation of motion for this field has exactly tsame form~6! as was derived by Sen’s action.

Following the procedure described in Sec. II, now applto the Lagrangian~45!, one gets the following potential corresponding to the equation of statep5k« ~with k.0):

W~T!54

9

Ak~11k!

1

T2. ~48!

The exact solution of the field equations that reproducesdynamics of the perfect fluid is

T~ t !5A11kt ~49!

~we restrict our attention to the region of the phase spwhereT>0 and Ṫ>1). There are two other obvious solutions for this model, corresponding to other choices ofinitial conditions; they also give rise to linearly growinfields: T(t)5t andT(t)5A11(1/k)t.

We would like to point out an interesting fact: the Lagrangian~45!, which, together with the explicit form~48! ofthe potential, describes the field theory corresponding tpositive value ofk, is actually the same as Sen’s Lagrangi~3! with the potential~24! itself considered for positivek.Indeed, it is true, on the one hand, that in this case thetential ~24! becomes imaginary. However, this can be copensated for by considering the kinetic term in the region2Ṫ2,0 so that the action as a whole remains real. It canreinterpreted as the product of two real terms,

L52V~T!A12Ṫ25~A21!2V~T!A12Ṫ2

5W~T!AṪ221. ~50!

This model is introduced here as a pedagogical introducto the model with the potential~31!, which we discuss indetail in the following section. The properties of the presemodel can be recovered in the limitL→0 and we will notcomment further on it.

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V. DYNAMICS OF THE TOY TACHYONIC MODEL

We now provide the analysis of the dynamics of thechyonic model based on the potential~31!. In this case, Eq.~6! is equivalent to the following system of two first-orddifferential equations:

Ṫ5s, ~51!

ṡ523AV~12s2!3/4s2~12s2!V,TV

, ~52!

where using the Friedmann equation~1! we have expressethe Hubble variableh as a function of the variablesT ands.The model has the following two exact solutions~we takeT050 without loss of generality!:

T1~ t !52

3AL~11k!arctan sinh

3~11k!ALt2

, 0,t,`,

~53!

T2~ t !52

3AL~11k!S p2arctan sinh3~11k!ALt2 D

0,t,`. ~54!

By inserting Eq.~31! into Eq. ~52! and by eliminating thetime, we obtain an equation for the phase-space trajectos5s(T),

ds

dT52

3~12s2!AL

sin3AL~11k!T

2

3S 12~k11!cos2 3AL~11k!T212s2

D 1/42

3AL~11k!2

12s2

scotS 3AL~11k!T

2D

3

~k11!cos23AL~11k!T

21~k21!

12~k11!cos23AL~11k!T

2

. ~55!

In the phase plane (T,s), the solutions~53!, ~54! correspondto arcs of the curves ~see Figs. 1, 2, and 4!,

s5A11k cos3AL~11k!T

2. ~56!

The behavior of the cosmological radius for both solutio~53! and ~54! is the following:

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a~ t !5a0S sinh3AL~11k!t2 D2/[3(11k)]

. ~57!

As expected, we get back the cosmological evolution~28!determined by the equation of state~27! which was the start-ing point for constructing the potential~31!. To study thecosmological evolutions corresponding to all possible initconditions, we need to distinguish two different cases. Whthe parameterk

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ḣ52 32 h2s2, ~64!

by substituting into Eq.~63! it follows that

R53h2~423s2!53V~T!~423s2!

A12s2. ~65!

Thus the scalar curvatureR tends to infinity when approaching the boundary of the rectangle~with the exception of thecorners, which will be treated separately!.

All integral curves end up in the node; let us see how thbehave close to the boundary and begin with a right neborhood ofT50. There, Eq.~55! takes the form

ds

dT'

2~12s2!

sT S 12 ~2k!1/4sA11k~12s2!1/4D 5 F~s!T . ~66!If s,A11k, then ṡ→1` asT→0. Therefore, the integracurves ats,A11k, which get close to theT50 axis, risealmost vertically, climbing leftwards fors,0 and rightwardsfor s.0 until they get close tos, at which point they reacha maximum and thereafter approach the de Sitter node~60!~see Fig. 1!. These are the curves of bundlef. If A11k,s,1, then ṡ→2` as T→0. Therefore, the correspondincurves ats.A11k which get close to theT50 axis dropalmost vertically until they get close tos, at which point

h

l

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they also approach the de Sitter node. These are the curvbundle f 8. On the separatrixs one attains the poin(0,A11k) where ṡ50. Symmetric considerations apply tthen-reflected curves, i.e., those which lie to the right of tseparatrixs8.

But where do all the curves originate from? We first shothat apart froms, none of them can touch any point of thesaxis. Indeed, let us consider a point (T,s) close to thes axis.Equation~66! can be integrated backwards to give

T~s!5T0~s0!expEs0

s dx

F~x!. ~67!

This equation shows that, ifF(s0)Þ0, it is impossible torealize the conditionT0(s0)50 and therefore touch thes50 axis on a given trajectory. The roots of the equatiF(s0)50 ares5A11k,61. A closely similar reasoning excludes the point (0,1). The point (0,21) can also be ex-cluded since we should haveṪ>0 in the neighborhood ofsuch a point; buts is negative, and this contradicts the eqution Ṫ5s. We are therefore left with the point (0,A11k),where the exact solutions originates.

Let us examine now the upper boundary of Fig. 1. Insmall neighborhood of the point (T* ,1) @whereT* is in thedomain~58!#, Eq. ~55! can be replaced by the following approximate equation:

ds

dT52

3@2~12s!#3/4ALS 12~11k!cos2 3AL~11k!T*2

D 1/4sin

3AL~11k!T*2

. ~68!

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l

This equation is not Lipschitzian. The upper integral is ttrivial solution s51 while the lower integral is

s→H 51 for T,T*'12C~T* !~T2T* !4 for T>T* , ~69!where

C~T* !581

32

L2S 12~11k!cos2 3AL~11k!T*2 Dsin4

3AL~11k!T*2

. ~70!

The intermediate solutions stay constant ats51 for a whileand then leave thes51 line at a valueT** .T* . Therefore,from each point (T* ,1) there originates only one integracurve behaving as Eq.~69!. In particular, the separatrixs8originates at a point (Ts8,1) ~the value ofTs8 being un-known!.

e The condition of cosmic acceleration is

2ä

a52~«13p!.0. ~71!

For the tachyon cosmological model, using formulas~4! and~5! this condition can be reexpressed as

s2, 23 . ~72!

Therefore, ifk

he


FIG. 1. Potential and phase portrait fork,0 ~herek520.36). The horizontal liness56A2/3 separate the central region where texpansion of the Universe is accelerated from the two external regions where it decelerates.

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3A~11k!Larccos

1

A11k,

T452

3A~11k!L S p2arccos 1A11kD . ~73!Together with the condition21,s,1, Eq.~73! defines therectangle where we study the model at first. There are nthree fixed points. One of them is the attractive de Sinode~60!, whereas the other two are saddles correspondto the maxima of the potential at coordinates

T152

3A~11k!LarccosA12k

11k,

T252

3A~11k!LS p2arccosA12k11kD ~74!

(s50). They give rise to an unstable de Sitter regime wHubble parameter

H15A~11k!L2Ak

.H0 . ~75!

We first analyze the behavior of the trajectories in tvicinity of the line T5T3 and setT5T31T̃, with T̃ smalland positive andsÞ61. With these conditions the modeldescribed by the approximate equation

ṡ'212s2

2T̃, ~76!

which implies that the trajectories passing close toboundaryT5T3 drop steeply down without crossing it. Th‘‘physical’’ reason for this behavior is the vanishing of th

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potential atT5T3 ~to our knowledge this is a novel featurof our model!; indeed, the structure of the tachyonic actioimplies that the ‘‘force’’ is proportional to the logarithmiderivative of the potential, and this is infinite atT5T3. Theimpossibility of crossingT5T3 remains true also withoucoupling the tachyon to gravity. On the other hand, in cotrast with the situation encountered before, the geometrregular atT5T3: the vertical boundaries of the rectangle anot curvature singularities, because the potential doesdiverge there. Actually, the curvature scalarR vanishes therebecause of the vanishing of the potential~see Fig. 3!.

Instead, the horizontal sides are still singular. The sittion is exactly as before and there is one integral curve whoriginates at (T* ,1), whose behavior is again given by Eq~69!, ~70!. Now, however, due to the positivity of the parameterk, the non-negative functionCk(T* ) vanishes atT5T3and atT5T4, is maximal atT5T1 and atT5T2, and has apositive local minimum atT5T0.

Now consider the behavior of the trajectories in the upleft vertex of the rectangleP5(T3,1). Settings512 s̃ andT5T31T̃, with T̃ and s̃ small but not zero, from Eq.~55!we get the approximate equation

ds̃

dT̃2

s̃

T̃5As̃3/4T̃1/4, ~77!

whereA535/423/4L5/8(11k)5/8k23/8. The general solution is

s̃51

256~AT̃1B!4T̃, ~78!

where B is an arbitrary constant. WhenBÞ0, the leadingbehavior is

s̃5DT̃, D.0. ~79!

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FIG. 2. Potential and phase portrait fork50. Sincek.2 13 , there are trajectories which undergo two epochs of deceleration andepochs of acceleration.

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This means that there are trajectories which gush out fthe pointP in all possible directions, except of course vercally ~see Fig. 4!. If B50, Eq. ~78! becomes

s̃5A4

256T̃5. ~80!

This equation describes the leading behavior of a curvrwhich separates the trajectories of type~79! from those origi-nating at the points (T* ,1). We point out thatP is a cosmo-logical singularity for the curver ~which therefore originatesat P), whereas it is regular for the curves~79!. Indeed, wesee from Eq.~1! that whenT̃→0, h2 behaves as 1/T̃2 alongr, while it is finite along trajectories~79!.

Now consider the upper right vertexQ5(T4,1). By set-ting as befores512 s̃ and T5T42T̃, we get the approxi-mate equation

ds̃

dT̃2

s̃

T̃52As̃3/4T̃1/4, ~81!

whereA is the same as above. The general solution is

s̃5 1256 ~2AT̃1B!4T̃, ~82!

whereB is an arbitrary positive constant. Therefore, the tjectories enter pointQ along all possible directions excepvertically and horizontally. We now classify the behaviorthe trajectories in the interior of the rectangle. First note tthere are five distinguished trajectories~separatrices!: s,which connectsP with the node;t, which connectsP withthe saddle (T1,0); j, which is the curve which enters thsaddle (T2,0) from above@we are unable to say whetherjoriginates at some point (T* ,1) or if it belongs to the family~79!, or if it coincides with the curver defined by Eq.~80!:for this reason we have not tried to draw curver in Fig. 4#;c, which originates from the saddle (T1,0) and enters the

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node with tangent defined by the eigenvaluel25232 AL(1

2k) @see Eq.~61!#; and finally x, which connects (T2,0)with Q ~see Fig. 4!. Each of these separatrices has its on-reflected counterpart~denoted by the same symbol!.

Corresponding to the separatricess, t, j, c, andx onecan distinguish four bundles of qualitatively different trajetories.~i! The bundlef I of trajectories limited byt, c, ands: they originate fromP and enter the node alongc andfrom above.~ii ! The bundle f II of trajectories limited bys, j, andc: they enter the node alongc from below. ~iii !The bundlef III of trajectories limited byj, x, and the hori-zontal lines51; they stream intoQ. ~iv! The bundlef IV ofthe curves which are limited byt, x, and the vertical lineT5T3: they gush out ofP and stream intoQ8.

Now we have to face a problem that we have not mtioned so far. The question is the following: it takes a finproper time for the fields~and the universe! to get from any-

FIG. 3. Potential fork.0 ~herek50.44).

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where to the cornerQ or Q8 on a trajectory of bundlef III orof bundlef IV. But these corners are not critical points of tdynamical system and, furthermore, the universe doesexperience any singularity by getting there along thesejectories. Similar remarks apply to the past of the trajectooriginating from P and P8. If the model could not be extended to follow these trajectories outside the rectanwhere it has been originally defined, it would be useleHowever, we now show that this extension is actually psible. Indeed, one can see by inspection that the field etions are well defined also inside the four semi-infinite strdefined by the following inequalities: 0,T,T3, with s.1or s,21; T4,T,2p/@3AL(11k)#, with s.1 or s,21. Then, following the strategy sketched in Sec. IV, wintroduce a ‘‘new’’ Lagrangian,

L5W~T!AṪ221, ~83!

where the ‘‘new’’ potential is given by

W~T!5L

A~k11!cos2 3AL~11k!T2

21

sin23AL~11k!T

2

. ~84!

The ‘‘new’’ Lagrangian itself comes out of Sen’s action@Eqs.~3!,~31!# by the previous trick,

V~T!→W~T!5 iV~T!, ~85!

A12Ṫ2→ iA12Ṫ25AṪ221, ~86!

i.e., both the ‘‘old’’ kinetic term and the potential becomimaginary but their product remains real. It follows from thdynamics~55! that the expressions under the square roothe potential and the kinetic term change sign simuneously. In other words, in the phase diagram no trajeccan cross any side of the rectangle. Thus, all the quant

FIG. 4. Phase portrait evolution fork.0 (k50.44).

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characterizing the model stay real during the dynamical elution. White regions in the phase diagram, where thegrangian and other quantities would become imaginary,forbidden. The product~83!, which amounts exactly to the‘‘old’’ Lagrangian, can be interpreted in terms of the ‘‘newkinetic term and potential that are both real. This makeclear why the ‘‘new’’ Lagrangian gives rise to field equatioin the above four strips, which are the same as in the inteof the rectangle. At first glance, one might have the imprsion that there is a freedom of choice of sign for the nLagrangian~83!, or, in other words, that one may choosopposite signs in Eqs.~85!, ~86!. However, this is not so: thechoice of sign in Eq.~83! is determined by the requiremenof continuity of the Einstein~Friedmann! equations whenpassing from the rectangle to the stripes. As anticipatedSec. IV, both the energy density and the pressure are posin the considered strips.

However, there is an important difference betweenpresent situation and the one described in Sec. IV. Hereare dealing with one single model (k is fixed!. One is com-pelled to make the extension of the model and, contrarytheL50 case, the two different ‘‘phases’’~i.e., negative andpositive pressure! are found within the same model, at diferent stages of the cosmic evolution~one phase in the rectangle, the other in the strips!. In the following, we give theprecise mathematical meaning of this extension. This opthe way to the study of a new class of tachyon field theor

We start by describing the behavior of the trajectoriesthe lower left strip~see Fig. 4!. SinceṪ,21, the evolutionalong any given trajectory will lead us in a finite amounttime to either hit the vertical lineT50 or to approach avertical asymptoteT5TB , with 0

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neighborhood ofs521. As before, we can divide the trajectories inside the strip into three families according to thleading behavior,

s'211E~T2T3!, E.0, ~89!

s'211A4

256~T2T3!

5, ~90!

s'211C~T* !~T2T* !4, 0,T* ,T3 .

~91!

Trajectories of type~89! fan out from pointQ8 into the stripat all possible angles. Trajectories of type~91! can bethought of as originating from (T* ,21). Note that, just asbefore, the functionC(T* ) approaches~minus! infinity asT* →0. It approaches zero asT* →T3. Curve~90! separatesthe families~89! and ~91!. The coordinateTB of the asymp-tote is an increasing function ofT* . As T* →T3, it attains avalue characterizing the asymptote of curve~90!, beyondwhich it becomes an increasing function ofE.

Let us go back to physics and consider the behavior ofcosmological radiusa(t) when the tachyon fieldT(t) tendsto TB along the solution of the field equations. The Friemann equation implies thath2→0 and thatḣ→2` @see Eq.~64!#. Therefore, the scalar curvature~63! diverges and theuniverse reaches a cosmological singularity in a finite tim

This is an unusual type of singularity which we callbigbrake. Indeed, sinceä/a5ḣ1h2, in a big brake we have tha

ä→2`,

ȧ→0,

a→aB,`. ~92!

In other words, the evolution of the universe comes tscreeching halt in a finite amount of time and its ultimascale depends on the final valueTB of the tachyon field.

We now turn to the behavior of the trajectories in tupper left strip. In the vicinity ofT50, the equation for thetrajectories takes the form~87!. The coefficient ofT21 van-ishes at the values

s51, A11k, A11 1k. ~93!

As explained earlier, it is only at these values ofs that thetrajectories can leave the lineT50. In addition, an analysissimilar to the one performed earlier shows that the onlyjectory starting at (0,1) is the lines51, and that the onlytrajectory starting at (0,A11k) is the curves. All othertrajectories start from„0,A11(1/k)…. Equation~87! showsthat as T→0, the derivativeds/dT approaches2` forA11k,s,A(11k)/k, whereas it approaches1` for 1,s,A11k and s.A(11k)/k ~see Fig. 4!. To study thebehavior in the neighborhood of„0,A(11k)/k…, we sets5A@(11k)/k#1 s̃ with s̃ small. In the approximate equatio

12351

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-

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a

-

for s̃(T), it is necessary, besides the leading term, to keepterm proportional toT. Thus we get

ds̃

dT5

2~12k!

11k

s̃

T2

9L

8A11k

k S k13k DT. ~94!The general solution of this equation is

s̃5S D2 9L~11k!3/2~k13!32k5/2

T4k/(11k)D T[2(12k)]/(11k),~95!

whereD is an arbitrary constant. IfDÞ0, we have the lead-ing behavior

s'A11 1k

1DT[2(12k)]/(11k). ~96!

Therefore, ifk, 13 the trajectories start from„0,A11(1/k)…with horizontal tangent, whereas they start with vertical tagent whenk. 13 . If k5

13 , they are born with any possibl

tangent. WhenD50, we must go to the next order,

s'A11 1k

29L~11k!3/2~k13!

32k5/2T2. ~97!

This curve acts as a separatrix for the curves having posand negativeD, respectively. Regarding the curves of thstrip which lie belows, they depart from (T* ,1) and behavein the neighborhood of this point as

s512C~T* !~T2T* !4, 0,T* ,T3 , ~98!

where the functionC(T* ) is the same as in Eq.~91!. Re-garding the behavior of the trajectories in the neighborhoof T5T3, a simple analysis shows that they stream intoP atall possible angles~except vertically and horizontally!. Theseproperties show that each curve of type~96! with positiveDattains a maximum somewhere betweenT50 and T5T3whose height is an increasing function ofD which tends toinfinity as D→1`.

So far, we have analyzed the behavior of the trajectoin two distinct regions: the rectangle and the four stripNow, it has to be noted that the trajectories in the rectanwhich leaveP ~or P8) at all possible angles in the opeinterval (0,p/2) and the trajectories which enterQ8 ~or Q),again at all possible angles, are incomplete, since the verof the rectangle are not cosmological singularities for thcurves. The same is true for the trajectories in the strwhich enterP ~or P8) and leaveQ ~or Q8). This circum-stance, and the fact that the equation of motion forT is thesame in the rectangle and in the strips, indicates that it mbe possible to extend and complete the above set of tratories by continuation through the vertices of the rectangPrecisely, the trajectories enteringP ~or P8) from the upperleft ~or lower right! strip shall be continued into the trajectories entering into the rectangle fromP ~or P8). Similarremarks apply to the cornersQ and Q8. The uniqueness othis continuation procedure can be proved by applying

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s-process method of resolution of singularities~see, e.g.,@48#!. This amounts to blowing up~unfolding! the vertices ofthe rectangle by transition to suitable projective coordinathe use of which removes the degeneracy of the vector fiat these points. We do not give the mathematical details

VI. COSMOLOGY

The rich mathematical structure that we have exhibitedthe previous section gives rise to the possibility that cmologies that have very different features coexist withinsame model. In this sense tachyonic models are richerthe ‘‘corresponding’’ standard scalar field models.

We now characterize all the evolutions in our model thare portrayed in Fig. 4. We start from the trajectories orinating from the cosmological singularity„0,A11(1/k)… andwhich are characterized in the neighborhood of the latterformula ~96! with the parameterD positive and very largeOne such trajectory, as soon as it leaves the singularity, rsteeply upwards until it attains some maximum valuHenceforth it turns steeply downwards, enters the rectanat P at some small anglea with the vertical axisT5T3, andmoves towardsQ8. Upon reaching this point it enters thlower left strip and eventually ends up in a finite timetB in abig brake corresponding to some field valueTB very close toT3 ~inf tB50).

As D decreases, the height of the maximum decreaaccordingly,a andtB increase, andTB decreases. Eventuallyat some critical valueDc of the parameterD, the trajectorydegenerates inside the rectangle with the separatricest andxentering and, respectively, leaving the saddle point (T1,0).The curves for whichD.Dc belong to the bundlef

IV. ForD,Dc we get the bundlef

I. These trajectories correspondevolutions which are asymptotically de Sitter with a HubbparameterH05AL. The upper bound ofD for the curves ofbundle f I is Dc and corresponds to the separatrixt and c.This means that a tiny difference in the initial conditions wresult in dramatically different evolutions: one universe gointo an accelerating expansion of the de Sitter type andother ends with a big brake. This should not be confuwith a chaotic behavior: the two evolutions are almost indtinguishable for a very long time and then suddenly divefrom each other. AsD approaches2` we end up with theseparatrix s originating at the cosmological singularit(0,A11k).

Proceeding further, we encounter those evolutions whstart at (T* ,1), T* ,T3. In the neighborhood of the startinpoint they behave according to Eq.~98!, then enter the rectangle throughP passing above the curves from below. Inthe limit T* 5T3 we obtain the trajectoryr. As T* increasesfurther beyondT3 the curves detach themselves from taxis s51 according to Eq.~98! now with T3,T* ,T4. Atsome critical valueT

*c of T* ,0,T*

c ,T4, the trajectory de-generates into the separatricesj,c, and x. The curves forwhich 0,T* ,T*

c form the family f II and they asymptoti-cally arrive at the de Sitter node (T0,0) from below the axiss50. Those for whichT

*c ,T* ,T4 form the family f

III ;they enter the upper right strip throughQ and eventually end

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up in finite time in a big brake. Finally, the trajectories thdetach themselves from the axiss51 at T* >T4 bend againupwards and they form another bundle that we denote byf V.They also end up in a big brake.

The evolutions corresponding to the trajectories ofdifferent bundles can also be studied in terms of the quative behavior of the Hubble parameterh(t) as a function oftime ~see Fig. 5!. In our model, using Eqs.~1!, ~31!, and~84!we get

h~ t !5AL

sinS 3AL~11k!2

T~ t !D

3U12~11k!cos2S 3AL~11k!2 T~ t !D12s2~ t !

U 1/4.~99!

Since ḣ52 32 h2s2, h(t) is positive and strictly decreas

ing, except at those values oft for which s(t)50.Close to the initial singularity, the trajectories can be

vided into two classes depending on the singular point frwhich each of them starts: the class~A! of trajectories origi-nating at the point„0,A(11k)/k… and the class~B! of trajec-tories which start from the points (T* ,1). The two classesare separated by the curves. The leading behavior ofh(t) inthe neighborhood of the singularity att50 for the trajecto-ries of class~A! is

h~ t !52k

3~11k!

1

t2

Dk

3 S 11kk D(123k)/[2(11k)]

3t (123k)/(11k)1••• for DÞ0, ~100!

FIG. 5. Time evolution of the Hubble parameterh(t).

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h~ t !52k

3~11k!

1

t2

k222k19

32kLt1••• for D50.

~101!

For trajectories of class~B! the leading behavior is universa~it does not depend onk),

h~ t !52

3t1•••. ~102!

To study the behavior ofh(t) at later times, one needs texamine each bundlef separately, which, in turn, requirefirst plotting h(t) for the different separatricess, t, j, c,andx.

We have from Eq.~57! that h(t) for the separatrixs isgiven by

hs~ t !5AL cothS 3AL~11k!t2 D . ~103!In particular, in the neighborhood oft50,

hs~ t !52

3~11k!

1

t1

L~11k!

2t1••• ~104!

to be compared with Eqs.~100!, ~101!, and ~102!. Further-more, hs(t) is strictly decreasing and, ast→`, it ap-proaches a stable de Sitter expansion with Hubble paramAL.

Regarding the unstable cosmological evolutionht(t), forsmall t we have

ht~ t !52k

3~11k!

1

t2

Dck

3 S 11kk D(123k)/[2(11k)]

3t (123k)/(11k)1••• ~105!

and limt→`ht(t)5H1 @see Fig. 5 and Eq.~75!#. Qualita-tively, ht(t) behaves similarly tohs(t) but the asymptoticvalue of the Hubble parameter is higher. As for the separac, the corresponding cosmological evolution is unstablenonsingular: the Hubble parameter decreases steadilyan unstable de Sitter regimehc(2`)5H1 at large negativetimes to a stable onehc(`)5AL at large positive timeshj(t) andht(t) have qualitatively similar behaviors at largtimes while they are different at small times. Finally,hx(t)decreases steadily from an unstable asymptotic de Svalue hx(2`)5H1 at large negative times to the final bbrake singularityhx(tB)50, with ḣx(tB)52`, for somesuitabletB .

For a trajectory of bundlef I which lies close to the sepratricest andc, h(t) behaves according to Eq.~100! close tothe initial singularity, with D,Dc (Dc2D) small. Thenh(t) decreases steadily and remains close toht(t) for a verylong time during whichs ~and henceḣ) gets close to zeroTherefore, in this regime the evolution simulates

12351

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asymptotic de Sitter expansion with Hubble parameter clto the valueH1. Eventually, however,s starts increasingagain and the graph ofh(t) bends downwards away fromht(t) and approaches asymptotically the stable de Sittergime with Hubble parameterAL. Instead, for a trajectory obundle f I which lies very close tos, the behavior~100! ofh(t) at small times is characterized by a value of the consD which is negative and very large; the graph ofh(t) partsonly slightly from the graph ofhs(t) and the asymptoticvalueAL is approached much earlier. Other elements of Idisplay behaviors which are intermediate between thosescribed above. The time dependence ofh for the trajectoriesof bundlef II is qualitatively similar to the one relative to thcurves of f I, the differences being the following: the smatime behavior is given by Eq.~102!, and for each trajectoryof f II there is a valuet0 of t ~which depends on the particulatrajectory! for which s(t0)50 so thatḣ(t0)50.

Now consider a curve of bundlef III which lies very closeto the separatricesj and x. For such a curveh(t) remainsvery close tohj(t) for a very long time,s(t) decreases,getting close to zero, and the evolution simulates againasymptotic de Sitter expansion with Hubble parameter. Hoever, eventuallys(t) starts increasing indefinitely, and infinite ~though long! time the cosmology ends up in a bibrake. Moving to curves that are farther and farther awfrom j and x, the valueT* of T at the initial singularitymoves to the right towards the value 2p/@3AL(11k)#, h(t)decreases more and more steeply, and the big brake timtBtends to zero~when T* gets larger thanT4 the trajectoriesswitch from bundlef III to bundlef V).

Like those of bundlesf III and f V, the evolutions of bundlef IV likewise give rise to a big brake and the behavior ofh(t)is qualitatively similar in the two cases. However, contrarywhat happens for bundlesf III and f V, for each trajectory ofbundle f IV there is a timet5t0 ~which depends on the particular trajectory and spans the whole open half-line! forwhich s(t0)50 and henceḣ(t0)50. Therefore, even thoughthe big brake time can get as close to zero as one likes forcurves of type IV@for such curves,tB is a decreasing function of the parameterD in Eq. ~100!, with limD→DctB(D)5` and limD→`tB(D)50] and henceh(t) may decrease tozero very steeply, there is always some intermediate timwhich h(t) is momentarily stationary. Note that, as one csee from Fig. 5, there is nothing peculiar in the behaviorh(t) at the times when vertices of the rectangle are cross

The possibility of cosmological singularities characterizby the divergence of the second time derivative of the cmological scale factor has also been considered in@49# in thebrane cosmology context.

We conclude this section by another simple examplecosmology sharing this property. Let us consider theFriedmann universe filled with a perfect fluid with a staequation similar to Eq.~40!,

p5A

«, ~106!

where A is positive. We can call this fluid the ‘‘anti-

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Chaplygin gas.’’ This equation of state arises in the studythe so-called wiggly strings@50,51#. The dependence of thenergy density on the cosmological radius is given by

«5ABa6

2A, ~107!

whereB is a positive constant. At the beginning of the comological evolution«;AB/a3, as in the dust-dominatecase. Now there is a maximal value possible for the coslogical scale

aF5S BAD1/6

~108!

that is attained in a finite cosmic timetF . The behavior ofa(t) in the vicinity of the maximum is the following:

a~ t !'aF2C~ tF2t !4/3, C5227/335/3~AB!1/6.

~109!

Since ȧ(tF)50 while ä(tF)52`, we are back into a bigbrake cosmological singularity.

Thus, a big brake singularity can be found in an elemtary cosmological model~though based on an exotic fluid!.The difference with the tachyonic model is that in the latthere are evolutions culminating in a big brake which coexwith other evolutions giving rise to an infinite acceleratexpansion. These two types of evolutions correspond toferent classes of initial conditions.

It may be worth mentioning that also fork.0 one canconstruct a scalar field model; indeed the potential displain Eq. ~29! is not restricted tok,0. But this model has amuch poorer dynamics: all the trajectories tend to theSitter attractive node point.

VII. FINAL CONSIDERATIONS: FATE OF THE UNIVERSE

The study of the fate or the future of the universe is ratpopular nowadays@52–79#, and, as was predicted more tha20 years ago by Dyson@52#, ‘‘eschatology’’ has now becomepart of the cosmological studies. These studies includecourse also biological aspects of and the fate of conscioness in the different cosmological scenarios~see, for ex-ample,@52,61,80#!, but our goal is much more modest anwe shall concentrate on the purely geometrical facet oftopic.

There are three mainly studied possible scenarios forfuture of the universe, intensively discussed in the literatuan infinite expansion, an expansion followed by a contractending in a ‘‘big crunch,’’ and an infinitely bouncing or recycling universe.

The present set of observational data seems to favorfirst scenario: the data are quite compatible with a flat uverse with a positive cosmological constant.

On the other hand, a negative value of the cosmologconstant fits better with string theory~see, e.g.@22,62,63#!.The presence of a small negative cosmological conscould be made compatible with the present-day cosmic

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celeration, provided there are some fields or types of maresponsible for this acceleration. However, in this contextcosmological radius sooner or later will start decreasingthe expansion will be followed by a contraction which wounormally lead the universe to a cosmological singularitythe big crunch type.

The third scenario of an infinitely bouncing or recyclinuniverse appears to many to be very attractive becausopens an opportunity to escape the ‘‘frying’’ in the bcrunch and the ‘‘freezing’’ in the infinite expansion case.this scenario, during the process of contraction there willtwo opportunities: collapse or bounce with subsequentpansion. The choice of one of these opportunities dependthe initial conditions and this dependence has usually a cotic character@71,74–77#.

Furthermore, one can show@63# that for every model ofdark energy describing an eternally expanding universecan construct many closely related models which descthe present stage of acceleration of the universe followedits global collapse. However, these models correspondineternally expanding and collapsing universes are differand have different values of their basic parameters.

One interesting feature of our toy tachyon model is ththe first and the second scenario coexist in its context.pending on initial conditions, some correspond to eternalpansions of the universe which approach asymptoticallpure de Sitter regime, while others end their evolution atcosmological singularity.

The second distinguishing feature of this model is thatsingularity at which the universe ends its evolution is nostandard big crunch singularity. Instead, it corresponds tfinite nonzero cosmological radius at which the Hubble prameter is finite and the deceleration parameter is infinitehas a positive sign. We have called this fate the ‘‘big brakThe prospect of hitting the cosmological singularity duriexpansion has been also discussed in Ref.@57#, where thesingularity corresponds to an infinite value of the Hubbvariable and the cosmological radius. This scenario is knoas ‘‘big rip’’ or ‘‘phantom cosmology’’ ~see, e.g.@65,67–69#!.

The third distinguishing aspect of our model is the fathat the regions of the phase space corresponding to diffetypes of trajectories are well separated and the dependenthe cosmology on the choice of initial conditions is quregular~nonchaotic!. A remark is in order here: the chaoticitof the classical dynamics hinders the application of the Wapproximation and, hence, undermines the basis of thejority of results of quantum cosmology@76#. In our model,quantum-cosmological schemes of the traditional type@81–84# can be attempted. The corresponding wave functshould describe a probability distribution over different intial conditions for the classical evolution of the universe. Tquantum evolution of the universe in our toy model mightexpressed in the language of the many-worlds interpretaof quantum mechanics@85,86#. In this framework one cansay that the wave function of the universe describesquantum birth of the universe and subsequently the proc

2-14

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of the so-called ‘‘classicalization’’~see, e.g., Ref.@87#!. Dur-ing this process, the wave function splits into differebranches or Everett worlds corresponding to different psible classical histories of the universe. The peculiarity oftoy model consists in the fact that some of these brancdescribe eternally expanding universes while other branccorrespond to universes which end their evolution hittingcosmological singularity.

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ACKNOWLEDGMENTS

A.K. is grateful to CARIPLO Science Foundation andthe University of Insubria for financial support. His worwas also partially supported by the Russian FoundationBasic Research under Grant No. 02-02-16817 and byscientific school Grant No. 2338.2003.2 of the Russian Mistry of Science and Technology.

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y

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