Phantom inflation and the “Big Trip”

b

fying anpose

ooth exitrimordialWe callr

Physics Letters B 596 (2004) 16–25

www.elsevier.com/locate/physlet

Phantom inflation and the “Big Trip”

Pedro F. González-Díaz, José A. Jiménez-Madrid

Colina de los Chopos, Instituto de Matemáticas y Física Fundamental, Consejo Superior de Investigaciones Científicas,Serrano 121, 28006 Madrid, Spain

Received 16 May 2004; accepted 4 June 2004

Editor: P.V. Landshoff

Abstract

Primordial inflation is regarded to be driven by a phantom field which is here implemented as a scalar field satisequation of statep = ωρ, with ω < −1. Being even aggravated by the weird properties of phantom energy, this willa serious problem with the exit from the inflationary phase. We argue, however, in favor of the speculation that a smfrom the phantom inflationary phase can still be tentatively recovered by considering a multiverse scenario where the pphantom universe would travel in time toward a future universe filled with usual radiation, before reaching the big rip.this transition the “BigTrip” and assume it to take place with the help of some form of anthropic principle which chooses oucurrent universe as being the final destination of the time transition. 2004 Elsevier B.V. All rights reserved.

PACS:04.60.-m; 98.80.Cq

Keywords:Phantom energy; Inflation; Wormholes

theex-tiond

hentionac-

ence

usingng

ioan-isnt-

p ineif-

that

Phantom energy might be currently dominatinguniverse. This is a possibility which is by no meanscluded by present constraints on the universal equaof state[1]. If phantom energy currently dominateover all other cosmic components in the universe, twe were starting a period of super-accelerated inflawhich would even be more accelerated than thecelerating process that corresponded to the exist

E-mail address:[email protected](P.F. González-Díaz).

0370-2693/$ – see front matter 2004 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2004.06.080

of a positive cosmological constant. One could thbe tempted to look at the primordial inflation as beoriginated by a phantom field. In fact, Piao and Zhahave already considered[2] an early-universe scenarwhere the inflationary mechanism is driven by phtom energy. The main difficulty with that scenariothat it is difficult to figure out in it how the universe casmoothly exit from the inflationary period, as an inflaing phantom universe seems to inexorably ends ua catastrophic big rip singularity. Now, however, wseem to have a tool which might help to solve that dficulty. It is based on the rather astonishing effects

.

http://www.elsevier.com/locate/physletb

mailto:[email protected]

P.F. González-Díaz, J.A. Jiménez-Madrid / Physics Letters B 596 (2004) 16–25 17

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accretion of phantom energy can induce in the evotion of Lorentzian wormholes[3].

Since phantom energy violates dominant enecondition, wormholes can naturally occur[4] in a uni-verse dominated by phantom energy[5]. It can infact be thought that if Planck-sized wormholesquantum-mechanically stable and were also allowed texist in the primordial spacetime foam of a phantdominated early universe, then before occurrencthe big rip[6], the throat of the wormholes will rapidlincrease and become larger than the size of the ining universe itself, to blow up before the big rip[3].The moment at which all theoriginally Planck-sizedwormholes of the spacetime foam become simultaously infinite could thus be dubbed as the “Big Tri(Fig. 1), as a sufficiently inflated early universe wouthen be inside the throat of the wormholes and cothus be itself converted into a “time traveller” whicmay be instantly transferred either to its past or tofuture. Conditions for the Big Trip does not just ocur at the time when the size of the wormhole thrblows up, but they extend backwards, down to the tat which the wormhole throat size just overcomesphantom universe size, all the way along the timetervaltnonchronalshown inFig. 1.

In what follows we shall implement the abovpicture by using a semi-quantitative scenario whwe first describe how a set of parallel quantum uverses (which would include both the phantom uverse and a universe filled with usual radiation) canquantum-mechanically derived, and then discussway through which a time travel from the phantouniverse to the Friedmann universe with usual radtion may lead to a smooth exit from inflation takinplace in the former universe. Let us thus considerspacetime manifoldM for a little flat FRW universewith metric

(1)ds2 = −N2 dt2 + a(t) dΩ23,

whereN is the lapse function resulting from foliatinthe manifoldM, dΩ2

3 is the metric on the unit threesphere, anda(t) is the scale factor. Such a univerwill be filled with dark energy and equipped withgeneral equation of statep = ωρ, whereω is a pa-rameter which is here allowed to be time-dependWe shall then formulate the quantum-mechanicalscription of such a universe starting with its actiintegral. For this we shall take the most gene

Fig. 1. Evolution of the radius of the wormhole throat,b0, inducedby accretion of phantom energy. At timet = t , the negative ex-otic mass becomes infinite and then changes sign, so converting thwormhole into an Einstein–Rosen bridge whose associated mascreases down to zero at the big rip att = t∗ . During the time intervaltnonchronal there will be a disruption of the causal evolutionthe whole universe. The evolution of the phantom universe att = t is fully noncausal and all sorts of time travel are then allowTherefore, the moment att = t is here dubbed as the “Big Trip”.

Hilbert–Einstein action where, besides consideringscale factora(t) and the scalar fieldφ(t) to be time-dependent, and hence appearing in the actiongiven combination ofa, φ, a and φ (with ˙ = d/dt),we will generally regard the state equation paramω to be initially time-dependent, too, even thoughrestrict ourselves to the case where we always hω = 0.

Differentiating then the Friedmann equation (i.Eq. (14) below which will be in principle defined foa flat FRW universe with constantω) with respect totime one can see that, if theω in that equation is nevertheless regarded to be time-dependent then the scurvature can be generalized to a new expressioR

18 P.F. González-Díaz, J.A. Jiménez-Madrid / Physics Letters B 596 (2004) 16–25

ar.ed

venn

ing-ond

x-,

e

te-ionsoingif-

ate,e

as-in

n-

ua-e to

eldi-

-

c-

dlthen

r-eld

e

d-

othe

s-

which thereby is suggested to be given by

R = R − aω lna9

a,

where R is the conventional Ricci curvature scalIn obtaining this expression we have first assum

ρ = ρ0a−3(1+ω), letting thenω ≡ ω(t) in it. Thus,

since the scalar curvature for a flat universe is giby 6(a + a2/a2), it can be checked that differentiatioof Eq. (14)directly produces the extra term appearin the expression forR. To this generalized Ricci curvature scalar one should then associate a correspingly generalized extrinsic curvatureK . In such a casethe action integral of the manifoldM with boundary∂M can be written as

S =∫M

d4x√−g

(R

16πG+ Lφ

)

− 1

8πG

∫∂M

d3x√−hTr K

=∫M

d4x√−g

[1

16πG

(R − aω lna9

aN2

)

+ 1

2(∂µφ)2 − V (φ)

]

(2)− 1

8πG

∫∂M

d3 x√−h

(TrK − ω lna3/2

N

),

in which K is the conventional expression for the etrinsic curvature, andg and h denote, respectivelythe determinants of the general four-metricgij on M

and three-metrichij on the given hypersurface at thboundary∂M, characterized by a lapseN and a shiftNi functions. We note that (i) the above action ingral becomes the conventional Hilbert–Einstein actfor ω = const, (ii) in the surface integral we have aladded a new non-conventional extra term dependon ω which would represent a transition between dferent universes each with a fixed equation of stand together with TrK would make the trace of thgeneralized extrinsic curvature TrK , and finally that(iii) the momenta conjugate toa(t) andω(t) are notseparable from each other even when we havesumedω = 0. In the case that we consider that,principle, no particular constant value forω is spec-ified, from this action integral the Hamiltonian co

-

straint can be derived by takingδS/δN and this canthen be converted into a Wheeler–DeWitt wave eqtion by applying a suitable correspondence principlthe momenta conjugated to both the scale factora(t)

and the state equation parameterω(t). We do not in-clude here a momentum conjugate to the scalar fiφ because, ifω is also promoted to a dynamic varable andp = ωρ, then φ2 and V (φ) can always beexpressed as a single function ofω and the energy density, that is in terms of the two dynamic variablesω anda (see, e.g., Eqs.(12) and (13)given below). Using amost covenient Euclidean manifold wheret → iτ forour flat geometry, the final form of the Euclidean ation can be reduced to

I = −∫

N dτ

(−aa2

N2+ 1

2

ωaa2

N2+ 2

pωa3ρ(a,ω)

),

(3)

in which p = √8πG/3 is the Planck length, an

ρ(a,ω) (= ρ0a−3(1+ω) in the explicit simple mode

considered below) is the dark energy density. Ingauge whereN = 1 we have then for the Hamiltoniaconstraint

H = δI

δN− (1+ ω)2

pa3ρ

(4)= aa2 − 1

2aωa2 − 2

pa3ρ(a,ω) = 0,

where the term(1+ ω)2pa3ρ should be added to co

rect the effect of replacing the Lagrangian of the fiLφ for pressurep = ωρ initially in the actionS. Hadwe keptLφ = 1

2φ2 − V (φ) in the actionS, then wehad obtained Eq.(4) again by simply applying just thoperatorH = δI/δN . It is worth noticing that in thecaseω = 0, Eq. (4) reduced to the customary Friemann equation for flat geometry (i.e., Eq.(14)below),and that ifω = 0, even though Eq.(4) looks formallydifferent of Eq.(14), they should ultimately turn out tbe fully equivalent to each other once the fact thatenergy density is given by

ρ = ρ0a−3 exp

(−3

∫ω(t)

a

adt

)if ω = 0,

and by

ρ = ρ0a−3(1+ω) if ω = 0,

is taken into account. These expressions forρ(a) comefrom direct integration of the conservation law for co


ntr,

otthe

cor-ads

nt,itt

ve

rns.torn-er,heuct

de-f-

c-full

esnalingd-ing

d

-of

t

e

-

mic energy,

ρ = −3(p + ρ)a

a= −3(1+ ω)

ρa

a,

in the case that eitherω is taken to be time-dependeeven before integrating, orω = const, respectively. Fothe momenta conjugate toa andω we have, moreover

(5)πa = i

(1

2ωa2 − 2aa

), πω = i

2aa2.

In terms of the momentaπa andπω, the Hamiltonianconstraint can be cast in the form

(6)H = π2ω − a

2πaπω − 2

p

4a6ρ(a,ω) = 0.

As it was anticipated before, this Hamiltonian is nseparable in the two considered components ofmomentum space.

The necessary quantum description requires arespondence principle which in the present case leto introducing the following quantum operators

πa → πa = −i2p

∂

∂a,

(7)πω → πω = −i2p

∂

∂ω,

which, when brought into the Hamiltonian constraiallow us to obtain the following Wheeler–DeWequation[

∂2

∂ω2 − 1

2a

∂2

∂ω∂a+ 1

4−2p a6ρ(a,ω)

]Ψ (a,ω) = 0,

(8)

whereΨ (a,ω) is an in principle nonseparable wafunctional for the original universe.

Solving this equation is very difficult even fothe simplest conceivable initial boundary conditioThere could be, moreover, a problem with the operaordering, which would not be alien to the kind of quatum cosmological framework we are using. Howevif we tentatively assume for a moment that initially twave functional can be written as a separable prodof the formΨ (a,ω) = exp(−a/p)Ψ (ω), with whichthe Wheeler–DeWitt equation would reduce to

(9)

[∂2

∂ω2 + a

2p

∂

∂ω+ 1

4−2p a6ρ(a,ω)

]Ψ (ω) = 0,

then, if conservation of the total dark energyET = a3ρ

is also assumed, this differential equation wouldscribe an oscillator with a damping force with coeficient λ = a/(2p) for the squared frequencyν2 =a3ET /(42

p), which admits the general solutions:

1. ET < 1/(4a) (Overdamped regime)

(10a)Ψ (ω) = e− aω

4p

(C1e

ξ1aω

4p + C2e− ξ1aω

4p

).

2. ET = 1/(4a) (Critically damped regime)

(10b)Ψ (ω) = e− aω

4p (C1 + C2ω).

3. ET > 1/(4a) (Underdamped regime)

Ψ (ω) = e− aω

4p

[C1 cos

(ξ2aω

4p

)

(10c)+ C2 sin

(−ξ2aω

4p

)],

whereC1 and C2 are arbitrary constants,ξ1 =√1− 4ET a andξ1 = √

4ET a − 1.1

The condition Ψ = exp(−a/p)Ψ (ω), or Ψ =exp(−a2/22

p)Ψ (ω), is nevertheless too much restritive and therefore one should then have to solve theWheeler–DeWitt equation with the operatorsi∂/∂ω

andi∂/∂a that correspond to a classical momentaπω

andπa for suitable initial conditions (including, e.g.,putting upper and lower limits to the allowed valuof ω). Once one have got a suitable wave functioΨ (a,ω), one should Fourier or Laplace (dependon whether we work in the Lorentzian or Eucliean formalism) transform it into the correspondwave functional in theK-representationΦ(a, ω) (i.e.,extending also toω-space theK-representation ina-space[7] which, in the Euclidean manifold, woulbe obtained by path-integratingΨ (a,ω)exp(aK +ωω lna3/2) overa andω) to obtain a set of (presumably discrete) states for the little universe in termsallowed quantum eigenvalues of parameterω. Thus,if the parameterω of the equation of state is no

1 Had we chosen forΨ (a) a square-integrable function of th

form Ψ (a) = exp(−a2/22p) satisfying the no-boundary initial con

dition [7] we had obtained the same solutionsΨ (ω) given byEqs.(10), but replacing in these all ratiosa/p for a2/2

p , so ad-mitting a similar interpretation.


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ere

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fixed, one can always rearrange things so that ithis parameterω which can be quantum-mechanicadescribed and take on a set positive and/or negadistinct values, interpretable as being predicted by thmany-worlds interpretation of a cosmological quatum mechanics, eachω-eigenvalue describing the typof radiation that characterizes a different parallel uverse in a given multiverse scenario[8].

At this point, let us recapitulate. Variability of parameterω has been assumed as an intrinsic propaf a generic primordial universe in such a way thwhile initially the size of that universe increases,ω cantake on a set of quantum eigenvalues. The many-winterpretation of the resulting quantum cosmologiscenario is then adopted so that eachω-eigenstatewould describe a differently expanding, parallel uverse characterized by a givenω-dependent kind ofilling radiation. Finally, as the set of parallel primodial universes enters the classical evolution regithey continue to keep theω-values fixed by the ini-tial quantum dynamics, and therefore all their charteristic properties remain settled down forever, whnoncausal connections among the classical paruniverses are allowed to occur. Then, among thetinct future destinations of the time travel of a pmordial phantom universe, we shall conjecture tanthropic principle[9] will choose that time travelling which leads to a universe evolving according tothe flat Friedmann–Robertson–Walker dictum, fillwith radiation characterized by a positive parameof the cosmic equation of stateω = 1/3, which willbe assumed to correspond to one of the eigenstatthe ω-representation obtained from solving the aboWheeler–DeWitt equation. All those different destintions of the time travel of the phantom universe whtook place before the big rip, or after it, for the samsmaller than−1 equation of state parameter (the saeigenstate) in the future of the same phantom univeor all of those corresponding to the future of other native or positive values of that parameter in differeparallel universes (other than theω = 1/3 eigenstate)would be aborted by the anthropic principle, relatto observers of our present civilization.

We shall discuss next a rather simple classmodel where it will be considered how inflation cbe implemented in an early universe dominatedphantom energy according to the above lines. Thwhereas in the quantum cosmological initial era th

f

is a set of eigenstates (parallel universes), each chterized by a particular eigenvalue ofω, in the regimethat follows that initial era, we shall consider the clasical evolution of each of such parallel universes invidually. We in this way interpret the initial quanturegime as being characterized by the dynamic qutum variablesω(t) anda(t), and the subsequent clasical regime by a classicala(t) and given constanvalues ofω. This is the way through which the quatum cosmological evolution is linked to the classicscalar field evolution. In what follows, we shall firregard an early universe as being one of the abmentioned parallel universes filled with a dominatingeneric dark energy fluid with constant equationstatep = ωρ, and then particularize in the special cawhere the universe is filled with phantom energywhich ω < −1, so ensuring a super-acceleratedpansion interpretable as an inflationary phase. Letherefore take for the Lagrangian of a dark energy fiφ in a FRW universe the customary general exprsion

(11)L = 1

2φ2 − V (φ),

whereV (φ) is the dark-energy field potential and whave chosen the fieldφ to be defined in terms ofpressurep and an energy densityρ such that

(12)ρ = 1

2φ2 + V (φ), p = 1

2φ2 − V (φ).

So, for the equation of statep = ωρ with constantω,we have

(13)φ2 = (1+ ω)ρ.

Now, in our dark-energy-dominatedearly universe,Friedmann equation for a flat geometry with scale ftor a(t) and constantω can be given by

(14)

(a

a

)2

= 8πG

3ρ;

thus, by integrating the equation for cosmic eergy conservation,ρ + 3(p + ρ)a/a = 0, and usingEq.(14), we can obtain for the scale factor

a(t) =(

C + 3

2(1+ ω)t

)2/[3(1+ω)]≡ T (t)2/[3(1+ω)],

(15)

whereC ≡ C(ω) is a constant for every value ofω.From Eqs.(13) and (15)we can then derive for th


e

the

-re-ke

sles

ac-tion

ratein-

rger

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ustant

,

s ain arerse,fa-

ers-m

uldisom--asing

i-y

an-that

beal

bylue

ler, atthel

potential of the dark energy field

(16)V (φ) = 1

2ρ0(1+ ω)exp

−3

√1+ ω

ρ0φ

,

whereρ0 is the initial arbitrary constant value of thenergy density, and the scalar field is given by

(17)φ = 2

3

√ρ0

1+ ωlnT (t).

We now specialize in the phantom region. Forphantom energy regimeω < −1 andφ → iΦ [10],Eqs.(16) and (17)become

(18)V (Φ) = −1

2ρ0

(|ω| − 1)exp

3

√1+ ω

ρ0Φ

,

(19)Φ = −2

3

√ρ0

|ω| − 1lnT ′,

with

(20)T ′ = C(ω < −1) − 3

2

(|ω| − 1)t .

Using the phantom potential(18), the phantom fieldwill initially (at t = 0) be at the bottom of the potential, being driven to climb up alone the potential theafter. That evolution of the phantom field would taplace while the universe rapidly inflates according tothe lawa = (T ′)−2/[3(|ω|−1)]. The inflationary procescan only stop before reaching the big rip if wormhowith a Planck-sized throats, originally placed in thequantum spacetime foam, are allowed to exist andcrete phantom energy. As a result of such an accreprocess the wormhole throats will increase at aquite greater than the one at which the universeflates, so that the wormholes eventually become lathan the universe itself andfinally reach an infinite sizeat a finite time before the big rip given by[3]

(21)t = t∗1+ b0i

C(ω<−1)b0i

,

with b0i the initial wormhole throat radius,t∗ the big-rip time,

(22)t∗ = 2C(ω < −1)

3(|ω| − 1),

and

(23)b0i = 32 ,

4π Dρ0

in whichD is a constant of order unity. Thus, inflatioof the universe would stop due to nonchronal evotion once the causal evolution reaches a timet , beforethe big trip, at which time the wormhole throat radijust overcomes the size of the universe. For a consequation of state parameterω = −5/3, by instancethat time will be given by

t = C(ω = −5/3) −A +

√A2 − 4b2

0iC(ω=−5/3)

b0i t∗2b0i

,

(24)

where

(25)A = 1+ b0i

b0i t∗.

After t = t , the phantom universe as a whole enternoncausal phase where it can travel through timenonchronal way, back to its origin or toward its fututo get, relative to present observers in our univeinto the observable Friedmann cosmology we aremiliar with. Among all the presumably infinite numbof possible future destinations of the primordial comic time travel, which are allowed by the quantuparallel universe picture, the anthropic principle woin this way choose only that future evolution whichgoverned by the observable Friedmann scenario dinated by radiation withω = 1/3, leaving the remaining potentially evolutionary cosmological solutionsaborted possibilities, relative to present human-becivilization (Fig. 2).

Since the generalω-dependent expressions for radation temperature and energy density are respectivelgiven by [11] Θ ∝ (1 + ω)a−3ω and ρ ∝ [Θ/(1 +ω)](1+ω)/ω, and hence the temperature of the phtom universe is negative and therefore hotter thanof the host universe, time travelling from aω < −1universe to aω = 1/3 universe (both assumed todifferent eigenstates of the same quantum-mechaniccosmic wave equation, seeFig. 3) will naturally fol-low the natural flow of energy andglobally converta radiation field which was initially characterizeda negative temperature with very large absolute vaand a very large energy density, atω < −1, into a ra-diation field at positive temperature with quite smalabsolute value and quite smaller energy densityω = 1/3 (because cosmological effects dictated byeigenvalue ofω will prevail over microscopic, loca


-the

hedargu--nics

os-an-

ini-icll

n-stivei-

yal-

af-ke

v-

ich

rep-oleitself

therttomthatopt.seated,are

he

n-meredonnd

e-at-

Fig. 2. Pictorial representation of the smooth exit from the primordial inflationary process driven by phantom energy. It is based ontime-travelling of a whole, sufficiently inflated, universe from tphantom-energy dominated regimeto the usual radiation-dominateregime. Such a nonchronal process is chosen by anthropicments over all other possible cosmological final configurations allowed by the many-worlds interpretation of the quantum mechaof the early universe.

effects globally). At the same time that such a cmological conversion takes place, however, the phtom stuff shouldlocally interact with the stuff of thehost Friedmann universe, as these stuffs shouldtially preserve their microscopic and thermodynamoriginal properties locally. In fact, within a given smavolumeδV filled with phantom energy the internal eergy, δU = (1 + ω)ρδV , is definite negative and itabsolute value quite larger than that for the posiinternal energy of the radiation initially in the host unverse, within the same volumeδV . As a result fromthat initial local interaction, all of the energy initiallpresent in the host universe was annihilated, whilemost all phantom radiation remained practically unfected microscopically, but would globally behave lithough if cosmologically it was theω = 1/3 radiationinitially contained in the Friedmann universe after ha

Fig. 3. Pictorial representation of the time travel process by wha phantom universe withω < −1 gets in a universe withω = 1/3in its future. That process is carried out by using the topologyresented in upper part of this figure by which an inflated wormhwhose size has exceeded the radius of the phantom universeinserts one of its mouths in the phantom universe while its omouth is inserted in the host universe. In the inset at the boof the figure we also show the other two possible topologiesthe larger wormhole throat and the phantom universe may adOn the left it is the topology involving just the phantom univerand the wormhole, and on the right we show the most complictopology where, besides the phantom universe and the wormholethere are two extra universes in which the two wormhole mouthsrespectively inserted.

ing undergone an inflationary period, in spite of tfact that a universe withω = 1/3 by itself could neverundergo primordial inflation or reheating without icluding an extra inflaton field, which we do not assuto exist. Thus, for a current observer, the transferphantom radiation left after microscopic interactiwould just be the customary microwave backgrouradiation characterized by a parameterω = 1/3.

It follows that, relative to current observers, bsides transferring a subdominant proportion of m


eattering,ct toours,

ino-

thee isceusalrse

nts,ity

at-se’so-d atrthkeoat

-idly

olesgy.ect-. Inblackolesleft

valisroatk-y

”ob-we

mo-ntsuni-

Fig. 4. Uniformity is transferredfrom the inflated phantom universto our universe when the former universe is absorbed by the luniverse after the time travel. Before the moment of time travellour universe was completely causally disconnected with respecurrent observers. As the phantom universe was hotter thanwhen they came in contact all the uniformly distributed energyit was transferred into our universe, annihilating all its inhomgeneities and thereby disappearing.

ter, the overall neat observable effect induced onFriedmann universe when a whole phantom universtransferred by a time travel act into it would in practibe converting the relative-to-current-observers cadisconnectedness owing to the inflationless univeinto full causal connectedness of all of its componerelative to such observers, so solving any uniformand horizon problems (Fig. 4). In this way, we wouldno longer expect ourselves to be made just of the mter and positive energy created at our own univerbig bang, but rather out of a mixture of such compnents with the matter and phantom energy createthe origin of a universe other than ours. It is woremarking that for the considered time travel to taplace it must be performed before the wormhole thr

Fig. 5. Upper part: After timet = t a single wormhole is immediately converted into a Einstein–Rosen bridge whose throat rappinches off, leaving a black hole plus a white hole. These hwill then fade rapidly off by continuing accreting phantom enerLower part: The same regime for the case of a wormhole conning a phantom universe with another distinct universe like oursthis case, each of these two universes is first enclosed in a gianthole or white hole which are mutually disconnected. Then, the hfade rapidly off and finally the two disconnected universes areas the sole result of the whole process.

radius becomes infinity (that is along the time intertnonchronalin Fig. 1), since otherwise the wormholeconverted into a Einstein–Rosen bridge whose thwould immediately pinches off, leaving a pair blacwhite holes which accrete phantom energy and rapidlvanish[3] (seeFig. 5).

We shall finally notice that, relative to a “quantumobserver (i.e., an observer able to simultaneouslyserve all the eigenuniverses), the whole processhave just discussed violates the second law of therdynamics. In fact, if we disregard the matter conteof the universes, then the entropy of each of theseverses is a given universal constantσ [11], and there-fore the total initial entropy would benσ , with n the


sedto

dourd inionse.t toav-tiallytivech aclea

iteandra-ndedeseure

in-is-

n-

inrto

w-

, therms

een

ell,y-

-

f

w-it-selith

-

B

-

d-n-ornly

number of eigenuniverses. After the above discustime travel, the total entropy would decrease downa value(n − 1)σ . Moreover, a violation of the seconlaw would also take place relative to an observer inuniverse due to the increase of coherence inducethe host universe by exchanging its original radiatcontent for that carried into it by the phantom univerThese violations of the second law can be thoughbe the consequence from the fact that any stuff hing phantom energy must be regarded as an essenquantum stuff with no classical analog, where negatemperatures and entropies are commonplace, suhas been seen to occur in entangled-state and nuspin systems.

The whole scenario discussed in this Letter is quspeculative and will of course pose some problemsdifficulties which would deserve thorough considetion. The most important problem refers to density agravitational wave fluctuations which are originatin the phantom inflationary epoch, and the way thpredicted fluctuations compare with the temperatfluctuations currently observed in WMAP CMB[12].Though we do not intend to perform a completevestigation on such subjects in this Letter, a brief dcussion appears worth considering. From Eqs.(15),(18)–(20), it can be shown that the slow-climb coditions for our phantom inflation model are

εph = − H

H 2 = 3

2

(|ω| − 1) 1,

(26)δph = − Φ

ΦH= εph 1,

whereH = a/a. When conditions(26) are satisfiedthe scale factor will approximately evolve with timean exponential, de Sitter like way, while the parameteω < −1 must take on values which are very close−1. Metric perturbations can then be studied folloing, e.g., the procedure in Ref.[2]. Thus, in the longi-tudinal gauge and absence of anisotropic stressesscalar metric perturbations can be expressed in teof a perturbed metric as a function of both the Bardpotential,VB , and the conformal time

η =∫

dt

a(t), with a(η) =

[−(|ω| − 1)

η

2

] −22(|ω|−1)

.

As expressed in terms of the Bardeen potential as wthe curvature perturbation on uniform comoving h

sr

persurface is then given by

(27)ς = VB + (3|ω| − 1)ηV ′B − 2VB

3(|ω| − 1),

where′ = d/dη. It follows that the differential equation of motion for the perturbationς can be writtenas

(28)ϑ ′′k +

(k2 − 2(1+ 3|ω|)

(3|ω| − 1)2η2

)ϑk = 0,

where

ϑ = a2Φς/a′ = zς,

with z = √ρ0(|ω| − 1)

(−(3|ω| − 1)

η

2

) −23|ω|−1

.

Eq.(28)admits an exact analytical solution in terms othe Bessel functionB [13]:

(29)ϑk = η1/2B 3(1+|ω|)2(3|ω|−1)

(kη) η1/2B 32+εph

(kη),

the last approximate equation being valid if the sloclimb conditions are satisfied. The inclusion of suable initial conditions will choose the precise Besfunction we should use. The amplitude associated wthis perturbation can then be given by[2]

(30)As = 2

3(|ω| − 1)(T ′)2k=aH

,

while the spectral index becomes

ns = 3

2

(5|ω| − 3

3|ω| − 1

).

The amplitude for tensor perturbation can be analogously derived. It reads

(31)At =(

1

T ′

)2

k=aH

,

with a spectral indexnt −3(|ω| − 1).The ratioAt/As = 3(|ω| − 1)/2 εph should now

be compared with the tensor/scalar ratio of CMquadrupole contributions,r ∼ εph, to low order. Ourphantom scenario withεph = δph and exponential potential differs from other phantom inflation models[2],and a thorough join comparation of all of these moels with the usual scalar fieldinflationary scenarios othe r−ns plane is left for a future publication. Actually, just like it happens with scalar field models finflation, the phantom inflation scenarios could o


thethe

fulaso.

ces

, in

5;tt.

v.

rares,

hen

ci-

04)

.

al

be taken seriously if they succeeded in predictingtypes of temperature fluctuations discovered fromWMAP CMB data.

Acknowledgements

The author thanks Carmen L. Sigüenza for useinformation exchange and comments. This work wsupported by DGICYT under Research Project NPB97-1218.

References

[1] J.S. Alcaniz, Phys. Rev. D 69 (2004) 083521, and referentherein.

[2] Y.-S. Piao, Y.-Z. Zhang, astro-ph/0401231.[3] P.F. González-Díaz, astro-ph/0404045, Phys. Rev. Lett.

press.[4] M. Visser, Lorentzian Wormholes, AIP, Woodbury, NY, 199

S. Nojiri, O. Obregon, S.D. Odintsov, K.E. Osetrin, Phys. LeB 449 (1999) 173.

[5] P.F. González-Díaz, Phys. Rev. D 68 (2003) 084016.[6] R.R. Caldwell, Phys. Lett. B 545 (2002) 23;

R.R. Caldwell, M. Kamionkowski, N.N. Weinberg, Phys. ReLett. 91 (2003) 071301;P.F. González-Díaz, Phys. Rev. D 68 (2003) 021303;J.D. Barrow, Class. Quantum Grav. 21 (2004) L79;M. Bouhmadi, J.A. Jiménez-Madrid, astro-ph/0404540;S. Nojiri, S.D. Odintsov, Phys. Lett. B 595 (2004) 1.

[7] J.B. Hartle, S.W. Hawking, Phys. Rev. D 28 (1983) 2960;S.W. Hawking, Phys. Rev. D 32 (1985) 2489;S.W. Hawking, Quantum cosmology, in: B. DeWitt, R. Sto(Eds.), Relativity Groups and Topology, Les Houches LectuNorth-Holland, Amsterdam, 1984.

[8] See the contributions in: B.S. DeWitt, N. Graham (Eds.), TMany-Worlds Interpretation of Quantum Mechanics, PrincetoUniv. Press, Princeton, NJ, 1973.

[9] J.D. Barrow, F.J. Tipler, The Anthropic Cosmological Prinple, Clarendon, Oxford, UK, 1986.

[10] ] P.F. González-Díaz, Phys. Rev. D 69 (2004) 063522.[11] P.F. González-Díaz, C.L. Sigüenza, Phys. Lett. B 589 (20

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Functions, Dover, New York, 1965.

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Phantom inflation and the “Big Trip”