Recycling the universe using scalar fields

9 Pages·0.28 MB·English

by  Nissim Kanekar

Recycling the universe using scalar fields
Nissim Kanekar1,⋆, Varun Sahni 2,† and Yuri Shtanov 3,‡
1National Centre for Radio Astrophysics, Pune 411007, India
2Inter-University Centre for Astronomy & Astrophysics, Post Bag 4, Pune 411007, India
3Bogolyubov Institute for Theoretical Physics, Kiev 03143, Ukraine
thermodynamicsnecessarilyrequireauniversewhichwas
Weexamine thebehaviourof a closed oscillating universe
created at a finite time in the past and which is fated
filledwith ahomogeneousscalar fieldandfindthat,contrary
for stagnation and death in the future” [5]. Thus, the
to naive expectations, such a universe expands to larger vol-
oscillating universe was seen to present a credible alter-
umesduringsuccessiveexpansionepochs. Thisintriguingbe-
nativeto the idea ofthe thermodynamicheatdeathpos-
haviourintroducesanarrowoftimeinasystemwhichistime-
tulated by nineteenth century physicists and popular in
1 reversible. The increase in themaximum size of the universe
0 isclosely relatedtotheworkdoneon/bythescalar fielddur- this century as well. (The latter may still be possible in
0 a flat/open universe.)
ing one complete oscillatory cycle which, in turn, is related
2 (ii)AsdemonstratedbyZeldovichandNovikov[6],the
to the asymmetry in the scalar field equation of state dur-
increase in entropy from cycle to cycle suggests that an
n ing expansion and collapse. Our analysis shows that scalar
a fields with polynomial potentials V(φ) = λφq, q > 1 lead to oscillating universe could not have had an infinitely long
J agrowing oscillation amplitudefortheuniverse: theincrease totaldurationsince,giventhepresent(finite)valueofits

5 in amplitude between successive oscillations is more signifi- total entropy and postulating that the entropy increase
2 cant for smaller values of q. Such behaviour allows for the fromcycletocycleisfinite,oneisledtoconcludethatthe
effectiverecyclingoftheuniverse. Arecycleduniversecanbe numberofcyclespreceedingthepresentoneisalsofinite.
1
quiteold and can resolve the flatness problem. These results An oscillating universe cannot therefore be infinitely old
v
havestrongbearingoncosmological modelsinwhichtherole and must have been created, perhaps quantum mechan-
8
of dark matter is playedby ascalar field. They are also rele- ically, at some point in the past. Since the total mass
4
4 vantforchaoticinflationarymodelsoftheearlyuniversesince (energy) of a closed universe is zero, its creation from
1 theydemonstratethat,eveniftheuniversefailstoinflatethe the vacuum does not violate any known laws of conser-
0 first timearound, it will eventually doso duringfutureoscil- vation and is therefore possible, in principle [7].
1 latory cycles. Thus,thespaceof initialconditions favourable (iii) An important consequence of an oscillating uni-
0 for chaotic inflation increases significantly. verse with an increasing expansion maximum at every
/
h cycle,is thatthe horizonandflatnessproblemsaregrad-
p uallyamelioratedastheuniversegrowsolder,larger,and
- I. INTRODUCTION flatterduringeachsuccessiveexpansioncycle. Theoscil-
o
r latinguniversemaythuspresentacrediblealternativeto
t The idea of a cyclical oscillating universe—one that the inflationary scenario in this respect.
s
a is continuously reborn from the ashes of a previous An oscillating universe can also have other impor-
v: existence—finds expression in the philosophical and cul- tant cosmological implications. For instance, relics of
i tural beliefs of many ancient civilizations [1–4]. anearlierexpansionepochmaybe measurabletoday[8].
X
Within the framework of modern relativistic cosmol- Starobinsky, in his seminal paper on graviton produc-
r ogy, oscillating models (analytically continued through tion in an inflationary universe [9], showed that gravity
a
the big bang singularity) arise naturally as exact solu- waves would also be created in an oscillatory universe;
tions ofthe Einsteinfieldequations fora spatiallyclosed further,hederivedanexpressionfortheiramplitudeand
universeconsistingofaperfectfluid. Sinceallexpansion- spectrum which allows us to infer details of previous ex-
contractioncyclesinsuchmodelsareidentical,onemight pansion epochs from measurements made today.
feel that an oscillating universe containing an infinite Finally,the ideaofanoscillatinguniversealsoappears
number of cycles would be infinitely old, somewhat re- in the quasi-steady-state cosmology of Hoyle, Burbidge,
sembling, on the average,a steady-state model. Dissipa- and Narlikar [10,11], in which alternate cycles of expan-
tiveprocessesleadingtoentropygrowth,however,change sionand contractionmodulate an exponentially expand-
thispictureradically. AsoriginallydemonstratedbyTol- ing background geometry, with the creation of matter
man[5],thegrowthofentropyincreasesthetotalvolume being most intense at the minimum of each cycle.
of the universe at the maximum of each expansion cy- A key issue, which remains at present unresolved, re-
cle;this observationhasseveralimportantconsequences, lates to physical mechanisms which can cause the uni-
some of which are summarised below. verse to bounce at the end of each cycle. The singu-
(i)Tolmanstronglyfeltthatthe possibility ofthermo- larity theorems of Penrose and Hawking suggest that a
dynamically recycling the universe would have a “liber- singularstatenecessarilyarisesatthe endofcontraction
alizing action on our general thermodynamic thinking” inaclosedFriedman–Robertson–Walker(FRW)universe
since it would dispel the notion that “the principles of
1
described by general relativity, if matter satisfies cer- II. SCALAR FIELDS IN A CLOSED UNIVERSE.
tain ‘energy conditions’ [12]. However, it is not clear
whether, at very high curvatures l−2, matter would
P
R∼
behave as it does under ‘normal’ conditions when the
space-time curvature is much smaller than the Planck
value, lP−2. Furthermore, at lP−2, particle a[t]
R ≪ R ∼
production and vacuum polarisation effects are likely to
be significant; the resulting vacuum expectation value of
the energy-momentum tensor T need not satisfy the
ik
h i
above energy conditions, due to which the space-time
metric could very well ‘bounce’ without ever reaching a
singular state [6,13]. Finally, we should mention that a
‘bouncing’ universe may also arise in quantum cosmol-
ogy or as a consequence of the duality conditions which
t
are generic features of superstring-inspired cosmological
models. The limiting curvature hypothesis of Markov
[14] (see also [15]) might also lead to a time symmetric
bounce. Other physical mechanisms leading to singular-
ity avoidance are discussed in [16,17].
The bouncing of the universe in the vicinity of the
singularity, a = 0, is perceived less dramatically by ex-
tending the range of the scale factor to negative values.
Since only the absolute value of the scale factor, a, has FIG.1. The‘bounce’ata=0,t=0(solidline)isreplaced
| |
direct physicalmeaning, such anextensionis quite legit- by a continuous transition of the scale factor through a = 0
imate(see,forinstance,[18]andFig.1). Thenotionofa towards negative values of a (dotted line). Both representa-
‘bounce’isthenreplacedbythemorenaturalnotionofa tions are physically equivalent since only the absolute value
continuous‘passage’ofthescalefactorthroughthevalue of a hasphysical significance.
ofa=0to the regionofnegativevaluesa<0. This also
canbe viewed as the changeof the (unphysical?) spatial The recent detection of anisotropy in the cosmic mi-
orientationof the universe after its passagethroughzero crowave background on degree scales appears to favour
value of the scale factor [19]. We will use this notion in a closed FRW universe with Ωtotal =1.11 0.07 [20]. A
±
ourformulationof‘natural’conditionsatthebounce. Of spatially closed space-time is described by the metric
course,the dynamics of the universe in the close vicinity
dr2
of the point a = 0 is beyond classical description, even ds2 =dt2 a2(t) +r2dθ2+r2sin2θdφ2 , (1)
in this model. − 1 r2
(cid:18) − (cid:19)
Thispaperpresentsaradicaldeparturefrommostpre-
where a(t)is the cosmic expansionfactor;units inwhich
vious work on oscillating models in which entropy pro-
the speed of light c = 1 are used throughout this paper.
duction associated with dissipative processes led to the
In a closed universe, the Einstein equations acquire the
growthoftheexpansionmaximumofeachcycle. Ourre-
well-known form
sults show that an alternate mechanism which does not
requireentropyproductionexists,alsoleadingtoincreas- 2
a˙ 8πG 1
ing oscillatorycycles. We demonstratethat the presence = ρ , (2)
a 3 − a2
of a massive scalar field (in a closed FRW universe), un- (cid:18) (cid:19)
der certain reasonable conditions at the bounce, gives
rise to growing expansion cycles, the increase in expan- a¨ 4πG
sionamplitude being relatedtothe workdone by/onthe = (ρ+3P), (3)
a − 3
scalar field during the expansion/contraction of the uni-
verse. (Thepresenceofothermatterfields,inadditionto where ρ and P are,respectively,the totalenergydensity
thescalarfield,doesnotaffectourconclusions,aslongas and pressure of the various components of the universe.
interactions between such fields and the scalar are suffi- For a radiation-dominated universe, we have P = ρ/3,
ciently weak.) Our results have important consequences and the field equations can be solved exactly in terms of
fortheinflationaryuniversescenario,oneofthestandard theconformaltimecoordinateη = dt/a(t),toyieldthe
paradigms of cosmology. solution
R
a(η)=Asinη, t=A(1 cosη), (4)
−
which describes a semicircle in the a–t plane. The solu-
tion for a matter dominated universe (with P =0) is
2
a(η)=A(1 cosη), t=A(η sinη), (5) the asymmetry in pressure leads to a significantincrease
− −
intheenergyofthescalarfieldandresultsinanincrease
which describes a cycloid. After solution (4) is extended of the maximum volume of the oscillating universe.
periodicallyin time, both(4)and(5)describe a periodic The Lagrangiandensity for a scalar field has the form
evolutionwithaninfinitenumberofidenticalexpansion–
contraction cycles, as shown in Fig. 2. 1
= gij∂ φ∂ φ V(φ). (7)
i j
L 2 −
The 0-0Einsteinequationfora homogeneousscalarfield
5
in the closed FRW universe (1) becomes
8πG 1 1
4 H2 = φ˙2+V(φ) , (8)
3 2 − a2
(cid:20) (cid:21)
and the energy density and pressure of the scalar field
3 are, respectively,
]
t
a[ ρ 1φ˙2+V(φ), P 1φ˙2 V(φ). (9)
φ φ
2 ≃ 2 ≃ 2 −
The scalar-field equation of motion is
1 dV
φ¨+3Hφ˙+ =0. (10)
dφ
0 The motion of φ arises in response to the dual action of
0 5 10 t 15 20 25 theacceleratingforcedV/dφandthedampingterm3Hφ˙.
In the chaotic inflationary scenario with V φn (n =
FIG. 2. The cycloid solution for a matter dominated uni-
∝
verse. Note that the amplitude of the cycloid does not in- 2,4),forsufficientlylargevaluesofφ,namely,forφ>mP,
crease from one cycle to thenext. where mP is the Planck mass, the damping caus∼ed by
the expansion of the universe (H > 0) settles the scalar
The presence of a bulk viscosity ζ results in a modifi- field into a ‘slow-roll’regime during which 1φ˙2 V and
2 ≪
cation of the cosmologicalfluid pressure P to P ρ. Since the equation of state of the scalar field
≃ −
mimics that of the cosmological constant, the expansion
P =P0 3ζH. (6) ofthe universeis inflationary,a exp H(t)dt. Exactly
− ∝
the reverse situation arises when the universe contracts.
Here, P0 is the equilibrium pressure and H = a˙/a is In this case, we have H < 0, and thRe term 3Hφ˙ now
the Hubble parameter. (An example of bulk viscosity is
accelerates the motion of φ instead of damping it, as it
provided by a fluid in which energy is easily exchanged
did during expansion. As a result, the kinetic energy of
between translational and rotational/internal degrees of
the scalar field becomes much larger than its potential
freedom, an example being a gas of rough spheres [21].) energy, 1φ˙2 V, and the resulting equation of state
FromEq.(6)onefindsthat,duringexpansion,H >0and 2 ≫
becomesP =ρ(sometimescalledtheequationofstateof
P < P0, whereas during collapse, H < 0 and P > P0. ‘stiff’ matter). Consequently, the scalar field in a closed
This asymmetry during the expanding and contracting
universe satisfies two generic regimes [22]
phases results in the growthof both energy and entropy,
in the words of Tolman [5], “if the pressure tends to be
P ρ for H >0 (expansion), (11)
greater during a compressionthan during a previous ex- ≃−
pansion,aswouldbe expectedwith alagbehind equilib-
riumconditions,anelementoffluidcanreturntoitsorig- P ρ for H <0 (contraction). (12)
≃
inal volume with increased energy and hence also with
increased entropy.” The increase in energy makes the For polynomial potentials, V φn, these two regimes
∝
amplitude of successive expansion cycles larger enabling are separated by an epoch during which the scalar field
theuniversetospend“agreaterandgreaterproportionof oscillates about its minimum value while its equation of
its periodinaconditionoflowerdensity...eventhough state mimics that of dust: P =0 (n=2), orradiation:
h i
a return to higher densities would always occur.” P = ρ /3 (n = 4). (The time average is taken over
h i h i
Although Tolman linked the asymmetry in pressure many oscillations of the field.) It is interesting that the
during expansion and collapse to the production of en- quantityP/ρ,whenplottedasafunctionoftheexpansion
tropy, we will show that such an asymmetry also arises factor, resembles a hysteresis curve. The area enclosed
for non-dissipative Lagrangians such as those describing bythe curveisrelatedtotheworkdoneby/onthe scalar
amassivescalarfieldinanFRWspace-time. Inthiscase, field during the expansion of the universe δW = PdV.
H
3
If one postulates that the universe ‘bounces’ during In the introduction, we mentioned the possibility of ex-
contraction, then one can expect the ‘work done’ δW tending the range of the scale factor to negative values
during a given expansion cycle to be converted into ‘ex- and then replacing the idea of a ‘bounce’ by the some-
pansion energy’, resulting in the growth in amplitude of what more appealing idea of ‘passage’to negative values
each successive expansion cycle. An estimate of the in- of the scale factor. This viewpoint leads to the following
crease in the expansion maximum can be obtained from reasonable conditions at the bounce: a a, a˙ a˙,
the following elementary considerations: setting H = 0 φ φ, φ˙ φ˙, which is equivalent to th→e ‘c−onventi→onal’
in (2) we obtain pr→escriptio→n a a, a˙ a˙, φ φ, φ˙ φ˙ adopted
→ → − → →
inthis paper. Further,we assumethatthe bounce arises
3 at scales at which quantum gravitational effects become
ρ = , (13)
∗ 8πGa2max important, i.e., at Planck scales. Two possibilities then
exist for the location of the bounce:
where ρ is the density of the universe at the expansion
∗ (i) The bounce occurs when the curvature of the uni-
maximum. Since the total mass of a closed universe is
verse becomes comparable to the Planck scale, in other
M =2π2a3ρ, this gives
words, the bounce takes place at a fixed value amin lP
≃
of the scale factor.
3π
M = amax. (14) (ii) The bounce occurs when the energy density in the
4G scalar field crosses the Planck energy, i.e., when 1φ˙2 +
2
Equating the increase in energy, δE = δM, to the work V(φ) m4.
P
≃
done during a single expansion-contraction cycle, δE = Both the above possibilities are considered in the cur-
δW, we obtain rent work, with results shown in Figs. 3 and 4, respec-
tively, for the boundary conditions (i) and (ii), and for
∆amax = 4G PdV , (15) thepotentialV(φ)= 21m2φ2. InFig.3,wefollowtheex-
3π pansion factor through one expansion-contraction cycle;
I
it can be seen that the universe inflates at the beginning
which relates the increase in the expansion maximum
ofthesecondcycle,resultinginaverylargeuniverseand
between two successive cycles ∆amax to the work done
a correspondingly large value of amax.
δW = PdV. (The increase in energy δE is brought
Next,inimplementingcondition(ii),wewillworkwith
about at the expense of the gravitational field energy,
H what we feel is the ‘worst case scenario’ for inflation,
since the total energy of a closed universe is identically
namely, we set
zero [5,23,6].)
We probe further implications of this analysis by 1
φ˙2 m4 V(φ) , (16)
scthuadoytiicngpeoqteunattiiaolnssV(8()φ–)(1=0) λnuφmq,erqica>lly0,forantdheVc(laφs)s o=f 2 (cid:12)a=amin ≃ P ≫ |a=amin
(cid:12)
V0 exp( µφ). The potential V(φ) φq has been dis- (cid:12)
− ∝
cussedbothinthecontextofinflation[24]andasacandi- φ|a=amin <φ∗ ≃mP Ω/6π (17)
dateforcolddarkmatter[25]. Exponential-basedclasses
p
of potentials are known to have important cosmological atthestartofthefirstexpansioncycle. NotethatV(φ)=
consequencesbothwithintheinflationaryframework[26] 1m2φ2 and Ω=8πGρ /3H2 >1.
2 φ
and as candidates for quintessence and cold dark mat- Theinitialconditions(16)and(17)guaranteethatthe
ter [27]. (An early use of the exponential potential in condition for accelerated expansion φ˙2 < V(φ) is not
the context of a closed universe may be found in [28].) met and thus ensure that the inflationary regime does
Before describing our results, it is necessary to first dis- not commence. Our results, shown in Fig. 4, are inter-
cussthetypeof‘bouncing’conditionswhichareimposed esting; we find that, although the universe is prevented
at the start of each expansion-contraction cycle. As re- frominflatingduringthefirstcycle,itsamplitudeduring
marked earlier, it is likely that a unified theory of the subsequent cycles increases,resulting finally in inflation!
weak, strong, and gravitational interactions will provide The physical reasons for a growing oscillation amplitude
a key to understanding physical processes which oper- are simple to understand: the largekinetic energy of the
ate during the strong-curvature regime of a contraction scalar field during the collapse phase of a cycle drives φ
cycle. In the absence of such a theory, we will do the to regions higher up on the potential at the commence-
nextbestthingandsetconditionswhichwefeelareboth ment of the next expansion cycle, until the field finally
simple and natural and may therefore arise in a future reaches an amplitude φ > φ , which is large enough to
∗
‘physically complete’ theory of the early universe. make the universe inflate.
The conditions imposed at the bounce are, a a,
a˙ a˙, φ φ, and φ˙ φ˙, (the bounce is assum→ed to
→ − → →
occur instantaneously). Thus, the effect of the bounce is
to reverse the direction of motion of the universe (since
a˙ a˙),withallotherquantitiesremainingunchanged.
→ −
4
videdtheuniversebounceswhenitreachesahighdensity.
25 (It should be pointed out that, for scalar fields originat-
ing fromlargerinitial valuesφ|a=amin >φ∗, there will be
greater hysteresis in the equation of state and a larger
20 associated growth in the volume of the universe at max-
imum expansion.)
Wethusfindthatagrowthinthemaximumexpansion
15 amplitudecanbeachievedwithoutanyrecoursetoanen-
tropy generating mechanism. In fact, the field equations
n[a] (8)–(10) are non-dissipative and time-reversible; more-
l
over, the bouncing conditions imposed are also time-
10
reversible, in the sense that the time-reversed evolution
will respect the same bouncing conditions. One there-
fore arrives at the following important conclusion: time-
5
asymmetry in the evolution of a closed universe can be
achieved even with time-reversible field equations and
bouncing conditions! This result may appear counter-
0
0 5 10 15 20 25 intuitive, especially when compared with the behaviour
ln[t]
of the periodic solutions (4) and (5), which are exactly
FIG. 3. The expansion factor for an oscillating universe
time-symmetric withrespectto the expansionmaximum
satisfying theboundary condition (i).
of each cycle. The reason behind the growing ampli-
tude of successive expansion cycles has to do with the
fact that convex potentials (which have so far been ex-
amined) have a well defined minimum about which the
20
scalar field oscillates. Rapid oscillations of φ with fre-
18 quency m d2V/dφ2 H cause the field to ‘mix’ in
≡ ≫
its phase-space φ,φ˙ so that, at the time of recollapse,
16 its location in {φ,φ˙ }is almost completely uncorrelated
{ }
14 with its location in φ,φ˙ before the onset of oscilla-
{ }
tions. As a result, the chance that the scalar field will
12
roll up its potential during recollapse along exactly the
n[a] 10 same trajectory down which it rolled during expansion
l is essentially zero. This accounts for the fact that the
8 equation of state during contraction is P = ρ and not
P = ρ, which it would have been for exactly time-
6
−
symmetricexpansion–collapse. Thetime-asymmetricbe-
4 haviour of φ is clearly shown in Figs. 5 and 6, in which
the solid line showing the equation of state P/ρ during
2
expansion is displaced with respect to the broken line
0 showing P/ρ during collapse. For exact time reversal,
0 5 10 15 20
ln[t] nosuchdifferencebetweenexpansionandcollapsewould
FIG. 4. The expansion factor for an oscillating universe havebeenpresent;thesolidandbrokenlineswouldover-
satisfying the boundary condition (ii). Note the monotonic lap, and the ‘hysteresis’ seen in the figures would not
increase in successive expansion maxima (plotted on the log- occur. The global time-asymmetry of the evolution of
arithmic scale). an oscillatory universe in our model is also a direct con-
sequence of the time-asymmetric physical conditions at
The hysteresis curves corresponding to the first oscil- the bounce, in which the sign of a˙ is reversed while the
lationofthe expansionfactorin Figs. 3 and4 areshown sign of φ˙ is not. It should be noted at this point that,
in Figs. 5 and 6, respectively. Suprisingly, we find that had we adopted the bouncing condition φ˙ φ˙ for
→ −
the phenomenon of ‘hysteresis’ is present even when the the scalar-field velocity, then every two subsequent cy-
conditions (11) and (12) are no longer met. The result- cles would obviously be time-symmetric with respect to
ing increase in oscillation amplitude emphasises the fact the corresponding bouncing point, and the cosmological
that, although the universe may not inflate during its evolution as a whole would thus be periodic with period
‘firstattempt’, itwilleventuallydosoduringsubsequent comprising two subsequent cycles.
cycles! This result considerably enhances the scope and
appeal of inflationary models by showing that inflation
could beginfrom a much broadersetof initial data, pro-
5
periodic andhysteresisis obviouslyabsent. Interestingly
1 enough,hysteresis reappearswhen we modify the poten-
0.8 tial to V = V0[cosh(λφ)−1], which has been proposed
as a candidate for cold dark matter in [29]. This poten-
0.6 tial has the following limiting forms: V exp( λφ) for
λφ 1,φ<0;andV λ2φ2for λφ ∝1. Int−hiscase,
0.4 | |≫ ∝ | |≪
althoughtheamplitudeofsuccessivecyclesincreases,the
0.2 increase is much smaller than for a purely λ2φ2 poten-
tial. This observation supports our conclusion that hys-
w[t] 0 teresis, and the accompanying increase in the amplitude
of successive expansion maxima, depends crucially upon
-0.2
the ability of the scalar field to oscillate. The absence
-0.4 of a well-defined minimum in the exponential potential
preventsthe fieldfromoscillating,its behaviouris there-
-0.6
foremonotonicand,intheabsenceof‘mixing’,thescalar
-0.8 field equationof state P/ρ does not show any hysteresis.
Forpotentials characteristicofchaoticinflation,V φq,
-1 ∝
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 q > 0, the extent of ‘hysteresis’ and the accompanying
ln a[t]
increase in successive expansion maxima depend sensi-
FIG.5. Thescalar-fieldequationofstateω=P/ρisshown
tively upon the exponent q—reflecting the steepness of
for the first cycle of Fig. 3. Solid/broken lines correspond the ‘chaotic’ potential φq. For larger q, the amount of
to the expanding/collapsing epochs. Note that during the
hysteresis is smaller, as is the increase in the value of
pre-oscillation and post-oscillation epochs, the behaviour of
successive maxima of the expansion factor.
P/ρ resembles a hysteresis curve, which contributes signifi-
It is interesting to follow the value of the cosmological
cantly to the total ‘work done’ δW = PdV.
density parameterΩ as the universe oscillates. Fromthe
H definition
Ω 1=(aH)−2, (18)
1 −
we find that a larger value of the expansion factor a (at
0.8
identical H) will result in a value of Ω which is closer
0.6 to unity (during successive cycles). This is borne out by
the results of our numerical analysis shown in Fig. 7, in
0.4
which the value of Ω measured at identical values of t
0.2 tmin H−1isshownforsuccessiveexpansion-contractio−n
∼
cycles [the boundary condition is (ii); the corresponding
w[t] 0 expansion factor is shown in Fig. 4]. We find that Ω
approaches unity more closely and for a longer duration
-0.2
during successive cycles thereby gradually ameliorating
-0.4 the flatness problem. (During the third expansion cycle,
the universe inflates and Ω approaches unity to great
-0.6
accuracy.)
-0.8
-1
0 0.5 1 1.5 2 2.5 3
ln a[t]
FIG.6. The hysteresiscurvecorresponding tothefirst cy-
cle of Fig. 4.
Theaboveargumentissupportedbyouranalysisofex-
ponential potentials V =V0exp( µφ) for which we find
−
nohysteresisintheequationofstateP/ρandnoincrease
in the amplitude ofthe expansionfactor throughsucces-
siveoscillations. Theexampleofanexponentialpotential
iseasytoanalysebecausetheevolutionofthescalarfield
proceedswithtimetowardslargepositivevaluesofφand,
atsuchvaluesofthescalarfield,thepotentialapproaches
anidenticalzero—atrivialcaseforwhichtheevolutionis
6
Ωtotal = 1.11 0.07 [20]. By substituting this into (20),
10 ±
we get 258 < V/V < 2467. The total volume of our
H
universecouldthereforebeseveralthousandtimeslarger
than its causal horizon! We also obtain an estimate of
8 thevalueoftheexpansionfactoratrecollapse(amax)and
the correspondingrecollapse‘redshift’ (zrecoll) by setting
to zero the (future) value of the Hubble parameter [27]
6
1/2
H(z)=H0(1+z) 1 Ωtotal+ Ωα(1+z)γα ,
" − #
4 α
X
(21)
2 whereΩα isthe fractioncorrespondingtomatteroftype
α in Ωtotal, Ωtotal = αΩα, γα = 1+3wα, and 1+z =
a0/a. Specialising to a CDM dominated universe and
0 assumingforsimpicitPythatall thematterintheuniverse
0 5 10 15
ln [t] is pressureless so that Ωtotal =Ωm, we get
(ΩF→IG∞. 7.asTthheeudneinvseirtsyerpeacroalmlaeptseers)Ω. Naosteathfuantctthioentimofetaimxies H(z)=H0(1+z)[1+Ωmz]1/2 =0, (22)
is given in logarithmic units; the oscillations seen (in Ω) in
the first cycle are thus not visible in the later cycles due to which leads to zrecoll =−1/Ωm or
thecompression of thescale.
amax Ωm
= . (23)
The time derivative of equation (18) yields a0 Ωm−1
Substituting Ω = 1.11 0.07, we ob-
m
±
Ω˙ =HΩ(1 Ω)[1+3ω(t)], (19) tain 6.5 < amax/a0 < 26, i.e., the size of the universe
− at recolla∼pse is 10∼times larger than its present size.
∼
The corresponding age of the universe at recollapse is
where ω(t) is the general time-dependent equation of
state ω = P(t)/ρ(t). Equation (19) shows that the ∞ dz
behaviour Ω = const arises if either (i) Ω = 1 or (ii) trecoll = , (24)
ω(t) 1/3. The value Ω = 1 is the well-known crit- Z−1/Ωm H(z)(1+z)
≡ −
ical point of the Friedmann equations. The fact that
giving
the equation of state ω 1/3 also results in a uni-
≡ −
verse evolving with an unchanging value of Ω is not so
π √Ω Ω
wellknown,eventhoughthisresultiscompletelygeneral trecoll = m m (25)
and independent of any assumptions about the curva- 2H0[Ωm−1]s−(cid:18)1−Ωm(cid:19)
ture of the universe. It is interesting that oscillations of
the scalar field [with ω(t) varying between ω = +1 and i.e. trecoll 400 4000 billion years (for h=0.5).
∼ −
ω = 1]induceoscillationsinΩ(t),sinceeverytimeω(t) The late-time behaviour of a ΛCDM model suggested
passe−s through the critical value ω = 1/3, the value by recent supernova observations [30] is more complex
c
of Ω˙ changes sign. Thus, close to the tim−e of recollapse, since, if Λ is a constant, the universe need not recol-
lapseatallbut couldcontinue expandingforever[31,27].
a large increase in the value of Ω is modulated by small
Quintessence fields which generate a time-dependent Λ-
oscillations, as shown in Fig. 7. The growth in Ω is due
term result either in an ever-expanding universe (if w
to the increase in the importance of the curvature term Q
alwaysremains < 1/3)orin recollapse(if the current
relative to the scalar field density in (2), whereas small −
acceleration of th∼e universe is a transient phenomenon).
modulations in Ω are due to changes in the equation of
state of the oscillating scalar field.
From (18) we find the following simple expression for
III. DISCUSSION AND CONCLUSIONS.
the total volume of a closed universe
2π2 We have analysed the behaviour of a massive scalar
V = V , (20)
(Ωtotal 1)3/2 H field in a closed FRW universe and shown that oscilla-
− tions of the field about the minimum of its potential can
whereV =H−3isthecurrentHubblevolume. Observa- leadto anasymmetry inits equationof stateω =P /ρ
H φ φ
tionsofanisotropiesinthecosmicmicrowavebackground duringtheexpansionandcollapseepochs. Thisasymme-
on degree scales appear to favour a closed universe with tryisreflectedbythepresenceofahysteresis-likefeature
7
in ω which is, in turn, related to the amount of ‘work Hindu philosophy narrates thefollowing description of a
done’ by/on the scalar field during a given expansion- cosmiccycle: “Onethousandmahayugas—4,320,000,000
contraction cycle. An important consequence of this ef- years of human reckoning—constitute a single day of
fect is that the imposition of bouncing conditions at an Brahma,asinglekalpa... Ihaveknownthedreadfuldis-
appropriate early time causes the work done during a solution of theuniverse.I haveseen all perish again and
givenexpansioncycle to be convertedinto expansionen- again, at every cycle. At that terrible time, every single
ergy,resultinginthegrowthinamplitudeofeachsucces- atomdissolvesintotheprimeval,purewatersofeternity,
sive expansion maximum in an oscillating universe. The whence all originally arose” ... and a new cycle begins
afresh; see, for instance, H. Zimmer, Myths and Symbols
increase in the value of the expansion maximum results
in Indian Art and Civilization, Bollingen Foundation,
in a recycled universe which is considerably longer-lived
Washington,D.C.(1946);paperbackreprint,Harperand
and more flat during each successive expansion epoch.
Row,NewYork(1962);C.W.Misner,K.S.Thorne,and
Thus, the flatness problem is gradually ameliorated in
J.A.Wheeler,Gravitation, W.H.FreemanandCo.,NY
this model.
(1973), p.752.
Our results also have a direct bearing on the issue of
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9