8 March 2001

Physics Letters B 501 (2001) 257–263www.elsevier.nl/locate/npe

Can we predict the fate of the Universe?

P.P. Avelinoa,b, J.P.M. de Carvalhoa,c, C.J.A.P. Martinsd,1

a Centro de Astrofísica, Universidade do Porto, Rua das Estrelas s/n, 4150 Porto, Portugalb Dep. de Física da Faculdade de Ciências da Univ. do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal

c Dep. de Matemática Aplicada da Faculdade de Ciências da Univ. do Porto, Rua das Taipas 135, 4050 Porto, Portugald Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge,

Wilberforce Road, Cambridge CB3 0WA, UK

Received 25 November 2000; received in revised form 9 January 2001; accepted 21 January 2001Editor: P.V. Landshoff

Abstract

We re-analyze the question of the use of cosmological observations to infer the present state and future evolution of ourpatch of the universe. In particular, we discuss under which conditions one might be able to infer that our patch will enter aninflationary stage, as aprima facie interpretation of the type Ia supernovae and CMB data would suggest. We then establisha ‘physical’ criterion for the existence of inflation, to be contrasted with the more ‘mathematical’ one recently proposed byStarkman et al., Phys. Rev. Lett. 83 (1999) 1510. 2001 Published by Elsevier Science B.V.

PACS: 98.80.Es; 98.80.Cq; 98.62.PyKeywords: Gravitation; Cosmology; Inflation; Observational tests

1. Introduction

The issue of the present state, future dynamics andfinal fate of the universe, or at least our patch of it,has been recently pushed to the front line of researchin cosmology. This is mostly due to observationsof high redshift type Ia supernovae, performed bytwo independent groups (the “Supernova CosmologyProject” and the “High-Z Supernova Team”), whichallowed accurate measurements of the luminosity-redshift relation out to redshifts up to aboutz ∼ 1 [2–4]. It should be kept in mind that these measurementsare done on the assumption that these supernovae are

E-mail addresses: pedro@astro.up.pt (P.P. Avelino),mauricio@astro.up.pt (J.P.M. de Carvalho),c.j.a.p.martins@damtp.cam.ac.uk (C.J.A.P. Martins).

1 Also at C.A.U.P., Rua das Estrelas s/n, 4150 Porto, Portugal.

standard candles, which is by no means demonstratedand could conceivably be wrong. There are concernsabout the evolution of these objects and the possibledimming caused by intergalactic dust [5,6], but we willignore these for the purposes of this Letter, and assumethat the quoted results are correct.

The supernovae data, when combined with theever growing set of CMBR anisotropy observations,strongly suggest an accelerated expansion of the uni-verse at the present epoch, with cosmological para-metersΩΛ ∼ 0.7 andΩm ∼ 0.3. A further cause ofconcern here is the model dependence of the CMBRanalysis, but we shall again accept the above resultsfor the purpose of this Letter.

Taken at face value, these results would seemto show that the universe will necessarily enter aninflationary stage in the near future. However, aspointed out by Starkman, Trodden and Vachaspati [1]

0370-2693/01/$ – see front matter 2001 Published by Elsevier Science B.V.PII: S0370-2693(01)00151-4

258 P.P. Avelino et al. / Physics Letters B 501 (2001) 257–263

this is not necessarily so. We could be living in a small,sub-horizon bubble, for example. And even if we wereindeed inflating, it would not be trivial to demonstrateit. In the above work, these authors looked at thecrucial question of ‘How far out must we look to inferthat the patch of the universe in which we are livingis inflating?’ Their analysis is based on previous workby Vachaspati and Trodden [7] which shows that theonset of inflation can in some sense be identified withthe comoving contraction of our minimal anti-trappedsurface (MAS).2 They then argue that if one canconfirm cosmic acceleration up to a redshiftzMAS anddetect the contraction of our MAS, then our universemust be inflating. Unfortunately, even if we can do theformer (forΩΛ = 0.8 the required redshift iszMAS ∼1.8), it turns out that there is no way to presentlyconfirm the latter, because the accelerated expansionhasn’t been going for long enough for the MAS tocontract. Only if we hadΩΛ 0.96 would we be ableto demonstrate inflation today.

As in the proverbial mathematicians joke, the meth-od outlined by Starkman et al. provides an answer thatis completely accurate but will take a long time tofind, and hence is of no immediate use to us. In thisLetter, however, we will explore a different possibility.Our main aim is to provide what could be called aphysicists version of the “mathematical” question ofStarkman et al. [1]. In other words, we are asking,‘If we can’t know for sure the fate of the universe atpresent, what is our best guess today?’. As we willdiscuss, we can answer this question, although it willinvolve making some crucial additional assumptions.

In order to answer the above question we comparethe particle and event horizons. We show that for aflat universe withΩΛ 0.14 the particle horizon isgreater than the distance to the event horizon meaningthat today we may be able to observe a larger portionof the universe than that which will ever be ableto influence us. We argue that if we find evidencefor a constant vacuum density up to a distance fromus equal to the event horizon then our universe willnecessarily enter an inflationary phase in the not too

2 The MAS of each comoving observer is a sphere centered onhim/her, on which the velocity of comoving objects isc. For theparticular case of an homogeneous universe, the MAS has a physicalradiuscH−1.

distant future,assuming that the potential of the scalarfield which drives inflation is time-independent andthat the content of the observable universe will remain‘frozen’ in comoving coordinates.

Note that Starkman et al. argue that inflation canonly take place if the vacuum energy dominates theenergy density on a region with physical radius notsmaller than that of the MAS at that time. However,they did not assume that the content of the observ-able universe would remain frozen in comoving coor-dinates and so they found that the larger is the contri-bution of a cosmological constant for the total densityof the universe, the larger is the redshift out to whichone has to look in order to infer that our portion ofthe universe is inflating. This result seems paradoxi-cal, until one realizes that the size of the MAS at agiven timedoes not, by itself, say anything about in-flation. The main reason whyzMAS grows withΩΛ issimply because the scale factor has grown more.

The plan of this Letter is as follows. In the next sec-tion we introduce the various length scales that arerelevant to our discussion, and provide a qualitativediscussion of our test for inflation. We also discussthe assumptions involved and compare our ‘physical’test with the ‘mathematical’ one recently proposed byStarkman et al. [1]. In Section 3 we provide a morequantitave analysis of our criterion. We also discussin more detail our crucial assumption of an energy–momentum distribution which remains frozen in co-moving coordinates. Finally, in Section 4 we summa-rize our results and discuss some other outstanding is-sues.

2. A ‘physical’ test for inflation

The dynamical equation which describes the evolu-tion of the scale factora in a Friedmann–Robertson–Walker (FRW) universe containing matter, radiationand a cosmological constant can be written as

(1)H 2 = H 20

(Ωm0a

−3 + Ωr0a−4 + ΩΛ0 + Ωk0a

−2),whereH = a/a and the density parametersΩm, Ωr

andΩΛ express respectively the densities in matter,radiation and cosmological constant as fractions of the

P.P. Avelino et al. / Physics Letters B 501 (2001) 257–263 259

critical density.3 Naturally one hasΩk = 1 − Ωm −Ωr − ΩΛ.

The distanced , to a comoving observer at a redshiftz is given by

d(z) = c

t0∫t (z)

dt ′

a(t ′)

(2)

= cH−10

z∫0

dz′[Ωm0(1+ z′)3 + Ωr0(1+ z′)4

+ ΩΛ0 + Ωk0(1+ z′)2]−1/2,

and is related to the ‘radius’ of the local universewhich we can in principle observe today. The distanceto theevent horizon can be defined as

de = c

∞∫t0

dt ′

a(t ′)

(3)

= cH−10

∞∫1

da

(Ωm0a + Ωr0 + ΩΛ0a4 + Ωk0a2)1/2

and represents the portion of the universe which willever be able to influence us.4 On the other hand, theparticle horizon, dp, is defined by (from Eq. (2))

(4)dp ≡ limz→∞d(z),

and it represents the maximum distance which we canobserve today.

If today the distance to the event horizon is smallerthan the particle horizon (de < dp) this means thattoday we are able to observe a larger portion of theuniverse than that which will ever be able to influenceus. We can do this if we look at a redshift greater thanz∗ defined by (see also [8])

(5)d(z∗) = de.

3 A dot represents a derivative with respect to the cosmic timet .The subscript ‘0’ means that the quantities are to be evaluated atpresent epoch, and we have also takena0 = 1.

4 In writing the upper integration limit as infinity we are ofcourse assuming that the universe will keep expanding forever; ananalogous formal definition could be given for an universe endingin a ‘big crunch’.

In a flat universe solutions to this equation are onlypossible forz∗ 1 and forΩΛ0 0.14. Hence, as-suming that the energy–momentumdistribution withinthe patch of the universe which we are able to see re-mains unchanged in comoving coordinates, our uni-verse will necessarily enter an inflationary phase in thefuture if there is a uniform vacuum density permeatingthe universe up to a redshiftz∗.

This assumption obviously requires some furtherdiscussion. One can certainly think of a universe madeup of different ‘domains’, each with its own values ofthe matter and vacuum energy density. Furthermore,by cleverly choosing the field dynamics, one canalways get patches with time-varying vacuum energydensities, or patches where the vacuum energy densityis non-zero for only short periods. In all such cases, thedomain walls separating these patches can certainlyhave a very complicated dynamics, and in particularit is always possible that a domain wall will suddenlyget inside our horizon sometime between the epochcorresponding to our observations and the present day.On the other hand, it should also be pointed out thata certain amount of fine-tuning would be required tohave a bubble coming inside our horizon right afterwe have last observed it. In these circumstances, thebest that can be done is to impose constraints on thecharacteristics of any bubble wall that could plausiblyhave entered the patch of the universe we are currentlyable to observe, given that we have so far seen none.We shall analyse this point in a more quantitativemanner in the following section.

We think that the results obtained in this way,even if less robust from a formal point of view, areintuitively more meaningful than those obtained in [1]in the sense that, among other things, in this case theminimum redshiftz∗ out to which one must observein order to be able to predict an inflationary phase(subject to the conditions mentioned earlier) decreasesasΩΛ increases (see Fig. 1). In other words, the largerthe present value of the cosmological constant, theeasier it should be to notice it.

It is perhaps instructive to compare our test withthat of [1] in more detail. Starkman et al. require thecontraction of the MAS. Now, in order to see the MASone has to look at a redshift defined by:

(6)a(zMAS)d(zMAS) − cH−1(zMAS) = 0.

260 P.P. Avelino et al. / Physics Letters B 501 (2001) 257–263

Fig. 1. The solution of Eq. (5) for cosmologies withΩm + ΩΛ = 0.7,1.0,1.3 (solid curves, bottom to top) andΩm = 0.3 for illustration (dotted curve). Observing a uniformvacuum energy density up to a redshiftz∗ will imply that our uni-verse will enter an inflationary phase in the future, subject to theconditions specified in the text.

This finds the redshift,zMAS, for which the physicaldistance to a comoving observer at that redshift,evaluated at the corresponding timetMAS, is equalto the Hubble radius at that time. However, if thevacuum density already dominates the dynamics of theuniverse at the redshiftz∗ then Eq. (5) reduces to:

(7)d(z∗) − cH−1(z∗) = 0;(recall thata0 = 1) because during inflation the physi-cal size of the event horizon is simply equal5 to cH−1

(this is ultimately the reason for the choice of criteriumfor inflation by Vachaspati and Trodden [7]). As hasbeen discussed above, Eqs. (6) and (7) have totally dif-ferent solutions (z∗ = 1 while zMAS → ∞).

To put it in another way, the main differencebetween our approach and that of Ref. [1] lies onthe fact that we assume that the energy–momentumcontent of the observable universe does not changesignificantly in comoving coordinates. This allowsus to use the equation of state of the local universeobserved for a redshiftz (looking back at a physicaltime t (z)) to infer the equation of state of the local

5 This is only exactly true when the vacuum energy density isthe only contributor to the energy density, in which case exponentialinflation occurs.

universe at the present time. In the following section,we shall discuss these points in somewhat more detail.

We should also point out that if we were to relaxthe assumption of a co-movingly frozen content of theobservable universe, then the equation — analogous to(5) — specifying the redshift out to which one shouldlook in order to be able to predict the future of theuniverse would be

(8)d(z+) = de(z+).

This equation has no solution, so a stronger test of thiskind is not feasible in practice.

3. Discussion

Here we go through some specific aspects of ourtest in more quantitative detail. To begin with, wehave solved numerically Eq. (5); the numerical resultswere obtained for choices of cosmological parameterssuch thatΩm + ΩΛ = 0.7,1.0,1.3, with an additionalΩm = 0.3 for illustration. We are interested onlyin a matter-dominated orΛ-dominated epoch of theevolution of the universe, and therefore we havedropped the radiation density parameterΩ r

0 of Eq. (1),in the calculations.

These results are displayed in Fig. 1 as a function ofΩΛ0. The cases with constant total density are shownin solid curves (with the top curve corresponding to thehigher value of the density), while the case of a fixedΩm, is shown, for comparison purposes, by a dottedcurve.

As expected, as the universe becomes moreΛ-dominated and/or less matter-dominated, the comov-ing distance to the event horizon decreases, which isreflected in the decrease of the redshiftz∗ of a comov-ing source located at that distance.6 For the observa-tionally preferred values ofΩm = 0.3 andΩΛ = 0.7,the required redshift will bez∗ ∼ 1.8.

For comparison, the redshift,zMAS, defined byEq. (6), which is the analogous relevant quantity forthe criterion of Starkman et al. is shown, for the samechoices of cosmological parameters, in Fig. 2. Notethat in this case, as the universe becomes moreΛ-dominated and/or less matter-dominated, the redshift

6 Note that pushingΩΛ0 down to zero, the value ofz∗ tends toinfinity, since in such universes an event horizon does not exist.

P.P. Avelino et al. / Physics Letters B 501 (2001) 257–263 261

Fig. 2. The redshift of the MAS, relevant for the inflationarycriterion of Starkman et al., for the same cosmological models asin Fig. 1. Note that the solid curves forΩm + ΩΛ = 0.7,1.0,1.3now appear in the graph from top to bottom.

of the MAS will increase. As we already pointed outour test will not be applicable for very low values ofthe vacuum energy density, and for intermediate val-ues, it requires a higher redshift thanzMAS. However,for high values of the cosmological constant and/orlow matter contents, the fact that the universe willbe expanding much faster makes the redshift of theMAS increase significantly, and even become largerthanz∗ for some combinations of cosmological para-meters. For the same observationally preferred valuesof Ωm andΩΛ quoted above, the required redshift willbezMAS ∼ 1.6. A comparison of the values ofz∗ andzMAS for the spatially flat model is shown in Fig. 3.

We now return to our assumption about the energy–momentum content of the universe, considering thepossibility that different regions of space may havedifferent values for the vacuum energy density, whichare separated by domain walls. This means that weare assuming the existence of a scalar field, sayφ,which within each region sits in one of a number ofpossible minima of a time-independent potential. Itis obvious that if the potential depends on time orif the scalar field did not have time to roll to theminimum of the potential, then it is not possible topredict the fate of the universe without knowing moreabout the particle physics model which determinesits dynamics. For simplicity, we shall assume thatwe live in a spherical domain with constant vacuum

Fig. 3. Comparing the values of the critical redshiftsz∗ andzMAS,as a function of the vacuum energy density, for the spatially flatmodels (Ωm + ΩΛ = 1.0).

energy density (effectively a cosmological constant)that is surrounded by a much larger region in which thevacuum density has a different value — for the presentpurposes we will assume it to be zero. Note that this isthe case where the dynamics of the wall will be faster(more on this below).

Is it possible that a region with a radiusd(z), saycentred on a nearby observer, can be inside a givendomain at the conformal timeη(z), but outside thatdomain at the present time,η0? This problem can pro-vide some measure of how good the assumption ofa frozen energy–momentum distribution in comovingcoordinates is. In other words, is it likely that a do-main wall may have entered this region at a redshiftsmaller thanz? In order to provide a more quantitativeanswer to this question, we have performed numericalsimulations of domain wall evolution using the PRS[9] algorithm, in which the thickness of the domainwalls remains fixed in comoving coordinates for nu-merical convenience. See also [10] for a description ofthe simulations.

We assume that the domain wall has spherical sym-metry, thereby reducing a three-dimensional problemto a one-dimensional one. We perform simulations ofthis wall in a flat universe on a one-dimensional 8192grid. The comoving grid spacing isx = cηi whereηi = 1 is the conformal time at the beginning of thesimulation. The initial comoving radius of the spher-

262 P.P. Avelino et al. / Physics Letters B 501 (2001) 257–263

ical domain was chosen to beR = 2048x, and thecomoving thickness of the domain wall was set to be10x. In these simulations we neglect the gravita-tional effect induced by the different domains and do-main walls on the dynamics of the universe, and wealso do not consider the possibility of an open uni-verse. We must emphasise, however, that both these ef-fects would slow down the defects, thereby helping tojustify our assumption of a constant equation of statein comoving coordinates even more.

We have obtained the following fit for the radius ofthe domain wall as a function of the conformal timeη

(9)R(η) = R∞(

1−(

cη

αR∞

)n)2/n

,

where

(10)α = 2.5, n = 2.1

andR∞ is the initial comoving radius of the domainwall (with R∞ cηi ). This fit is accurate to betterthan 5%, except for the final stages of collapse. In a flatuniverse with no cosmological constant the comovingdistance to a comoving object at a redshiftz is givenby

(11)d(z) = cη0(1− 1/

√1+ z

),

whereas the radius of the spherical domain wall can bewritten as a function of the redshiftz, given its initialradiusR∞, as follows

(12)R(z) = R∞(

1−(

cη0

αR∞√

1+ z

)n)2/n

.

Now, by solving the equation

(13)R(z = 0) = d(z)

we can find the initial comoving radius of our do-main (in units of the present conformal time, that isRi(z)/η0) which it would be required to have so thatits comoving size today is equal to the comoving dis-tance to an object at a redshiftz. Finally, we can cal-culate the radius of this domain at the present timeη0and at the redshiftz (call it Rmax(z)) with the valueof Ri(z)/η0 obtained from the previous equation. InFig. 4 we plot the value ofRmax(z)/d(z) as a functionof the redshift,z.

If the radius of our domain at a redshiftz wassmaller thand(z) the domain wall would be in causal

Fig. 4. The comoving radius of a domain wall at redshiftz whosepresent comoving size equals the comoving distance to an object atredshiftz — denotedd(z), see (11) — in units ofd(z), as a functionof redshift.

contact with us at the present time and we couldin principle detect the gravitational effect both ofthe domain wall and of the different vacuum densityoutside our bubble. On the other hand, if the radius ofour domain at a redshiftz was greater thanRmax thenit would not have time to enter the sphere of radiusd(z) before today.

When the redshift of the cosmological object weare looking at is small, that is (z → 0), its comovingdistance from us,d(z), is much smaller than thecomoving horizon,η(z), at the time at which thelight was emitted. Consequently, a domain wall witha comoving size equal tod(z) at the present timewould already have a velocity very close to the speedof light by the redshiftz. It is easy to calculate themaximum comoving size,Rmax, which our domainwould need to have at the redshiftz, in order for thedomain wall to enter a sphere of comoving radiusd(z)

centred on a nearby observer sometime between todayand redshiftz. This is simply given by

(14)Rmax(z)

d(z)→ 2

whenz → 0, becaused(z) is the distance travelled bylight from a redshiftz until today (see Fig. 4).

If we assume that the comoving radius of our bubbleat a redshiftz is larger thanη(z), then it will remainfrozen in comoving coordinates until its size gets

P.P. Avelino et al. / Physics Letters B 501 (2001) 257–263 263

smaller than the horizon. This means that in this casethe value ofRmax/d(z) is even smaller, approaching

(15)Rmax(z)

d(z)→ α−2 × (

√1+ 4α−n − 1)−2/n

2−2/n,

whenz → ∞ (see Fig. 4). For a spherical domain wehave

(16)Rmax(z)

d∞≈ 1.12.

We thus see that for the purposes of predicting thefate of the universe it may be a plausible assumptionto assume a fixed content in comoving coordinates.The above discussion also suggests, in particular, thatone may finda posteriori that it is indeed a reasonableassumption if we can observe the dynamical effects ofa uniform vacuum density up to a redshiftz 1.

4. Conclusions

We have provided a simple analysis of the use ofcosmological observations to infer the state and fateof our patch of the universe. In particular, in thesame spirit of Starkman et al. [1], we have discussedpossible criteria for inferring the present or futureexistence of an inflationary epoch in our patch of theuniverse.

We have presented a ‘physical’ criterion for the ex-istence of inflation, and contrasted it with the ‘mathe-matical’ one that has been introduced in [1]. Ours hasthe advantage of being able to provide (in principle)

a definite answer at the present epoch, but the disad-vantage of ultimately relying on assumptions on thecontent of the local universe and on field dynamics.We consider our assumptions to be plausible, but wecan certainly conceive of (arguably contrived or fine-tuned) mechanisms that would be capable of violatingit.

Acknowledgements

C.M. is funded by FCT (Portugal) under ‘ProgramaPRAXIS XXI’ (grant no. PRAXIS XXI/BPD/11769/97). We thank Centro de Astrofísica da Universidadedo Porto (CAUP) for the facilities provided.

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