PHYSICAL REVIEW D, VOLUME 61, 023502

Causality and cosmic inflation

Tanmay Vachaspati* and Mark Trodden†

Department of Physics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106-7079~Received 11 November 1998; published 16 December 1999!

In the context of inflationary models with a pre-inflationary stage, in which the Einstein equations areobeyed, the null energy condition is satisfied, and spacetime topology is trivial, we argue that homogeneity onsuper-Hubble scales must be assumed as an initial condition. Models in which inflation arises from fielddynamics in a Friedmann-Robertson-Walker background fall into this class but models in which inflationoriginates at the Planck epoch may evade this conclusion. Our arguments rest on causality and general rela-tivistic constraints on the structure of spacetime. We discuss modifications to existing scenarios that may avoidthe need for initial large-scale homogeneity.

PACS number~s!: 98.80.Cq

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It is well recognized that an early inflationary epoch cexplain several of the observed features of the presentverse@1,2#. The remarkable homogeneity of the universemeasured by the Cosmic Backgrounnd Explorer~COBE!, theflatness of the universe indicated by some of the currentmic data, the distribution of structure, and the absencemagnetic monopoles may all be simultaneously explainedinvoking about 60e-folds of cosmic inflation. This remarkable fact has spurred considerable effort in building modthat realize an inflationary phase of the universe. The goainflationary models is to explain how an assumed ninflationary universe after the big bang develops into anflationary universe at some epoch. Eventually, after somee-folds, the universe must gracefully exit the inflationastage and enter the radiation epoch of standard cosmolo

There exist alternative explanations for some of the cmological observations that inflation so naturally explaiThe distribution of structure may follow from topologicadefects@3#; the absence of magnetic monopoles from detof particle physics@4#, or the interaction of domain walls anmagnetic monopoles@5#. Other observed features of the unverse are harder to explain by non-inflationary means. Ifuniverse is indeed flat, it would be hard to explain this oservation without invoking inflation.~It is known that certaininflationary models can lead to a non-flat universe, andflatness is not a generic prediction of inflation but onecertain models.! Finally, the homogeneity of the universevirtually impossible to explain without invoking inflation anthis is a key compelling feature of the theory.

The ability of inflation to smooth out the universe on sperhorizon scales is an effective mechanism to explainobserved homogeneity only if the inflation itself does nrequire large-scale homogeneity. This means that we massume a pre-inflationary epoch of the universe from whicsmall patch of the universe underwent inflation entirelycausal processes. Note that causality dictates that the ition must be ‘‘local.’’ In other words, any spacelike sectioof the boundary of the inflating region must not extend byond the causal horizon of the pre-inflationary spacetime

*Email address: txv7@po.cwru.edu†Email address: trodden@theory1.phys.cwru.edu

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The question we address here is: under what conditionit possible to havelocal inflation?

The embedding of an inflating region~not necessarily un-dergoingexponentialinflation! within an exterior cosmologyis constrained by the nature of matter in the universe. Thibest seen by employing the Raychaudhuri equation fordivergence of a congruence of future directed, affinelyrametrized null geodesics. This congruence is taken tonormal to a two dimensional sphere centered at the origincoordinates and may be in- or out-going~i.e. directed to-wards or away from the origin of coordinates, respective!.Let us denote the tangent vector field to the congruenceNa. Then the divergenceu is defined by

u5¹aNa. ~1!

The Raychaudhuri equation is

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wheret is the affine parameter,sab is the shear tensor,vabthe twist tensor andRab the Ricci tensor.~We follow theconventions of Wald@6#.! The shear tensor is purely spatiand hence its contribution to the right-hand side is positiThe twist tensor vanishes since the congruence of null raytaken to be hypersurface orthogonal. Then,

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If Einstein’s equations hold then

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and if the null energy condition is satisfied

TabNaNb>0. ~5!

~Note that this mild condition is implied by, for example, thweak energy condition:Tabj

ajb>0 for any timelike vectorja.! Putting these conditions together we obtain

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TANMAY VACHASPATI AND MARK TRODDEN PHYSICAL REVIEW D 61 023502

du

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For our purposes, however, it proves sufficient to useweaker condition

du

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Regions of a spherically symmetric spacetime in whthe divergences of both in- and out-going rays, normalspatial two dimensional spheres centered at the origin,negative ~positive! will be referred to as trapped~anti-trapped! regions. Regions in which in-going rays have negtive divergence~that is, are converging! but out-going rayshave positive divergence will be called ‘‘normal,’’ since this the behavior in flat spacetime. Then the condition~7! saysthat a converging null geodesic cannot start to diverge pto having reached the origin of coordinates, orfocussed. Inother words, in-going null rays cannot start out in normregions and then enter an antitrapped region. This becothe constraint in patching together an inflationary region ibackground cosmology.

Consider a topologically trivial universe such as shownFig. 1. The universe starts out in a big bang and containnormal region and an antitrapped region at distances lathan some distance that depends on the details of the cosogy. Now consider a patch of this region that starts to inflaThe patch is denoted by the horizontal line OQ, and haphysical size that we will denote byxQ . The section OP ofthe line OQ denotes a spatial patch equal to the size ofinflationary horizonHin f

21 . For inflation to occur, one assumes that vacuum energy must dominate over a relarger than the inflationary horizon distance, and so

xQ>xP5Hin f21 . ~8!

Further, a straightforward calculation for ingoing null raysspacetimes with the metric of the inflating region given byflat Friedmann-Robertson-Walker~FRW! metric

FIG. 1. A Penrose diagram for local inflation. The arrow dnotes a future directed, radial, affinely parametrized null geodfrom the exterior spacetime into the inflating region. Shaded regare antitrapped; unshaded regions are normal.

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ds252dt21a2~ t !@dr21r 2dV2#, ~9!

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x5a~ t !r ~11!

wherer is the coordinate of a null ray. Equation~10! showsthat the region with physical distance larger thanHin f

21 in theinflating region is antitrapped. Hence the region PQRSFig. 1 is antitrapped. The crucial question is: what areallowed positions of the point P?

In Fig. 1 we show the situation where P is not locatedthe antitrapped region of the background cosmology. Thlight rays such as shown in Fig. 1~from point a to point b)can enter the inflating antitrapped region from the externormal region~unshaded in the figure!. While the ingoingrays are in the normal region,u is negative, but once theyenter the antitrapped inflationary regionu must become posi-tive. This is forbidden by the condition in Eq.~7!. Hence, wemust conclude that the outer boundary PQR of the atrapped inflating region must lie entirely inside the antrapped region of the background cosmology as shownFig. 2. This is the key constraint on inflating spacetimderived in this paper and, except for spherical symmetryindependent of the background cosmology.

To appreciate the constraint, it is useful to think of tsituation when the background cosmology is a flFriedmann-Robertson-Walker universe with a scale faca(t). Then, the boundary of the antitrapped region of tbackground universe is given by

xFRW~ t !5HFRW21 ~ t !. ~12!

Now, since the point P must lie within the background antrapped region

xP>xFRW~ tP!, ~13!

which yields

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FIG. 2. A Penrose diagram for local inflation in which ingoinnull geodesics that enter the inflating region emanate from atrapped regions.

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CAUSALITY AND COSMIC INFLATION PHYSICAL REVIEW D 61 023502

xQ>xP5Hin f21>HFRW

21 ~ tP! . ~14!

This says that the size of the initial inflationary patch mustgreater than the inflationary horizon, which must be larthan the background FRW inverse Hubble size at the tinflation starts. That is, the conditions appropriate for infltion to occur must be satisfied over a patch that is larger tthe FRW inverse Hubble scale.1

Note that the inverse Hubble distance,H21, can be dif-ferent from the causal horizon~the distance light has propagated from the big bang!. However,H21 is in most casesstill a large patch compared to length scales over which pticle physics processes occur that can homogenize theverse. Also, for a flat, radiation dominated FRW cosmoloH21 coincides with the causal horizon. We conclude thinflationary model building must assume homogeneitysuper-Hubble scales. In this sense, inflationary modelsattempt to obtain inflation within a background FRW unverse cannot explain the homogeneity of the observedverse.

Our result is consistent with the result due to Farhi aGuth @10# who found that it is impossible to create an infltionary universe in the laboratory subject to the Einstequations, the weak energy condition and the absence ofgularities. On small enough scales in an expanding univeit should be possible to ignore the background expansionthen the Farhi-Guth result should be applicable. This is csistent with our result since we find that the Hubble scalethe background spacetime provides a lower bound onsize of the inflating patch. If one admits the possibilityinflating false vacuum bubbles born at a singularity, tspacetime diagrams drawn by Blau, Guendelman and G@11# show that the inflating region emerges from a white hinterior in which all two spheres are antitrapped. Then, oagain, the boundary of the inflating region borders an atrapped region.

In @12#, Goldwirth and Piran numerically solved the Einstein equations together with a scalar field and foundinflation is obtained only if homogeneous initial conditioare assumed over a length scale that encompasses sehorizons.~Similar numerical analyses were also performby Kung and Brandenberger@13#.! Our result generalizes anproves this numerical finding.

It is also worth pointing out that chaotic inflation@2# doesnot fall within the purview of our result since, in this modeinflation starts at the Planck epoch with homogeneitysumed on the Planck scale.

1Based on homogeneous FRW cosmologies, the conclus

HFRW<0, was reached by Linde and collaborators@7–9#. Thoughapparently similar to our result in Eq.~14!, there are importantdifferences in the two results since these authors compareHubble scales for homogeneous universes at two different tiwhile we are considering inhomogeneous universes and the Huscales are compared at the same time. Our result also has theof being independent of the particle physics model under consiation.

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Hence we conclude that, within the conditions describabove, local inflation is not possible. However, observatioindicate homogeneity of the early universe on super-horiscales and this needs an explanation. It is indeed possthat the initial homogeneity required by inflation — or evethe homogeneity of the entire visible universe — occurrjust by chance. Whether this is a satisfactory resolutionthe observed homogeneity of the universe is at presentclear, since, among other things, such arguments dependform of the anthropic principle.

Let us now discuss the conditions under which localflation can occur without assuming accidental homogenon large scales. The first possibility is that the weak enecondition ~for diagonal, spherically symmetric fluid energymomentum tensors, this amounts to assumingr>0 and r1P>0) may be classically violated. For this one wouneed exotic forms of matter in the early universe. An attrtive alternative is that quantum effects could give riseeffective violations of the weak energy condition.~Howeverthese violations are constrained by the Ford-Roman inequties @14#.! Whether quantum effects can be sufficient to leto local inflation is an interesting question that has notbeen answered~an early related attempt was made in@15#!.The second possibility is that the Einstein equations maymodified, leading to changes in Eqs.~4! and ~6!. This ispossible, for example, if we have a non-minimally couplscalar field in the model. To us, this way out seems to bparticularly attractive possibility in view of modern partictheories in which such scalar fields are abundant. A thpossibility may be to have a topologically non-trivial bacground universe. Such universes have attracted signifiattention recently@16# and should be investigated further.fourth possibility is the one that occurs in topological infltion within magnetic monopoles as we discussed in detai@17#. Here the inflation is manifestly local and causal butpreceded by a singularity or topology change~see Fig. 3!.

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TANMAY VACHASPATI AND MARK TRODDEN PHYSICAL REVIEW D 61 023502

Conflict with our constraint is avoided because there arenull rays that enter the inflating region from the externregion. The singularity or topology change plays the role omini big bang for the inflating spacetime, though is somwhat different in character from an FRW big bang since itimelike. In any case, predictability in the inflating univeris lost because of signals that can emanate from the silarity or topology changing event. It is not possible to evolto the inflating region from data on a spacelike hypersurfin the pre-inflationary epoch. Instead, initial data mustprovided on a spacelike surface (S) within the inflating re-gion. However, it may be possible for quantum effectsspecify the extra boundary condition required to restore pdictability @see@18# for a discussion of this in the context othe AdS conformal field theory~CFT! correspondence#.

To summarize, we have argued that inflationary modbased on the classical Einstein equations, the null eneconditions, and trivial topology, require initial homogene

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on super-Hubble scales. Inflation with no requirementsinitial large-scale homogeneity might be achieved with oor more of the following conditions:~1! violations of theclassical Einstein equations, say due to non-minimacoupled scalar fields,~2! violations of the weak energy condition in the early universe,~3! non-trivial topology of theuniverse,~4! the birth of the universe directly into an inflaing phase, that is, the absence of a pre-inflationary eposuch as might be invoked in specific inflationary models, echaotic inflation, or arise out of quantum cosmology@19–21#.

We would like to thank Arvind Borde for guidance, AlaGuth and Andrei Linde for extensive discussions, JauGarriga, Lawrence Krauss, Glenn Starkman, Alex Vilenkand Bob Wald for comments. This work was supportedthe Department of Energy~D.O.E.!.

ett.

y

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