inflationary space times are not past-complete

8/9/2019 Inflationary Space Times Are Not Past-complete

1/4

arXiv:g

r-qc/0110012v214Jan2003

MIT-CTP-3183

Inflationary spacetimes are not past-complete

Arvind Borde,1, 2 Alan H. Guth,1, 3 and Alexander Vilenkin1

1Institute of Cosmology, Department of Physics and AstronomyTufts University, Medford, MA 02155, USA.

2Natural Sciences Division, Southampton College, NY 11968, USA.3Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics,

Massachusetts Institute of Technology, Cambridge, MA 02139, USA.

(Dated: January 11, 2003)Many inflating spacetimes are likely to violate the weak energy condition, a key assumption of

singularity theorems. Here we offer a simple kinematical argument, requiring no energy condition,that a cosmological model which is inflating or just expanding sufficiently fast must be incompletein null and timelike past directions. Specifically, we obtain a bound on the integral of the Hubbleparameter over a past-directed timelike or null geodesic. Thus inflationary models require physicsother than inflation to describe the past boundary of the inflating region of spacetime.

PACS numbers: 98.80.Cq, 04.20.Dw

I. Introduction. Inflationary cosmological models [1, 2, 3]are generically eternal to the future [4, 5]. In these mod-els, the Universe consists of post-inflationary, thermal-

ized regions coexisting with still-inflating ones. In co-moving coordinates the thermalized regions grow in timeand are joined by new thermalized regions, so the comov-ing volume of the inflating regions vanishes as t .Nonetheless, the inflating regions expand so fast thattheir physical volume grows exponentially with time. Asa result, there is never a time when the Universe is com-pletely thermalized. In such spacetimes, it is naturalto ask if the Universe could also be past-eternal. If itcould, eternal inflation would provide a viable model ofthe Universe with no initial singularity. The Universewould never come into existence. It would simply exist.

This possibility was discussed in the early days of in-flation, but it was soon realized [6, 7] that the idea couldnot be implemented in the simplest model in which theinflating universe is described by an exact de Sitter space.More general theorems showing that inflationary space-times are geodesically incomplete to the past were thenproved [8]. One of the key assumptions made in thesetheorems is that the energy-momentum tensor obeys theweak energy condition. Although this condition is satis-fied by all known forms of classical matter, subsequentwork has shown that it is likely to be violated by quan-tum effects in inflationary models [9, 10]. Such viola-tions must occur whenever quantum fluctuations result

in an increase of the Hubble parameter H i.e., whendH/dt > 0 provided that the spacetime and the fluc-tuation can be approximated as locally flat. Such upwardfluctuations in H are essential for the future-eternal na-ture of chaotic inflation. Thus, the weak energy condi-tion is generally violated in an eternally inflating uni-verse. These violations appear to open the door again tothe possibility that inflation, by itself, can eliminate theneed for an initial singularity. Here we argue that this isnot the case. In fact, we show that the general situation

is very similar to that in de Sitter space.The intuitive reason why de Sitter inflation cannot be

past-eternal is that, in the full de Sitter space, expo-

nential expansion is preceded by exponential contraction.Such a contracting phase is not part of standard inflation-ary models, and does not appear to be consistent withthe physics of inflation. If thermalized regions were ableto form all the way to past infinity in the contractingspacetime, the whole universe would have been thermal-ized before inflationary expansion could begin. In ouranalysis we will exclude the possibility of such a con-tracting phase by considering spacetimes for which thepast region obeys an averaged expansion condition, bywhich we mean that the average expansion rate in thepast is greater than zero:

Hav > 0. (1)

With a suitable definition ofH and the region over whichthe average is to be taken, we will show that the averagedexpansion condition implies past-incompleteness.

It is important to realize that the terms expansion andcontraction refer to the behavior of congruences of time-like geodesics (the potential trajectories of test particles).It is meaningless to say that a spacetime is expanding ata single point, since in the vicinity of any point one canalways construct congruences that expand or contract atany desired rate. We will see, however, that nontrivialconsequences can result if we assume the existence of a

single congruence with a positive average expansion ratethroughout some specified region.

While the past of an inflationary model is a matter ofspeculation, the attractor nature of the inflationary equa-tions implies that many properties of the future can bededuced unambiguously. According to the standard pic-ture of inflation, all physical quantities are slowly vary-ing on the scale of H1. In the vicinity of any point Pin the inflating region, we can choose an approximatelyhomogenous, isotropic and flat spacelike surface which
http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2http://arxiv.org/abs/gr-qc/0110012v2


2/4

2

can serve as the starting point for the standard analy-sis of stochastic evolution [11]. A simple pattern of ex-pansion is established, in which the comoving geodesicsx = const in the synchronous gauge form a congruencewith H >

(8/3)G0, where 0 is the minimum energy

density of the inflationary part of the potential energyfunction. This congruence covers the future light coneof P. While large fluctuations can drive H to negative

values, such fluctuations are extremely rare. Once in-flation ends in any given region, however, many of thegeodesics are likely to develop caustics as the matterclumps to form galaxies and black holes. If we try todescribe inflation that is eternal into the past, it wouldseem reasonable to assume that the past of P is like theinflating region to the future, which would mean that acongruence that is expanding everywhere, except for rarefluctuations, can be defined throughout that past.

For the proof of our theorem, however, we find that it issufficient to adopt a much weaker assumption, requiringonly that a congruence with Hav > 0 can be continu-ously defined along some past-directed timelike or nullgeodesic.

In Section II, we illustrate our result by showing howit arises in the case of a homogeneous, isotropic, and spa-tially flat universe. In the course of the argument we shedsome light on the meaning of an incomplete null geodesicby relating the affine parameterization to the cosmologi-cal redshift. In Section III we present our main, model-independent argument. In Section IV we offer some re-marks on the interpretation and possible extensions ofour result.

II. A simple model. Consider a model in which the metrictakes the form

ds2 = dt2 a2(t)dx2. (2)

We will first examine the behavior of null geodesics, andthen timelike ones.

From the geodesic equation one finds that a nullgeodesic in the metric (2), with affine parameter , obeysthe relation

d a(t) dt. (3)

Alternatively, we can understand this equation by consid-ering a physical wave propagating along the null geodesic.

In the short wavelength limit the wave vector k is tan-gential to the geodesic, and is related to the affine pa-rameterization of the geodesic by k dx/d. This al-lows us to write d dt/, where k0 is the physicalfrequency as measured by a comoving observer. In an ex-panding model the frequency is red-shifted as 1/a(t),so we recover the result of Eq. (3).

From Eq. (3), one sees that if a(t) decreases suffi-ciently quickly in the past direction, then

a(t) dt can be

bounded and the maximum affine length must be finite.

To relate this possibility to the behavior of the Hubbleparameter H, we first normalize the affine parameter bychoosing d = [a(t)/a(tf)] dt, so d/dt 1 when t = tf,where tf is some chosen reference time. Using H a/a,where a dot denotes a derivative with respect to t, we canmultiply Eq. (3) by H() and then integrate from someinitial time ti to the reference time tf:

(tf )

(ti)

H() d =a

(tf )

a(ti)

daa(tf)

1, (4)

where equality holds if a(ti) = 0. Defining Hav to be anaverage over the affine parameter,

Hav 1

(tf) (ti)

tfti

H() d 1

(tf) (ti),

(5)we see that any backward-going null geodesic with Hav >0 must have a finite affine length, i.e., is past-incomplete.

A similar argument can be made for timelike geodesics,parameterized by the proper time . For a particle of

mass m, the four-momentum P

m dx

/d, so we canwrite d = (m/E) dt, where E P0 is the energy ofthe particle as measured by a comoving observer. If wedefine the magnitude of the three-momentum p by p2 gij P

i Pj, where i and j are summed 1 to 3, then E =p2 + m2. For a comoving trajectory we have Pi = 0,

and therefore d = dt. For all others, p 1/a(t) [17], sowe can write p(t) = [a(tf)/a(t)]pf, where pf denotes thevalue of the three-momentum p at the reference time tf.Combining all this, we find

tfti

H() d =

a(tf )a(ti)

m da

m2 a2 + p2f

a2(tf)

ln

Ef + m

pf

=

1

2ln

+ 1

1

, (6)

where the inequality becomes an equality when a(ti) = 0.

Here Ef

p2f + m2 and 1/

1 v2rel, where

vrel pf/Ef is the speed of the geodesic relative tothe comoving observers at time tf. Since the integralis bounded, the argument used for null geodesics can berepeated, with the average taken over proper time.

III. The main argument. In this section we show thatthe inequalities of Eqs. (4) and (6) can be established in

arbitrary cosmological models, making no assumptionsabout homogeneity, isotropy, or energy conditions. Toachieve this generality, we need a definition of the Hubbleparameter H that applies to arbitrary models, and whichreduces to the standard one (H = a/a) in simple models.

Consider a timelike or null geodesic O (the observer).We assume that a congruence of timelike geodesics (co-moving test particles) has been defined along O [18], andwe will construct a definition for H that depends only onthe relative motion of the observer and test particles.


3/4

3

FIG. 1: The observers worldline O and two test particles.

In order to motivate what we do, we first considerthe case of nonrelativistic velocities in Minkowski space.Suppose that the observer measures the velocities of thetest particles as a function of the time on his own clock.

At time 1 particle 1 passes with velocity u(1), and attime 2 = 1 + particle 2 passes with velocity u(2).What expansion rate could he infer from these measure-ments? The separation vector between the positions r1and r2 of the two particles at 2 is r = r1 r2 = u ,and its magnitude is r |r |. Their relative velocityis u = u1 u2 = (du/d). The Hubble expansionrate is defined in terms of the rate of separation of theseparticles, which in turn depends on the radial compo-nent of their relative velocity, ur = u r/r. Theinferred Hubble parameter H is then

H ur

r=

u r

|r |2=

u (du/d)

|u |2, (7)

which will equal the standardly defined Hubble parame-ter for the case of a homogeneous, isotropic universe. Theexpression for H may be simplified to H = d ln vrel/d,where vrel = |u |. The fact that H is the total derivativeof a function of vrel implies that the variation of vrel isdetermined completely by the local value of H, even ifthe universe is inhomogeneous and anisotropic.

We can now generalize this idea to the case of arbitraryvelocities in curved spacetime. Let v = dx/d be thefour-velocity of the geodesic O, where we take to bethe proper time in the case of a timelike observer, or an

affine parameter in the case of a null observer. Let u()denote the four-velocity of the comoving test particle thatpasses the observer at time . We define uv , so inthe timelike case may be viewed as the relative Lorentzfactor (1/

1 v2rel) between u

and v. In the null case, = dt/d, where t is the time as measured by comovingobservers, and is the affine parameter of O.

Consider observations made by O at times 1 and2 = 1 + , as shown in Fig. 1, where is infinites-imal. Let r be a vector that joins the worldlines of

the two test particles at equal times in their own restframe. Such a vector is perpendicular to the worldlines,and can be constructed by projecting the vector vto be perpendicular to u: r = v + u.In the rest frame of the observer this answer reducesfor small velocities to its nonrelativistic counterpart,r = (0, ui ). This is a spacelike vector of length

r |r| = 2 , where vv is equalto 1 for the timelike case and 0 for the null case. Theseparation velocity will be u = (Du/d) , whereD/d is the covariant derivative along O. The covariantderivative allows us to compare via parallel transport vec-tors defined at two different points along O, and can be

justified by considering the problem in the free-fallingframe, for which the affine connection vanishes on O at = 1. The radial component of this velocity will beur = (ur) /r, where the sign arises from theLorentz metric. We define the Hubble parameter as [19]

H urr

=v(Du

/d)

2 . (8)

Since O is a geodesic, we have (Dv/d) = 0, andtherefore

H =d/d

2 =

d

dF

()

, (9)

where

F() =

1 null observer ( = 0)

1

2ln

+ 1

1timelike observer ( = 1)

(10)

As in Section II, we now integrate H along O from some

initial i to some chosen f:fi

H d = F(f) F(i) F(f) . (11)

In the null case F(f) = 1f , which is equal to the value

of d/dt at tf, normalized in Section II to unity.Eq. (11) therefore reproduces exactly the results of

Eqs. (4) and (6), but in a much more general context.Again we see that if Hav > 0 along any null or non-comoving timelike geodesic, then the geodesic is neces-sarily past-incomplete.

IV. Discussion. Our argument shows that null and time-

like geodesics are, in general, past-incomplete in infla-tionary models, whether or not energy conditions hold,provided only that the averaged expansion conditionHav > 0 holds along these past-directed geodesics. Thisis a stronger conclusion than the one arrived at in pre-vious work [8] in that we have shown under reasonableassumptions that almost all causal geodesics, when ex-tended to the past of an arbitrary point, reach the bound-ary of the inflating region of spacetime in a finite propertime (finite affine length, in the null case).


4/4

inflationary space times are not past-complete

Documents