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«Abstract Black holes came into existence together with the universe through the quantum process of pair creation in the inflationary era. We present ...»

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Pair Creation of Black Holes During Inflation

Raphael Bousso∗ and Stephen W. Hawking†

Department of Applied Mathematics and

Theoretical Physics

University of Cambridge

Silver Street, Cambridge CB3 9EW

DAMTP/R-96/33

Abstract

Black holes came into existence together with the universe through the

quantum process of pair creation in the inflationary era. We present the

instantons responsible for this process and calculate the pair creation rate

from the no boundary proposal for the wave function of the universe. We find that this proposal leads to physically sensible results, which fit in with other descriptions of pair creation, while the tunnelling proposal makes unphysical predictions.

We then describe how the pair created black holes evolve during inflation.

In the classical solution, they grow with the horizon scale during the slow roll–down of the inflaton field; this is shown to correspond to the flux of field energy across the horizon according to the First Law of black hole mechanics.

When quantum effects are taken into account, however, it is found that most black holes evaporate before the end of inflation. Finally, we consider the pair creation of magnetically charged black holes, which cannot evaporate. In standard Einstein–Maxwell theory we find that their number in the presently observable universe is exponentially small. We speculate how this conclusion may change if dilatonic theories are applied.

∗ R.Bousso@damtp.cam.ac.uk † S.W.Hawking@damtp.cam.ac.uk 1 Introduction It is generally assumed that the universe began with a period of exponential expansion called inflation. This era is characterised by the presence of an effective cosmological constant Λeff due to the vacuum energy of a scalar field φ. In generic models of chaotic inflation [1, 2], the effective cosmological constant typically starts out large and then decreases slowly until inflation ends when Λeff ≈ 0. Correspondingly, these models predict cosmic density perturbations to be proportional to the −1 logarithm of the scale. On scales up to the current Hubble radius Hnow, this agrees well with observations of near scale invariance. However, on much larger length −1 scales of order Hnow exp(105 ), perturbations are predicted to be on the order of one.

Of course, this means that the perturbational treatment breaks down; but it is an indication that black holes may be created, and thus warrants further investigation.

An attempt to interpret this behaviour was made by Linde [3, 4]. He noted that in the early stages of inflation, when the strong density perturbations originate, the quantum fluctuations of the inflaton field are much larger than its classical decrease per Hubble time. He concluded that therefore there would always be regions of the inflationary universe where the field would grow, and so inflation would never end globally (“eternal inflation”). However, this approach only allows for fluctuations of the field. One should also consider fluctuations which change the topology of space–time. This topology change corresponds to the formation of a pair of black holes. The pair creation rate can be calculated using instanton methods, which are well suited to this non-perturbative problem.

One usually thinks of black holes forming through gravitational collapse, and so the inflationary era may seem an unlikely place to look for black holes, since matter will be hurled apart by the rapid cosmological expansion. However, there are good reasons to expect black holes to form through the quantum process of pair creation. We have already pointed out the presence of large quantum fluctuations during inflation. They lead to strong density perturbations and thus potentially to spontaneous black hole formation. But secondly, and more fundamentally, it is clear that in order to pair create any object, there must be a force present which pulls the pair apart. In the case of a virtual electron–positron pair, for example, the particles can only become real if they are pulled apart by an external electric field. Otherwise they would just fall back together and annihilate. The same holds for black holes: examples in the literature include their pair creation on a cosmic string [5], where they are pulled apart by the string tension; or the pair creation of magnetically charged black holes on the background of Melvin’s universe [6], where the magnetic field prevents them from recollapsing. In our case, the black holes will be separated by the rapid cosmological expansion due to the effective cosmological constant. So we see that this expansion, which we na¨ ıvely expected to prevent black holes from forming, actually provides just the background needed for their quantum pair creation.

Since inflation has ended, during the radiation and matter dominated eras until the present time, the effective cosmological constant was nearly zero. Thus the only time when black hole pair creation was possible in our universe was during the inflationary era, when Λeff was large. Moreover, these black holes are unique since they can be so small that quantum effects on their evolution are important. Such tiny black holes could not form from the gravitational collapse of normal baryonic matter, because degeneracy pressure will support white dwarfs or neutron stars below the Chandrasekhar limiting mass.





In the standard semi–classical treatment of pair creation, one finds two instantons: one for the background, and one for the objects to be created on the background. From the the instanton actions Ibg and Iobj one calculates the pair creation

rate Γ:

Γ = exp [− (Iobj − Ibg)], (1.1) where we neglect a prefactor. This prescription has been very successfully used by a number of authors recently [7, 8, 9, 10, 11, 12, 13] for the pair creation of black holes on various backgrounds.

In this paper, however, we will obtain the pair creation rate through a somewhat more fundamental, but equivalent procedure: since we have a cosmological background, we can use the Hartle–Hawking no boundary proposal [14] for the wave function of the universe. We will describe the creation of an inflationary universe by a de Sitter type gravitational instanton, which has the topology of a four–sphere, S 4.

In this picture, the universe starts out with the spatial size of one Hubble volume.

After one Hubble time, its spatial volume will have increased by a factor of e3 ≈ 20.

However, by the de Sitter no hair theorem, we can regard each of these 20 Hubble volumes as having been nucleated independently through gravitational instantons.

With this interpretation, we are allowing for black hole pair creation, since some of the new Hubble volumes might have been created through a different type of instanton that has the topology S 2 × S 2 and thus represents a pair of black holes in de Sitter space [15]. Using the framework of the no boundary proposal (reviewed in Sec. 2), one can assign probability measures to both instanton types. One can then estimate the fraction of inflationary Hubble volumes containing a pair of black holes by the fraction Γ of the two probability measures. This is equivalent to saying that Γ is the pair creation rate of black holes on a de Sitter background. We will thus reproduce Eq. (1.1).

In Sec. 3.1 we follow this procedure using a simplified model of inflation, with a fixed cosmological constant, before going to a more realistic model in Sec. 3.2. In Sec. 3.3 we show that the usual description of pair creation arises naturally from the no boundary proposal, and Eq. (1.1) is recovered. We find that Planck size black holes can be created in abundance in the early stages of inflation. Larger black holes, which would form near the end of inflation, are exponentially suppressed.

The tunnelling proposal [16], on the other hand, predicts a catastrophic instability of de Sitter space and is unable to reproduce Eq. (1.1).

We then investigate the evolution of black holes in an inflationary universe.

In Sec. 4 their classical growth is shown to correspond to energy-momentum flux across the black hole horizon. Taking quantum effects into account, we find in Sec. 5 that the number of neutral black holes that survive into the radiation era is exponentially small. On the other hand, black holes with a magnetic charge can also be pair created during inflation. They cannot decay, because magnetic charge is topologically conserved. Thus they survive and should still be around today. In Sec. 6, however, we show that such black holes would be too rare to be found in the observable universe. We summarise our results in Sec. 7. We use units in which mP = ~ = c = k = 1.

2 No Boundary Proposal We shall give a brief review; more comprehensive treatments can be found elsewhere [17]. According to the no boundary proposal, the quantum state of the universe is defined by path integrals over Euclidean metrics gµν on compact manifolds M. From this it follows that the probability of finding a three–metric hij on a spacelike surface Σ is given by a path integral over all gµν on M that agree with hij on Σ. If the spacetime is simply connected (which we shall assume), the surface Σ will divide M into two parts, M+ and M−. One can then factorise the probability of finding hij into a product of two wave functions, Ψ+ and Ψ−. Ψ+ (Ψ− ) is given by a path integral over all metrics gµν on the half–manifold M+ (M−) which agree with hij on the boundary Σ. In most situations Ψ+ equals Ψ−. We shall therefore drop the suffixes and refer to Ψ as the wave function of the universe. Under inclusion of

matter fields, one arrives at the following prescription:

–  –  –

where R is the Ricci-scalar, Λ is the cosmological constant, and K is the trace of Kij, the second fundamental form of the boundary Σ in the metric g.

The wave function Ψ depends on the three–metric hij and on the matter fields Φ on Σ. It does not however depend on time explicitly, because there is no invariant meaning to time in cosmology. Its independence of time is expressed by the fact that it obeys the Wheeler–DeWitt equation. We shall not try to solve the Wheeler– DeWitt equation directly, but we shall estimate Ψ from a saddle point approximation to the path integral.

We give here only a brief summary of this semi–classical method; the procedure will become clear when we follow it through in the following section. We are interested in two types of inflationary universes: one with a pair of black holes, and one without. They are characterised by spacelike sections of different topology. For each of these two universes, we have to find a classical Euclidean solution to the Einstein equations (an instanton), which can be analytically continued to match a boundary Σ of the appropriate topology. We then calculate the Euclidean actions I of the two types of solutions. Semiclassically, it follows from Eq. (2.1) that the wave function is given by Ψ = exp (−I), (2.3) where we neglect a prefactor. We can thus assign a probability measure to each type

of universe:

P = |Ψ|2 = exp −2I Re, (2.4) where the superscript ‘Re’ denotes the real part. As explained in the introduction, the ratio of the two probability measures gives the rate of black hole pair creation on an inflationary background, Γ.

3 Creation of Neutral Black Holes The solutions presented in this section are discussed much more rigorously in an earlier paper [18]. We shall assume spherical symmetry. Before we introduce a more realistic inflationary model, it is helpful to consider a simpler situation with a fixed positive cosmological constant Λ but no matter fields. We can then generalise quite easily to the case where an effective cosmological “constant” arises from a scalar field.

3.1 Fixed Cosmological Constant 3.1.1 The de Sitter solution First we consider the case without black holes: a homogeneous isotropic universe.

Since Λ 0 its spacelike sections will simply be round three–spheres. The wave function is given by a path integral over all metrics on a four–manifold M+ bounded by a round three–sphere Σ of radius aΣ. The corresponding saddle point solution is the de Sitter space–time. Its Euclidean metric is that of a round four–sphere of

radius 3/Λ:

ds2 = dτ 2 + a(τ )2dΩ2, (3.1)

–  –  –

This describes a Lorentzian de Sitter hyperboloid, with τ Im serving as a Lorentzian time variable. One can thus construct a complex solution, which is the analytical continuation of the Euclidean four–sphere metric. It is obtained by choosing a contour in the complex τ –plane from 0 to τ Re = Λ π and then parallel to the imaginary τ –axis. One can regard this complex solution as half the Euclidean four– sphere joined to half of the Lorentzian de Sitter hyperboloid (Fig. 1).

–  –  –

Figure 1: The creation of a de Sitter universe. The lower region is half of a Euclidean four{sphere, embedded in ve{dimensional Euclidean at space. The upper region is a Lorentzian four{hyperboloid, embedded in ve{dimensional Minkowski space.

–  –  –

3.1.2 The Schwarzschild–de Sitter solution We turn to the case of a universe containing a pair of black holes. Now the cross sections Σ have topology S 2 ×S 1. Generally, the radius of the S 2 varies along the S 1.

This corresponds to the fact that the radius of a black hole immersed in de Sitter space can have any value between zero and the radius of the cosmological horizon.

The minimal two–sphere corresponds to the black hole horizon, the maximal two– sphere to the cosmological horizon. The saddle point solution corresponding to this topology is the Schwarzschild–de Sitter universe. However, the Euclidean section of this spacetime typically has a conical singularity at one of its two horizons and thus does not represent a regular instanton. This is discussed in detail in the Appendix.

There we show that the only regular Euclidean solution is the degenerate case where the black hole has the maximum possible size. It is also known as the Nariai solution

and given by the topological product of two round two–spheres:

–  –  –



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