Topological Inflation^{*}^{*}This work is
supported in part by funds
provided by the U. S. Department of Energy (D.O.E.) under contract
#DE-AC02-76ER03069.

Alexander Vilenkin^{†}^{†} On leave from
Tufts University

Center for Theoretical Physics

Laboratory for Nuclear Science

and Department of Physics

Massachusetts Institute of Technology

Cambridge, Massachusetts 02139 U.S.A.

ABSTRACT

Inflation can occur in the cores of topological defects, where the scalar field is forced to stay near the maximum of its potential. This topological inflation does not require fine-tuning of the initial conditions.

CTP#xxxx February 1994

Inflation is a state of very rapid cosmological expansion driven by the potential energy of a scalar field (called the “inflation”). The inflationary scenario was originally proposed[1] as an explanation for some very unnatural features of the initial state that was required in the standard cosmological model. Subsequent analysis has shown, however, that inflation itself requires a certain amount of fine-tuning of the initial conditions[2-4]. In models of “new inflation”[5,6] the universe has to have a region, a few horizons across, where the field is relatively smooth and its average value is very close to a local maximum of the potential . In “chaotic” inflation scenario[3], a similar region should have a value of greater than (few), where is the Planck mass. Since the latter condition is less restrictive, chaotic inflation appears to be more generic than new inflation[4]. The purpose of this letter is to make a simple observation that there exists a wide class of models where the field is forced to stay near the maximum of for topological reasons, and thus inflation of the “new” type can occur without fine-tuning of the initial state.

I begin with a simple model where is a one-component scalar field with a double-well potential, such as

Let us suppose, for the sake of argument, that the universe emerged from the quantum era in some kind of a random state and that the field is initially given by some stochastic function with a dispersion . As the universe expands, the spatial variation of will tend to be smoothed out and the magnitude of will tend to “roll” towards one of the minima of the potential at . Hence, one could expect that after a while the universe will split into domains with and , while all the variation between these two values will be confined into the walls separating the domains[2].

This is indeed what would happen in cases when the scalar field model (1) has domain wall solutions of sufficiently small thickness. The wall thickness in flat spacetime, , is determined by the balance of the gradient and potential energy, , where . This gives

and for the model (1), . Now, the horizon size corresponding to the vacuum energy in the interior of the wall is

where is the Planck mass.

If , then gravity does not substantially affect the wall structure in the transverse direction[7]. In particular, the wall thickness is not much different from its flat-space value . However, for the size of the false vacuum region inside the wall is greater than in all three directions, and it is natural to assume that this region will undergo inflationary expansion.

The condition , combined with Eqs. (2), (3), implies

We expect, therefore, that with gravity taken into account, models like (1) with a symmetry breaking scale have no domain wall solutions of fixed thickness. Instead, the walls will be smeared by the expansion of the universe, and the false vacuum regions inside the walls will serve as sites of inflation. With random initial conditions, the formation of such inflating regions appears to be inevitable. All one needs is that sufficiently large parts of the universe do not recollapse before reaching the densities .

Condition (4) does not represent a significant constraint on the parameters of the model. In fact, the same condition is necessary for a slow-rollover inflation to occur (regardless of initial conditions). To a good accuracy, the slow rollover of the field is described by the equation

where

is the Newton’s constant, and the metric is given by

The slow rollover regime assumes the conditions

which with the aid of (5), (6) can be expressed [8] as requirements for the potential ,

With from Eq. (1), the first of these requirements implies , and thus Eq. (4) does not impose any additional constraints. More generally, for values of not too close to and , one expects that , , which again leads to Eq. (4).

Cosmic strings and monopoles can also serve as sites of topological inflation. For example, the potential

with gives rise to global strings for and to global monopoles for [9]. The corresponding flat-space solutions have core radii ( is substantially smaller than within the core). As for domain walls, the condition requires that . The same mechanism could in principle work for gauge-symmetry defects. However, if had a gauge charge , the radiative corrections to the self-coupling would be , and very small values of needed to explain the isotropy of the microwave background would require unnatural fine-tuning.

The conjecture that static defect solutions in model (9) do not exist for is known to be true in the case of strings[9]. The gravitational field of a gauged U(1) string is described by an asymptotically conical metric. For the conical deficit angle is given by , where is the mass per unit length of string. As the deficit angle increases and becomes greater than , the space develops a singularity[10]. The corresponding critical value of is and has a weak dependence on the relative magnitude of scalar and gauge couplings, . The case of a global string, , is somewhat different in that its spacetime is always singular[11]. For the singularity is at a very large distance from the string core and is, therefore, unrelated to our discussion. But for the singularity encroaches upon the core and its nature is similar to that for a supermassive gauge string[10].

In the case of monopoles, the mass of the core can be estimated as , and the ratio of the core size to the Schwarzschild radius is . For the core is inside its Schwarzschild sphere, and one expects the solution to be singular. This expectation is confirmed by detailed analysis, as well as by numerical calculations[12]. For a global monopole, the solid deficit angle is[13] . This exceeds for , suggesting again that non-singular static solutions do not exist in this regime.

A related problem of the existence of static defect solutions in de Sitter space has been studied in Ref.[14], disregarding the gravitational back-reaction of the defects on the background spacetime. There, it is shown that domain walls, strings and monopoles in models (1), (9) can exist as coherent objects only if , where is the de Sitter horizon. As the flat space core size approaches its critical value , the size in de Sitter space diverges as . For , the defects are smeared by the expansion of the universe.

Global symmetries that give rise to inflating walls, strings, or monopoles do not have to be exact. An approximate discrete symmetry would result in the formation of regions of unequal vacuum energy separated by domain walls. Strings resulting from an approximate symmetry breaking get attached to domain walls, and monopoles get attached to strings. In models with this can drastically alter the cosmological evolution of these defects[8]. However, for inflation starts as soon as the defects are formed, and the approximate nature of the symmetry is unimportant.

Once started, topological inflation never ends. Although the field is driven away from the maximum of the potential, the inflating core of the defect cannot disappear for topological reasons. In fact, it can be shown that the core thickness grows exponentially with time. Taking the double-well model (1) as an example, let us consider a small region of space around the surface , where changes sign, at some (after the onset of inflation). We can choose the coordinates so that the surface lies locally in the -plane. Then we can expand the function in powers of and the potential (1) in powers of ,

where and . The following evolution of the field and of the metric is determined by Eqs. (5), (6):

where I have set at . We see from Eq. (12) that exponentially grows with time. When it becomes comparable to , the approximation (11) breaks down and Eqs. (12), (13) no longer apply.

The range of validity of Eqs. (12), (13) can be specified as , where is comfortably smaller than , say, . From (10) and (12), the boundary of this range is

The corresponding physical distance is given by

and is an exponentially growing function of time.

It is well known that “new” and “chaotic” inflation can also be eternal[15]. This is due to quantum fluctuations of the field , which can cause it to stay at large values of instead of rolling down towards the minimum. A remarkable feature of topological inflation is that it is eternal even at the classical level. As in the “new” inflationary scenario, quantum fluctuations will dominate the scalar field dynamics at sufficiently small values of . This will cause the formation of a multitude of thermalized regions inside the inflating domain. As a result, the geometry of this domain will be that of a self-similar fractal. The corresponding fractal dimension can be calculated using the technique of Ref. [16].

What do inflating defects look like from the outside? One could expect that in a model like (1), inflating walls would appear at the boundaries of thermalized regions with and . One could also expect that an observer may be able to get into the false vacuum region if she moves towards the boundary sufficiently fast. However, it can be shown [17] that the boundaries of thermalized regions are spacelike hypersurfaces. This appears to be a general feature of slow-rollover inflationary models. In our example, it is easily seen from Eq. (14) that as , indicating that the surface (which can be thought of as defining the boundary of the defect “core”) is asymptotically spacelike. Hence, the wall will appear to a “thermalized” observer not as a boundary that can be crossed, but as a spacelike hypersurface in her past.

Let us now consider the same question for an external observer in a region that never inflated. To make the question more specific, suppose that the initial expansion rate of the universe is high, so that the geometry of spacetime outside the defects is rapidly approaching a locally flat regime. We want to know what inflating defects will look like to an observer in a flat region. In the case of gauge (magnetic) monopoles, a plausible answer is that, when viewed from the outside, an inflating monopole has the appearance of a magnetically charged black hole. Solutions of Einstein’s equations describing inflating universes contained in black hole interiors have been discussed in Ref. [18]. The situation with global strings is more puzzling. For example, in the case of strings with , the static solutions of Einstein’s equations contain naked singularities, and the formation of such singularities from a non-singular initial configuration would contradict the cosmic censorship hypothesis [19]. Thus, the evolution of the exterior region of superheavy defects remains an interesting problem for future research.

After this work was completed, I noticed that the possibility of inflation in a domain wall interior has been mentioned in a recent preprint by Linde et. al.[20]. They considered a “hybrid” inflationary model with two scalar fields, and . The first period of inflation, driven by the field , results in a locally homogeneous universe with the second field nearly constant on scales comparable to the horizon, but inhomogeneous on much larger scales (due to quantum fluctuations). The second period of inflation occurs in regions where takes values near the maximum of the potential , that is, in the interiors of domain walls. The main contribution of the present paper is to give a quantitative criterion, Eq. (4), for topological inflation and to point out that it can occur with a very generic initial state.

Although plausible, the topological inflation scenario outlined in this paper requires further justification. In particular, it would be interesting to test it by numerical simulations with various initial conditions. Numerical simulations of the onset of inflations have been performed by a number of authors [21]. However, most of this work focussed on the question of whether or not cosmological expansion had enough time to smooth out the inhomogeneities of the scalar field before the domain structure would develop. It is possible that some of these simulations were terminated exactly when topological inflation was about to begin.

I am grateful to Robert Brandenberger, Larry Ford, Alan Guth and Miguel Ortiz for discussions and to Alan Guth for his hospitality at MIT where this work was completed. I also acknowledge partial support from the National Science Foundation and from the U.S. Department of Energy. After this paper was written, I learned that similar ideas have been recently suggested by Andrei Linde.

References

1A.H. Guth, Phys. Rev. D23, 347 (1981). For a review of inflation see A.D. Linde, Ref. 4; K.A. Olive, Phys. Rep. 190, 307 (1990).

2G.F. Mazenko, W.G. Unruh and R.M. Wald, Phys. Rev. D31, 273 (1985).

3A.D. Linde, Phys. Lett. B129, 177 (1983).

4A.D. Linde, Particle Physics and Inflationary Cosmology (Harwood Academic, Chiv, Switzerland, 1990).

5A.D. Linde, Phys. Lett. B108, 389 (1982).

6A. Albrecht and P.J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).

7L.M. Widrow, Phys. Rev. D39, 3571 (1989).

8P.J. Steinhardt and M.S. Turner, Phys. Rev. D29, 2162 (1984).

9For a review of topological defects, see, e.g., A. Vilenkin and E.P.S. Shellard, Cosmic Strings and Other Topological Defects (Cambridge University Press, Cambridge, 1994).

10J.R. Gott, Ap. J. 288, 422 (1985);

P. Laguna and D. Garfinkle, Phys. Rev. D40, 1011 (1989);

M.E. Ortiz, Phys. Rev. D43, 2521 (1991).

11A. Cohen and D. Kaplan, Phys. Lett. B215, 67 (1988);

R. Gregory, Phys. Lett. B215, 663 (1988);

G.W. Gibbons, M.E. Ortiz, and F.R. Ruiz, Phys. Rev. D39, 1546 (1989).

12K.-Y. Lee, V.P. Nair and E. Weinberg, Phys. Rev. Lett. 68, 1100 (1992); Phys. Rev. D45, 2751 (1992);

M.E. Ortiz, Phys. Rev. D45, R2586 (1992);

P. Breitenlohner, P. Forgács and D. Maison, Nucl. Phys. B383, 375 (1992).

13M. Barriola and A.Vilenkin, Phys. Rev. Lett. 63, 341 (1989).

14R. Basu and A. Vilenkin, Tufts University Preprint #TUTP-94-2.

15P.J.Steinhardt, in The Very Early Universe, ed. by G.W. Gibbons, S.W. Hawking and S.T.C. Siklos (Cambridge University Press, Cambridge, 1983)

A. Vilenkin, Phys. Rev. D27, 2848 (1983);

A.A. Starobinsky, in Field Theory, Quantum Gravity and Strings, ed. by M.J. de Vega and N. Sanchez (Springer-Verlag, New York, 1986);

A.D. Linde, Phys. Lett. B175, 395 (1986).

16M. Aryal and A. Vilenkin, Phys. Lett. B199, 351 (1987).

17A. Vilenkin, unpublished.

18K. Sato, M. Sasaki, H. Kodama and K. Maeda, Prog. Theor. Phys. 65, 1443 (1981); 66, 2025 (1981); Phys. Lett. B108, 103 (1982);

S.K. Blau, E.I. Guendelman and A.H. Guth, Phys. Rev. D35, 1774 (1987).

19I am grateful to Larry Ford for pointing this out to me.

20A.D. Linde, D.A. Linde and A. Mezhlumian, Stanford University Preprint #SU-ITP-93-13.

21A. Albrecht, R.H. Brandenberger and R.A. Matzner, Phys. Rev. D32, 1280 (1985); D35, 429 (1987). For an up to date review, see D.S. Goldwirth and T. Piran, Phys. Rep. 214, 223 (1992).