PHYSICAL REVIEW D VOLUME 52, NUMBER 2 15 JULY 1995

Towards quantum cosmology without singularities

Klaus Behrndt* and Thomas T. Burwickt Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309

(Received 13 February 1995)

In this Brief Report we investigate a cosmological solution with singular scalar fields (a moduli field and the dilaton). We show that after quantizing the scalar fields the quantum corrections smooth out the singularities. In addition, we argue that the incorporation of nonperturbative quantum corrections form a dilaton potential.

PACS number(s): 98.80.Hw, 04.20.Jb, 11.25.-w

We consider a cosmological string solution, which has classical singularities (big bang). Near these singularities, the theory factorizes into a smooth spherical part and a singular two-dimensional (2D) part. This singular part is the well-known dilaton gravity (see, e.g., Refs. [I-4]), and as a first step we are going to quantize this part with the result that all singularities disappear. This procedure is also known as s-wave reduction and has so far been used for 4D black hole (BH) physics.

Classical theory. Our 4D classical model is given by

where 4 is a dilaton field, HPux = d[,Bux1 is the torsion corresponding to the antisymmetric tensor field B,,, and p is a modulus field. A cosmological solution to this model is given by [5]

After a time reparametrization, one obtains the standard Friedmann-Robertson-Walker (FRW) metric with dR2 as the three-dimensional (3D) volume form corresponding to the spatial curvature k (= 0, -1, +I ) . The parameter t - is the minimal extension and, for k = 1, the parameter t+ denotes the maximal extension of the Universe. This is obvious after transforming the solution to the conformal time for which the metric is

(3)

Unfortunately, in Ref. [5] no analytic results for the world radius a ( ~ ) in the standard parametrization of the FRW

metric ds2 = - d ~ ' + a ( ~ ) ~ d n ; could be found. We have plotted numerical results in Fig. 1. Figure l ( a ) shows the oscillating solution for k = 1 and Fig. l ( b ) shows the wormhole solution for k = -1. For k = 0, the so- lution has again the geometry of Fig. l ( b ) but with the difference that there are no asymptotic flat regions as for k = -1. Remarkably, the scalar fields p and 4 have diver- gences, although the metric behaves completely smooth for all times. To understand this phenomena, one has to go back to the 5D origin. In the sense of a Kaluza-Klein approach solution, Eq. (2) can be obtained by dimen- sional reduction of a 5D theory. Then, the modulus field p corresponds to a time-dependent compactification ra- dius of the fifth coordinate. The corresponding action is given by the effective string action

and the 5D solution can be written as

ds2 = (A/ tan f i ) 2 d ~ 2

+ i t > (t: - kt?)(sin fi /&)2)[-dr12 + dn;],

The 5D dilaton is related to 4 by 21) = 24 + l n x and p = G,,. For k = 1 and after switching the signature of the metric (dw2 t -dw2 and dV2 t dV2), this 5D solution is just the 5D BH solution (in the conformal time q) discussed by Horowitz and Strominger in Ref. [6]. Here, t i define the two horizons of the theory and our cosmological solution lives between these horizons.

s-wave reduction. We are particularly interested in the fate of the singularities if one starts to quantize the the- ory. Therefore, it is reasonable to restrict ourselves to

'Electronic address: behrndt@jupiter.slac.stanford.edu FIG. 1. (a) Plot of the closed oscillating solution for k = 1; t~lectronic address: burwick@slacvm.slac.stanford.edu (b) is the wormhole solution for k = -1.

0556-2821/95/52(2)/1292(4)/$06.00 52 1292 01995 The American Physical Society

52 - BRIEF REPORTS 1293

the region near the singularities (sin &q 2 0). In this region, one can assume that quantum corrections become important. Furthermore, as one can see from Eq. (5), in this limit the 5D solution decouples in a 3D spherical part ( N do:) and a 2D (w, q) part, which is the known dual 2D BH [7]. In the figure, it is just the region of minimal extension, e.g., inside the wormhole of Fig. l (b ) .

Before we can start to quantize the 2D part, we have to reduce the 5D action (in Ref. [4]) down to a 2D the- ory. This procedure is motivated by the assumption that the quantum corrections respect the spherical symmetry. Generally, this is not the case, but it is sufficient for a first approximation. In BH physics, this procedure is also known as s-wave reduction. For the 5D metric we make the ansatz

where 9 2 ) is the 2D metric part. In what follows, we

quantize only 9 2 ) and the dilaton q5 (or 4, respectively, see below). The remaining fields are assumed to be clas- sical backgrounds given by Eqs. (2) or (6). After inte- grating out the angular degrees of freedom and using the H field from Eq. (2) we obtain, for Eq. (4),

with 4 = $ + g X and V(X) = 6keP2X - 2t:t5ep6X. Near the singularity (q E O), the background field x is smooth, d,xlo = 0 [see Eqs. (5) and (6)] and up to the second order in q we can approximate the x terms by a constant

with X = (2/t?)[3k - (t+/t-)2]. A classical solution in conformal coordinates is given by Ref. [7]:

where u is constant. This solution can be transformed to the 2D (w,q) part of Eq. (5), where q 2 0 corresponds to u E Xz+z-.

Quantization. The quantization of Eq. (8) has been studied in various papers (see Refs. [l-4,8,9]). Each of these models will be discussed but, before we do so in detail, let us come back to the classical solution once more. The fact that Eqs. (5) and (2) are independent of the fifth coordinate leads to the dual solution

ds2 = (tan d Z q / ~ ' E ) ~ d w ~

+{t> (t: - kt?) (sin f i / h ) 2 ) [ - d q 2 + do:],

Both solutions have a significant difference. Singular

points of Eq. (5) are regular in Eq. (1) and vice versa. Furthermore, in the region that we are interested in (sin &q 2 0) the solution (5) is in the strong-coupling region (e2@ -+ co), whereas the dual solution (10) is in the weak-coupling region (e2@ << 1; if we assume that t+ >> t- , which is reasonable, since t+ corresponds to the maximal and/or minimal extension of the Universe). Again, the 2D part decouples and can be transformed into Eq. (9), where q .v 0 corresponds to z+z- E 0.

When quantizating this theory, we are especially in- terested in what happens in the strong-coupling region, i.e., the fate of the singularity in Eq. (5). As a con- sistency condition of this procedure, we have to ensure that in the classical limit (weak-coupling region) we get back our classical result Eq. ( lo ) , which is nonsingular for q Y 0. We are following the procedure of de Al- wis [8]; later we discuss the modification concerning the other models. After choosing the conformal gauge (gab = e20gab) we can rewrite (8) as a general 2D a model:

with X, = (4, a ) . Thus, the quantization of the dila- ton gravity is reduced to the quantization of a 2D a model with the target space spanned by 4 and a. This model, however, is well defined only if the background fields G,,, 9, and T define a 2D conformal field the- ory. This symmetry is a consequence of the fact that the original theory depends only on g and not on g, and thus has to respect the symmetry j -+ e2Pg and a -+ a - p. We transform the theory to an exact model and define the quantum theory by this (exact) conformal field the- ory (see, e.g., [8,9]). Following this approach, we first note that the target space metric G,, has the general structure

dS2 = -4eC2@[1 + h(4)]dqh2

+4eC2@[1 + L(4)lda d4 + K do2, (12)

where h and h are model-dependent functions of 4 or X1. For h = h = 0 we have the Callan-Giddings-Harvey- Strominger (CGHS) model [I]; for 2h = L = -e2@, we have the model from Strominger [3]; h = 0 and h = -(n/4)e2@ describes the Russo-Susskind-Thorlacius (RST) model [4]. The parameter K = (24 - N ) / 6 orig- inates from the definition of the functional integration measure and N corresponds to additional conformal mat- ter. As the next step, we introduce new target space coordinates

x = ( 2 1 6 ) J d4ep2@ J ( l+ L)Z + ~ e Z @ ( l + h),

and obtain a flat metric (for negative K , we have to per- form a Wick rotation in x and y). For a flat metric, it is easy to find the dilaton 9 and tachyon T that define a conformal field theory. The general solution of the cor- responding /3 equations is

BRIEF REPORTS

@(x) = ax + by with a2 - b2 = -K, (14)

T(x) - with + ( a 2 - ,B2) - aa + b,B - 2 = 0.

The demand to get the classical model (8) in the weak- coupling limit yields a further restriction to 4 and T. Following the suggestion of de Alwis, we set a = 0 and a = -0 = -2/J K and get the known Liouville theory (y as Liouville field) that couples to the matter field x:

This describes a well-defined 2D gravity theory on the classical as well as on the quantum level. The strategy is to define the quantum theory in terms of this actian and to regard Eq. (8) as the classical limit.

Solving the equations of motion, we have to restrict ourselves to solutions that reproduce the BH solution (9) in the classical limit. Therefore, we are interested in a solution depending on z+z- only and find

(u=const). Using the transformation in Eq. (13), we can express this solution in 4 and a. In doing so, we have to fix the up-to-now arbitrary functions h(4) and A($). Let us start with parametrization suggested by de Alwis: h = 0, TL = -l/ce24. 2 This choice is motivated by the fact that for all values of 4 and u the transformation in Eq. (13) is nonsingular and also that the range of x and y goes from -oo to +oo if 4 and u do so. For x and y, one gets

In terms of Eq. (16), one finds in the weak-coupling limit e24 << 1 the desired classical solution Eq. (9):

Since we are in the weak-coupling region, this solution corresponds to our dual solution in Eq. ( lo) , which is nonsingular for 77 = 0. In the strong-coupling limit e2" 1, we obtain

Therefore, after incorporation of quantum corrections [= O(e24)] the black hole solution gets smooth also in the strong-coupling region. Note that in dilaton grav- ity, a singularity in the metric has to be accompanied by a singularity in the dilaton; i.e., singularities can only appear in the strong- or weak-coupling region. For the other models, the picture is qualitatively the same. In the CGHS model (h = 71 = 0) one obtains, for Eq. (13),

x = - ( 1 / f i ) e P 2 4 d m

- ( 4 / 2 ) In [x + 2eP2"1 + J-)] , (20)

y = -&[u - ( l / ~ ) e - ~ + ] ,

and in strong-coupling region this model gives

e-@ N u - XZ+Z- , u = ( ~ / K ) ( u - Xz+z-). (21)

For the Strominger model (2h = TL = -e24) we find

F [4] = d e - 4 + - (2 - ~)e-~@' + (2 - ~ ) / 2 , which gives, in the strong-coupling region,

(23)

And, finally, in the RST model [h = 0, = -(K/4) e24], the general solution is given by

In the strong-coupling region, this model behaves as

Therefore, we find that all models have no singularities in the strong-coupling region and yield the classical result in the weak-coupling region.

One may now study the influence of this quantization procedure for the further evolution of the universe. For the derivation of our results, we used that the solution de- couples in a 2D (dilaton gravity) part and a 3D spherical part. This was only the case, e.g., inside the wormhole of Fig. l (b ) . Extending this procedure to the region away from the wormhole seems to be difficult, but neverthe- less, quantum correction inside the wormhole can form a dilaton potential that could be a source of an inflationary period a t later times. A dilaton potential in our original action in Eq. (1) or Eq. (4) corresponds to an additional tachyon contribution in the 2D action, which is indepen- dent of X [since X was correlated to the constant x field in the wormhole; see Eq. (7)]. The tachyon that we dis- cussed so far is only one possibility. The most general tachyon field, however, is a combination of contributions given by Eq. (14). But there is also another parametriza- tion for the tachyon. Using the mass-shell condition, we can replace a or and then make a power expansion. In

52 BRIEF REPORTS - 1295

order to get the right classical limit, we set a = 0. Since the tachyon /3 function is a linear equation in T every term in this expansion does also obey the /3 equation. This procedure was described by Kawai and Nakayama [ll] for constructing new vertex operators. We find an infinite set of vertex operators or tachyon fields, which are parametrized by two integers, m and n. If we restrict ourselves to n = (24 - N) /6 = 4 (i.e., N = 0) these additional terms are

Instead of Eq. (14), we have now, as general tachyon field T(x ,Y) ,

T (x ,y )=Xexp - x - y ) (A( )

where the functions x and y are given by Eq. (13) (the term Tz(0) was already discussed e.g., in Refs. [8,10]).

A remarkable property of these terms is that in the classical limit (q5 t -co) they show the typical nonper- turbative structure

T2,3 e-e-20/2 e-1/(29:), (28)

where g, = e+ is the string-coupling constant [we have used that h, h = O(e2+) because they are quantum cor- rections]. On the other hand, in the strong-coupling re- gion (q5 + co) we have

T2 N e4+ + co, T3 N e-'+ -+ 0. (29)

Therefore, these terms vanish very rapidly in the weak- coupling (classical) region and become important in the strong-coupling region. Furthermore, since x and y are functions of the dilaton and moduli field these addi- tional tachyon terms represent a dilaton-moduli poten- tial created by nonperturbative quantum corrections in the strong-coupling region. If we insert x and y for the several models Eqs. (17), (20), (22), and (24), we obtain different potentials.

Discussion. Starting with a classical solution of the low-energy string effective action, we investigated the quantization near the singularity. Via a 5D Kaluza-Klein approach, this solution was obtained as a dimensional reduced theory. Near the singularity, the 5D theory de- couples into a 3D nonsingular (spherical) part and a sin- gular 2D part. As a first step, we have quantized only this singular 2D part (s-wave reduction). The results

in Eqs. (17)-(25) show that, for all models, the singu- larity disappears after the quantization of the theory, i.e., the 2D metric part and the dilaton remains finite. An interpretation of this result is that the wormhole be- comes traversable via quantum corrections. In addition, we have shown that the incorporation of nonperturba- tive quantum corrections form a dilaton/moduli poten- tial. We presented the general structure of the possible potential. It remains an open question for further inves- tigations to discuss whether this potential can drive an inflation.

We used the 5D theory to get contact with the known dilaton gravity, but it is also possible to consider the 4D theory in Eq. (1) directly. Our approximation to quan- tize only the divergent 2D part in five dimensions is ef- fectively the same as to quantize the dilaton and moduli matter fields only. Thus, from the 4D point of view, we replaced the dilaton and moduli contributions in the Einstein equation by its vacuum expectation value

where GE) = e-'+G,, is the metric in the Einstein frame. Therefore, our discussion is a semiclassical ap- proximation but a full quantum theory with respect to the scalar fields. Classically, the 4D string metric was smooth, but the Einstein metric was singular (caused by the dilaton and moduli). However, after quantiza- tion the singularities in the scalar fields disappeared and thereby also the Einstein metric turned out to be non- singular. This implies that, similar to the string frame, the Einstein metric describes a universe. which starts and ends (for k = 1) not with a singularity 'but with a mini- mal (nonzero) extension (wormhole). Therefore, in both frames the spatial part of the universe is qualitatively given in Fig. 1.

An open question is to investigate the stability against fluctuation in the scalar fields and additional matter. Moreover, the quantization of the scalar fields near the singularity can only be a first step and future investiga- tions have to show whether a complete quantum theory will leave the qualitative feature of the described solution intact.

We would like to thank S. Forste and J. Garcia-Bellido for useful comments. The work of K.B. was supported by a DAAD grant and of T.T.B. by the Swiss National Science Foundation. The work of both authors was also supported by U.S. Department of Energy, Contract No. DE-AC03-76SF00515.

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