Russian Physics Journal, Vol. 44, No. 12, 2001

1346 1064-8887/01/4412-1346$25.00 2001 Plenum Publishing Corporation

BRIEF COMMUNICATIONS

QUANTUM INFLATION UNIVERSE IN DILATON GRAVITATION

Yu. A. Shaido UDC 530.12

The Bruns-Dicke theory is one of the simplest examples of scalar-tensor (dilaton) gravitation, where the

background manifold is characterized by the metrics and the dilaton (i.e., by a mere scalar field). Of special interest is the

quantum dilaton gravitation (see, for review [2]). The quantum actions of the matter non-minimally associated with the

dilaton were studied in [1–5]. In particular, the quantum model of the expanding (inflation) universe with a time-dependent

dilaton was constructed in [1]. In this paper, we continue the study of the kind for the case where a dilaton potential is also

present in the classical action of the dilaton gravitation.

The standard 4D Bruns-Dicke action in the Gordan form is

4

BD

1

16

S d x g R S ,

где is the scalar Bruns–Dicke field, is the interaction constant, and S is the effect of the matter.

On performing certain matched conformal transformations of the metrics and the scalar, we may get the action in

the Einstein form

4

1exp

16 2M

R

S d x g x A L g

G

�� �� � �� � . (1)

Here 8

2 3

G

A and M

L is the Lagrangian.

We find the solution to the field equations in the form of the Friedmann Universe whose metric tensor is

determined by the following interval:

2 2 2

dS a dS , (2)

2

dS is the Minkovski’s space-time interval.

The classical action (1) on the metrics (2) is of the following form:

2 2

3

6 1exp 2 exp 2

16 2

S V d

G

. (3)

Let us add the dilaton potential V (from the metrics (2)) to the classical action (3). Hence,

Tomsk State Pedagogical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12,

pp. 86–87, December 2001. Original article submitted June 19, 2001.

1347

2 2

3

6 1exp 2 exp 2 exp 4

16 2

S V d V

G

.

Then we add the quantum action [1, 2, 5] to the above action to give

22

13 1 1 1

2 2W V d b b b .

The action (S+W) is varied over and . Note, that the quantum action is induced by quantum spinors N .

In varying, we derive the following equations of motion:

2 3

2 4

12exp 4 0 ,

3 16

exp 0 ,

3 3

A

C a a V a

G

VA A

Ca a a

�

�

where

3

42 1 ,

exp exp .

3 3

b a b a

C

a a b a

AA

a a a

� ��

� � �

�

Combining the equations of motion, we have

2 3 4 2

124 0

3 16

VA

a a a V a a a

G

. (4)

Allowing for the solution to the inflation Universe, we can obtain a partial solution as

1

a

H

,

1

1

H

.

Here H and 1

H are certain unknown constants,

2

1a

H

, 3

2a

H

, and

1

1ln

H

.

Let us choose the dilaton potential in the following form: expV B , where B is a specified constant and

is an infinitesimal parameter.

It follows from the definition V that

1348

exp 1 ,

exp 1 .

V B B B

V

B B B

Substituting the relations used in Eq. (4), we get

2

1

2

11

8

3 2 18

AB GH

H

H A GA GH

,

1

3 3

,

, ,

2 2

H D F

q q

D Q F Q

3 22 281 32

, , ,

864 3 2 3

A B b b p q a

c Q p b

ABb

3 2 2

2

64 4 27 32 2432 , ,

3 3 64 288

A B G b ba ab A Bb

q c a b

AB Gb A Bb

.

Thus, H and 1

H are expressed in terms of the initial parameter theories, in particular, the dilaton potential,

quantum field number, etc., as follows

1 1

, , , , , ,

, , , , , .

H H A B G a b b

H H A B G a b b

In the case where the dilaton potential is absent 0 , 1

H and H are reduced to the value obtained in 1 .

Thus, the quantum solution satisfying the inflation Universe with the time-dependent dilaton is derived.

Interestingly, that in expanding the Universe, the dilaton drastically (exponentially) decreases, which agrees with the well

known fact about the absence of the dilaton at currently achievable energies.

The author thanks Prof. S. D. Odintsov for the statement of the problem and fruitful discussions.

REFERENCES

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2. S. Nojiri and S. D. Odintsov, Int. J. Mod. Phys. A, hep-th 0009202 (2001).

3. S. Nojiri and S. D. Odintsov, Mod. Phys. Lett., A12, 2083 (1997), Phys. Rev., D57, 2363 (1998).

4. S. Nojiri, O. Obregon O, S. D. Odintsov, and K. E. Osetrin, Phys. Lett., B449, 173 (1999).

5. P. van Nieuwenhuizen, S. Nojiri, and S. D. Odintsov, Phys. Rev., D60, 084014 (1999).