725

Russian Physics Journal, Vol. 56, No. 7, December, 2013 (Russian Original No. 7, July, 2013)

ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY

NEW CLASS OF COSMOLOGICAL SOLUTIONS FOR A SELF-INTERACTING SCALAR FIELD

A. A. Chaadaev and S. V. Chervon UDC 530.12

New cosmological solutions are found to the system of Einstein scalar field equations using the scalar field φ as the argument. For a homogeneous and isotropic Universe, the system of equations is reduced to two equations, one of which is an equation of Hamilton–Jacobi type. Using the hyperbolically parameterized representation of this equation together with the consistency condition, explicit dependences of the potential V of the scalar field and the Hubble parameter H on φ are obtained. The dependences of the scalar field and the scale factor a on cosmic time t have also been found. It is shown that this scenario corresponds to the evolution of the Universe with accelerated expansion out to times distant from the initial singularity.

Keywords: cosmology, scalar fields, exact solutions, accelerated expansion of the Universe.

INTRODUCTION

Cosmological inflation models have served as a new stimulus for research into gravitating scalar fields with a self-interacting potential. As pioneering works in this direction we can cite studies by G. G. Ivanov [1], who in fact was the first to obtain an exact inflationary solution 0

Hta a e= . Further steps in this direction were made in the works of J. D. Barrow [2] and A. G. Muslimov [3] (where, by the way, Ivanov’s results [1] are cited). It can be stated that in the 1990s new directions for constructing exactly solvable inflation models arose, which can be arbitrarily divided according to the following methods of posing the problem: 1) The direct problem, where it is assumed that the potential of the self-interacting scalar field is given, based on predictions of elementary particle physics and quantum field theory. The solution of problems in such an approach was achieved in [1–4]. 2) The method of exact adjustment of the potential and evolution of the scalar field, in which the scale factor is treated as an initial datum that is predicted by astrophysical observations. It should be noted that this method was first suggested in [5] in 1991, then reopened in [6] and later in [7]. 3) The method of finding the scale factor and adjusting the potential for a given evolution of the scalar field [2]. 4) The method of comparative analysis of approximate and exact solutions [8], which, in particular, led to the possibility of investigating models with a potential of unbounded growth [9]. 5) The superpotential method [10, 11], based on the method of the total energy potential. The present paper proposes a fundamentally new method for constructing exact cosmological solutions, based on representing the equations of scalar cosmology in Hamilton–Jacobi form and investigating the dependence of the Hubble parameter H on the scalar field φ.

SCALAR COSMOLOGY IN THE HAMILTON–JACOBI REPRESENTATION

In cosmological models with scalar fields, solutions exist with logarithmic and linear time dependence [12, 13]. Therefore the idea of using the scalar field as the evolutionary variable for the equations of cosmological dynamics is

Ul’yanovsk State Pedagogical University, Ul'yanovsk, Russia, e-mail: alexandr308mail.ru; chervon.sergey@gmail.com. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 3–7, July, 2013. Original article submitted March 28, 2013.

1064-8887/13/5607-0725 ©2013 Springer Science+Business Media New York

726

entirely justified. We consider a homogeneous and isotropic Universe in the class of Friedmann–Robertson–Walker metrics:

( )2

2 2 2 2 2 2 22 sin

1drds dt a r d d

r

⎛ ⎞= − + θ + θ ϕ⎜ ⎟

− ε⎝ ⎠. (1)

The system of Einstein equations and a self-interacting scalar field in the case of a spatially flat Universe

(ε = 0), taking into account the definition of the Hubble parameter ,d daH da dt

⎛ ⎞= =⎜ ⎟⎝ ⎠

, can be represented in the form

2

2 2

1 , (2.1)2

13 ( ) , (2.2)2

3 0, (2.3)

H

H V

H V

⎧ = − κϕ⎪⎪⎪ ⎛ ⎞+ Λ = κ ϕ + ϕ⎨ ⎜ ⎟

⎝ ⎠⎪⎪ ′ϕ + ϕ + =⎪⎩

where Н and φ are functions of cosmic time t, and the prime denotes the derivative with respect to φ. System of equations (2.1)–(2.3) is overdetermined: any one of these equations is a consequence (or differential consequence) of the other two [15].

Having H depend on φ leads to transformation of Eq. (2.1) to the following form:

12

H ′ = − κϕ . (3)

Equation (2.2) then acquires the form

2 2( )H H V Λ′α − β + =κ

, (4)

where 22

α =κ

, 3β =

κ, Λ is the cosmological constant, and κ is Einstein’s gravitational constant.

Thus, the problem of finding the Hubble parameter H for a cosmological scalar field reduces to solving equation of Hamilton–Jacobi type (4) and subsequent recovery of the dependence of H on cosmic time t using Eq. (3).

Solution of the equation of Hamilton–Jacobi type

According to the weak energy condition, the potential energy should be positive, regardless of the choice of its constant component, i.e.,

0 00,V V V Λ− > =

κ.

Here, for purposes of simplification, we have set the cosmological constant Λ as proportional to the constant component of the potential ( ).V ϕ

We reduce Eq. (4) to the form

727

2 2

0 0

( ) 1H HV V V V

′β − α =

− −. (5)

We introduce new notation:

2 20 0( ) ( ), , ( ) , ( )V V V VdHy x H b ad

ϕ − ϕ −= = ϕ = ϕ =

ϕ α β.

As a result, Eq. (5) transforms to the canonical equation of a hyperbola with variable coefficients which depend on the parameter φ:

2 2

1( ) ( )x y

a b⎛ ⎞ ⎛ ⎞− =⎜ ⎟ ⎜ ⎟ϕ ϕ⎝ ⎠ ⎝ ⎠

. (6)

In parameterized form, Eq. (6) has the standard form

cosh ,

sinh ,

x a

y b

= λϕ⎧⎪⎨⎪ = λϕ⎩

or, reverting to the initial quantities, we obtain

0

0

cosh , (7.1)

sinh , (7.2)

V VH

V VdHd

⎧ −= λϕ⎪ β⎪⎪

⎨⎪ −⎪ = λϕ

ϕ α⎪⎩

where λ is a constant (a parametric constant), analogous to the constant used in the dependence of fields on cosmic time t [9].

This system is consistent only for a certain value of ( )V ϕ . The necessary form of the potential is found from the consistency condition for system (7.1)–(7.2). Differentiating Eq. (7.1) with respect to φ and equating the result to expression (7.2), we obtain

2(1 )0( ) (cosh )V V C − −δϕ = + λϕ , (8)

where 1 3 const.δ = =λ κ

The obtained form (8) of the potential ( )V ϕ ensures consistency of system (7.1)–(7.2). It is easy to show that substituting expression (8) into Eq. (4) with allowance for Eqs. (7.1)–(7.2) also gives a valid equality. It is noteworthy that such an expression for potential (8) is encountered in a number of works in which this form is introduced into the system as given (see, e.g., [16]). In our case, the solution was obtained on the basis of an analysis of system of equations (2.1)–(2.3), transformed to a dependence of quantities on the scalar field.

Taking into account that violation of the weak energy condition is possible, in particular for wormholes and in the inflation stage of the evolution of the Universe, let us consider the case

0 0V V− < .

728

In an analogous way, the consistency conditions for a system similar to (7.1)–(7.2) lead to the potential

2(1 )0( ) (sinh )V V C − −δϕ = − λϕ . (9)

Note that such a form is also mentioned in some other works (see, e.g., [14]). We have the possibility of comparing our solution with results obtained earlier. Starkovich and Cooperstock

[15] obtained a solution for the potential ( )V V= ϕ of the following form:

1/( ) ( )V C e e−λϕ λϕ ωϕ = + χ . (10)

It can be shown that expression (9) reduces to expression (10) if we assume that

010, 2(1 ) const, 1V = = − − δ = χ =ω

.

Thus we have shown that for the cosmological constant Λ equal to zero or, setting 0 0,V = our solutions coincide with those obtained earlier in [15]. Note that Eq. (7.2), allowing for solution (8), yields

( ) (cosh )3CH δκ

ϕ = λϕ . (11)

Now our problem consists in recovering the evolution of the scalar field in time. Introducing the notation sinh ,ξ = λϕ , Eq. (3) yields the relation

2 2

2

(1 )

d Ctδξ

= −λ

− ξ ξ

∫ . (12)

Calculating the integral on the left-hand side of Eq. (12) without any special choice of the parametric constant leads to a solution in terms of the hyperbolic function:

22

2 1 * *22 2

1 11, ; 1; 2 ( ), const

2 2 2( 1)

GF C t t t

δ

δ

⎛ ⎞+⎜ ⎟

⎛ ⎞δ δ δξ⎝ ⎠− + − = −λ − =⎜ ⎟ξ⎝ ⎠

δ ξ +

. (13)

The latter expression gives a general relation between the scalar field and time.

In the particular case when δ = 2 1 3in this case2 2

⎛ ⎞κλ =⎜ ⎟

⎝ ⎠, integral (12) is taken in terms of elementary

functions. Thus, the dependence of the field on time has the form

*2 32 arccosh exp ( )

3 2C t t

⎡ ⎤⎛ ⎞κϕ = −⎜ ⎟⎢ ⎥κ ⎝ ⎠⎣ ⎦

. (14)

The evolution of the scale factor with allowance for the definition of the Hubble parameter and Eq. (11) is written as

729

0 exp (cosh ( )3Ca a t dtδ⎡ ⎤κ

= λϕ⎢ ⎥⎣ ⎦

∫ . (15)

Integrating expression (15), it is possible to obtain the form of the explicit dependence of ( )a ϕ :

[ ]2

*30 sinha a −= λϕ . (16)

The dependence on cosmic time t is expressed by the formula

1/3

*0 *

3exp ( ) 12

Ca a t t⎛ ⎞⎧ ⎫κ

= − −⎜ ⎟⎨ ⎬⎩ ⎭⎝ ⎠

. (17)

Analogous calculations in regard to potential (9) lead to the following results. The Hubble parameter and the solution of Eq. (3) are in general expressed as

( ) (sinh )3CH δκ

ϕ = λϕ , (18)

22

2 1 *22 2

111, ; 1; 2 ( )

2 2 2( 1)

GF C t t

δ

δ

⎛ ⎞−⎜ ⎟

⎛ ⎞δ δ δη⎝ ⎠− + = −λ −⎜ ⎟η⎝ ⎠

δ η −

. (19)

In the particular case, for δ = 2, we have the same solution for the field φ:

*2 32 arccosh exp ( )

3 2C t t

⎛ ⎞⎛ ⎞κϕ = −⎜ ⎟⎜ ⎟κ ⎝ ⎠⎝ ⎠

. (20)

The scale factor is given by the following formulas:

[ ]2

*30 cosha a −= λϕ , (21)

1/3

*0 *

31 exp ( )2

Ca a t t⎛ ⎞⎧ ⎫κ

= − −⎜ ⎟⎨ ⎬⎩ ⎭⎝ ⎠

. (22)

CONCLUSIONS

We have found new classes of exact cosmological solutions corresponding to two types of potentials, the form of which was dictated by the continuity condition for the equations of the dynamics of the scalar field, represented by an equation of Hamilton–Jacobi type. Requiring positivity of the scale factor, we observe that solution (17) is valid for

*t t> while solution (22) is valid for *.t t< Combining these solutions, it is possible to describe the behavior of the

Universe in the following way. At the zero moment of time, the Universe has a finite radius ** 0( )a t a< and it

monotonically falls to zero at *t t= according to solution (22). Then, according to solution (17), it grows from zero to infinity with deceleration of expansion of the Universe changing over into acceleration at the inflection point

730

exp *2 4ln 1

3 3 3t t

C C⎡ ⎤= + +⎢ ⎥⎣ ⎦κ κ

. Considering the case when * 0,t = we discard solution (22) as unphysical. Thus, the

Universe evolves, according to Eq. (17), from a singular state at 0t = and embarks upon an accelerated expansion after the time expt , whereby the scale factor a →∞ as t →∞ . Such a solution corresponds to the scenario that includes

accelerated expansion of the Universe, where the role of dark matter is played by the scalar field with self-action potential 2(1 )

0( ) (cosh ) .V V C − −δϕ = + λϕ This work was carried out within the framework of a State Order of the Ministry of Education and Science of

the Russian Federation in accordance with Project No. 2.7621.2013.

REFERENCES

1. G. G. Ivanov, in: Gravitation and the Theory of Relativity [in Russian], V. R. Kaigorodov, ed., Kazan’ State University Press, Kazan’ (1981), Vol. 18, pp. 54–60.

2. J. D. Barrow, Phys. Lett., B187, 12–16 (1987); A. B. Burd and J. D. Barrow, Nucl. Phys., B308, 929–945 (1988); J. D. Barrow, Phys. Rev., D49, 3055 (1994).

3. A. G. Muslimov, Class. Quantum Grav., 7, 231 (1990). 4. J. J. Halliwell, Phys. Lett., B187, 341–344 (1987). 5. G. Ellis and M. Madsen, Class. Quantum Grav., 8, 667–676 (1991). 6. S. V. Chervon and V. M. Zhuravlev, Russ. Phys. J., 39, No. 8, 776–789 (1996). 7. T. Padmanabhan, аrXiv:hep-th/0204415. 8. S. V. Chervon and V. M. Zhuravlev, Russ. Phys. J., 43, No. 1, 11–17 (2000); V. M. Zhuravlev and

S. V. Chervon, Zh. Eksp. Teor. Fiz., 118, No. 2, 259–272 (2000). 9. S. V. Chervon, Gen. Relativ. Gravit., 36, 1547–1553 (2004).

10. S. V. Chervon, O. G. Panina, and M. Sami, Vestnik Samarsk. Gosud. Tekhn. Univ., Ser. Fiz-Mat. Nauki, No. 3, 221–226 (2010).

11. A. V. Yurov, V. A. Yurov, S. V. Chervon, and M. Sami, Teor. Mat. Fiz., 166, No. 2, 258–268 (2011). 12. S. V. Chervon, Nonlinear Fields in Gravitation Theory and Cosmology [in Russian], Ul’yanovsk State

University Press, Ul’yanovsk (1997). 13. L. P. Chimento and A. S. Jakubi, arXiv: gr-qc/950615 v1 7 Jun 1995. 14. L. A. Urena-Lopez and T. Matos, arXiv: astro-ph/0003364 v1 23 Mar 2000. 15. S. P. Starkovich and F. I. Cooperstock, Astrophys. J., 398, 1–11 (1992).