Arbitrary scalar field and quintessence cosmological models

Arbitrary scalar field and quintessence cosmological models

Tiberiu Harko t.harko@ucl.ac.uk Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom    Francisco S. N. Lobo flobo@cii.fc.ul.pt Centro de Astronomia e Astrofísica da Universidade de Lisboa, Campo Grande, Edificío C8, 1749-016 Lisboa, Portugal    M. K. Mak mkmak@vtc.edu.hk Department of Computing and Information Management, Hong Kong Institute of Vocational Education, Chai Wan, Hong Kong, P. R. China
July 25, 2019
Abstract

The mechanism of the initial inflationary scenario of the universe and of its late-time acceleration can be described by assuming the existence of some gravitationally coupled scalar fields , with the inflaton field generating inflation and the quintessence field being responsible for the late accelerated expansion. Various inflationary and late-time accelerated scenarios are distinguished by the choice of an effective self-interaction potential , which simulates a temporarily non-vanishing cosmological term. In this work, we present a new formalism for the analysis of scalar fields in flat isotropic and homogeneous cosmological models. The basic evolution equation of the models can be reduced to a first order non-linear differential equation. Approximate solutions of this equation can be constructed in the limiting cases of the scalar field kinetic energy and potential energy dominance, respectively, as well as in the intermediate regime. Moreover, we present several new accelerating and decelerating exact cosmological solutions, based on the exact integration of the basic evolution equation for scalar field cosmologies. More specifically, exact solutions are obtained for exponential, generalized cosine hyperbolic, and power law potentials, respectively. Cosmological models with power law scalar field potentials are also analyzed in detail.

pacs:
04.50.+h, 04.20.Jb, 04.20.Cv, 95.35.+d

I Introduction

Scalar fields are assumed to play a fundamental role in cosmology, where one of the first major mechanisms for which scalar fields are thought to be responsible is the inflationary scenario 1 (); 2 (). Although originally inflationary models were proposed in cosmology to provide solutions to the issues of the singularity, flat space, horizon, homogeneity problems and absence of magnetic monopoles, as well as to the problem of large numbers of particles Li90 (); Li98 (), by far the most useful property of inflation is that it generates both density perturbations and gravitational waves. These can be measured in a variety of different ways including the analysis of microwave background anisotropies, velocity flows in the Universe, clustering of galaxies and the abundance of gravitationally bound objects of various types Li98 (). The possibility that a canonical scalar field with a potential, dubbed quintessence, may be responsible for the late-time cosmic acceleration, was also explored caldwell (). Contrary to the cosmological constant, the quintessence equation of state changes dynamically with time 11 (). In fact, a plethora of exotic fluids have been proposed to explain the accelerated expansion of the Universe, which include amongst many others essence models, in which the late-time can be driven by the kinetic energy of the scalar field kessence (); coupled models where dark energy interacts with dark matter coupledDE (); and unified models of dark matter and dark energy DM_DE ().

In a wide range of inflationary models the underlying dynamics is that of a single scalar field, with the inflaton rolling in some underlying potential 1 (); 2 (); Li90 (); Li98 (). In order to study the inflationary dynamics, the usual strategy is an expansion in the deviation from the scale invariance, formally expressed as the slow-roll approximation, which arises in two separate contexts. The first is in simplifying the classical inflationary dynamics of expansion and the lowest order approximation ignores the contribution of the kinetic energy of the inflation to the expansion rate. The second is in the calculation of the perturbation spectra, where the standard expressions are valid to lower order in the slow roll approximation Co94 (). Exact inflationary solutions have also been found for a large number of inflationary potentials, and the respective potentials allowing a graceful exit have been classified Mi (). In fact, many quintessential potentials have been proposed in the literature, which may be crudely classified as “freezing” models and “thawing” models Caldwell:2005tm (). Note that in the former class freezing (), the field rolls along the potential in the past, and the movement gradually slowing down as the system enters the phase of cosmic acceleration. In the latter, “thawing” models, the scalar field, possessing a mass of , has been frozen by the Hubble friction term until recently, and eventually starts to evolve as drops below thawing (). Another interesting model involves a double exponential potential, which requires that the potential becomes shallow, or has a minimum in order to slow the movement of the scalar field Barreiro:1999zs (); the latter behavior has also been exhibited by more general potentials general_pot ().

More recently, the released Planck data of the 2.7 full sky survey P1 (); P2 () have shown a number of novel and unexpected features, whose explanation will certainly require a deep change in our standard understanding of the Universe. These recent observations have measured the Cosmic Microwave Background to an unprecedented precision. Even though generally the Planck data confirm the foundations of the CDM model, the observational data show some tension between the fundamental principle of the model and observations. For example, the Planck data combined with the WMAP polarization data show that the index of the power spectrum is given by P1 (), which rules out the exact scale-invariance () at more than level. Hence Planck data “severely limits the extensions of the simplest paradigm” P1 (). On the other hand Planck data do not require the consideration of inflationary models beyond the simplest canonical single field scenarios 3 (). More specifically, a chaotic inflationary model, based on a quartic potential, is highly disfavored by the observations. The inflationary model based on a quadratic potential is marginally consistent with the observation at 2 level, and models with a linear or fractional power potential lie outside the 1, but within the 2 allowed region P1 (); 4 ().

The observations of high redshift supernovae, and the WMAP/Planck data, showing that the location of the first acoustic peak in the power spectrum of the microwave background radiation is consistent with the inflationary prediction , have provided compelling evidence for a net equation of state of the cosmic fluid lying in the range acc (). To explain these observations, two dark components are invoked: pressureless cold dark matter (CDM), and dark energy (DE) with negative pressure. CDM contributes P2 (), and is mainly motivated by the theoretical interpretation of the galactic rotation curves and large scale structure formation. DE is assumed to provide , and is responsible for the acceleration of the distant type Ia supernovae acc (). There are a huge number of proposed candidates for DE (see, for instance, PeRa03 (); Pa03 ()). One possibility is cosmologies based on a mixture of cold dark matter and quintessence, a slowly-varying, spatially inhomogeneous component 8 (). An example of implementation of the idea of quintessence is the suggestion that it is the energy associated with a scalar field , with a self-interaction potential . If the potential energy density is greater than the kinetic one, then the pressure associated to the -field is negative. Quintessential cosmological models have been intensively investigated in the physical literature (for a recent review see Tsu ()). The interaction between dark energy and dark matter in the framework of irreversible thermodynamics of open systems with matter creation/annihilation has also been recently explored Harko:2012za (), where dark energy and dark matter are considered as an interacting two component (scalar field and “ordinary” dark matter) cosmological fluid in a homogeneous spatially flat and isotropic Friedmann-Robertson-Walker (FRW) Universe. The possibility of cosmological anisotropy from non-comoving dark matter and dark energy have also been proposed Harko:2013wsa ().

Models with nonstandard scalar fields, such as phantom scalar fields and Galileons, which can have bounce solutions and dark energy solutions with have also been extensively investigated in the literature. In the Galileon theory one imposes an internal Galilean invariance, under which the gradient of the relativistic scalar field , with peculiar derivative self-interactions, and universally coupled to matter, is shifted by a constant term gal1 (). The Galilean symmetry constrains the structure of the Lagrangian of the scalar field so that in four dimensions only five terms that can yield sizable non-linearities without introducing ghosts do exist. Different extensions of the Galileon models were considered in gal2 (). In gal3 () a new class of inflationary models was proposed, in which the standard model Higgs boson can act as an inflaton due to Galileon-like non-linear derivative interaction. The generated primordial density perturbation is consistent with the present observational data. Generalized Galileons as a framework to develop the most general single-field inflation models, i.e., Generalized G-inflation, were studied in gal4 (). As special cases this model contains k-inflation, extended inflation, and new Higgs inflation. The background and perturbation evolution in this model were investigated, and the most general quadratic actions for tensor and scalar cosmological perturbations was obtained. The stability criteria and the power spectra of primordial fluctuations were also presented. For a recent review of scalar field theories with second-derivative Lagrangians, whose field equations are second order see Rub (). Some of these theories admit solutions violating the null energy condition and have no obvious pathologies.

In order to explain the recent acceleration of the Universe, in which , scalar fields that are minimally coupled to gravity with a negative kinetic energy, and which are known as “phantom fields”, have been introduced in phan1 (). The energy density and pressure of a phantom scalar field are given by and , respectively. The properties of phantom cosmological models have been investigated in phan2 (). The phenomenon of the phantom divide line crossing in the scalar field models with cusped potentials was considered in phan3 (). Cosmological observations show that at some moment in the past the value of the parameter of the dark energy equation of state may have crossed the value , corresponding to the cosmological constant . This phenomenon is called phantom divide line crossing phan4 (). Non-phantom dark energy is described by a minimally coupled scalar field, having a kinetic term with the positive sign. Therefore in order to describe the phantom divide line crossing it seems natural to use two scalar fields, a phantom field with the negative kinetic term, and a standard one phan3 (). Another possible way of explaining the phantom divide line crossing is to use a scalar field nonminimally coupled to gravity phan3 ().

The mathematical properties of the Friedmann-Robertson-Walker (FRW) cosmological models with a scalar field as matter source have also been intensively investigated. In Mus (); Sal () a simple way of reducing the system of the gravitational field equations to one first order equation was proposed, namely, to the Hamilton-Jacobi-like equation for the Hubble parameter considered as a function of the scalar field , . The gravitational collapse and the dynamical properties of scalar field models were considered in Giambi (). The solution of the field equations for a cosine hyperbolic type scalar field potential for the case of an equation of state equivalent to the nonrelativistic matter plus a cosmological term was derived in Kis (). The relation between the inflationary potential and the spectra of density (scalar) perturbations and gravitational waves (tensor perturbations) produced during inflation, and the possibility of reconstructing the inflaton potential from observations, was considered in Lid0 (). If inflation passes a consistency test, one can use observational information to constrain the inflationary potential. The key point in the reconstruction procedure is that the Hubble parameter is considered as a function of the scalar field, and this allows to reconstruct the scalar field potential, and determine the dynamics of the field itself, without a priori knowing the Hubble parameter as a function of time or of the scale factor Kam1 (); Kam2 (). General solutions for flat Friedmann universes filled with a scalar field in induced gravity models and models including the Hilbert-Einstein curvature term plus a scalar field conformally coupled to gravity were also derived in Kam3 (). The corresponding models are connected with minimally coupled solutions through the combination of a conformal transformation and a transformation of the scalar field. The explicit forms of the self-interaction potentials for six exactly solvable models was also obtained. In Nunes:2000yc (), a phase-plane analysis was performed of the complete dynamical system corresponding to a flat FRW cosmological models with a perfect fluid and a self-interacting scalar field and it was shown that every positive and monotonous potential which is asymptotically exponential yields a scaling solution as a global attractor. The dynamics of models of warm inflation with general dissipative effects, was also extensively analyzed Mimoso:2005bv (), and a mechanism that generates the exact solutions of scalar field cosmologies in a unified way was also investigated.

The connections between the Korteweg-de Vries equation and inflationary cosmological models were explored in Lid1 (). The relation between the non-linear Schrödinger equation and the cosmological Friedmann equations for a spatially flat and isotropic Universe in the presence of a self-interacting scalar field has been considered in Lid2 (). A Hamiltonian formalism for the study of scalar fields coupled to gravity in a cosmological background was developed in Ber (). A number of integrable one–scalar spatially flat cosmologies, which play a natural role in the inflationary scenarios, were studied in Fre1 (). Systems with potentials involving combinations of exponential functions, and similar non–integrable cases were also studied in detail. It was shown that the scalar field emerges from the initial singularity while climbing up sufficiently steep exponential potentials (“climbing phenomenon”), and that it inevitably collapses in a Big Crunch, whenever the scalar field tries to settle at the negative extremals of the potential. The question whether the integrable one scalar-field cosmologies can be embedded as consistent one-field truncations into Extended Gauged Supergravity or in supergravity gauged by a superpotential without the use of terms was considered in Fre2 ().

Therefore, the theoretical investigation of scalar field models is an essential task in cosmology. It is the purpose of the present paper to consider a systematic analysis of scalar field cosmologies, and to derive a basic evolution equation describing flat, isotropic and homogeneous scalar field cosmological models. The evolution equation is a first order, strongly non-linear differential equation, which, however, allows the possibility of considering analytical solutions in both the asymptotic limits of scalar field kinetic or potential energy dominance and in the intermediate domain, respectively. Moreover, a large number of exact solutions can also be obtained. The cases of the exponential, hyperbolic cosine, and power law potentials are explicitly considered.

The present paper is organized as follows. The basic evolution equation for scalar field cosmologies with an arbitrary self-interaction potential is derived in Section II. Several classes of exact scalar field solutions are considered in Sections III and IV. The general formalism is used in Section V to obtain some approximate solutions of the gravitational field equations. In Section VI we consider in detail the case of the simple power law potential. We discuss and conclude our results in Section VII.

Ii Scalar field cosmologies with arbitrary self-interaction potential

Let us consider a rather general class of scalar field models, minimally coupled to the gravitational field, for which the Lagrangian density in the Einstein frame reads

(1)

where is the curvature scalar, is the scalar field, is the self-interaction potential and is the gravitational coupling constant, respectively. In the following, we use natural units with , and we adopt as our signature for the metric , as is common in particle physics.

For a flat FRW scalar field dominated Universe with the line element

(2)

where is the scale factor, the evolution of a cosmological model is determined by the system of the field equations

(3)
(4)

and the evolution equation for the scalar field

(5)

where is the Hubble expansion rate function, the overdot denotes the derivative with respect to the time-coordinate , while the prime denotes the derivative with respect to the scalar field , respectively. In the following we will restrict our study to expansionary cosmological models, which satisfy the condition that the scale factor is a monotonically increasing function of time. For expanding cosmological models the condition is always satisfied. Cosmological models with correspond to collapsing scalar field configurations, in which the scale factor is a monotonically decreasing function of time.

By adding Eqs. (3) and (4), we obtain the Riccati type equation satisfied by , of the form

(6)

By substituting the Hubble function from Eq. (3) into Eq. (5), we obtain the basic equation describing the scalar field evolution as

(7)

Now, in order to deduce a basic equation describing the dynamics of the scalar fields in the flat FRW Universe, which will be useful throughout this work, we consider several transformations. First, by defining a new function so that , and changing the independent variable from to , Eq. (7) becomes

(8)

which may be reorganized into the following form:

(9)

Next, by introducing a new function , so that , Eq. (9) takes the form

(10)

Thus, we introduce now the function defined as , which transforms Eq. (10) to

(11)

With the help of the transformation , we obtain the basic equation describing the dynamics of the scalar fields in the flat FRW Universe as

(12)

For we obtain , leading to . As a function of time satisfies the equation

(13)

Note that the function can be obtained from the scalar field with the use of the equation

(14)

Furthermore, as a function of the scalar field, the scale factor is given by the equation

(15)

or alternatively, as a function of , the latter scale factor can be obtained from

(16)

An important observational quantity, the deceleration parameter , can also be expressed in the form

(17)

As a function of the potential and of , the deceleration parameter is given by

(18)

By inserting Eq. (12) into Eq. (18), the latter yields the result

(19)

and by substituting Eq. (14) into Eq. (19), yields the deceleration parameter in the following useful form:

(20)

respectively. If the potential energy dominates the kinetic energy of the scalar field, , from Eq. (20) it follows that , and it is important to note that this property is independent of the form of the scalar field self-interaction potential .

By using the new variable and from Eqs. (3) and (4) we obtain

(21)

and

(22)

respectively. Eqs. (21) and (22) show that the functions and are related to the Hubble function, and its time derivative. A similar approach, in which the Hubble function is assumed to be a function of the scalar field , was considered in Sal ().

Iii Exact scalar field models

As mentioned in the Introduction, scalar fields are considered to play a central role in current models of the early Universe. The self-interaction potential energy density of such a field is undiluted by the expansion of the Universe, and hence can behave as an effective cosmological constant, driving a period of inflation, or of a late-time acceleration. The evolution of the Universe is strongly dependent upon the specific form of the scalar field potential . A common form for the self-interaction potential is the exponential type potential. Note that Eq. (12) can be integrated immediately in the case of potentials satisfying the condition . Therefore, for this class of potentials the general solution of the gravitational field equations can be obtained in an exact analytical form. Other classes of exact solutions can be constructed by assuming that is some function of , i.e., . For a large number of choices of the function , the first order evolution equation, given by Eq. (12), can be solved exactly, and the solution corresponding to a given potential can be obtained in an exact form. In the following, we consider some exact analytical classes of scalar field cosmologies.

iii.1 The exponential potential scalar field

If is a constant, that is, constant, the scalar field self-interaction potential is of the exponential form,

(23)

where is an arbitrary constant of integration. The cosmological behavior of the Universe filled with a scalar field, with a Liouville-type exponential potential, has been extensively investigated in the physical literature for both homogeneous and inhomogeneous scalar fields exp1 (); K1 (); K2 (); Cui (). In particular, an exponential potential arises in four-dimensional effective Kaluza-Klein type theories from compactification of the higher-dimensional supergravity or superstring theories CaMaPeFr85 (). In string or Kaluza-Klein theories the moduli fields associated with the geometry of the extra-dimensions may have effective exponential potentials due to the curvature of the internal spaces or to the interaction of the moduli with form fields on the internal spaces. Exponential potentials can also arise due to non-perturbative effects such as gaugino condensation CaCaMu93 (). The integrability of the gravitational field equations for exponential type scalar potentials was considered in Sch (); Sch1 (); Sch2 (); pow1 ().

Taking into account Eq. (23), then Eq. (12) takes the form

(24)

We analyze below several cases of interest.

iii.1.1 The case

For , Eq. (24) gives immediately

(25)

where is an arbitrary constant of integration. The time dependence of the physical parameters can be obtained from Eq. (13) as

(26)

With the use of Eq. (25), we obtain the following integral representation for the time ,

where is an arbitrary constant of integration. Thus, Eqs. (26) and (III.1.1) give a parametric representation of the time evolution of the scalar field with an exponential type self-interaction potential, with taken as a parameter. For the scale factor, we obtain

(28)

where is an arbitrary constant of integration, while the deceleration parameter is given by

(29)

The time integral given by Eq. (III.1.1) can be obtained in an exact form for some particular values of . Thus, if , we have

(30)

For we obtain

(31)

while for we find

(32)

Therefore, in these cases the exact solution of the gravitational field equations in the presence of a scalar field with exponential potential can be obtained in an exact parametric form, and there is no need to resort to numerical integration.

A particular solution of the field equations corresponds to the case . In this case Eq. (24) is identically satisfied, with given by

(33)

From Eq. (15) it follows that the scale factor can be obtained as a function of the scalar field as

(34)

while the time variation of the scalar field is determined from Eq. (14) by the equation

(35)

with the general solution given by

(36)

where is an arbitrary integration constant. With the help of Eq. (34) we obtain the scale factor in the form

(37)

The time variation of the scalar field potential is given by

(38)

with the constants , and satisfying the consistency condition

(39)

which follows immediately from the comparison of the second and the last term in Eq. (38).

Simple power law solutions for cosmological models with scalar fields with exponential potentials have been obtained, and studied, in pow1 ().

The time variations of the scale factor, scalar field, scalar field potential, and deceleration parameter for the exponential potential scalar field filled Universe are represented, for different values of , in Figs. 1-2. For the considered range of the parameter , the scale factor is a monotonically increasing function of time, and therefore the solution represents an expanding Universe. The Universe starts its evolution from a decelerating phase, with , but after a finite interval it enters in an accelerated era, with . In the large time limit , as can be seen immediately from Eq. (25), for the chosen values of the parameters, the scalar field is an increasing function of time, becoming a constant in the large time limit, as well as the scalar field potential. In the limit of large times the Universe enters a de Sitter type accelerated phase, with .

Figure 1: Depicted is the time variation of the scale factor (in arbitrary units), in the left plot, and the time variation of the cosmological scalar field, in the right plot, with an exponential potential for different values of : (solid curve), (dotted curve), (short dashed curve), (dashed curve), and (long dashed curve), respectively. The arbitrary integration constants and have been normalized so that .
Figure 2: Plots of the time variation of the exponential scalar field potential, depicted in the left figure, and the time variation of the deceleration parameter of the Universe filled with an exponential potential scalar field, depicted in the right figure, for different values of : (solid curve), (dotted curve), (short dashed curve), (dashed curve), and (long dashed curve), respectively. The arbitrary integration constants and have been normalized so that .

iii.1.2 The case

In the specific case of , from Eq. (24) we obtain the following dependence of the scalar field on ,

(40)

and

(41)

respectively, where and are arbitrary constants of integration. For the time dependence of the cosmological model we obtain the integral representations

(42)

and

(43)

respectively, where and are arbitrary constants of integration, giving the explicit dependence of the physical time on the parameter as

(44)

and

(45)

respectively. The parametric dependence of the scale factor is given by

(46)

and

(47)

respectively, where and are arbitrary constants of integration. The deceleration parameter is given in parametric form by Eq. (29).

iii.2 Generalized hyperbolic cosine type scalar field potentials

As a second example of an exact integrability of the evolution equation, given by Eq. (12), we consider the case in which the scalar field potential can be represented as a function of in the form

(48)

where is an arbitrary constant. With this choice, the evolution equation takes the simple form

(49)

with the general solution given by

(50)

where is an arbitrary constant of integration. With the use of this form of , we immediately obtain the self-interaction potential of the scalar field as

(51)

The time dependence of the scalar field can be obtained in a parametric form as

(52)

Finally, the scale factor is given by

(53)

while the deceleration parameter is expressed as

(54)

Hence, the exact solution of the field equations for a hyperbolic cosine type scalar field potential can be obtained in an exact parametric form. The variations of the scalar field potential as a function of and , respectively, of the scale factor of the Universe, and of the deceleration parameter, are represented, for different values of , in Figs. 3-4, respectively.

Figure 3: Depicted is the variation of the generalized hyperbolic cosine scalar field potential as a function of , in the left plot, and as the variation in time, in the right plot, for different values of : (solid curve), (dotted curve), (short dashed curve), (dashed curve), and (long dashed curve), respectively.
Figure 4: Depicted is the time variation of the scale factor, in the left plot, and of the deceleration parameter, in the right plot, of the Universe filled with a scalar field with a generalized hyperbolic cosine self-interaction potential for different values of : (solid curve), (dotted curve), (short dashed curve), (dashed curve), and (long dashed curve), respectively.

In all of the considered models the Universe shows an expansionary, accelerated behavior, starting with an initial value of the deceleration parameter. The scalar field potential is increasing in time, leading, in the long time limit, to accelerated expansions, with .

iii.3 Power-law type scalar field potential

A simple solution of the gravitational field equations for a power-law type scalar field potential can be obtained by assuming for the function the following form

(55)

With this choice of , then Eq. (12) immediately provides the scalar field potential given by

(56)

where is an arbitrary constant of integration. The time dependence of the scalar field is given by a simple power law,

(57)

The scale factor can be obtained from Eq. (15), , and has an explicit exponential dependence on the scalar field and the time, namely,

with an arbitrary constant of integration. The deceleration parameter is given by

(59)

Iv Further integrability cases for scalar field cosmologies

In the present Section, we will consider several general integrability cases of Eqs. (3) and (5), describing the dynamics of a scalar field filled homogeneous and isotropic space-time. By introducing a new set of variables, the basic equation (12) can be separated into two ordinary first order differential equations. The resulting compatibility condition can be integrated exactly for two different forms of the scalar field potential, thus leading to some exactly integrable classes of the field equations.

iv.1 The general integrability condition for the field equations

We rewrite the hyperbolic function in the form , where we have introduced a new function defined as . By substituting , into Eq. (12), and by denoting and , then Eq. (12) takes the form

(60)

where we have introduced a new separation function . Therefore, we have obtained two linear differential equations for and , given by

(61)

and

(62)

respectively. Equation (61) can be integrated to provide

(63)

where is an arbitrary constant of integration. Equation (62) can be integrated to give

(64)

where is an arbitrary constant of integration. Using Eqs. (63) and (64), we obtain a consistency integral relation between the separation function and the self-interaction potential , given by

(65)

In the following, we shall solve the integral equation (65) for two particular cases.

iv.1.1 Specific case I:

First we assume that the separation function takes the form . By substituting into Eq. (65), the latter gives an integral equation for the potential

(66)

where we have denoted . In order to solve the integral Eq. (66), we rewrite it as a linear first order differential equation for

(67)

with the general solution given by

(68)

where is an arbitrary constant of integration. Now by inserting into Eq. (68) yields the expression of the scalar field potential as

(69)

Therefore we have obtained the following:

Theorem 1. If the scalar field self-interaction potential is given by Eq. (69), then the general solution of Eq. (12) is given by

(70)

where

(71)

iv.1.2 Specific case II:

Now we assume that the separation function takes the form . By substituting into Eq. (65), the latter gives an integral equation for the potential ,

(72)

In order to solve Eq. (72), we rewrite it as a linear first order differential equation for

(73)

Equation (73) can be easily integrated, and yields the following solution

(74)

where is an arbitrary constant of integration.

Therefore we have obtained the following:

Theorem 2. If the scalar field self-interaction potential is given by Eq. (74), then the general solution of Eq. (12) is given by

(75)

where

(76)

V Approximate solutions for scalar fields with arbitrary self-interaction potentials

Generally, for a given scalar field potential , Eq. (12) cannot be solved exactly. However, there are several limiting cases in which approximate general solutions can be obtained for arbitrary potentials. These cases correspond to the approximation of the function

(77)

in the asymptotic limits of small and large by some simple analytical expressions.

v.1 The limit of large

One such important case is the limit of large , , when . From Eq. (77) it follows that this case corresponds to scalar fields satisfying the condition . Therefore, Eq. (12) takes the form

(78)

with the general solution given by

(79)

where is an arbitrary constant of integration. Then we obtain the following solutions

(80)
(81)

respectively.

By using the relation between and , , we obtain

(82)

The scale factor can be obtained as

(83)

where is an arbitrary constant of integration. As one can see from Eq. (83), in the limit of large , , the scale factor can be represented as an exponential function of the scalar field,

(84)

Equations (82) and (83) give a parametric representation of the time variation of the scale factor, with the scalar field taken as parameter. The deceleration parameter is given in this limit by

(85)

In the limit of large , corresponding to the limit of , we have .

v.2 The limit of small G

A second case in which an approximate general solution of Eq. (12) can be found for arbitrary potentials corresponds to the limit of small , , when . This condition is satisfied for potential dominated scalar fields, with . In this case, one can neglect the small term of the order of unity in the equation (12), thus obtaining

(86)

The general solution of Eq. (86) is

(87)

where is an arbitrary constant of integration.

As in the previous case we obtain

(88)

This gives immediately

(89)

and

(90)

The deceleration parameter is obtained as

(91)

v.3 Power series solution of the field equations

The hyperbolic function is given by

(92)

where we have introduced the Bernoulli number , defined in terms of the Riemann zeta function, and given by PoZa ()