# The method of generating functions in exact scalar field inflationary cosmology

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## Abstract

The construction of exact solutions in scalar field inflationary cosmology is of growing interest. In this work, we review the results which have been obtained with the help of one of the most effective methods, viz., the method of generating functions for the construction of exact solutions in scalar field cosmology. We also include in the debate the superpotential method, which may be considered as the bridge to the slow roll approximation equations. Based on the review, we suggest a classification for the generating functions, and find a connection for all of them with the superpotential.

## 1 Early inflation and the implemented scalar field

Inflationary expansion of the universe during very early times, once the universe emerged from the quantum gravity (Planck) era, has been proposed in the late 1970’s and, mainly in the beginning of the 1980’s and is becoming more accepted as a necessary stage of the standard Big Bang theory model. In the work of Starobinsky [1], nonsingular isotropic cosmological models with a massive scalar field were investigated. Although this work was concerned more with a bouncing cosmology rather than inflation, the field equations and the corresponding slow-roll solutions were already derived in this paper. These were later employed in the chaotic inflation model of Linde [9]. The works by Starobinsky [2], Guth [3], Linde [4] and Albrecht and Steinhardt [5] include the physical mechanism based on quantum corrections and phase transitions during the very early stage of the universe. Exponential (de Sitter) expansion is the feature of inflationary models which helped to solve the long standing problems of the standard Big Bang theory model: the horizon, flatness, homogeneity, isotropy and some other problems.

In Sato’s work [6], the first-order phase transition model of the early Universe that leads to an exponential expansion which stretches domains much greater than the horizon scales was considered. Also, in [7] it was pointed out that fluctuations associated with the phase transition are exponentially stretched and then may play the role of seed fluctuations for large-scale structures. Further, the monopole problem was also shown to be resolved by exponential expansion by Einhorn and Sato [8]. The chaotic inflation scenario proposed by Linde [9] differs from other previous versions since it is not based on the theory of high-temperature phase transition in the very early universe, but contains the locally homogeneous scalar field which is slowly rolling down to the minimum of the scalar field potential.

After that proposal, many investigations took place of the inflationary universe connected with a self-interacting scalar field as the source of gravitation in the Friedmann world. Let us briefly mention some interesting works concerning the study of a scalar field in inflationary cosmology.

Homogeneous isotropic cosmological models with a massive scalar field have been studied in the works [10, 11]. It was shown that inflationary stages are a fairly general property of most solutions in the considered model. The general conditions for inflation were investigated in the work [12]. It was found that under the lower limit for the amplitude of a scalar field, the universe naturally enters into and exits out of an inflationary phase. What is important is that such behavior takes place under a large variety of scalar potentials which are polynomial, logarithmic or exponential. It was also stated that a scalar field is essential for inflation [12]; it is unlikely that a vector or other non-scalar field will lead to inflation. The difference between scalar potentials in particle physics and those in cosmology has been stressed in the work [13]. The author wrote: ”... we do not really know which theory of particle physics best describes the very early universe. One should therefore keep an open mind as to the form of \(V(\phi )\)”. Halliwell chose the exponential potential and showed that it leads to a solution with power-law inflation, and that this solution is an attractor. Detailed investigations of power-law inflation have been carried out in the work [14]. The authors found the constraints on the model coming from the requirement of solving the horizon, flatness, reheating and perturbation-spectrum problems. It was stated also that these constraints can be suitably satisfied. An exact power-law inflationary solution possessing an exponential potential was given in the work [15]. The generic inhomogeneous generalization of this solution having both scalar and tensor superhorizon “hairs” was derived in the paper [16].

Let us mention also the investigation carried out by Ivanov [17] where he found exact solutions for a nonlinear scalar field in cosmology. The solutions he obtained included polynomial, trigonometric and exponential potentials. The method he used for searching for exact solutions was subsequently called the Hamilton-Jacobi-like approach.

From the observational point of view, most results which can be related to observational data have been obtained from the so-called slow roll approximation of the cosmological dynamical equations [4, 5]. Detailed investigations of various physical phenomena from particle physics and GUT theories for the period until the 1990’s can be studied from the reviews [18, 19, 20]. Our attention will be concentrated on exact solutions of inflationary models, the study of which started about ten years later, after inflationary cosmology had been proposed.

Thus we are going to present a brief review of the construction of exact solutions in the inflationary universe, i.e., the solutions of self-consistent Einstein and scalar field equations in Friedmann cosmology. The direct connection between scalar field cosmology and cosmology based on the perfect fluid stress-energy tensor needs to be mentioned. This connection is always valid except in the case of dust matter. Therefore we included the case of exact solutions for perfect fluid as the source of gravitation.

The construction of exact solutions in inflationary cosmology started with the work by Muslimov [21]. The results presented in that article will be discussed in Sect. 3. Here we would like to mention that the very method and many interesting exact solutions presented in [17] have been reproduced and generalized in [21]. New methods and new sets of exact solutions have been developed in the work [21] as well.

Barrow [22] found a simple way to solve exactly the cosmological dynamic equations in terms of a pressure-density relationship. In this way he obtained the known power-law and de Sitter forms of inflation and new classes of behavior in which the expansion scale factor increases as the exponent of some power of the cosmic time coordinate. The double-exponential law solution was obtained as well.

The work by Ellis and Madsen [23] was the first where “the inverse problem” was considered in the framework of cosmology. Usually one suggests that we know the scalar potential in the very early universe from HEP, and our task is to find the scale factor and the scalar field as functions of time. However Ellis and Madsen [23] suggested starting from the given scale factor! Indeed, it is clear that the scale factor may be found from observational data. Then we may take into account this fact to find the potential and scalar field from the cosmological equations. This work was done and examples of exact solutions have been presented for the pure scalar field (without taking into account radiation which is also considered there). Further this approach was developed in the works [24, 25, 26].

Reconstruction of models with a scalar field (quintessence) and dust-like particles (baryons and dark matter) from observational data was further developed by Starobinsky [27], Huterer and Turner [28], Nakamura and Chiba [29] using the luminosity data and in Starobinsky [27] from the growth factor of inhomogeneities.

Our paper is organised as follows: in Sect. 2, we present the basic equations of scalar field cosmology. In Sect. 3, we discuss generating functions for finding solutions, and Sect. 4 is devoted to the classification of the generating functions. The superpotential method is presented in Sect. 5 and we conclude with Sect. 6.

## 2 Basic equations of scalar field cosmology

*R*is the curvature scalar, \(\phi \) the scalar field, \(\phi _\mu =\partial _\mu \phi \) the short representation of the partial derivative \( d\phi /dx^\mu \), \(\kappa \) is Einstein’s gravitational constant, and \(\Lambda \) is the cosmological constant, which will mainly be included in the scalar field potential \(V(\phi )\) as the constant part of it.

*the Scalar Cosmology Equations (SCEs)*.

The representation above, Eqs. (10)–(12) has some advantages for the derivation of any of the three Eqs. (10)–(12) from the other two, and differential consequences of them.

*H*on the scalar field \(\phi \), the transformation of the Eqs. (10)–(12) for the spatially-flat universe (\(\epsilon =0\)) to the form, which was called later the Hamilton–Jacobi-like form, was made. Equation (11) is transformed to

*the Ivanov–Salopek–Bond (ISB) equation*.

## 3 Generating function for solving the Ivanov–Salopek–Bond equation

*the generating function*. Comparing Eqs. (14) with (15), it is easy to find the solution of (14)

### 3.1 Potential in polynomial form

Also, for the first time, solution (24) was obtained in [9] by using the slow-roll approximation for the potential \(V(\phi )=m^{2}\phi ^{2}/2\) in contrast to the shifted potential (21).

It is interesting to note that the same solution and its application for the calculation of the number of e-folds and scalar spectral parameter were found and developed later by Wang [31].

*H*and \(H'\) expressed through \(\phi \)

*a*(

*t*)

*a*(

*t*)

### 3.2 Trigonometric potential

The solution for the potential which leads to the Sine-Gordon type equation was obtained in [17]. Such a setting used the special choice of the additional parameter. Let us consider this point in detail.

*t*, we obtain the scale factor

### 3.3 Exponential potential

*t*has a logarithmic character

### 3.4 The solution with an inverse potential

### 3.5 The solution with an intermediate (hyperbolic) function

*y*(

*x*), we obtain

*u*in the following way

*y*(

*x*) is performed by inverse substitution of (71) and by introducing the derivative

### 3.6 Kim’s exact solutions

### 3.7 Exact solutions for constant-roll inflation

*f*(

*R*) gravity on the basis of conformal transformations from the Jordan frame to the Einstein frame which lead to Ivanov–Salopek–Bond (Hamilton–Jacobi-like) Eqs. (13)–(14) [36]. In the works [36, 37], it was shown that these types of models satisfy the latest observational constraints.

\(\bullet \) The solutions for \(\alpha >-3\) only:

^{1}which was already considered in Sect. (3.3)

*M*is an integration constant.

\(\bullet \) The solutions for both cases \(\alpha >-3\) and \(\alpha <-3\):

## 4 The classification of generating functions

We propose a classification of generating functions as they appeared in the literature in chronological order. As the first class, we name the Ivanov generating function \(F(\phi )\) (15)–(16). It was not represented in direct form in [17], but in the present publication for the first time, using this generating function, we recover and generalise all solutions of Ivanov’s work [17].

Now we continue the classification of the generating functions that occur in the literature.

### 4.1 The second class of generating functions

*F*(

*a*) is a new type of generating function. The scalar field equation can be integrated and it yields

*C*is an arbitrary integration constant. Thus, the problem of generation of exact solutions has reduced to the quadratures:

*n*are constants and \(s(n+1)=6\). If we take \(C=0\), the potential is simplified to

*s*:

### 4.2 The third class of generating functions

*g*(

*H*), the exact solutions of the SCES can be generated.

*n*is real and

*A*is a positive constant.

### 4.3 The fourth class of generating functions

*D*is a constant. The form of the potential suggests that \(\phi (t)\) be a function that decreases from an initial maximum value similar to the chaotic inflation model. We can choose \(D=0\) and so (148) can be written as,

### 4.4 The fifth class of generating functions

### 4.5 The sixth class of generating functions

*G*with dynamical equations

*G*can be defined via the scalar field from the Eq. (159) as

*G*in the form

*G*, one can obtain the self-interaction potential of the scalar field and the scale factor

*G*the following form

*G*, Eq. (158) immediately provides the scalar field potential given by

## 5 The superpotential method

As we know, any one from the presented exact SCEs in the slow roll form (177)–(179) can be derived as a consequence (or a differential consequence) of the remaining two. In our approach, we exclude from consideration equation (178) in the first step.

*H*, and inserting it into (179), taking into account the superpotential definition (176), we have the consequence of (177) and (179) in the form

*W*, we obtain the relation:

*W*, one can find

*H*from the Friedmann equation (177) with the following relation

*a*(

*t*).

Thus, the proposed method presents some combination of the two methods: slow roll-like presentation of the exact equations [47, 48] and obtaining cosmological solutions for the given scalar field evolution [50].

The advantage of the proposed method lies in the essential simplification of the integration procedure: one needs to calculate only one integral for obtaining a superpotential and Hubble parameter. Then the potential *V* (176), as well as the scale factor *a*(*t*), are calculated from related definitions. The exhibition of the simplicity of the procedure, and its effectiveness can be found in the applications of the method to cosmology on the brane, in phantom and tachyon fields [51, 52, 53]. The two last have a very restricted number of exact solutions which can be essentially extended by virtue of the superpotential method.

We can have a look at the superpotential method from another position. The system (170)–(172) has three unknowns, \(\phi (t),V(\phi )\) and *H*(*t*) (or *a*(*t*)). To solve it, one of these variables has to be given a priori. It is customary to look for the solution for a given \(V(\phi )\), but as it is known, it is very difficult to solve the SCEs for a given potential exactly.

*H*or the scale factor. To solve them, note that Eq. (177) defines \(\dot{a}/a\) as a function of \(\phi ,H(\phi )\), which when inserted into Eq. (179), gives the scalar field \(\phi (t)\) as a function of

*t*, at least in quadratures

*a*(

*t*), respectively, and the solution is completed.

*H*=\(H(\phi )\), but it is usually desirable to have some description of the potential instead, and for this reason it is preferable to give \(W(\phi )\). One could also use

*H*(

*t*) to determine \(\phi (t)\), since

*H*(

*t*) fully determines the solution to the problem.

Types of generating functions and connections between them through superpotential *W*

References | Kind of generating function | Connection with |
---|---|---|

\(H(\phi )=\frac{1}{\sqrt{3}}\left( F(\phi )+F_{*}\right) \) | \(F(\phi )=\sqrt{W}-F_{*}\) | |

(II) Chimento et al. [38] | \(F(a)=V[\phi (a)]a^{6}\) | \(F(a(\phi ))=a^{6}\left[ W-\frac{W_{\phi }'^{2}}{6W}\right] \) |

(III) Schunck and Mielke [40] | \(g(H)=V(H)-3H^2\) | \(g(H(\phi ))=-\frac{W_{\phi }'^{2}}{6W}\) |

(IV) Kruger and Norbury [41] | \(F(\phi )=1+\left( \frac{\dot{\phi }^{2}}{V(\phi )}\right) \) | \(F(\phi )=\frac{6W^{2}+W_{\phi }'^{2}}{6W^{2}-W_{\phi }'^{2}}\) |

(V) Charters and Mimoso [42] | \(x(\phi )=\dot{\phi }/H\) | \(x(\phi )=-\frac{W_{\phi }'}{W}\) |

(VI) Harko et al. [44] | \(\dot{\phi }=2V(\phi )\sinh ^{2}G(\phi )\) | \(\coth G(\phi )=\frac{\sqrt{6}W}{W_{\phi }'}\) |

### 5.1 Examples of exact solutions

*W*are represented in the Table 1.

## 6 Conclusion

The space of exact solutions in scalar field cosmology is very huge. In this work, we review exact solutions in inflationary cosmology and the method of finding such solutions. There are basically three methods of constructing such solutions. For any scale factor *a*(*t*), we can find the potential and kinetic energy to be satisfied in the self-consistent system of Einstein and scalar field equations. Secondly, from the evolution of the scalar field \(\phi =\phi (t)\), we can find the scale factor *a*(*t*) and the potential *V*(*t*), thus defining the exact solution. Thirdly, choosing the generation function, represented in this given review, one can, once again, obtain a great number of exact solutions.

Thus, as the next step of the investigation, we suggest analysing the method of confrontation of theoretical predictions from exact solutions with observational data.

## Footnotes

## Notes

### Acknowledgements

The authors are grateful to an anonymous referee for qualitative comments and suggestions, which contributed to the improvement of this work. I.V. Fomin was supported by RFBR Grants 16-02-00488 A and 16-08-00618 A.

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