# Building analytical three-field cosmological models

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## Abstract

A difficult task to deal with is the analytical treatment of models composed of three real scalar fields, as their equations of motion are in general coupled and hard to integrate. In order to overcome this problem we introduce a methodology to construct three-field models based on the so-called “extension method”. The fundamental idea of the procedure is to combine three one-field systems in a non-trivial way, to construct an effective three scalar field model. An interesting scenario where the method can be implemented is with inflationary models, where the Einstein–Hilbert Lagrangian is coupled with the scalar field Lagrangian. We exemplify how a new model constructed from our method can lead to non-trivial behaviors for cosmological parameters.

## 1 Introduction

Since the 1970s, topological defects have been investigated as promising analytical solutions in high energy physics and in ferromagnet models [1, 2, 3]. In the previous decades, these defects were applied in several different scenarios, like braneworld models, condensate matter, besides Einstein–Hilbert and generalized cosmology [2, 4, 5, 6, 7]. This enhanced applicability was accompanied by the emergence of new mathematical methods to treat topological defects, especially when we talk about models composed of two or more scalar fields.

A relevant methodology which should be highlighted is the so-called BPS (Bogolmon’y–Prasad–Sommerfield) method [8, 9], which enables one to determine analytical solutions for one or more real scalar field models from first-order differential equations, instead standard second order equations of motion. Moreover, BPS solutions, or BPS states, are associated with the minimal energetic solutions of static physical systems. However, when we deal with Lagrangian densities composed of two or more real fields, even the BPS first-order differential equations are very hard to integrate, as they are generally coupled. Thus, we need specific methodologies to find analytical defects for two or more scalar field models.

An interesting method to solve two-field systems was proposed by Bazeia et al. [10]; here, inspired by Rajaraman’s trial orbit method [11], the authors introduced an approach to finding analytical solutions for the coupled first-order differential equations of such systems. One of the most popular models solved by this method is due to Bazeia, Nascimento, Ribeiro, and Toledo (BNRT) [12] and it has been applied in several different contexts [4, 12, 13, 14, 15]. As an alternative for the trial orbit presented in [10], de Souza Dutra [17] constructed new orbits for the BNRT models through the so-called “integrating factor method”. Despite the success of such methodologies, the challenge of finding new analytical models formed by one, two or more real scalar fields remains tricky.

In scenarios composed of one scalar field, new analytical models can be generated with the deformation method proposed by Bazeia, Losano and Malbouisson [18]. Such a method is based on a connection between two one-field models, via the so-called deformation function. So, if we know the deformation function and an analytical one-field model, we are able to generate several families of new scalar field systems [18, 19, 20].

Inspired by the deformation method, Bazeia, Losano and Santos [21] introduced the extension method to construct analytical two scalar field models, starting from one-field ones. The advantage of such a methodology is that a possible set of solutions for the equations of motion of the two-field model is exactly formed by the solutions of the one-field systems used in the construction process. The last method was applied in the quintessence cosmology, leading to new sets of analytical cosmological parameters [6].

As a motivation to apply our methodology in the context of cosmological models, we may point out that multi-field inflation is able to yield to a proper relation between the tensor-to-scalar ratio and the spectral index, as pointed out by Ellis et al. [22]. Moreover, multiple fields allow us to address new features for the physical systems and for the cosmological parameters, which cannot be derived from single field models, as pointed out very recently in [23, 24].

Our aim in the present work is to increase the amount of analytical three scalar field models inspired by the extension method. We believe that this approach can overcome several difficulties related with the integration process of coupled first-order differential equations. In order to show the applicability of our procedure as well as its robustness, we will use it to build a three-field quintessence model.

The article is organized as follows: Sect. 2 shows some generalities of the deformation method and of the BPS approach for three scalar field models. In Sect. 3 we present a new version for the extension method, while its applicability is discussed carefully in Sect. 4, where we construct several examples. In Sect. 5 we establish the bases for our quintessence model, we apply one of our examples in this context and we analyze the cosmological features of the effective model. Final remarks and perspectives of this methodology are reported in Sect. 6.

## 2 Generalities

*V*and

*U*are their respective potentials and \(\mu =0,1\) if we are working in \(1+1\) space-time. The equations of motion for both theories can be derived in a straightforward way, yielding

*x*-coordinate.

*f*is named the “deformation function”. Therefore, replacing the deformation function in the first-order differential equation for the field \(\phi \), we find the constraints

*V*can be defined as

## 3 The method

The method to construct new analytical three-field models will be a generalization of the extension method for two scalar fields, introduced by Bazeia, Losano and Santos [21]. In order to establish the three-field version for the extension method, we are going to use two deformation functions to rewrite the left-hand side of Eq. (5) as the first differential equation presented in (14). An analogous procedure can be repeated to generate the other two first-order differential equations shown in (14).

This mechanism means that we can build a three scalar field model combining three one-field systems. Moreover, once we know the solutions of the three one-field models and the deformation functions, the effective three-field model is going to be analytically solvable. Such a nice feature agrees with the results derived in [21].

*g*are responsible for connecting the fields \(\phi \) and \(\chi \) in this effective three-field model via the additional constraint

*g*, \(\tilde{f}\) and \(\tilde{g}\), we need to establish some restrictions for constraints (20), (24) and (26). Such restrictions may yield different forms for the effective three-field model. Below we are going to show two different scenarios which can be generated from these restrictions.

*g*, \(\tilde{f}\) and \(\tilde{g}\).

*g*. After the calculation of

*g*, \(\tilde{f}\) and \(\tilde{g}\), we can substitute all the ingredients into Eqs. (18), (21) and (22) to derive \(W_{\phi }\), \(W_{\chi }\) and \(W_{\xi }\). The next section exemplifies the applicability of our methodology and unveils new analytical three scalar field models.

## 4 Examples

### 4.1 Example I: \(\phi ^{\,4}\) versus \(\chi ^{\,4\,I}\) versus \(\xi ^{\,4\,I}\)

*I*stands for “inverted”. The first-order differential equations for each one of these models are

*a*and

*b*real constants whose solutions are

*g*, \(\tilde{f}\) and \(\tilde{g}\). In order to do it, let us consider the two possible scenarios that we pointed out in the last section, then we can compare the similarities or the differences between these approaches.

### 4.2 First approach

*g*. So, integrating the last with respect to \(\chi \) we find

*g*, \(\tilde{f}\), \(\tilde{g}\), (38), (39) and (40) back into Eqs. (18), (21) and (22) yields

*a*and

*b*are

### 4.3 Second approach

*W*if one chooses the first or the second approach or even other possible scenarios for Eqs. (19), (23) and (25). Based on this, we are going to consider just the first scenario approach for the next examples.

### 4.4 Example II: \(\phi ^{\,4}\) versus \(\chi ^{\,4\,I}\) versus \(\xi ^{\,4}\)

An interesting feature of this new model is that it represents two domain walls with an internal structure. Models like this emerged before in the work of Shifmann et al., where the authors investigated localization of gauge fields inside of domain walls [25]. Besides, the model here derived complements the discussions presented by Bazeia et al. in [26], where the authors worked with an analytical model composed of one domain wall with an internal structure formed by two other fields. Another special aspect about both, our model and the one from [26], is that they represent a natural bridge for the four-field model introduced by Callen and Volkas [27], which has two domains walls plus an internal structure composed of two other fields.

The resultant potential can be derived combining (70) with (9), and one can see that it has \(Z_2\) symmetry (\(\phi \rightarrow -\phi , \chi \rightarrow -\chi , \xi \rightarrow -\xi \)), securing the stability of this topological configuration of fields [27]. The \(Z_2\) symmetry implies that our model has the same features of the Dirichlet domain walls introduced by Carroll and Trodden [28]. Such a symmetry also matches with the behavior of the model studied by Bazeia et al. in [26], moreover, we can follow the ideas presented in the mentioned work to discuss the physical features of the internal structure of our model.

*V*at this region. Furthermore, the projections of

*V*inside and outside both walls are

## 5 Analytical three-field cosmological model

*p*are the density and the pressure related with the scalar field model. From the previous equation, we are able to compute

*H*is the Hubble parameter.

*H*and \(\dot{H}\) into (82), we find the first-order differential equations

*a*,

*b*and \(\tau \) are real constants.

*a*,

*b*, \(\kappa \), and \(\tau \) are not extremely constrained, or, in another words, small variations of these values do not change too much the physical aspects of the cosmological parameters.

*V*in Eq. (87) and the EoS (83), respectively, as

*V*(91) together with

*W*(71) and the solutions (89) to prove that the equations of motion (88) are satisfied.

## 6 Conclusion

A mechanism to generate new models with three scalar fields was presented in this work. We started the method coupling three analytical one-field models via a deformation procedure, introduced in [21]. The non-trivial combinations of these one-field systems unveiled effective three-field models. As a first example we were able to derive a well-known three-field version for the BNRT model [4]. Another interesting feature is that the new models are automatically satisfied by the solutions of the one-field systems, corroborating the results derived in [21].

The superpotentials here derived are all polynomial, but we also can use this methodology to build three-field models with functional potentials, like combinations involving sine-Gordon potentials, for instance. Besides, with this superpotential in hands we are able to find the total energy related with the correspondent defects solutions, as well as the potential \(V(\phi ,\chi ,\xi )\).

The mechanism has shown to provide an interesting cosmological scenario, able to predict two accelerated eras, including a late time accelerated one, which simulates the dark energy era. The acceleration and EoS parameters for the decelerated stages of the Universe have also been obtained, which leads us to conclude that from the deformation procedure applied to a scenario with three scalar fields, one is able to obtain a complete cosmological scenario, with the transition stages being described continuously.

We believe that the method here presented can be applied in compactons-like defects [30], in braneworld [4, 30] and in other cosmological scenarios [6, 7, 31]. It is going to be an interesting task to observe the consequence of such new analytical three-field systems for the physical parameters like the cosmological ones or the warp factor. Some of these ideas are under investigation and we hope to report on them in the near future.

## Notes

### Acknowledgements

D. A. Ferreira and D. C. Vilar Neta would like to thank Capes (Brazilian agency) for financial support. P. H. R. S. Moraes would like to thank São Paulo Research Foundation (FAPESP), grant 2015/08476-0, for financial support. The authors also would like to thank both anonymous referees for their criticism, which enhanced the potential of this work.

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