Painlevé Analysis of the Cosmological Field Equations in Weyl Integrable Spacetime

Abstract

We apply a singularity analysis to investigate the integrability properties of the gravitational field equations in Weyl Integrable Spacetime for a spatially flat Friedmann–Lemaître–Robertson–Walker background spacetime induced by an ideal gas. We find that the field equations possess the Painlevé property in the presence of the cosmological constant, and the analytic solution is given by a left Laurent expansion.

1. Introduction

The Weyl Integrable Spacetime (WIS) is a natural way to extend Einstein’s General Relativity, in which a scalar field is introduced in the natural space by geometrical degrees of freedom [1]. Scalar fields play an important role in the description of gravitational phenomena at large scales [2,3]. Indeed, it has been proposed that the late-time and early-time acceleration phases of the universe could be attributed to scalar fields [4,5,6,7,8]. Another novelty of the WIS is that interaction is introduced geometrically between the matter components [9,10,11]. Cosmological models with interaction in the dark sector of the universe have been widely studied before. Furthermore, the fact that interacting models survive the constraints is in line with the cosmological observations [12,13,14]. More specifically, the interaction in the dark sector of the universe is a mechanism for solving the cosmic coincidence problem, and it has been used to explain the discrepancy in the cosmological constant [15,16,17,18,19]. In addition, WIS is related to chameleon cosmology [20].

There are various studies in the literature on cosmological models on the WIS. Multi-dimensional spacetimes were investigated in [21], while lower-dimensional gravitational models were studied in [22]. Inflation in the WIS was investigated in [23,24], where it was found that non-singular inflationary solutions are provided by the specific theory. In [25,26,27], WIS was considered as the dark energy mechanism which drives the present acceleration of the universe, while recently, an extension of WIS in the Einstein–Aether theory was studied in [28]. The dynamical analysis of the field equations with different fluid sources for the construction of the cosmological history was the subject of study in [29].

The field equations in WIS are nonlinear, ordinary differential equations of second order. Because of the new degrees of freedom provided by the scalar field, there are not many known solutions in the literature. Note that the cosmological field equations in WIS admit a minisuperspace description the Noether symmetry analysis, which was applied in [30] for the construction of conservation laws and the derivation of analytic solutions. The Noether symmetry analysis is a powerful and systematic approach for the study of the integrability properties of dynamical systems that follow from a variational principle, with many applications in cosmological studies [31,32,33,34,35,36]. For the cosmological model in WIS, where the matter source is an ideal gas, it was found that the introduction of the cosmological constant term into the gravitational action integral leads to a dynamical system with fewer Noetherian conservation laws, from which we are not able to infer the integrability properties. In this work we investigate the same problem with the use of another important tool of analysis for the construction of conservation laws and analytic solutions; this tool is known as Painlevé analysis or singularity analysis [37]. The singularity analysis used for the field equations in gravity for a wide range of applications; for instance, see [38,39,40,41,42,43,44,45]. The determination of the integrability properties of dynamical systems is essential in physics and in all areas of applied mathematics. The novelty of a physical system being described by an integrable dynamical system is that we know that an actual solution exists when we apply numerical methods for the study of the system, or that there exists a closed-form function which solves the dynamical system. For a recent discussion on the integrable cosmological models, we refer the reader to [46]. In our study, the singularity analysis leads to the construction of analytic solutions expressed as Laurent expansions around a movable singularity. The plan of this paper is as follows.

In Section 2, we present the cosmological model of our interest, which is that of the WIS in a homogeneous and isotropic universe, and where the matter source is an ideal gas. In Section 3, we review previous results wherein we show the existence of additional conservation laws for the field equations in the absence of the cosmological constant term. Section 4 includes the main analysis of this study, in which the singularity analysis is applied for the study of the integrability properties of the field equations. We find that the field equations possess the Painlevé property with or without a cosmological constant term, and the analytic solution is expressed by Laurent expansions. Finally, in Section 5, we summarize our results and draw our conclusions.

2. Field Equations

In WIS or in Weyl Integrable Geometry (WIG), the modified Einstein–Hilbert action is expressed as follows:

$$S_W=\int d{x}^4\sqrt{-g}\left(\tilde{R}+\xi \left({\tilde{\nabla }}_{\nu }\left({\tilde{\nabla }}_{\mu }\varphi \right)\right)g^{\mu \nu }-\mathsf{\Lambda }+{\mathcal{L}}_m\right),$$

(1)

in which $g_{\mu \nu }$ is the metric tensor for the physical space, ${\tilde{\nabla }}_{\mu }$ denotes the covariant derivative defined by the symbols ${\tilde{\mathsf{\Gamma }}}_{\mu \nu }^{\kappa }$, where ${\tilde{\mathsf{\Gamma }}}_{\mu \nu }^{\kappa }$ are the Christoffel symbols for the conformally related metric ${\tilde{g}}_{\mu \nu }=\varphi g_{\mu \nu }$. The scalar field $\varphi$ is the coupling function, $\mathsf{\Lambda }$ is the cosmological constant term, the parameter $\xi$ is an arbitrary coupling constant, and ${\mathcal{L}}_m$ is the Lagrangian function for the matter source.

When ${\mathcal{L}}_m$ describes a perfect fluid with an energy density $\rho$ and a pressure component p, the field equations in the Einstein–Weyl theory can be derived as follows:

$${\tilde{G}}_{\mu \nu }+{\tilde{\nabla }}_{\nu }\left({\tilde{\nabla }}_{\mu }\varphi \right)-\left(2\xi -1\right)\left({\tilde{\nabla }}_{\mu }\varphi \right)\left({\tilde{\nabla }}_{\nu }\varphi \right)+\xi g_{\mu \nu }{g}^{\kappa \lambda }\left({\tilde{\nabla }}_{\kappa }\varphi \right)\left({\tilde{\nabla }}_{\lambda }\varphi \right)-\mathsf{\Lambda }{g}_{\mu \nu }=-\left(\tilde{\rho }+\tilde{p}\right)u_{\mu }{u}_{\nu }-\tilde{p}{g}_{\mu \nu },$$

(2)

where ${\tilde{G}}_{\mu \nu }$ is the Einstein tensor with respect the metric ${\tilde{g}}_{\mu \nu }$, and $u^{\mu }$ is the comoving observer. The parameters $\tilde{\rho },\tilde{p}$ are the energy density and pressure components for the matter source multiplied by the factor $e^{-\frac{\varphi }{2}}$.

Field Equation (2) can be written in the following equivalent form:

$$G_{\mu \nu }-\lambda \left({\varphi }_{,\mu }{\varphi }_{,\nu }-\frac{1}{2}{g}_{\mu \nu }{\varphi }^{,\kappa }{\varphi }_{,\kappa }\right)-\mathsf{\Lambda }{g}_{\mu \nu }=-\left(\tilde{\rho }+\tilde{p}\right)u_{\mu }{u}_{\nu }-\tilde{p}{g}_{\mu \nu },$$

(3)

where $G_{\mu \nu }$ is the Einstein tensor for the background space $g_{\mu \nu }$ and $\lambda =2\xi -\frac{3}{2}$.

For the homogeneous, isotropic, and spatially flat Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime,

$$d{s}^2=-N^2\left(t\right)d{t}^2+a^2\left(t\right)\left(d{r}^2+r^2\left(d{\theta }^2+{sin}^2\theta d{\phi }^2\right)\right),$$

(4)

and the modified Friedmann equations are as follows:

$$3H^2-\frac{\lambda }{2N^2}{\dot{\varphi }}^2-\mathsf{\Lambda }-e^{-\frac{\varphi }{2}}\rho =0,$$

(5)

$$\dot{H}+H^2+\frac{1}{6}{e}^{-\frac{\varphi }{2}}\left(\rho +3p\right)+\frac{\lambda }{3N^2}{\dot{\varphi }}^2-\frac{\mathsf{\Lambda }}{3}=0,$$

(6)

$$\ddot{\varphi }-\frac{\dot{N}}{N}\dot{\varphi }+3H\dot{\varphi }+\frac{1}{2\lambda }{N}^2{e}^{-\frac{\varphi }{2}}\rho =0$$

(7)

and

$$\dot{\rho }+3NH\left(\rho +p\right)-\rho \dot{\varphi }=0$$

(8)

in which we considered the comoving observer $u^{\mu }=\frac{1}{N}{\delta }_t^{\mu }$, and $H=\frac{1}{N}\frac{\dot{a}}{a}$ is the Hubble function.

In this study, we consider that the matter source is an ideal gas, that is $p=\left(\gamma -1\right)\rho$, $\gamma <2$. For $\gamma =2$, a two-scalar-field model is recovered [47]. From Equation (8), it follows that $\rho ={\rho }_0{a}^{-3\gamma }{e}^{\varphi }$. Thus, we end with the set of differential Equations (5)–(7).

The cosmological history for this specific cosmological model has been investigated in [29]. From the analysis of the dynamics, it was found that the cosmological history consists of two matter epochs and two acceleration phases. The matter epochs correspond to unstable asymptotic solutions, while for the two accelerated phases, the one that is always unstable can be associated with the early acceleration phase of the universe; the second accelerated asymptotic solution describes the future de Sitter attractor.

3. Lagrangian Description

In the case of an ideal gas, the cosmological field Equations (5)–(7) can be reproduced by the variation of the point-like Lagrangian:

$$\mathcal{L}\left(N,a,\dot{a},\varphi ,\dot{\varphi }\right)=\frac{1}{N}\left(-3a{\dot{a}}^2+\frac{\lambda }{2}{a}^3{\dot{\varphi }}^2\right)-N\left(a^3\mathsf{\Lambda }+{\rho }_{m0}{e}^{\frac{\varphi }{2}}{a}^{3-3\gamma }\right).$$

(9)

The Lagrangian function (9) is a singular Lagrangian because $\frac{\partial \mathcal{L}}{\partial \dot{N}}=0.$ The constraint Equation (5) is the Euler–Lagrange equation with respect to the lapse function $N\left(t\right)$, i.e., $\frac{\partial \mathcal{L}}{\partial N}=0$, while the second-order differential Equations (6) and (7) are the Euler–Lagrange equations with respect to the variables $a\left(t\right)$ and $\varphi \left(t\right)$, respectively, i.e., $\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{a}}\right)-\frac{\partial \mathcal{L}}{\partial a}=0$ and $\frac{d}{dt}\left(\frac{\partial \mathcal{L}}{\partial \dot{\varphi }}\right)-\frac{\partial \mathcal{L}}{\partial \varphi }=0$.

The existence of the singular Lagrangian (9) and the minisuperspace description of the gravitational model are important characteristics for the study of the integrability properties of the field equations. Without loss of generality, the lapse function can be assumed to be $N\left(t\right)=N\left(a\left(t\right),\varphi \left(t\right)\right)$. Under such consideration, the Lagrangian (9) is regular, while Equation (5) is the conservation law of “energy”, that is, the Hamiltonian of the dynamical system.

Normal Coordinates

We follow [29], and without loss of generality, we select $N\left(t\right)=a{\left(t\right)}^3$. Furthermore, we define the new field $\mathsf{\Phi }$ as follows:

$$\varphi =\frac{1}{\lambda \left(2-\gamma \right)}\left(2\lambda ln\mathsf{\Phi }-lna\right).$$

(10)

In the new variables $\left\{a,\mathsf{\Phi }\right\}$, the Lagrangian of the field equations is

$$\mathcal{L}\left(a,\dot{a},\mathsf{\Phi },\dot{\mathsf{\Phi }}\right)=\frac{1}{2}\left(6-\frac{1}{{\left(\gamma -2\right)}^2\lambda }\right){\left(\frac{\dot{a}}{a}\right)}^2+\frac{2}{{\left(\gamma -2\right)}^2}\frac{\dot{a}\mathsf{\Phi }}{a\mathsf{\Phi }}-\frac{2\lambda }{{\left(\gamma -2\right)}^2}{\left(\frac{\dot{\mathsf{\Phi }}}{\mathsf{\Phi }}\right)}^2+\mathsf{\Lambda }{a}^6+a^{3\left(2-\gamma \right)-\frac{1}{2\lambda \left(2-\gamma \right)}}{\mathsf{\Phi }}^{\frac{1}{2-\gamma }}.$$

(11)

The field equations are

$$\frac{1}{2}\left(6-\frac{1}{{\left(\gamma -2\right)}^2\lambda }\right){\left(\frac{\dot{a}}{a}\right)}^2+\frac{2}{{\left(\gamma -2\right)}^2}\frac{\dot{a}\mathsf{\Phi }}{a\mathsf{\Phi }}-\frac{2\lambda }{{\left(\gamma -2\right)}^2}{\left(\frac{\dot{\mathsf{\Phi }}}{\mathsf{\Phi }}\right)}^2-\mathsf{\Lambda }{a}^6-a^{3\left(2-\gamma \right)-\frac{1}{2\lambda \left(2-\gamma \right)}}{\mathsf{\Phi }}^{\frac{1}{2-\gamma }}=0,$$

(12)

$$\ddot{a}+\frac{1}{2}{a}^{7-3\gamma -\frac{1}{2\lambda \left(2-\gamma \right)}}{\mathsf{\Phi }}^{\frac{1}{2-\gamma }}-\frac{{\dot{a}}^2}{a}-\mathsf{\Lambda }{a}^7=0$$

(13)

and

$$\ddot{\mathsf{\Phi }}-\frac{{\dot{\mathsf{\Phi }}}^2}{\mathsf{\Phi }}-\frac{\mathsf{\Lambda }}{2\lambda }{a}^6\mathsf{\Phi }=0.$$

(14)

When $\mathsf{\Lambda }=0$, Equation (14) provides the conservation law $I_0=\frac{d}{dt}ln\mathsf{\Phi }$. This conservation law was also derived in [29,30]. The set of variables $\left\{a,\mathsf{\Phi }\right\}$ constitutes the normal coordinates for the field equations. However, in the presence of the cosmological constant, the conservation law does not exist. The existence of the second conservation law for the field equations is essential in order to be able to reach a conclusion about the integrability of the field equations. Indeed, in [30], the Hamilton–Jacobi equation was solved, and the field equations were reduced to a system of two first-order ordinary differential equations.

In the presence of $\mathsf{\Lambda }$, we applied a symmetry analysis for the point and contact symmetries, and we found that the field equations do not admit any Noether symmetry. Furthermore, we considered the polynomial functions of $I\left(a,\dot{a},\mathsf{\Phi },\dot{\mathsf{\Phi }}\right)$ to be a conservation law. However, we were not able to determine any function I with this specific requirement.

The concept of integrability is not limited to the existence of invariant functions and conservation laws. According to the Painlevé approach, a dynamical system is integrable if it admits a movable pole and if its solution is expressed in terms of a Laurent expansion around the movable pole. In the following, we apply the singularity analysis in order to investigate the integrability properties of the dynamical system of our consideration.

4. Singularity Analysis

The modern treatment of a singularity analysis is described by the ARS algorithm. The algorithm has three main steps. They are (a) the derivation of the leading-order behavior, (b) the derivation of the resonances, and (c) the consistency test. For more details and examples on the application of the ARS algorithm, we refer the reader to [37]. In the first step of the algorithm, we should show that there exists a movable singularity for the dynamical system at which the solution is approximated by a singular expression. For instance, for the power-law expressions ${\left(\tau -{\tau }_0\right)}^p$, the exponent p should be a negative number. However, it has been shown that p can also be a rational number. The resonances should be the same in number as the degrees of freedom of the problem, and one of the resonances has to be $-1$; in addition, the resonances should be rational numbers. If the requirements of steps (a) and (b) are satisfied, we can write the solution as a Laurent expansion; in our case, this is a Puiseux series, which we can use in the original equation to check if it is an actual solution for the original dynamical system.

We then follow [48], and we apply the singularity analysis for the equivalent dynamical system in the dimensionless variables. Indeed, we follow the H-normalization approach, and we define the new variables as follows:

$$x=\sqrt{\frac{1}{6}}\frac{\dot{\varphi }}{H},{\mathsf{\Omega }}_{\mathsf{\Lambda }}=\frac{\mathsf{\Lambda }}{3H^2},{\mathsf{\Omega }}_m=\frac{{\rho }_m}{3H^2}{e}^{-\frac{\varphi }{2}},d\tau =NHdt,$$

(15)

where the latter parameter $\tau$ is the new independent variable.

Consequently, field Equations (5)–(8) for the ideal gas are written as the following equivalent algebraic differential system:

$$\frac{d{\mathsf{\Omega }}_{\mathsf{\Lambda }}}{d\tau }=-{\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(\left(\gamma -2\right)\lambda x^2+\gamma \left({\mathsf{\Omega }}_{\mathsf{\Lambda }}-1\right)\right)$$

(16)

and

$$\frac{dx}{d\tau }=\frac{1}{12\lambda }\left(\left(\lambda x^2-1\right)\left(\sqrt{6}-6\left(\gamma -2\right)\lambda x\right)+\left(\sqrt{6}-6\gamma \lambda x\right){\mathsf{\Omega }}_{\mathsf{\Lambda }}\right),$$

(17)

with the constraint equation

$${\mathsf{\Omega }}_m=1-\lambda x^2-{\mathsf{\Omega }}_{\mathsf{\Lambda }}.$$

(18)

Therefore, we continue with the investigation of the singularity analysis for the system of first-order ordinary differential Equations (16) and (17).

4.1. Case $\mathsf{\Lambda }=0$

We focus now on the case of $\mathsf{\Lambda }=0$. Equation (17) is reduced to the simple form of

$$\frac{dx}{d\tau }=\frac{1}{12\lambda }\left(\left(\lambda x^2-1\right)\left(\sqrt{6}-6\left(\gamma -2\right)\lambda x\right)\right).$$

(19)

The closed-form solution of the this equation is

$$\tau -{\tau }_0=\frac{6\lambda \left(\gamma -2\right)}{1-6\lambda {\left(\gamma -2\right)}^2}ln\frac{\left(\lambda x^2-1\right)}{{\left(\sqrt{6}-6\lambda \left(\gamma -2\right)x\right)}^2}-\frac{2\sqrt{\lambda }}{1-6\lambda {\left(\gamma -2\right)}^2}arctanh\left(\lambda x\right).$$

(20)

However, we are interested in finding out if Equation (19) possesses the Painlevé property and if the analytic solution can be expressed as a Laurent expansion around a movable pole.

According to the ARS algorithm, we replace $x\left(t\right)=x_0{\left(\tau -{\tau }_0\right)}^p$ in (19), that is,

$$p{x}_0{\left(\tau -{\tau }_0\right)}^{-1+p}-\frac{1}{12\lambda }\left(\left(\lambda x_0^2{\left(\tau -{\tau }_0\right)}^{2p}-1\right)\left(\sqrt{6}-6\left(\gamma -2\right)\lambda x_0{\left(\tau -{\tau }_0\right)}^p\right)\right).$$

(21)

Hence, the dominant terms provide the algebraic equation $-1+p=3p$, that is, $p=-\frac{1}{2}$ and $\lambda \left(\gamma -2\right)x_0^2-1=0$.

For the determination of the resonances, we substitute

$$x\left(t\right)=x_0{\left(\tau -{\tau }_0\right)}^{-\frac{1}{2}}+m{\left(\tau -{\tau }_0\right)}^{-\frac{1}{2}+S},\lambda x_0^2+1=0$$

(22)

into (19), and we expand around $m^2\to 0$. Hence, we end with the algebraic equation $S+1=0$, which means that $S=-1$. That is in agreement with the singularity analysis because one of the resonances should be $-1$. This resonance indicates that the leading-order behavior describes a movable singularity and that $t_0$ is an integration constant.

For the third step of the ARS algorithm, which is known as the consistency test, we substitute

$$x\left(\tau \right)=x_0{\left(\tau -{\tau }_0\right)}^{-\frac{1}{2}}+x_1{\left(\tau -{\tau }_0\right)}^0+x_2{\left(\tau -{\tau }_0\right)}^{\frac{1}{2}}+x_3\left(\tau -{\tau }_0\right)+\cdots$$

(23)

into (19), where we find

$$x_1=\frac{x_0^2}{3\sqrt{6}},x_2=\frac{x_0\left(x_0^2+18\left(\gamma -2\right)\right)}{72},\dots .$$

We therefore conclude that Equation (19) possesses the Painlevé property, and the analytic solution is described by the Puiseux expansion.

4.2. Case $\mathsf{\Lambda }\ne 0$

We focus now on the case for which there exists a nonzero cosmological constant term. In order to determine the leading-order term for the systems (16) and (17), we substitute the following equations:

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(\tau \right)& ={\mathsf{\Omega }}_{\mathsf{\Lambda }0}{\left(\tau -{\tau }_0\right)}^q\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill x\left(\tau \right)& =x_0{\left(\tau -{\tau }_0\right)}^p,\hfill \end{array}$$

(25)

and as above, we determine $p=-\frac{1}{2}$ and $p=-1$ with the constraint equation $\lambda \left(\gamma -2\right)x_0^2-1=\gamma {\mathsf{\Omega }}_{\mathsf{\Lambda }0}$. We can easily see that for ${\mathsf{\Omega }}_{\mathsf{\Lambda }}=0$, the previous leading-order term is recovered.

For the resonances, we replace

$$\begin{array}{cc}\hfill {\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(\tau \right)& ={\mathsf{\Omega }}_{\mathsf{\Lambda }0}{\left(\tau -{\tau }_0\right)}^{-1}+n{\left(\tau -{\tau }_0\right)}^{-1+S},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill x\left(\tau \right)& =x_0{\left(\tau -{\tau }_0\right)}^{-\frac{1}{2}}+m{\left(\tau -t_0\right)}^{-\frac{1}{2}+S}\hfill \end{array}$$

(27)

in (16) and (17).

We then expand around $m^2\to 0$, $n^2\to 0$, and $mn\to 0$. The first-order perturbed terms provide the matrix

$$A=\left(\begin{array}{cc}-\frac{2}{x_0}\left({\mathsf{\Omega }}_{\mathsf{\Lambda }0}\left(\gamma {\mathsf{\Omega }}_{\mathsf{\Lambda }0}-1\right)\right)& -\left(S+\gamma {\mathsf{\Omega }}_{\mathsf{\Lambda }0}\right)\\ \left(\left(1+S\right)-\gamma {\mathsf{\Omega }}_{\mathsf{\Lambda }0}\right)& \frac{1}{2}{x}_0\gamma \end{array}\right),$$

(28)

where the zeros of the determinant $det\left(A\right)=0$ give the resonances; they are

$$S_1=-1,S_2=0.$$

(29)

The resonances indicate that the there exists a movable singularity and that one of the coefficients, $x_0$ or ${\mathsf{\Omega }}_{\mathsf{\Lambda }0}$, is arbitrary and is the second integration constant of the dynamical system.

We then write the Puiseux series as follows:

$${\mathsf{\Omega }}_{\mathsf{\Lambda }}\left(\tau \right)={\mathsf{\Omega }}_{\mathsf{\Lambda }0}{\left(\tau -{\tau }_0\right)}^{-1}+{\mathsf{\Omega }}_{\mathsf{\Lambda }1}{\left(\tau -{\tau }_0\right)}^{-\frac{1}{2}}+{\mathsf{\Omega }}_{\mathsf{\Lambda }2}{\left(\tau -{\tau }_0\right)}^0+{\mathsf{\Omega }}_{\mathsf{\Lambda }3}{\left(\tau -{\tau }_0\right)}^{\frac{1}{2}}+\dots ,$$

(30)

$$x\left(\tau \right)=x_0{\left(\tau -{\tau }_0\right)}^{-\frac{1}{2}}+x_1{\left(\tau -{\tau }_0\right)}^0+x_2{\left(\tau -{\tau }_0\right)}^{\frac{1}{2}}+x_3\left(\tau -{\tau }_0\right)+\dots$$

(31)

which we replace in (16) and (17). Hence, the latter Puiseux series solves the dynamical system when

$${\mathsf{\Omega }}_{\mathsf{\Lambda }1}=\frac{4x_1{\mathsf{\Omega }}_{\mathsf{\Lambda }0}\left(\gamma {\mathsf{\Omega }}_{\mathsf{\Lambda }0}-1\right)}{x_0\left(1+2\gamma {\mathsf{\Omega }}_{\mathsf{\Lambda }0}\right)},x_1=\frac{x_0\left(\sqrt{6}{x}_0\left(1-2{\mathsf{\Omega }}_{\mathsf{\Lambda }0}\right)+6\gamma {\mathsf{\Omega }}_{\mathsf{\Lambda }1}\left(\gamma {\mathsf{\Omega }}_{\mathsf{\Lambda }0}-1\right)\right)}{6\left(\gamma {\mathsf{\Omega }}_{\mathsf{\Lambda }0}-1\right)\left(2\gamma {\mathsf{\Omega }}_{\mathsf{\Lambda }0}-3\right)},\dots .$$

(32)

We summarize our results in the following proposition.

Proposition 1.

The cosmological field equations in the Weyl Integrable theory for a spatially flat FLRW background geometry, which is induced by a cosmological constant term and an ideal gas, form a dynamical system which possesses the Painlevé property, that is, the field equations are integrable and the analytic solution is expressed by the Puiseux expansions (30) and (31).

5. Conclusions

In this study, we investigated the integrability properties of the field equations in Weyl Integrable theory. In particular, we studied the existence of analytic solutions in the cosmological scenario for a universe whose matter source is an ideal gas in a spatially flat FLRW geometry with a nonzero cosmological constant term. For our analysis we applied the singularity analysis, which has been widely applied in gravitational physics and has many interesting results.

The field equations for the model of our consideration have the property to admit a minisuperspace. That means that the field equations follow from the variation of the dynamical variables of a point-like Lagrangian. For the case in which there is no cosmological constant term, we know from previous results that there exists a second conservation law which indicates the integrability properties of the dynamical system. However, in the presence of the cosmological constant, this conservation law does not exist. Thus, we were not able to infer the integrability of the field equations.

For simplicity of analysis, we wrote the field equations into the equivalent system with the use of dimensionless variables. For the equivalent system, we applied the singularity analysis and found that the field equations possess the Painlevé property, that is, the cosmological model of our analysis is integrable with or without the cosmological constant term. This is a very interesting result because after the loss of the additional integration constant provided by Noether’s analysis, we were not able to make an inference on the integrability, and someone could infer that the introduction of the cosmological constant into the field equations may violate the integrability property.

As we have discussed before, this cosmological model describes important eras of cosmological history. The existence of an analytic solution indicates that there exist real solutions which correspond to the numerical simulations of the field equations. This is an important feature because we can understand the evolution of the dynamical system according to the free parameters. In addition, the above analysis can be used for the analytic reconstruction of the cosmographic parameters [49,50] and the relation of these parameters with the initial value problem.

This work contributes to the subject of the derivation of exact and analytic solutions in gravitational physics. Moreover, it shows that the Noether symmetry analysis and the singularity analysis are two complementary methods that can provide interesting results.