Cosmological solutions in Ho\v{r}ava-Lifshitz gravity

TL;DR

This paper investigates the integrability of Hořava-Lifshitz scalar field cosmology using Singularity Analysis, successfully deriving explicit solutions for exponential potentials through Painlevé series expansion.

Contribution

It applies Singularity Analysis to Hořava-Lifshitz cosmology, providing explicit solutions and demonstrating integrability for exponential scalar field potentials.

Findings

Successfully performed Painlevé analysis on the field equations.

Derived explicit solutions in terms of Laurent and Painlevé series.

Confirmed integrability for exponential potential cases.

Abstract

We perform a detailed study on the integrability of the Ho\v{r}% ava-Lifshitz scalar field cosmology in a Friedmann--Lema\^{\i}% tre--Robertson--Walker background spacetime. The approach we follow to determine the integrability is that of the Singularity Analysis. More specifically, we test if the gravitational field equations possesses the Painlev\'{e} property. For the exponential potential of the scalar field we are able to perform an analytic explicit integration of the field equations and write the solution in terms of Laurent expansion and more specifically write the solution in terms of Right Painlev\'{e} series.

Equations

$$S_g$$

$$d{s}^2=-N^{2}d{t}^2+g_{ij}(d{x}^i+N^{i}dt)(d{x}^j+N^{j}dt),$$

$$K_{ij}=\left(\dot{g_{ij}}-{\nabla }_i{N}_j-{\nabla }_j{N}_i\right)/2N$$

$$C^{ij}=\epsilon^{ijk}\nabla_{k}\bigl{(}R_{i}^{j}-R\delta_{i}^{j}/4\bigr{)}/\sqrt{g}.$$

$${\gamma }_{ij}d{x}^{i}d{x}^j=\frac{d{r}^2}{1-k{r}^2}+r^{2}d{\Omega }_2^2,$$

$$S_{\varphi }=\int dt{d}^{3}x\sqrt{g}N\left[\frac{3\lambda -1}{4}\frac{{\dot{\varphi }}^2}{N^2}-V(\varphi )\right],$$

$$3H^2$$

$$2\dot{H}+3H^2=-\frac{1}{2(3\lambda -1)}\left[\frac{3\lambda -1}{4}{\dot{\varphi }}^2-V(\varphi )\right]$$

$$-\frac{1}{16(3\lambda -1{)}^2}\left[-\frac{{\mu }^2{k}^2}{a^4}+3{\mu }^2{\Lambda }^2-\frac{2{\mu }^2\Lambda k}{a^2}\right],$$

$$\ddot{\varphi }+3H\dot{\varphi }+\frac{2V^{\prime }(\varphi )}{3\lambda -1}=0.$$

$$3H^2={\rho }_{\varphi }+{\rho }_{HL},$$

$$-2\dot{H}-3H^2=p_{\varphi }+p_{HL},$$

$${\dot{\rho }}_{\varphi }+3H\left({\rho }_{\varphi }+p_{\varphi }\right)=0,$$

$${\rho }_{\varphi }=\frac{1}{2(3\lambda -1)}\left[\frac{3\lambda -1}{4}{\dot{\varphi }}^2+V(\varphi )\right],$$

$$p_{\varphi }=\frac{1}{2(3\lambda -1)}\left[\frac{3\lambda -1}{4}{\dot{\varphi }}^2-V(\varphi )\right],$$

$${\rho }_{HL}=\frac{3}{16(3\lambda -1{)}^2}\left[-\frac{{\mu }^2{k}^2}{a^4}-{\mu }^2{\Lambda }^2+\frac{2{\mu }^2\Lambda k}{a^2}\right],$$

$$p_{HL}=\frac{1}{16(3\lambda -1{)}^2}\left[-\frac{{\mu }^2{k}^2}{a^4}+3{\mu }^2{\Lambda }^2-\frac{2{\mu }^2\Lambda k}{a^2}\right].$$

$$x=\frac{\dot{\varphi }}{2\sqrt{6}H},y=\frac{\sqrt{V(\varphi )}}{\sqrt{6}H\sqrt{3\lambda -1}},$$

$$z=\frac{\mu }{4(3\lambda -1)a^{2}H},u=\frac{\Lambda \mu }{4(3\lambda -1)H},$$

$$s=-\frac{V^{\prime }(\varphi )}{V(\varphi )},f\left(s\right)\equiv \frac{V^{\prime \prime }(\varphi )}{V(\varphi )}-\frac{V^{\prime 2}(\varphi )}{V(\varphi {)}^2},$$

$$\frac{dx}{d\tau }=\left(3x-\sqrt{6}s\right)\left(x^2-1\right),$$

$$\frac{dz}{d\tau }=\left(3x^2-2\right)z,$$

$$\frac{ds}{d\tau }=-2\sqrt{6}xf(s),$$

$$\ln \left({\left(x\left(\tau \right)+1\right)}^{\frac{1}{6+2\sqrt{6}s}}{\left(x\left(\tau \right)+1\right)}^{\frac{1}{6-2\sqrt{6}s}}\right)+$$

$$\frac{3}{9-6s^2}\ln \left(\sqrt{6}s-3x\left(\tau \right)\right)=\tau -{\tau }_0,s\ne \pm \frac{\sqrt{6}}{2},$$

$$\frac{1}{12}\ln \left(\frac{x+1}{x-1}\right)-\frac{1}{6\left(x+1\right)}=\tau -{\tau }_0,s=-\frac{\sqrt{6}}{2},$$

$$\frac{1}{12}\ln \left(\frac{x-1}{x+1}\right)+\frac{1}{6\left(x+1\right)}=\tau -{\tau }_0,s=-\frac{\sqrt{6}}{2},$$

$$\ln z\left(\tau \right)=3\int x{\left(t\right)}^{2}d\tau -2\left(\tau -{\tau }_1\right)\text{.}$$

$$z(x)=z_0{\left(\frac{3x-\sqrt{6}s}{3x_0-\sqrt{6}s}\right)}^{\frac{2(s^2-1)}{2s^2-3}}{\left(\frac{x^2-1}{x_0^2-1}\right)}^{\frac{1}{6-4s^2}}\cdot$$

$$\cdot \exp \left\{\frac{\sqrt{6}s\left[{\tanh }^{-1}(x)-{\tanh }^{-1}(x_0)\right]}{6s^2-9}\right\},$$

$$\tau -{\tau }_0=\ln \left(\frac{\sqrt{6}-3x_0}{\sqrt{6}-3x}\sqrt{\frac{1-x^2}{1-x_0^2}}\right)$$

$$-\sqrt{\frac{2}{3}}\left({\tanh }^{-1}(x)-{\tanh }^{-1}(x_0)\right).$$

$$x\left(\tau \right)=x_0{\tau }^p,z\left(\tau \right)=z_0{\tau }^q$$

$$p{x}_0{\tau }^{-1+p}-3x_0^3{\tau }^{3p}+3x_0{\tau }^p+\sqrt{6}s{x}_0^2{\tau }^{2p}-\sqrt{6}s=0,$$

$$\left(q{\tau }^{-1+q}+2t^q-3x_0{\tau }^{2p+q}\right)z_0=0.$$

Full text

Hořava-Lifshitz; Scalar field; Cosmology; Integrability; Painlevé analysis

I Introduction

Modified theories of gravity have been a subject of special interest over recent years because they provide a geometric approach to the description of observable phenomena. In modified theories of gravity new geometric quantities are introduced into the Einstein-Hilbert Action of General Relativity (GR). The new terms in the gravitational action provide new components of geometric origin in the field equations which change the dynamics such as the solutions of the latter to describe the phenomena observed. In this work, we are interested specifically in the Hořava-Lifshitz (HL) gravity hor3 . HL gravity is a power-counting renormalization theory with consistent ultra-violet behaviour exhibiting an anisotropic Lifshitz scaling between time and space at the ultra-violet limit. HL theory can provide Einstein’s GR as a critical point.

The Einstein-Aether theory DJ ; DJ2 ; Carru is related with HL theory in the classical limit esf . In Einstein-Aether theory the kinematic quantities of a time-like vector field (the Aether) are introduced into the gravitational Action Integral. The theory preserves locality and covariance while it contains GR. Now when the aether field is hypersurface-orthogonal then the classical limit of HL is recovered. Hence every hypersurface-orthogonal Einstein-æther solution is a solution of HL gravity jac01 .

Lorentz=violated theories have been applied in various models of gravitational physics Barrow ; in1 ; in2 . Applications of HL theory cover a wide range of subjects, from compact stars to cosmological studies Wang:2017brl ; Nilsson:2018knn ; or1 ; or2 . In kir0 ; kir1 it was found that  HL cosmology provides an alternative to inflation and that the universe can be singularity-free. For reviews of HL cosmology we refer the reader to kir2 ; kir3 . Very few closed=form solutions exist in HL cosmology with or without any matter source hsol1 ; hsol2 ; hsol3 ; hsol4 ; hsol5 . However, according to our knowledge there are no known analytical solutions in HL cosmology when a minimally=coupled scalar field contributes to the total evolution of the universe.

Recently, a detailed study on the dynamics of the HL scalar field cosmology was performed in genlyhl . More specifically the phase space of HL cosmology was examined for a wide range of scalar field potentials by means of the powerful method of $f$-devisors. Applications were presented for the exponential, the power-law and other potentials. Singular power-law solutions or de Sitter solutions were found to be described at the critical points. However, these are peculiar solutions as they do not exist for any initial conditions and they describe only an approximation of the generic solution for arbitrary initial conditions.

In general, physical theories provide a large number of free parameters or boundary conditions to be involved so that numerical solutions, even if practicable, give no real idea for the properties of the differential equations, which are defined by the theory. This means that analytic techniques are essential for the study of real-world problems. Hence, the knowledge for the existence and the determination of analytical or exact solutions for a given dynamical system is important for the detailed study and understanding of the dynamical system. More specifically, when a dynamical system is integrable, then know that trajectories of (numerical) solutions correspond to “real” solutions for the given dynamical system, or when it is feasible, we can write the analytic solution by using closed-form functions. There are many methods to study the integrability of a dynamical system. In terms of Hamiltonian system usually in physics we refer to the Liouville integrability; where the Hamiltonian system admits sufficient number of conservation laws in order to be solved by quadratures arnold1 . There is a plethora of cosmological models which are Liouville integrable, for instance see ns01 ; ns02 ; ns03 ; ns04 ; ns05 ; ns06 and references therein. The importance of the Liouville integrable cosmological models, apart that the analytical solution can be determined, it is that it is possible that the conservation laws to be applied to perform a canonical quantization which is a specific approach of quantum cosmology q01 ; q02 ; q03 ; q04 .

Although, there are dynamical systems where they are not Liouville integrable, but there is an analytical way to perform an explicit integration of the dynamical system. One of the alternative methods is the singularity analysis which is the main mathematical tool that we apply in this work. The modern treatment of singularity analysis is summarized in the Ablowitz, Ramani and Segur (ARS) algorithm Abl1 ; Abl2 ; Abl3 which provide us with the information if a given differential equation passes the Painlevé test and consequently if the solution of the differential equation can be written as a Laurent expansion around a movable singularity. Singularity analysis is a powerful method which has lead to the determination of analytic solutions of various cosmological models in GR miritzis ; helmi ; cots1 ; cots2 ; pcosm ; bun1 ; bun2 or in modified theories of gravity cots ; sinFR ; sinFR2 ; sinFT ; sinS01 ; sinr01 ; sinr02 . The plan of the paper is as follows.

In Section II, the formal theory of HL cosmology under the detailed-balance condition is presented. While the field equations of HL scalar field cosmology are given. Moreover we derive the equivalent field equations in the dimensionless variables by using the $H$-normalization approach Leon:2009rc ; genlyhl . By using the $H$-normalization we present the dynamical systems of our consideration which we study in terms of integrability. In Section III, we briefly discuss the ARS algorithm. The main results of our analysis and the new analytic solutions in HL cosmology are presented in Section IV. In Section V, we discuss our results and draw our conclusions.

II Hořava-Lifshitz scalar field cosmology

The gravitational Action Integral in Hořava-Lifshitz gravity under the detailed-balance condition is given by the following expression kir1

[TABLE]

where the underlying geometry is written as

[TABLE]

in which the lapse and shift functions are respectively $N$ and $N_{i}$. The spatial metric is given by $g_{ij}$, and roman letters indicate spatial indices. Tensor $K_{ij}$ is the extrinsic curvature defined as

[TABLE]

and $C^{ij}$ denotes the Cotton tensor

[TABLE]

The covariant derivatives are defined with respect to the spatial metric $g_{ij}$. Finally, $\epsilon^{ijk}$ is the totally antisymmetric unit tensor, $\lambda$ is a dimensionless constant and the quantities $w$ and $\mu$ are constants. For simplicity we select to work with units where $\kappa^{2}=8\pi G=1$.

According to the cosmological principle, the universe in large scales is homogeneous and isotropic, consequently the line element (2) reduces to the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime where $N^{i}=0$ and $g_{ij}=a^{2}(t)\gamma_{ij}$ where now $\gamma_{ij}$ denotes the maximally symmetic space of constant curvature $k$, i.e.

[TABLE]

where $k=-1,0,+1$ and $d\Omega_{2}$ is the two-sphere.

In addition, we assume the contribution of a scalar field in the universe with Action Integral

[TABLE]

where for the scalar field it has been considered that $\phi$ inherits the symmetries of the FLRW spacetime.

Variation with respect to the metric in the Action Integrals provides the second-order differential equations

[TABLE]

[TABLE]

where $H=\frac{\dot{a}}{a}$.

On the other hand, variation with respect to the scalar field provides the Klein-Gordon equation

[TABLE]

The latter field equations can be written equivalently as follows

[TABLE]

[TABLE]

and

[TABLE]

where we have defined

[TABLE]

and

[TABLE]

[TABLE]

In this work we focus on the integrability of the dynamical system (7)-(8). However, to follow a similar singularity analysis as the one performed for the Einstein-Aether theory in pallatta we prefer to work with $H$-normalized variables.

$H$-normalized variables defined as ${x,y,z,u}$ in (16) were used in Leon:2009rc to analyze the dynamics of HL cosmology in presence of an scalar field with exponential potential. That analysis can be extended with the variable $s,$ and the function $f(s)$ to analyze potentials beyond the exponential potential like in genlyhl by introducing the quantities $s$ and $f$ defined in (16). Hence, the dimensionless dynamical system genlyhl defined by the new variables $\left\{x,y,z,u,s\right\}$

[TABLE]

where function $f\left(s\right)$ is defined by the specific form of the potential $V\left(\phi\right).$ As a new independent variable it is assumed the number of e-fold $\tau=\ln\left(\frac{a}{a_{0}}\right)$ allows to recast the cosmological equations for arbitrary potentials in a closed form. We remark, that for the exponential potential $V\left(\phi\right)=V_{0}e^{\sigma\phi}$, function $f\left(s\right)$ vanishes and $s=\sigma$, while for the power-law potential $V\left(\phi\right)=V_{0}\phi^{2n}$  function $f\left(s\right)$ is defined as $f(s)=-\frac{s^{2}}{2n}$.

At this point it is important to mention that while the analysis of critical points in genlyhl for the HL scalar field cosmology is for arbitrary potential, in order to study the integrability of the field equations by applying the ARS algorithm the potential function has to be specified. The exponential and power-law potentials are two well-known and well-studied potentials in GR while they can approximate the functional behaviour of other nonlinear potentials in their limits; for instance, the hyperbolic potential $V\left(\phi\right)=V_{0}\sinh\left(\sigma\phi\right)^{2n}$ where for small values of $\phi$ is approximated by the power law potential $V\left(\phi\right)\simeq\phi^{2n}$ and for large values of $\phi$ the hyperbolic potential has an exponential behaviour.

III Singularity analysis

The integration of a differential equation is performed by the global knowledge of the general solution independently from the local solutions provided by Cauchy’s existence theorem. A differential equation which passes the Painlevé test we shall say that it is integrable. However, because not all the integrable differential equations have the Painlevé property, the latter property when it is exists it is better to be referred as the uniformizability of the general equation for the given differential equation Rconte .

In order to study if a given differential equation admits the Painlevé property we apply the ARS algorithm which is summarized in the three main steps: (a) determine the leading-order term which describe a movable singularity; (b) determine the resonances which gives the position of the integration constants and (c) write a Painlevé Series with exponent and step as given in steps (a), (b) and test if solves the differential equation, the latter it is called the consistency test.

There are various criteria and conditions which should be satisfied in the ARS algorithm in order a differential equation to have the Painlevé property, for instance, the number of the resonances should be equal with the number of the degrees of freedom of the differential equation, while one of the resonances should be the $r=-1$, otherwise the singularity determined at step (a) is not movable. Moreover, if the resonances are positive, the Painlevé Series is written by a Right Laurent expansion, when the resonances are negative the Painlevé Series is given by a Left Laurent expansion, otherwise the Painlevé Series is given by a mixed Laurent expansion.  For more details on the criteria of the ARS algorithm and the interpretation of the algorithm in the complex plane we refer the reader to Rconte ; buntis .

IV Explicit integration in Hořava-Lifshitz Cosmology

In this section we study if the field equations of HL scalar field cosmology (7)-(9) possess the Painlevé property, when the latter is true we perform an explicit integration around the movable singularity and we present the generic solution in terms of Laurent expansion. We divide our analysis in the four different cases as studied in genlyhl for arbitrary potentials, and previously in Leon:2009rc for exponential potential.

In case A we assume that the underlying geometry is spatially flat $\left(k=0\right)$ and there is no cosmological constant term $\left(\Lambda=0\right)$, case B is with $k\neq 0$ and $\Lambda=0$. Cases C and D are with $\Lambda\neq 0$ while for the spatial curvature it holds $k=0$ and $k\neq 0$ respectively. Last but not least for the scalar field potential we consider the exponential potential in where $f\left(s\right)\equiv 0$ and the power law potential with $f(s)=-\frac{s^{2}}{2n}$.

IV.1 Case A: $k=0,~{}\Lambda=0$

For the first case of our analysis where the spatial curvature of the FLRW spacetime vanishes and there is not a cosmological constant term, the gravitational field equations in the dimensionless variables (16) form the following three dimensional first-order differential equations genlyhl

[TABLE]

defined on the phase space $\{(x,z,s)\in\mathbb{R}^{3}:-1\leq x\leq 1\}.$

We continue our analysis by presenting the application of the ARS algorithm for the dynamical system.

IV.1.1 Exponential potential

For the exponential potential because the rhs of equation (19) is identically zero the dynamical system is reduced to a two-dimensional first-order differential equations where $s\left(\tau\right)=const$.

We observe that the dynamical system (17), (18) for $s=const$. can be easily integrated by quadratures. Indeed, from equation (17) it follows that

[TABLE]

or

[TABLE]

or

[TABLE]

while equation (18)  gives

[TABLE]

Notice that this system was already integrated at Leon:2009rc , in a rather different way:

[TABLE]

for $x_{0}-\sqrt{6}s\neq 0,x_{0}^{2}\neq 1,s\neq\pm\frac{\sqrt{6}}{2}$.

Let us now apply the ARS algorithm in the system (17), (18). For the first step of the algorithm we substitute

[TABLE]

and we find

[TABLE]

[TABLE]

In order to find the leading-order term we should balance at least two of the exponents of powers of $\tau$. Note that in order a singularity to be occurred parameters $p,q$ should be negative111This is not absolute, in the modern treatment the exponent $p$ of the leading-order term can be also be a positive fractional number., while in general $\tau\rightarrow\tau-\tau_{0}$ where $\tau_{0}$ denotes the position of the singularity where without loss of generality we consider $\tau_{0}=0$.

Hence, from (28) it follows that $-1+q=2p+q$ or $q=2p+q$. The second case provide $p=0$ which we reject, while from the other case we derive $p=-\frac{1}{2}$ where the leading order terms have exponents $\tau^{-\frac{3}{2}}$ which they cancel when $x_{0}^{2}=\sqrt{\frac{q}{3}}$. Moreover from expression (27) we find that the leading-order terms are that with power $\tau^{-\frac{3}{2}}$ and coefficient $-\frac{x_{0}}{2}\left(1+6x_{0}^{2}\right)$ where by demanding the latter to be zero we find that $q=0$ or $q=-\frac{1}{2}$. We conclude, that the leading order behaviour is

[TABLE]

In order to find the resonances $r$ we substitute

[TABLE]

in (17), (18) from where we get

[TABLE]

[TABLE]

where $x_{0}=\pm\frac{i}{\sqrt{6}}$.

From the latter system we define the two-dimensional matrix

[TABLE]

whose requirement $\det A=0$ says that parameters $m$ and $n$ are arbitrary. We find $\det A=r\left(r+1\right)$ from where we calculate the two resonances to be

[TABLE]

The existence or $r=-1$ indicate that the singularity is movable while $r=0$ says that the second integration constant is the coefficient parameter $z_{0}~{}$ in (29). Since the two integration constants have been determined we can say that the system (17), (18) possess the Painlevé property. The algebraic solution is expressed by the Laurent expansions

[TABLE]

where the first coefficient terms for the $x_{0}=+\frac{i}{\sqrt{6}}$ coefficient are determined to be

[TABLE]

[TABLE]

Therefore, the Laurent expansions (35) pass the third-step of the ARS algorithm and we conclude that the two-dimensional dynamical system (17), (18) possess the Painlevé property. In the following we avoid the presentation of the calculations and we give directly the main results which follow from our analysis.

IV.1.2 Power-law potential

We continue with the study of the three-dimensional dynamical system (17)-(19) where $f(s)=-\frac{s^{2}}{2n}$. The leading-order terms are found to be

[TABLE]

in which

[TABLE]

where $n\neq 0,-1,-3$.  The resonances are the zeros of the polynomial equation $\left(1+2r\right)\left(r+1\right)r=0$ which are

[TABLE]

At this case the generic solution is given by the following mixed Laurent expansions

[TABLE]

[TABLE]

[TABLE]

where $x_{0},~{}z_{0}$ and $s_{0}$ given in (41).

In order to perform the consistency test we select $n=1$, and for $x_{0}=\frac{i}{\sqrt{3}}~{}$we find that $x_{j}=0~{},~{}s_{j}=0$ for every value of $j$ and the third integration constant is parameter $\bar{s}_{1}$, recall that the other two integration constants are the position of the movable singularity $\tau_{0}$ and the coefficient $z_{0}$. For negative values of $n$ the consistency test fails. Hence the system posses the Painlevé property only for $n>0$.

We proceed with the second case of our analysis.

IV.2 Case B: $k\neq 0,~{}\Lambda=0$

In the presence of curvature the dimensionless field equations form the following three-dimensional dynamical system genlyhl

[TABLE]

which is defined on the phase space $\{(x,z,s)\in\mathbb{R}^{3}:x^{2}-z^{2}\leq 1\}.$

IV.2.1 Exponential potential

For the exponential potential and for the two-dimensional system (44), (45) the leading-order behaviour is found to be

[TABLE]

in which $z_{0}$ is an arbitrary constant. The resonances are determined to be again $r=-1$ and $r=0$ as in the spatially flat case. The solution is expressed in the right Laurent expansions

[TABLE]

where now the first coefficients are defined as

[TABLE]

We continue with the three-dimensional system defined by the power-law potential.

IV.2.2 Power-law potential

For the power-law potential the first step of the ARS algorithm for the dynamical system (44)-(46) provides the leading-order behaviour

[TABLE]

for $n\neq 0,-1$.

As far as the resonances are concerned they are derived

[TABLE]

where we conclude that the given dynamical system does not possess the Painlevé property.

IV.3 Case C: $k=0,~{}\Lambda\neq 0$

For the spatially flat background space and in the presence of the cosmological constant $\Lambda$, the field equations reduce to the following dimensionless system genlyhl

[TABLE]

defined on the phase space $\{(x,u,s)\in\mathbb{R}^{3}:x^{2}-u^{2}\leq 1\}.$

IV.3.1 Exponential potential

The two-dimensional dynamical system (52), (53) with $s=const$. passes the Painlevé test. The solution is given by the right Laurent expansions

[TABLE]

where the constants of integrations are $u_{0}$ and the position of the movable singularity $\tau_{0}$.

By replacing (55) in the dynamical system the first coefficients are calculated (for $x_{0}=\frac{i}{\sqrt{6}}$ )

[TABLE]

IV.3.2 Power-law potential

In the case of the power-law potential and for the three-dynamical system (52)-(54) with $n\neq-1,0$; from the first step of the ARS algorithm we determine the leading order behaviour

[TABLE]

The resonances are found to be the solutions of the polynomial equation $\left(r+1\right)\left(2r+1\right)\left(r-n\right)=0$, which gives

[TABLE]

In order to write the Laurent expansions and perform the consistency test the power $n$ should be defined. We have studied various values of $n$, positives and negatives, integers and fractional, and we found that in the possible cases the Laurent expansions fail at the consistency test.  Hence, we conclude that the dynamical system (52)-(54) does not pass the Painlevé test.

IV.4 Case D: $k\neq 0,~{}\Lambda\neq 0$

The dimensionless dynamical system which describe the field equations in the last case of our consideration consists by the following four first-order differential equations genlyhl

[TABLE]

defined on the phase space $\{(x,z,u,s)\in\mathbb{R}^{3}:x^{2}-(u-kz)^{2}\leq 1\}.$

IV.4.1 Exponential potential

For the three-dimensional dynamical system (59)-(61) where parameter $s$ is constant, we find the leading-order behaviour

[TABLE]

where $x_{0},~{}z_{0}$ are arbitrary constants and $u_{0}=\frac{4z_{0}^{2}-1-6x_{0}^{2}}{4z_{0}}$, with $4z_{0}^{2}-1-6x_{0}^{2}\neq 0$. Because there are already three arbitrary constants, including the position of the singularities someone will expect to find two resonances with value zero.

Indeed, by replacing

[TABLE]

in the dynamical system (59)-(61) we derive the resonances are the zeros of the polynomial equation $r^{2}\left(r+1\right)$. We conclude that the dynamical system (59)-(61) with $s$ constant   possess the Painlevé property. The algebraic solution is given by the following Laurent expansions

[TABLE]

IV.4.2 Power-law potential

For the power-law potential the leading-order behaviour are found to be

[TABLE]

with

[TABLE]

and $x_{0}$ arbitrary. The second step of the ARS algorithm provide the four resonances

[TABLE]

In a similar way with case C we have to define power-index $n$ in order to perform the consistency test. We have performed the consistency test for various rational numbers of $n$ and we found can conclude that the four-dimensional dynamical system (59)-(62) does not posses the Painlevé property.

V Conclusion

In this work we studied the integrability of the HL scalar field cosmology in a FLRW background spacetime for the exponential and power-law potentials. We performed our study for the dimensionless dynamical system under the $H$-normalization which is usually applied in the fixed point analysis of gravitational dynamical systems. We categorized our study in for cases of study according to the existence of the cosmological constant term $\Lambda$ and if the spatially curvature $k~{}$vanishes or not. More specifically the four cases of study are: Case A: $k=0,~{}\Lambda=0,~{}$Case B: $k\neq 0,~{}\Lambda=0$, Case C: $k=0,~{}\Lambda\neq 0$ and Case D: $k\neq 0,~{}\Lambda\neq 0$.

The main mathematical tool that we applied for the study of the integrability of the dimensionless field equations for the abovementioned cases of study is that of the singularity analysis. In particular we examined if the given gravitational dynamical system possess the Painlevé property which tell us that an explicit analytic integration can be performed where the solution is expressed in Laurent expansion.

As far as the power-law potential is concerned we found that only Case A provides an integrable system. In contrary to the exponential potential where the field equations always possesses the Painlevé property. Our results are summarized in the following proposition:* “The gravitational field equations for the HL scalar field cosmology in a FLRW background (7)-(9) expressed in the dimensionless variables in the $H-$normalization (16) pass always the Painlevé test when $V\left(\phi\right)=V_{0}e^{-\sigma\phi}$, and the equations can be explicitly integrated by Laurent expansions, where in all cases the resonances are $r=-1$ and $r=0$, where the rank of $r=0$ is greater of equal to one”. *At this point we want to mention that by applying the same analysis for the field equations beyond the detailed-balance condition  genlyhl we found that the field equations do not posses the Painlevé property.

The results of this analysis complement the dynamical study of HL scalar field equations genlyhl and the analysis in Leon:2009rc . Last but not least, this work contributes to the subject of integrability of gravitational field equations in cosmological studies.

Acknowledgements.

The authors want to thank the anonymous referee for valuable comments and suggestions which helped to improve the presentation of this work. This work was supported by Comisión Nacional de Investigación Científica y Tecnológica (CONICYT) through FONDECYT Iniciación 11180126. GL thanks to Departmento de Matemática and to Vicerrectoría de Investigación y Desarrollo Tecnológico at Universidad Católica del Norte for financial support.