∎

e1e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>e2e-mail:
<EMAIL_ADDRESS>e3e-mail<EMAIL_ADDRESS>
11institutetext:  Mathematics Division, Department of Basic Sciences and
Social Sciences, North Eastern Hill University, Shillong, Meghalaya 793022,
India 22institutetext: Inter University Centre for Astronomy and Astrophysics,
Pune 411 007, India 33institutetext: Department of Mathematics, North Eastern
Hill University, Shillong, Meghalaya 793022, India 44institutetext: Department
of Mathematics, St. Anthony’s College, Shillong, Meghalaya 793001, India

# Cosmological dynamics of the general non-canonical scalar field models

Jibitesh Duttae1,addr1,addr2  Wompherdeiki Khyllepe2,addr3,addr4  Hmar
Zonunmawiae3,addr3

(Received: date / Accepted: date)

###### Abstract

We extend the investigation of cosmological dynamics of the general non-
canonical scalar field models by dynamical system techniques for a broad class
of potentials and coupling functions. In other words, we do not restrict the
analysis to exponential or power-law potentials and coupling functions. This
type of investigation helps in understanding the general properties of a class
of cosmological models. In order to better understand the phase space of the
models, we investigate the various special cases and discuss the stability and
viability issues. Performing a detailed stability analysis, we show that it is
possible to describe the cosmic history of the universe at the background
level namely the early radiation dominated era, intermediate matter dominated
era and the late time dark energy domination. Moreover, we find that we can
identify a broad class of potentials and coupling functions for which it is
possible to get an appealing unified description of dark matter and dark
energy. The results obtained here, therefore, enlarge the previous analyses
wherein only a specific potential and coupling functions describes the
unification of dark sectors. Further, we also observe that a specific scenario
can also possibly explain the phenomenon of slow-roll inflationary exit.

††journal: Eur. Phys. J. C

##  1 Introduction

The source for the observed accelerated expansion of the Universe still
remains to be an obscured problem in modern cosmology. It is widely believed
that this phenomenon can be theoretically explained by an exotic quantity
commonly known as dark energy (DE) with large negative pressure Riess:1998cb ;
Perlmutter:1998np . In the literature, there are various candidates of DE and
the debate over the fittest candidate of DE, is yet to be resolved. The
details and status of various candidates of DE so far are summarised in an
excellent review by Brax Brax:2017idh  recently. Scalar field plays a
significant role in cosmology as they are sufficiently complicated to produce
the observed dynamics. They are used to model DE and characterise inflation
Liddle:2000cg . Motivated by high energy physics, scalar field models play an
important role to explain the nature of DE due to its simple dynamics
Copeland:2006wr ; Tsujikawa:2013fta . The simplest form of a scalar field is
the canonical field also known as quintessence field with potential. However,
the canonical scalar field cannot fully explain several complex cosmological
dynamics of the Universe. For instance, the quintessence model cannot explain
the crossing of the phantom divide line, bouncing solutions. This leads to a
more general description of a scalar field known as non-canonical scalar field
Fang:2006yh ; Mukhanov:2005bu . Another advantage of the non-canonical setting
over canonical is that it can resolve the coincidence problem without any
fine-tuning issues Lee:2014bwa .

The most general non-canonical form of a scalar field which involves higher
order derivatives of a scalar field falls under the well-known Horndeski
Lagrangian Horndeski:1974wa . Within this class, a simple form of a non-
canonical scalar field model is collectively known as $k$-essence, the first
term of the Horndeski Lagrangian. Basically, $k$-essence models are the
generalisation of the canonical scalar field and its Lagrangian is a function
of both the scalar field and its kinetic term ArmendarizPicon:2000ah . An
important feature of the $k$-essence is that, unlike the canonical scalar
field, its kinetic energy term can also source acceleration. In the context of
cosmology, the $k$-essence field was first used to describe early inflationary
epoch in ArmendarizPicon:1999rj  and was later found that it can also describe
DE by Chiba et al. Chiba:1999ka . In literature, there are several forms of
$k$-essence models depending on the type of its Lagrangian Mukhanov:2005bu ;
ArmendarizPicon:2005nz ; Unnikrishnan:2008ki .

In the present work, we will consider a general form of $k$-essence Lagrangian
introduced by Melchiorri et al. Melchiorri:2002ux . It has been found that
while the potential energy of this type of non-canonical scalar field models
behaves as DE, its kinetic energy plays the role of dark matter (DM)
Kamenshchik:2001cp ; Padmanabhan:2002cp ; Scherrer:2004au ; DiezTejedor:2006qh
; DeSantiago:2011qb ; Sahni:2015hbf ; Mishra:2018tki . This leads to a
unification of DE and DM, allowing structure formation and avoiding the strong
integrated Sachs-Wolfe effect in the CMB anisotropies which usually afflict
the unification between dark sectors Bertacca:2008uf . Interestingly, the
Chaplygin gas and its generalised form can also describe the DE-DM unification
Kamenshchik:2001cp ; bil ; ben . But the Chaplygin gas is associated with
gravitational clustering problems and therefore cannot describe the real
universe bar ; sen . Another advantage of the non-canonical models is that the
adiabatic sound speed is sensitive to the values of a non-canonical parameter
which have an effect on the gravitational clustering Sahni:2015hbf . Moreover,
cosmological observations such as SNIa, CMB and BAO favour the general
$k$-essence modelsMamon:2015qai .

Dynamical system techniques are useful tools to investigate the complete
asymptotic behaviour of a given cosmological model which allows us to bypass
the difficulty in solving non-linear cosmological equations. These tools also
provide a description of the global dynamics of the universe by analysing the
local asymptotic behaviour of critical points of the system and relate them
with the main cosmological epochs of the universe. For instance, while a late-
time DE domination would typically correspond to a stable point, the radiation
and matter dominated eras correspond to saddle points. Dynamical system
techniques had been extensively used in literature to analyse the evolution of
various cosmological models of DE as well as modified gravity based models
Copeland:1997et ; Carloni:2007eu ; Leon:2012mt ; Leon:2013qh ; Carloni:2015lsa
; Boehmer:2015kta ; Boehmer:2015sha ; Dutta:2017fjw ; Dutta:2017kch ;
Dutta:2017wfd ; Dutta:2016bbs ; Zonunmawia:2017ofc ; Hohmann:2017jao . For a
recent and comprehensive review on the applications of dynamical system tools
to various cosmological models see reference Bahamonde:2017ize . However, one
limitation of the dynamical system approach is the dependence on the choice of
variables which characterise the solution associated with critical points. It
is worth mentioning that the absence of some cosmological epoch of the
Universe does not always imply the inability of the theory to explain such
epoch, but it may be due to the inability of the associated dynamical system
to exhibit the presence of an epoch.

In the non-canonical scalar field settings, the dynamical system tools have
been applied extensively to various subclasses of $k$-essence Lagrangian
Bahamonde:2017ize . A dynamical system analysis for a $k$-essence Lagrangian
introduced in Melchiorri:2002ux  has been analysed in DeSantiago:2012nk  for a
power-law and exponential potential. Further, dynamical system analysis of an
interacting DE within the framework of the $k$-essence Lagrangian introduced
in Melchiorri:2002ux  with power-law potential has been performed in
Das:2014yoa . In DeSantiago:2012nk ; Das:2014yoa , the choice of potential is
restricted only to power-law and exponential case due to the choice of
dynamical variables considered. However, dynamical analysis of the general
non-canonical scalar field beyond specific potentials have never been studied
before. On the other hand, there are various studies of the canonical scalar
field beyond exponential potential syc ; fang ; mat ; lop . The important
reason for studying the dynamical properties of DE model beyond a specific
potential is that it helps in understanding general properties of a DE model
which in turn helps in predicting properties of a class of DE models.
Therefore, it is more economic and scientific to study for arbitrary
potentials. Furthermore, the generalisation to various scalar field potentials
is not only important from the mathematical point of view but also to connect
these phenomenological models with the low energy limit of high energy
physical theories. Therefore, it is interesting to extend the analysis to
encompass the various class of potentials.

With this motivation, here we extend the work of DeSantiago:2012nk  by
performing a dynamical analysis for a more general $k$-essence Lagrangian over
a wider class of scalar field potentials. Moreover, we shall consider a
different choice of dynamical variables from the one defined in
DeSantiago:2012nk  so as to encompass the desired class of potentials. The
main goal of this work is to determine the choice of model parameters which
leads to interesting background cosmological dynamics such as the well-known
unification of DE and DM within the non-canonical scenario. The present work
might also provide a preliminary test of the general non-canonical models
which are of interest for further investigation.

The organisation of the paper is as follows: In Sect. 2, we briefly review the
basic equations of the non-canonical scalar field models. In Sect. 3, we
perform a dynamical system analysis for the autonomous system of differential
equations obtained from the basic cosmological equations. Within this section,
we also discuss the cosmological dynamics by considering a different class of
scalar fields by taking various types of potentials and non-canonical coupling
functions in various subsections. Finally, the conclusion is given in Sect. 4.

Notation: In this work, we shall assume the $(+,-,-,-)$ signature convention
for the metric. We shall also adopt units where $8\pi G=c=1$.

##  2 The general non-canonical scalar field model and its basic cosmological
equations

The general action of a minimally coupled scalar field model is given by

$S=\int d^{4}x\sqrt{-g}\Big{(}\frac{R}{2}+\mathcal{L}(\phi,X)\Big{)}+S_{m},$
(1)

where $\mathcal{L}(\phi,X)$ is the Lagrangian density which is an arbitrary
function of a scalar field $\phi$ and its kinetic term $X$
($X=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi$), $R$ is the Ricci scalar
and $g$ is the determinant of the metric $g_{\mu\nu}$. The last term $S_{m}$
represents the action of the background matter field.

Varying the action (1) with respect to the metric yields the Einstein field
equations as

$G_{\mu\nu}=T^{\phi}_{\mu\nu}+T^{m}_{\mu\nu},$ (2)

where $G_{\mu\nu}$ is the Einstein tensor, $T^{\phi}_{\mu\nu}$ is the energy-
momentum for $\phi$ given by

$T^{\phi}_{\mu\nu}=\frac{\partial\mathcal{L}}{\partial
X}\partial_{\mu}\phi\partial_{\nu}\phi-g_{\mu\nu}\mathcal{L}.$ (3)

The energy-momentum tensor $T^{m}_{\mu\nu}$ for the matter component which is
modeled as a perfect fluid is given by

$T^{m}_{\mu\nu}=(\rho_{m}+p_{m})u_{\mu}u_{\nu}+p_{m}\,g_{\mu\nu},$ (4)

where $\rho_{m}$ and $p_{m}$ are the energy density and pressure of the matter
component and $u_{\mu}$ is the four velocity of the fluid.

In the present work, we shall consider a homogeneous, isotropic and spatially
flat Friedmann Robertson Walker (FRW) universe which is characterised by the
line element

$ds^{2}=dt^{2}-a^{2}(t)[dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta
d\phi^{2}],$ (5)

and the following non-canonical Lagrangian density of the scalar field
Melchiorri:2002ux

$\mathcal{L}(\phi,X)=f(\phi)F(X)-V(\phi),$ (6)

where $a(t)$ is the scale factor, $t$ is the cosmic time, $V(\phi)$ is a self-
interacting potential for the scalar field $\phi$, $f(\phi)$ is an arbitrary
function of $\phi$ with $f(\phi)\geq 0$ and $F(X)$ is an arbitrary coupling
function of $X$. For a spatially homogeneous FRW metric (5),
$X=\frac{1}{2}\dot{\phi}^{2}$. The positive semi-definiteness of function $f$
is required to explain various cosmological observations as discussed in
Melchiorri:2002ux . It can be seen that Eq. (6) reduces to quintessence field
when $f(\phi)=1$ and $F(X)=X$, it reduces to phantom field when $f(\phi)=1$
and $F(X)=-X$. It means that each quintessence or phantom scalar field is
equivalent to a particular $k$-essence model. This type of scalar field model
(6) constitutes an alternative model of DE yielding late time accelerated
solutions and it is well motivated from high-energy physics Tsujikawa:2004dp ;
Piazza:2004df .

There are several functional forms of $F(X)$ proposed so far in the literature
see Mukhanov:2005bu ; ArmendarizPicon:2005nz ; Unnikrishnan:2008ki . However,
in order to obtain a concrete description of the cosmological dynamics, we
shall consider a case where $F(X)=X^{\alpha}$ i.e. the non-canonical scalar
field models whose Lagrangian density is given by Mukhanov:2005bu ;
Unnikrishnan:2008ki

$\mathcal{L}(\phi,X)=f(\phi)X^{\alpha}-V(\phi),$ (7)

where $\alpha$ is a dimensionless parameter. This class of $F(X)$ is a simple
generalisation of the canonical scalar field (for $\alpha=1$). It leads to
some rich phenomenology that is not present in the canonical case for e.g. the
existence of non-singular bouncing solutions DeSantiago:2012nk . This choice
of $F(X)$ also allows a non-canonical field to cluster and depending on values
of $\alpha$, it behaves either like warm or cold dark matter at small scales
Sahni:2015hbf . It has been found that the slow-roll conditions can be easily
realized for the non-canonical scalar field models ($\alpha\neq 1$) compared
to the corresponding canonical case ($\alpha=1$) Unnikrishnan:2012zu .
Further, non-canonical models reduce the tensor-to-scalar ratio than their
canonical counterparts, leading to a better agreement with CMB observations
Unnikrishnan:2012zu . These interesting features of non-canonical scalar
fields motivate us to further investigate the cosmological dynamics of such
fields in a more general context.

In terms of Lagrangian density $\mathcal{L}$, the expressions of the energy
density $\rho_{\phi}$ and pressure $p_{\phi}$ associated with the scalar field
are given by

$\displaystyle\rho_{\phi}$ $\displaystyle=$ $\displaystyle
2X\Big{(}\frac{\partial\mathcal{L}(\phi,X)}{\partial
X}\Big{)}-\mathcal{L}(\phi,X)\,,$ (8) $\displaystyle p_{\phi}$
$\displaystyle=$ $\displaystyle\mathcal{L}(\phi,X).$ (9)

Employing the line element (5), the Einstein field equations can be written as

$\displaystyle 3H^{2}$ $\displaystyle=$
$\displaystyle(2\alpha-1)f(\phi)\,X^{\alpha}+V(\phi)+\rho_{m},$ (10)
$\displaystyle\dot{H}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\Big{[}2\alpha
f(\phi)\,X^{\alpha}+\rho_{m}(1+w)\Big{]},$ (11)

where the over-dot denotes differentiation with respect to cosmic time $t$,
$H=\frac{\dot{a}}{a}$ is the Hubble parameter and $w$ is the equation of state
(EoS) of matter $(-1\leq w\leq 1)$ defined as $p_{m}=w\rho_{m}$.

Substituting for $\mathcal{L}$ from (7) into (8) and (9), the energy density
and pressure of the scalar field are respectively given by

$\displaystyle\rho_{\phi}$ $\displaystyle=$ $\displaystyle\rho_{X}+V,$ (12)
$\displaystyle p_{\phi}$ $\displaystyle=$ $\displaystyle p_{X}-V,$ (13)

where $\rho_{X}=(2\alpha-1)\ f(\phi)\,X^{\alpha}$ and
$p_{X}=f(\phi)\,X^{\alpha}$. Finally, we assume that the scalar field and
barotropic fluid energy momentum tensors are conserved separately i.e. there
is no exchange of energy between the scalar field and barotropic fluid. In
that case by employing line element (5), the energy conservation equations are
given by

$\displaystyle\dot{\rho}_{\phi}+3H(\rho_{\phi}+p_{\phi})$ $\displaystyle=$
$\displaystyle 0,$ (14) $\displaystyle\dot{\rho}_{m}+3H\rho_{m}(1+w)$
$\displaystyle=$ $\displaystyle 0.$ (15)

Using Eq. (12) in Eq. (14), the evolution equation of the scalar field can be
expressed as

$\displaystyle\ddot{\phi}$ $\displaystyle=$
$\displaystyle-\Big{[}\frac{1}{2\alpha}\frac{df}{d\phi}\frac{\dot{\phi}^{2}}{f(\phi)}+\frac{1}{\alpha(2\alpha-1)}\frac{dV}{d\phi}\Big{(}\frac{2}{\dot{\phi}^{2}}\Big{)}^{\alpha-1}\frac{1}{f(\phi)}$
(16) $\displaystyle+\frac{3H\dot{\phi}}{2\alpha-1}\Big{]}.$

As expected, the above equation reduces to the case of a canonical field,
$\ddot{\phi}+3H\dot{\phi}+\frac{dV}{d\phi}=0$, when $\alpha=1$ and
$f(\phi)=1$. It is worth noting that for constant potential, the kinetic part
$\rho_{X}$ and the potential part $V(\phi)$ behave as two non-interacting
fluids as they separately satisfy the conservation equation (14).

For theoretical consistency of the Horndeski’s most general scalar-tensor
theories, the conditions $Q_{s}>0$, $C_{s}^{2}\geq 0$, $Q_{t}>0$,
$C_{t}^{2}\geq 0$ are required for the avoidance of ghosts and Laplacian
instabilities associated with a scalar and tensor perturbations
DeFelice:2011bh . In this model, we have verified that the conditions
$Q_{t}>0$, $C_{t}^{2}\geq 0$ holds for any choices of model parameters.
However, the quantities $Q_{s}$ and $C_{s}^{2}$ are given by

$\displaystyle Q_{s}$ $\displaystyle=$ $\displaystyle
2^{\alpha}\alpha(2\alpha-1)H^{-2}X^{\alpha}f\,$ (17) $\displaystyle C_{s}^{2}$
$\displaystyle=$
$\displaystyle\frac{1}{2\alpha-1}+\frac{3(1+w)\rho_{m}}{2^{\alpha}\alpha(2\alpha-1)X^{\alpha}f}$
(18)

Therefore in the absence of barotropic matter component, the Eq. (18) reduces
to the case obtained in Mishra:2018tki  given by

$\displaystyle C_{s}^{2}=\frac{1}{2\alpha-1}\,.$ (19)

It is important to note here that we have $Q_{s}>0$ and $C_{s}^{2}\geq 0$ for
$\alpha>\frac{1}{2}$. However, for $\alpha=\frac{1}{2}$ or at a point where
$X=0$ or $f=0$, the speed of sound $C_{s}^{2}$ diverges. This result is
expected as discussed in Ref. Chimento:2004ha  for a typical form of
$k$-essence Lagrangian. The divergence of speed of sound is a well known
result in $k$-essence model and therefore violates causality Bonvin:2006vc .
Further the existence of superluminal solutions i.e. $C_{s}^{2}>1$ is a
familiar characteristic of Horndeski model in the presence of external normal
matter vik  and does not necessarily imply pathologies of a cosmological
solution. Hence, to avoid a possibility of ghost instabilities
($C_{s}^{2}<0$), in what follows, we shall investigate only for the case of
$\alpha>\frac{1}{2}$. In the next section, we shall convert these cosmological
equations into an autonomous system of equations and perform a dynamical
system analysis for various types of $f(\phi)$ and $V(\phi)$.

##  3 Dynamical system analysis

In order to write the above set of cosmological equations as an autonomous
system of ordinary differential equations, we introduce the following set of
normalised phase space variables

$\displaystyle x=\frac{\dot{\phi}}{\sqrt{6}H},\qquad
y=\frac{\sqrt{V}}{\sqrt{3}H},\qquad z=H^{2(\alpha-1)}f,$ $\displaystyle
s=-\frac{1}{V}\frac{dV}{d\phi},\qquad
v=-\frac{1}{f}\frac{df}{d\phi}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(20)

Using these variables (3), the above cosmological equations can be converted
to the following autonomous system as

$\displaystyle x^{\prime}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\Big{[}\frac{3x^{2}v}{\alpha}+\frac{3y^{2}s}{zx^{2(\alpha-1)}\alpha(2\alpha-1)3^{(\alpha-1)}}-\frac{3\sqrt{6}x}{(2\alpha-1)}\Big{]}$
(21) $\displaystyle+\frac{x}{2}\Big{[}2\alpha
3^{\alpha}x^{2\alpha}z+3(1+w)\\{1-3^{(\alpha-1)}x^{2\alpha}z(2\alpha-1)$
$\displaystyle-y^{2}\\}\Big{]},~{}$ $\displaystyle y^{\prime}$
$\displaystyle=$ $\displaystyle-\frac{xys}{2}+\frac{y}{2}\Big{[}2\alpha
3^{\alpha}x^{2\alpha}z+3(1+w)\\{1-3^{(\alpha-1)}x^{2\alpha}z(2\alpha-1)$ (22)
$\displaystyle-y^{2}\\}\Big{]},~{}$ $\displaystyle z^{\prime}$
$\displaystyle=$
$\displaystyle-\sqrt{6}xzv-\frac{2z(\alpha-1)}{2}\Big{[}2\alpha
3^{\alpha}\,x^{2\alpha}z$ (23)
$\displaystyle+3(1+w)\\{1-3^{(\alpha-1)}x^{2\alpha}z(2\alpha-1)-y^{2}\\}\Big{]},~{}$
$\displaystyle s^{\prime}$ $\displaystyle=$ $\displaystyle-\sqrt{6}xg(s),~{}$
(24) $\displaystyle v^{\prime}$ $\displaystyle=$
$\displaystyle-\sqrt{6}xh(v),~{}$ (25)

where $g(s)=s^{2}(\Gamma_{1}-1)$ and $h(v)=v^{2}(\Gamma_{2}-1)$ with
$\Gamma_{1}=V\frac{d^{2}V}{d\phi^{2}}\Big{/}\Big{(}\frac{dV}{d\phi}\Big{)}^{2}$,
$\Gamma_{2}=f\frac{d^{2}f}{d\phi^{2}}\Big{/}\Big{(}\frac{df}{d\phi}\Big{)}^{2}$
and the prime denotes differentiation with respect to $N$= $\ln a$. Note that
if $s=s(\phi)$ is invertible, then we can also express $\phi$ as function of
$s$. Since the parameter $\Gamma_{1}$ is a function of $\phi$ only, hence
$\Gamma_{1}$ can also be expressed as $\Gamma_{1}=\Gamma_{1}(s)$. Similarly,
we can write $\Gamma_{2}$ as function of $v$. This does not include any
arbitrary $V$ or $f$ but it includes a specific class of potential $V$ and
coupling function $f$. Thus, for such choices of $V$ and $f$, the system
(21)-(25) constitutes an autonomous system of equations. Here, we observed
that in the context of cosmological evolution, there are four basic variables
$H$, $\phi$, $\dot{\phi}$ and $\rho_{m}$, which are related by a Friedmann
constraint (10). Therefore, there are only three independent variables
altogether. However, the introduction of two extra variables allows us to
track the effect of the coupling and potential functions on the overall
dynamics. In the literature, these types of variables are often introduced to
study the class of unknown functions fang ; leyva . It is worth noting that
the variables (3) are different from those defined in DeSantiago:2012nk  and
this choice of variables helps in obtaining cosmologically viable critical
points which however cannot be captured in DeSantiago:2012nk . More
importantly, our choice of variables leads to a viable cosmological sequence:
radiation era $\rightarrow$ matter era $\rightarrow$ DE era and also can
possibly explain a unified description of DM and DE. Moreover, the above
system (21)-(25) reduces to the corresponding system of a canonical case for
$\alpha=1$ and $f=1$ fang .

It can be seen from the above system that $y=0$ is an invariant sub-manifold
111The simplest approach to determine the existence of an invariant sub-
manifold of $\mathbb{R}^{n}$ (here $n=5$) is to look at the form of the right-
hand side (rhs) of a dynamical equation for a given variable (for e.g. $y$).
If the dynamical equation is of the form $y^{\prime}=(y-\xi)g$, where
$g:\mathbb{R}^{n}\rightarrow\mathbb{R}$ is a continuous function, then $y=\xi$
is an invariant sub-manifold with respect to a flow corresponding to an
autonomous vector field given by (21)-(25) tava .. Similarly, $z=0$ is an
invariant sub-manifold, however, it is singular as $z$ appears as the
denominator in Eq. (21). For constant potential $V$, $x=0$ also represents an
invariant sub-manifold of the system. Depending on the choice of $V$ and $f$,
$s=0$ and $v=0$ may be invariant sub-manifolds. Invariant sub-manifolds play
an important role in the characterisation of the phase space as these
manifolds separate the phase space into independent sections. The presence of
invariant sub-manifolds tells us that there is no global attractor except
those stable points which lie on the intersection of all the invariant sub-
manifolds Bahamonde:2017ize . The existence of accelerated global attractors
guarantees late time evolution of the universe, irrespective of the choice of
model parameters. Throughout the work, $s_{*}$ and $v_{*}$ denote the zeroes
of $g(s)=0$ and $h(v)=0$ respectively. Further, $dg(s)$, $dh(v)$ denote the
derivative of $g$ and $h$ with respect to $s$ and $v$ respectively. It is
worth noting from the definition (3), $y>0$ corresponds to $H>0$ i.e.
expanding universe and $y<0$ corresponds to $H<0$ i.e. contracting universe.
However, as the system (21)-(25) is invariant under the transformation
$y\rightarrow-y$, we shall only analyse the case of expanding universe which
is also cosmologically viable. Further, as we consider expanding universe,
therefore we have $z\geq 0$.

In terms of variables (3), the relative energy density due to the scalar field
$\Omega_{\phi}$, the relative energy density due to the barotropic matter
$\Omega_{m}$ and the scalar field EoS $w_{\phi}$ can be expressed as

$\displaystyle\Omega_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\rho_{\phi}}{3H^{2}}=(2\alpha-1)3^{(\alpha-1)}x^{2\alpha}z+y^{2},$
(26) $\displaystyle\Omega_{m}$ $\displaystyle=$
$\displaystyle\frac{\rho_{m}}{3H^{2}}=1-(2\alpha-1)3^{(\alpha-1)}x^{2\alpha}z-y^{2},$
(27) $\displaystyle w_{\phi}$ $\displaystyle=$
$\displaystyle\frac{p_{\phi}}{\rho_{\phi}}=\frac{3^{(\alpha-1)}zx^{2\alpha}-y^{2}}{(2\alpha-1)3^{(\alpha-1)}zx^{2\alpha}+y^{2}}.$
(28)

Note that the relative energy density parameters are related by the Friedmann
constraint (10) as

$\Omega_{m}+\Omega_{\phi}=1.$ (29)

Here we see that for $\alpha>0$ if $x\approx 0$ and $y\approx 1$, then
$w_{\phi}\approx-1$. Therefore, when the potential energy of the scalar field
dominates, the scalar field behaves as cosmological constant (i.e. DE).
However, when $y\,\approx\,0$ and
$(2\alpha-1)3^{(\alpha-1)}x^{2\alpha}z\approx 1$ i.e. the contribution from
the potential is negligible and maximum contribution comes from the kinetic
part, we have $w_{\phi}\approx\frac{1}{2\alpha-1}$. This implies that purely
kinetic non-canonical models cannot describe the cosmic acceleration for
$\alpha>\frac{1}{2}$ as reviewed in sari . Further, we see that the scalar
field scales as radiation for $\alpha=2$, stiff matter for $\alpha=1$ and as
dust for very large $\alpha$. This does not mean that the matter component
comes from the DE, but both of them are sourced by a single scalar field.
Additionally, we can define the effective energy density and pressure as

$\displaystyle\rho_{\rm eff}$ $\displaystyle=$
$\displaystyle\rho_{\phi}+\rho_{m},$ (30) $\displaystyle p_{\rm eff}$
$\displaystyle=$ $\displaystyle p_{\phi}+p_{m},$ (31)

where we can obtain the effective EoS as

$\displaystyle w_{\rm eff}$ $\displaystyle=$
$\displaystyle-1+\frac{2}{3}\,\alpha\,{3}^{\alpha}{x}^{2\,\alpha}z+\left(w+1\right)[1-{3}^{\alpha-1}{x}^{2\,\alpha}z\left(2\,\alpha-1\right)$
(32) $\displaystyle-{y}^{2}],$

which is usually related to the deceleration parameter $q$ through

$\displaystyle q=-1-\frac{\dot{H}}{H^{2}}=\frac{1+3w_{\rm eff}}{2}.$ (33)

Hence for accelerated universe, one must obtain $q<0$ i.e. $w_{\rm
eff}<-\frac{1}{3}$. From Eq. (29), the physical requirement $\rho_{m}\geq 0$
yields the constraint

$\Omega_{\phi}\leq 1.$ (34)

Hence the five dimensional phase space of the system (21)-(25) in terms of
variables (3) is given by

$\displaystyle\Psi$ $\displaystyle=$
$\displaystyle\left\\{(x,y,z)\in\mathbb{R}^{3}:(2\alpha-1)3^{(\alpha-1)}x^{2\alpha}\,z+y^{2}\leq
1,y,z\geq 0\right\\}$ (35)
$\displaystyle\times\left\\{(s,v)\in\mathbb{R}^{2}\right\\}.$

To determine the evolution of relevant energy densities numerically, we
estimate the evolution of the Universe in such a way that it is consistent
with the present observational data ($\Omega_{m}\simeq 0.3$, $w_{\rm
eff}\simeq-0.7$) ade:2015xua . This corresponds to points in the phase space
where $x_{0}=0$, $y_{0}=\pm 0.83$, $z_{0}$ is any value.

In order to extract the dynamics of the above system, first, we need to find
the critical points of the system by equating the right-hand side of the
system (21)-(25) to zero. This is followed by perturbing the system near the
critical points from which the stability and type of critical points can be
determined from the eigenvalues of the corresponding perturbed matrix. In the
present work, since we have arbitrary functions $f$ and $V$, in what follows,
we shall analyse various cases separately for different choices of $f$ and
$V$.

###  3.1 $f(\phi)$=Constant, $V(\phi)=$Constant

We start our study by considering the simplest choice of $f$ and $V$ where
both are constant. In this case the variables $s=0$ and $v=0$, hence, the
system (21)-(25) reduces to a three dimensional system in $x$, $y$ and $z$. It
is worth noting that for this choice of $V$ and $f$, the functions
$\Gamma_{1}$, $\Gamma_{2}$ cannot be expressed as explicit functions of $s$
and $v$ respectively but the three dimensional system is still autonomous as
the equations involving $\Gamma_{1}$, $\Gamma_{2}$ are not involved. We note
here that the system has three invariant sub-manifolds $x=0,y=0$ and $z=0$.
Under this case, we have only one critical point $A_{1}=(0,0,0)$ and one set
of critical points $A_{2}=(0,1,z)$.

  * •

The critical point $A_{1}$ corresponds to a matter dominated universe
$[\Omega_{m}=1$, $\Omega_{\phi}=0$, $w_{\rm eff}=w$,
$w_{\phi}=\frac{1}{2\alpha-1}]$ with eigenvalues of the Jacobian matrix
$\Big{\\{}\lambda_{1}=\frac{3(1+w)}{2},\lambda_{2}=-3(\alpha-1)(1+w),\lambda_{3}=\frac{3(2\alpha
w+2\alpha-w-3)}{2(2\alpha-1)}\Big{\\}}$ of the perturbed matrix at $A_{1}$.
Therefore, this point behaves as unstable node for
$\frac{1}{2}+\frac{1}{w+1}<\alpha<1$, otherwise it behaves as a saddle. Thus,
even though point $A_{1}$ lies on the intersection of all invariant sub-
manifolds, it cannot be a global attractor.

  * •

The set $A_{2}$ corresponds to an accelerated scalar field dominated solution
$[\Omega_{m}=0$, $\Omega_{\phi}=1$, $w_{\rm eff}=-1,w_{\phi}=-1]$. It is a
non-hyperbolic set with eigenvalues
$\Big{\\{}\lambda_{1}=0,\lambda_{2}=-\frac{3}{2\alpha-1},\lambda_{3}=-3(1+w)\Big{\\}}$.
However, as the dimension of the set is same as the number of vanishing
eigenvalues, it is therefore a normally hyperbolic set coley . Therefore,
points on this set behave as stable points for $\alpha>\frac{1}{2}$ but this
set cannot be global attractor as points on it corresponds to $y\neq 0$.

Thus, in this case, the overall dynamics is very simple where the universe
after evolving from matter domination (point $A_{1}$), it evolves towards an
accelerated DE dominated solution (set $A_{2}$). Hence, at the background
level, this model can successfully explain the structure formation as well as
the late time evolution of the universe for $\alpha>\frac{1}{2}$. This is more
visible from Fig. 1, where we plot the time evolution of different energy
density parameters along with the effective EoS against the redshift
$z=\frac{a_{0}}{a}-1$ (with $a_{0}=1$ the present scale factor) with
$\alpha=2$. Here, while $z=0$ (which corresponds to $a=a_{0}$) represents the
present time of the universe, $z\approx-1$ (which corresponds to
$a\rightarrow\infty$) represents an asymptotic future of the universe.

Figure 1: The time evolution of the relative matter energy density
$\Omega_{m}$, the relative scalar field energy density $\Omega_{\phi}$ and the
effective EoS $w_{\rm eff}$ for the case when $f=$ constant, $V=$constant.
Here, $w=0$, $\alpha=2$.

Figure 2: Projection of the time evolution of phase space trajectories along
the (a) $x$-axis, (b) $y$-axis, (c) $z$-axis and (d) $s$-axis which determine
the stability of set $B_{2}$. Here we take $w=0$ and $\alpha=0.6$,
$V=\frac{V_{0}}{\phi^{n}}$ with $n=4$ and $f=$ constant.

###  3.2 $f(\phi)=$Constant

In this subsection, we investigate the case where only the function $f$ is
constant. In this case variable $v$ vanishes and thus the system (21)-(25)
reduces to a four-dimensional system in $x,y,z,s$. The system contains one
invariant sub-manifold $y=0$ and one singular invariant sub-manifold $z=0$.
Depending on the choice of potential, the system may possess another invariant
sub-manifold $s=0$. For $\alpha=1$, we have checked that the behaviour of the
system coincides with that of the canonical scalar field Copeland:1997et ,
therefore, we will not further analyse this case. However for $\alpha\neq 1$,
there are only two set of critical points: $B_{1}=(0,0,z,s)$ and
$B_{2}=(0,1,z,s)$. Both sets are independent of the choice of the potential
for their existence. From the second term of Eqs. (21) and (18), it can be
seen that both sets demand $\alpha<1$ for their existence.

  * •

The set $B_{1}$ corresponds to an unaccelerated DM dominated solution
$[\Omega_{m}=1$, $\Omega_{\phi}=0$, $w_{\rm
eff}=w,w_{\phi}=\frac{1}{2\alpha-1}]$. It is a normally hyperbolic set of
points with eigenvalues
$\Big{\\{}\lambda_{1}=\frac{3(1+w)}{2},\lambda_{2}=3(1-\alpha)(1+w),\lambda_{3}=0,\lambda_{4}=\frac{3(2\alpha
w+2\alpha-w-3)}{2(2\alpha-1)}\Big{\\}}$. Therefore, points on this set are
behaving as unstable node for $\frac{1}{2}+\frac{1}{w+1}<\alpha<1$, otherwise
they are behaving as a saddle.

  * •

The set $B_{2}$ corresponds to an accelerated scalar field dominated solution
$[\Omega_{m}=0$, $\Omega_{\phi}=1$, $w_{\rm eff}=-1,w_{\phi}=-1]$. It is also
non-hyperbolic in nature with eigenvalues
$\\{\lambda_{1}=0,\lambda_{2}=0,\lambda_{3}=0,\lambda_{4}=-3(1+w)\\}$ but not
normally hyperbolic. Usually, the centre manifold theory is employed to
determine the stability of points on this set wiggins . However, in this work,
we will simply refer to a numerical computational method of perturbation plot
to determine the stability near points of this set. This numerical method has
been found to be quite successful in literature for determining the stability
behaviour of critical points of the system Dutta:2017fjw ; Dutta:2017kch ;
Dutta:2017wfd ; Dutta:2016bbs ; Zonunmawia:2017ofc . For this, we perturb the
system near points of this set and plot the projections of phase trajectories
along the $x$, $y$, $z$ and $s$ axes as given in Fig. 2. We note here that for
these plots, we choose the values of initial conditions that are very close to
the points of the set. It is evident from Figs. 2 and 2, that the phase space
trajectories approach $x=0$ and $y=1$ respectively as $N\rightarrow\infty$.
Further, it can be seen in Fig. 2, 2, that trajectories starting from any
values of $z$ and $s$ remains almost constant. From the behaviour of the
perturbed system near the set $B_{2}$, we can conclude that the set $B_{2}$
behaves as a stable set but points on this set cannot be global attractors as
they do not lie on the intersection of invariant sub-manifolds.

Hence, for a restricted choice of $\alpha$ ($\frac{1}{2}<\alpha<1$), this
class of models describes the simple evolution of the universe where after the
universe passes through matter domination (set $B_{1}$), it eventually evolves
towards an accelerated DE dominated solution (set $B_{2}$).

###  3.3 $V(\phi)=$Constant

In this subsection, we investigate the case where only the potential is
constant. For this choice of potential, the variable $s$ vanishes, so the
system (21)-(25) reduces to a four dimensional system in $x,y,z$ and $v$. The
system thus contains three invariant sub-manifolds $x=0,y=0$ and $z=0$,
however, depending on the choice of $h(v)$, the system may contains another
invariant sub-manifold $v=0$. Under this case, we have two sets of critical
points $C_{1}=(0,0,0,v)$, $C_{2}=(0,1,z,v)$ and three critical points
$C_{3}=\Big{(}-\frac{\alpha\sqrt{6}(2\alpha
w+2\alpha-w-3)}{2v_{*}(2\alpha-1)},0,0,v_{*}\Big{)}$,
$C_{4}=\Big{(}\frac{\sqrt{6}\alpha}{v_{*}(2\alpha-1)},1,0,v_{*}\Big{)}$ and
$C_{5}=\Bigg{(}-\frac{\alpha\sqrt{6}(\alpha-1)}{v_{*}(2\alpha-1)},0,$
$\frac{3^{(1-\alpha)}\Big{[}-\frac{\alpha\sqrt{6}(\alpha-1)}{v_{*}(2\alpha-1)}\Big{]}^{-2\alpha}}{(2\alpha-1)},v_{*}\Bigg{)}$.
Note that from the second term of the equation (21), the existence of point
$C_{4}$ specifically demands $s=0$ i.e. the constant potential only. Further,
points $C_{3}$, $C_{4}$, $C_{5}$ do not exists when $v_{*}=0$.

  * •

The set of critical points $C_{1}$ corresponds to a matter dominated universe
$[\Omega_{m}=1$, $\Omega_{\phi}=0$, $w_{\rm
eff}=w,w_{\phi}=\frac{1}{2\alpha-1}]$ with eigenvalues
$\Big{\\{}\lambda_{1}=\frac{3(1+w)}{2},\lambda_{2}=-3(\alpha-1)(1+w),\lambda_{3}=\frac{3(2\alpha
w+2\alpha-w-3)}{2(2\alpha-1)},\lambda_{4}=0\Big{\\}}$. Therefore, points on
this set behaving as unstable node for $\frac{1}{2}+\frac{1}{w+1}<\alpha<1$,
otherwise they are behaving as saddle points. This again implies that points
on this set cannot be global attractor.

  * •

The set $C_{2}$ corresponds to an accelerated scalar field potential dominated
solution $[\Omega_{m}=0$, $\Omega_{\phi}=1$, $w_{\rm eff}=-1,w_{\phi}=-1]$. It
is a normally hyperbolic set with eigenvalues
$\Big{\\{}\lambda_{1}=0,\lambda_{2}=0,\lambda_{3}=-3(1+w),\lambda_{4}=-\frac{3}{2\alpha-1}\Big{\\}}$.
Therefore, it is always stable but it cannot be global attractor as $y\neq 0$.

  * •

The point $C_{3}$ corresponds to a matter dominated solution with
$[\Omega_{m}=1$, $\Omega_{\phi}=0$, $w_{\rm
eff}=w,w_{\phi}=\frac{1}{2\alpha-1}]$. The eigenvalues of the corresponding
perturbed matrix are
$\Big{\\{}\lambda_{1}=\frac{3(1+w)}{2},~{}~{}\lambda_{2}=\frac{3(2\alpha
w-w-1)}{2\alpha-1},~{}\lambda_{3}=-\frac{3(2\alpha
w+2\alpha-w-3)}{2(2\alpha-1)},~{}\\\ \lambda_{4}=\frac{3\alpha(2\alpha
w+2\alpha-w-3)dh(v_{*})}{v_{*}(2\alpha-1)}\Big{\\}}$. Since $\lambda_{1}>0$
but $\lambda_{2}$, $\lambda_{3}$ are not both positive, therefore this point
is a saddle for any choice of function $f$.

  * •

The point $C_{4}$ corresponds to an accelerated solution dominated by the
potential term of the scalar field $[\Omega_{m}=0,\Omega_{\phi}=1,w_{\rm
eff}=-1,w_{\phi}=-1]$ with eigenvalues
$\Big{\\{}\lambda_{1}=-3(1+w),\lambda_{2}=\frac{3}{2\alpha-1},\lambda_{3}=-\frac{6\alpha}{2\alpha-1},\lambda_{4}=-\frac{6\alpha
dh(v_{*})}{v_{*}(2\alpha-1)}\Big{\\}}$. It is therefore saddle for $\alpha>0$
and also for $\alpha<0$ with $v_{*}dh(v_{*})<0$, however, it is stable for
$\alpha<0$ and $v_{*}dh(v_{*})>0$. This point corresponds to a slow roll
behaviour where potential part of the scalar field dominates and its
derivative vanishes (as its existence demands $s=0$). Its saddle property is
interesting as it provides an explanation of how the Universe could possibly
evolve towards a slow roll regime and eventually escape. It is also worth to
note that this type of critical point is also obtained in DeSantiago:2012nk .

  * •

The critical point $C_{5}$ corresponds to a solution dominated by the kinetic
term of a scalar field $[\Omega_{m}=0,\Omega_{\phi}=1,w_{\rm
eff}=\frac{1}{2\alpha-1},w_{\phi}=\frac{1}{2\alpha-1}]$. The eigenvalues of
the corresponding perturbed matrix are
$\Big{\\{}\lambda_{1}=\frac{3\alpha}{2\alpha-1},\lambda_{2}=\frac{3(1-\alpha)}{2\alpha-1},$
$\lambda_{3}=\frac{3\left(1-w(2\alpha-1)\right)}{2\alpha-1},\lambda_{4}=\frac{6\alpha(\alpha-1)}{2\alpha-1}\frac{dh(v_{*})}{v_{*}}\Big{\\}}$.
This point is an unstable node if $\frac{1}{2}<\alpha<1$ and
$v_{*}\,dh(v_{*})<0$, otherwise it is a saddle. It is worth mentioning that
the existence of this point is cosmologically very interesting as it describes
the early radiation domination for $\alpha=2$, the stiff matter solution for
$\alpha=1$ and the matter domination for very large values of $\alpha$.

Hence, from the above analysis it can be seen that for those function $f$ in
which $v_{*}\neq 0$, the system evolves from the point $C_{5}$ which behaves
as a radiation (i.e. $w_{\rm eff}=\frac{1}{3}$ for $\alpha=2$) or DM ($w_{\rm
eff}\approx 0$ for large $\alpha$) or stiff matter ($w_{\rm eff}=1$ for
$\alpha=1$) towards matter domination describe by either a set $C_{1}$ or
point $C_{3}$ with $w_{\rm eff}=0$ and then eventually settle in a DE
dominated epoch described by a set $C_{2}$ with $w_{\rm eff}=-1$. Therefore, a
viable sequence of cosmological eras (radiation $\rightarrow$ matter
$\rightarrow$ dark energy) is achieved by choosing the trajectory:
$C_{5}\,\rightarrow C_{1}\,\rightarrow C_{2}$ or $C_{5}\,\rightarrow
C_{3}\,\rightarrow C_{2}$ for $\alpha=2$. Further, the early times radiation
(for $\alpha=2$) or the DM behaviour (for large $\alpha$) is specifically due
to the non-canonical scalar field. This explains the well-known unified DE-DM
behaviour in non-canonical setting where the kinetic term of the scalar field
plays the role of DM and the potential plays the role of DE Kamenshchik:2001cp
; Padmanabhan:2002cp ; Scherrer:2004au ; DiezTejedor:2006qh ;
DeSantiago:2011qb ; Sahni:2015hbf ; Mishra:2018tki . Similar unified
description is obtained in the context of generalized Galileon theories but
without any instabilities at the perturbative level at all times
Koutsoumbas:2017fxp . In order to clearly see the background cosmological
sequence in detail, we plot the time evolution of different relative energy
densities along with the overall effective EoS by considering a specific
choice of the function $f$ viz. $f=f_{0}e^{\beta\phi}$ with $\beta=5$ and
$\alpha=1,2$ (see Fig. 3). It is worth mentioning that this choice of $f$ is
specifically taken so that the function $\Gamma_{2}$ can be written as
function of $v$. Moreover, it has been found that this choice of $f$ fit with
data from the combination of various datasets such as SNIa, BAO and CMB
Mamon:2015qai . From Fig. 3, it can be seen that the background evolution of
the Universe resembles that of the $\Lambda$CDM model, with an early time
contribution arising from the non-canonical scalar field term. However, the
presence of a non-canonical parameter $\alpha$ in the speed of sound produces
a possible deviation of this model from $\Lambda$CDM at the perturbation level
Sahni:2015hbf .

Figure 3: The time evolution of the relative matter energy density
$\Omega_{m}$, the relative scalar field energy density $\Omega_{\phi}$ and the
effective EoS $w_{\rm eff}$. Here we take $f(\phi)=f_{0}e^{\beta\phi}$,
$\beta=5$ and $V=$constant, $w=0$ with $\alpha=2$ in (a) and $\alpha=1$ in
(b).

Figure 4: Projection of the time evolution of phase space trajectories along
the (a) $x$-axis, (b) $y$-axis, (c) $z$-axis, (d) $s$-axis and (e) $v$-axis
which determine the stability of set $D_{2}$. Here, we take $w=0$,
$\alpha=0.8$, $V(\phi)=V_{0}(1-e^{-\lambda\phi})^{2}$ with $\lambda=4$ and
$f(\phi)=f_{0}\phi^{-n}$ with $n=2$.

Figure 5: (a) Regions of stability of point $D_{3}$: The red shaded region
represents the region of stability for $dg(s_{*})>0$ and $dh(v_{*})>0$, the
blue shaded region of stability for the case when $dg(s_{*})<0$ and
$dh(v_{*})<0$. (b) Regions of instability of point $D_{4}$: The green shaded
region represents the region of unstable node for $dg(s_{*})>0$ and
$dh(v_{*})>0$, the blue shaded region represents the region of unstable node
for the case when $dg(s_{*})<0$ and $dh(v_{*})<0$. In both panels, the
remaining unshaded regions represent regions where points are saddle. Here, we
have taken $w=0$.

Figure 6: Projection of the time evolution of phase space trajectories along
the (a) $x$-axis, (b) $y$-axis, (c) $z$-axis and (d) $s$-axis which determine
the stability of set $E_{2}$. Here we take $w=0$ and $\alpha=1$,
$V=V_{0}\phi^{-n}$ with $n=4$ and $f(\phi)=f_{0}e^{\beta\phi}$ with $\beta=2$.

###  3.4 $f(\phi)=$General function, $V(\phi)=$General potential

In this subsection, we shall investigate the case where both function $f$ and
potential $V$ are kept general. By general, we mean $V$ and $f$ are such that
functions $\Gamma_{1}$, $\Gamma_{2}$ can be written as functions of $s$ and
$v$ respectively. It is worth mentioning that constant potential and constant
function $f$ do not belongs to this category as both $\Gamma_{1}$ and
$\Gamma_{2}$ cannot be defined. For this case, the system possess an invariant
sub-manifold $y=0$ and a singular invariant sub-manifold $z=0$. For some
choice of $V$ and $f$, $s=0$ and $v=0$ may be invariant sub-manifolds. In what
follows, we split the analysis of system (21)-(25) into two cases $\alpha\neq
1$ and $\alpha=1$.

####  3.4.1 $\alpha\neq 1$

In this case, we have two sets of critical points: $D_{1}=(0,0,0,s,v),\\\
D_{2}=(0,1,z,s,v)$ and two critical points:
$D_{3}=\Big{(}-\frac{\alpha\sqrt{6}(2\alpha
w+2\alpha-w-3)}{2v_{*}(2\alpha-1)},0,0,s_{*},v_{*}\Big{)}$ and
$D_{4}=\Bigg{(}-\frac{\sqrt{6}\alpha(\alpha-1)}{v_{*}(2\alpha-1)},0,\frac{3\Big{(}-\frac{3\alpha(\alpha-1)}{v_{*}(2\alpha-1)}\Big{)}^{-2\alpha}2^{-\alpha}}{2\alpha-1},s_{*},v_{*}\Bigg{)}$.
It can be seen from the second term of Eq. (21) that the existence of set
$D_{1}$ demands $0<\alpha<\frac{1}{2}$ and $D_{2}$ is $0<\alpha<1$. Therefore,
the set $D_{1}$ is not physically viable as it existence implies
$C_{s}^{2}<0$. Further, the existence of points $D_{3}$ and $D_{4}$ demands
those functions $f$ for which $v_{*}\neq 0$. In what follows, we discuss the
stability conditions of each physically viable critical points.

  * •

The set $D_{2}$ corresponds to an accelerated scalar field dominated solution
$[\Omega_{m}=0$, $\Omega_{\phi}=1$, $w_{\rm eff}=-1,w_{\phi}=-1]$. It is non-
hyperbolic in nature with eigenvalues
$\Big{\\{}\lambda_{1}=-3(1+w),\lambda_{2}=0,\lambda_{3}=0,\lambda_{4}=0,\lambda_{5}=0\Big{\\}}$.
The stability behaviour of points on this set can be determined numerically
from the behaviour of perturbed trajectories near this set by taking a
specific example of $f$ and $V$. It can be seen that trajectories near this
set approach to different points on the set as $N\rightarrow\infty$ (see Fig.
4). Therefore, this set behaves as a late time attractor but not global as
$y\neq 0$.

  * •

The point $D_{3}$ corresponds to a matter dominated solution $[\Omega_{m}=1$,
$\Omega_{\phi}=0$, $w_{\rm eff}=w,w_{\phi}=\frac{1}{2\alpha-1}]$. The
eigenvalues of the corresponding perturbed matrix are
$\Big{\\{}\lambda_{1}=\frac{3}{2}{\frac{2\,s_{*}\,\left(w+1\right){\alpha}^{2}+\left(\left(2\,w+2\right)v_{*}-s_{*}\,\left(w+3\right)\right)\alpha-
v_{*}\,\left(w+1\right)}{v_{*}\,\left(2\,\alpha-1\right)}},\\\
\lambda_{2}=\frac{3\alpha(2\alpha
w+2\alpha-w-3)dg(s_{*})}{v_{*}(2\alpha-1)},\lambda_{3}=-\frac{3(2\alpha
w+2\alpha-w-3)}{2(2\alpha-1)},\\\ \lambda_{4}=\frac{3(2\alpha
w-w-1)}{2\alpha-1},\lambda_{5}=\frac{3\alpha(2\alpha
w+2\alpha-w-3)dh(v_{*})}{v_{*}(2\alpha-1)}\Big{\\}}$. Due to the complicated
expressions of the eigenvalues on model parameters, we analysed its region of
stability for a physically interesting case $w=0$. It can be seen that this
point is either stable or saddle by determining the region of stability in
$(\alpha,s_{*},v_{*})$ parameter space for the case $w=0$ (see Fig. 5). In
fact the stability behaviour of this point is not cosmologically viable to
describe the accelerated late time DE dominated universe, but its saddle
behaviour is of cosmological interest as it corresponds to an intermediate
unaccelerated matter domination.

  * •

The critical point $D_{4}$ also corresponds to a solution dominated by the
kinetic energy part of the scalar field $[\Omega_{m}=0,\Omega_{\phi}=1,w_{\rm
eff}=\frac{1}{2\alpha-1},w_{\phi}=\frac{1}{2\alpha-1}]$. The eigenvalues of
the corresponding perturbed matrix are
$\Big{\\{}\lambda_{1}=\frac{3\alpha\left(s_{*}(\alpha-1)+v_{*}\right)}{v_{*}\left(2\alpha-1\right)},\\\
\lambda_{2}=\frac{3(1-\alpha)}{2\alpha-1},\lambda_{3}=\frac{3\left(1-w(2\alpha-1)\right)}{2\alpha-1},\lambda_{4}=\frac{6\alpha(\alpha-1)}{2\alpha-1}\frac{dg(s_{*})}{v_{*}},\lambda_{5}=\frac{6\alpha(\alpha-1)}{2\alpha-1}\frac{dh(v_{*})}{v_{*}}\Big{\\}}$.
It can be either a stable point or saddle or unstable node depending on the
choice of $\alpha$, $w$, the function $f$ and potential $V$. However, for the
case $w=0$, this point is either saddle or unstable node only as one of the
eigenvalue $\lambda_{3}$ is positive. In Fig. 5, we plot the region of
instability of this point in $(\alpha,s_{*},v_{*})$ parameter space for the
case $w=0$. The saddle behaviour of this point is cosmologically rich as it
shows the contribution of a non-canonical term to describe two important
intermediate epochs in the history of the universe: the radiation dominated
epoch for $\alpha=2$ ($w_{\rm eff}=\frac{1}{3}$) and the matter domination for
very large values of $\alpha$ ($w_{\rm eff}=0$).

From the above analysis, we see that depending on the choice of function $f$
and the value of parameters $\alpha$, this class of non-canonical scalar field
models can successfully describe the behaviour of the Universe at the
background level. For $f$ in which $v_{*}\neq 0$ and $\alpha<1$, the models
can describe the evolution of the Universe from a matter domination (point
$D_{3}$) towards a deSitter attractor (set $D_{2}$). For $\alpha=2$, the
models can explain the radiation domination (point $D_{4}$) to matter
domination (point $D_{3}$) only, but these cannot be connected to a DE finite
late time attractor. Thus, in summary, the class of models in which $f$ is
such that $v_{*}\neq 0$ can only describe either the transition of radiation
$\rightarrow$ DM only without a DE epoch or DM $\rightarrow$ DE but without a
radiation epoch. However, for $f$ in which $v_{*}=0$, the models cannot
successfully describe the complete evolution of the Universe as there is no
matter dominated critical point which is required to explain the large-scale
structure formation at the background level.

####  3.4.2 $\alpha=1$

Here, we have two sets of critical points $E_{1}=(0,0,z,s,v)$ and
$E_{2}=(0,1,z,0,v)$. The set $E_{1}$ corresponds to a matter dominated
solution $[\Omega_{m}=1$, $\Omega_{\phi}=0$, $w_{\rm
eff}=w,w_{\phi}=\frac{1}{2\alpha-1}]$ with eigenvalues
$\Big{\\{}\lambda_{1}=\frac{3(w-1)}{2},\lambda_{2}=\frac{3(1+w)}{2},\lambda_{3}=0,\lambda_{4}=0,\lambda_{5}=0\Big{\\}}$.
Therefore, points on this set always behave as saddle points. The set $E_{2}$
corresponds to an accelerated scalar field dominated solution $[\Omega_{m}=0$,
$\Omega_{\phi}=1$, $w_{\rm eff}=-1,w_{\phi}=-1]$. It is non-hyperbolic in
nature with eigenvalues
$\Big{\\{}\lambda_{1}=0,\lambda_{2}=0,\lambda_{3}=-\frac{3}{2}\Big{[}1+\sqrt{1-\frac{4}{3}\frac{g(0)}{z}}\Big{]},\lambda_{4}=-\frac{3}{2}\Big{[}1-\sqrt{1-\frac{4}{3}\frac{g(0)}{z}}\Big{]},\\\
\lambda_{5}=-3(1+w)\Big{\\}}$. However, this set is normally hyperbolic set if
$\frac{g(0)}{z}\neq 0$. Therefore, points on this set behave as stable node
for $0<\frac{g(0)}{z}\leq\frac{3}{4}$, they behave as stable focus for
$\frac{g(0)}{z}>\frac{3}{4}$ and behave as saddle for $\frac{g(0)}{z}<0$. For
the case $\frac{g(0)}{z}=0$, the set $E_{2}$ corresponds to a non-hyperbolic
set. In order to confirm the stability behaviour of this set, we numerically
perturbed the solution near the set and determine the behaviour of
trajectories by considering a specific choice of $f$ and $V$. It is evident
from Figs. 6, 6 and 6 that the phase space trajectories approach to $x=0$,
$y=1$ and $s=0$ respectively as $N\rightarrow\infty$. Further it can be seen
from Fig. 6, that trajectories starting from any values of $z$ remains almost
constant. From the behaviour of the perturbed system near the set $E_{2}$, we
can conclude that the set $E_{2}$ behaves as a stable set even for
$\frac{g(0)}{z}=0$. Note that for the choice of potential considered in Fig.
6, we have $g(0)=0$. From the above analysis, we see that in this case for
some choice of potential and initial conditions, the universe passes through a
matter domination epoch and evolve towards an accelerated DE dominated epoch.
Hence, the model can successfully describe the late time transition of the
universe.

Finally, we end this section with some comments on the choice of coupling and
potential functions. In the context of inflation, there are huge classes of
potentials which give the same dynamics. Thus in order to connect observations
with theories, it is important to identify possible degeneracy between two
classes of potentials. For a detail discussion of degeneracy between non-
canonical and canonical models in the context of inflation see gwyn ; jarv .
Usually, the general $k$-essence Lagrangian of the form
$\mathcal{L}(\phi,X)=f(\phi)X^{\alpha}-V(\phi)$ can always be recast into a
canonical kinetic term with a modified potential for $\alpha=1$ by a field
redefinition gwyn . This is, however, non-trivial for a higher value of the
exponent $\alpha$. In general, by field redefinition, out of $f$ and $V$, one
of them can be reduced to a trivial form for $\alpha=1$. This indicates that
there may be some kind of correspondence between the results of $f=$ constant
and $V=$ constant cases for $\alpha=1$. This is also suggested from the
cosmological evolution of both the cases at $\alpha=1$, where the universe
evolves from a stiff matter to a DM domination and then eventually evolves to
a DE domination.

##  4 Conclusion

In the present work, we have performed a dynamical system analysis of a
general $k$-essence Lagrangian. The general $k$-essence models give
interesting cosmological dynamics such as a viable cosmological transition
from radiation to matter domination and eventually to DE domination. More
importantly, we get a scenario in hand which gives the appealing unified
description of DE and DM for a broad class of coupling functions. The main aim
of this work is to analyse the background cosmological dynamics for a broad
class of coupling and potential functions. Dynamical system techniques allowed
us to determine the broad class of coupling and potential functions which can
lead to interesting cosmological dynamics.

In the analysis presented above, we have first analysed the case when both the
function $f$ and potential $V$ are constant (see Sect. 3.1). In this case, the
system contains only a matter dominated solution ($\Omega_{m}=1,w_{\rm
eff}=w$) and a stable deSitter solution ($\Omega_{\phi}=1,w_{\rm eff}=-1$).
Therefore, the dynamics is very simple and resembles that of the $\Lambda$CDM
where the Universe evolves from matter domination towards a DE domination.
Secondly, we have considered the case when function $f$ is still constant but
the potential is not constant (Sect. 3.2). In both the cases, for all
physically viable critical sets (or points), we have $x=0$ which implies that
there is no contribution from the kinetic component of the scalar field during
the main cosmological eras.

Then we have analysed the case when the function $f$ is assumed to be such
that $\Gamma_{2}$ is a function of $v$. Here, the DM contribution comes from
the extra barotropic fluid alone except for the case when the function $f$ is
such that $v_{*}\neq 0$ ($v_{*}$ is a root of $v^{2}(\Gamma_{2}(v)-1)=0$)
along with a constant potential. In the latter case, the early cosmic matter
behaviour is mainly due to the kinetic part of the non-canonical scalar field
(point $C_{5}$ in Sect. 3.3). Depending on the value of the non-canonical
parameter $\alpha$, the Universe evolves either from a radiation or DM or
stiff matter solution described by the point $C_{5}$ towards a DM dominated
epoch and eventually settles as a cosmological constant (see Fig. 3 in Sect.
3.3). In particular for $\alpha=2$, the model provides a viable cosmological
sequence: radiation [$w_{\rm eff}=\frac{1}{3}$] $\rightarrow$ DM [$w_{\rm
eff}=0$] $\rightarrow$ DE [$w_{\rm eff}=-1$]. This result extends the
expansion history of the Universe obtained in Sahni:2015hbf . Note that the
early radiation behaviour is solely due to the contribution of the scalar
field without a need for normal radiation. Such interesting dynamics cannot be
achieved in the case of a canonical scalar field within the GR framework which
indeed requires additional normal radiation for identical dynamics. It is
worth mentioning, the identical sequence of cosmological background dynamics
can also be obtained in the context of mimetic gravity Dutta:2017fjw .
Moreover, as the non-canonical parameter $\alpha$ affects the dark energy
speed of sound, therefore depending on the clustering property of a scalar
field, the early non-canonical scalar field domination behaves either as cold
or warm dark matter Sahni:2015hbf . This might somehow circumvent the problem
of $\Lambda$CDM in fitting CMB and weak lensing data simultaneously
Kunz:2015oqa . Further, the early dark matter behaviour of the kinetic
component of a scalar field could also explain the formation of seeds for
super-massive black holes Sawicki:2013wja . Interestingly, the scenario in
hand can also explain a possible inflationary exit problem (point $C_{4}$) for
a constant potential. This behaviour is also achieved in $k$-essence model
discussed in DeSantiago:2012nk . To have a clear picture on this, one requires
to investigate the dynamical behaviour of the potential $V$ and determine the
conditions for evolving towards a slow-roll regime. It deserves mentioning
that a similar form of a solution is however also obtained in a well-known
$\alpha$-attractors inflationary models with non-constant potential
Alho:2017opd . Here, we should remark that in $\alpha$-attractors inflationary
models, the slow roll conditions are usually satisfied when the coupling
function $f$ is very large, irrespective of the form of the potential $V$. In
particular, a singular $f$ may correspond to a critical point, which however
cannot be captured by variables (3). The critical point for such behaviour
might be pushed towards infinity and compactification techniques may give such
a point. Further, it is worth noting that due to the absence of accelerated
global attractors, the late time evolution from decelerated to an accelerated
phase is not guaranteed but strictly depends on fine-tuning of initial
conditions and model parameters. In addition to this, the non-compact form of
the phase space demands the compactification technique of the phase space, so
as to have a global picture on the cosmological dynamics. However, such an
analytical investigation is beyond the scope of the present work.

It is worth noting that in the presence of the photons, the radiation
dominated era would comprise of two components i.e. the scalar field kinetic
component and photons that cause the space to expand. However, only one
component i.e. photons that are emitted and absorbed by matter. Despite tight
constraints from standard big bang nucleosynthesis, this may signal
interesting observational characteristics and perhaps could be connected with
the puzzle of the 21 cm line anomaly. The discussion on the implications of
this phenomenon will require further investigations in the future.

In summary, we conclude that on employing dynamical system techniques, the
non-canonical scalar field models give contrasting dynamics for different
choices of $f$ and $V$. More precisely, our approach identified a broad class
of coupling and potential functions which give interesting cosmological
dynamics at the background level. In particular, our analysis reveals that a
class of coupling functions $f$ where the root of an equation
$v^{2}(\Gamma_{2}(v)-1)=0$ does not vanish and with a constant potential lead
to interesting cosmological dynamics viz. a viable history of the Universe
(radiation $\rightarrow$ DM $\rightarrow$ DE dominated era) as well as the
unified description of DM and DE. It would be of interest in the future to
further analyse the behaviour using an alternative choice of dynamical
variables which might give a more broad class of $f$ and $V$ to describe the
dark sectors unification. This would corroborate the work of Mishra:2018tki .
Further, for a various choice of $f$ and $V$, this model can mimic the
$\Lambda$CDM model in late times at the background level which supports the
work of Sahni and Sen Sahni:2015hbf . However, there may be a possible
deviation at the perturbation level, which might lead to interesting
signatures towards the present and future observations Sawicki:2013wja . By
employing dynamical system techniques, analysis at the perturbation level
might give a general conclusion over a wide range of initial conditions for
perturbations. Finally, in the light of precise cosmological data, it would be
of interest to investigate the degree of degeneracy of this class of models
for higher order kinetic terms.

###### Acknowledgements.

The authors thank Laur Järv for useful discussions. JD was supported by the
Core research grant of SERB, Department of Science and Technology India (File
No.CRG/2018/001035) and the Associate program of IUCAA. The authors also thank
the anonymous reviewer for constructive suggestions which lead to the
improvement of the work.

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