# New dynamical features of pure k-essential cosmologies

Ugo Moschella1,2,3 and Mario Novello4

1 Institut des Hautes Études Scientifiques,
35 Route de Chartres, 91440, Bures-sur-Yvette (France)
2 DiSAT, Università degli Studi dell’Insubria,
Via Valleggio 11, 22100 Como (Italy)
3 INFN, Sez di Milano, Via Celoria 16, 20146, Milano (Italy )
4 Centro de Estudos Avançados de Cosmologia/CBPF
Rua Dr. Xavier Sigaud 150, Urca 22290-180 Rio de Janeiro, RJ-Brazil

###### Abstract

We come back on the dynamical properties of $k$-essential cosmological models
and show how the interesting phenomenological features of those models are
related to the existence of boundaries in the phase surface. We focus our
attention to the branching curves where the energy density has an extremum and
the effective speed of sound diverges. We discuss the behaviour of solutions
of a general class of cosmological models exhibiting such curves and give two
possible interpretations; the most interesting possibility regards the arrow
of time that is reversed in trespassing the branching curve. This study
teaches to us something new about general FLRW cosmologies where the fluids
driving the cosmic evolution have equations of state that are multivalued
functions of the energy density and other thermodynamical quantities.

##  1 Introduction

Models based on a classical scalar field with non-canonical kinetic terms,
also called $k$-essential models, were introduced in [1] and have since then
attracted a considerable amount of attention in cosmology and in modified
theories of gravity [2, 3, 4]. Because of their innate ability to simulate the
behaviour of the cosmological constant, they can be used to model both the
early and the late accelerating phases of the cosmic history and this is why
they are widely studied. Also, they have a remarkable relation with the Born-
Infeld theory [5] and, most of all, with the Chaplygin gas and the associated
family of integrable models of fluidodynamics [6], relation that enrolls some
specific $k$-essential models in the attempts at unifying the dark components
of the cosmic fluid.

There are however various types of instabilities that might render some, if
not most, of the models in this class pathological. One particular instability
may described by making use of the analogy between the scalar field and a
fluid, analogy which is valid under the assumption of homogeneity and
isotropy, i.e. when the scalar field depends only on the cosmic time. One may
associate to the scalar field an energy density $\rho$ and a pressure $p$ and
derive from the Lagrangian a relation between $\rho$, $p$ and the field
$\varphi$ that may be called an ”equation of state” (eos):

$p=p(\rho,\varphi).$ (1)

The above function is in general ramified: there are many possible values for
the pressure compatible with a given value of the field and of the energy
density. Alternatively, the eos may be introduced in a parametric form:

$p=p(\varphi,\dot{\varphi}),\ \ \ \ \rho=\rho(\varphi,\dot{\varphi}).$ (2)

We may define an effective squared speed of sound as usual, by taking the
derivative

$c_{s}^{2}=\frac{\partial p}{\partial\rho}$ (3)

or else by the ratio [7] $c_{s}^{2}={\frac{\partial
p}{\partial\dot{\varphi}}}/{\frac{\partial\rho}{\partial\dot{\varphi}}}.$ The
two definitions are in fact equivalent.

Points where the squared speed of sound diverges define the ”critical curves”
on the eos surface. Typically they separate regions where the squared speed of
sound is positive from regions where it is negative; the latter are usually
disregarded [1, 7] because of a catastrophic perturbative instability over
there.

The state of things is the same also in the simpler purely kinetic case, where
the Lagrangian depends only on the derivatives of the field (actually, only on
the time derivative) and not on the field itself. Purely kinetic models have
been be used to describe low energy dynamics of zero-temperature superfluids
[8, 9, 10, 11]. Similarly to hydrodynamics these models are subject to
formation of caustic singularities [12, 13]. This caustics formation can be
avoided by a UV completion. [14, 15]. This UV completion would also break down
around our critical points. Here the equation of state is just a curve in the
($\rho,p$)-plane and critical points are the ramification points where the
tangent to the curve $p=p(\rho)$ is vertical (see Fig. 1). These features were
already noticed in the seminal paper [1] but its explicit focus was only on
solutions which describe an expanding universe and therefore the analysis of
the dynamical features of the models around critical points was left out.

In what follows we focus on the simplest nontrivial example of a scalar field
minimally coupled to gravity whose dynamics is described by a quartic
Lagrangian density

$F(\varphi,\partial\varphi)=\lambda(\varphi)(\partial\varphi)^{2}+\mu(\varphi)(\partial\varphi)^{4}.$
(4)

This model has been introduced in the context of $k$-inflation in [1] and
could also possibly describe the late universe where dark energy is dominant;
our prinicpal aim is contributing to elucidate the role of critical points and
surfaces.

We will at first in Sect. 2 exactly solve the purely kinetic case
$F(\partial\varphi)$ by explicitly constructing the solutions for all possible
choices of the relative signs of the coupling constants $\lambda$ and $\mu$.
All the interesting phenomena such as the dynamical generation of either a
positive or a negative cosmological constant or the bounce in the cosmological
evolution arise as consequences of the existence of boundaries in the phase
space; these phenomena are indeed quite model independent and are essentially
the same for any choice of a purely kinetic Lagrangian. These facts are
somehow obvious; the minor new points here are the formulae giving the
explicit solutions and a cursory look at a model mimicking a negative
cosmological constant.

In Sect. 3, we focus on the study of branching points always in the simpler
purely kinetic case. The presence of a branching point in the physical region
of the phase diagram renders the dynamical behaviour much subtler: having an
explicit solution at hand now is not just an academic pastime but is crucial
to understand what is going on.

What is puzzling at the branching point is the fact that the second time
derivative of the field $\ddot{\varphi}$ diverges while there is no
discontinuity in the curvature invariants. We propose two possible ways out of
this paradoxical situation. The more conservative way requires a discontinuity
in $\dot{a}(t)$ which, however, is compatible with the absence of
discontinuities of the curvature invariants. The evolution of the universe
makes the radius shrink with a decreasing negative velocity $\dot{a}(t)$ till
$t=0$ when the universe hits the branching point.

The branching point acts like a wall: the universe undergoes a sort of elastic
collision against a wall where the velocity is reverted and becomes positive.
Then it enters in a short-lived phase of decelerated expansion; after a while
the expansion stops and the universe shrinks again until it hits the branching
point for a second time. Between the first and the second passage at the
critical point the squared speed of sound is negative. However, since the
interval is finite the instability may be not catastrophic.

There is however another interpretation that the field equations seem to
dictate: it points towards the existence of time loops related to the two
adjacent branches of the equation of state. In passing from one branch into
the other the sign of time is reversed and the universe bounces and bulges ;
once back in the first branch the sign of time is reversed again and the
original arrow of time is restored.

We devote the last section of the paper to investigate whether the behaviour
of the universe around a critical point we have just described is a
peculiarity of purely kinetic models which are very constrained or else it is
a feature that may be found also in more general situations. We show that
indeed the latter is the case under quite general condition on the coupling
functions $\lambda(\varphi)$ and $\mu(\varphi)$.

The new phenomena described in this paper are at the moment mathematical
features of pure $k$-essence models. Actually they teach to us something new
about general FLRW cosmologies where the fluids driving the cosmic evolution
have equations of state that are a multivalued functions of the energy density
and other thermodynamical quantities. Whether there is any observable
consequence for the universe we live in is a separate question that requires a
better understanding of perturbations around critical points and the presence
of other components of the cosmic fluid. We leave this question for further
research.

###  1.1 A few general facts

We consider a noncanonical scalar field $\varphi$ described by the Lagrangian
density

$\displaystyle L(\varphi,\partial\varphi)=F(\varphi,X),\ \ \ \
X=(\partial\varphi)^{2}=g^{\mu\nu}\,\partial_{\mu}\varphi\,\partial_{\nu}\varphi,$
(5)

minimally coupled to gravity in a flat FLRW geometry

$\displaystyle ds^{2}=dt^{2}-a^{2}(t)\,(dx^{2}+dy^{2}+dz^{2}).$ (6)

Homogeneity calls for $\varphi=\varphi(t)$ (so that $X=\dot{\varphi}^{2}\geq
0$); the energy-momentum tensor is

$T_{\mu\nu}=F_{X}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}Fg_{\mu\nu}$
(7)

(in the above formula $F_{X}=\partial F/\partial X$). One may think of the
field as a perfect fluid whose equation of state is parametrically written in
terms of $\varphi$ and its time derivative $\dot{\varphi}$ as follows:

$\displaystyle
T_{00}=\rho(\varphi,X)=XF_{X}-\frac{1}{2}F=\frac{1}{2}\left(\dot{\varphi}\frac{\partial
F}{\partial\dot{\varphi}}-F\right),$ (8) $\displaystyle
T_{ii}=p(\varphi,X)=\frac{1}{2}F.$ (9)

The field equations coupling the field to the scale factor are the following:

$\displaystyle\rho_{X}\ddot{\varphi}+\sqrt{3\rho}\,p_{X}\dot{\varphi}+\frac{1}{2}\frac{\partial\rho}{\partial\varphi}=0,$
(10) $\displaystyle 3\left(\frac{\dot{a}}{a}\right)^{2}=\rho,$ (11)

where $\rho_{X}=XF_{XX}+\frac{1}{2}F_{X};$ units are such that $8\pi G=1$.

We will at first focus on the purely kinetic case. Here the functions $p=p(X)$
and $\rho=\rho(X)$ define a parametric equation of state that can be
represented graphically as a plane curve as we do in Fig. 1 for a polynomial
self-interaction. The dashed part of the phase curve is unphysical: it is the
arc where $X<0$. Also unphysical is the arc of the curve where $\rho$ is
negative.

Figure 1: Plot of the equation of state for the polynomial self-interaction
$F(X)=-X-X^{2}+X^{3}+X^{4}$.

The solid red lines in Fig. 1 discriminate three regions :

  1. 1.

Region I, where the Weak Energy Condition (wec) holds:

$\rho\geq 0,\;\;\ \ \rho+p\geq 0.$ (12)

Region I is subdivided into two subregions: Region Ia where the acceleration
$\ddot{a}(t)$ is negative (i.e. $\rho+3p>0$) and Region Ib where it is
positive.

  2. 2.

Region II, where the Ultra Weak Energy Condition (uwec) holds:

$0\leq\rho\leq-p,\ \ \ p<0.$ (13)

  3. 3.

The unphysical region III, where the energy density is negative:

$\rho<0.$ (14)

Region I and Region II are dynamically disconnected. Region III is unphysical
but dynamically inaccessible since we supposed a flat spatial geometry and the
energy density cannot become negative, see Eq. (11).

There are in general several algebraic branching points where the speed of
sound $c_{s}^{2}=\partial p/\partial\rho$ becomes infinite, the energy density
has an extremum and the tangent to the phase curve is vertical (see Fig. 1).
They can be situated either in the physical regions or in the unphysical one.

For purely kinetic models the field equations can be then solved by a
quadrature. Since

$F_{X}\,\sqrt{X}=\pm\left(\frac{a_{0}}{a}\right)^{3},\ \ \ a_{0}\not=0,$ (15)
$\frac{dX}{dt}=\mp\frac{\sqrt{6}\,X\,F_{X}\,\sqrt{XF_{X}-\frac{1}{2}F}}{2XF_{XX}+F_{X}}=\mp\frac{(\rho(X)+p(X))\,\sqrt{3\rho(X)}}{\rho^{\prime}(X)},$
(16)

we get

$t(X)=\mp\int\frac{\rho_{X}(X)dX}{(\rho(X)+p(X))\,\sqrt{3\rho(X)}}.$ (17)

The only subtlety is about how to glue the solutions around branching points:
at a branching point the density has an extremum, $\dot{X}$ diverges and its
sign may jump. We will clarify this point in the examples.

##  2 The quartic purely kinetic model with and without a cosmological
constant

As a warm-up, let us begin to provide solutions of our prototypical model (4)
in the purely kinetic case by supposing that all the coupling constants be
positive;

$F=-\Lambda+X+X^{2};$ (18)

we set with no loss of generality $\mu=1$ and $\lambda=1$ and added a
cosmological constant term. The physical parts of the phase curves (where
$X>0$) are all contained in region I where the wec (12) is satisfied (see Fig.
2); also, there are no branching points in the physical region. For
$\Lambda>0$ the solution can be explicitly written in parametric form as
follows:

$\displaystyle a(X)$ $\displaystyle=$
$\displaystyle{a_{0}}{(({2X+1)\sqrt{X}})^{-\frac{1}{3}}},$ (19) $\displaystyle
t(X)$ $\displaystyle=$
$\displaystyle\frac{\log\left(\frac{2\Lambda+2\sqrt{\Lambda}\sqrt{\Lambda+3X^{2}+X}+X}{\left(2\sqrt{3}\sqrt{\Lambda}+1\right)X}\right)}{\sqrt{6\Lambda}}+\frac{4\log\left(\frac{4\Lambda+2\sqrt{4\Lambda+1}\sqrt{\Lambda+3X^{2}+X}-4X-1}{\left(\sqrt{12\Lambda+3}-2\right)(2X+1)}\right)}{\sqrt{24\Lambda+6}}.$
(20)

The pressure becomes negative for $X<\frac{1}{2}(\sqrt{1+4\Lambda}-1)$. After
the initial singularity at $t=0$ (i.e. at $X=\infty$) the universe starts
decelerating behaving as as if it were radiation dominated. The cosmological
constant stops the deceleration at
$X_{\Lambda}=\frac{1}{3}\left(\sqrt{3\Lambda+1}-1\right)$ and it is no
surprise that at late times, when $X$ tends to zero, the cosmological constant
dominates: $\rho\sim-p\sim\Lambda/2$. The universe evolves between a radiation
dominated phase and an asymptotically de Sitter spacetime. By switching off
$\Lambda$ the universe is always decelerating and the model interpolates
between radiation at early times and stiff matter $\rho\sim a^{-6}$ at late
times (see Fig. 2) .

Figure 2: Plots of the eos and the scale factors for the three possible
choices of the cosmological constant. The dashed parts of the phase curves
correspond to $X<0$. For $\Lambda\geq 0$ (lower two curves) the wec is
respected all the way down to zero, the field $X$ takes all the values between
infinity and zero and the universe is driven towards the attractor
$p=-\rho=-\frac{\Lambda}{2}$. For $\Lambda<0$ (upper curve) not all the
positive values of $X$ are attainable. The universe gets to a minimal value of
the pressure with zero density and zero velocity; the acceleration is negative
and the universe bounces back and recollapses in a finite cosmic time
interval.

If $\Lambda=-\Delta<0$ the energy density takes negative values in the
interval

$0<X<{X_{\Delta}}=\frac{1}{6}\left(\sqrt{12\Delta+1}-1\right);$ (21)

on the other hand the quantity $\rho+p$ is always positive for $X>0$ . The
parametric solution, $a(X)$ does not depend on $\Lambda$ and is the same as in
Eq. (19); $t(X)$ may be obtained by analytic continuation of Eq. (20) in the
cosmological constant.

It takes a finite interval $t_{\Delta}$ of cosmic time to get to
$X={X_{\Delta}}$ starting from $X=\infty$; at $X=X_{\Delta}$ the density
vanishes and the velocity vanishes as well (this is of course a consequence of
the first Friedmann’s equation). For a negative cosmological constant the
acceleration is always negative and therefore the universe bounces back and
collapses after $t_{\Delta}$ more seconds. The bounce forbids the energy
density to become negative but the universe is short-lived.

Now we get rid of the cosmological constant and consider the case $\mu<0$ and
$\lambda>0$. As before, we may set with no loss of generality $\mu=-1$ and
$\lambda=1$ so that

$F=-X+X^{2}.$ (22)

This case is discussed in the seminal paper [1] and indeed partly explains the
interest for $k$-essential models as it catches the most important feature of
these models: the cosmic acceleration. The eos in parametric form is given by

$\rho(X)=\frac{1}{2}\left(3X^{2}-X\right),\ \ \ \
p(X)=\frac{1}{2}\left(X^{2}-X\right).$ (23)

The energy density is bounded from below but becomes negative for
$X<\frac{1}{3}$. The wec is violated for $X<\frac{1}{2}$ as $\rho+p$ becomes
negative there. There is a branching point at $X_{c}=1/6$ but it lies in the
exclusion region. There is however a warning [7]: the squared speed of sound

$\displaystyle c_{s}^{2}=\frac{1-2X}{1-6X}$ (24)

is now negative in the region $\frac{1}{6}<X<\frac{1}{2}$. This includes the
arc of the phase curve lying in Region II.

A glance at the phase curve in Region I ($X>1/2$, see Fig. 3) shows that the
dynamical behaviour of this model at the unperturbed level  should be similar
to the one described above with a positive cosmological constant. However the
concavity of the undashed part of the phase curve is opposite to that of Fig.
2.

Figure 3:  Plot of the equation of state (23). The branching point is in the
exclusion Region III. The dashed part of the curve is excluded by the
condition $X>0$. In Region I, after the initial singularity the universe is
driven towards the de Sitter attractor with $p=-\rho=-1/8$ [1] (compare with
Fig. 2, $\Lambda>0$). On the other hand in Region II (the uwec region) the
universe undergoes a bounce between two asymptotic de Sitter geometries of the
same curvature.

The solution of this model is written as follows:

$\displaystyle a(X)$ $\displaystyle=$
$\displaystyle{a_{0}}{((2X-1)\sqrt{X})^{-\frac{1}{3}}},$ (25) $\displaystyle
t(X)$ $\displaystyle=$
$\displaystyle\frac{8\sqrt{X(3X-1)}\tanh^{-1}\left(\sqrt{\frac{X}{3X-1}}\right)+2-6X}{\sqrt{6X(3X-1)}}+\sqrt{2}-4\sqrt{\frac{2}{3}}\coth^{-1}\sqrt{3}.$
(26)

$a(X)$ and $t(X)$ are both defined and monotonically decreasing in Region I
where they are invertible. They both go to infinity at $X=1/2$; values of $X$
smaller than $1/2$ are dynamically inaccessible (see Fig. 3). The universe
begins with a singularity at $t=0$ ($\dot{a}=\infty$ and $\ddot{a}=-\infty$)
and immediately enters a radiation dominated decelerating phase that ends at
$X=2/3$. Then the accelerating epoch starts.

Both the models (18) and (22) interpolate between radiation at early times and
the cosmological constant at late times, but the action (22) contains no
cosmological term; at late times (i.e. when $X\sim 1/2$) the quantity $p+\rho$
tends to zero while both the density and the pressure remain different from
zero:

$\displaystyle\rho(a)\sim\frac{1}{8}+\frac{1}{\sqrt{2}a^{3}}\ \ \ \ \
p(a)\sim\frac{1}{8}\ \ \ \ \ \ {\rm for}\ a\to\infty.\ \ \ \ $ (27)

The effective cosmological constant arises dynamically by protection of the
wec. Note also that the speed of sound goes to zero when the universe
approaches the attractor.

The (unperturbed) dynamics becomes a little subtler when the initial condition
are given in Region II (where $\frac{1}{3}<X<\frac{1}{2}$). This case was left
out in [1] because the focus was on expanding universes and most of all
because the squared speed of sound here is negative. To construct the
parametric solution we should at first choose $\dot{X}<0$:

$\displaystyle a(X)$ $\displaystyle=$
$\displaystyle{a_{0}}{((1-2X)\sqrt{X})^{-\frac{1}{3}}}$ (28) $\displaystyle
t(X)$ $\displaystyle=$
$\displaystyle\frac{6X-8\sqrt{X(3X-1)}\coth^{-1}\left(\frac{\sqrt{X}}{\sqrt{3X-1}}\right)-2}{\sqrt{6X(3X-1)}}.$
(29)

It takes an infinite amount of cosmic time to the universe to get at $X=1/3$
starting from $X=1/2$; in Eq. (29) we have set the arbitrary integration
constant so that $t(1/3)=0$. The function $t(X)$ may be continued as a
bivalued function of $X$ by the same formula (29) but with opposite sign (the
dotted branch in Fig. 4). Inversion provides the function $X(t)$ which is
smooth at $t=0$ and therefore provides a smooth scale factor $a(X(t))$.

The universe starts in a quasi de Sitter contracting phase at $t\to-\infty$,
gets to the minimal value of the scale factor at $t=0$ and then expands
asymptotically reaching the same expanding de Sitter phase at $t\to\infty$. It
is maybe worthwhile to remark that, at variance with the true de Sitter case
which would have a spherical geometry, the spatial geometry here is flat. This
model is however perturbatively unstable because, as we already said, the
squared speed of sound is negative.

Figure 4:  Plot of the scale factor as function of $X$ and of the time $t$.
Here the initial conditions are given in the uwec region $1/3<X<1/2$. $t(X)$
is a bivalued function of $X$. When it reaches the minimal value $X=1/3$ the
universe bounces.

Figure 5:  Plot of the equation of state (31). The two branches are
disconnected. The branching point is imaginary. The dashed branch of the curve
is excluded by the condition $X>0$.

Is it possible to mimic the dynamical features of a negative cosmological
constant in a purely kinetic $k$-essence model? The answer is yes but it is
necessary to go beyond polynomial self-interactions. The simplest model is as
follows

$F=X+\frac{1}{X}$ (30)

(see [5] for a discussion of a similar model of nonlinear electrodynamics).
The eos in parametric form is given by

$\rho(X)=\frac{X^{2}-3}{2X},\ \ \ \
p(X)=\frac{1}{2}\left(X+\frac{1}{X}\right).$ (31)

The energy density is unbounded from below; it becomes negative for
$X<\sqrt{3}$. The wec is violated for $X<1$ as $\rho+p$ becomes negative
there. The parametrical solution now reads

$\displaystyle a(X)$ $\displaystyle=$
$\displaystyle\frac{{a_{0}}\sqrt{X}}{\sqrt[3]{X^{2}-1}},$ (32) $\displaystyle
t(X)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{6}}\int_{\sqrt{3}}^{X}\frac{x^{2}+3}{\sqrt{x^{3}-{3}{x}}\left(x^{2}-1\right)}dx.$
(33)

The integral expressing $t(X)$ may be written in terms of hypergeometric and
elliptic functions but its explicit expression does not help very much. We
want just to mention that, as in the negative cosmological constant case
discussed in Sect. 2, the lifetime of the universe is finite; here it is given
by

$\displaystyle
T=\frac{\sqrt[4]{3\pi^{2}}\Gamma\left(\frac{1}{4}\right)\left(\sqrt{2}+\sqrt[4]{3}B_{\frac{1}{3}}\left(\frac{3}{4},\frac{1}{2}\right)\right)}{8\Gamma\left(\frac{7}{4}\right)}.$
(34)

A comment maybe in order here: the finite lifetime arises from the conflict
between the tendency of an effective negative cosmological constant to curve
the time-like directions and the flatness of the spatial sections. In this
respect, it may be useful to recall that the purely anti-de Sitter geometry
may written as a FLRW geometry only by choosing hyperbolic spatial sections
and not as a flat FLRW metric.

##  3 Phenomenology of a branching point in the purely kinetic case and the
arrow of time

###  3.1 The energy density has a maximum value

Here we start describing to the new features that are investigated in this
paper i.e. the dynamical behaviour of a model universe around branching
points. These features are at first described in a purely kinetic model whose
solution can be explicitly displayed:

$\displaystyle F=X-X^{2},\ \ \ \ \rho=\frac{1}{2}\left(X-3X^{2}\right),\ \ \ \
p=\frac{1}{2}\left(X-X^{2}\right),\ \ \ \ c_{s}^{2}=\frac{1-2X}{1-6X}$ (35)

(we set $\mu=1>0$ and $\lambda=-1<0$).

In this model the energy density has a maximum value, becomes negative and is
not bounded from below. In flat spacetime the model would be considered
pathological but the curvature may improve its status.

The squared speed of sound is the same as in Eq. (24), being invariant w.r.t.
a global change of sign of the Lagrangian. As before it is negative when
$\frac{1}{6}<X<\frac{1}{2}$. This now includes the arc of the upper branch of
the phase curve contained in the first quadrant of Fig. 6, where the energy
density is positive.

The branching point is at $X_{c}=1/6$ where the speed of sound diverges and
the energy density reaches its maximum (see Fig. 6). The condition of
criticality $c_{s}^{2}=\infty$ is thus equivalently written as follows:

$\frac{\partial\rho}{\partial X}=0,\ \ \ \ \ \ \frac{\partial p}{\partial
X}\not=0.$ (36)

The case where the above derivatives are both zero (with the coupling
constants that may also depend on the field $\varphi$, see below) plays a
central role to explore the possibility of a transition to a phantom regime in
the model (4) in [16]; in our case however $p_{X}$ does not vanish at the
critical point.

Figure 6: The branching point is in Region Ia. At the branching point the
energy density reaches its maximum but $p_{X}\not=0$; the tangent to the phase
curve is vertical.

The critical point is a ramification point for the equation of state; the
upper and lower branches are respectively given by

$p=\frac{1}{18}\left(6\rho+1\pm\sqrt{1-24\rho}\right).$ (37)

The presence of a branching point in the physical region renders the dynamical
behaviour much subtler and care is required to understand it. Here, having an
explicit solution at hand is not just an academic play but a crucial help in
understanding the dynamical behaviour of the model. The equations are

$\displaystyle a(X)$ $\displaystyle=$ $\displaystyle
a_{0}\left({{\sqrt{X}(1-2X)}}\right)^{-1/3},$ (38) $\displaystyle\dot{X}$
$\displaystyle=$ $\displaystyle\pm\frac{\sqrt{6}X(1-2X)\sqrt{X-3X^{2}}}{1-6X}$
(39)

and the task is to describe the scale factor $a(t)$ as a function of the
cosmic time.

#### Lower branch of the state equation

Let us start at $X\sim 0$ (which means $a\sim\infty$). $X$ cannot become
negative and therefore at this stage we have to choose $\dot{X}>0$ (i.e. the
plus sign in Eq. (39)). With this choice $\dot{a}$ is a negative decreasing
function of $X$ in the interval $0<X<X_{c}$; the acceleration is also negative
since these points are in region Ia:

$\displaystyle\dot{a}(X)$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{X-3X^{2}}}{\sqrt{6}\sqrt[3]{(1-2X)\sqrt{X}}}=-\sqrt{\frac{\rho(X)}{3}}a(X)$
(40) $\displaystyle\ddot{a}(X)$ $\displaystyle=$
$\displaystyle-\frac{3X(2-3X)}{2\sqrt[3]{(1-2X)\sqrt{X}}}=-\frac{a(X)}{6}(\rho+3p)<0$
(41)

(here we set $a_{0}=1$). The universe gets to the branching point with finite,
strictly negative velocity and acceleration:

$\displaystyle a(X_{c})=\frac{\sqrt{3}}{\sqrt[6]{2}},\ \ \ \
\dot{a}(X_{c})=-\frac{1}{2^{5/3}\sqrt{3}},\ \ \ \
\ddot{a}(X_{c})=-\frac{1}{8\times{2}^{\frac{1}{6}}\sqrt{3}}.$ (42)

All the curvature invariants are finite and nothing special seems to happen
there. Integrating Eq. (39) up to the branching point gives:

$\displaystyle
t(X)=\frac{6X-2-8\sqrt{3X^{2}-X}\tanh^{-1}\left(\frac{\sqrt{X}}{\sqrt{3X-1}}\right)}{\sqrt{6X-18X^{2}}}+\frac{2}{3}\sqrt{\frac{2}{3}}\pi+\sqrt{2},\
\ \ 0<X<1/6,$ (43) (44)

where we set $t(X_{c})=0$. The tangent to the curve $t(X)$ at $X_{c}$ is
horizontal (see Fig. 7). We may also give initial conditions at the branching
point by reverting the velocity in Eq. (42) i.e.

$a(X_{c})=\frac{\sqrt{3}}{\sqrt[6]{2}},\ \ \ \
\dot{a}(X_{c})=\frac{1}{2^{5/3}\sqrt{3}};$ (45)

in this case we should choose the negative sign in Eq. (39) and the solution
is given by $t_{1}(X)=t_{0}-t(X),\ 0<X<1/6,$ where a suitable value of the
constant $t_{0}$ has still to be chosen. These functions and the corresponding
scale factors $a(t)$ are plotted in Fig. 7.

Figure 7:  Approaching the branching point and stepping away from it from the
left $X<X_{c}$.

Figure 8:  Stepping away from the branching point, bouncing at the boundary of
the allowed region and approaching the branching point from the right
$X>X_{c}$.

The question that is not completely trivial remains unanswered yet: once at
the branching point, what happens next? To give a possible solution to this
problem we need to solve the model for the (unstable, since $c^{2}_{s}<0$ on
it) upper branch of the equation of state.

#### Upper branch of the state equation

We may set the same initial conditions (45) at the branching point but
consider instead a positive $\dot{X}$; this amounts to the negative sign in
Eq. (39) because we are exploring the region at the right of the critical
point $X>1/6$. Integration gives

$t^{\prime}(X)=\frac{8\sqrt{X-3X^{2}}\tan^{-1}\left(\frac{\sqrt{X}}{\sqrt{1-3X}}\right)+2-6X}{\sqrt{6X-18X^{2}}}-\frac{2}{3}\sqrt{\frac{2}{3}}\pi-\sqrt{2}$
(46)

The universe expands from $X_{c}=1/6$ to $X=1/3$ in a finite time:

$\Delta
t^{\prime}=t^{\prime}(1/3)=\frac{1}{9}\sqrt{2}\left(4\sqrt{3}\pi-9\right).$
(47)

At $X=1/3$ the scale function has a maximum, the velocity vanishes and the
acceleration is strictly negative. The solution is smoothly continued by
$t^{\prime}_{1}(X)=2\Delta t^{\prime}-t(X).$ The universe start recollapsing
and get back to the branching point in another $\Delta t^{\prime}$ seconds; at
$X_{c}$ the scale factor and its derivatives have the values given in Eq. (42)
(see Fig. 8).

#### Matching of the lower and upper solutions

There is a conservative way to match the upper and lower solutions described
above: they are glued together simply by choosing111We set aside the more
artificial and less interesting possibility to chose $t_{0}=0$. $t_{0}=2\Delta
t^{\prime}$ This choice amounts to thinking of the branching point as behaving
like a wall. When the field gets at $X_{c}$ the density reaches a maximum and
the universe cannot shrink anymore. Therefore, like in the elastic collision
of a ball against a wall, the sign of $\dot{a}$ is reverted; on the other hand
the sign of $\dot{X}$ does not change while traversing the critical point. The
expansion following the collision keeps decelerating as the acceleration in
this model is aways negative. The scale factor $a(t)$ is continuous but not
differentiable at the the branching point (see Fig. 9). The left and right
derivatives have the same modulus but opposite signs. As regards the
acceleration, the left and right derivatives of the velocity w.r.t. $X$ vanish
at the branching point but, when multiplied by the diverging function
$\dot{X}$, they have the same limit and Eq. (41) is still valid. Also the
curvature invariants, which depend only on $\dot{a}^{2}$, have no
discontinuity at the branching point.

Figure 9:  Plot of the field $X(t)$ and the scale factor $a(t)$ as functions
of the cosmic time. The are both irregular at $t=0$ and $t=2\Delta
t^{\prime}$.

#### History of the universe

  1. 1.

The Universe starts at $t=-\infty$ with infinite radius and zero velocity. It
is a flat Minkowski spacetime.

  2. 2.

The evolution of the universe makes the radius shrink with a negative and
decreasing velocity $\dot{a}$ up to $t=0$ when the universe hits the branching
point.

  3. 3.

The (unstable) bulged bounce. The branching point acts like a wall and the
universe undergoes an elastic collision where the velocity is reverted and
becomes positive. The universe enters in a phase of decelerated expansion. At
$t=\Delta t^{\prime}$ the expansion stops and the universe starts again to
shrink up to $t=2\Delta t^{\prime}$ where it hits the branching point for a
second time. In this phase the squared speed of sound is negative. However it
lasts a finite interval of time that may be very short; therefore the
instability may be not catastrophic.

  4. 4.

At the branching point the velocity is reverted again. The universe enters in
a phase of everlasting decelerated expansion that will drive it back to
Minkowski space at $t=\infty$.

  5. 5.

There is no singularity at $t=\infty$.

#### Back to the future. Matching of the lower and upper solutions reloaded

There is however a drawback in the previous construction due to the stretch we
made in gluing the solutions as we did. Indeed, the non-differentiability of
the function $a(t)$ w.r.t. to the cosmic time arises because of the
conservative choice we made in insisting that the time variable unfolds
regularly and always in one direction. This choice should however imply the
presence of distributional delta contributions at the branching point,
something that we avoided by joining the left and the right limits of the
scale function and its derivatives at the branching point. This would be a
junction condition different from the Darmois and Lichnerowicz’s junction
conditions; the comforting aspect of this procedure is that in doing so no
discontinuity arises in the curvature invariants.

There is however an alternative viewpoint that automatically emerges by taking
the solution of the field equations at its face value without any stretch: the
solution (44) is actually well defined in the whole interval $0<X<1/3$ and no
effort of imagination is needed in taking it as such and joining it with the
reflected solution $t_{1}(X)=-2\Delta t^{\prime}-t(X)$, also valid in the full
region $0<X<1/3$.

Figure 10:  Plot of the field $X(t)$ and the scale factor $a(t)$ as functions
of the cosmic time. They are multivalued but regular at the branching times
$t=0$ and $t=-2\Delta t^{\prime}$.

The solution $X(t)$ and $a(t)$ are plotted in trompe l’œil  w.r.t. the time
variable in Fig. 10. What is the meaning of such diagrams? They represents the
clever solution that the universe gives to apparently unsolvable problem it
has to face when arriving at the branching point: how could it go from
$a(X_{c})=\sqrt{3}/{\sqrt[6]{2}}$ to $a(1/3)=\sqrt{3}$ with a negative
velocity and from $\dot{a}(X_{c}=1/6)=-{\sqrt{3}}/{2^{5/3}}$ to
$\dot{a}(1/3)=0$ with a negative acceleration? Running backward in time!

The ramification point in the equation of state is encountered twice during
the time evolution. When the universe gets at the ramification point the time
starts flowing backward till the universe gets again to the ramification
point, when the usual forward orientation of time is recovered. The
phenomenology is essentially the same as in the previous conservative
description but the inversion of the velocity is caused by an inversion of the
sense of the flow of time, inversion that is short-lived.

Borrowing relativistic quantum mechanics ideas of Dirac, Stuckelberg and
Feynman, it is tempting to say that a universe pair is annihilated when the
ramification point is first encountered; a pair of universes is created when
the ramification point is encountered a second time.

Here the time runs backward for a finite interval that may be very short; the
issue of the instability due to the negative squared velocity of sound needs
therefore to be reconsidered. We leave this investigation for future work.

###  3.2 Energy has a minimum

For the sake of completeness we discuss also a case where the energy density
has a minimum value. This is most simply obtained by changing a sign in the
Lagrangian (30):

$F=X-\frac{1}{X},\ \ \ \ \rho=\frac{X^{2}+3}{2X},\ \ \ \
p=\frac{1}{2}\left(X-\frac{1}{X}\right),\ \ \ \ \rho+3p=2X.$ (48)

The energy density is always positive and has a minimum at $X_{c}=\sqrt{3}$;
the acceleration always negative (see Fig. 11). The upper and lower sheets of
the phase curve are given by

$p=\frac{1}{3}\left(\rho\pm 2\sqrt{\rho^{2}-3}\right).$ (49)

The cosmic evolution is now ruled by the following equations $(a_{0}=1)$:

$\displaystyle a(X)=\frac{\sqrt{X}}{\sqrt[3]{X^{2}+1}},\ \ \
\dot{X}=\pm\frac{\sqrt{6}X\sqrt{X+\frac{3}{X}}\left(X^{2}+1\right)}{X^{2}-3}.$
(50)

Figure 11: At the branching point the energy density reaches its minimum. The
physical part of the phase curve is in Region Ia so the acceleration is always
negative.

By giving initial conditions at $X\sim\infty$ on the stable upper branch,
$\dot{X}$ has to be chosen negative and therefore

$\displaystyle\dot{a}(X)=\frac{\sqrt{X^{2}+3}}{\sqrt{6}\sqrt[3]{X^{2}+1}}>0,\
\ \ \ \ \ddot{a}(X)=\frac{X^{3/2}}{3\sqrt[3]{X^{2}+1}}<0.$ (51)

The universe starts from an initial singularity; it expands decelerating and
gets to the branching point $X_{c}=\sqrt{3}$ after a finite interval of cosmic
time

$\displaystyle
T=\int_{\sqrt{3}}^{\infty}\frac{X^{2}-3}{\sqrt{6X}\sqrt{X^{2}+3}\left(X^{2}+1\right)}dX.$
(52)

At $X_{c}$ the energy density has a minimum so the universe cannot expand
anymore. But the velocity $\dot{a}(X_{c})={1}/{\sqrt[3]{4}}$ is positive so
the universe cannot stay at the branching point as well. Again the function
$X(t)$ obtained by inverting

$\displaystyle
t(X)=2\sqrt{2X}F_{1}\left(\frac{1}{4};-\frac{1}{2},1;\frac{5}{4};-\frac{X^{2}}{3},-X^{2}\right)-\sqrt{2X}\,_{2}F_{1}\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\frac{X^{2}}{3}\right)$
(53)

is ramified (see Fig. 12).

Figure 12: The energy density has a minimum when of a pair of universes
annhilates.

In the second branch the direction of time is inverted and the universe starts
contracting to get to the singularity at $X=0$ in a finite interval of cosmic
time.

##  4 General models

At this point one may ask whether the strange behaviour of a Friedmann-
Lemaître universe going trough a ramification point of an equation of state is
just a peculiarity of the purely kinetic $k$-essential models that have not
enough degrees of freedom to avoid it. To show that the features described in
Sec. 3 are indeed generic we turn now our attention to the Lagrangian (4) in
its full generality. By suitable field redefinitions [1] we are left with the
following three possibilities:

$\displaystyle{\rm I.}$ $\displaystyle
F_{1}=\lambda(\varphi)X+X^{2}=\lambda(\varphi)\dot{\varphi}^{2}+\dot{\varphi}^{4},$
(54) $\displaystyle{\rm II.}$ $\displaystyle
F_{2}=\lambda(\varphi)X-X^{2}=\lambda(\varphi)\dot{\varphi}^{2}-\dot{\varphi}^{4},\
\ \ \ \lambda(\varphi)>0.$ (55) $\displaystyle{\rm III.}$ $\displaystyle
F_{3}=X+\mu(\varphi)X^{2}=\dot{\varphi}^{2}+\mu(\varphi)\dot{\varphi}^{4}.$
(56)

In models of the first and the third classes the coupling functions
$\lambda(\varphi)$ and $\mu(\varphi)$ are allowed to vanish and to change
their signs [1, 16]; $\lambda(\varphi)$ is bound to be strictly positive in
models of the second class. The first class of models was introduced in the
seminal paper [1] to describe $k$-inflation for various choices of the
coupling $\lambda(\varphi)$. Here the critical curve is outside the physical
region with the only exception of the origin $\lambda=0,\rho=0$ (see Fig. 13).
Let us focus on the second class to which the model (35) belongs as a very
special simple case. Energy and pressure are given by

$\displaystyle\rho=\frac{1}{2}\left(\lambda(\varphi)\dot{\varphi}^{2}-3\dot{\varphi}^{4}\right),\
\ \ \
p=\frac{1}{2}\left(\lambda(\varphi)\dot{\varphi}^{2}-\dot{\varphi}^{4}\right).$
(57)

The energy density becomes negative for
$\dot{\varphi}^{2}>\lambda(\varphi)/3$. Given $\varphi$, the maximum of the
energy density is attained at at
$X_{c}(\varphi)=\dot{\varphi}^{2}_{c}=\lambda(\varphi)/6$:

$\rho_{max}(\varphi)=\lambda(\varphi)^{2}/24.$ (58)

The condition $\rho+p\geq 0$ is violated for $X>\lambda(\varphi)/2$.

The ramified equation of state (1) is here written as follows:

$p(\rho,\varphi)=\frac{1}{18}\left(6\rho+\lambda(\varphi)^{2}\pm\lambda(\varphi)\sqrt{\lambda(\varphi)^{2}-24\rho}\right).$
(59)

The maxima of $\rho(\varphi)$ (at given $\varphi$) constitute the ramification
curve for the surface (59) (the red curve in Fig. (13)); the upper and lower
sheets are respectively given by the upper and lower signs at the rhs of Eq.
(59). At the maxima, the effective squared speed of sound

$\displaystyle c_{s}^{2}=\frac{\partial
p}{\partial\rho}=\frac{1}{3}\mp\frac{2\lambda(\varphi)}{3\sqrt{\lambda(\varphi)^{2}-24\rho}}$
(60)

diverges. The effective squared speed of sound has also been given the
following expression [7] which is identical to (60):

$\displaystyle c_{s}^{2}=\frac{\frac{\partial p}{\partial
X}}{\frac{\partial\rho}{\partial
X}}=\frac{\lambda(\varphi)-2\dot{\varphi}^{2}}{\lambda(\varphi)-6\dot{\varphi}^{2}}.$
(61)

$c_{s}^{2}$ negative for
$\frac{\lambda(\varphi)}{6}<\dot{\varphi}^{2}<\frac{\lambda(\varphi)}{2}$;
this contains the ”physical” part of the upper sheet of the phase surface (59)
that lies in the physical region where both $\rho$ and $\lambda$ are positive
(see Fig. (13)) and of course diverges at $X=X_{c}(\varphi)$.

Figure 13: At the left the equation of state for models of class I at the
right for models of class II. The branching curve is the red curve. At the
branching curve the energy density reaches its maxima and the effective speed
of sound diverges.

The field equations are

$\displaystyle\ddot{\varphi}+3H\frac{\lambda(\varphi)-2\dot{\varphi}^{2}}{\lambda(\varphi)-6\dot{\varphi}^{2}}\dot{\varphi}+\frac{1}{2(\lambda(\varphi)-6\dot{\varphi}^{2})}\frac{\partial\lambda}{\partial\varphi}\dot{\varphi}^{2}=0.$
(62) (63) (64) $\displaystyle
H=\frac{\dot{a}}{a}=\pm\sqrt{\frac{\rho}{3}}=\pm\sqrt{\frac{1}{6}\dot{\varphi}^{2}(\lambda(\varphi)-3\dot{\varphi}^{2}),}$
(65)
$\displaystyle\frac{\ddot{a}}{a}=-\frac{1}{6}(\rho+3p)=\frac{1}{2}\dot{\varphi}^{4}-\frac{1}{3}\lambda(\varphi)\dot{\varphi}^{2},$
(66)

They might be used to reverse-engineering the coupling $\lambda(\varphi)$ for
a given time evolution of the universe but we will not consider this
possibility here. In any case, solving them analytically is out of the
question even for very simple choices of the coupling $\lambda(\varphi)$.
However, the exact solution for the case $\lambda=1$, discussed at length in
the previous section, will guide us in performing a qualitative analysis.

#### Lower sheet of the phase surface

First of all we notice that setting the initial condition $\dot{\varphi}=0$ at
$t=t_{i}$ gives the trivial constant solution $\varphi=\varphi_{i}$. This
implies that the value $\dot{\varphi}=0$ cannot be attained at finite values
of the cosmic time but only in the limits $t\to\pm\infty$ and, unless a
discontinuity is traversed, $\dot{\varphi}$ does not change its sign. We shall
therefore set

$\varphi(t_{i})=\varphi_{i}>0,\ \ \ \ \ \dot{\varphi}(t_{i})=\epsilon>0\ \ \
at\ \ t=t_{i}>-\infty.$ (67)

Since $\dot{\varphi}(t_{i})^{2}\sim\epsilon^{2}<\lambda(\varphi_{i})/6$ and
because $\lambda(\varphi)$ is strictly positive, the above initial conditions
locate the system in the lower sheet of the phase surface (59). As in Eq. (40)
let us suppose at first that the universe is contracting and choose the minus
sign at the rhs of Eq. (65) :

$\displaystyle\ddot{\varphi}=\sqrt{\frac{3}{2}(\lambda(\varphi)-3\dot{\varphi}^{2})}\frac{\lambda(\varphi)-2\dot{\varphi}^{2}}{\lambda(\varphi)-6\dot{\varphi}^{2}}\dot{\varphi}^{2}-\frac{1}{2(\lambda(\varphi)-6\dot{\varphi}^{2})}\frac{\partial\lambda}{\partial\varphi}\dot{\varphi}^{2}.$
(68)

From Eq. (68) one sees that $\ddot{\varphi}(t_{i})>0$ provided that
${\partial_{\varphi}\lambda}(\varphi_{i})\leq\sqrt{6}(\lambda(\varphi_{i}))^{3/2}.$
The slightly more restrictive condition

${\partial_{\varphi}\lambda}(\varphi)\leq\frac{5}{2\sqrt{6}}\lambda(\varphi)^{3/2}\simeq
1.02062\,\lambda(\varphi)^{3/2}$ (69)

guarantees that $\ddot{\varphi}(t)>0$ as long as the dynamical system stays in
the lower sheet of the phase surface; this implies that $\dot{\varphi}(t)$ is
strictly increasing as the cosmic time and consequently also $\varphi(t)$
increases. From

$\frac{1}{2}\partial_{t}({\lambda(\varphi)-6\dot{\varphi}^{2}})^{2}=\dot{\varphi}{\lambda(\varphi)\lambda^{\prime}(\varphi)}-6\sqrt{6}\dot{\varphi}^{3}{\sqrt{\lambda(\varphi)-3\dot{\varphi}^{2}}\left(\lambda(\varphi)-2\dot{\varphi}^{2}\right)}$
(70)

we deduce that, if

${\partial_{\varphi}\lambda}(\varphi)<\frac{2\,\lambda(\varphi)^{3/2}}{\sqrt{3}},$
(71)

the quantity $({\lambda(\varphi)-6\dot{\varphi}^{2}})$ is positive and
strictly decreasing on lower sheet of the phase surface (59) and eventually
vanishes at a certain time $t_{c}$; at that point $X$ reaches the critical
value and $\dot{X}$ diverges.

Altogether, if the sufficient condition (69) holds, as the cosmic time
approaches $t_{c}$ the quantity $\ddot{\varphi}(t)$ tends to infinity and the
curve $\dot{\varphi}(t)$ becomes vertical exactly as in the purely kinetic
case (see Eq. (44) and Figs. 9 and 10). At the critical point

$\lambda(\varphi(t_{c}))=6\dot{\varphi}(t_{c})^{2},\ \ \ \
\frac{\dot{a}(t_{c})}{a(t_{c})}=-\frac{\dot{\varphi}(t_{c})}{\sqrt{2}},\ \ \ \
\frac{\ddot{a}(t_{c})}{a(t_{c})}=-\frac{3\dot{\varphi}(t_{c})^{2}}{2}.$ (72)

#### Upper sheet of the state equation

Let us now give initial conditions at $\lambda(\varphi)=3\dot{\varphi}^{2}$
i.e. at the boundary of the upper sheet of the phase surface (59) where
$\rho=0$:

$\varphi(t_{i})=\varphi_{i},\ \ \ \ \
\dot{\varphi}(t_{i})=\sqrt{\frac{\lambda(\varphi_{i})}{3}}\ \ \ at\ \
t=t_{i}.$ (73)

Since $\dot{a}(t_{i})=0$ and
${\ddot{a}(t_{i})}/{a(t_{i})}=-{\lambda{(\varphi}_{i})}/{54}<0$, from Eq. (62)
it follows that

$\displaystyle\ddot{\varphi}(t)=\left(\mp\sqrt{\frac{3}{2}(\lambda(\varphi(t))-3\dot{\varphi}(t)^{2})}\frac{\lambda(\varphi(t))-2\dot{\varphi}(t)^{2}}{\lambda(\varphi(t))-6\dot{\varphi}(t)^{2}}-\frac{1}{2(\lambda(\varphi(t))-6\dot{\varphi}(t)^{2})}\frac{\partial\lambda}{\partial\varphi(t)}\right)\dot{\varphi}(t)^{2},$
(74) $\displaystyle t\ {\lessgtr}\ t_{i}.$ (75)

Note that $\ddot{\varphi}$ is continuous at $t=t_{i}$:

$\displaystyle\ddot{\varphi}(t_{i})=\frac{1}{6}\frac{\partial\lambda}{\partial\varphi}(\varphi_{i}).$
(76)

When $\lambda(\varphi)$ is a constant, as in the purely kinetic case,
$\dot{\varphi}$ has an extremum on the boundary
$\lambda(\varphi)=3\dot{\varphi}^{2}$ of the phase surface (see Figs. 8, 9 and
10). In general this is not true when $\lambda(\varphi)$ is not a constant.
This behaviour is exemplified in the plots of Fig. 14 where $\dot{\varphi}$ is
numerically solved and plotted for the couplings
$\lambda(\varphi)=\varphi^{2}$ , $\lambda(\varphi)=1/(1+\varphi^{2})$ and
$\lambda(\varphi)=\exp\varphi)$. It is seen that the dynamics changes very
little and is eesntiaky driven by the boundaries of the phase surface and by
the critical curve.

However, since

$\partial_{t}\rho(t)=\mp\sqrt{\frac{3}{2}}\dot{\varphi}(t)^{3}\sqrt{\lambda(\varphi(t))-3\dot{\varphi}(t)^{2}}\left(\lambda(\varphi(t))-2\dot{\varphi}(t)^{2}\right),\
\ \ \ \ t\ {\lessgtr}\ t_{i},$ (77)

it follows that

$\rho(t_{i})=0,\ \ \ \partial_{t}\rho(t_{i})=0,\ \ \
\partial^{2}_{t}\rho(t_{i})=\frac{1}{54}\lambda(\phi(t))^{4};$ (78)

this means that the energy density cannot become negative whatever is the
choice of the coupling $\lambda(\varphi)$ and the boundary of the upper sheet
where the density vanishes cannot be trespassed; the crucial fact is that Eqs.
(77) do not depend on ${\partial\lambda}/{\partial\varphi}$.

Figure 14: Generic solutions for three different choices of the coupling
$\lambda(\varphi)$. The chosen initial condition are
$\varphi(0)=1,\dot{\varphi}(0)=0.01$. The maxima of $\dot{\varphi}$ are
attained in the interior of the upper sheet of the phase surface and therefore
the time loops are no more symmetric. The positivity of the energy density is
protected by the boundary $\lambda(\varphi)=3\dot{\varphi}^{2}$. The dynamics
changes very little for quite different choices of potential and is
essentially driven by the boundaries of the phase surface and by the critical
curve. Of course the conservative interpretation of the time evolution without
time loops is also available.

##  5 Summary

We have examined the flat FLRW cosmological models based on the quartic
$k$-essence Lagrangian (4) and discussed features that have stayed uncovered
to date, as they cannot be qualitatively guessed by inspecting the equation of
state even in the simpler purely kinetic case. They are related to the
existence of boundaries and branching curves in the phase surfaces of the
models, a (mathematical) feature that seems to be completely overlooked in the
literature.

The dynamical properties of a cosmological model traversing a branching point
are quite strange; there is also relation to the arrow of time that we found
interesting and new; it is described in Sections 3 and 4 . Actually these
features are not specific to $k$-essential models but are expected to always
show up when the equation of state is a multivalued function of the
thermodynamical parameters. It remains to understand whether our universe has
ever met a branching point and, if yes, what would be the observable
consequences of this fact. This would require to understand how branching
points may be reconciled with the standard wisdom of linear cosmological
perturbations, a question that we leave for future work.

## Acknowledgments

U. M. is indebted to Thibault Damour for many illuminating discussions and
suggestions and to the IHES for its generous hospitality during the writing of
this paper.
M.N. thanks CNPq and FAPERJ for financial support.

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