#  Unification: Emergent universe followed by inflation and dark epochs from
multi-field theory.

Eduardo Guendelman<EMAIL_ADDRESS>Department of Physics, Ben-Gurion
University of the Negev, Beer-Sheva, Israel.
Frankfurt Institute for Advanced Studies (FIAS), Ruth-Moufang-Strasse 1, 60438
Frankfurt am Main, Germany.
Bahamas Advanced Study Institute and Conferences, 4A Ocean Heights, Hill View
Circle, Stella Maris, Long Island, The Bahamas.  Ramón Herrera
<EMAIL_ADDRESS>Instituto de Física, Pontificia Universidad Católica de
Valparaíso, Avenida Brasil 2950, Casilla 4059, Valparaíso, Chile.

###### Abstract

A two scalar field model that incorporates non Riemannian Measures of
integration or usually called Two Measures Theory (TMT) is introduced, in
order to unify the early and present universe. In the Einstein frame a
K-essence is generated and as a consequence for the early universe, we can
have a Non Singular Emergent universe followed by Inflation and for the
present universe dark epochs with consistent generation of dark energy (DE),
dark matter (DM) and stiff matter. The scale invariance is introduced and then
is spontaneously broken from the integration of the degrees of freedom
associated with the modified measures. The resulting effective potentials and
K-essence in the Einstein frame produce three flat regions corresponding to
the different epochs mentioned before. For the first flat region we can
associate an emergent and an inflationary universe. Here for a parameter-space
region this flat plateau possesses a non singular stable emergent universe
solution which characterizes an initial epoch of evolution that precedes the
inflationary scenario. Also assuming this first plateau, we study the
inflation in the framework of the slow-roll approximation. In this scenario
under the slow roll approximation we obtain a linear combination that is a
constant. The corresponding cosmological perturbations in our model are
determined and we also obtain the different constrains on the parameter-space
from the Planck data.

In the following flat region DE and also the DM, which does not need to be
introduced separately, it is instead a result of a K-essence induced by the
multi measures, multi field theory. Also stiff matter component is
automatically generated from the K-essence theory from two scalar fields. From
the perturbative analysis associated to the perturbation solution of
background, we find a correlation between the two scalars. Besides, we obtain
that our model during the dark epoch has a behaviour of tracking freezing
model.

##  I Introduction

In the most popular paradigm for the early universe, it is postulated that the
universe suffers a period of exponential expansion called “inflation”
introduced by Guth Guth  , Starobinsky starobinsky  and others (cf. the books
early-univ ; primordial  and references therein). However these models do not
address the big-bang singularity problem. Nevertheless a scenario that while
incorporating inflation, takes into account of an even earlier period, called
the emergent period or emergent universe was proposed in Refs.emerg ; emerg2 .
On the other hand, the late universe after the discovery of the accelerating
universe accel-exp ; accel-exp-2 , also shows a period that some common
features with the inflationary period, although the relevant scales are very
different. In this framework, the late universe “standard cosmological”
description for the late universe, is now the $\Lambda$CDM model lambdaCDM .
This model incorporates a cosmological constant and a Dark Matter (DM)
consisting of pressure-less dust and ordinary visible matter (which is also
dust) and in this scenario the universe being now dominated by the
Cosmological Constant $\Lambda$ or Dark Energy (DE). The $\Lambda$CDM model is
now being challenged by the discovery of some observational tensions, the
$H_{0}$ and the $\sigma_{8}$ tensions H0 ; sigma8 .

In the inflationary period also primordial density perturbations are generated
(Ref.primordial  and references therein). The “inflation” is followed by
particle creation, where the observed matter and radiation were generated
early-univ , and finally the evolution arrives to a present phase of slowly
accelerating universe accel-exp ; accel-exp-2 . In this standard model,
however, at least three fundamental questions remain unanswered:

  * •

The inflationary epoch, although solving many cosmological puzzles, like the
horizon, flatness problems etc. and also providing a mechanism for primordial
density perturbations, cannot address the initial singularity problem;

  * •

In this model there is no reason for the existence of two scenarios of
exponential expansion with such wildly different scales the inflationary epoch
and the current era of slowly accelerated expansion of the universe.

  * •

There is no explanation for the existence of the Dark Matter, that represents
an invisible part and most of the dust in the universe that must be different
from the baryonic matter, the visible part of the dust component of our
universe.

The best known mechanism for producing a phase of accelerated expansion is
from some vacuum energy or a cosmological constant. In the framework of a
standard scalar field theory, vacuum energy density arises when the scalar
field acquires an effective potential $U_{\rm eff}$ which has flat plateaus so
that the field (inflaton) can “slowly roll” slow-roll ; slow-roll-param  and
its kinetic energy can be disregarded with which the energy-momentum tensor
$T_{\mu\nu}\simeq-g_{\mu\nu}U_{\rm eff}$, a contribution that mimics that of a
cosmological constant. We will make use of this mechanism to explain the
accelerated expansion of the universe associated to the inflation phase of the
universe. In the late phases of the universe we will also find a contribution
from an effective scalar field potential, but there will be also a
contribution to the DE from a K-essence background configuration that also
gives rise to a contribution that satisfies the equation of state $p=-\rho$.
Besides, in the emergent universe the effective potential contribution also
can combine with the K-essence to produce an equation of state $p=-\rho/3$
which produces no acceleration and in particular a static Einstein universe,
which is stable for a range of parameters, unlike the original Einstein
universe. Therefore the interplay of an effective potential and K-essence will
be crucial for many effects studied in this paper along many continuously
connected phases of the universe.

The connection between the inflationary epoch to a slowly accelerating
universe through the evolution of a single scalar field denominated “the
quintessential inflation scenario” has been first analyzed in Ref.peebles-
vilenkin . Besides, a quintessential inflation mechanism developed on the
K-essence model, was studied in Ref.saitou-nojiri . Also quintessential
inflation founded on the “variable gravity” model was developed in
Ref.wetterich  and for another list of references, see Ref.murzakulov-etal .
Additionally other approaches founded on the so called $\alpha$ attractors,
which utilizes non canonical kinetic terms have been analyzed in
Ref.alphaatractors . Also a quintessential inflation developed on a Lorentzian
slow- roll approximation which automatically gives two flat regions was
studied in Lorentzian . In addition, the $F(R)$ models can connect both
periods an early time inflationary era and a late time de Sitter epoch from
different values of effective vacuum energies, see Ref.starobinsky-2 . For a
review of $f(R)$ model and another modified gravity, see Refs.Re1 ; Re2 .

The framework for this research is the use of the metric independent non
Riemannian measures for the construction of modified gravity theories
Refs.TMT-orig-1 -TMT-orig-3  (see also Refs.TMT-recent-1-a -TMT-recent-2 ), in
some instances we have included the standard measure as well, where the
standard Riemannian integration measure might also contain a Weyl-scale
symmetry preserving $R^{2}$-term TMT-orig-3 . Some applications have been, (i)
$D=4$-dimensional models of gravity and matter fields containing the new
measure of integration appear to be promising candidates for resolution of the
dark energy and dark matter problems, the fifth force problem, and a natural
mechanism for spontaneous breakdown of global Weyl-scale symmetry TMT-orig-1
-TMT-recent-2 , (ii) To study of reparametrization invariant theories of
extended objects (strings and branes) based on employing of a modified non-
Riemannian world-sheet/world-volume integration measure mstring , nishino-
rajpoot , leads to dynamically induced variable string/brane tension and to
string models of non-abelian confinement, interesting consequences from the
modified measures spectrum mstringspectrum , and construction of new
braneworld scenarios mstringbranes , (iii) To study in Ref.susy-break  of
modified supergravity models with an alternative non-Riemannian volume form on
the space-time manifold .

Directly connected to the present research are our papers ourquintessence
where we have studied a unified scenario where both an inflation and a slowly
accelerated phase for the universe can appear naturally from the existence of
two flat regions in the effective scalar field potential which we derive
systematically from a Lagrangian action principle. Namely, we started with a
new kind of globally Weyl-scale invariant gravity-matter action within the
first-order (Palatini) approach formulated in terms of two different non-
Riemannian volume forms (integration measures) quintess . Also we have studied
the reheating scenario in this theory assuming the curvaton field in order to
decay to the hot big bang modelcurv ; curv2 . In this new theory there is a
single scalar field with kinetic terms coupled to both non-Riemannian
measures, and in addition to the scalar curvature term $R$ also an $R^{2}$
term is included (which is similarly allowed by global Weyl-scale invariance).
Scale invariance is spontaneously broken upon solving part of the
corresponding equations of motion due to the appearance of two arbitrary
dimensionfull integration constants. Furthermore, in a subsequent paper we
generalized this to a two-field caseourquintessencewithEDE  where three flat
regions appear, one for inflation and the remaining two for early DE and a
late DE phase. In order to describe the early DE and DM, we have introduced a
matter action defined as a scale invariant form, which is independent of the
scalar fields.

In this paper we will add a new aspect, that is, without introducing any new
additional matter action, therefore not introducing the DM separately, as
compared to ourquintessencewithEDE , we consider instead more generic scale
invariant couplings, like involving $R^{2}$ and others, and we produce as a
result a theory that is also able to provide a non singular emergent phase for
the universe where the effective potential contribution combines with the
K-essence to produce a $p=-\rho/3$ equation of state which produces no
acceleration and in particular a static Einstein Universe, which is stable for
a range of parameters, unlike the original Einstein universe. This emergent
phase is then followed by the inflationary phase as well as, in addition after
the emergent and inflation eras we have a second flat region that provide an
explanation for the DE and DM of the late universe. This is a consequence of
the K-essence produced by the additional kinetic terms and $R$ square terms
introduced into the action. In this context, we have a contribution to the DE
that comes from an effective scalar field potential and also there is a
contribution to the DE from a K-essence background configuration. Aditionally
we have a DM component and an additional stiff matter component which
originate from the perturbation of the background K-essence configuration.
Here we obtain a consistency condition that correlates the perturbations of
the two fields in the late universe. Also, the different constraints from each
phase are shown to be consistent with the constraints on the same parameters
from the other phases.

The plan of the present article is as follows. In the next Section we
discussion the general formalism for the new class of gravity-matter
considering two independent non Riemannian volume forms. In Section III we
analyze the three flat regions from the effective scalar potential in the
Einstein framework. In Section IV we study the non singular emergent universe.
Also we determine the stability of the solution in a region of the parameter-
space.

In Section V we describe the inflationary epoch. Here we analyze the slow-roll
approximation during the background scenario and also we determine the
perturbations cosmological. In this context we find the constraints of the
different parameters from the Planck data. In Section VI, we analyze how our
model of K-essence can describe the late universe. From the second flat
plateau we find different components assuming the perturbations K-essence
background configuration. Also we determine the equation of state associated
to the dark sector together with the derivative of this parameter. In Section
VII we end up with a discussion section concerning generalizations of the
model, additional mechanisms for inducing DE and the role of the third flat
region and its possible relation to early DE. For simplicity we will consider
units where the Newton constant is taken as $G_{\rm Newton}=1/16\pi$.

##  II Theory using Two Independent Non-Riemannian Volume-Forms

We shall assume the action of the general form involving two independent non-
metric integration measure densities generalizing the model analyzed in
quintess  is given by

$S=\int d^{4}x\,\Phi_{1}(A)\Bigl{[}R+L^{(1)}\Bigr{]}+\int
d^{4}x\,\Phi_{2}(B)\Bigl{[}L^{(2)}+\epsilon
R^{2}+\frac{\Phi(H)}{\sqrt{-g}}\Bigr{]}\;.$ (1)

Here the following definitions are used:

  * •

The quantities $\Phi_{1}(A)$ and $\Phi_{2}(B)$ are two densities and these are
independent non-metric volume-forms defined in terms of field-strengths of two
auxiliary 3-index antisymmetric tensor gauge fields

$\Phi_{1}(A)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}A_{\nu\kappa\lambda}\quad,\quad\Phi_{2}(B)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}B_{\nu\kappa\lambda}\;.$
(2)

  * •

The scalar curvature $R=g^{\mu\nu}R_{\mu\nu}(\Gamma)$ and the Ricci tensor
$R_{\mu\nu}(\Gamma)$ are defined in the first-order (Palatini) formalism, in
which the affine connection $\Gamma^{\mu}_{\nu\lambda}$ is a priori
independent of the metric $g_{\mu\nu}$. Let us recall that $R+R^{2}$ gravity
within the second order formalism was originally developed in starobinsky .

  * •

The two different Lagrangians $L^{(1,2)}$ correspond to two scalar matter
fields $\varphi_{1}$ and $\varphi_{2}$ such that

$\displaystyle
L^{(1)}=-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi_{1}\partial_{\nu}\varphi_{1}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi_{2}\partial_{\nu}\varphi_{2}-V(\varphi_{1},\varphi_{2}),\quad\;$
(3) $\displaystyle
L^{(2)}=-\frac{b_{1}}{2}e^{-\alpha_{1}\varphi_{1}}g^{\mu\nu}\partial_{\mu}\varphi_{1}\partial_{\nu}\varphi_{1}-\frac{b_{2}}{2}e^{-\alpha_{2}\varphi_{2}}g^{\mu\nu}\partial_{\mu}\varphi_{2}\partial_{\nu}\varphi_{2}+U(\varphi_{1},\varphi_{2}),\quad$
(4)

where the potentials $V(\varphi_{1},\varphi_{2})$ and
$U(\varphi_{1},\varphi_{2})$ are defined as

$V(\varphi_{1},\varphi_{2})=f_{1}\exp\\{-\alpha_{1}\varphi_{1}\\}+g_{1}\exp\\{-\alpha_{2}\varphi_{2}\\},\,\,\mbox{and}\,\,\,\,U(\varphi_{1},\varphi_{2})=f_{2}\exp\\{-2\alpha_{1}\varphi_{1}\\}+g_{2}\exp\\{-2\alpha_{2}\varphi_{2}\\}.$
(5)

Here $\alpha_{1},\alpha_{2},f_{1},g_{1},f_{2},\epsilon$ and $g_{2}$ are
positive parameters, whereas $b_{1}$ and $b_{2}$ are dimensionless and their
signs are to be discussed. The parameters $\alpha_{1}$ and $\alpha_{2}$ have
dimensions of $M_{Pl}^{-1}$, instead the parameters $f_{1}$, $f_{2}$, $g_{1}$
and $g_{2}$ have units of $M_{Pl}^{4}$ and the parameter $\epsilon$ has units
of $M_{Pl}^{-2}$. Let us recall that since we are considering units in which
$G_{\rm Newton}=1/16\pi$ then the Planck mass $M_{Pl}=\sqrt{2}=\sqrt{1/8\pi
G_{\rm Newton}}$.

  * •

The density $\Phi(H)$ denotes the dual field strength of a third auxiliary
3-index antisymmetric tensor

$\Phi(H)=\frac{1}{3!}\varepsilon^{\mu\nu\kappa\lambda}\partial_{\mu}H_{\nu\kappa\lambda}\;.$
(6)

The action given by Eq.(1) is invariant under the global Weyl-scale
transformations:

$\displaystyle g_{\mu\nu}\to\lambda
g_{\mu\nu}\;\;,\;\;\Gamma^{\mu}_{\nu\lambda}\to\Gamma^{\mu}_{\nu\lambda}\;\;,\;\;\varphi_{1}\to\varphi_{1}+\frac{1}{\alpha_{1}}\ln\lambda\;\;,\varphi_{2}\to\varphi_{2}+\frac{1}{\alpha_{2}}\ln\lambda$
$\displaystyle A_{\mu\nu\kappa}\to\lambda
A_{\mu\nu\kappa}\;\;,\;\;B_{\mu\nu\kappa}\to\lambda^{2}B_{\mu\nu\kappa}\;\;,\;\;H_{\mu\nu\kappa}\to
H_{\mu\nu\kappa}\;.$ (7)

The variation of the action (1) w.r.t. affine connection
$\Gamma^{\mu}_{\nu\lambda}$ we have

$\int
d^{4}\,x\,\sqrt{-g}g^{\mu\nu}\Bigl{(}\frac{\Phi_{1}}{\sqrt{-g}}+2\epsilon\,\frac{\Phi_{2}}{\sqrt{-g}}\,R\Bigr{)}\left(\nabla_{\kappa}\delta\Gamma^{\kappa}_{\mu\nu}-\nabla_{\mu}\delta\Gamma^{\kappa}_{\kappa\nu}\right)=0,$
(8)

and following Ref.TMT-orig-1 , its solution $\Gamma^{\mu}_{\nu\lambda}$
becomes a Levi-Civita connection

$\Gamma^{\mu}_{\nu\lambda}=\Gamma^{\mu}_{\nu\lambda}({\bar{g}})=\frac{1}{2}{\bar{g}}^{\mu\kappa}\left(\partial_{\nu}{\bar{g}}_{\lambda\kappa}+\partial_{\lambda}{\bar{g}}_{\nu\kappa}-\partial_{\kappa}{\bar{g}}_{\nu\lambda}\right)\;,$
(9)

where the Weyl-rescaled metric ${\bar{g}}_{\mu\nu}$ is given by

${\bar{g}}_{\mu\nu}=(\chi_{1}+2\epsilon\chi_{2}R)g_{\mu\nu}\;\;,\;\;\mbox{where}\,\,\,\,\chi_{1}\equiv\frac{\Phi_{1}(A)}{\sqrt{-g}}\;\;,\;\;\mbox{and}\,\,\,\,\chi_{2}\equiv\frac{\Phi_{2}(B)}{\sqrt{-g}}\;.$
(10)

On the other hand, the variation of (1) w.r.t. auxiliary tensors
$A_{\mu\nu\lambda}$, $B_{\mu\nu\lambda}$ and $H_{\mu\nu\lambda}$ becomes

$\partial_{\mu}\Bigl{[}R+L^{(1)}\Bigr{]}=0\quad,\quad\partial_{\mu}\Bigl{[}L^{(2)}+\epsilon
R^{2}+\frac{\Phi(H)}{\sqrt{-g}}\Bigr{]}=0\quad,\quad\partial_{\mu}\Bigl{(}\frac{\Phi_{2}(B)}{\sqrt{-g}}\Bigr{)}=0\;,$
(11)

whose solutions are

$\frac{\Phi_{2}(B)}{\sqrt{-g}}\equiv\chi_{2}={\rm
const}\;\;,\;\;R+L^{(1)}=-M_{1}={\rm const}\;\;,\;\;L^{(2)}+\epsilon
R^{2}+\frac{\Phi(H)}{\sqrt{-g}}=-M_{2}={\rm const}\;.$ (12)

Here the parameters $M_{1}$ and $M_{2}$ are arbitrary dimensionful and the
quantity $\chi_{2}$ corresponds to an arbitrary dimensionless integration
constant. Note that the constant $\chi_{2}$ in (12) preserves global Weyl-
scale invariance (7), whereas the appearance of the second and third
integration constants $M_{1},\,M_{2}$ implies breakdown of global Weyl-scale
invariance under (7).

Now the variation of the action (1) w.r.t. $g_{\mu\nu}$ and considering
relations (12) yields

$\chi_{1}\Bigl{[}R_{\mu\nu}+\frac{1}{2}\left(g_{\mu\nu}L^{(1)}-T^{(1)}_{\mu\nu}\right)\Bigr{]}-\frac{1}{2}\chi_{2}\Bigl{[}T^{(2)}_{\mu\nu}+g_{\mu\nu}\left(\epsilon
R^{2}+M_{2}\right)-2R\,R_{\mu\nu}\Bigr{]}=0\;,$ (13)

in which $T^{(1,2)}_{\mu\nu}(\varphi_{1},\varphi_{2})=T^{(1,2)}_{\mu\nu}$ are
the energy-momentum tensors of the scalar fields Lagrangians with the standard
definitions:

$T^{(1,2)}_{\mu\nu}=g_{\mu\nu}L^{(1,2)}-2\frac{{\partial{}}}{{\partial{g^{\mu\nu}}}}L^{(1,2)}\;.$
(14)

By considering the trace of Eq.(13) and using again second relation (12) we
find for the scale factor $\chi_{1}$ the expression

$\chi_{1}=2\chi_{2}\frac{T^{(2)}/4+M_{2}}{L^{(1)}-T^{(1)}/2-M_{1}}\;,$ (15)

where the trace $T^{(1,2)}=g^{\mu\nu}T^{(1,2)}_{\mu\nu}$.

From Eqs.(12) and (13) we obtain the Einstein-like form:

$\displaystyle
R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{1}{2}g_{\mu\nu}\left(L^{(1)}+M_{1}\right)+\frac{1}{2\Omega}\left(T^{(1)}_{\mu\nu}-g_{\mu\nu}L^{(1)}\right)+\frac{\chi_{2}}{2\chi_{1}\Omega}\Bigl{[}T^{(2)}_{\mu\nu}+g_{\mu\nu}\left(M_{2}+\epsilon(L^{(1)}+M_{1})^{2}\right)\Bigr{]}\;,$
(16)

in which the function $\Omega$ is defined as

$\Omega=1-\frac{\chi_{2}}{\chi_{1}}\,2\epsilon\left(L^{(1)}+M_{1}\right)\;.$
(17)

By combining Eqs.(10), (12) and (17), the relation between
${\bar{g}}_{\mu\nu}$ and $g_{\mu\nu}$ can be written as

${\bar{g}}_{\mu\nu}=\chi_{1}\Omega\,g_{\mu\nu}\;.$ (18)

In this way, the Eq.(16) for the rescaled metric ${\bar{g}}_{\mu\nu}$ (18),
i.e., the Einstein-frame can be rewritten as

$R_{\mu\nu}({\bar{g}})-\frac{1}{2}{\bar{g}}_{\mu\nu}R({\bar{g}})=\frac{1}{2}T^{\rm
eff}_{\mu\nu},$ (19)

where the effective energy-momentum tensor in the Einstein-frame becomes

$T^{\rm eff}_{\mu\nu}=g_{\mu\nu}L_{\rm
eff}-2\frac{{\partial{}}}{{\partial{g^{\mu\nu}}}}L_{\rm eff},$ (20)

in which the effective Einstein-frame scalar fields Lagrangian results

$L_{\rm
eff}(\varphi_{1},\varphi_{2})=\frac{1}{\chi_{1}\Omega}\Bigl{\\{}L^{(1)}+M_{1}+\frac{\chi_{2}}{\chi_{1}\Omega}\Bigl{[}L^{(2)}+M_{2}+\epsilon(L^{(1)}+M_{1})^{2}\Bigr{]}\Bigr{\\}}\;.$
(21)

For the effective Lagrangian $L_{\rm eff}$ in terms of the Einstein-frame
metric ${\bar{g}}_{\mu\nu}$ (18) we can consider the short-hand notation for
the scalar kinetic terms $X_{1}$ and $X_{2}$ such that

$X_{1}\equiv-\frac{1}{2}{\bar{g}}^{\mu\nu}\partial_{\mu}\varphi_{1}\partial_{\nu}\varphi_{1},\,\,\,\,\mbox{and}\,\,\,\,X_{2}\equiv-\frac{1}{2}{\bar{g}}^{\mu\nu}\partial_{\mu}\varphi_{2}\partial_{\nu}\varphi_{2},$
(22)

and the two Lagrangians $L^{(1,2)}$ become

$L^{(1)}=\chi_{1}\Omega\,\left[X_{1}+X_{2}\right]-V\quad,\quad
L^{(2)}=\chi_{1}\Omega\,\left[b_{1}e^{-\alpha_{1}\varphi_{1}}X_{1}+b_{2}e^{-\alpha_{2}\varphi_{2}}X_{2}\right]+U\;,$
(23)

with $V$ and $U$ given by Eq.(5).

From Eqs.(15) and (17), taking into account (3), we get

$\frac{1}{\chi_{1}\Omega}=\frac{(V-M_{1})}{2\chi_{2}\Bigl{[}U+M_{2}+\epsilon(V-M_{1})^{2}\Bigr{]}}\,\left[1-\chi_{2}\left[\left(\frac{b_{1}e^{-\alpha_{1}\varphi_{1}}}{V-M_{1}}-2\epsilon\right)X_{1}+\Bigl{(}\frac{b_{2}e^{-\alpha_{2}\varphi_{2}}}{V-M_{1}}-2\epsilon\Bigr{)}X_{2}\right]\right]\;.$
(24)

Upon substituting expression (24) into (21) we arrive at the explicit form for
the Einstein-frame scalar Lagrangian:

$L_{\rm
eff}=A_{1}(\varphi_{1},\varphi_{2})X_{1}+A_{2}(\varphi_{1},\varphi_{2})X_{2}+B_{1}(\varphi_{1},\varphi_{2})X_{1}^{2}+B_{2}(\varphi_{1},\varphi_{2})X_{2}^{2}+B_{12}(\varphi_{1},\varphi_{2})X_{1}X_{2}-U_{\rm
eff}(\varphi_{1},\varphi_{1})\;,$ (25)

where the functions $A_{1}(\varphi_{1},\varphi_{2})$ and
$A_{2}(\varphi_{1},\varphi_{2})$ are given by

$A_{1}(\varphi_{1},\varphi_{2})=1+\Bigl{[}\frac{1}{2}b_{1}e^{-\alpha_{1}\varphi_{1}}-\epsilon(V-M_{1})\Bigr{]}\frac{V-M_{1}}{U+M_{2}+\epsilon(V-M_{1})^{2}}=1+$
$\Bigl{[}\frac{1}{2}b_{1}e^{-\alpha_{1}\varphi_{1}}-\epsilon\left(f_{1}e^{-\alpha_{1}\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1}\right)\Bigr{]}\,\frac{f_{1}e^{-\alpha_{1}\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1}}{f_{2}e^{-2\alpha\varphi_{1}}+g_{2}e^{-2\alpha\varphi_{2}}+M_{2}+\epsilon(f_{1}e^{-\alpha\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1})^{2}}\;,$
(26)

and

$A_{2}(\varphi_{1},\varphi_{2})=1+\Bigl{[}\frac{1}{2}b_{2}e^{-\alpha_{2}\varphi_{2}}-\epsilon(V-M_{1})\Bigr{]}\frac{V-M_{1}}{U+M_{2}+\epsilon(V-M_{1})^{2}}$
$=1+\Bigl{[}\frac{1}{2}b_{2}e^{-\alpha_{2}\varphi_{2}}-\epsilon\left(f_{1}e^{-\alpha\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1}\right)\Bigr{]}\,\frac{f_{1}e^{-\alpha_{1}\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1}}{f_{2}e^{-2\alpha\varphi_{1}}+g_{2}e^{-2\alpha\varphi_{2}}+M_{2}+\epsilon(f_{1}e^{-\alpha\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1})^{2}}.$
(27)

The coefficient $B_{1}(\varphi_{1},\varphi_{2})$ is defined as

$\displaystyle
B_{1}(\varphi_{1},\varphi_{2})=\chi_{2}\frac{\epsilon\Bigl{[}U+M_{2}+(V-M_{1})b_{1}e^{-\alpha_{1}\varphi_{1}}\Bigr{]}-\frac{1}{4}b_{1}^{2}e^{-2\alpha\varphi_{1}}}{U+M_{2}+\epsilon(V-M_{1})^{2}}$
$\displaystyle=\chi_{2}\frac{\epsilon\Bigl{[}f_{2}e^{-2\alpha\varphi_{1}}+g_{2}e^{-2\alpha\varphi_{2}}+M_{2}+(f_{1}e^{-\alpha\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1})b_{1}e^{-\alpha_{1}\varphi_{1}}\Bigr{]}-\frac{1}{4}b_{1}^{2}e^{-2\alpha_{1}\varphi_{1}}}{f_{2}e^{-2\alpha_{1}\varphi_{1}}+g_{2}e^{-2\alpha_{2}\varphi_{2}}+M_{2}+\epsilon(f_{1}e^{-\alpha_{1}\varphi_{1}}+g_{2}e^{-\alpha_{2}\varphi_{2}}-M_{1})^{2}}\;,$
(28)

and for the function $B_{2}(\varphi_{1},\varphi_{2})$ we obtain

$\displaystyle
B_{2}(\varphi_{1},\varphi_{2})=\chi_{2}\frac{\epsilon\Bigl{[}U+M_{2}+(V-M_{1})b_{2}e^{-\alpha_{2}\varphi_{2}}\Bigr{]}-\frac{1}{4}b_{2}^{2}e^{-2\alpha\varphi_{2}}}{U+M_{2}+\epsilon(V-M_{1})^{2}}$
$\displaystyle=\chi_{2}\frac{\epsilon\Bigl{[}f_{2}e^{-2\alpha\varphi_{1}}+g_{2}e^{-2\alpha\varphi_{2}}+M_{2}+(f_{1}e^{-\alpha\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1})b_{2}e^{-\alpha_{2}\varphi_{2}}\Bigr{]}-\frac{1}{4}b_{2}^{2}e^{-2\alpha_{2}\varphi_{2}}}{f_{2}e^{-2\alpha_{1}\varphi_{1}}+g_{2}e^{-2\alpha_{2}\varphi_{2}}+M_{2}+\epsilon(f_{1}e^{-\alpha_{1}\varphi_{1}}+g_{2}e^{-\alpha_{2}\varphi_{2}}-M_{1})^{2}}\;,$
(29)

and the coefficient $B_{12}(\varphi_{1},\varphi_{2})$ becomes

$B_{12}(\varphi_{1},\varphi_{2})=\chi_{2}E_{0}\Big{[}-E_{1}b_{2}e^{-\alpha_{2}\varphi_{2}}-E_{2}b_{1}e^{-\alpha_{1}\varphi_{1}}+2E_{0}E_{1}E_{2}[(M_{2}+U)+$
$\epsilon(M_{1}-V)^{2}]+2\epsilon[(1/E_{0})-(E_{1}+E_{2})(M_{1}-V)]\Big{]},$
(30)

where the quantities $E_{0}$, $E_{1}$ and $E_{2}$ are defined as

$E_{0}=\frac{(V-M_{1})}{2\chi_{2}[U+M_{2}+\epsilon(V-M_{1})^{2}]},\,\,\,E_{1}=\chi_{2}\left[\frac{b_{1}e^{-\alpha_{1}\varphi_{1}}}{V-M_{1}}-2\epsilon\right],\,\,\,\mbox{and}\,\,\,\,E_{2}=\chi_{2}\left[\frac{b_{2}e^{-\alpha_{2}\varphi_{2}}}{V-M_{1}}-2\epsilon\right].$

The effective scalar field potential as a function of the scalar fields yields

$U_{\rm
eff}(\varphi_{1},\varphi_{2})=\frac{(V-M_{1})^{2}}{4\chi_{2}\Bigl{[}U+M_{2}+\epsilon(V-M_{1})^{2}\Bigr{]}}=$
$\frac{(f_{1}e^{-\alpha_{1}\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1})^{2}}{4\chi_{2}\,\Bigl{[}f_{2}e^{-2\alpha_{1}\varphi_{1}}+g_{2}e^{-2\alpha_{2}\varphi_{2}}+M_{2}+\epsilon(f_{1}e^{-\alpha_{1}\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1})^{2}\Bigr{]}}\;,$
(31)

where we have used for $V$ and $U$, the expressions given by Eq.(5).

##  III Three Flat Regions from an Effective Scalar Potential

The crucial feature of $U_{\rm eff}(\varphi_{1},\varphi_{2})$ is the presence
of three infinitely large flat regions. In this sense, we have one for large
positive values of the scalar fields $\varphi_{1}$ and $\varphi_{2}$ and two
others for the limits $\varphi_{1}\rightarrow-\infty$ and
$\varphi_{2}\rightarrow-\infty$.

For large negative values of $\varphi_{1}$, which we will choose to describe
the very early phase of the universe, meaning the emergent phase and inflation
we have for the effective potential and the coefficient functions in the
Einstein-frame scalar Lagrangian (25)-(31):

$U_{\rm eff}(\varphi_{1},\varphi_{2})\simeq U_{\rm
eff}{(-\infty,\varphi_{2})}=U_{eff}=\frac{f_{1}^{2}/f_{2}}{4\chi_{2}(1+\epsilon
f_{1}^{2}/f_{2})}\;,\\\ $ (32) $A_{1}(\varphi_{1},\varphi_{2})\simeq
A_{1}{(-\infty,\varphi_{2})}=A_{1}=\frac{1+\frac{1}{2}b_{1}f_{1}/f_{2}}{1+\epsilon
f^{2}_{1}/f_{2}}\;\;,\;\;B_{1}(\varphi_{1},\varphi_{2})\simeq
B_{1}{(-\infty,\varphi_{2})}=B_{1}=-\chi_{2}\frac{b_{1}^{2}/4f_{2}-\epsilon(1+b_{1}f_{1}/f_{2})}{1+\epsilon
f^{2}_{1}/f_{2}}\;.$ (33)

For the terms $A_{2}$ and $B_{2}$ in the limit in which
$\varphi_{1}\rightarrow-\infty$ we have

$A_{2}(\varphi_{1},\varphi_{2})\simeq
A_{2}{(-\infty,\varphi_{2})}=A_{2}=\frac{1}{1+\epsilon
f^{2}_{1}/f_{2}},\,\,\,\mbox{and}\,\,\,\,\,B_{2}(\varphi_{1},\varphi_{2})\simeq
B_{2}{(-\infty,\varphi_{2})}=B_{2}=\frac{\chi_{2}\epsilon}{1+\epsilon
f^{2}_{1}/f_{2}}.$ (34)

For the coefficient $B_{12}(\varphi_{1},\varphi_{2})$ in the limit in which
the scalar field $\varphi_{1}\rightarrow-\infty$ becomes

$B_{12}(\varphi_{1},\varphi_{2})\simeq
B_{12}{(-\infty,\varphi_{2})}=B_{12}=\chi_{2}\tilde{E}_{0}\Big{[}-\tilde{E}_{2}b_{1}+\tilde{E}_{0}f_{2}+\frac{2\epsilon}{\tilde{E}_{0}}+\epsilon\tilde{E}_{0}f_{1}^{2}+2\epsilon
f_{1}(\tilde{E}_{1}+\tilde{E}_{2})\Big{]},$ (35)

where

$\tilde{E}_{0}=\frac{f_{1}}{2\chi_{2}[f_{2}+\epsilon
f_{1}^{2}]},\,\,\,\,\tilde{E}_{1}=\chi_{2}\left[\frac{b_{1}}{f_{1}}-2\epsilon\right],\,\,\mbox{and}\,\,\,\,\tilde{E}_{2}=-2\chi_{2}\epsilon.$

For the second flat region we will consider that the scalar field
$\varphi_{2}\rightarrow-\infty$ such that the effective potential in the
second flat region is given by

$U_{\rm eff}(\varphi_{1},\varphi_{2})\simeq U_{\rm
eff}{(\varphi_{1},-\infty)}=U_{\rm
eff\,g}=\frac{g_{1}^{2}/g_{2}}{4\chi_{2}(1+\epsilon g_{1}^{2}/g_{2})}\;,\\\ $
(36)

and the kinetic coefficients $A_{2}$ and $B_{2}$ in the limit in which
$\varphi_{2}\rightarrow-\infty$ result

$A_{2}{(\varphi_{1},-\infty)}=A_{2g}=\frac{1+\frac{1}{2}b_{2}g_{1}/g_{2}}{1+\epsilon
g^{2}_{1}/g_{2}},\;\;B_{2}{(\varphi_{1},-\infty)}=B_{2g}=-\chi_{2}\frac{b_{2}^{2}/4g_{2}-\epsilon(1+b_{2}g_{1}/g_{2})}{1+\epsilon
g^{2}_{1}/g_{2}}\;,$ (37)

and the terms $A_{1}$ and $B_{1}$ in this limit result

$A_{1}(\varphi_{1},\varphi_{2})\simeq
A_{1}{(\varphi_{1},-\infty)}=A_{1g}=\frac{1}{1+\epsilon
g^{2}_{1}/g_{2}}\;\;,\;\;B_{1}(\varphi_{1},\varphi_{2})\simeq
B_{1}{(\varphi_{1},-\infty)}=B_{1g}=\frac{\chi_{2}\epsilon}{1+\epsilon
g^{2}_{1}/g_{2}}\;.$ (38)

For the coefficient $B_{12}(\varphi_{1},\varphi_{2})$ in the limit in which
the scalar field $\varphi_{2}\rightarrow-\infty$ we have

$B_{12}(\varphi_{1},\varphi_{2})\simeq
B_{12}{(\varphi_{1},-\infty)}=B_{12g}=\chi_{2}\tilde{E}_{3}\Big{[}-\tilde{E}_{2}b_{2}+\tilde{E}_{3}g_{2}+\frac{2\epsilon}{\tilde{E}_{3}}+\epsilon\tilde{E}_{3}g_{1}^{2}+2\epsilon
g_{1}(\tilde{E}_{4}+\tilde{E}_{2})\Big{]},$ (39)

where

$\tilde{E}_{3}=\frac{g_{1}}{2\chi_{2}[g_{2}+\epsilon
g_{1}^{2}]},\,\,\,\mbox{and}\,\,\,\,\,\tilde{E}_{4}=\chi_{2}\left[\frac{b_{2}}{g_{1}}-2\epsilon\right].$

In the third flat region for large positive $\varphi_{1}$ and also
$\varphi_{2}$, we find that the effective potential reduces to

$U_{\rm eff}(\varphi_{1},\varphi_{2})\simeq U_{\rm
eff}{(+\infty,+\infty)}=U_{\rm
eff(+)}=\frac{M_{1}^{2}/M_{2}}{4\chi_{2}(1+\epsilon M_{1}^{2}/M_{2})}\;,\\\ $
(40)

and the kinetic coefficients are

$A_{1}(\varphi_{1},\varphi_{2})=A_{2}(\varphi_{1},\varphi_{2})\simeq
A_{(+)}\equiv\frac{M_{2}}{M_{2}+\epsilon M_{1}^{2}}\quad,\quad
B_{1}(\varphi_{1},\varphi_{2})=B_{2}(\varphi_{1},\varphi_{2})\simeq
B_{(+)}\equiv\epsilon\chi_{2}\frac{M_{2}}{M_{2}+\epsilon M_{1}^{2}}\;,$ (41)

and for this limit we find that the coefficient
$B_{12}(\varphi_{1},\varphi_{2})$ becomes

$B_{12}(\varphi_{1},\varphi_{2})=B_{12}(+\infty,+\infty)=B_{12(+)}=2\epsilon\chi_{2}\frac{M_{2}}{M_{2}+\epsilon
M_{1}^{2}}=2\,B_{(+)}.$ (42)

We will consider that the flat region (32) corresponds to the evolution of the
early universe (emergent and inflation). On the other hand, the flat regions
(36) and (40) concern to the evolution of the late universe with a two phase
structure.

In particular, if we assume the order of magnitude of the coupling parameters
in the effective potential given by Eq.(32), are $f_{1}\sim
f_{2}\sim(10^{-2}M_{Pl})^{4}$, then the order of magnitude of the vacuum
energy density of the early universe during the inflationary epoch yields

$U_{\rm eff}{(-\infty,\varphi_{2})}=U_{eff}\sim f_{1}^{2}/f_{2}\sim
10^{-8}M_{Pl}^{4}\;,$ (43)

here we have assumed that the parameter $\epsilon$ is small and the
integration constant $\chi_{2}\sim\mathcal{O}(1)$.

In order to study the evolution of the universe from an emergent and
inflationary scenarios to dark epoch, we consider that the standard Friedman-
Lemaitre-Robertson-Walker space-time metric is given by

$ds^{2}=-dt^{2}+a^{2}(t)\Bigl{[}\frac{dr^{2}}{1-Kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\Bigr{]},$ (44)

where $a(t)$ denotes the scale factor and $K$ corresponds to the space
curvature.

By assuming that the matter is described by a perfect fluid with an energy
density and pressure $\rho$ and $p$, we have that the associated Friedmann
equations are

$\frac{\stackrel{{\scriptstyle..}}{{a}}}{a}=-\frac{1}{12}(\rho+3p),\quad
H^{2}+\frac{K}{a^{2}}=\frac{1}{6}\rho,\;\;\;\,\,\,\mbox{and}\,\,\,\,\;\,\;\dot{\rho}+3H(\rho+p)=0,$
(45)

where $H=\frac{\stackrel{{\scriptstyle.}}{{a}}}{a}$ is the Hubble parameter.
Also, here the energy density and pressure associated to the scalar fields
$\varphi_{1}=\varphi_{1}(t)$ and $\varphi_{2}=\varphi_{2}(t)$ are defined as

$\rho=A_{1}(\varphi_{1},\varphi_{2})X_{1}+A_{2}(\varphi_{1},\varphi_{2})X_{2}+3B_{1}(\varphi_{1},\varphi_{2})X_{1}^{2}+3B_{2}(\varphi_{1},\varphi_{2})X_{2}^{2}+3B_{12}(\varphi_{1},\varphi_{2})X_{1}X_{2}+U_{\rm
eff}(\varphi_{1},\varphi_{2})\;,$ (46)

and

$p=A_{1}(\varphi_{1},\varphi_{2})X_{1}+A_{2}(\varphi_{1},\varphi_{2})X_{2}+B_{1}(\varphi_{1},\varphi_{2})X_{1}^{2}+B_{2}(\varphi_{1},\varphi_{2})X_{2}^{2}+B_{12}(\varphi_{1},\varphi_{2})X_{1}X_{2}-U_{\rm
eff}(\varphi_{1},\varphi_{2})\;.$ (47)

Henceforth the dots indicate derivatives with respect to the time $t$ and we
have assumed that the scalar fields are homogeneous.

##  IV Non-Singular Emergent Universe

In this section we will exhibit that under specific conditions on the
parameters-space there exist an era preceding the inflationary epoch. In this
sense, we obtain a cosmological solution of the Einstein-frame system in our
model from Lagrangian (25) describing a non-singular “emergent universe” emerg
; emerg2  when one the scalar field $\varphi_{1}$ evolves on the first flat
region for large negative, where the effective potential is described by
Eq.(5). We mention that for previous analysis of “emergent universe” epoch
considering modified-measure gravity-matter theories with one non-Riemannian
and one standard Riemannian integration measures but with one scalar field was
studied in ourquintessence , see also Refs.TMT-recent-1-a ; TMT-recent-1-c .

The emergent universe, is characterized through the standard Friedmann-
Lemaitre-Robertson-Walker space-time metric (44) as a solution of Eq.(45)
under the condition on the Hubble parameter $H=0$ such that

$a(t)=a_{0}={\rm
const}\,,\;\;\rho+3p=0\quad,\quad\mbox{and}\,\,\,\,\frac{K}{a_{0}^{2}}=\frac{1}{6}\rho~{}(={\rm
const})\;,$ (48)

with $\rho$ and $p$ given by Eqs. (46)-(47). From Eq.(48) and noticing that
the combination
$\rho-3p=-2(A_{1}(\varphi_{1},\varphi_{2})X_{1}+A_{2}(\varphi_{1},\varphi_{2})X_{2})+4U_{\rm
eff}(\varphi_{1},\varphi_{2})$ does not contain the non linear terms and since
$\rho-3p=2\rho-(\rho+3p)=\frac{12K}{a_{0}^{2}}$, then we can find that the
relation between $X_{10}$ and $X_{20}$, is given by

$X_{20}=\frac{\dot{\varphi}_{20}^{2}}{2}=\frac{C_{0}-A_{1}\,X_{10}}{A_{2}}>0,$
(49)

where $C_{0}=2U_{eff}-\frac{6K}{a_{0}^{2}}$. Here we note that the parameter
$C_{0}>0$, since $A_{1}$ and $A_{2}$ are positive and $C_{0}>A_{1}X_{10}$. In
this sense, in multi-fields model the spatial curvature of the universe cannot
be too big, as opposed to the single field TMT model studied before in
Gravityassisted , where no such restriction was found.

Using this we now find that the emergent universe condition (48) implies that
the $\varphi_{1}$-velocity
$\stackrel{{\scriptstyle.}}{{\varphi}}_{1}\equiv\stackrel{{\scriptstyle.}}{{\varphi}}_{10}$
and $\varphi_{2}$-velocity
$\stackrel{{\scriptstyle.}}{{\varphi}}_{2}\equiv\stackrel{{\scriptstyle.}}{{\varphi}}_{20}$
are time-independent and the kinetic term $X_{10}=\dot{\varphi}_{10}^{2}/2$
satisfies the algebraic equation:

$X_{10}^{2}[B_{1}+B_{2}A_{1}^{2}/A_{2}^{2}-B_{12}A_{1}/A_{2}]+X_{10}[-6B_{2}A_{1}C_{0}/A_{2}^{2}+B_{12}C_{0}/A_{2}]+[2C_{0}/3+B_{2}^{2}C_{0}^{2}/A_{2}^{2}-U_{eff}/3]=0.$
(50)

Defining the quantities $F_{1}$,$F_{2}$ and $F_{3}$ as
$F_{1}=B_{1}+B_{2}A_{1}^{2}/A_{2}^{2}-B_{12}A_{1}/A_{2}$,
$F_{2}=-6B_{2}A_{1}C_{0}/A_{2}^{2}+B_{12}C_{0}/A_{2}$ and
$F_{3}=2C_{0}/3+B_{2}^{2}C_{0}^{2}/A_{2}^{2}-U_{eff}/3$, the solution of the
kinetic term $X_{01}$ yields

$X_{10}=\frac{1}{2F_{1}}\Bigr{[}-F_{2}\pm\sqrt{F_{2}^{2}-4F_{1}F_{3}}\Bigr{]}=\frac{1}{2}\left[-\frac{F_{2}}{F_{1}}\pm\sqrt{\left(\frac{F_{2}}{F_{1}}\right)^{2}-\frac{4F_{3}}{F_{1}}}\right]>0\;.$
(51)

Here the ratio $F_{2}/F_{1}$ becomes

$\frac{F_{2}}{F_{1}}=-C_{0}\,\left(\frac{-A_{2}B_{12}+6A_{1}B_{2}}{A_{2}^{2}B_{1}-A_{1}A_{2}B_{12}+A_{1}^{2}B_{2}}\right),$

and assuming that the parameter $\epsilon\ll 1$, then we have that the
coefficients $A_{1}>0$, $A_{2}>0$, $B_{1}<0$, $B_{2}\sim 0$ and $B_{12}>0$
with which the ratio $F_{2}/F_{1}$ is negative. Also, we find that the
quantity

$\left(\frac{F_{2}}{F_{1}}\right)^{2}-\frac{4F_{3}}{F_{1}}=\frac{4(A_{2}^{2}B_{1}-A_{1}A_{2}B_{12}+A_{1}^{2}B_{2})[A_{2}^{2}(12k/a_{0}^{2}-3U_{eff})-3B_{2}^{2}C_{0}^{2}]+3(A_{2}B_{12}-6A_{1}B_{2})^{2}C_{0}^{2}}{3A_{2}^{4}\left(B_{1}+\frac{A_{1}(A_{1}B_{2}-A_{2}B_{12})}{A_{2}^{2}}\right)^{2}}.$

As before considering that the parameter $\epsilon\ll 1$, we find the the
quantity $(F_{2}/F_{1})^{2}-4F_{3}/F_{1}$ is positive. Here we have considered
that $3U_{eff}/2>6k/a_{0}^{2}$ i.e., $C_{0}>2k/a_{0}^{2}$. Also, we note that
the term $F_{3}/F_{1}$ is a negative quantity for $\epsilon\ll 1$, then in the
following we will consider the positive sign of Eq.(51). Thus, we require that
these solutions for $X_{10}$ and $X_{20}$ be reals and positives.

In order to study the stability of this solution, we can analyze the
perturbation associated to the energy density $\delta\rho$ in which

$\frac{\delta\stackrel{{\scriptstyle..}}{{a}}}{a_{0}}+\frac{1}{12}(\delta\rho+3\delta
p)=0\,\,\,,\quad\mbox{with}\,\,\,\,\,\,\,\delta\rho=-\frac{2\rho_{0}}{a_{0}}\delta
a,\,$ (52)

where we need to express $\delta\rho$ in terms of $\delta\varphi_{1}$ and
$\delta\varphi_{2}$. Notice that due to the shift symmetries that exist in the
flat region, $\varphi_{1}\rightarrow\varphi_{1}+constant$ and
$\varphi_{2}\rightarrow\varphi_{2}+constant$, we have conserved quantities
that should be preserved in these perturbations, and this ends up providing us
with an important relation between $\delta\varphi_{1}$ and
$\delta\varphi_{2}$. The conservation law related to the symmetry
$\varphi_{1}\rightarrow\varphi_{1}+constant$ is

$C_{1}=a^{3}\left[A_{1}\dot{\varphi}_{1}+B_{12}\dot{\varphi}_{1}X_{2}+2B_{1}\dot{\varphi}_{1}X_{1}\right],$
(53)

and the conservation law associated to the symmetry
$\varphi_{2}\rightarrow\varphi_{2}+constant$ results

$C_{2}=a^{3}\left[A_{2}\dot{\varphi}_{2}+B_{12}\dot{\varphi}_{2}X_{1}+2B_{2}\dot{\varphi}_{2}X_{2}\right],$
(54)

where $C_{1}$ and $C_{2}$ are two arbitrary constants. Also, the quantities
$a_{0}$ and $\rho_{0}$ are defined as

$a_{0}^{2}=\frac{6K}{\rho_{0}}\quad,\quad\rho_{0}=A_{1}X_{10}+A_{2}X_{20}+3B_{1}X_{10}^{2}+3B_{2}X_{20}^{2}+3B_{12}X_{10}X_{20}+U_{\rm
eff}(-\infty,\varphi_{2})\;,$ (55)

with $X_{10}$ and $X_{20}$ given by Eqs.(49) and (51).

On the other hand, we perturb Friedmann equations given by (45) and the
expressions for $\rho,\,p$ (46)-(47) w.r.t. $a(t)=a_{0}+\delta a(t)$ and
$\stackrel{{\scriptstyle.}}{{\varphi}}_{1}(t)=\stackrel{{\scriptstyle.}}{{\varphi}}_{10}+\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{1}(t)$,
$\stackrel{{\scriptstyle.}}{{\varphi}}_{2}(t)=\stackrel{{\scriptstyle.}}{{\varphi}}_{20}+\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{2}(t)$
but keep the effective potential on the flat region in which
$\varphi_{1}\rightarrow-\infty$, $U_{\rm eff}=U(-\infty,\varphi_{2})$:

$\frac{\delta\stackrel{{\scriptstyle..}}{{a}}}{a_{0}}+\frac{1}{12}(\delta\rho+3\delta
p)=0\quad,\quad\delta\rho=-\frac{2\rho_{0}}{a_{0}}\delta a\;,$ (56)
$\delta\rho=\tilde{A}\,\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{1}+\tilde{B}\,\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{2}=-\frac{2\rho_{0}}{a_{0}}\delta
a,$ (57)

where the quantities $\tilde{A}$ and $\tilde{B}$ are given by

$\tilde{A}=\left(A_{1}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}+3B_{1}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}^{3}+\frac{3}{2}B_{12}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}^{2}\right),\,\,\,\mbox{and}\,\,\,\tilde{B}=\left(A_{2}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}+3B_{2}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}^{3}+\frac{3}{2}B_{12}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}^{2}\right)\;\;,\;\;$
(58)

and for the perturbation associated to pressure we have

$\delta
p=\tilde{C}\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{1}+\tilde{D}\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{2},$
(59)

with $\tilde{C}$ and $\tilde{D}$ are defined as

$\tilde{C}=\left(A_{1}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}+B_{1}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}^{3}+\frac{1}{2}B_{12}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}^{2}\right),\,\,\,\mbox{and}\,\,\,\,\tilde{D}=\left(A_{2}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}+B_{2}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}^{3}+\frac{1}{2}B_{12}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}^{2}\right).$
(60)

Also, the perturbation equation for the scalar fields from Eq.(53) becomes

$\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{2}=D_{0}\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{1}+D_{1}\delta
a,$ (61)

where the coefficients $D_{3}$ and $D_{4}$ are defined

$D_{0}=-\Big{[}\frac{A_{1}+B_{12}X_{20}+6B_{1}X_{10}}{B_{12}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}}\Big{]},\,\,\,\,\,\mbox{and}\,\,\,\,\,\,D_{1}=-\frac{3}{a_{0}}\Big{[}\frac{A_{1}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}+B_{12}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}X_{20}+B_{1}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}X_{10}}{B_{12}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}}\Big{]}.$

However, from Eq.(54) we have

$\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{2}=D_{2}\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{1}+D_{3}\delta
a,$ (62)

where the terms $D_{2}$ and $D_{3}$ are given by

$D_{2}=-\Big{[}\frac{B_{12}\stackrel{{\scriptstyle.}}{{\varphi}}_{10}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}}{A_{2}+B_{12}X_{10}+6B_{2}X_{20}}\Big{]},\,\,\,\,\,\mbox{and}\,\,\,\,\,\,D_{3}=-\frac{3}{a_{0}}\Big{[}\frac{A_{2}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}+B_{12}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}X_{10}+B_{2}\stackrel{{\scriptstyle.}}{{\varphi}}_{20}X_{20}}{A_{2}+B_{12}X_{10}+6B_{2}X_{20}}\Big{]}.$

In order to have consistency between Eqs.(61) and (62) we must have
$D_{0}=D_{2}$ and $D_{1}=D_{3}$, respectively.

By expressing $\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{1}$ and
$\delta\stackrel{{\scriptstyle.}}{{\varphi}}_{2}$ in terms of the $\delta a$
and substituting into the first Eq.(56) we get a harmonic oscillator type
equation for $\delta a$:

$\delta\stackrel{{\scriptstyle..}}{{a}}+\omega^{2}\delta
a=0\quad,\quad\omega^{2}=\frac{1}{12(\tilde{A}+\tilde{B}D_{0})}\left[(\tilde{B}a_{0}D_{1}-2\rho_{0})(\tilde{A}+3\tilde{C})+a_{0}D_{1}\tilde{A}+2D_{0}\tilde{B}a_{0}D_{1}-2D_{0}\rho_{0}\right]\;.$
(63)

To obtain the stability of the emergent universe solution we need to consider
the parameter space for which $\omega^{2}>0$. Here we have that
$(\tilde{A}+\tilde{B}D_{0})\neq 0$.

Figure 1:  The figure depicts the frequency square as a function of the
parameter $b_{1}$, for different values of $f_{1}$ and $f_{2}$. Here we have
fixed the values $\chi_{2}=1$, $\epsilon=10^{-4}$ (in units of $M_{Pl}^{-2}$)
and $\rho_{0}=(3/2)U_{\rm eff}$.

Because of the complexity of the above expression for $\omega^{2}$, we perform
a numerical analysis of the dependence of $\omega^{2}$ versus the parameter
$b_{1}$ which has been plotted in Fig.1. In this context, in Fig.1 we show the
dependence of the frequency square in terms of the parameter $b_{1}$, for
various values of the parameters $f_{1}$ and $f_{2}$. In order to satisfy the
stable emergent universe solution, the parameter $b_{1}$ is negative and it
has to be in the interval $-2<b_{1}<0$. Here we observe that this range for
the parameter $b_{1}$ does not depend of the values of $f_{1}$ and $f_{2}$.

##  V  Inflationary epoch

In this section we will analyze the inflationary universe in the situation in
which the scalar field $\varphi_{1}\rightarrow-\infty$ where the effective
potential corresponds to first flat region and the kinetic coefficients are
given by Eqs.(33)-(35).

The full equations for the scalar fields $\varphi_{1}$ and $\varphi_{2}$ are
given by

$\ddot{\varphi}_{1}\left(A_{1}+6B_{1}X_{1}+B_{12}X_{2}\right)+3H\dot{\varphi}_{1}\left[A_{1}+2B_{1}X_{1}+B_{12}X_{2}+\frac{B_{12}}{3H}\dot{\varphi_{2}}\ddot{\varphi}_{2}\right]$
$=-U_{eff,\varphi_{1}}+A_{1,\varphi_{1}}X_{1}+A_{2,\varphi_{1}}X_{2}+B_{1,\varphi_{1}}X_{1}^{2}+B_{2,\varphi_{1}}X_{2}^{2}+B_{12,\varphi_{1}}X_{1}\,X_{2},$
(64)

and

$\ddot{\varphi}_{2}\left(A_{2}+6B_{2}X_{2}+B_{12}X_{1}\right)+3H\dot{\varphi}_{2}\left[A_{2}+2B_{2}X_{2}+B_{12}X_{1}+\frac{B_{12}}{3H}\dot{\varphi_{1}}\ddot{\varphi}_{1}\right]$
$=-U_{eff,\varphi_{2}}+A_{2,\varphi_{2}}X_{2}+A_{1,\varphi_{2}}X_{1}+B_{2,\varphi_{2}}X_{1}^{2}+B_{1,\varphi_{2}}X_{1}^{2}+B_{12,\varphi_{2}}X_{1}\,X_{2},$
(65)

where the notation $,\varphi_{1}$ and $,\varphi_{2}$ denote
$\partial/\partial\varphi_{1}$ and $\partial/\partial\varphi_{2}$,
respectively.

Introducing the “slow-roll” parameters for the scalar fields slow-roll-param

$\varepsilon_{H}=-\frac{\stackrel{{\scriptstyle.}}{{H}}}{H^{2}},\quad\eta_{1}=-\frac{\stackrel{{\scriptstyle..}}{{\varphi}}_{1}}{H\stackrel{{\scriptstyle.}}{{\varphi}}_{1}}\;,\,\,\,\,\,\mbox{and}\,\,\,\,\,\,\eta_{2}=-\frac{\stackrel{{\scriptstyle..}}{{\varphi}}_{2}}{H\stackrel{{\scriptstyle.}}{{\varphi}}_{2}},$
(66)

and assuming that these parameters are $\varepsilon_{H}$, $\eta_{1}$ and
$\eta_{2}\ll 1$, then we can ignore the terms $\ddot{\varphi}_{1,2}$ and non-
linear, such that the Eqs.(64) and (65) together with the Friedmann equation
reduce to

$3A_{1}H\stackrel{{\scriptstyle.}}{{\varphi}}_{1}+U_{eff,\varphi_{1}}\simeq
0,\,\,\,\,\,\,3A_{2}H\stackrel{{\scriptstyle.}}{{\varphi}}_{2}+U_{eff,\varphi_{2}}\simeq
0,\,\,\,\mbox{and}\,\quad H^{2}\simeq\frac{1}{6}U_{\rm eff}\;.$ (67)

From Eq.(31) and considering the region in which
$f_{1}e^{-\alpha_{1}\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}}\gg M_{1}$ and
$f_{2}e^{-2\alpha_{1}\varphi_{1}}+g_{2}e^{-2\alpha_{2}\varphi_{2}}\gg M_{2}$,
the effective potential becomes

$U_{eff}(\varphi_{1},\varphi_{2})=\frac{(f_{1}e^{-\alpha_{1}\,\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}})^{2}}{4\chi_{2}(f_{2}e^{-2\alpha_{1}\varphi_{1}}+g_{2}e^{-2\alpha_{2}\varphi_{2}}+\epsilon(f_{1}e^{-\alpha_{1}\varphi_{1}}+g_{1}e^{-\alpha_{2}\varphi_{2}})^{2})}.$
(68)

Note that from the above expression we have that the effective potential can
be rewritten in term of a single scalar field $\hat{\phi}$ in which

$U_{eff}(\hat{\phi})=\frac{(f_{1}e^{-\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}\hat{\phi}}+g_{1})^{2}}{4\chi_{2}(f_{2}e^{-2\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}}\hat{\phi}+g_{2}+\epsilon(f_{1}e^{-\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}\hat{\phi}}+g_{1})^{2})},$
(69)

where

$\hat{\phi}=\frac{\alpha_{1}\varphi_{1}-\alpha_{2}\varphi_{2}}{\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}}.$

For large negative value of the scalar field $\varphi_{1}$, which means also
that $\hat{\phi}$ is also large and negative we have that we can neglect
$g_{1}$ and $g_{2}$ in the denominator and keeping the leading contribution
beyond the constant $\frac{f_{1}^{2}}{4\chi_{2}(f_{2}+\epsilon f_{1}^{2})}$ of
the infinite plateau, we get that the effective potential becomes (going back
to the $\varphi_{1},\varphi_{2}$ variables)

$U_{eff}\simeq\left(\frac{f_{1}^{2}+2f_{1}g_{1}e^{\sqrt{\alpha_{1}^{2}+\alpha_{2}^{2}}\hat{\phi}}}{4\chi_{2}(f_{2}+\epsilon
f_{1}^{2})}\right)=\left(\frac{f_{1}^{2}+2f_{1}g_{1}e^{\alpha_{1}\varphi_{1}-\alpha_{2}\varphi_{2}}}{4\chi_{2}(f_{2}+\epsilon
f_{1}^{2})}\right)\;.$ (70)

The potential (69) interpolates between the vacuums with asymptotic values
$\frac{f_{1}^{2}}{4\chi_{2}(f_{2}+\epsilon f_{1}^{2})}$ and
$\frac{g_{1}^{2}}{4\chi_{2}(g_{2}+\epsilon g_{1}^{2})}$ independent of the
choice of the relative signs of $g_{1}$, and $f_{1}$. Beyond, this the more
detailed structure of the potential is different if the relative sign is the
same or if it is opposite. In the case signs of $g_{1}$, and $f_{1}$ are the
same, the potential does not have a minimum where the potential is zero, and
although the potential still connects the two vacuum, the potential shows a
bump (as it is apparent also by the fact that the $f_{1}g_{1}$ term in (70)
raises the energy density over the energy density of the plateau ) that pushes
slow roll solutions in the opposite to the desired direction, i.e., not in the
direction necessary to interpolate with the desired vacuum, If the case the
signs of $g_{1}$, and $f_{1}$ are the opposite is considered, the potential
does have a minimum where the potential is zero, and now the potential does
not show a bump and the slow roll solutions causes the scalar field to move to
the desired direction, i.e., in the direction necessary to interpolate with
the desired vacuum. We can choose therefore the case where the signs of
$g_{1}$, and $f_{1}$ are the opposite. In the following, we will choose the
parameter $f_{1}>0$ and then the parameter $g_{1}$ is negative.

Now, in order to study the inflationary epoch, we will consider the slow roll
approximation. In this framework, from Eq.(67) we find that the relation
between the scalar field $\varphi_{1}$ and $\varphi_{2}$, under the slow roll
approximation yields

$\varphi_{1}=-\left(\frac{A_{2}\alpha_{1}}{A_{1}\alpha_{2}}\right)\varphi_{2}+C,$
(71)

where $C$ denotes an integration constant and here we have used the effective
potential given by Eq.(70).

By defining two new scalar fields $\phi_{1}$ and $\phi_{2}$ as a linear
combination of the $\varphi_{1}$ and $\varphi_{2}$, such that we have a
transformation orthogonal between these fields

$\phi_{1}=\frac{A_{1}\alpha_{2}\varphi_{1}+A_{2}\alpha_{1}\varphi_{2}}{\sqrt{(A_{1}\alpha_{2})^{2}+(A_{2}\alpha_{1})^{2}}},\,\,\,\,\mbox{and}\,\,\,\,\,\phi_{2}=\frac{A_{1}\alpha_{2}\varphi_{1}-A_{2}\alpha_{1}\varphi_{2}}{\sqrt{(A_{1}\alpha_{2})^{2}+(A_{2}\alpha_{1})^{2}}}.$
(72)

Note that from Eq.(71) the new scalar field $\phi_{1}$ becomes an arbitrary
constant, such that $\dot{\phi_{1}}=0$. Thus we can rewrite the new field
$\phi_{2}$ in terms of the $\varphi_{2}$ as

$\phi_{2}=\frac{C-2\Big{(}\frac{A_{2}\alpha_{1}}{A_{1}\alpha_{2}}\Big{)}\varphi_{2}}{\sqrt{\Big{(}\frac{A_{2}\alpha_{1}}{A_{1}\alpha_{2}}\Big{)}^{2}+1}}.$
(73)

In this way the effective potential can be write as a function of the new
scalar field $\phi_{2}$ as

$U_{eff}(\phi_{2})=\left(\frac{f_{1}^{2}+2f_{1}g_{1}e^{\delta_{0}\phi_{2}+\delta_{1}}}{4\chi_{2}f_{2}(1+\epsilon
f_{1}^{2}/f_{2})}\right)\;,$ (74)

where the constants $\delta_{0}$ and $\delta_{1}$ are given by

$\delta_{0}=d_{0}d_{1},\,\,\,\,\mbox{and}\,\,\,\delta_{1}=C[\alpha_{1}-d_{0}],\,\,\,\,\mbox{where}\,\,\,d_{0}=\frac{A_{1}\alpha_{2}^{2}}{2A_{2}\alpha_{1}}\Big{[}\frac{A_{2}\alpha_{1}^{2}}{A_{1}\alpha_{2}^{2}}+1\Big{]},\,\,\,\,\,\mbox{and}\,\,\,\,\,d_{1}=\sqrt{\Big{(}\frac{A_{2}\alpha_{1}}{A_{1}\alpha_{2}}\Big{)}^{2}+1}\,.$

Notice that the shift $\phi_{2}\rightarrow\phi_{2}+\Delta$ has the same effect
of just changing the value of $g_{1}$ according to $g_{1}\rightarrow
g_{1}e^{\delta_{0}\Delta}$, so, we would expect physical quantities not to
depend on $g_{1}$, except for its sign, if we restrict to inflation
observables. Independence of the absolute value of $g_{1}$ of physical
quantities is equivalent to the independence of physical quantities under the
shifts $\phi_{2}\rightarrow\phi_{2}+\Delta$, that is that we do not care from
where in the infinite plateau we start the slow roll inflation, invariance of
physical quantities under $\phi_{2}\rightarrow\phi_{2}+\Delta$ also means that
the integration $C$ in (73) should not affect any physical quantity relevant
to the inflationary period.

Under slow roll approximation the motion equation for the new field becomes

$3\,H\,A_{2}\,\dot{\phi}_{2}\simeq-\frac{4\Big{(}\frac{A_{2}\alpha_{1}}{A_{1}\alpha_{2}}\Big{)}^{2}}{d_{1}^{2}}\,U_{eff,\phi_{2}}.$
(75)

From Eqs.(66) and (75), we find that the slow roll parameter $\epsilon_{H}$
results

$\epsilon_{H}\simeq\frac{1}{A}\left(\frac{U_{eff,\phi_{2}}}{U_{eff}}\right)^{2}$
(76)

where the constant $A$ is defined as

$A=\frac{A_{2}\,d_{1}^{2}}{4\left(\frac{A_{2}\alpha_{1}}{A_{1}\alpha_{2}}\right)^{2}}.$

Note that the two second slow roll parameters
$\eta_{1}=\eta_{2}=\eta=-\ddot{\phi_{2}}/{(H\dot{\phi_{2}})}$ are equivalent,
since $\dot{\varphi_{1}}\propto\dot{\varphi_{2}}\propto{\dot{\phi_{2}}}$.

Therefore, we obtain that the slow roll parameter $\epsilon_{H}$ can be
written as

$\epsilon_{H}\simeq\frac{16g_{1}^{2}\delta_{0}^{2}\,e^{2(\delta_{0}\,\phi_{2}+\delta_{1})}}{Af_{1}^{2}}=\Big{(}\frac{16g_{1}^{2}\alpha_{2}^{2}}{A_{2}f_{1}^{2}}\Big{)}\left(\frac{A_{2}\alpha_{1}^{2}}{A_{1}\alpha_{2}^{2}}+1\right)^{2}\,\,e^{2(\delta_{0}\,\phi_{2}+\delta_{1})}.$
(77)

Here we have used that the effective potential
$U_{eff}\simeq(f_{1}^{2}/f_{2})/(4\chi_{2}[1+\epsilon f_{1}^{2}/f_{2}])$. In
order to find the scalar field at the end of the inflationary scenario, we
consider that the slow roll parameter $\epsilon_{H}(\phi_{2}=\phi_{2end})=1$
(or equivalently $\ddot{a}=0$) with which

$e^{2\delta_{0}\,\phi_{2end}}=\Big{(}\frac{A_{2}f_{1}^{2}}{16g_{1}^{2}\alpha_{2}^{2}}\Big{)}\left(\frac{1}{\Big{[}\Big{(}\frac{A_{2}\alpha_{1}^{2}}{A_{1}\alpha_{2}^{2}}\Big{)}+1\Big{]}^{2}}\right)e^{-2\delta_{1}}.$
(78)

Introducing the number of folds $N$ between two values of times $t_{*}$ and
$t_{end}$ (or analogously between two different values $\phi_{2*}$ and
$\phi_{2end}$) we have

$N=\int_{t_{*}}^{t_{\rm
end}}Hdt=\int_{\phi_{2*}}^{\phi_{2end}}\frac{H}{\dot{\phi}_{2}}d\phi_{2}\simeq-A\int_{\phi_{2*}}^{\phi_{2end}}\frac{3H^{2}}{U_{eff,\phi_{2}}}d\phi_{2}\simeq-A\int_{\phi_{2*}}^{\phi_{2end}}\frac{U_{eff}}{2U_{eff,\phi_{2}}}d\phi_{2}\;,$
(79)

and then we obtain that the number $N$ becomes

$N=\left(\frac{A_{2}f_{1}}{16g_{1}\alpha_{2}^{2}\Big{[}\Big{(}\frac{A_{2}\alpha_{1}^{2}}{A_{1}\alpha_{2}^{2}}\Big{)}+1\Big{]}^{2}}\right)\,\left[e^{-\delta_{0}\phi_{2end}-\delta_{1}}-e^{-\delta_{0}\phi_{2*}-\delta_{1}}\right].$
(80)

Here we mention that the observational quantity $N$ should be evaluated when
the cosmological scale exits the horizon. In the following the subscript $*$
is utilized to denote the epoch in which the cosmological scale exits the
horizon.

On the other hand, we will determine the scalar and tensor perturbations
during the inflationary era for our model of a single field $\phi_{2}$ (since
$\phi_{1}=$ cte). It is well known that for multifield inflation, the
expressions for the scalar and tensor perturbations are modified in relation
to a single field, see Ref.Kaiser:2013sna . However from the relation between
the scalar fields given by Eq.(71), the inflationary dynamics in our model is
reduced to a single field $\phi_{2}$ and then the scalar and tensor
perturbations correspond to the standard formulas. Thus, from Ref.p1  the
power spectrum of the scalar perturbation ${{\mathcal{P}}_{S}}$ under the
slow-roll approximation for the new scalar field $\phi_{2}$ is given by

${{\mathcal{P}}_{S}}=\left(\frac{H^{2}}{2\pi\,\dot{\phi_{2}}}\right)^{2}\simeq\left(\frac{A^{2}}{96\pi^{2}}\,\frac{U_{eff}^{3}}{U_{eff,\phi_{2}}^{2}}\right),$
(81)

and the observational scalar spectral index $n_{s}$ is defined as

$n_{s}-1=\frac{d\ln{\mathcal{P}}_{S}}{d\ln k}=-6\epsilon_{H}+2\eta\;,$ (82)

where the slow roll parameters $\epsilon_{H}$ is defined by Eq.(76) and $\eta$
is given by $\eta\simeq(2/A)(U_{eff,\phi_{2}\phi_{2}}/U_{eff})$.

Also, it is well known that the generation of tensor perturbations during the
epoch of inflation would generate gravitational waves. In this sense, the
spectrum of the tensor perturbations ${\mathcal{P}}_{T}$ is given byp1

${\mathcal{P}}_{T}=\left(\frac{H}{\pi}\right)^{2}\simeq\frac{U_{eff}}{6\pi^{2}}\;,$
(83)

and its tensor spectral index $n_{T}$ can be described in terms of the slow
parameter $\epsilon_{H}$ as $n_{T}=\frac{d\ln{\mathcal{P}}_{T}}{d\ln
k}=-2\epsilon_{H}$. Besides, an important observational quantity corresponds
to the tensor-to-scalar ratio $r=\frac{{\mathcal{P}}_{T}}{{\mathcal{P}}_{S}}$.
Here we emphasize that these quantities should be evaluated during the epoch
in which the cosmological scale exits the horizon and in particular at
$\phi_{2}=\phi_{2*}$.

In this way, from Eq.(81) the power spectrum of the scalar perturbation
${{\mathcal{P}}_{S}}$ under the slow-roll approximation in terms of the number
of $e-$folds yields

${{\mathcal{P}}_{S}}(N)\simeq
k_{0}\,\left[F_{1}-\frac{N}{F_{2}}\right]^{2},\,\,\,\,\mbox{where}\,\,\,\,\,\,k_{0}=\frac{A^{2}f_{1}^{4}}{96\times(4\pi)^{2}\chi_{2}f_{2}g_{1}^{2}\delta_{0}^{2}(1+\epsilon
f_{1}^{2}/f_{2})},$ (84)
$F_{1}=\frac{4g_{1}\alpha_{2}}{f_{1}\sqrt{A_{2}}}\left[\frac{A_{2}\alpha_{1}^{2}}{A_{1}\alpha_{2}^{2}}+1\right],\,\,\,\,\,\mbox{and}\,\,\,\,\,F_{2}=\left(\frac{A_{2}f_{1}}{16g_{1}\alpha_{2}^{2}\Big{[}\Big{(}\frac{A_{2}\alpha_{1}^{2}}{A_{1}\alpha_{2}^{2}}\Big{)}+1\Big{]}^{2}}\right),$

respectively.

Also, the scalar spectral index $n_{s}$ in terms of the number of $e-$folds
$N$ becomes

$n_{s}=1-\frac{8g_{1}\delta_{0}^{2}}{Af_{1}}\,\left(F_{1}-\frac{N}{F_{2}}\right)^{-1}\,\left[3g_{1}\left(F_{1}-\frac{N}{F_{2}}\right)^{-1}+1\right].$
(85)

From this scalar spectral index $n_{s}$ we find that the solution for the
parameter $\alpha_{2}$ is given by

$\alpha_{2}=\frac{\sqrt{A_{2}}(1+4(n_{s}-1)N)}{16NL_{1}}+\frac{1}{16}\left[+\sqrt{L_{2}}+\sqrt{L_{3}}\right],$
(86)

where the values of $L_{1}$, $L_{2}$ and $L_{3}$ are defined as

$L_{1}=[1+2(n_{s}-1)N],\,\,\,\,\,\,\,\,\,L_{2}=\frac{A_{2}(1-12f_{1}N-24f_{1}(n_{s}-1)N)}{N^{2}L_{1}^{2}},$

and

$L_{3}=\frac{-256\alpha_{1}^{2}A_{2}N^{2}L_{1}^{2}+2A_{1}A_{2}N[1-2(2+3f_{1}-2n_{s})L_{1}]-2A_{1}\sqrt{A_{2}}N[L_{1}(3+4(n_{s}-1)N)]\,L_{4}}{A_{1}N^{2}L_{1}^{2}},$

with $L_{4}$ given by

$L_{4}=\sqrt{\frac{A_{2}(1-12f_{1}N-24f_{1}(n_{s}-1)N^{2})}{N^{2}L_{1}^{2}}}.$

We mention that in the solution for the parameter $\alpha_{2}$ given by
Eq.(86), we have considered the solution with the positive signs.

By using that the tensor to scalar ratio $r$ under the slow roll approximation
is given by $r\simeq 16\epsilon_{H}$ we find

$r\simeq\frac{64g_{1}^{2}\delta_{0}^{2}}{Af_{1}^{2}}\,\left[F_{1}-\frac{N}{F_{2}}\right]^{-2}.$
(87)

From this expression for the tensor to scalar ratio, we find that the
parameter $\alpha_{1}$ becomes

$\alpha_{1}=\frac{1}{2}\sqrt{\frac{A_{1}\,\alpha_{2}}{\sqrt{A_{2}}}\left[\frac{2}{N\sqrt{r}}+\frac{1}{N}-\frac{4\alpha_{2}}{\sqrt{A_{2}}}\right]},$
(88)

where we have considered the positive sign for the solution of $\alpha_{1}$.
Also, as we anticipated, physical quantities cannot depend on the magnitude of
$g_{1}$ and in fact we note that this result obtained for the parameter
$\alpha_{1}$ does not depend of the parameter $g_{1}$ analogously as occurred
for the $\alpha_{2}$ and neither $n_{s}$ as given by (85), $r$ as given by
(87), etc.

In particular assuming the values $f_{1}=f_{2}=10^{-7}$, $\epsilon=10^{-4}$
and $\chi_{2}=1$, together with ${{\mathcal{P}}_{S}}\simeq 10^{-9}$,
$n_{s}=0.967$ and $r=0.039$ at $N=60$, then numerically we find that the
parameters $\alpha_{1}$, $\alpha_{2}$ and $b_{1}$ result $\alpha_{1}\simeq
10^{-9}$, $\alpha_{2}\simeq 10^{-16}$ and $b_{1}\simeq-1$, respectively. Here
we note that the special value of the parameter $b_{1}\simeq-1$ we have found
from inflation is also in the allowed values found for a consistent stable
emergent universe condition, see Fig.1.

In order to find the consistency relation i.e., the relation between the
tensor to scalar ratio $r$ and the scalar spectral index $n_{s}$, we can
combine Eqs.(85) and (87) obtaining

$n_{s}(r)=n_{s}=1-\delta_{0}\sqrt{\frac{r}{A}}\left[\frac{3f_{1}\sqrt{A\,r}}{8\delta_{0}}+1\right].$
(89)

Figure 2:  The plot shows the tensor to scalar ratio $r$ versus the scalar
spectral index $n_{s}$ (consistency relation $r=r(n_{s})$) for three values of
the dimensionless parameter $b_{1}$. For $b_{1}=-0.5$ solid line, $b_{1}=-1$
dash-dot line and $b_{1}=-1.5$ dots line, respectively. Here we have fixed the
values $\chi_{2}=1$, $\epsilon=10^{-4}$ (in units of $M_{Pl}^{-2}$),
$f_{1}=f_{2}=10^{-8}$ (in units of $M_{Pl}^{4}$), $\alpha_{1}=4\times 10^{-9}$
and $\alpha_{2}=10^{-16}$ (in units of $M_{Pl}^{-1}$). Here we have used the
two-dimensional marginalized joint confidence contours for $(n_{s},r)$ at 68%
and 95% levels of confidence from the latest BICEP/Keck results BICEP:2021xfz
.

In figure 2 we show the dependence of the tensor to scalar ratio on the scalar
spectral index for different values of the parameter $b_{1}$ studied during
the emergent epoch (see Fig.1) from the latest BICEP/Kerck data BICEP:2021xfz
. From the Ref.BICEP:2021xfz , two-dimensional marginalized constraints on
inflationary parameters $r$ and the scalar spectral index $n_{s}$ defined at
$k_{0}=0.002$Mpc-1. The latest BICEP/Keck results places stronger limits on
the tensor to scalar ratio shown in blue (at 68% blue region and 95% light
blue region levels of confidence) and the green color corresponds to the two-
dimensional marginalized constraints obtained in Ref.Planck:2018vyg . We noted
from Fig.(2) that the parameter $b_{1}$, which lies in the range $-2<b_{1}<0$
in order to satisfy the stable emergent universe solution, it is well
supported by the the latest BICEP/Keck data.

##  VI First Dark region with consistent generation of DE, DM and stiff matter

In this section we will analyze the second flat region in which the scalar
field $\varphi_{2}\rightarrow-\infty$ and it corresponds to the dark energy
sector region and the effective potential in this region is given by Eq.(36).
In order to study this scenario that includes dark matter together with dark
energy, we use the $k-$essence model in which dark matter appears naturally as
was shown in Ref.Scherrer:2004au , see also Ref.guendelmanKineticDEDM  for
other examples of this effect in TMT.

In this context, the equation of motion for the scalar field $\varphi_{1}$ is
reduced to

$\frac{d}{dt}\left[a^{3}\dot{\varphi}_{1}(A_{1g}+2B_{1g}X_{1}+B_{12g}X_{2})\right]=0.$

The obvious solution corresponds to $\dot{\varphi}_{1}=0$, together with the
term $(A_{1g}+2B_{1g}X_{1}+B_{12g}X_{2})\neq 0$. In this way we can consider
that the term $X_{1}=0$. Thus, under a small perturbation we can assume that

$X_{1}=0+\delta_{1},\,\,\,\,\,\,\mbox{then}\,\,\,\,\,\,\,\frac{d}{dt}(a^{3}\delta_{1}^{1/2})=0,$

and the solution for $X_{1}$ can be written as

$X_{1}=\delta_{01}\left(\frac{a}{a_{0}}\right)^{-6},$ (90)

where $\delta_{01}$ corresponds to a positive constant.

On the other hand, the equation of motion for the scalar field $\varphi_{2}$
is given by

$\frac{d}{dt}\left[a^{3}\dot{\varphi}_{2}(A_{2g}+2B_{2g}X_{2}+B_{12g}X_{1})\right]=0.$
(91)

In this situation we do not consider the solution $\dot{\varphi}_{2}=0$,
instead we assume the case where
$(A_{2g}+2B_{2g}X_{2}+B_{12g}X_{1})=(A_{2g}+2B_{2g}X_{2})=0$, where we have
used that $X_{1}=0$. Thus the obvious solution for $\dot{\varphi}_{2}$ can be
written as

$X_{2}=-\frac{A_{2g}}{2B_{2g}}=X_{20}=\mbox{positive constant}.$ (92)

In this case we must have $b_{2}^{2}>4g_{2}\epsilon(1+b_{2}g_{1}/g_{2})$ with
which the term $B_{2g}$ is negative and $A_{2g}>0$ (or $-2<b_{2}g_{1}/g_{2}$).

Following Ref.Scherrer:2004au  we can show that the solution given by Eq.(92)
is stable under a small perturbation. In this form, we can perturb this
solution as

$X_{2}=-\frac{A_{2g}}{2B_{2g}}+\delta_{2}=X_{20}+\delta_{2},$ (93)

with $\delta_{2}\ll X_{20}$. In this way, replacing Eq.(93) into Eq.(91) and
expanding to order one in $\delta_{2}$ yields

$\frac{d}{dt}\,(a^{3}\,\delta_{2})=0,\,\,\,\,\mbox{then}\,\,\,\,\,\delta_{2}\propto\,a^{-3}.$

Thus, the perturbative solution given by Eq.(93) results

$X_{2}(a)=X_{20}\left[1+\delta_{02}(a/a_{0})^{-3}\right],$ (94)

where $\delta_{02}$ and $a_{0}$ are two new constants. Besides, as
$\delta_{2}\ll X_{20}$, then we have $\delta_{02}(a/a_{0})^{-3}\ll 1$.

Replacing Eqs.(90) and (94) into the energy density $\rho$ we have

$\rho(a)\simeq\rho_{0}+\rho_{1}\,\left(\frac{a}{a_{0}}\right)^{-3}+\rho_{2}\,\left(\frac{a}{a_{0}}\right)^{-6},$
(95)

where $\rho_{0}$ corresponds to the energy density associated to dark energy
and it is given by

$\rho_{0}=\frac{1}{4}\frac{A_{2g}^{2}}{B_{2g}}+U_{\rm
eff\,g}=-\frac{1}{\chi_{2}}\left[\frac{b_{2}g_{1}+g_{2}}{b_{2}^{2}-4\epsilon(b_{2}g_{1}+g_{2})}\right]>0.$

Note that in Eq.(95) we have neglected the term $(a/a_{0})^{-12}$ since during
the evolution of present universe this term is highly suppressed.

Also, we note that as the term $b_{2}^{2}>4\epsilon(b_{2}g_{1}+g_{2})$ (see
Eq.(92)) then we have that $b_{2}g_{1}+g_{2}<0$, since $\epsilon>0$ and from
Eq.(37) we have that the term $A_{2g}>0$, then we get that
$b_{2}g_{1}/2+g_{2}>0$. In this form, assuming that the parameter $g_{2}$ is
positive, we find that the constraint for the ratio $b_{2}g_{1}/g_{2}$ becomes

$-1>\frac{b_{2}g_{1}}{g_{2}}>-2.$ (96)

This result suggest that the parameter $b_{2}$ is a positive quantity, since
during the inflationary epoch we consider that the effective potential does
not have bump when $g_{1}<0$.

Now the second term of Eq.(94) represents to the dark matter and the quantity
$\rho_{1}$ is defined as

$\rho_{1}=\frac{\delta_{02}\,A_{2g}^{2}}{B_{2g}}.$

We note that as $B_{2g}$ is a negative quantity, then we impose that the
constant $\delta_{02}<0$, in order to obtain that the dark matter to be
positive and also $|\delta_{02}|\ll 1$.

The last term corresponds to stiff equation of state where the quantity
$\rho_{2}$ is given by

$\rho_{2}=\,\frac{3\,A_{2g}^{2}}{4\,B_{2g}}\,\,\delta_{02}^{2}+A_{1g}\delta_{01}-\frac{3B_{12g}\delta_{01}A_{2g}}{2B_{2g}},$

and this parameter corresponds to a positive quantity.

Also, we find that the pressure in terms of the scale factor results

$p(a)\simeq
p_{0}+p_{1}\left(\frac{a}{a_{0}}\right)^{-3}+p_{2}\left(\frac{a}{a_{0}}\right)^{-6}\simeq
p_{0}+p_{2}\left(\frac{a}{a_{0}}\right)^{-6},$ (97)

where the quantities $p_{0}$, $p_{1}$ and $p_{2}$ are defined as

$p_{0}=-\frac{A_{2g}^{2}}{4B_{2g}}-U_{\rm
eff\,g}=-\rho_{0},\,\,\,\,\,\,\,\,\,p_{1}=X_{20}\delta_{20}[A_{2g}+2B_{2g}X_{20}]=0,\,\,\,\,$

and

$p_{2}=\delta_{01}\left[A_{1g}-\frac{B_{12g}A_{2g}}{2B_{2g}}\right]+\frac{A_{2g}^{2}\delta_{02}^{2}}{4B_{2g}}.$

As before in Eq.(97) we have neglected the term $(a/a_{0})^{-12}$. Also, in
order to obtain an adequate stiff equation of state, which is necessary from
energy momentum conservation (and therefore for the consistency of the
equations of motion), in which the coefficients $\rho_{2}=p_{2}$. In this
form, we find that the relation between the constants $\delta_{01}$ and
$\delta_{02}$ from the condition given by stiff equation of state yields

$\delta_{01}=\left(\frac{A_{2g}}{2B_{12g}}\right)\,\delta_{02}^{2},$ (98)

and then we have that $\rho_{2}=p_{2}=\delta_{01}A_{1g}$. Note that in order
to obtain $\delta_{01}>0$, we have that from Eq.(39) the coefficient

$B_{12g}=\frac{g_{1}^{2}+8\epsilon\chi_{2}^{2}[g_{1}(b_{2}-\epsilon
g_{1})+g_{2}]}{4\chi_{2}(\epsilon g_{1}^{2}+g_{2})}>0.$ (99)

We see first that for $\epsilon=0$ the above Eq.(99) is of course satisfied.
Now taking $\epsilon>0$, when we start to increase $\epsilon$ towards bigger
positive values, we will reach a critical value of $\epsilon$ beyond which the
condition breaks down i.e., when $B_{12g}=0$. Thus, we have then that the
above condition holds for $0<\epsilon<\epsilon_{c}$, where the critical value
of $\epsilon$ becomes

$\epsilon_{c}=\frac{2g_{2}(\frac{b_{2}g_{1}}{g_{2}}+1)\chi_{2}^{2}\pm\sqrt{2g_{1}^{4}\chi_{2}^{2}+4(b_{2}g_{1}+g_{2})^{2}\chi_{2}^{4}}}{4g_{1}^{2}\chi_{2}^{2}}.$

Since $-1>\frac{b_{2}g_{1}}{g_{2}}>-2$, the first term in the numerator of
$\epsilon_{c}$ is negative and then we must take the plus sign in the second
positive term in the above equation so that $\epsilon>0$ is satisfied.

In conclusion, we can obtain definite ranges in the parameter-space for a
consistent generation of DE, DM and stiff component from multi fields
dynamics.

On the other hand, the equation of state (EoS) or EoS parameter $w=p/\rho$ in
our model as a function of the scale factor becomes

$w(a)=\frac{-\Omega_{0}+(1-\Omega_{0}-\Omega_{1})(a/a_{0})^{-6}}{\Omega_{0}+\Omega_{1}(a/a_{0})^{-3}+(1-\Omega_{0}-\Omega_{1})(a/a_{0})^{-6}},$
(100)

where the parameters $\Omega_{0}$ and $\Omega_{1}$ associated to the dark
energy and dark matter at the present time are defined as

$\Omega_{0}=\frac{\rho_{0}}{6H_{0}^{2}},\,\,\,\,\mbox{and}\,\,\,\,\,\,\Omega_{1}=\frac{\rho_{1}}{6H_{0}^{2}},$

with $H_{0}$ the Hubble parameter at the present epoch. Here we have used the
Friedmann equation $6H^{2}=\rho$, in which
$1-\Omega_{0}-\Omega_{1}=\Omega_{2}=\rho_{2}/6H_{0}^{2}$ at the present era.
From Eq.(100) we note that in the far future in which the scale factor is
large, the EoS parameter $w\rightarrow-1$, i.e., the universe is dominated by
a cosmological constant.

Figure 3:  The left panel shows the evolution of the EoS parameter as a
function of the normalized scale factor $x=a/a_{0}$. The right panel shows the
behaviour of $dw/da$ versus $x$. Here we have considered different values of
the density parameter associated to the dark matter $\Omega_{1}$ and also we
have fixed the density parameter of the dark energy to $\Omega_{0}=0.7$.

In Fig.3 (left panel) we show the evolution of the EoS parameter $w(x)$ versus
the normalized scale factor $x=a/a_{0}$ defined by Eq.(100) for same values of
the parameter $\Omega_{1}$. Here we have fixed that the density parameter
associated to dark energy $\Omega_{0}=0.7$. Additionally, we have added
different horizontal lines associated to the constant values of the EoS
parameter $w=1/3,w=0,w=-1/3$ and $w=-1$, respectively. This help us
distinguish the radiation dominated era, the matter domination era and the
dark energy domination era in terms of the parameter $\Omega_{1}$. In this
sense, from Fig.3 (left panel) we observe that the end of the radiation
dominated, matter dominated universe and dark energy occurs later for the
situation in which the parameter associated to dark matter $\Omega_{1}$
decrease. Thus, for example in the case $\Omega_{1}=0.25$ the EoS $w=0$
(matter domination) takes place for a scale factor $x\simeq 0.67$ while for
the value $\Omega_{1}=0.29$ occurs before at $x\simeq 0.5$. From the plot we
note that when we decrease the density parameter associated to the dark matter
$\Omega_{1}$ the EoS parameter increase. Also, we observe that for
$\Omega_{1}<0.29$ (more stiff matter) and for values of the scale factor
$x<0.7$, the universe does not present an accelerated phase, since the EoS
parameter becomes positive. Thus, we note that when we have a very small
quantity of stiff matter ($\Omega_{1}\geq 0.29$) the universe always presents
an accelerated scenario in the plot region for the scale factor. In order to
account of the transition from deceleration expansion to acceleration
expansion, we have that from Eq.(100), the situation in which the pressure can
be considered negligible or matter domination ($\omega\sim 0$) occurs when the
scale factor $a/a_{0}=x\geqslant(\Omega_{0}/(1-\Omega_{0}-\Omega_{1}))^{1/6}$
up to the value
$a/a_{0}=x<(\frac{\Omega_{1}[1+\sqrt{1+32\Omega_{0}(1-\Omega_{0}-\Omega_{1})/\Omega_{1}^{2}}]}{4\Omega_{0}})^{1/3}$
where $w>-1/3$, see Fig.3. During this period the energy density given by
Eq.(95) can be approximated to $\rho\propto a^{-3}$ and the scale factor
evolve as $a\propto t^{1/2}$. However, for values of the scale factor
$a/a_{0}=x\gtrsim(\frac{\Omega_{1}[1+\sqrt{1+32\Omega_{0}(1-\Omega_{0}-\Omega_{1})/\Omega_{1}^{2}}]}{4\Omega_{0}})^{1/3}$,
the universe presents an accelerated scenario ($\ddot{a}>0$) in which the
energy density can be considered as a constant. In this sense, from the EoS
parameter found in our model (Eq.(100)) we can account of the matter dark
energy domination transition may precede the deceleration to acceleration
transition. The study of this interrelation of the epochs of deceleration-
acceleration transition from explicit EoS parameters was analyzed in detail
Nojiri:2006jy  and references therein.

In the right panel of Fig.3 we shows the evolution of the derivative $dw/da$
versus the normalized scale factor $x$. From this plot we observe that the
model corresponds to a tracking freezing model where the derivative $dw/da$ is
negative. Also we note that when we decrease the values of the density
parameter related to dark matter the quantity $dw/da$ results more negative.

It is interesting to study the behaviour of the EoS parameter $w(a)$ and in
particular the derivative $dw/da$ related to the dark epoch into the second
flat region. In this way from Eq.(100) we find that the quantity $dw/da$
becomes

$\frac{dw}{da}=-3x^{2}\left(\frac{(1-\Omega_{0}-\Omega_{1})(\Omega_{1}+4\Omega_{0}x^{3})+\Omega_{0}\Omega_{1}x^{6}}{[1+\Omega_{1}(x^{3}-1)+\Omega_{0}(x^{6}-1)]^{2}}\right)<0,\,\,\,\,\,\mbox{with}\,\,\,\,x=\frac{a}{a_{0}}.$
(101)

This result shows that the behaviour of our model corresponds to a tracking
freezing model since $dw/da<0$, see Refs.Ar1 ; Ar2 .

On the other hand, from Friedmann equation $6H^{2}=\rho$, we can obtain that
the scale factor as a function of the cosmological time $t$ i.e., $a(t)$
results Chavanis:2014lra

$\frac{a(t)}{a_{0}}=\left[\left(\frac{\Omega_{1}}{\Omega_{0}}+2\sqrt{\frac{(1-\Omega_{0}-\Omega_{1})}{\Omega_{0}}}\right)\sinh^{2}\left[\frac{3\sqrt{\Omega_{0}}(H_{0}t+C)}{2}\right]+\sqrt{\frac{(1-\Omega_{0}-\Omega_{1})}{\Omega_{0}}}\left(1-e^{-3\Omega_{0}^{1/2}(H_{0}t+C)}\right)\right]^{1/3},$
(102)

where $C$ corresponds to an integration constant. Note that in the absence of
stiff matter ($1-\Omega_{0}-\Omega_{1}=\Omega_{2}=0$) the scale factor
$a(t)\propto\sinh^{2/3}(3\sqrt{\Omega_{0}}(H_{0}t+C)/2)$, see
Ref.Frieman:2008sn

Also, we can find the scalar fields in terms of the scale factor. Thus, for
the scalar field $\varphi_{1}$ we have that from Eq.(90) that the speed of the
scalar field $\varphi_{1}$ can be written as

$\dot{\varphi}_{1}=Hx\frac{d\varphi_{1}}{dx}=\pm\frac{\sqrt{2\delta_{01}}}{x^{3}},$

and its solution yields

$\varphi_{1}(x)=C_{1}\mp\frac{\sqrt{2\delta_{01}}}{3H_{0}\sqrt{1-\Omega_{0}-\Omega_{1}}}\mbox{Arctanh}\left[\frac{\Omega_{1}x^{3}+2(1-\Omega_{0}-\Omega_{1})}{2\sqrt{1-\Omega_{0}-\Omega_{1}}\sqrt{x^{3}(\Omega_{1}+\Omega_{0}\,x^{3})+(1-\Omega_{0}-\Omega_{1})}}\right].$
(103)

Analogously for the scalar field $\varphi_{2}$, we find that the solution as a
function of the normalized scale factor $x=a/a_{0}$ can be expressed as

$\varphi_{2}(x)=C_{1}\pm\frac{2\sqrt{2X_{02}}}{3H_{0}}\sqrt{\frac{\bar{\Omega}-\Omega_{1}-2\Omega_{0}x^{3}}{\bar{\Omega}}}\left(\frac{g_{1}(x)-g_{2}(x)}{g_{3}(x)}\right),$
(104)

where $C_{1}$ is another integration constant and the functions $g_{1}(x)$,
$g_{2}(x)$ and $g_{3}(x)$ are defined as

$g_{1}(x)=\sqrt{\frac{\Omega_{0}(x^{3}+\delta_{02})}{2\delta_{02}-\Omega_{1}-\bar{\Omega}}}\,(\bar{\Omega}^{2}+\Omega_{1}\bar{\Omega}+\Omega_{0}\Omega_{1}x^{3}+\Omega_{0}\bar{\Omega}x^{3})\,{{\mathcal{F}}}[\bar{x},m_{1}],$
$g_{2}(x)=\delta_{02}\Omega_{0}(\Omega_{1}+\bar{\Omega}-2\delta_{02}\Omega_{0})\,(\Omega_{1}+\bar{\Omega}+2\Omega_{0}x^{3})\,\sqrt{\frac{1}{\Omega_{1}+\bar{\Omega}-2\Omega_{0}\delta_{02}}}\,\sqrt{\frac{(\delta_{02}+x^{3})\Omega_{0}}{2\Omega_{0}\delta_{02}-\Omega_{1}-\bar{\Omega}}}\,\Pi[m_{2},\bar{x},m_{1}],$

and

$g_{3}(x)=\Omega_{0}(\Omega_{1}+\bar{\Omega})\,\sqrt{\delta_{02}+x^{3}}\,\sqrt{\frac{\Omega_{1}+\bar{\Omega}+2\delta_{02}\Omega_{0}}{\Omega_{1}+\bar{\Omega}-2\delta_{02}\Omega_{0}}}\sqrt{(1-\Omega_{0}-\Omega_{1})+\Omega_{1}x^{3}+\Omega_{0}x^{6}}.$

Here the functions ${\mathcal{F}}$ and $\Pi$ correspond to the Elliptic
functions of the first and second kind, respectively, see Libro . Also, the
quantities $\bar{x}$, $m_{1}$, $m_{2}$, and $\bar{\Omega}$ are given by

$\bar{x}=\mbox{Arcsin}\left[\sqrt{\frac{\Omega_{1}+\bar{\Omega}+2\Omega_{0}x^{3}}{\Omega_{1}+\bar{\Omega}-2\delta_{02}\Omega_{0}}}\right],\,\,\,\,\,\,m_{1}=\frac{\Omega_{1}+\bar{\Omega}-2\delta_{02}\Omega_{0}}{2\bar{\Omega}},\,\,\,m_{2}=\frac{\Omega_{1}+\bar{\Omega}-2\delta_{02}\Omega_{0}}{\Omega_{1}+\bar{\Omega}},$

and $\bar{\Omega}=\sqrt{\Omega_{1}^{2}-4\Omega_{0}(1-\Omega_{0}-\Omega_{1})}$.

We mention that in order to obtain the scalar fields as a function of the
cosmological time, we need to replace the normalized scale factor given by
Eq.(102) into Eqs.(103) and (104) to find $\varphi_{1}(t)$ and
$\varphi_{2}(t)$, respectively.

##  VII Summary and Discussion of Possible future lines of research

In summary, in this paper we have explored a new approach to study all the
most relevant aspects of the evolution of the universe using a scale invariant
theory with metric independent measures which employs two scalar fields. In
this sense, we have analyzed in the early universe the scenarios of the
emergent and inflationary epochs. Also we have studied the dark sector
considering the DE, DM and stiff matter during the late universe. In relation
to the scale invariance is spontaneously broken by the integration of the
equations of motion and going to the Einstein frame, we find a theory with an
effective potential for the scalar fields that displays three flat regions and
corresponding K-essence terms. We use each of theses flat regions to describe
the different phases of the universe. The higher energy density describes the
very early universe, first a non singular emergent universe which is then
followed by an inflationary era.

For the emergent universe we have analyzed the stability of the emergent
solution from the perturbative analysis. In this context because of the non
triviality of the equation associated to the parameter $\omega^{2}$ we have
considered a numerical treatment. In particular for different values of the
parameter-space, we have found that the parameter $b_{1}$ is negative and its
value is in the range $-2<b_{1}<0$, see Fig.1. Here we have obtained that this
range for the parameter $b_{1}$ does not depend of the values of $f_{1}$ and
$f_{2}$.

In the context of the inflationary epoch, we have considered the slow roll
approximation. Under this approximation we have defined two new scalar fields
$\phi_{1}$ and $\phi_{2}$ from a transformation orthogonal in which
$\dot{\phi}_{1}=0$ and then the effective potential reduces to a single scalar
field $\phi_{2}$, see Eq.(69). Also, we have noted that the effective
potential (69) interpolates between the vacuums with asymptotic values
$\frac{f_{1}^{2}}{4\chi_{2}(f_{2}+\epsilon f_{1}^{2})}$ and
$\frac{g_{1}^{2}}{4\chi_{2}(g_{2}+\epsilon g_{1}^{2})}$ independent of the
choice of the relative signs of $g_{1}$, and $f_{1}$. However, we have chosen
the sign of $f_{1}g_{1}$ negative, so that the potential smoothly rolls down
from one vacuum to the other without an intermediate bump. From the
observational parameters; scalar power spectrum, scalar spectral index and the
upper bound of the ratio tensor to scalar, we have found numerically the
values of the parameters $\alpha_{1}$, $\alpha_{2}$ and $b_{1}$. In this
context, we have obtained that the constraint imposes for the inflationary
epoch for the parameter $b_{1}$ is in the range found from the stability
analysis given for the emergent universe.

The second flat region describes a dark sector with DE and DM and a quickly
decaying stiff component as well. The DE, DM and stiff components owe its
existence to the K-essence induced by multi measure theory. It is interesting
that a consistency condition for the generation of the DE, DM and stiff
components which correlates the perturbations of the two scalar fields with
respect to a certain background solution is obtained in (98) from the
condition found by stiff equation of state $\rho_{2}=p_{2}$. From the
evolution of the EoS parameter $w$ versus the normalized scale factor
$x=a/a_{0}$, we have noted that when we decrease the density parameter
associated to dark matter $\Omega_{1}$, the EoS parameter increase for the
same value of $x$, see Fig.3 (left panel). Also, we have found that the
derivative $dw/da$ is negative with which this model corresponds to a tracking
freezing model, see Fig.3 (right panel). Besides, we have determined the
solution for the scale factor as a function of the cosmological time together
with the solutions for the scalar fields in term of the normalized scale
factor.

There is also a third flat region, which could represent a future DE sector,
or may be represents the present state of the universe. Among the issues that
should be studied further, we mention first whether the universe makes use of
the third flat region with asymptotic behavior (40), (41) and (42). In this
case, we should connect the first dark region described in section VI with the
one satisfying the asymptotic behavior (40), (41) and (42), in order to study
the early dark epoch and its transition to a late DE era.

For this purpose we can construct a simplified effective potential that
interpolates between these two vacuums, that could be simply achieved by
considering the effective potential given by Eq.(31) and ignoring the
$f$-terms, such that

$U_{\rm
eff}=\frac{(g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1})^{2}}{4\chi_{2}\,\Bigl{[}g_{2}e^{-2\alpha_{2}\varphi_{2}}+M_{2}+\epsilon(g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1})^{2}\Bigr{]}}\;.$
(105)

By considering the analysis described in the inflationary epoch (section V),
we know that the potential (105) may connect the two vacuums through a slow
roll process, and will also contain a minimum at zero if
$(g_{1}e^{-\alpha_{2}\varphi_{2}}-M_{1})^{2}=0$ at some point, which would
require $M_{1}<0$ , since we have determined that $g_{1}<0$. However, we will
assume that the parameter $M_{1}>0$ in which case the potential will contain a
bump and the only way to connect these two vacuums will be through tunnelling,
which reminds us of the original (or old) inflationary universe by Guth Guth .
This scenario, where two DE states are connected through bubble nucleation was
considered in sloth , sloth2  to formulate a possible resolution of the
$H_{0}$ problem and we could consider it also in a future research for our
model.

In this case one may ask what may happen to the dark matter generated in the
first DE region, could the DM remain as islands or impurities of the previous
vacuum?, or may be this DM collapses into black holes before the transition to
the final vacuum takes place, and then of course the black holes are resilient
and should remain even after the transition to the final vacuum state. These
issues could be studied in a future research.

The second subject concerning we want to discuss concerning possible future
lines of research is a very simple possible variation of the model that could
lead to interesting results in the unification of the early and current
universe. As we have seen in the previous sections, the best results are
obtained for $\epsilon$ small, so it is worthwhile to study $\epsilon=0$ .
Furthermore, it is quite interesting that for $\epsilon=0$, we can also
include a term, that which when considered in Einstein frame, produces just a
shift in the effective potential, that is it just produces a the same effect
of a cosmological term in Einstein frame.

The slightly modified theory, where the $\epsilon$ is eliminated and a
$\Lambda_{0}$ term is added and then the new action can be written as

$S=\int
d^{4}x\,\Phi_{1}(A)\Bigl{[}R-2\Lambda_{0}\frac{\Phi_{1}(A)}{\sqrt{-g}}+L^{(1)}\Bigr{]}+\int
d^{4}x\,\Phi_{2}(B)\Bigl{[}L^{(2)}+\frac{\Phi(H)}{\sqrt{-g}}\Bigr{]}\;,$ (106)

which is still invariant under the global scale transformations (7). To see
application of this type of term in questions also related to DE, applied to a
Gravity-Assisted Emergent Higgs Mechanism in the Post-Inflationary Epoch in
Gravityassisted .

In this context now the Weyl-rescaled metric ${\bar{g}}_{\mu\nu}$ that defines
the Einstein frame is given by ${\bar{g}}_{\mu\nu}=\chi_{1}g_{\mu\nu}$ where
$\chi_{1}\equiv\frac{\Phi_{1}(A)}{\sqrt{-g}}$ one can indeed suspect that the
extra term just induces a cosmological term in the Einstein frame, since using
the previous definition we can check that
$-2\Lambda_{0}\frac{\Phi_{1}(A)^{2}}{\sqrt{-g}}=-2\Lambda_{0}\sqrt{-{\bar{g}}}$.

This expectation is indeed confirmed by a more detailed analysis, and we can
see that the effective Lagrangian (25) is still valid, but now the
corresponding $A$ and $B$ coefficients are evaluated for $\epsilon=0$ and the
effective potential is shifted by $2\Lambda_{0}$.

This has some consequences, like it is easier to guarantee a positive DE, just
choose a big enough value for $2\Lambda_{0}$. Apart from that there is not
much change.

Notice also that $\Phi_{1}(A)$ is now not just a measure of integration, since
it appears also square, and the theory with $\Phi_{1}(A)$ appearing just
linearly has the highly non conventional infinite dimensional additional
symmetry, valid for any regular set of four functions $f^{\delta}$, such that,
$A_{\mu\nu\gamma}\rightarrow
A_{\mu\nu\gamma}+\epsilon_{\mu\nu\gamma\delta}f^{\delta}(R+L^{(1)})$ (where
$\epsilon_{\mu\nu\gamma\delta}$ is the totally antisymmetric symbol taking
values zero, one or minus one), which is absent in the theory where
$\Phi_{1}(A)$ appears also squared, i.e. for $\Lambda_{0}\neq 0$, so that can
be an argument for $\Lambda_{0}=0$, or a symmetry that protects $\Lambda_{0}$
from becoming big after quantum corrections.

A stiff era, as the one obtained in this work can have interesting
observational consequences. This era could occurs before the BBN or after. In
both cases the stiff era can affect the light element abundances and perhaps
the CMB, if the modes correspond to linear modes. Such an issue has been
discussed for example in 2309.04850 and references therein. The study of this
issue concerning the consequences of component with a stiff equation of state
and how would be the special features that would be a consequence this stiff
state in our scenario would be a very interesting subject for future research
concerning our model.

###### Acknowledgements.

E.G. want to thank the Pontificia Universidad Católica de Valparaíso, Chile,
for hospitality during this collaboration, FQXi for great financial support
for work on this project at BASIC in Ocean Heights, Stella Maris, Long Island,
Bahamas and CA16104 - Gravitational waves, black holes and fundamental physics
and CA18108 - Quantum gravity phenomenology in the multi-messenger approach
for additional financial support.

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