# Constant-roll inflation with a complex scalar field

Ramón Herrera<EMAIL_ADDRESS>Instituto de Física, Pontificia
Universidad Católica de Valparaíso, Avenida Brasil 2950, Casilla 4059,
Valparaíso, Chile.  Mehdi Shokri<EMAIL_ADDRESS>School of
Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran  Department
of Physics, University of Tehran, North Karegar Ave., Tehran 14395-547, Iran
Canadian Quantum Research Center, 204-3002 32 Ave, Vernon, BC V1T 2L7, Canada
Jafar Sadeghi<EMAIL_ADDRESS>Department of Physics, University of Mazandaran,
P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

We consider inflation with a constant rate of rolling in which a complex
scalar field plays the role of inflaton during the inflationary epoch. We
implement the inflationary analysis for an accredited angular speed
$\dot{\theta}$ which satisfies our dynamical equations. Scalar and tensorial
perturbations generated in the framework of constant roll inflation with a
complex field are studied. In this respect, we find analytically solutions to
the gauge invariant fluctuations, with which an expression for the scalar
power spectrum together with its scalar index spectral in this scenario were
found. By comparing the obtained results with the observations coming from the
cosmic microwave background anisotropies, the constraints on the parameters
space of the model and also its predictions are analyzed and discussed.

##  I Introduction

Inflation theory as an unavoidable part of modern cosmology attempts to
describe the phenomena at the early time through the primordial perturbations
coming from the quantum fluctuations. The scalar perturbations as the seed of
the universe form the large-scale structures of the universe. Moreover, they
are the main ones responsible for the temperature anisotropies of the cosmic
microwave background (CMB) at the last scattering surface. On the other hand,
monitoring the B-mode polarized CMB photons discloses the existence of
primordial gravitational waves generated by the tensor perturbations of
inflation Guth:1980zm ; Linde:1981mu ; Albrecht:1982wi ; Lyth:1998xn . In the
standard cosmology, a single scalar field, so-that called inflaton, is the
dominant matter of the universe during cosmic inflation so that it decays to
the particles at the last step of inflation through the reheating process
Kofman:1994rk ; Shtanov:1994ce . Non-minimal coupling (NMC) model is another
broadly-used inflationary model considering a coupling between the Ricci
scalar and the inflaton field coming from the quantum corrections of scalar
field Lucchin:1985ip ; Futamase:1987ua ; Fakir:1990eg ; Kallosh:2013maa ;
Edwards:2014poa ; Shokri:2017mpl ; Shokri:2019rfi ; Shokri:2021rhy ;
Shokri:2021jxh ; Upadhyay:2023wcq . Besides the mentioned models, modified
theories of gravity introduce different mechanisms in order to explain the
early-time accelerating phase of the universe. For instance, in $f(R)$
gravity, modifying the Ricci scalar can describe the inflationary era instead
of considering a scalar field Starobinsky:1980te ; DeFelice:2010aj ;
Capozziello:2011et ; Nojiri:2010wj ; Ben-Dayan:2014isa ; Sadeghi:2015nda ;
Motohashi:2012tt ; Renzi:2019ewp . Also, in the extended versions of
teleparallel gravity, i.e. modified teleparallel gravity $f(T)$ and modified
symmetric teleparallel gravity $f(Q)$, two other geometrical objects, tension
and non-metricity, are responsible to demonstrate the inflationary epoch
Bahamonde:2021gfp ; BeltranJimenez:2019tme ; Capozziello:2022tvv . Apart from
the inflationary criteria, these models could be also tested in the context of
weak gravity conjecture (WGC), in particular, in the swampland region where
the low-energy effective field theories are incompatible with string theory
Vafa:2005ui ; Obied:2018sgi ; Agrawal:2018own ; Blumenhagen:2018hsh ;
Shokri:2021iqp ; Gashti:2022hey ; Sadeghi:2023cxh .

A wide range of inflationary literature has been dedicated to compere the
predictions of single field models with the CMB observations. Consequently,
some of them are excluded while some of them are still compatible with the
Planck datasets Starobinsky:1980te ; Barrow:1988xh ; Martin:2016ckm . As a
shortcoming of these models, they don’t predict any non-Gaussianity in their
spectrum Chen:2010xka  while the future CMB observations propose the existence
of the non-Gaussianity in the detected spectrum of the inflationary
perturbations. In such a case, single field models will be put into question.
To escape from this, constant-roll inflation has been suggested in which
inflaton rolls down with a constant rate from the maximum point of the
potential to the minimum point at the end of inflation as

$\ddot{\varphi}=-(3+\alpha)H\dot{\varphi},$ (1)

where $\alpha$ is a non-zero parameter Motohashi:2014ppa ; Motohashi:2017aob .
Deviation from the slow-roll approximation is also traced in the ultra slow-
roll regime where we assume a non-negligible
$\ddot{\varphi}=-3H\dot{\varphi}$. This class of inflationary models shows a
finite value of the non-decaying mode in the spectrum of curvature
perturbations Inoue:2001zt . Although the solutions of ultra models are
situated in the non-attractor phase of inflation, sometimes they present an
attractor-like behaviour Pattison:2018bct . Moreover, the large $\eta$
predicted by the ultra models can’t guarantee to solve the $\eta$ problem
introduced in supergravity Kinney:2005vj . Assuming a fast rate of rolling at
the beginning of inflation is comprehended as another kind of deviation from
the slow-roll inflation Contaldi:2003zv ; Lello:2013awa ; Hazra:2014jka .

Recently, the idea of constant-roll has been engaged broadly for different
inflationary models Odintsov:2017yud ; Nojiri:2017qvx ; Motohashi:2017vdc ;
Ito:2017bnn ; Odintsov:2017hbk ; Odintsov:2017qpp ; Awad:2017ign ; Lin:2019fcz
; Motohashi:2019tyj ; Odintsov:2019ahz ; Shokri:2021aum ; Shokri:2021iqp ;
Shokri:2021jxh ; Shokri:2021rhy ; Shokri:2021zqw ; Herrera:2023ywx ;
Mohammadi:2022tmk ; Ahmadi:2023qcw . In this work, we focus on a type of
inflationary models in which a complex scalar field is assumed as inflaton to
drive inflation. Complex scalar fields in quantum and classical cosmology were
first introduced in Khalatnikov:1992sj ; Kamenshchik:1995ib  and then were
developed in Amendola:1994xf ; Scialom:1994uq ; Kamenshchik:1997dmk . Complex
fields are comprehended as an alternative to the quintessence field in order
to explain the late-time accelerating phase of the universe, so-that called
dark energy (DE) Liu:2020bmp ; Carvente:2020aae ; Gu:2001tr . Besides this, a
complex scalar field can play the role inflaton in the early universe
Yurov:2002nu . Consequently, complex inflation naturally presents a graceful
exit from the inflationary era with a very small number of e-folds
Yurov:2001ud . Hence, a complex scalar field can be considered as an assistant
field of a real inflaton in the context of hybrid inflation in order to
terminate inflation Buchmuller:2014epa .

The main aim of the present manuscript is to study a constant-roll
inflationary scenario by considering a complex scalar field as the scalar
field driven inflation. To achieve this, we arrange the paper as follows. In
Section II, we review the foundation of complex scalar field theory and its
dynamical equations, briefly. We investigate the complex inflation in the
context of the constant-roll approach by using the accredited form of the
angular speed $\dot{\theta}$ in Section III. In Section IV, we present the
cosmological perturbations of the complex constant-roll inflation. Also, we
attempt to find the constraints on the parameter space of the model using the
observational values of the spectral parameters. Conclusions and remarks are
drawn in Section V. In the following, we chose units so that
$c=\hbar=\kappa^{2}=8\pi G=1.$

##  II Cosmology with a Complex Scalar Field

Let us start with the action of complex scalar theory Khalatnikov:1992sj ;
Kamenshchik:1995ib

$S=\int{d^{4}x\sqrt{-g}\bigg{(}\frac{R}{2}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\Psi^{*}\partial_{\nu}\Psi-V(|\Psi|)+\mathcal{L}_{m}\bigg{)}},$
(2)

that considers a complex scalar field $\Psi$ minimally coupled to the Ricci
scalar $R=g^{\mu\nu}R_{\mu\nu}$ in the presence of ordinary matter described
by a Lagrangian density $\mathcal{L}_{m}$. Here, we assume that the self-
interacting potential $V$ just depends on the absolute value or amplitude of
the complex scalar field Gu:2001tr  and the quantity $g$ corresponds to the
determinant of the metric $g_{\mu\nu}$. We define the complex scalar field in
terms of its amplitude $\varphi(x)$ and phase $\theta(x)$ by

$\Psi(x)=\varphi(x)e^{i\theta(x)}.$ (3)

This configuration allows us to describe both early and late-time accelerating
phases of the universe only by using a complex field without any cosmological
constant Liu:2020bmp ; Gu:2001tr ; Yurov:2002nu ; Yurov:2001ud . Moreover, the
usefulness of using the $\varphi(x)$ and $\theta(x)$ will advantage the
obtaining of the motion equations (as well as the solutions) that relate the
background variables, in particular the effective potential $V$ and the Hubble
parameter $H$. In this form, we can rewrite the action (2) as

$S=\int{d^{4}x\sqrt{-g}\bigg{(}\frac{R}{2}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\varphi\partial_{\nu}\varphi-\frac{1}{2}\varphi^{2}g^{\mu\nu}\partial_{\mu}\theta\partial_{\nu}\theta-V(\varphi)+\mathcal{L}_{m}\bigg{)}}.$
(4)

By variation of the action with respect to the metric and then considering a
spatially flat universe described by the Friedmann-Robertson-Walker (FRW)
metric, the dynamical equations of the model are given as

$H^{2}\equiv\Big{(}\frac{\dot{a}}{a}\Big{)}^{2}=\frac{1}{3}\rho=\frac{1}{3}\Big{(}\rho_{m}+\frac{1}{2}\big{(}\dot{\varphi}^{2}+\varphi^{2}\dot{\theta}^{2}\big{)}+V(\varphi)\Big{)},$
(5)
$\frac{\ddot{a}}{a}=-\frac{1}{6}(\rho+3p)=-\frac{1}{3}\Big{(}\frac{1}{2}(\rho_{m}+3p_{m})+\big{(}\dot{\varphi}^{2}+\varphi^{2}\dot{\theta}^{2}\big{)}-V(\varphi)\Big{)},$
(6)

where the fluid filling the universe satisfies the continuity equation

$\dot{\rho}+3H(\rho+p)=0,$ (7)

where $H$ is the Hubble parameter, $\rho=\rho_{m}+\rho_{\Phi}$ corresponds to
the total energy density, $p=p_{m}+p_{\Phi}$ denotes the total pressure and
the dot depicts the derivative with respect to cosmic time $t$. Here we have
considered that a spatially homogeneous complex scalar field $\Phi(t)$, such
that $\Phi(t)=\varphi(t)e^{i\theta(t)}$ and the tensor energy momentum
$T^{\text{o.m.}}_{\mu\nu}$ associated to the ordinary matter Lagrangian can be
described by a perfect fluid of the form
$T^{\text{o.m}}_{\mu\nu}=$diag$(-\rho_{m},p_{m},p_{m},p_{m})$ in which
$\rho_{m}$ and $p_{m}$ denote the effective energy density and pressure
associated to ordinary matter, respectively. Analogously, we have assumed that
the energy density and pressure of the complex scalar field are given by
Gu:2001tr

$\rho_{\Phi}=\frac{1}{2}\big{(}\dot{\varphi}^{2}+\varphi^{2}\dot{\theta}^{2}\big{)}+V(\varphi),\hskip
28.45274ptp_{\Phi}=\frac{1}{2}\big{(}\dot{\varphi}^{2}+\varphi^{2}\dot{\theta}^{2}\big{)}-V(\varphi).$
(8)

By varying the action (4) with respect to the scalar field $\varphi$, the
equation of motion (EoM) for the scalar field $\varphi$ becomes

$\ddot{\varphi}+3H\dot{\varphi}-\dot{\theta}^{2}\varphi+V^{\prime}(\varphi)=0,$
(9)

and for phase we have the motion equation and its solution (for angular speed
$\dot{\theta}$) given by

$\ddot{\theta}+(2\dot{\varphi}/\varphi+3H)\dot{\theta}=0,\,\,\,\,\,\,\Rightarrow\,\,\,\dot{\theta}\propto\frac{1}{\varphi^{2}\,a^{3}}.$
(10)

In the following we will consider that the prime denotes the derivative with
respect to the scalar field $\varphi$.

In the context of the scenario dominated by the dark sector (DE) and in order
to explain late time accelerating phase of the universe, the authors in
Ref.Gu:2001tr  considered a non-relativistic matter with pressure $p_{m}=0$
together with a complex quintessence theory in which the angular velocity
$\dot{\theta}$ is proportional to ${a^{-3}\varphi^{-2}}$, see Eq.(10). For a
review of the complex field in the framework of DE, see Refs.Liu:2020bmp ;
Carvente:2020aae ; Gu:2001tr .

On the other hand, during the early universe the classical and quantum
description of the a inflationary epoch from the complex scalar inflaton field
was studied in different articles Khalatnikov:1993vs ; Amendola:1994xf ;
Kamenshchik:1997dmk ; Lee:1994qb  and during the reheating and the primordial
black hole production in Carrion:2021yeh , see also Ref.Scialom:1996yd . The
importance of considering a complex scalar field during this epoch lies in the
fact that such fields, the non-Abelian multiplets of scalar fields and other
fields arise naturally in the new theories of particle physics, string etc. In
particular in Ref.Yurov:2001ud  was utilized the Barrow’s method to solve the
inflationary dynamics associated to the complex field as a natural exist of
inflation but with a small quantity of expansion under an angular velocity
given by the expression $\dot{\theta}=\frac{M}{\varphi^{2}},$ where the
quantity $M$ corresponds to a constant. In particular, in the case in which
the constant $M=0$ the inflationary model reduces to, inflation with a real
scalar field. Despite the mentioned success, the complex inflaton field
suffers from a problem that is the number of $e$-folds related to the
expansion of the universe is very small. Hence, using a single complex field
as inflaton field will be put into doubt Yurov:2001ud ; Yurov:2002nu .

In the following we will assume that the ordinary matter Lagrangian
$\mathcal{L}_{m}=0$ ($\rho=\rho_{\Phi}$ and $p=p_{\Phi}$), in order to study
the constant roll inflation in the framework of the complex field. Thus, we
find that the dynamical equations are given by

$3H^{2}=\frac{1}{2}\big{(}\dot{\varphi}^{2}+\varphi^{2}\dot{\theta}^{2}\big{)}+V(\varphi),\hskip
28.45274pt2\dot{H}=-\dot{\varphi}^{2}-\varphi^{2}\dot{\theta}^{2},$ (11)

and

$\ddot{\varphi}+3H\dot{\varphi}+V^{\prime}(\varphi)=\varphi\dot{\theta}^{2}.$
(12)

From Eq.(11), we find that the effective potential can be written as

$V(\varphi)=3H^{2}+\dot{H}.$ (13)

In this form, knowing the Hubble parameter in terms of the scalar field
together with the speed of the scalar field $\dot{\varphi}$, we can
reconstruct the effective potential $V(\varphi)$ using Eq.(13).

##  III Complex Constant-Roll Inflation

To study the complex constant-roll inflation, we start with the constant-roll
condition (1) for the absolute value of the complex scalar field as

$\ddot{|\Psi|}=-(3+\alpha)H\dot{|\Psi|},$ (14)

where as before the parameter $\alpha\neq 0$. By using the definition of the
complex scalar field (3), the above constant-roll condition is reduced to the
familiar form (1). From Eq.(11) and considering the definition
$\dot{H}=\dot{\varphi}H^{\prime}$, we find that the speed of the scalar field
$\dot{\varphi}$ becomes

$\dot{\varphi}=-H^{\prime}\pm\sqrt{H^{\prime 2}-\varphi^{2}\dot{\theta}^{2}}.$
(15)

In order to find Eq.(15), we have considered that the Hubble parameter is only
function the scalar field $\varphi$ and not of $\theta$ (or $\dot{\theta}$),
since the solution given by Eq.(10) allows eliminating the dependency of the
Hubble parameter of this variable (see Eq.(11)). Thus, from Eq.(10), we can
write $\dot{H}=\varphi H^{\prime}$ and then the relation given by Eq.(15)
becomes useful to determine $\dot{\varphi}$.

As we see, in order to find a real solution for $\dot{\varphi}$, we have that
during the constant roll scenario it is necessary that $H^{\prime
2}>\varphi^{2}\dot{\theta}^{2}$. Also, from Eq.(13), the potential and the
speed of the scalar field can be driven as

$V(\varphi)=3H^{2}-H^{\prime 2}\pm H^{\prime}\sqrt{H^{\prime
2}-\varphi^{2}\dot{\theta}^{2}}.$ (16)

Now, taking the derivative of Eq.(15) with respect to cosmic time $t$ and
using the constant-roll condition given by (1), we obtain a differential
equation for the Hubble parameter in terms of the scalar field given by

$-(3+\alpha)H=-H^{\prime\prime}\pm\left[\frac{H^{\prime}H^{\prime\prime}-(\varphi^{2}\dot{\theta}^{2}/2)^{\prime}}{\sqrt{H^{\prime
2}-\varphi^{2}\dot{\theta}^{2}}}\right].$ (17)

From Eq.(10), the phase speed $\dot{\theta}$ can be written as
$\dot{\theta}=\sqrt{w}/\varphi^{2}a^{3}$, with $w=constant>0$. Then, the speed
of the real scalar field (15) and the corresponding potential (16) are
rewritten as

$\dot{\varphi}=-H^{\prime}\pm\sqrt{H^{\prime 2}-w/\varphi^{2}a^{6}},\hskip
28.45274ptV(\varphi)=3H^{2}-H^{\prime 2}\pm H^{\prime}\sqrt{H^{\prime
2}-w/\varphi^{2}a^{6}}.$ (18)

Moreover, we find that the differential equation for the Hubble parameter $H$
(17) becomes

$-(3+\alpha)H=-H^{\prime\prime}\pm\left[\frac{H^{\prime}H^{\prime\prime}-(w/2\varphi^{2}a^{6})^{\prime}}{\sqrt{H^{\prime
2}-w/\varphi^{2}a^{6}}}\right].$ (19)

To obtain the analytical solutions for the background variables, we can
consider that during inflation $1\gg w/\varphi^{2}a^{6}H^{\prime 2}$ with
which

$-(3+\alpha)HH^{\prime}\simeq-H^{\prime}H^{\prime\prime}\pm[H^{\prime}H^{\prime\prime}-(w/2\varphi^{2}a^{6})^{\prime}][1+\frac{w}{2\varphi^{2}a^{6}H^{\prime
2}}+....],$ (20)

and for the speed of scalar field in this approximation results

$\dot{\varphi}\simeq-H^{\prime}\pm
H^{\prime}\left[1-\frac{w}{2\varphi^{2}a^{6}H^{\prime 2}}+...\right].$ (21)

By considering the positive sign and keeping the first term into the expansion
given by Eq.(20), we have the reduced differential equation

$(3+\alpha)HH^{\prime}\simeq(\frac{w}{2\varphi^{2}a^{6}})^{\prime},$ (22)

with the solution

$H^{2}+C\simeq\frac{w_{1}}{\varphi^{2}a^{6}},$ (23)

where we have defined $w_{1}=w/(3+\alpha)$ and $C$ corresponds to an
integration constant. Also, in this case the quantity $\dot{\varphi}$ (21)
yields

$\dot{\varphi}\simeq-\frac{w}{2\varphi^{2}a^{6}H^{\prime}}=-\frac{w}{2w_{1}}\,\frac{(H^{2}+C)}{H^{\prime}}=-\frac{(3+\alpha)}{2}\,\frac{(H^{2}+C)}{H^{\prime}}.$
(24)

In the particular case in which the integration constant $C=0$ and in order to
find the Hubble parameter in terms of the scalar field, we take the derivative
of Eq.(23) with respect to scalar field results

$H^{\prime}=-H\left[\frac{1}{\varphi}+3\frac{H}{\dot{\varphi}}\right],$ (25)

where we have considered that $H=\tilde{H}\dot{\varphi}$ where
$\tilde{H}=a^{\prime}/a$. Then, using Eq.(24), the differential equation (25)
reduces to

$H^{\prime}\simeq-H\left[\frac{1}{\varphi}-\frac{6}{(3+\alpha)}\,\frac{H^{\prime}}{H}\right],$
(26)

and its solution is given by

$H(\varphi)=C_{1}\,[(3-\alpha)\varphi]^{-\beta},\,\,\,\,\mbox{with}\,\,\,\,\,\,\beta=\left(\frac{3+\alpha}{\alpha-3}\right),$
(27)

where $C_{1}$ corresponds to another integration constant. Note that Eqs.(22)
and (25) are equivalent, since we have only utilized the solution given by
Eq.(23) (with $C=0$).

For the case in which the integration constant $C\neq 0$, we can introduce the
variable change $\mathcal{H}^{2}=H^{2}+C$ into Eq.(23) and using Eq.(24), we
find the same differential equation (26) for the variable $\mathcal{H}$. Thus,
we obtain that the solution for the Hubble parameter as a function of the
scalar field in the case in which the integration constant $C\neq 0$ yields

$H(\varphi)=\sqrt{C_{1}^{2}[(3-\alpha)\varphi]^{-2\beta}-C},$ (28)

where as before $C_{1}$ denotes an integration constant and $\beta$ is defined
by Eq.(27). Clearly, when the integration constant $C=0$, the solution given
by Eq.(28) is reduced to the Eq.(27).

For simplicity, in the following we will focus our analysis on the stage in
which the integration constant $C$ is set to zero. In this form, plugging
Eq.(27) into Eq.(24), the scalar field as a function of the cosmological time
becomes

$\varphi(t)=\frac{1}{(3-\alpha)}\left[\frac{C_{1}(3+\alpha)}{2}\,t+C_{2}\right]^{1/\beta},\,\,\,\,\,\,\mbox{with}\,\,\,\,\,\,\,\alpha\neq
3.$ (29)

The range of the validity for the scalar field during inflation can be
determined from the condition $1\gg w/(\varphi^{2}a^{6}H^{\prime 2})$ results

$\varphi^{-2}\gg\frac{(3+\alpha)}{\beta^{2}},\,\,\,\,\,\mbox{with
which}\,\,\,\,\,\,\,0>\varphi>-\frac{|\beta|}{\sqrt{(3+\alpha)}},\,\,\,\,\mbox{or}\,\,\,\,\,\,0<\varphi<\frac{|\beta|}{\sqrt{(3+\alpha)}},$
(30)

where we have used Eqs.(23) and (27), respectively.

Besides, we find that the phase speed $\dot{\theta}$ in terms of the scalar
field becomes

$\dot{\theta}=\frac{\sqrt{w}}{\varphi^{2}\,a^{3}}=\dot{\theta}_{0}\,\varphi^{-(\beta+1)},\,\,\,\,\,\,\mbox{where}\,\,\,\,\,\dot{\theta}_{0}=C_{1}(3+\alpha)^{1/2}\,(3-\alpha)^{-\beta}.$
(31)

Thus, combining Eqs.(29) and (31) we obtain that the phase as a function of
the time is given by

$\theta(t)=\theta_{1}\,\left[\frac{C_{1}(3+\alpha)}{2}\,t+C_{2}\right]^{-1/\beta}+C_{3},$
(32)

where the quantity $\theta_{1}$ is a constant and it is defined as

$\theta_{1}=\frac{-2\beta\,\dot{\theta}_{0}}{C_{1}\,(3+\alpha)\,(3-\alpha)^{-(\beta+1)}},$
(33)

and $C_{3}$ corresponds to another integration constant. Also, we find that
the effective potential (13) as a function of the scalar field becomes

$V(\varphi)=\left(\frac{3-\alpha}{2}\right)C_{1}^{2}[(3-\alpha)]^{-2\beta}\,\varphi^{-2\beta}=V_{0}\,\varphi^{-2\beta},$
(34)

where we have defined
$V_{0}=\frac{(3-\alpha)}{2}C_{1}^{2}[(3-\alpha)]^{-2\beta}$. Here we observe
that from the approximation $w(\varphi\,a^{3}\,H^{\prime})^{-2}\ll 1$, as much
as the solution for the Hubble parameter $H(\varphi)$ and the reconstructed
potential $V(\varphi)$ do not depend on the parameter $w$ associated to the
angular speed $\dot{\theta}$. This is due to the fact that we have considered
the first term into the expansion (20) and then Eq.(24) does not depend on
$w$. In the situation in which we keep the first and second term of the
expansion given by Eq.(20), the differential equation (22) is modified to

$\left[\frac{2(3+\alpha)}{w}\right]\,HH^{\prime}\simeq\left(\frac{1}{\varphi^{2}a^{6}}\right)^{\prime}\left[1+\frac{w}{2\varphi^{2}a^{6}H^{\prime
2}}\right]-\frac{H^{\prime\prime}}{\varphi^{2}a^{6}H^{\prime}},$ (35)

and together with the equation
$\dot{\varphi}\simeq-w/(2\varphi^{2}a^{6}H^{\prime})$ obtained of (21) are
part of the system to be solved (recall that $H=(a^{\prime}/a)\dot{\varphi}$).
Thus, in order to solve (numerically) this coupled system of differential
equations we would need to specify the value of the parameter $w$.

Figure 1: The plot shows the comparison between the analytic expression given
by Eq.(27) and the numerical solution for the Hubble parameter $H(\varphi)$ in
terms of the scalar field $\varphi$. Here we have used the positive signs of
Eqs.(18) and (19) together with the values of $\alpha=-2.99$, $w=0.01$ and
$C_{1}=0.99$, respectively.

The Fig.1 shows the comparison between the analytic solution given by Eq.(27)
where the approximation $1\gg w/\varphi^{2}a^{6}H^{\prime 2}$ is used and the
numerical solution for the Hubble parameter as a function of the scalar field.
Here we have utilized the values $\alpha=-2.99$, $w=0.01$ and $C_{1}=0.99$. We
note that the comparison is suitable between the numerical solution and the
analytical solution inside of the validity range defined by Eq.(30) for the
scalar field.

Besides, the number of $e-$folds in this case results

$N=-\int_{t_{*}}^{t_{f}}Hdt=-\int_{\varphi_{*}}^{\varphi_{f}}\frac{H}{\dot{\varphi}}d\varphi=\frac{2w_{1}}{w}\int_{\varphi_{*}}^{\varphi_{f}}\frac{H^{\prime}}{H}d\varphi=-\frac{2}{(\alpha-3)}\,\ln\left[\frac{\varphi_{f}}{\varphi_{*}}\right].$
(36)

In the following, the subscripts $f$ and $*$ are used to describe to the epoch
the end of inflation and when the cosmological scales exit the horizon,
respectively.

By assuming that the end of inflation occurs when the slow roll parameter
$\epsilon_{1}\simeq\frac{1}{2}(V^{\prime}/V)^{2}=1$, then we find that the
value of the scalar field at the end of inflation is given by

$\epsilon_{1}(\varphi=\varphi_{f})\simeq\frac{2\beta^{2}}{\varphi_{f}^{2}}=1,\,\,\,\,\,\,\Rightarrow\,\,\,\,\,\,\,\varphi_{f}=\pm\sqrt{2}\beta.$
(37)

In this way, we obtain that the value of the scalar field when the
cosmological scales exit the horizon yields

$\varphi_{*}=-\sqrt{2}\,\beta\,e^{(\alpha-3)N/2},$ (38)

where we have considered the negative sign of $\varphi_{f}$ (see Eq.(37)),
since as we will see later the parameter $\beta$ is a negative quantity.

##  IV Cosmological Perturbations

In this section, we will analyze the scalar and tensor perturbations, in which
the former will be characterized by considering the longitudinal gauge on the
metric. Thus, by considering the longitudinal gauge in the perturbed FRW
metric we write

$ds^{2}=(1+2\Phi)dt^{2}-a^{2}(t)(1-2\Phi)\delta_{ij}dx^{i}dx^{j},$ (39)

where the function $\Phi=\Phi(t,\bf{x})$ corresponds to the Bardeen’s gauge
invariant variable Bardeen:1980kt . Assuming the spatial dependence
$e^{i\bf{kx}}$, where the quantity $k$ is the wavenumber, each Fourier mode
satisfies the following perturbed equations of motion given by

$\dot{\Phi}+H\Phi=\frac{1}{2}\left[\dot{\varphi}\delta\varphi+\varphi^{2}\dot{\theta}\delta\theta\right],$
(40)
$\delta\ddot{\varphi}+3H\delta\dot{\varphi}+\left[\frac{k^{2}}{a^{2}}+V^{\prime\prime}-\dot{\theta}^{2}\right]\delta\varphi-2\varphi\dot{\theta}\delta\dot{\theta}=4\dot{\varphi}\dot{\Phi}-2V^{\prime}\Phi,$
(41)

and

$\delta\ddot{\theta}+\left[3H+\frac{2}{\varphi}\dot{\varphi}\right]\delta\dot{\theta}+\frac{k^{2}}{a^{2}}\,\delta\theta-\frac{2\dot{\varphi}}{\varphi}\delta\dot{\varphi}=4\dot{\theta}\dot{\Phi}.$
(42)

Here, the quantities $\delta\varphi$ and $\delta\theta$ correspond to the
gauge invariant fluctuation variables associated to the respective fields
$\varphi$ and $\theta$, respectively. In order to establish that the
quantities $\delta\phi$ and $\delta\theta$ are gauge invariant fluctuation
variables, we can consider an infinitesimal gauge transformation on the
coordinates $\tilde{x^{\nu}}=x^{\nu}+\delta x^{\nu}$ together with the most
generic perturbed metric; $ds^{2}=a^{2}[(1+2\Phi)d\tau^{2}-2\partial_{i}Bd\tau
dx^{i}-[(1-2\psi)\delta_{ij}+(\partial_{i}\partial_{j}-(1/3)\delta_{ij}\nabla^{2})E]dx^{i}dx^{j}]$,
with $\tau$ the conformal time and a detailed explication see Refs.
Mukhanov:1990me ; Riotto:2002yw . Since $\varphi$ and $\theta$ are scalar
fields (amplitude and phase), these satisfy the transformation laws for gauge
invariant (GI);
$\delta\varphi_{GI}=\delta\varphi-\varphi^{\prime}(E^{\prime}/2-B)$ and
$\delta\theta_{GI}=\delta\theta-\theta^{\prime}(E^{\prime}/2-B)$, where a
prime now indicates differentiation wrt to the conformal time Mukhanov:1990me
; Riotto:2002yw . Thus, from the longitudinal gauge on the metric given by
Eq.(39), we have $B=E=0$ and $\Phi=\psi$ and then the quantities
$\delta\varphi$ and $\delta\theta$ are gauge invariant fluctuation variables.

By considering large scale perturbations in which $k\ll aH$ and neglecting the
term $\dot{\Phi}$ and those terms which include second order time derivative,
then the above equations of motion reduce to

$H\Phi\simeq\frac{1}{2}\left[\dot{\varphi}\delta\varphi+\varphi^{2}\dot{\theta}\delta\theta\right],$
(43)
$3H\delta\dot{\varphi}+V^{\prime\prime}\delta\varphi\simeq-2V^{\prime}\Phi,$
(44)

and

$3H\delta\dot{\theta}\simeq
0,\,\,\,\,\,\Rightarrow\,\,\,\delta\theta=\mathcal{C}.$ (45)

Here we have considered that as we need the non-decreasing modes on large
scale in our model, which are determined to be weakly time-dependent
Polarski:1992dq ; Polarski:1994rz , then we can consistently ignore the term
associated to $\dot{\Phi}$, including those involving two time derivatives. In
relation to the existence of growing and decaying modes is attributed to the
the observation outlined in Ref.Starobinsky:1982ee , with a more detailed
explication provided in Ref.Polarski:1992dq .

Thus, we find that the Eq.(44) can be written as

$\delta\varphi^{\prime}+\frac{1}{3H\dot{\varphi}}\left[V^{\prime\prime}+\frac{V^{\prime}}{H}\dot{\varphi}\right]\delta\varphi+\left(\frac{V^{\prime}\varphi^{2}\dot{\theta}}{3H\dot{\varphi}}\right)\mathcal{C}=0,$
(46)

where we have used $\delta\dot{\varphi}=\dot{\varphi}\delta\varphi^{\prime}$.

Figure 2: The plot shows the numerical solution for the gauge invariant
fluctuation variable $\delta\varphi$ versus the scalar field $\varphi$ given
by Eq.(46), for different values of the constant $\mathcal{C}$. In this plot
we have utilized the values $C_{1}=0$, $C_{2}=0$ and $\alpha=-2.99$.

We note that the relation between the real variables $\delta\varphi$ and
$\delta\theta$ with the variation of the complex field $\delta\Psi$ from
Eq.(3) becomes $\delta\Psi=(\Psi/\varphi)[\delta\varphi+i\,\delta\theta]$.
Under the large scale approximation, we find that the relation between the
Bardeen’s variable $\Phi$ and the variation of the complex field $\delta\Psi$
can be written as
$2H\Phi\simeq\delta\Psi[(\varphi/\Psi)\dot{\Psi}-i\,\dot{\theta}](\varphi/\Psi)-\mathcal{C}[\dot{\theta}(1-\varphi^{2})+i\,(\varphi/\Psi)\dot{\Psi}]$,
where we have utilized Eqs.(43) and (45), respectively. Thus, this relation
indicates that the metric perturbation is determined when the evolution of the
fields are known.

In order to find an analytical solution to the gauge invariant variable
$\delta\varphi$, we consider for the integration constant as $\mathcal{C}=0$.
In this way, we find that the solution of Eq.(46) can be written as

$\delta\varphi=\mathcal{C}_{0}\,\exp\left[-\int_{\varphi}^{\varphi_{f}}\,\frac{1}{3H\dot{\varphi}}\left(V^{\prime\prime}+\frac{V^{\prime}}{H}\dot{\varphi}\right)d\varphi\right],$
(47)

where $\mathcal{C}_{0}$ corresponds to an integration constant. Therefore, the
variable $\Phi$ becomes

$\Phi\simeq\frac{1}{2}\frac{\dot{\varphi}}{H}\,\mathcal{C}_{0}\,\exp\left[-\int_{\varphi}^{\varphi_{f}}\,\frac{1}{3H\dot{\varphi}}\left(V^{\prime\prime}+\frac{V^{\prime}}{H}\dot{\varphi}\right)d\varphi\right].$
(48)

Thus, from Eq.(24), we find that Eq.(47) can be rewritten as

$\delta\varphi\simeq\,\mathcal{C}_{0}\exp\left[\int_{\varphi}^{\varphi_{f}}\,\frac{2w_{1}H^{\prime}}{3wH(H^{2}+C)}\left(V^{\prime\prime}-\frac{w(H^{2}+C)V^{\prime}}{2w_{1}H\,H^{\prime}}\right)d\varphi\right],$
(49)

and analogously for $\Phi$, we get

$\Phi\simeq-\mathcal{C}_{0}\,\left(\frac{w(H^{2}+C)}{4w_{1}H\,H^{\prime}}\right)\,\exp\left[\int_{\varphi}^{\varphi_{f}}\,\frac{2w_{1}H^{\prime}}{3wH(H^{2}+C)}\left(V^{\prime\prime}-\frac{w(H^{2}+C)V^{\prime}}{2w_{1}H\,H^{\prime}}\right)d\varphi\right].$
(50)

The Fig.2 shows the numerical solution given by Eq.(46) for the fluctuation
variable $\delta\varphi$ in terms of the scalar field $\varphi$, for different
values of the constant $\mathcal{C}$ defined by Eq.(45). Here we have used the
values $C_{1}=1$, $C_{2}=0$ and $\alpha=-2.99$. In order to write down values
for the numerical solution of Eq.(46), we have utilized Eqs.(27), (29), (31)
and (34) for the background variables. From this plot, we observe that for
values of integration constant $\mathcal{C}\ll 1$, the numerical solutions of
$\delta\varphi(\varphi)$ exhibit behaviors closely resembling those observed
when $\mathcal{C}=0$. Hence, given that the gauge-invariant fluctuation
variable $\delta\varphi$ remains significantly smaller than unity
($\delta\varphi\ll 1$), it is pertinent to consider the scenario where
$\mathcal{C}=0$ as a suitable approximation.

Following Refs.Starobinsky:1994mh ; Langlois:2008mn ; Starobinsky:2001xq , we
assume that gauge invariant density fluctuation
$\delta\rho^{c}/\rho=\delta_{H}$ is defined as
$\delta_{H}=-(2/3)(k/aH)^{2}\Phi$ and considering that the curvature
perturbation due to primordially adiabatic fluctuation
$\Phi=[1+2/(3(1+p/\rho))]^{-1}\mathcal{C}_{0}$, then from Eq.(49) we find that
the density perturbation at the horizon crossing in which $k=aH$, results

$\delta_{H}=f\left(\delta\varphi\,\,\,\exp\left[-\int_{\varphi}^{\varphi_{f}}\,\frac{2w_{1}H^{\prime}}{3wH(H^{2}+C)}\left(V^{\prime\prime}-\frac{w(H^{2}+C)V^{\prime}}{2w_{1}H\,H^{\prime}}\right)d\varphi\right]\right)\Big{|}_{\varphi=\varphi_{*}}=f\left(\delta\varphi\,e^{{-\mathcal{F}}(\varphi)}\right)\Big{|}_{\varphi=\varphi_{*}},$
(51)

where the function $\mathcal{F}(\varphi)$ is defined as

$\mathcal{F}(\varphi)=\int_{\varphi}^{\varphi_{f}}\,\frac{2w_{1}H^{\prime}}{3wH(H^{2}+C)}\left(V^{\prime\prime}-\frac{w(H^{2}+C)V^{\prime}}{2w_{1}H\,H^{\prime}}\right)d\varphi.$
(52)

Further, the quantity $f$ is a constant and in the specific case associated to
the radiation domination $f$ takes value $f=4/9$ and for matter domination we
have $f=2/5$. Here the constant $f=(2/3)[1+2/(3(1+p/\rho))]^{-1}$ is
introduced by normalization at the second crossing after inflation during the
radiation or matter domination Refs.Starobinsky:1994mh ; Langlois:2008mn ;
Starobinsky:2001xq . Also the minus sign can be omitted, since this sign may
be absorbed into the stochastic variable $\delta\varphi$. Thus, the large-
angular-scale anisotropy of background due to the Sachs-Wolfe effect is well
defined as $\delta T/T=\Phi/3$ Sachs:1967er .

Besides, the fluctuations $\delta\varphi$ are generated by small scale
perturbations and then they can be assumed as free massless scalar field in
which are described by independent random variables such that at the horizon
crossing we have $\delta\varphi=H/2\pi$, see e.g. Refs.Lyth:1998xn ;
Liddle:2000cg . By considering that the scalar power spectrum
$\mathcal{P_{S}}=(25/4)\delta_{H}^{2}$ Bassett:2005xm , then we find that the
spectrum at the horizon crossing can be written as

$\mathcal{P_{S}}(\varphi=\varphi_{*})=\left(\frac{25}{4}\right)\,\left(\frac{f}{2\pi}\right)^{2}\,\left[H^{2}\,e^{-2\mathcal{F}(\varphi)}\right]\Big{|}_{\varphi=\varphi_{*}}.$
(53)

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

$n_{s}-1=\frac{d\ln\mathcal{P_{S}}}{d\ln k},$ (54)

in which the wavenumber $k$ is related to the number of $e$-folds $N$ by the
relationship $d\ln k\simeq d\,N$. Thus, from Eq.(53) we find that the index
$n_{s}$ can be written as

$n_{s}\simeq
1+2\frac{\dot{\varphi}}{H}\,\left[\frac{H^{\prime}}{H}-\mathcal{F}\,^{\prime}(\varphi)\right]\simeq
1-\left(\frac{w}{w_{1}}\right)\,\frac{(H^{2}+C)}{H\,H^{\prime}}\,\left[\frac{H^{\prime}}{H}-\mathcal{F}\,^{\prime}(\varphi)\right].$
(55)

Besides, it is well known that the production of tensor perturbations during
the inflationary epoch would produce gravitational wave. In this context, the
spectrum of the tensor perturbations $\mathcal{P_{T}}$ after Hubble exist can
be approximated to Bassett:2005xm ; Liddle:2000cg

$\mathcal{P_{T}}\simeq\,4\left(\frac{H}{\pi}\right)^{2}\Big{|}_{\varphi=\varphi_{*}}.$
(56)

Also, an important observational parameter corresponds to the tensor to scalar
ratio which is defined as

$r=\frac{\mathcal{P_{T}}}{\mathcal{P_{S}}},$ (57)

and in our case this ratio can be written as

$r\simeq\frac{64}{25}\frac{1}{f^{2}}e^{2\mathcal{F}(\varphi)}.$ (58)

For our model, from Eq.(23), we find that the function $\mathcal{F}(\varphi)$
defined by Eq.(52) yields

$\mathcal{F}(\varphi)=\frac{\beta(\alpha-3)}{3}\Bigg{\\{}\ln(\varphi)+\frac{\beta(1+2\beta)}{\varphi^{2}}\frac{w_{1}}{w}\bigg{(}1+{}_{2}F_{1}\Big{[}1,-\frac{1}{\beta},\frac{-1+\beta}{\beta},-\frac{C(\varphi(\alpha-3))^{2\beta}}{C_{1}^{2}}\Big{]}\bigg{)}\Bigg{\\}}\Bigg{|}^{\varphi}_{\varphi_{f}},$

where the quantity ${}_{2}F_{1}$ corresponds to the hypergeometric function.
In the particular case in which the integration constant $C=0$, the above
function reduces to

$\mathcal{F}(\varphi)=\frac{\beta(3-\alpha)}{3}\Bigg{\\{}\ln(\varphi)+\frac{\beta(1+2\beta)}{\varphi^{2}}\frac{w_{1}}{w}\Bigg{\\}}\Bigg{|}^{\varphi}_{\varphi_{f}}.$
(59)

By considering Eqs.(55) and (59) we obtain the that scalar spectral index
$n_{s}$ can be written as

$n_{s}(\varphi=\varphi_{*})=n_{s_{*}}\simeq-(2+\alpha)+\frac{(\alpha^{2}-9)}{3}\left[1-\frac{2\beta(1+2\beta)}{\varphi^{2}}\frac{w_{1}}{w}\right]\Bigg{|}_{\varphi_{*}}.$
(60)

Also, the tensor-to-scalar ratio of the model is given by

$r_{*}\simeq\frac{64}{25}\frac{1}{f^{2}}\exp\bigg{[}\frac{2\beta(3-\alpha)}{3}\Big{(}\ln(\frac{\varphi}{2\sqrt{\beta}})+\frac{1+2\beta}{3+\alpha}\big{(}\frac{\beta}{\varphi^{2}}-\frac{1}{4}\big{)}\Big{)}\bigg{]}\Bigg{|}_{\varphi_{*}}.$
(61)

Thus, considering that the scalar spectral index at the horizon crossing takes
the value $n_{s}{{}_{*}}=0.967$ and using Eqs.(37) and (38), we find
numerically that the parameter $\alpha$ has a negative value given by

$\alpha\simeq\,-2.989.$ (62)

Here we have considered that at the crossing the number of $e-$ folds $N=60$.

Besides, from the scalar power spectrum defined by Eq.(53) and considering
that at the crossing this quantity is
$\mathcal{P_{S}}(\varphi=\varphi_{*})\simeq 2.2\times 10^{-9}$ together with
$\alpha=-2.989$, we find that the integration constant $C_{1}=\pm\,1.02\times
10^{-4}$. However, as the Hubble parameter is a positive quantity we only have
to consider $C_{1}=\,1.04\times 10^{-4}$. This value of $C_{1}$ suggests that
the quantity $V_{0}=\frac{(3-\alpha)}{2}C_{1}^{2}(3-\alpha)^{-2\beta}$
(amplitude of the potential) associated to the reconstructed potential
$V(\varphi)=V_{0}\varphi^{-2\beta}$ becomes $V_{0}\simeq 3.1\times 10^{-8}$,
when $\alpha=-2.989$ and $C_{1}\simeq 10^{-4}$. In this form, we find that the
value of the effective potential at the end of inflation results
$V(\varphi=\varphi_{f})\simeq 3.03\times 10^{-8}$ (in units of $M_{p}^{4}$,
where $M_{p}$ the Planck mass ) and at the crossing in which the number of
$e-$folds $N=60$ the potential $V(\varphi=\varphi_{*})\simeq 5.88\times
10^{-8}$ (in units of $M_{p}^{4}$). In this context, we obtain that the energy
density of the universe during inflation in our model becomes $\rho\sim
V\sim\mathcal{O}(10^{-7})\sim\mathcal{O}(10^{-8})$ $M_{p}^{4}$ and this energy
scale is similar to that obtained in different inflationary models during the
early universe, see e.g., Guth:1980zm ; Linde:1981mu ; Albrecht:1982wi ;
Bassett:2005xm . As before we have used that the number of $e-$folds $N=60$
and also we have considered the value $f=4/9$.

Also, from Eq.(61) we find that the tensor to scalar ratio $r_{*}\sim
10^{-78}$, when we consider the values of $N=60$, $f=4/9$, together with the
value given by Eq.(62) for the parameter $\alpha$. In this sense, we find that
our model predicts that the tensor to scalar ratio ratio $r\simeq 0$, see
right panel of Fig.3.

In Fig.3 we show the evolution of the observational parameters $n_{s}$ and $r$
versus the parameter $\alpha$ associated to Eq.(14). The left panel shows that
the observational value for the scalar spectral index $n_{s}=0.967$ takes
place for the value $\alpha\simeq-2.989$. In order to write down the scalar
spectral index $n_{s}$ as a function of the parameter $\alpha$, we consider
Eq.(60) together with the Eq.(38) in which the number of $e-$folds $N=60$. The
right panel shows the evolution of the tensor to scalar ratio $r$ (logarithmic
scale) in terms of the parameter $\alpha$. Also, in order to write down the
observational parameter $r$ as a function of $\alpha$, we consider Eqs.(37),
(38) and (61), where we have assumed $N=60$ and the parameter $f=4/9$,
respectively. From this panel, we note that the tensor to scalar ratio
$r\simeq 0$, when the observational parameter $n_{s}\sim 0.97$.

Figure 3: Observational parameters: The left panel shows the behavior of the
scalar spectral index versus the parameter $\alpha$ and the right panel shows
the tensor to scalar ratio $r$ (logarithmic scale) versus the parameter
$\alpha$. From the left panel we observe that the value $n_{s}=0.967$ for the
scalar spectral index corresponds to $\alpha=-2.989$. Additionally, from the
right panel we note that the tensor to scalar ratio $r\sim 0$ for values of
$\alpha$ in which $n_{s}\simeq 0.97$. Here we have used that at the crossing
the number of $e-$folds $N=60$ and $f=4/9$.

On the other hand, in order to give an appropriate comparison and distinction
between the real inflaton field and the complex field, we have to consider the
situation in which the parameter $\alpha>-3$. In this context, when we analyze
the case in which $\alpha>-3$ for a real field, see Ref.Motohashi:2014ppa ,
the reconstructed potentials in terms of the scalar field are an exponential
potential associated to power-law inflation Abbott:1984fp  and a hyperbolic
cosine potential which is similar to that found in Ref.Barrow:1994nt
(inflationary solutions). For the complex field we have found that the
reconstructed potential corresponds to a power-law potential
$V(\varphi)\propto\varphi^{-2\beta}$ becoming different to the real field.
Here the evolution of the scalar field on the reconstructed potential in the
framework of complex field becomes similar to the hyperbolic cosine potential,
since both potentials present a minimum at $\varphi=0$, and the evolution of
the field takes place from large-$\varphi$ towards the minimum. In relation to
the observational parameters and in particular from the scalar spectral index
$n_{s}$, it is found that the parameter $\alpha$ takes the value
$\alpha=-3.02$ when the index $n_{s}=0.96$ Motohashi:2014ppa . In our case
using a complex field we have obtained the value $\alpha=-2.989$ and it
suggests that both constrains on the parameter $\alpha$ are similar from
observational parameter $n_{s}$. However, the analysis realized assuming a
real inflaton field predicts $\alpha<-3$ and from a complex inflaton field
suggests $\alpha>-3$ from the scalar spectral index.

##  V Conclusions and Remarks

In the present manuscript, we have studied the complex inflation in which a
complex field as inflaton is the main responsible to drive inflation. We have
worked with an expression for the angular velocity
$\dot{\theta}\propto\varphi^{-2}\,a^{-3}$ determined from the motion equation
for the phase $\theta$ that leads to some interesting inflationary solution.

By applying the constant-roll condition on the absolute value of the complex
scalar field, we have found a specific form for the Hubble parameter $H$ and
of the effective potential as a function of the scalar field, by assuming that
during inflationary era the term $w(\varphi\,a^{3}\,H^{\prime})^{-2}\ll 1$.
Here the solutions for the Hubble parameter and the effective potential have a
dependency power-law type with the scalar field $\varphi$. Also, we have found
that the evolution of the phase as a function of the cosmological time scale
as $\theta(t)\propto t^{-1/\beta}$. In this context, we have noted that in the
approximation $w(\varphi\,a^{3}\,H^{\prime})^{-2}\ll 1$ the parameter related
to the angular speed $w$ is not present in the solutions of $H(\varphi)$ and
$V(\varphi)$, then this parameter can not be constrained. Besides, from the
condition $w(\varphi\,a^{3}\,H^{\prime})^{-2}\ll 1$ we have found different
ranges of validity for the scalar field, in which an upper bound is obtained
for positive values of $\varphi$ or a lower bound for negative values of
$\varphi$, see Eq.(30). In this form, we have utilized these ranges for the
scalar field to constraint the different results found on parameter-space from
the observational parameters.

Additionally, we investigated the corresponding scalar perturbations where we
have obtained the perturbed equations of motion using the longitudinal gauge.
By considering large scale perturbations and neglecting some terms we have
reduced the equations system and then we have found solutions for the
Bardeen’s variable and the perturbation associated to the scalar field. A
general relation for the scalar power spectrum and the scalar spectral index
$n_{s}$ are given by Eqs.(53) and (55), respectively. In particular from the
background solutions for the Hubble parameter and the effective potential in
terms of the scalar field we have found analytical quantities for the
observational parameters such as; scalar spectral index and the tensor to
scalar ratio.

By comparing the obtained results with the datasets coming from CMB
anisotropies, we have attained the observational constraints on the parameters
space of the model, in particular, the constant-roll parameter $\alpha$ see
Eq.(62) and the integration constant associated to the Hubble parameter
$C_{1}\sim\mathcal{O}(10^{-4})$.

Also, from Eq.(61) we have obtained that the tensor to scalar ratio
$r_{*}\sim\mathcal{O}(10^{-78})$, when we consider the values of $N=60$ and
$\alpha=-2.989$. This value for the tensor to scalar ratio suggests that our
analysis done in the framework of constant roll with a complex inflaton,
predicts a ratio $r\sim 0$, see Fig.3. Additionally, from Fig.3 (left panel)
we have ratified that the observational parameter $n_{s}\sim\mathcal{O}(1)$
when the parameter $\alpha\sim-3$.

In relation to the condition give by Eq.(1), we see the difficult to
understand how the scalar field could begin its movement on the reconstructed
potential given by (34) with the appropriate initial condition
$\ddot{\varphi}=-(3+\alpha)H\varphi$, in which the potential satisfies the
relation $V^{\prime}(\varphi)=\alpha H\dot{\varphi}+\varphi\dot{\theta}^{2}$.
In this sense, the study of this issue concerning the consequences of the
constant roll initial condition from the dynamics of the inflaton and how
would be the special features that would be a consequence this initial
conditions in our scenario would be a very interesting subject for future
research concerning our model.

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