Complex Scalar elds in Scalar-Tensor and Scalar-Torsion theories Andronikos Paliathanasis1 2 3 1Institute of Systems Science Durban University of Technology

2025-04-27 0 0 754.18KB 19 页 10玖币

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Complex Scalar ﬁelds in Scalar-Tensor and Scalar-Torsion theories

Andronikos Paliathanasis1, 2, 3, ∗

1Institute of Systems Science, Durban University of Technology,

PO Box 1334, Durban 4000, South Africa

2Instituto de Ciencias F´ısicas y Matem´aticas,

Universidad Austral de Chile, Valdivia 5090000, Chile

3Mathematical Physics and Computational Statistics Research Laboratory,

Department of Environment, Ionian University, Zakinthos 29100, Greece

(Dated: October 11, 2022)

We investigate the cosmological dynamics in a spatially ﬂat Friedmann–Lemaˆıtre–

Robertson–Walker geometry in scalar-tensor and scalar-torsion theories where the

nonminimally coupled scalar ﬁeld is a complex ﬁeld. We derive the cosmological

ﬁeld equations and we make use of dimensionless variables in order to determine the

stationary points and determine their stability properties. The physical properties

of the stationary points are discussed while we ﬁnd that the two-diﬀerent theories,

scalar-tensor and scalar-torsion theories, share many common features in terms of

the evolution of the physical variables in the background space.

PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x

Keywords: Scalar ﬁeld; Complex ﬁeld; Scalar-tensor; Scalar-torsion; Dynamical analysis.

1. INTRODUCTION

Scalar ﬁelds in gravitational theory is a simple mechanism to introduce new degrees of

freedom which can play an important role in the description of observable cosmological

phenomena [1]. The early acceleration phase of the universe is attributed to a scalar ﬁeld

known as inﬂaton [2–5]. In addition, scalar ﬁelds have been used to describe also the late-time

acceleration phase of the universe attributed to dark energy, or other matter components

∗Electronic address: anpaliat@phys.uoa.gr

arXiv:2210.04177v1 [gr-qc] 9 Oct 2022

such as dark matter, see for instance [6–15] and references therein.

Multi-scalar ﬁelds have been widely studied in the literature. Some well-known two-scalar

ﬁeld models are the quintom [16] or the Chiral model which leads to hyperbolic inﬂation

[17], while other proposed multi-scalar ﬁelds theories can be found for instance in [18–23]

and references therein. A simple mechanism to introduce a multi-scalar ﬁeld theory is to

consider the existence of a complex scalar ﬁeld, the real and imaginary parts of which give

the equivalent of a two scalar-ﬁeld theory [24, 25].

An inﬂationary model with a complex scalar ﬁeld was proposed in [26]. Speciﬁcally it

was found that, when inﬂation occurs, the imaginary component of the complex scalar ﬁeld

does not contribute in the cosmological ﬂuid, that is, the phase of the complex scalar ﬁeld

is constant. The cosmological perturbations with a complex scalar ﬁeld were investigated

in [27]. Furthermore, in [28] the authors used cosmological observations to reconstruct

the quintessence potential for a complex scalar ﬁeld. A nonminimally coupled scalar ﬁeld

cosmological model has been studied in [29], while for some recent studies of complex scalar

ﬁeld cosmological models we refer the reader to [30–37].

In this study we consider the scalar-tensor and the scalar-torsion theories with a complex

scalar ﬁeld [38]. In scalar-tensor theory, the scalar ﬁeld is minimally coupled to gravity. The

scalar ﬁeld interacts with the gravitational Action Integral of Einstein’s General Relativity,

that is, the Ricci-scalar of the Levi-Civita connection, the scalar-tensor theory satisﬁes

the Machian Principle [39] and the theory is deﬁned in the so-called Jordan frame [40].

The Brans-Dicke Lagrangian [39] is the most common scalar-tensor theory. On the other

hand, the scalar-torsion theory is the equivalent of scalar-tensor model in teleparallelism.

In the latter, the fundamental geometric invariant is the torsion scalar determined by the

curvatureless Weitzenb¨ock connection [41, 42]. There are many important results in the

literature on the cosmological studies of the scalar-tensor and scalar-torsion theories, for an

extended discussion we refer the reader to [43–54] and references therein.

The purpose of this study is to investigate the eﬀects of a complex scalar ﬁeld in the

evolution of the cosmological dynamics for the background space for these two diﬀerent

gravitational theories and to compare the results. Such analysis provides us with important

results in order to understand the diﬀerences between the use of the Ricci-scalar and of

the torsion scalar in the background geometry. We use dimensionless variables to perform

a detailed analysis of the dynamical systems which describe the evolution of the physical

variables. Such an approach has been widely studied before with many interesting results

about the viability of proposed gravitational theories, see for instance [55–60]. The plan of

the paper is as follows.

In Section 2 we consider a spatially ﬂat Friedmann–Lemaˆıtre–Robertson–Walker geom-

etry in scalar-tensor gravitational theory with complex scalar ﬁeld. We derive the ﬁeld

equations and we write the point-like Lagrangian. Moreover, we determined the stationary

points for the ﬁeld equations and we investigate their stability properties. In Section 3 we

perform a similar analysis but now in the context of scalar-tensor theory. Finally, in Section

4 we summarize our results and compare the physical results provided by the two theories

and draw our conclusions.

2. SCALAR-TENSOR COSMOLOGY

Consider the complex scalar ﬁeld ψin the case of scalar-tensor theory [38], for which the

gravitational Action Integral is deﬁned as

ST ensor =Zd4x√−gF(|ψ|)R+1

2gµν ψ,µψ∗

,ν −V(|ψ|),(1)

where Ris the Ricci scalar related to the Levi-Civita connection for the metric tensor gµν ;

|ψ|is the norm of the complex ﬁeld ψ, that is, |ψ|2=ψψ∗,F(|ψ|) is the coupling function

between the gravitational and the scalar ﬁeld ψand V(|ψ|) is the potential function which

drives the dynamics. The Action Integral (1) admits the U(1) symmetry.

The Brans-Dicke theory with a complex ﬁeld is recovered for F(|ψ|) = F0|ψ|2, that is,

the Action Integral (1) is [38]

SBD =Zd4x√−gF0|ψ|2R+1

2gµν ψ,µψ∗

,ν −V(|ψ|).(2)

In the case of a spatially ﬂat FLRW universe with line element,

ds2=−N2(t)dt2+a2(t)dx2+dy2+dz2,(3)

we derive for the Ricciscalar

R= 6 1

N˙

H+ 12H2,(4)

where H=1

˙a

a, ˙a=da

dt , is the Hubble function.

We substitute (4) into (1) which by integration by parts gives the point-like Lagrangian

LT ensor N, a, ˙a, ψ, ˙

ψ=1

N6F(|ψ|)a˙a2+ 6 ˙

F(|ψ|)a2˙a+1

2a3˙

ψ˙

ψ∗−a3NV (|ψ|),(5)

or in the case of Brans-Dicke

LBD N, a, ˙a, ψ, ˙

ψ=1

N6F0|ψ|2a˙a2+ 6F0|ψ|2·a2˙a+1

2a3˙

ψ˙

ψ∗−a3NV (|ψ|).(6)

We focus now in the Brans-Dicke theory in which the ﬁeld equations are described by the

point-like Lagrangian (6). Moreover, the complex scalar ﬁeld is written with the use of the

polar form, ψ(t) = φ(t)eiθ(t), such that the Lagrangian (6) becomes

LBD N, a, ˙a, φ, ˙

φ, θ, ˙

θ=1

N6F0φ2a˙a2+ 12F φa2˙a˙

φ+1

2a3˙

φ2+φ2˙

θ2−a3NV (φ).

(7)

It is obvious that the Lagrangian function (7) describes a multi-scalar ﬁeld cosmological

model, where φis the Brans-Dicke ﬁeld and θis a second-scalar ﬁeld minimally coupled to

gravity but coupled to the Brans-Dicke ﬁeld φ. The U(1) for the point-like Lagrangian (7)

provides the invariant transformation θ=θ+ε, and the conservation law I0=1

Na3φ2˙

θ.

Variation with respect to the dynamical variables {N, a, φ, θ}of the Lagrangian (7) pro-

vides the cosmological ﬁeld equations which are

0=6F0φ2H2+ 12F φH ˙

φ+1

2˙

φ2+φ2˙

θ2+V(φ),(8)

0 = 2F0φ22˙

H+ 3H2+ 8F0Hφ ˙

φ−1

2˙

φ2+ 4F0˙

φ2−1

2˙

φ2˙

θ2+ 4F0φ¨

φ+V(φ),(9)

0 = ¨

φ+φ12F0˙

H−˙

θ2+ 3 ˙

φ˙

H+ 12F0φH2+V,φ (10)

and

0 = φ¨

θ+2˙

φ+ 3Hφ˙

θor I0=a3φ2˙

θ, (11)

where without loss of generality we have assumed N(t) = 1.

2.1. Cosmological dynamics

In order to reconstruct the cosmological history provided by this speciﬁc complex scalar-

tensor theory we make use of dimensionless variables in the context of H-normalization and

we investigate the dynamical evolution of the ﬁeld equations (8)-(11) by determining the

stationary points and their stability properties.

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摘要：

ComplexScalareldsinScalar-TensorandScalar-TorsiontheoriesAndronikosPaliathanasis1,2,3,1InstituteofSystemsScience,DurbanUniversityofTechnology,POBox1334,Durban4000,SouthAfrica2InstitutodeCienciasFsicasyMatematicas,UniversidadAustraldeChile,Valdivia5090000,Chile3MathematicalPhysicsandComputationa...

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Complex Scalar elds in Scalar-Tensor and Scalar-Torsion theories Andronikos Paliathanasis1 2 3 1Institute of Systems Science Durban University of Technology

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