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Eur. Phys. J. C (2019) 79:324

https://doi.org/10.1140/epjc/s10052-019-6833-1

Regular Article - Theoretical Physics

Generalized tachyonic teleparallel cosmology

Sebastian Bahamonde

1,2,3,4,a

, Mihai Marciu

5,b

, Jackson Levi Said

6,7,c

Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411 Tartu, Estonia

Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK

School of Mathematics and Physics, University of Lincoln, Brayford Pool, Lincoln LN6 7TS, UK

Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK

Faculty of Physics, University of Bucharest, 405 Atomi¸stilor, M˘agurele, POB MG-11, 077125 Bucharest, Romania

Institute of Space Sciences and Astronomy, University of Malta, Msida MSD 2080, Malta

Department of Physics, University of Malta, Msida MSD 2080, Malta

Received: 18 January 2019 / Accepted: 1 April 2019 / Published online: 10 April 2019

Abstract In this paper we propose a new dark energy model

in the teleparallel alternative of general relativity, by con-

sidering a generalized non-minimal coupling of a tachyonic

scalar ﬁeld with the teleparallel boundary term. Within the

framework of teleparallel gravity, the boundary coupling

term is associated with the divergence of the torsion vec-

tor. Considering the linear stability technique for various

potentials and couplings, we have analyzed the dynamical

properties of the present tachyonic dark energy model in the

phase space, uncovering the corresponding essential dynam-

ical features. Our study of the phase space structure revealed

that for a speciﬁc class of potential energy, this model exhibits

various critical points which are related to different cosmo-

logical behaviors, such as accelerated expansion and scaling

solutions, determining the existence conditions and the cor-

responding physical features.

1 Introduction

Einstein’s theory of General Relativity (GR) has experienced

unprecedented success in its power to explain astrophysical

phenomena ranging from Solar System tests to strong ﬁeld

gravitational wave physics [1]. However, this theory of grav-

ity has required important modiﬁcations due to observational

realities which have arisen over the past few decades. In terms

of the energy budget of the Universe, the ﬁrst modiﬁcation

comes from observations of galaxies and their dynamical

structure, which is only possible with the addition of approxi-

mately purely gravitational interacting particles, namely dark

e-mails: sbahamonde@ut.ee; sebastian.beltran.14@ucl.ac.uk

e-mail: mihai.marciu@drd.unibuc.ro

e-mail: jackson.said@um.edu.mt

matter, which may potentially be beyond the standard model

of particle physics [2,3]. The second and larger contribution

to the modiﬁcation of GR comes from the relatively recent

observation of the accelerating expansion of the Universe

[4,5] which is an observational fact, called dark energy.This

can be accounted for in GR through the introduction of the

cosmological constant however this poses its own problems

[6,7]. The CDM is the most successful model which can

explain the current accelerated expansion and the evolution

of the observable Universe at the level of background dynam-

ics, involving the superposition between the dark matter ﬂuid

and the cosmological constant. On the other hand, the early

period of the Universe also features several facets that need

remedy. Most prominently, for CDM to correctly produce

our current picture of the Universe, a period of cosmological

inﬂation must of taken place [8–11] which would allow for a

natural solution to the horizon problem. However, this may

also necessitate further particles beyond the standard model

[12]. Cosmologically, the time that should be best described

by the CDM model is the present or late-time period of the

Universe. However, recent releases by the Planck collabora-

tion have revealed a growing tension in the local and global

measurements of H

and f σ

[13].

Now, it may be the case that the fundamental and observa-

tional problems surrounding CDM may be resolved in the

coming years, or it may be the case that CDM needs to be

changed in some way. Over the previous decades there has

been concerted efforts in extending GR to account for certain

elements of these problems [14]. However, it may also be the

case that a new paradigm is needed to confront the grow-

ing requirements of constructing a viable theory of gravity.

One such treatment is the teleparallel gravity approach where

the Levi-Civita connection is replaced with the Weitzenböck

connection [15]. The connection plays a crucial role in grav-

123

324 Page 2 of 10 Eur. Phys. J. C (2019) 79 :324

itational physics in that the expression of curvature, torsion,

or nonmetricity is not a property of the manifold itself but

of the connection which relates the elemental tangent spaces

of the manifold [16]. In this way, one can choose to con-

sider gravitation in terms of the curvatureless Weitzenböck

connection which also observes the metricity property.

In teleparallel gravity, the gravitation is characterized by

the torsion tensor, T

μν

, instead of the Riemann tensor in GR

and its extensions. As in the GR framework, a Lagrangian

can be constructed to represent the gravitational ﬁeld. Of

particular interest, in teleparallel gravity, is that a Lagrangian

can be constructed such that it is equivalent to the Einstein–

Hilbert Lagrangian up to a total divergence or boundary term,

B, that is

R =−T + B, (1)

where T is called the torsion scalar and contains only sec-

ond order terms, while the boundary term, B, encapsulates

the higher order contributions to the Ricci scalar, R.This

is the so-called teleparallel equivalent of general relativity

(TEGR) which is equivalent to GR at the level of the ﬁeld

equations [17,18]. The natural consequence of this realiza-

tion is that every test of GR also becomes a test of TEGR

with the difference that in TEGR gravity is again observed

to act as a (Lorentz) force, and that the barrier with the quan-

tum regime seems to have less tension [15]. Moreover, due

to the second order nature of the torsion tensor, even in the

case where the theory is extended to an f (T ) Lagrangian,

the resulting ﬁeld equations remain second order which has

important consequences for the gravitational wave polariza-

tion modes of the theory. In fact, f (T ) gravity continues to

exhibit the equivalent polarization modes as in the GR and

TEGR settings [19].

Given the decomposition of the Ricci scalar into the sec-

ond and fourth order terms expressed in Eq. (1), we con-

sider the analysis of a tachyonic dark energy model non-

minimally coupled to the aforementioned separate contribu-

tions through different functionals. In particular, we choose

to study a tachyonic scalar ﬁeld which has been shown to pro-

duce an inﬂationary epoch and late-time accelerating solu-

tions that do not violate the strong energy condition [20,21].

These models are partially inspired by string theory [22–24]

and k-essence theory [25]. In the teleparallel setting, scalar

ﬁelds have been investigated to a moderate degree, with vari-

ous extensions having been investigated in the cosmological

context [26–31]. Furthermore, the effects of the boundary

couplings in scalar tensor theories have been discussed in

various papers [28,32–44]. A recent review on various stud-

ies related to dynamical analysis in different cosmological

constructions can be found in [45]. In Ref. [46], a tachyonic

ﬁeld is investigated for the modiﬁed teleparallel setting, late-

time accelerating attractor solutions are found with a ﬁeld

equation of state that realistically tends to the current dark

energy value. This approach was extended to more general

models in Ref. [47] where the attractor solution context is

further clariﬁed.

The work is divided as follows: in Sect. 2 the tachyonic

approach to extended teleparallel theories of gravity is intro-

duced with a focus on the cosmological consequences of the

treatment. In Sect. 3, the dynamical analysis of the system

is undertaken for speciﬁc choices of tachyonic ﬁeld. Finally,

in Sect. 4 the conclusions are summarized and discussed.

Unless stated otherwise geometric units are used throughout

the paper. In addition, e

and E

represent the tetrads and the

inverse of the tetrads respectively and the (+−−−) metric

signature is used.

2 Generalized tachyonic teleparallel theories of gravity

In this paper, we present a new teleparallel tachyonic model

based on the following action

S =





2κ

f (φ)T +

g(φ)B

−V (φ)



1 −

V (φ)

+ L



x, (2)

where κ

= 8π G, L

is a matter Lagrangian, T is the scalar

torsion, B = (2/e)∂

(eT

) is the boundary term, f (φ) and

g(φ) are scalar ﬁeld dependent coupling functions, V (φ) is

the potential and

X =

(∂

φ)(∂

φ) (3)

is the kinetic term. The torsion tensor is identiﬁed as

μν

= ∂

− ∂

+ ω

bμ

− ω

bν

, (4)

where e

form a tetrad ﬁeld of the gravitational system and

represent coordinate transformations between the general

manifold and the tangent space at any point, while ω

bμ

form

the spin connection components which are purely inertial and

sustain the local Lorentz invariance of the theory [17,48]. The

torsion scalar is then deﬁned through the contraction

T = T

μν

, (5)

where the superpotential is deﬁned as

μν

+ T

νμ

− T

μν

) − E

αμ

+ E

αν

(6)

This work considers an analogous generalization of other

tachyonic models studied in the literature [46]. By taking,

f (φ) =−g(φ), one recovers a tachyonic theory with a non-

minimally coupling between the scalar ﬁeld and the Ricci

scalar R due to the relation in Eq. (1). By taking g(φ) = 0,

123

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