The hidden flat like universe
Eur. Phys. J. C
The hidden flat like universe
W. El Hanafy 0 2
G. G. L. Nashed 0 1 2
0 Egyptian Relativity Group , Giza , Egypt
1 Mathematics Department, Faculty of Science, Ain Shams University , Cairo , Egypt
2 Centre for Theoretical Physics, The British University in Egypt , P.O. Box 43, El Sherouk 11837 , Egypt
We study a single-fluid component in a flat like universe (FLU) governed by f (T ) gravity theories, where T is the teleparallel torsion scalar. The FLU model, regardless of the value of the spatial curvature k, identifies a special class of f (T ) gravity theories. Remarkably, FLU f (T ) gravity does not reduce to teleparallel gravity theory. In large Hubble spacetime the theory is consistent with the inflationary universe scenario and respects the conservation principle. The equation of state evolves similarly in all models k = 0, ±1. We study the case when the torsion tensor consists of a scalar field, which enables to derive a quintessence potential from the obtained f (T ) gravity theory. The potential produces Starobinsky-like model naturally without using a conformal transformation, with higher orders continuously interpolate between Starobinsky and quadratic inflation models. The slow-roll analysis shows double solutions, so that for a single value of the scalar tilt (spectral index) ns the theory can predict double tensor-to-scalar ratios r of E -mode and B-mode polarizations.
1 Introduction
The general relativity (GR) theory explained the gravity as
spacetime curvature. This description of gravitation has
succeeded to confront astrophysical observations for a long time.
It has predicted perfectly the perihelion shift of mercury, time
delay in the solar system. Even in the strong field regimes
such as binary pulsars it has amazingly predicted their orbital
decay due to gravitational radiation by the system. While it
fails to predict the accelerating cosmic expansion which is
evidenced by the astronomical observations of high-redshift
Type Ia supernovae [1]. The teleparallel equivalent of general
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relativity (TEGR) theory has provided an alternative
description of Einstein’s gravity. The theory constructed from
vierbein (tetrad) fields1 {ha μ} instead of metric tensor fields gμν .
The metric space, however, can be constructed from the
vierbein fields, so the Levi-Civita symmetric connection ◦ αμν .
It is, also, possible to construct Weitzenböck nonsymmetric
connection αμν . The first connection has a non-vanishing
( ◦ )
curvature tensor Rαβμν = 0 but a vanishing torsion tensor
( ◦ )
Tαμν = 0, while the later is characterized by a vanishing
cur( )
vature tensor Rαβμν = 0 but a non-vanishing torsion tensor
Tα(μν) = 0. The combined picture could be encoded in the
contracted Bianchi identity
R( ◦ ) ≡ −T ( ) − 2∇α(◦ )T ναν ,
◦
where R( ) is the usual Ricci scalar, the scalar invariant T ( )
is called the teleparallel torsion scalar (it will be briefly
investigated in Sect. 2 and the covariant derivative ∇( ◦ ) is with
respect to (w.r.t.) the Levi-Civita connection. The variation
of the left hand side w.r.t. the metric tensor provides the GR
field equations. Since the last term in the identity is a total
derivative, it does not contribute to the field equations, and
the variation of the right hand side w.r.t. the vierbein fields
provides a set of field equations equivalent to the GR that is
called TEGR. Although the two theories are quantitatively
equivalent at their level of the field equations, they are
qualitatively different at the level of their actions! Indeed, the total
derivative term is scalar invariant under a diffeomorphism but
not invariant under a local Lorentz transformation (LLT). On
the other hand, the Ricci scalar in the Einstein–Hilbert action
leads to a theory invariant under a diffeomorphism as well
1 The Greek letters α, β, . . . denote the spacetime indices and the Latin
ones a, b, . . . denote Lorentz indices. Both run from 0 to 3.
as LLT. Consequently, the teleparallel torsion scalar T is not
invariant under LLT [2]. The absence of local Lorentz
symmetry in the TEGR action will not be reflected in the field
equations, so it does not seem worth to worry about it.
However, the presence of the total derivative term is crucial when
we consider the f (T ) extension of TEGR [3,4].
Teleparallel geometry has received attention in the last
decade. However, this geometry has been used very much
earlier, in the 1920s, to unify gravity and electromagnetism
by Einstein [5]. After this trial the geometry has been
developed [6–8]. Later, successful extensions to Einstein’s
work allowed a class of theories with a quadratic torsion in
Lagrangian density [9–11]. Other trials to obtain a gauge
field theory of gravity using the teleparallel geometry have
shown to be of great interest [12–16]. Recent
developments attempted a global approach by using arbitrary moving
frames instead of the local expressions in the natural basis
[17,18]. Also, it is worth to mention (...truncated)