Reconstruction of $f(T)$-gravity in the absence of matter

TL;DR

This paper derives an exact $f(T)$ gravity model in a matter-free FRW universe, revealing novel behaviors of torsion and its potential to explain inflation and dark energy phenomena.

Contribution

It introduces a new $f(T)$ gravity formulation without matter, exploring its implications for cosmology, including inflation, cosmological constant, and scalar torsion fields.

Findings

$f(T)$ acts like a cosmological constant in flat space.

Open universe models show a decaying $f(T)$, addressing the cosmological constant problem.

Six inflationary models, including Higgs-like potentials, are derived.

Abstract

We derive an exact $f(T)$ gravity in the absence of ordinary matter in Friedmann-Robertson-Walker (FRW) universe, where $T$ is the teleparallel torsion scalar. We show that vanishing of the energy-momentum tensor $\mathcal{T}^{\mu \nu}$ of matter does not imply vanishing of the teleparallel torsion scalar, in contrast to general relativity, where the Ricci scalar vanishes. The theory provides an exponential (inflationary) scale factor independent of the choice of the sectional curvature. In addition, the obtained $f(T)$ acts just like cosmological constant in the flat space model. Nevertheless, it is dynamical in non-flat models. In particular, the open universe provides a decaying pattern of the $f(T)$ contributing directly to solve the fine-tuning problem of the cosmological constant. The equation of state (EoS) of the torsion vacuum fluid has been studied in positive and negative Hubble regimes. We study the case when the torsion is made of a scalar field (tlaplon) which acts as torsion potential. This treatment enables to induce a tlaplon field sensitive to the symmetry of the spacetime in addition to the reconstruction of its effective potential from the $f(T)$ theory. The theory provides six different versions of inflationary models. The real solutions are mainly quadratic, the complex solutions, remarkably, provide Higgs-like potential.