Scalar field source Teleparallel Robertson-Walker F(T)-gravity solutions

TL;DR

The paper develops exact scalar-field sourced solutions in scalar-torsion gravity within the teleparallel Robertson–Walker framework. By deriving and solving the unified F(T) field equations for flat ($k=0$) and non-flat ($k=\pm1$) cosmologies, the authors obtain general $F(T)$ reconstruction formulas and explicit analytic solutions for several scalar-field ansatzes, with $V(\phi)$ largely decoupled from the equations. For $k=0$ they provide a compact solution $F(T)= -\Lambda_0 + B\sqrt{T} - \frac{\sqrt{6}\kappa}{2} \int dT'\,T'^{-1/2} \left[ \int^{t(T')} \dot{\phi}^2 dt' \right]$, and show additional closed forms for power-law, logarithmic, and exponential scalar fields. In the non-flat cases, analytic solutions arise only for specific expansion rates $n$ (e.g., $n=\tfrac{1}{2},1,2$) or in the large-$n$ limit, yielding expressions with exponentials and Ei functions. These new solutions broaden the toolbox for teleparallel cosmologies and offer pathways to model quintessence, phantom, and quintom dark-energy scenarios, with future work including data fitting and thermodynamic analyses.

Abstract

This paper investigates the teleparallel Robertson--Walker (TRW) $F(T)$ gravity solutions for a scalar field source. We use the TRW $F(T)$ gravity field equations (FEs) for each $k$-parameter value case added by a scalar field to find new teleparallel $F(T)$ solutions. For $k=0$, we find an easy-to-compute $F(T)$ solution formula applicable for any scalar field source. Then, we obtain, for $k=-1$ and $+1$ situations, some new analytical $F(T)$ solutions, only for specific $n$-parameter values and well-determined scalar field cases. We can find by those computations a large number of analytical teleparallel $F(T)$ solutions independent of any scalar potential $V(φ)$ expression. The $V(φ)$ independence makes the FE solving and computations easier. The new solutions will be relevant for future cosmological applications in dark matter, dark energy (DE) quintessence, phantom energy and quintom models of physical processes.

Figures (3)

• Figure 1: Plots of $F_1(T)=\frac{F(T)+\Lambda_0}{B}$ versus torsion scalar $T$, described by Equations \ref{['403']}--\ref{['405']}, for different types of scalar field sources and $n$-parameter with $\frac{\kappa\,p_0^2}{2B}=1$ and $p=1.3$. Note that the $n=20$ case in subfigure (d) represents Equations \ref{['403']}--\ref{['405']} for the $n\,\rightarrow\,\infty$ limit.
• Figure 2: Plots of $F_2(T)=F(T)+\Lambda_0$ versus torsion scalar $T$ described by the $F(T)$ from Equations \ref{['506a']}--\ref{['510b']} for different types of scalar field sources, $n$ and $p$ parameters with $\frac{\kappa\,p_0^2}{2}=1$ and $\frac{\delta\,\sqrt{-k}}{a_0}=2$. Note that the $n=20$ case in subfigure (d) represents Equations \ref{['403']}--\ref{['405']} for the $n\,\rightarrow\,\infty$ limit, where $B=1$ and $\delta_1=1$, which is exactly the same in Figure \ref{['figure1']}d when $B=1$ in $F_2(T)$.
• Figure 3: Plots of $F_2(T)=F(T)+\Lambda_0$ versus torsion scalar $T$ described by the $F(T)$ from Equations \ref{['558a']}--\ref{['560b']} for different types of scalar field sources and $p$ parameters with $\frac{\kappa\,p_0^2}{2}=1$ in the $n=1$ and $n=2$ cases.