Modified teleparallel gravity with higher-derivative torsion terms

We construct $F(T,{(\nabla T)}^2,\square T)$ gravitational modifications, which are novel classes of modified theories arising from higher-derivative torsional terms in the action and are different than their curvature analogue. Applying them in a cosmological framework, we obtain an effective dark energy sector comprised of the novel torsional contributions. We perform a detailed dynamical analysis for two specific examples, extracting the stable late-time solutions and calculating the corresponding observables. We show that the thermal history of the Universe can be reproduced, and it can result in a dark-energy-dominated, accelerating universe, where the dark-energy equation-of-state parameter lies in the quintessence regime, or may exhibit the phantom-divide crossing during the cosmological evolution. Finally, the scale factor behaves asymptotically, either as a power law or as an exponential, in agreement with observations.

Figure 1

The projection of the phase-space evolution on the $Z_1-Z_2$ plane, for model I: $F(T,{(\nabla T)}^2,\square T)=T+\frac{{\alpha }_1{(\nabla T)}^2}{T^2}+{\alpha }_2{e}^{\frac{\delta {(\nabla T)}^2}{T^4}}$, for $\gamma =1$, ${\alpha }_1=1.2$ and ${\alpha }_2=0$, in units where $8\pi G=1$. The Universe is attracted by the quintessencelike stable point $P_{2+}$, marked by the red bullet.

Figure 2

Evolution of various observables as a function of the redshift ($z=\frac{a_0}{a}-1$ and for implicitly we set $a_0=1$), for model I: $F(T,{(\nabla T)}^2,\square T)=T+\frac{{\alpha }_1{(\nabla T)}^2}{T^2}+{\alpha }_2{e}^{\frac{\delta {(\nabla T)}^2}{T^4}}$, for ${\alpha }_1=3\times {10}^{-7}$, ${\alpha }_2=-2\times {10}^{-20}$ and $\delta =-1\times {10}^{-19}$, in units where $8\pi G=1$. In the upper graph we depict the evolution of the various density parameters, namely ${\mathrm{\Omega }}_{\mathrm{DE}}$ (solid line), ${\mathrm{\Omega }}_m$ (dashed line), and ${\mathrm{\Omega }}_r$ (dotted line). In the lower graph we present the evolution of the dark-energy (dotted line) and total (solid line) equation-of-state parameters. The Universe is attracted by the de Sitter fixed point $P_3$. For the numerics we have imposed ${\mathrm{\Omega }}_{\mathrm{DE}0}\approx 0.72$, ${\mathrm{\Omega }}_{m0}\approx 0.28$, $w_{\mathrm{DE}0}\approx -0.94$ and $w_{\mathrm{tot}0}\approx -0.68$ at present ($z=0$), in agreement with observations.

Figure 3

The projection of the phase-space evolution on the $Z_1-Z_2$ plane, for model II: $F(T,{(\nabla T)}^2,\square T)=T+\frac{{\beta }_1\square T}{T}+\frac{{\beta }_2{(\square T)}^2}{T^3}+{\beta }_3{e}^{\frac{\sigma \square T}{T^3}}$, for $\gamma =1$, ${\beta }_1=1.2$, ${\beta }_2=\frac{7{\beta }_1}{34}\approx 0.25$ and ${\beta }_3=0$, in units where $8\pi G=1$. The Universe is attracted by the quintessencelike stable point $Q_{3+}$, marked by the red bullet.