Cosmological applications of $F(T,T_G)$ gravity

We investigate the cosmological applications of $F(T,T_G)$ gravity, which is a novel modified gravitational theory based on the torsion invariant $T$ and the teleparallel equivalent of the Gauss-Bonnet term $T_G$. $F(T,T_G)$ gravity differs from both $F(T)$ theories as well as from $F(R,G)$ class of curvature modified gravity, and thus its corresponding cosmology proves to be very interesting. In particular, it provides a unified description of the cosmological history from early-times inflation to late-times self-acceleration, without the inclusion of a cosmological constant. Moreover, the dark energy equation-of-state parameter can be quintessence or phantomlike, or experience the phantom-divide crossing, depending on the parameters of the model.

Figure 1

Four inflationary solutions for the ansatz $F(T,T_G)={\alpha }_1{T}^2+{\alpha }_{2}T\sqrt{|T_G|}-T+{\beta }_1\sqrt{T^2+{\beta }_2{T}_G}$, corresponding to (a) ${\alpha }_1=-2.8$, ${\alpha }_2=8$, ${\beta }_1=0.001$, ${\beta }_2=1$ (black-solid), (b) ${\alpha }_1=-2$, ${\alpha }_2=8$, ${\beta }_1=0.001$, ${\beta }_2=1$ (red-dashed), (c) ${\alpha }_1=8$, ${\alpha }_2=8$, ${\beta }_1=0.001$, ${\beta }_2=1$ (blue-dotted), (d) ${\alpha }_1=20$, ${\alpha }_2=5$, ${\beta }_1=0.001$, ${\beta }_2=1$ (green-dashed-dotted). All parameters are in Planck units.

Figure 2

Upper graph: The evolution of the dark energy density parameter ${\mathrm{\Omega }}_{DE}$ (black-solid) and the matter density parameter ${\mathrm{\Omega }}_m$ (red-dashed), as a function of the redshift $z$, for the ansatz $F(T,T_G)={\alpha }_1{T}^2+{\alpha }_{2}T\sqrt{|T_G|}-T+{\beta }_1\sqrt{T^2+{\beta }_2{T}_G}$ with ${\alpha }_1=0.001$, ${\alpha }_2=0.001$, ${\beta }_1=2.5$, ${\beta }_2=1.5$. Lower graph: The evolution of the corresponding dark energy equation-of-state parameter $w_{DE}$. All parameters are in units where the present Hubble parameter is $H_0=1$, and we have imposed ${\mathrm{\Omega }}_{m0}\approx 0.3$, ${\mathrm{\Omega }}_{DE0}\approx 0.7$ at present.

Figure 3

Upper graph: The evolution of the dark energy density parameter ${\mathrm{\Omega }}_{DE}$ (black-solid) and the matter density parameter ${\mathrm{\Omega }}_m$ (red-dashed), as a function of the redshift $z$, for the ansatz $F(T,T_G)={\alpha }_1{T}^2+{\alpha }_{2}T\sqrt{|T_G|}-T+{\beta }_1\sqrt{T^2+{\beta }_2{T}_G}$, with ${\alpha }_1=0.001$, ${\alpha }_2=0.001$, ${\beta }_1=2.6$, ${\beta }_2=2$. Lower graph: The evolution of the corresponding dark energy equation-of-state parameter $w_{DE}$. All parameters are in units where the present Hubble parameter is $H_0=1$, and we have imposed ${\mathrm{\Omega }}_{m0}\approx 0.3$, ${\mathrm{\Omega }}_{DE0}\approx 0.7$ at present.

Figure 4

Five inflationary solutions for the ansatz $F(T,T_G)=-T+{\beta }_1(T^2+{\beta }_2{T}_G)+{\beta }_3(T^2+{\beta }_4{T}_G{)}^2$, corresponding to (a) ${\beta }_1=-0.01$, ${\beta }_2=1$, ${\beta }_3=-1$, ${\beta }_4=-2$ (black-solid), (b) ${\beta }_1=-0.1$, ${\beta }_2=1$, ${\beta }_3=-2$, ${\beta }_4=-2$ (red-dashed), (c) ${\beta }_1=-0.01$, ${\beta }_2=1$, ${\beta }_3=-1$, ${\beta }_4=-5$ (blue-dotted), (d) ${\beta }_1=-0.01$, ${\beta }_2=1$, ${\beta }_3=-6$, ${\beta }_4=-6$ (green-dashed-dotted), (e) ${\beta }_1=-0.01$, ${\beta }_2=1$, ${\beta }_3=-10$, ${\beta }_4=-10$ (yellow-dashed-dotted-dotted). All parameters are in Planck units.