Born-Infeld gravity with a Brans-Dicke scalar

Recently proposed Born-Infeld (BI) theories of gravity assume a constant BI parameter ($\kappa$). However, no clear consensus exists on the sign and value of $\kappa$. Recalling the Brans-Dicke (BD) approach, where a scalar field was used to generate the gravitational constant $G$, we suggest an extension of Born-Infeld gravity with a similar Brans-Dicke flavor. Thus, a new action, with $\kappa$ elevated to a spacetime dependent real scalar field, is proposed. We illustrate this new theory in a cosmological setting with pressureless dust and radiation as matter. Assuming a functional form of $\kappa (t)$, we numerically obtain the scale factor evolution and other details of the background cosmology. It is known that BI gravity differs from general relativity (GR) in the strong-field regime but reduces to GR for intermediate and weak fields. Our studies in cosmology demonstrate how, with this new, scalar-tensor BI gravity, deviations from GR, as well as usual BI gravity, may arise in the weak-field regime too. For example, we note a late-time acceleration without any dark energy contribution. Apart from such qualitative differences, we note that fixing the sign and value of $\kappa$ is no longer a necessity in this theory, though the origin of the BD scalar does remain an open question.

Figure 1

Plot of (a) $a(t)$, (b) $\kappa (t)$, (c) $y(t)$, and (d) $\dot{y}(t)$ for a vacuum ($\rho =0$ and $p=0$) solution for ${\kappa }_0>0$, $\epsilon <0$, and $\mu >0$. The parameters used are ${\kappa }_0=1$, $\mu =0.1$. We choose $a(0)=1$, $y(0)=1.001$, $\kappa (0)=0.999$ for the numerical solution.

Figure 2

Plot of (a) $\kappa (t)$, (b) $a(t)$, (c) $q(t)$, (d) $H(t)$, (e) $\rho (t)$, and (f) $U(t)$ (dotted line), $V(t)$ (dashed line) for ${\kappa }_0>0$, $\epsilon <0$, and $\mu >0$. The parameters used are ${\kappa }_0=1$, $\mu =0.1$, and ${\rho }_0=0.1$. We choose $a(0)=1$, $y(0)=1.001$, $\kappa (0)=0.999$ for the numerical solution. The dashed black curve in (b) corresponds to the EiBI solution with $\kappa ={\kappa }_0$ (constant). Equation of state (EOS), $p=0$.

Figure 3

Plot of (a) $\kappa (t)$, (b) $a(t)$, (c) $q(t)$, (d) $H(t)$, (e) $\rho (t)$, and (f) $U(t)$ and $V(t)$ for ${\kappa }_0<0$, $\epsilon <0$, and $\mu >0$. The parameters used are ${\kappa }_0=-1$, $\mu =0.1$, and ${\rho }_0=0.01$. We choose $a(0)=(-{\kappa }_0{\rho }_0{)}^{1/3}$, $y(0)=0.9999$, $\kappa (0)=-1.00001$ for the numerical solution. EOS, $p=0$. Evolution of $a(t)$ and $H(t)$ near the bounce are shown in the insets of (b) and (d). $q$ and $U$ diverge at the bounce.

Figure 4

Plot of (a) $\kappa (t)$, (b) $a(t)$, (c) $q(t)$, (d) $H(t)$, (e) $\rho (t)$, and (f) $U(t)$ and $V(t)$ for ${\kappa }_0>0$, $\epsilon <0$, and $\mu <0$. The parameters used are ${\kappa }_0=1$, $\mu =-0.1$, and ${\rho }_0=0.1$. We choose $a(0)=1$, $y(0)=0.99$, $\kappa (0)=0.99$ for the numerical solution. EOS, $p=0$. Evolution of $a(t)$ about the bounce is shown clearly in inset of (b). The inset of (d) shows that $H(t)$ approaches a constant negative value as $t\to -\infty .q$ diverges to $-\infty$ at bounce.

Figure 5

Plot of (a) $\kappa (t)$, (b) $a(t)$, (c) $q(t)$, (d) $H(t)$, (e) $\rho (t)$, and (f) $U(t)$ and $V(t)$ for ${\kappa }_0<0$, $\epsilon <0$, and $\mu <0$. The parameters used are ${\kappa }_0=-1$, $\mu =-0.1$, and ${\rho }_0=0.01$. We choose $a(0)=1$, $y(0)=1.001$, $\kappa (0)=-1.001$ for the numerical solution. EOS, $p=0$. $q$ diverges to $-\infty$ at bounce.

Figure 6

Plot of (a) $\kappa (t)$ and (b) $a(t)$ for ${\kappa }_0>0$, $\epsilon <0$, and $\mu >0$. The parameters used are ${\kappa }_0=1$, $\mu =0.1$, and ${\rho }_0=0.1$. We choose $a(0)=1$, $y(0)=1.001$, $\kappa (0)=0.999$ for the numerical solution. EOS, $p=\rho /3$. The inset of (b) shows the loitering phase where $a(t)$ approaches a nonzero minimum value asymptotically in the past.

Figure 7

Plot of (a) $\kappa (t)$ and (b) $a(t)$ for ${\kappa }_0<0$, $\epsilon <0$, and $\mu >0$. The parameters used are ${\kappa }_0=-1$, $\mu =0.1$, and ${\rho }_0=0.01$. We choose $a(0)=(-{\kappa }_0{\rho }_0{)}^{1/4}$, $y(0)=0.9999$, $\kappa (0)=-1.00001$ for the numerical solution. Evolution of $a(t)$ near the bounce is shown in the inset of (b). EOS, $p=\rho /3$.