Cosmological dynamics of a Dirac-Born-Infeld field

We analyze the dynamics of a Dirac-Born-Infeld (DBI) field in a cosmological setup which includes a perfect fluid. Introducing convenient dynamical variables, we show that the evolution equations form an autonomous system when the potential and the brane tension of the DBI field are arbitrary power law or exponential functions of the DBI field. In particular we find scaling solutions can exist when powers of the field in the potential and warp factor satisfy specific relations. A new class of fixed-point solutions are obtained corresponding to points which initially appear singular in the evolution equations, but on closer inspection are actually well defined. In all cases, we perform a phase-space analysis and obtain the late-time attractor structure of the system. Of particular note when considering cosmological perturbations in DBI inflation is a fixed-point solution where the Lorentz factor is a finite large constant and the equation of state parameter of the DBI field is $w=-1$. Since in this case the speed of sound $c_s$ becomes constant, the solution can be thought to serve as a good background to perturb about.

Figure 1

Demonstration of the attractive property of ($\alpha 1$) in the regime $\tilde{\lambda }+\tilde{\mu }>0$ and $q=1/2$. Note that for all the initial values of $x$, $y$, and $\tilde{\gamma }$, the solutions approach that of $\alpha 1$ where $(x,y,\tilde{\gamma })=(0,1,1)$. Note also that the physical region is confined to $x^2+y^2\le 1$ in the three-dimensional phase space. Although here we have chosen $q=1/2$, $r/p=-0.5$, $\tilde{\lambda }=10$ [or $p=-4$, $r=2$, $(\sigma \nu {)}^{1/2}=2/5$], $w_m=0$, the same results are obtained for $q=1/2$ with $r/p>-1$.

Figure 2

Late-time attractor structure in the models where the potential and the brane tension are power-law functions of the DBI field with $q=1/2$ and $w_m\ge 0$. We show the late-time attractor solution for a given model parameter in $\tilde{\lambda }\mathrm{\text{−}}\tilde{\mu }$ space with $\tilde{\lambda }>0$. In the case with $w_m<0$, the region where (a5) is the late-time attractor appears for $\tilde{\lambda }+\tilde{\mu }<0$ and $\tilde{\lambda }\ge \sqrt{3}(1+w_m)/\sqrt{-w_m}$. It is worth noting that $\tilde{\lambda }$ and $\tilde{\mu }$ are related with the parameters of the potential and brane tension as $\tilde{\lambda }=|p|/\sqrt{\sigma \nu }$ and $\tilde{\mu }=-(2+p)|p|/(p\sqrt{\sigma \nu })$.

Figure 3

Demonstration of the attractive property of ($\alpha 2$) in the regime $\tilde{\lambda }+\tilde{\mu }<0$ and $q=0$. Note that for all the initial values of $x$, $y$, and $\tilde{\gamma }$, the solutions approach that of ($\alpha 2$) where $(x,y,\tilde{\gamma })=(0,1,0)$. Note also that the physical region is confined to $x^2+y^2\le 1$ in the three-dimensional phase space. Although here we have chosen $q=0$, $r/p=-1.5$, $\tilde{\lambda }=10$, and $w_m=0$, the same results are obtained for $q=0$ with $r/p<-1$. Notice also that we chose the directions of the axis differently from Fig. 1.

Figure 4

Late-time attractor structure in the models where the potential and brane tension are power-law functions of the DBI field with $q=0$ and $w_m>0$. We show which fixed-point solution is the late-time attractor solution for a given set of model parameters in $\tilde{\lambda }\mathrm{\text{−}}\tilde{\mu }$ space with $\tilde{\lambda }>0$. In the case with $w_m\le 0$, the region where (c1) is the late-time attractor disappears. This figure can be also applied to the models where the potential and brane tension are exponential functions of the DBI field with $q=0$ through the replacements $\tilde{\lambda }(=|p|)\to \lambda$, $\tilde{\mu }(=r|p|/p)\to \mu$. For this class of models, it is possible to consider the case with $\mu =-\lambda$ where (c4) or (c5) can be the late-time attractor. Although it is complicated to write down, there is a critical value of $\lambda$ depending on $w_m$, $\sigma$, and $\nu$, below which (c4) is the late-time attractor and above which (c5) is the late-time attractor.

Figure 5

Demonstration of the attractive property of (c3) in the regime $\tilde{\lambda }+\tilde{\mu }<0$ and $q=-1/2$. For every initial value of $x$, $y$, and $\tilde{\gamma }$, the solution approaches $(x,y,\tilde{\gamma })=(0,1,\sqrt{3}/\sqrt{{\tilde{\lambda }}^2+3})$. Notice that the physical region is confined to $x^2+y^2\le 1$ in the three-dimensional phase space. Although we choose $q=-1/2$, $r/p=-1.5$, $\tilde{\lambda }=10$ [or $p=-4$, $r=2$, $(\sigma \nu {)}^{1/2}=2/5$] and $w_m=0$ in this example, the same results are obtained for $q=-1/2$ with $-1>r/p$. Notice that for $\tilde{\lambda }=10$, $\sqrt{3}/\sqrt{{\tilde{\lambda }}^2+3}\simeq 0.17$ as expected.

Figure 6

Demonstration of the attractive property of ($\alpha 1$) in the regime $\tilde{\lambda }+\tilde{\mu }>0$. Note that for all the initial values of $x$, $y$, and $\tilde{\gamma }$, the solutions approach that of $\alpha 1$ where $(x,y,\tilde{\gamma })=(0,1,1)$. Note also that the physical region is confined to $x^2+y^2\le 1$ in the three-dimensional phase space. Although here we have chosen $q=1/4$, $r/p=-0.5$, $\tilde{\lambda }=10$ [or $p=-8$, $r=4$, $(\sigma \nu {)}^{1/4}=4/5$], $w_m=0$, the same results are obtained for $q>0$ with $r/p>-1$ and $-1/2>q$ with $r/p<-1$.

Figure 7

This figure shows the attractive nature of the fixed point ($\alpha 2$) in the regime $\tilde{\lambda }+\tilde{\mu }<0$ and $q=\frac{1}{4}$. For all initial values of $x$, $y$, and $\tilde{\gamma }$, solutions approach the point ($\alpha 2$) where $(x,y,\tilde{\gamma })=(0,1,0)$. Notice that the physical region is confined to $x^2+y^2\le 1$ in the three-dimensional phase space. Although we have chosen $q=1/4$, $r/p=-1.5$, $\tilde{\lambda }=10$ [or $p=8$, $r=-12$, $(\sigma \nu {)}^{1/4}=4/5$], $w_m=0$ in this example, the same results are obtained for $1/2>q>-1/2$ with $-1>r/p$. Notice also that we chose the directions of the axis differently from Fig. 1.