Cosmological phase space analysis of the $F(X)\mathbf{-}V(\varphi )$ scalar field and bouncing solutions

We analyze the dynamical system defined by a universe filled with a barotropic fluid plus a scalar field with modified kinetic term of the form $\mathcal{L}=F(X)-V(\varphi )$. After a suitable choice of variables that allows us to study the phase space of the system, we obtain the critical points and their stability. We find that some of them reduce to the ones defined for the canonical case when $F(X)=X$. We also study the field energy conditions to have a nonsingular bounce.

Figure 1

Projection of the phase space along the $(x,y)$ plane for a Lagrangian of the form $\mathcal{L}=A{X}^2+B/{\varphi }^4$, with initial condition $\sigma =1.5$. The solutions tend to the critical points (c) and (d) studied in Table . As $\sigma$ changes, the critical points lie on the light green line for the (c) and the red segment of circle for (d). This behavior can be better seen in Figs. 2, 3 corresponding to different projections of the same system. The critical point curves are plotted with the same colors.

Figure 2

Projection of the phase space along the $(x,\sigma )$ plane for a Lagrangian of the form $\mathcal{L}=A{X}^2+B/{\varphi }^4$. The system is the same as in Figs. 1, 3. The solutions where chosen to start in $\sigma =1.5$ and for a constant $x$ they evolve in different directions due to the different values in $y$. See the explanation in Fig. 1.

Figure 3

Projection of the phase space along the $(x,\sigma )$ plane for a Lagrangian of the form $\mathcal{L}=A{X}^2+B/{\varphi }^4$. The system is the same as in Figs. 1, 2. The solutions where chosen to start in $\sigma =1.5$ and they evolve towards different directions depending on the values of $x$ and $y$. See the explanation in Fig. 1.

Figure 4

Phase space for the canonical scalar field with exponential potential and $\sigma =1.5$. The phase space for this system is two-dimensional, unlike in the case of nonexponential potentials or noncanonical kinetic terms with three dimensional phase spaces. The solutions tend to the critical point (d), scaling solution.

Figure 5

Phase space for the Lagrangian $\mathcal{L}=A{X}^2+B{\varphi }^2$. The solutions don’t tend to any critical point. They evolve towards high values of $\sigma$ because this parameter is proportional to ${\varphi }^{-1}$ for this Lagrangian.

Figure 6

Schematic projection of the phase space for a noncanonical nonphantom system with ${\rho }_k>0$ and ${\omega }_k=-5$ and $\sigma \sim \sqrt{2/3}$. The bounce occurs when the solutions cross the vertical (thick, red) line.

Figure 7

Schematic projection of the phase space for a field with ${\rho }_k<0$, ${\omega }_k=1/6$, $\sigma \sim -\sqrt{2/3}$ and ${\omega }_m=1/3$. The bounce occurs when the solutions cross the vertical (thick, red) line. There is no purely kinetic (${\rho }_V=0$) bounce.

Figure 8

Phase space $(\tilde{x},\tilde{y})$ for the special case of a phantom system $F(X)=-X$ and $V\propto e^{-\lambda \varphi /M_{\mathrm{Pl}}}$ (such that $\sigma$ is constant) plus a barotropic radiation component ${\omega }_m=1/3$. The bounce occurs when the solutions cross the vertical (thick, red) line. The spiral in the graph corresponds to the critical point (d) studied in the previous section.

Figure 9

Phase space for the special case of a canonical scalar field with potential $V\propto e^{-\lambda \varphi /M_{\mathrm{Pl}}}$ (such that $\sigma$ is constant) plus a barotropic radiation component ${\omega }_m=1/3$. All the solutions that cross $\tilde{x}=0$ have negative $d\tilde{x}/d\tilde{N}$ which corresponds to recollapse. The bounce is not possible.

Figure 10

The densities of the barotropic fluid (blue, dashed-dotted line), the scalar field (red, dotted line), and the total density of the Universe (green, continuous line), respectively, as a function of the scale factor. The total energy density tends to zero at the bounce and for smaller values of $a$ is negative, which is forbidden.

Figure 11

The bottom left region corresponds in the $\rho \mathrm{\text{−}}p$ plane to the part which can drive a bounce, with $\rho <0$ and ${\omega }_{\varphi }>{\omega }_m$ with ${\omega }_m=1/3$. The upper right region is the one that satisfies the null energy condition. The dashed-dotted lines cannot be crossed by $k$-essence Lagrangians like the ones considered here [43].