Dynamical analysis in scalar field cosmology

We give a general method to find exact cosmological solutions for scalar-field dark energy in the presence of perfect fluids. We use the existence of invariant transformations for the Wheeler De Witt (WdW) equation. We show that the existence of a point transformation under which the WdW equation is invariant is equivalent to the existence of conservation laws for the field equations, which indicates the existence of analytical solutions. We extend previous work by providing exact solutions for the Hubble parameter and the effective dark-energy equation of state parameter for cosmologies containing a combination of perfect fluid and a scalar field whose self-interaction potential is a power of hyperbolic functions. We find solutions explicitly when the perfect fluid is radiation or cold dark matter and determine the effects of nonzero spatial curvature. Using the Planck 2015 data, we determine the evolution of the effective equation of state of the dark energy. Finally, we study the global dynamics using dimensionless variables. We find that if the current cosmological model is Liouville integrable (admits conservation laws) then there is a unique stable point which describes the de-Sitter phase of the universe.

Figure 1

The evolution of the equation of state parameters $w_{\varphi }(a)$ for the scalar field and $w_{\text{tot}}(a)$ for the total fluid for the Lagrangian (32) with $\alpha =1,\beta =0$. In (46) we have taken the minus sign and $\mu =-1$ (dust fluid: $w_m=0$). For the numerical solution we use ${(x,y)}_{t\to 0}=(0.01,0.0099),\log (V_0)=4$, ${\rho }_{m0}=3{\mathrm{\Omega }}_{m0}{H}_0^2$ with ${\mathrm{\Omega }}_{m0}=0.308$ and $H_0=67.8\mathrm{Km}/\mathrm{s}/\mathrm{Mpc}$ (or ${\rho }_{m0}=4.25\times 10^3$ in units of $8\pi G\equiv 1$). The solid line is for $\log (c_1)=0$; the dashed line is for $\log (c_1)=1$; the dotted line is for $\log (c_1)=2$.

Figure 2

The evolution of the equation of state parameters $w_{\text{tot}}(a)$ for the total fluid for the Lagrangian (32) with $\alpha =1,\beta =0$ where in (46) we take the minus sign and $c_1=0$. For the numerical solution we use ${(x,y)}_{t\to 0}=(0.01,0.0099),\log (V_0)=4$ and ${\rho }_{m0}=4.25\times 10^3$. The solid line is for $\mu =-1$ [dust: $(w_m,\gamma )=(0,1)$], the dash-dotted line is for $\mu =-0.95$ and the dotted line is for $\mu =-1.05$.

Figure 3

Phase portrait in the $x-y$ plane and in the $E^3$ space for the potential $V(\varphi )=V_0{\cosh }^q(p\varphi )$. Left-hand figures are for the variables $(p,q,w_m)=(1,-1.5,0)$ and the right-hand figures are for the variables $(p,q,w_m)=(1,-3,0)$. For $(pq{)}^2>3,$ the stable points are the points $E_{(\pm )}$ (scaling solutions) while for $(pq{)}^2<3$ the two stable points are the points $D_{\pm }$. The solid lines are for the initial condition $\lambda =pq$, the dashed lines for $\lambda =-pq$, and the dotted lines for $\lambda =0$.

Figure 4

Phase portrait in the $x-y$ plane and in the $E^3$ space for the potential $V(\varphi )=V_0{\cosh }^q(p\varphi )$ with variables $(p,q,w_m)=(-\frac{\sqrt{6}}{4},2,0)$, which corresponds to the integrable case for $\mu =-1$. The point $C$ is the unique stable point. The points $D_{(\pm )}$ act as attractors in the plane $(x-y)$ for $\lambda =\pm \frac{\sqrt{6}}{2}$. From the right-hand plot we observe that the solutions reach a boundary where ${\mathrm{\Omega }}_{m0}=0$, $w_{\varphi }>-1$ and from there they move to the de Sitter point $C$. The solid lines are for initial condition $\lambda =pq$, the dashed lines for $\lambda =-pq$, and the dotted lines for $\lambda =0$.