Cosmological solutions with gravitational particle production and nonzero curvature

In a homogeneous and isotropic universe with nonzero spatial curvature we consider the effects of gravitational particle production in the dynamics of the universe. We show that the dynamics of the universe in such a background is characterized by a single nonlinear differential equation which is significantly dependent on the rate of particle creation and whose solutions can be dominated by the curvature effects at early times. For different particle creation rates we apply the singularity test in order to find the analytic solutions of the background dynamics. We describe the behavior of the cosmological solutions for both open and closed universes. We also show how the effects of curvature can be produced by the presence of a second perfect fluid with an appropriate equation of state. By combining that result with the analysis of the critical points we find that our consideration can be related with the preinflationary era. Specifically we find that for negative spatial curvature small changes of the Milne spacetime leads to a de Sitter universe.

Figure 1

Evolution of the energy density $\mathrm{\Omega }$ (left fig.) and ${\mathrm{\Omega }}_k$ (right fig.) for the constant matter creation rate ${\mathrm{\Gamma }}_0$ and negative spatial curvature, $k=-1$. The lines are for $\gamma =1$, i.e. dust fluid. Solid line is for ${\mathrm{\Gamma }}_0=1$, dash-dash line for ${\mathrm{\Gamma }}_0=0.3$, dash-dot line for ${\mathrm{\Gamma }}_0=0.2$ and dot-dot line for ${\mathrm{\Gamma }}_0=-0.1$.

Figure 2

Evolution of the energy density $\mathrm{\Omega }$ (left figs.) and ${\mathrm{\Omega }}_k$ (right figs.) for the constant matter creation rate ${\mathrm{\Gamma }}_A$. Upper figures are for ${\mathrm{\Gamma }}_1=1$ and lower figures are for ${\mathrm{\Gamma }}_1=-1$. The lines are for $\gamma =1$, i.e. dust fluid. Solid lines are for ${\mathrm{\Gamma }}_0=1$, dash-dash lines for ${\mathrm{\Gamma }}_0=0$, dot lines for ${\mathrm{\Gamma }}_0=-1$ and dash-dot lines for ${\mathrm{\Gamma }}_0=-2$.

Figure 3

Evolution of the energy density $\mathrm{\Omega }$ (left) and ${\mathrm{\Omega }}_k$ (right) for the constant matter creation rate ${\mathrm{\Gamma }}_0$ and negative spatial curvature, $k=-1$. The lines are for $\gamma =1$, i.e. dust fluid and for different initial condition $\mathrm{\Omega }(t_0)$. Solid lines are for ${\mathrm{\Gamma }}_0=1$, dash-dot lines for ${\mathrm{\Gamma }}_0=2$, dot-dot lines for ${\mathrm{\Gamma }}_0=3$.

Figure 4

Qualitative behavior of the deceleration parameter $q(a)$ for the constant matter creation rate ${\mathrm{\Gamma }}_0$ and negative spatial curvature, $k=-1$. The lines are for $\gamma =1$, i.e. dust fluid. Solid line is for ${\mathrm{\Gamma }}_0=1$, dash-dash line for ${\mathrm{\Gamma }}_0=0.3$, dash-dot line for ${\mathrm{\Gamma }}_0=0.2$ and dot-dot line for ${\mathrm{\Gamma }}_0=-0.1$.

Figure 5

Qualitative behavior of the deceleration parameter $q(a)$ for the varying matter creation rate $\mathrm{\Gamma }(H)={\mathrm{\Gamma }}_0+{\mathrm{\Gamma }}_1/H$ and $k=-1$. Left figure is for ${\mathrm{\Gamma }}_1=1$ and right figure is for ${\mathrm{\Gamma }}_1=-1$. The lines are for $\gamma =1$, i.e. dust fluid. Solid lines are for ${\mathrm{\Gamma }}_0=1$, dash-dash lines for ${\mathrm{\Gamma }}_0=0$, dot lines for ${\mathrm{\Gamma }}_0=-1$ and dash-dot lines for ${\mathrm{\Gamma }}_0=-2$.

Figure 6

Qualitative behavior of the deceleration parameter $q(a)$ for the varying matter creation rate $\mathrm{\Gamma }(H)={\mathrm{\Gamma }}_0+{\mathrm{\Gamma }}_1/H+{\mathrm{\Gamma }}_2{H}^2$ and $k=-1$. Left figure is for ${\mathrm{\Gamma }}_1=0.2$ and right figure is for ${\mathrm{\Gamma }}_1=-0.2$. The lines are for $\gamma =1$, i.e. dust fluid. Solid lines are for ${\mathrm{\Gamma }}_0=0.2$, dash-dash lines for ${\mathrm{\Gamma }}_0=0.1$, dot lines for ${\mathrm{\Gamma }}_0=-0.1$ and dash-dot lines for ${\mathrm{\Gamma }}_0=-0.2$. For the figures we considered ${\mathrm{\Gamma }}_2=-{10}^{-4}$.