Particle creation in a toroidal universe

We calculate the particle production rate in an expanding universe with a three-torus topology. We discuss also the complete evolution of the size of such a universe. The energy density of particles created through the nonzero modes is computed for selected masses. The unique contribution of the zero mode and its properties are also analyzed.

Figure 1

Size of the three-torus universe as a function of time, assuming $a(t_{\mathrm{univ}})=10R_H$.

Figure 2

Total energy density in the three-torus universe as a function of time.

Figure 3

Number density of the created particles for $m_1={10}^9\mathrm{GeV}$ (blue solid line). The green dashed line corresponds to the $mt\ll 1$ and $ma\ll 1$ approximation given by Eq. (41) and the orange dot-dashed line corresponds to the $mt\gg 1$ approximation given by Eq. (47).

Figure 4

Energy density of the created particles for $m_1={10}^9\mathrm{GeV}$ (thick blue solid line). The green dashed line corresponds to the constant density for $mt\ll 1$ given by Eq. (44) and the orange dot-dashed line corresponds to the $mt\gg 1$ approximation given by Eq. (48). For comparison, the total energy density of the universe is plotted in thin solid red.

Figure 5

Same as Fig. 3, but for $m_2={10}^3\mathrm{GeV}$.

Figure 6

Same as Fig. 4, but for $m_2={10}^3\mathrm{GeV}$.

Figure 7

Same as Fig. 3, but for $m_3={10}^{-3}\mathrm{GeV}$.

Figure 8

Same as Fig. 4, but for $m_3={10}^{-3}\mathrm{GeV}$.

Figure 9

Number density of the particles created through the zero mode for $m={10}^3\mathrm{GeV}$ (blue solid line), assuming $t_0=t_p$. The green dashed line corresponds to the $mt\ll 1$ approximation given by Eq. (58). The orange dot-dashed line denotes the $mt\gg 1$ approximation given by Eq. (60).

Figure 10

Energy density of the particles created through the zero mode for $m={10}^3\mathrm{GeV}$ (thick blue solid line), assuming $t_0=t_p$. The green dashed line corresponds to the $mt\ll 1$ approximation given by Eq. (58). The orange dot-dashed line corresponds to the $mt\gg 1$ approximation given by Eq. (60). The thin red solid line denotes the total energy density of the universe.

Figure 11

Ratio of the pressure $p_{ii}(t)$ and the energy density $\rho (t)$ of the particles created through the zero mode for $m={10}^3\mathrm{GeV}$ assuming the initial condition at $t_0=t_p$.