Cosmological evolution of finite temperature Bose-Einstein condensate dark matter

Once the temperature of a bosonic gas is smaller than the critical, density dependent, transition temperature, a Bose-Einstein condensation process can take place during the cosmological evolution of the Universe. Bose-Einstein condensates are very strong candidates for dark matter, since they can solve some major issues in observational astrophysics, like, for example, the galactic core/cusp problem. The presence of the dark matter condensates also drastically affects the cosmic history of the Universe. In the present paper we analyze the effects of the finite dark matter temperature on the cosmological evolution of the Bose-Einstein condensate dark matter systems. We formulate the basic equations describing the finite temperature condensate, representing a generalized Gross-Pitaevskii equation that takes into account the presence of the thermal cloud in thermodynamic equilibrium with the condensate. The temperature dependent equations of state of the thermal cloud and of the condensate are explicitly obtained in an analytical form. By assuming a flat Friedmann-Robertson-Walker geometry, the cosmological evolution of the finite temperature dark matter filled Universe is considered in detail in the framework of a two interacting fluid dark matter model, describing the transition from the initial thermal cloud to the low temperature condensate state. The dynamics of the cosmological parameters during the finite temperature dominated phase of the dark matter evolution are investigated in detail, and it is shown that the presence of the thermal excitations leads to an overall increase in the expansion rate of the Universe.

Figure 1

The density of the thermal cloud $\tilde{\rho }$ as a function of $\theta$ for different values of $T/T_{\mathrm{tr}}$: $T/T_{\mathrm{tr}}=0.95$ (solid curve), $T/T_{\mathrm{tr}}=0.90$ (dotted curve), $T/T_{\mathrm{tr}}=0.85$ (dashed curve), and $T/T_{\mathrm{tr}}=0.80$ (long dashed curve), respectively.

Figure 2

The pressure $\tilde{p}/p_0$ of the thermal cloud as a function of $\theta$ for different values of $T/T_{\mathrm{tr}}$: $T/T_{\mathrm{tr}}=0.95$ (solid curve), $T/T_{\mathrm{tr}}=0.90$ (dotted curve), $T/T_{\mathrm{tr}}=0.85$ (dashed curve), and $T/T_{\mathrm{tr}}=0.80$ (long dashed curve), respectively.

Figure 3

The pressure $\tilde{p}/p_0$ of the thermal cloud as a function of $\tilde{\rho }/{\rho }_{\mathrm{tr}}$ for different values of $T/T_{\mathrm{tr}}$: $T/T_{\mathrm{tr}}=0.95$ (solid curve), $T/T_{\mathrm{tr}}=0.90$ (dotted curve), $T/T_{\mathrm{tr}}=0.85$ (dashed curve), and $T/T_{\mathrm{tr}}=0.80$ (long dashed curve), respectively.

Figure 4

The normalized pressure $p_c/p_0$ of the condensate as a function of $\theta$ for $\kappa =50$ and for different values of $T/T_{\mathrm{tr}}$: $T/T_{\mathrm{tr}}=0.95$ (solid curve), $T/T_{\mathrm{tr}}=0.70$ (dotted curve), $T/T_{\mathrm{tr}}=0.60$ (dashed curve), and $T/T_{\mathrm{tr}}=0.40$ (long dashed curve), respectively.

Figure 5

Time variation of the dimensionless condensate density $\theta$ for $K_1=5\times {10}^{37}$ (solid curve), $K_1=4\times {10}^{37}$ (dotted curve), $K_1=2\times {10}^{37}$ (dash-dotted curve), and $K_1=0.6\times {10}^{37}$ (dashed curve), respectively. The numerical values of the initial conditions and of the parameters are ${\theta }_0={10}^{-9}$, $a_{\mathrm{tr}}={10}^{-4}$, ${\chi }_0=0.9999$, $\kappa ={10}^{15}$, and ${\kappa }_0={10}^{10}$.

Figure 6

Time variation of the reduced temperature $\chi =T/T_{\mathrm{tr}}$ of the finite temperature condensed dark matter, for $K_1=5\times {10}^{37}$ (solid curve), $K_1=4\times {10}^{37}$ (dotted curve), $K_1=2\times {10}^{37}$ (dash-dotted curve), and $K_1=0.6\times {10}^{37}$ (dashed curve), respectively. The numerical values of the initial conditions and of the parameters are ${\theta }_0={10}^{-9}$, $a_{\mathrm{tr}}={10}^{-4}$, ${\chi }_0=0.9999$, $\kappa ={10}^{15}$, and ${\kappa }_0={10}^{10}$.

Figure 7

Time variation of the total density ${\rho }_{\mathrm{DM}}$ of the finite temperature condensed dark matter, for $K_1=5\times {10}^{37}$ (solid curve), $K_1=4\times {10}^{37}$ (dotted curve), $K_1=2\times {10}^{37}$ (dash-dotted curve), and $K_1=0.6\times {10}^{37}$ (dashed curve), respectively. The numerical values of the initial conditions and of the parameters are ${\theta }_0={10}^{-9}$, $a_{\mathrm{tr}}={10}^{-4}$, ${\chi }_0=0.9999$, $\kappa ={10}^{15}$, and ${\kappa }_0={10}^{10}$. ${\rho }_0$ is the value of the condensate density corresponding to ${\theta }_0$ and ${\chi }_0$.

Figure 8

Time variation of the total pressure $p_{\mathrm{DM}}$ of the finite temperature condensed dark matter, for $K_1=5\times {10}^{37}$ (solid curve), $K_1=4\times {10}^{37}$ (dotted curve), $K_1=2\times {10}^{37}$ (dash-dotted curve), and $K_1=0.6\times {10}^{37}$ (dashed curve), respectively. The numerical values of the initial conditions and of the parameters are ${\theta }_0={10}^{-9}$, $a_{\mathrm{tr}}={10}^{-4}$, ${\chi }_0=0.9999$, $\kappa ={10}^{15}$, and ${\kappa }_0={10}^{10}$.

Figure 9

Time variation of the scale factor of the finite temperature condensed dark matter for $K_1=5\times {10}^{37}$ (solid curve), $K_1=4\times {10}^{37}$ (dotted curve), $K_1=2\times {10}^{37}$ (short dashed curve), $K_1=0.6\times {10}^{37}$ (long dashed curve), and of the pressureless dark matter $\Lambda \mathrm{CDM}$ model (dash-dotted curve). The numerical values of the initial conditions and of the physical parameters are ${\theta }_0={10}^{-9}$, $a_{\mathrm{tr}}={10}^{-4}$, ${\chi }_0=0.9999$, $\kappa ={10}^{15}$, and ${\kappa }_0={10}^{10}$.

Figure 10

Time variation of the scale factor of the Universe filled with dark energy, radiation, baryonic matter with negligible pressure, and Bose-Einstein condensed dark matter for $K_1=4\times {10}^{36}$ (solid curve), $K_1=3\times {10}^{36}$ (dotted curve), $K_1=2\times {10}^{36}$ (short dashed curve), $K_1=0.6\times {10}^{36}$ (long dashed curve), and of the standard $\Lambda \mathrm{CDM}$ model (dash-dotted curve). The numerical values of the initial conditions and of the physical parameters are ${\theta }_0={10}^{-9}$, $a_{\mathrm{tr}}={10}^{-4}$, ${\chi }_0=0.9999$, $\kappa ={10}^{15}$, and ${\kappa }_0={10}^{10}$.

Figure 11

The contour plot of Eq. (79), for $K_1=0.6\times {10}^{37}$, $\kappa ={10}^{15}$, and ${\kappa }_0={10}^{10}$.