Analog model of a Friedmann-Robertson-Walker universe in Bose-Einstein condensates: Application of the classical field method

Analog models of gravity have been motivated by the possibility of investigating phenomena not readily accessible in their cosmological counterparts. In this paper, we investigate the analog of cosmological particle creation in a Friedmann-Robertson-Walker universe by numerically simulating a Bose-Einstein condensate with a time-dependent scattering length. In particular, we focus on a two-dimensional homogeneous condensate using the classical field method via the truncated Wigner approximation. We show that for various forms of the scaling function the particle production is consistent with the underlying theory in the long wavelength limit. In this context, we further discuss the implications of modified dispersion relations that arise from the microscopic theory of a weakly interacting Bose gas.

Figure 1

Schematic of de Sitter expansion for the FRW analog model; for region I $(t<t_0)$, there is no expansion and the mode solutions are those of a Minkowski spacetime with $U=U_0$; for region II $(t_0<t<t_f)$, there is a de Sitter–type expansion and the mode solutions are nontrivial; finally, for region III $(t>t_f)$, the expansion is turned off and the mode solutions are those of a Minkowski spacetime with $U=U_0∕X$.

Figure 2

(Color online) Particle production for a de Sitter spacetime in the FRW analog model with $X=2000$ and a range of expansion rates ${\overline{t}}_s$ as shown. The dimensionless scaling unit is ${\overline{t}}_s=t_s\hslash ∕\left(m{L}^2\right)$ and the nonlinearity is $U_0{n}_0={10}^5{\hslash }^2∕\left(m{L}^2\right)$.

Figure 3

(Color online) de Sitter model: density plot for renormalized wave function at the beginning $(t=0)$ and end $(t=t_f)$ of expansion. Parameters are $t_s=1\times {10}^{-5}$, $C_{\mathrm{NL}}(t=0)=1\times {10}^5$, and $N_0={10}^7$.

Figure 4

(Color online) de Sitter expansion: time dependence of Bogoliubov mode populations. Parameters are $C_{\mathrm{NL}}(\overline{t}=0)=1\times {10}^5$, $N_0={10}^7$, and $X=2\times {10}^3$. The (blue) points on each curve show where each mode crosses over from phonon to free-particle behavior (as defined by ${\overline{k}}_c^2∕2=C_{\mathrm{NL}}$). The (green) dashed curve at the final time shows the analytic prediction in the acoustic approximation based on the Bogoliubov theory and Hamiltonian diagonalization as outlined in Sec. 5A. The (red) solid curve at the final time shows the analytic prediction (including quantum pressure) for a sudden transition from Eq. (125).

Figure 5

(Color online) tanh expansion: time dependence of Bogoliubov mode populations. Parameters are $C_{\mathrm{NL}}(\overline{t}=0)=1\times {10}^5$, $N_0={10}^7$, and $X={10}^3$. The (blue) points on each curve show where each mode crosses over from phonon to free-particle behavior (as defined by ${\overline{k}}_c^2∕2=C_{\mathrm{NL}}$). The (green) dashed curve at the final time shows the analytic prediction in the acoustic approximation from Eq. (99). The (red) solid curve at the final time shows the analytic prediction (including quantum pressure) for a sudden transition from Eq. (125).

Figure 6

(Color online) Equipartition of energy at the final time for de Sitter expansion with four different expansion rates. Parameters are $C_{\mathrm{NL}}(t=0)=1\times {10}^5$, $N_0={10}^7$, and $X=2\times {10}^3$.

Figure 7

(Color online) Cyclic universe: Time dependence of Bogoliubov mode populations for four different scenarios. Parameters are $C_{\mathrm{NL}}(\overline{t}=0)=1\times {10}^5$, $N_0={10}^7$ for all cases. The (red) solid line at the final time shows the position of the peak wave vector for parametric resonance from the analytic prediction (128). In subplots (a) and (b), the spike in population at $t\sim 0.5t_f$ is an artifact of the time-dependent quasiparticle projection.

Figure 8

(Color online) de Sitter expansion: Bogoliubov mode populations at four different times. Parameters are $t_s=1\times {10}^{-4}$, $C_{\mathrm{NL}}(\overline{t}=0)=1\times {10}^5$, $N_0={10}^7$, and $X=2\times {10}^3$ at $t_f$. The (blue) points are the simulation points; the vertical solid (red) line shows the crossover from phonon to free-particle behavior (as defined by ${\overline{k}}_c^2∕2=C_{\mathrm{NL}}$); and the (green) dashed curve shows the analytic prediction in the acoustic approximation based on the Bogoliubov theory and Hamiltonian diagonalization as outlined in Sec. 5A.