Numerical Studies of Back Reaction Effects in an Analog Model of Cosmological Preheating

We theoretically propose an atomic Bose-Einstein condensate as an analog model of backreaction effects during the preheating stage of the early Universe. In particular, we address the out-of-equilibrium dynamics where the initially excited inflaton field decays by parametrically exciting the matter fields. We consider a two-dimensional, ring-shaped BEC under a tight transverse confinement whose transverse breathing mode and the Goldstone and dipole excitation branches simulate the inflaton and quantum matter fields, respectively. A strong excitation of the breathing mode leads to an exponentially growing emission of dipole and Goldstone excitations via parametric pair creation: Our numerical simulations of the BEC dynamics show how the associated backreaction effect results not only in an effective friction of the breathing mode, but also in a quick loss of longitudinal spatial coherence of the initially in-phase excitations. Implications of this result on the validity of the usual semiclassical description of backreaction are finally discussed.

Figure 1

Bogoliubov spectrum of collective excitations around the ground state. The three curves correspond to modes with zero (Goldstone, black), one (dipole, blue), and two (breathing, green) nodes in the transverse direction. The red circle highlights the transverse breathing mode that is excited at early times to simulate the inflaton oscillations; the yellow (purple) circles highlight the dipole (Goldstone) modes of opposite momenta that are resonantly excited by the parametric processes indicated by the arrows. System parameters: gas of $N={10}^6$ atoms in an integration box of size $L_{x,y}/{\ell }_0=140$, 3.54 in units of the transverse harmonic oscillator length ${\ell }_0=\sqrt{\hslash /2m{\omega }_0}$, with $N_{x,y}=512$, 12 grid points; equilibrium chemical potential $\mu /\hslash {\omega }_0=2.38$. The independence of our conclusions from the specific numerical parameters is validated in Supplemental Material [55].

Figure 2

Numerical results of the TWA simulations. Stochastic averages are based on a sample of ${\mathcal{N}}_r=1000$ independent realization. (a) Momentum distribution of the population in the Bogoliubov modes at time ${\omega }_0^{b}t/2\pi =48$. The dot-dashed green, dashed blue, and solid black lines correspond to the breathing, dipole, and Goldstone branches, respectively. The dotted violet line shows the population in the breathing mode immediately after the excitation sequence. (b) Time evolution of the integrated population in the breathing, dipole, and Goldstone branches over the regions indicated by the shading in (a). The same color code as in (a). The dotted violet line shows the time evolution of the population in the single breathing mode at $k=0$.

Figure 3

Left panel: color plot of the time evolution of the spatial correlation function $C_w(X;t)$ of the transverse size of the condensate. Cuts at different times (indicated by the horizontal dashed lines) are shown in the middle panel. Right panel: time evolution of the spatial correlation function for coincident $C_w(0;t)$ and opposite $C_w(L/2;t)$ points around the ring. The same system and simulation parameters as in Figs. 1 and 2.