Sound Wave in the Backreaction Affected Spacetime in Analogue Gravity Based on Number-Conserving Approach

TL;DR

The paper addresses backreaction in analogue gravity by applying a number-conserving Bogoliubov expansion to a Bose gas, showing that sound perturbations propagate as a massive scalar field in a dynamical spacetime with a spacetime-dependent mass $m^2(t,\mathbf{x})$. The method yields amended fluid equations via $\Delta_C$ and $\Delta_E$, derives a generalized sound-wave equation that maps to a massive Klein–Gordon equation on an effective metric $\mathfrak{g}_{\mu\nu}$, and computes correlation functions in a finite-size quasi-1D Bose gas to reveal backreaction effects. Key findings include the backreaction-induced increase of the UV content of the equal-point correlation and a pattern where correlations are enhanced in a finite region but reduced at larger separations. The work provides a concrete framework for studying backreaction in analogue gravity and offers quantitative predictions for correlation patterns in low-dimensional Bose gases that could guide experiments.

Abstract

It is shown that the sound wave in the backreaction affected dynamical spacetime follows the equations for a massive scalar field in a analogue spacetime using number-conserving approach. Even with backreaction, the analogue metric is in the same form to the case without backreaction. The sound velocity, fluid density, and fluid velocity are defined with small correction to include the backreaction effect. Moreover, the modification of classical fluid dynamical equations by the backreaction introduces spacetime dependent mass. For a finite-size homogeneous quasi-one dimensional Bose gas, we find that the backreaction increase the UV divergence of the equal position correlation function. Moreover, in this model, we see that the backreaction increase the correlation in a finite region and decrease the correlation in far region.

Figures (3)

• Figure 1: Truncated difference between correlation function with and without backreaction, $\rho_0(C_{\delta\theta}(x,y)-C_{\delta\theta_0}(x,y))$ at $t=5$ with the lowest 1000 modes. Here and in the following plots, for the numerical simulation, we choose $g\rho_0 = 1$ with $\rho_0=150$. Note that the correlation function is $\mathcal{O}(1/N)$. Therefore, we rescale the correlation with $\rho_0$. All the units are chosen such that we have the scalings $x=x[\xi_0]$ and $t=t[\xi_0^2]$. Note that there is sharp large values at $x=y$.
• Figure 2: Comparing black lines between upper and lower panel, one can see the increase of $\rho_0C_{\delta\theta}(x,x)$ at the same point by including more modes. On the other hand, for the linear approximation $\rho_0C_{\delta\theta_0}(x,x)$, the UV divergent terms cancel and they have still finite values.
• Figure 3: Comparing upper and lower panel, one can see that both of the value of correlation functions $\rho_0C_{\delta\theta}(x,-x)$ and $\rho_0C_{\delta\theta_0}(x,-x)$ are saturated except small neighborhood near $x=0$. For $6.4\lesssim |x|$, $\rho_0C_{\delta\theta_0}(x,-x)<\rho_0C_{\delta\theta}(x,-x)$. On the other hand, for $|x| \lesssim 6.4$, $\rho_0C_{\delta\theta_0}(x,-x)>\rho_0C_{\delta\theta}(x,-x)$.