Correlation patterns from massive phonons in $1+1$ dimensional acoustic black holes: A toy model

Transverse excitations in analogue black holes induce a masslike term in the longitudinal mode equation. With a simple toy model we show that correlation functions display a rather rich structure characterized by groups of approximately parallel peaks. For the most part the structure is completely different from that found in the massless case.

Figure 1

Shown is a plot of the coefficient of $m^2$ in (2.11) for the sound speed profile (2.3) using the parameters (2.4)(dashed) and our approximation to that coefficient (2.12)(solid).

Figure 2

Penrose diagram with modes which are nonzero on $H_{-}$ in the interior, $L$ and exterior, $R$.

Figure 3

The real part of solutions for $J_{(\text{left}\text{−}\text{moving})}$ (dashed) and $J_{(\text{right}\text{−}\text{moving})}$ (solid). Note that when added together there is clear cancelation of the smaller peaks and only the single large peak that occurs when the points come together survives.

Figure 4

Density-density correlation function for $1+1\mathrm{D}$ BEC BH analog with no masslike term which shows 3 peaks. The figure is reproduced from [10] with the numbers superposed on that figure.

Figure 5

Correlations between the modes found in the massless case.

Figure 6

Plot for the two-point function for $m=4\times {10}^{-2}{m}_a$. The horizontal and vertical white stripes correspond to regions near the horizon which are excluded from the plot. The contrast has been arranged to show the primary structures that have been observed in this case. As a result regions which appear to be pure white aside from the exclusions are regions where the two-point function is larger than $1.7\times {10}^{-3}$. For the pure black regions the two-point function is smaller than $-2\times {10}^{-4}$.

Figure 7

Plot of the two-point function for $m=4\times {10}^{-4}{m}_a$. The region shown is the lower left quadrant of Fig. 6, where both points are in the interior of the analogue BH. Here the range of the plot has been extended to better illustrate the features in this region. The dashed lines indicate the correlation peaks in the massless case with locations specified by (4.9a). Note that the less pronounced set of peaks that is clearly visible in Fig. 6 is only visible here near the upper right corner of the plot. The reason is that the amplitudes of these peaks decrease as the separation between the points $x$ and $x^{\prime }$ increases.

Figure 8

Plot of the two-point function for $m=4\times {10}^{-4}{m}_a$. The region shown is the upper left quadrant of Fig. 6, where one point is in the interior and the other point is in the exterior of the analogue BH. Here the range of the plot has been extended to better illustrate the features in this region. The dashed lines indicate the locations of the correlation peaks in the massless case with locations specified by (4.7) and (4.8).

Figure 9

Plots of the two-point function for the case $m=4\times {10}^{-2}{m}_a$ for various frequency bands. Top: Low frequency contribution to the two-point function for $0\le \omega \le \frac{m_R}{100}$. Middle: Contributions from frequencies in the range $0\le \omega \le \frac{m_R}{10}$. Bottom: Contributions from all frequencies. Note that the scale of the color bar is larger than that in Fig. 6 which causes subtle differences in the appearance of the third plot with the lower left quadrant of Fig. 6.

Figure 10

Stationary phase result for part of the contribution to the two-point correlation function when both points are inside the horizon from products of the $f_H^L$ modes. One contribution consist of an upstream mode at point $x$ and a downstream mode at point $x^{\prime }$. The other has the point $x$ and $x^{\prime }$ for these modes reversed.

Figure 11

Shown are plots for a range of values of the mass $m$ of the phonon field. From left to right and top to bottom and starting in the upper left the masses are: $m/m_a=4\times {10}^{-2}$, $2\times {10}^{-2}$, ${10}^{-2}$, $8\times {10}^{-3}$, $6\times {10}^{-3}$, $4\times {10}^{-3}$, $2\times {10}^{-3},{10}^{-3}$ and ${10}^{-4}$.

Figure 12

Plot of the two-point function for $m={10}^{-4}{m}_a$, which is the lowest mass case shown in Fig. 11(lower right). The range of this plot has been extended significantly when compared to previous plot to show the structure due to the mass.

Figure 13

Plot of the two-point function for $m={10}^{-4}{m}_a$, which is the lowest mass case shown in Fig. 11(lower right). It is also the same mass as was used for the plot in Fig. 12. The region shown is where one point is in the interior and one point is in the exterior of the analogue BH. Near the horizon a peak can be seen at the same approximate location as peak (1) in the massless case, see Eq. (4.7) and Fig. 4. This peak disappears for points which are further from the horizon.