Spectral properties of acoustic black hole radiation: Broadening the horizon

The sensitivity of the black hole spectrum when introducing short distance dispersion is studied in the context of atomic Bose condensates. By considering flows characterized by several length scales, we show that, while the spectrum remains remarkably Planckian, the temperature is no longer fixed by the surface gravity. Rather it is determined by the average of the flow gradient across the horizon over an interval fixed by the healing length and the surface gravity, as if the horizon were broadened. This remains valid as long as the flow does not induce nonadiabatic effects that produce oscillations or some parametric amplification of the flux.

Figure 1

Upper panel: background flow $w_{\mathrm{B}}$ (solid line) for $n=3$, $D=0.5$ and two perturbations $w_p$ (amplified by a factor of 5, dashed and dotted lines), with $n_p=1$, and $({\kappa }_p/{\kappa }_{\mathrm{B}},D_p)$ equal to (0.4, 0.02) and $(-0.4,0.01)$. Central and lower panel: first and second derivative of $w_{\mathrm{B}}$ (solid line) and of $w=w_{\mathrm{B}}+w_p$ (dashed and dotted lines) for the same $w_p$. Narrower perturbations induce higher modifications of derivatives of $w$.

Figure 2

${log}_{10}(f_{\omega }/T_{\mathrm{H}})$ versus $\omega /{\kappa }_{\mathrm{B}}$ for different values of $D_p$, but the same surface gravity ${\kappa }_{\mathrm{K}}$. The solid line represents the Planck flux at temperature $T_{\mathrm{H}}={\kappa }_{\mathrm{K}}/2\pi$. When $D_p=0.1$, the phonon flux hardly differs from it. For smaller values of $D_p$, the flux is weaker, and for $D_p\to 0$, it is fixed by the background surface gravity ${\kappa }_{\mathrm{B}}$. The fixed parameters are $\Lambda /{\kappa }_{\mathrm{B}}=15$, $D=0.4$, ${\kappa }_p/{\kappa }_{\mathrm{B}}=0.5$, $n=n_p=1$.

Figure 3

$T_{\omega }/T_{\mathrm{B}}$ of Eq. (11) versus $\omega /{\kappa }_{\mathrm{B}}$ for various $D_p$ but the same value of ${\kappa }_{\mathrm{K}}=1.5{\kappa }_{\mathrm{B}}$. $T_{\mathrm{B}}={\kappa }_{\mathrm{B}}/2\pi$ is the Hawking temperature of the background flow. The values of the parameters are those of Fig. 2. For all values of $D_p$, $T_{\omega }$ hardly varies until $\omega \to {\omega }_{\max }$ of Eq. (a3), where the flux vanishes. Since ${\omega }_{\max }\gtrsim 10T_{\mathrm{B}}$, as can be seen from the horizontal upper scale, these spectra are, to a high precision, Planckian.

Figure 4

$T_0/T_{\mathrm{B}}$ versus $D_p$ for ${\kappa }_p/{\kappa }_{\mathrm{B}}$ from 0.2 to 1, and with $\Lambda /{\kappa }_{\mathrm{B}}=15$, $D=0.3$, $n=n_p=1$. Along each curve the surface gravity ${\kappa }_{\mathrm{K}}$ of Eq. (8) is constant. It fixes the temperature only for $D\gg D_p^{\mathrm{crit}}$. The vertical line gives the value of the healing length $\xi$.

Figure 5

$T_0/T_{\mathrm{B}}$ versus $D_p$ for $\Lambda /{\kappa }_{\mathrm{B}}$ from 12 to 20, and with ${\kappa }_p/{\kappa }_{\mathrm{B}}=0.5$, $D=0.3$, $n=n_p=1$. The vertical lines give the values of the healing length $\xi =\sqrt{2}{c}_0/\Lambda$. One can see that $D_p^{\mathrm{crit}}$ of Eq. (17) is not (directly) related to $\xi$.

Figure 6

Dots: $T_0/T_{\mathrm{B}}$ versus $D_p$ computed for $\Lambda /{\kappa }_{\mathrm{B}}=15$, $D=0.4$, $n_p=n=1$, ${\kappa }_p/{\kappa }_{\mathrm{B}}=0.5$. Solid line: best fit curve obtained by using $\overline{\kappa }/2\pi$ of Eq. (21). The perfect agreement establishes that the temperature $T_0$ equals $\overline{\kappa }/2\pi$. Dashed lines: asymptotic behaviors for $D_p\ll D_p^{\mathrm{crit}}$ and $D_p\gg D_p^{\mathrm{crit}}$ [see Eqs. (15, 16)].

Figure 7

Left panel: $T_0/T_{\mathrm{B}}$ versus $D_p$ for different values of $n_p$ of Eq. (6) with ${\kappa }_p/{\kappa }_{\mathrm{B}}=0.5$, $D=0.4$, $n=2$. $T_0$ monotonically approaches $T_{\mathrm{H}}$ for $n_p\le 1$. When $n_p>1$ instead, it exceeds $T_{\mathrm{H}}$ near $D_p^{\mathrm{crit}}$ and then approaches it with oscillations. Right panel: the relative difference $(n_p{}^{(\mathrm{fit})}-n_p)/n_p$ versus $n_p$. The difference decreases for smaller $n_p$, in a rather insensitive way with respect to $\Lambda /{\kappa }_{\mathrm{B}}$ (dots, $\Lambda /{\kappa }_{\mathrm{B}}=15$; squares, $\Lambda /{\kappa }_{\mathrm{B}}=20$).

Figure 8

On the left panel, $T_0/T_{\mathrm{B}}$ as a function of $D_p$ for various values of $\delta$, and, on the right, $T_0/T_{\mathrm{B}}$ as a function of $\delta$ with $D_p$ from $D_p=0.02$ to $D_p=0.3$. $\Lambda /{\kappa }_{\mathrm{B}}=15$, $D=0.4$, ${\kappa }_p/{\kappa }_{\mathrm{B}}=0.5$, $n=n_p=1$. On the left panel, for values of $\delta \ge -0.05$, $T_0(D_p)$ behaves as in Fig. 6, whereas, for more negative values of $\delta$, $T_0$ behaves differently. On the right panel, the asymmetry of $T_0$ under $\delta \to -\delta$ is clearly visible for $D_p<D_p^{\mathrm{crit}}\approx 0.06$.

Figure 9

The parameter ${\Delta }_T(\omega )$ of Eq. (13) versus $\omega /{\kappa }_{\mathrm{B}}$ for different values of $\delta$. For $\delta =0$, continuous line, the perturbation is symmetric with respect to the horizon, and ${\Delta }_T(\omega )$ is extremely small in agreement with Eq. (14). When $\delta >0$ and $(-\delta )\le d_{\xi }\approx 0.2c_0/{\kappa }_{\mathrm{B}}$, ${\Delta }_T(\omega )$ stays very small and no oscillations are found. Instead when $-\delta \gg d_{\xi }$, oscillations of large amplitudes appear even for low frequencies, i.e., $\omega \ll {\omega }_{\max }\simeq 13T_{\mathrm{B}}$, thereby indicating that the spectrum is no longer Planckian. The fixed parameters are $\Lambda /{\kappa }_{\mathrm{B}}=15$, $D=0.4$, $D_p=0.02$, ${\kappa }_p/{\kappa }_{\mathrm{B}}=0.5$, $n=n_p=1$.

Figure 10

Numerical data (dots) for the series $T_0(\delta )/T_{\mathrm{B}}$ with $D_p=0.15\approx 2.5D_p^{\mathrm{crit}}$ and $0.05\approx 0.8D_p^{\mathrm{crit}}$ of Fig. 8, right plot. The solid lines are the values of the fitted function of Eq. (20). The dotted line represents the surface gravity ${\kappa }_{\mathrm{K}}(\delta )/2\pi$ for $D_p=0.05$. The agreement between the temperature and $\overline{\kappa }/2\pi$ is quite good in the domains with no interference. The lowering of the peak and its shift to the left for small values of $D_p$ are rather well reproduced. None of these effects are found when using the surface gravity ${\kappa }_{\mathrm{K}}(\delta )$.

Figure 11

$T_0/T_{\mathrm{B}}$ as a function of $\theta$ for Eq. (29), varying $k_Z$ and $A$ so that $A{k}_Z=0.5{\kappa }_{\mathrm{B}}$ stays fixed. $\Delta =2D{c}_0/{\kappa }_{\mathrm{B}}$, $\Lambda /{\kappa }_{\mathrm{B}}=15$, $D=0.4$, $n=1$. For very high values of $k_Z$, i.e., $k_Z=100$ and 30 ${\kappa }_{\mathrm{B}}/c_0$, the temperature is approximately ${\kappa }_{\mathrm{B}}/2\pi$, i.e., equal to $\overline{\kappa }/2\pi$ of Eq. (20). For small values of $k_Z$, i.e., $k_Z=5{\kappa }_{\mathrm{B}}/c_0$, $T_0$ also approximately follows $\overline{\kappa }(\theta )/2\pi \sim {\kappa }_{\mathrm{K}}(\theta )/2\pi$ since the undulation wavelength is larger than $d_{\xi }$. In the intermediate regime instead, for $k_Z=10$ and $20{\kappa }_{\mathrm{B}}/c_0$, the temperature widely oscillates and no longer follows $\overline{\kappa }/2\pi$.

Figure 12

Graphical solution of the dispersion relation (9) for subsonic (left panel) and supersonic flow (right panel). Solid curves: $\pm \Omega (k)/\kappa$. Left panel, dashed line: $(\omega -vk)/\kappa$. Right panel, dashed, solid, dotted lines: $\omega -vk$, respectively, for $\omega <{\omega }_{\max }$, $\omega ={\omega }_{\max }$, $\omega >{\omega }_{\max }$. For $\omega <{\omega }_{\max }$, two extra (real) roots exist in the left lower quadrant. They cease to exist for $\omega >{\omega }_{\max }$ and this explains why there is no Hawking radiation above ${\omega }_{\max }$.