Hawking radiation in dispersive theories, the two regimes

We compute the black hole radiation spectrum in the presence of high-frequency dispersion in a large set of situations. In all cases, the spectrum diverges like the inverse of the Killing frequency. When studying the low-frequency spectrum, we find only two regimes: an adiabatic one where the corrections with respect to the standard temperature are small, and an abrupt one regulated by dispersion, in which the near-horizon metric can be replaced by step functions. The transition from one regime to the other is governed by a single parameter which also governs the net redshift undergone by dispersive modes. These results can be used to characterize the quasiparticles spectrum of recent and future experiments aiming to detect the analogue Hawking radiation. They also apply to theories of quantum gravity which violate Lorentz invariance.

Figure 1

$T_{\mathrm{eff}}/T_{\mathrm{H}}$ as a function of $\Lambda /\kappa$ and ${\omega }_{\max }/\kappa$ in logarithmic scales. The parameter $q$ of Eq. (8) is 0.3. Solid lines are loci of constant $T_{\mathrm{eff}}/T_{\mathrm{H}}$. One clearly sees that the departures from the Hawking regime are essentially governed by ${\omega }_{\max }/\kappa$. Roughly speaking, one leaves the Hawking regime (i.e., $T_{\mathrm{eff}}$ and $T_{\mathrm{H}}$ differ by 10%) for ${\omega }_{\max }/\kappa \lesssim 0.4$, irrespectively of the value of $\Lambda /\kappa$. Note that the upper white area is due to the fact that we restrict our analysis to $D<1$, whereas the lower one is due to our incapacity of numerically obtaining $n_{\omega }$ for $D\lesssim 0.02$.

Figure 2

The temperature ratios $T_{\mathrm{eff}}/T_{\mathrm{step}}$ (left panel) and $T_{\mathrm{eff}}^v/T_{\mathrm{step}}^v$ (right panel) as a function of $\Lambda /\kappa$ and ${\omega }_{\max }/\kappa$ for the same situations as in Fig. 1, where $T_{\mathrm{step}}$ and $T_{\mathrm{step}}^v$ are obtained from Eqs. (b16, b17). The departures from the steplike regime are essentially governed by ${\omega }_{\max }/\kappa$. Roughly speaking, one leaves this regime for ${\omega }_{\max }/\kappa \gtrsim 0.1$, irrespectively of the value of $\Lambda /\kappa$.

Figure 3

$R_{\mathrm{eff}}$ defined in Eq. (21) as a function of $\Lambda /\kappa$ and ${\omega }_{\max }/\kappa$ for the same situations as in Fig. 1. Solid lines give the various values of this ratio, whereas the dashed straight lines with slope $2/3$ are loci of constant $D$, see Eq. (12). Clearly, $R_{\mathrm{eff}}$ depends only on $D$.

Figure 4

In dark gray, we denote the validity region (defined by 10% difference in temperature) of the Hawking regime, and in light gray, that of the steplike regime. The thick horizontal line is the threshold value of Eq. (26). The dashed straight lines are loci of constant $D$. It is clear that ${\omega }_{\max }/\kappa$ governs the extension of the two domains and the transition between them.

Figure 5

$T_{\mathrm{eff}}/T_{\mathrm{av}}$ as a function of $\Lambda /\kappa$ and ${\omega }_{\max }/\kappa$ in logarithmic scale for the same profiles as in Fig. 1, where $T_{\mathrm{av}}$ is given by Eqs. (23, 24). Solid lines are loci of constant $T_{\mathrm{eff}}/T_{\mathrm{av}}$. The other lines are as in Fig. 4.

Figure 6

The number of $e$-folds of Eq. (7) evaluated for $\omega =T_{\mathrm{eff}}$ as a function of $\Lambda /\kappa$ and ${\omega }_{\max }/\kappa$ for the same profiles as in Fig. 1. The dotted lines bound the regions of validity (with 10% errors) of the Hawking regime (upper line) and the steplike regime (lower line). The thick line is the threshold value of Eq. (26). The Hawking and the steplike regime are, respectively, found for $N_{T_{\mathrm{eff}}}\gtrsim 2$ and $N_{T_{\mathrm{eff}}}\lesssim 1.3$.

Figure 7

Graphical solution of the dispersion relation (6) for subsonic (left panel) and supersonic flow (right panel). Solid curves: $\pm \Omega (k)$. In subsonic flows, for $\omega >0$, the dashed line ($\omega -vk$) crosses twice the dispersion relation with $\Omega >0$. In supersonic flows, two extra roots with $\Omega <0$ exist in the left lower quadrant for $\omega <{\omega }_{\max }$ (dashed line). When $v$ and $c$ take their asymptotic values in the limit $x\to -\infty$, the most (least) negative of these two roots corresponds to the wave number $k_{-\omega }^{\mathrm{in}}$ ($k_{-\omega }^{\mathrm{out}}$) of the ingoing (outgoing) wave, appearing in Eq. (7). For $\omega ={\omega }_{\max }$, the solid line is tangent to $\Omega (k)$, and for $\omega >{\omega }_{\max }$ (dotted line), these real roots no longer exist.

Figure 8

$T_{\omega }$ for 5 values of $\alpha$, with $D=0.5$ and $\Lambda /\kappa =10$ fixed. The horizontal dotted lines represent $T_{\mathrm{H}}={\kappa }_{\alpha }/2\pi$, with ${\kappa }_{\alpha }$ of Eq. (c3), respectively, for $|\alpha |=0$, 0.4, 0.8.

Figure 9

$T_{\mathrm{eff}}/T_{\mathrm{H}}$ (left panel) and $T_{\mathrm{eff}}/T_{\mathrm{step}}$ (right panel) as functions of $\Lambda /{\omega }_{\max }$ and ${\omega }_{\max }/\kappa$ evaluated in the asymmetric flow profile of Eq. (c1) for $\alpha =-0.5$. The validity domains of both the Hawking and the steplike regime are not affected by the introduction of the parameter $\alpha$ which governs the asymmetry of the sub- and supersonic regions.