Importance of being nonminimally coupled: Scalar Hawking radiation from regular black holes

TL;DR

This work demonstrates that a non-minimal scalar-curvature coupling, characterized by $\xi$, can substantially alter Hawking radiation from regular black holes with nonzero Ricci scalar $R$. By solving the non-minimally coupled Klein-Gordon equation on four RBH geometries (Bardeen, Hayward, Simpson-Visser, D'Ambrosio-Rovelli) and computing graybody factors $\Gamma_l^{s=0}(\omega)$, the authors show that the product $\xi fR$ (with $f=-g_{tt}$) determines whether the geometric barrier is raised or lowered, leading to enhanced or suppressed emission. In the conformal case $\xi=\tfrac{1}{6}$, deviations are generally modest, except for D'Ambrosio-Rovelli where suppression occurs; in the inflation-inspired case $\xi=-10^4$, emission can be enhanced by up to five orders of magnitude for D'Ambrosio-Rovelli while other RBHs are often suppressed and may display additional spectral features. These results imply that the evaporation history of light primordial RBHs—and hence their viability as dark matter candidates—can be markedly affected by scalar-curvature couplings, and they motivate extending the analysis to rotating RBHs and time-domain observables such as quasinormal modes and late-time tails.

Abstract

In curved space-time, a scalar field $φ$ is generically expected to couple to curvature, via a coupling of the form $ξφ^2R$. Yet in the study of Hawking emission from regular black holes (RBHs), where scalar fields are often introduced as simple probes of the geometry, and the Ricci scalar is generically non-zero, this non-minimal coupling is almost always ignored. We revisit this assumption by studying scalar Hawking emission from four representative RBHs (the Bardeen, Hayward, Simpson-Visser, and D'Ambrosio-Rovelli space-times), within two benchmark cases: the conformal case $ξ=1/6$, and a large negative value $ξ=-10^4$ motivated by Higgs inflation. We compute the graybody factors and emission spectra, showing that the latter can be either enhanced or suppressed, even by several orders of magnitude. A crucial role is played by the sign of the term $ξfR$, with $f(r)=-g_{tt}$ in Schwarzschild-like coordinates, as it determines whether the non-minimal coupling suppresses or enhances the geometric potential barrier. For the D'Ambrosio-Rovelli case with large negative $ξ$, the low-energy emission spectrum is enhanced by up to five orders of magnitude, since $ξfR<0$ throughout the space-time, leading to a deep potential well which broadens the transmissive window. The deviations we find can be particularly relevant in the case where primordial RBHs are dark matter candidates, given the impact of the non-minimal coupling on their evaporation history.

Figures (4)

• Figure 1: Each sub-panel shows the $f(r)R(r)$ factor (with $f(r)$ being minus the $g_{tt}$ metric element in Schwarzschild-like coordinates and $R$ the Ricci scalar) as a function of the radial coordinate normalized by the event horizon radius $r_H$, for the four regular black holes we consider: Bardeen (upper left, blue curve), Hayward (upper right, red curve), Simpson-Visser (lower left, green curve), and D'Ambrosio-Rovelli (lower right, purple curve) regular black holes. In all four cases, we consider a near-extremal value for the regularizing parameter, $\ell=0.99\ell_{\max}$. The $fR$ term is important because, once multiplied by the non-minimal coupling strength $\xi$, it is the one that modifies the standard geometric potential. For all regular black holes considered, $fR \to 0$ both at the event horizon ($r/r_H=1$) and at spatial infinity ($r/r_H \to +\infty$): this implies that the same set of boundary conditions which would be used when computing the graybody factors in the minimally coupled case can be used in the presence of the non-minimal coupling.
• Figure 2: Each sub-panel shows the graybody factors $\Gamma_{l=0}^{s=0}$, as a function of $\omega M$, for the four regular black holes we consider: Bardeen (upper left, blue curve), Hayward (upper right, red curve), Simpson-Visser (lower left, green curve), and D'Ambrosio-Rovelli (lower right, purple curve) regular black holes. In each sub-panel, we compare the GBFs for the four minimally coupled regular black holes ($\xi=0$, colored solid curves) to their conformally coupled counterparts ($\xi=1/6$, colored dashed curves), and the Schwarzschild GBF (black curve). The insets zoom into the regions where the deviations between the three GBFs are most significant. For illustrative purposes we only plot $\Gamma_{l=0}^{s=0}$, since we are interested in scalar emission ($s=0$) and the dominant emission mode is the $l=0$ one. We have fixed the regularizing parameter to the near-extremal value $\ell=0.99\ell_{\max}$ in all four cases. The features shown in this plot do not change sensibly for higher values of $l$ and other values of $\ell$.
• Figure 3: Graybody factors $\Gamma_{l=0}^{s=0}$ for the four regular black holes we consider, in the inflationary-inspired case where $\xi=-10^4$, and setting the regularizing parameter to $\ell=0.05\ell_{\max}$. The blue, red, green, and purple dashed curves correspond to the Bardeen, Hayward, Simpson-Visser, and D'Ambrosio-Rovelli regular black holes respectively, whereas the black solid curve corresponds to the Schwarzschild black hole, which here is taken as the comparison baseline instead of the respective minimally coupled counterparts: the reason is that the dynamical range of the resulting Hawking emission spectra is much larger than in the conformal case, and the corresponding minimally coupled spectra are nearly indistinguishable from each other and from the Schwarzschild spectrum across the plotting range we will later adopt.
• Figure 4: Scalar Hawking emission spectra resulting from the evaporation of a regular black hole of mass $M=10^{16}\,{\text{g}}$. Left panel: spectra computed in the conformal case, where $\xi=1/6$ and $\ell=0.99\ell_{\max}$, for the Bardeen (blue curves), Hayward (red curves), Simpson-Visser (green curves), and D'Ambrosio-Rovelli (purple curves) regular black holes. In all cases, the resulting spectra in the non-minimally coupled cases (dashed curves) are compared against their minimally coupled counterparts (solid curves). Right panel: spectra computed in the inflationary-inspired case, where $\xi=-10^4$ and $\ell=0.05\ell_{\max}$, for the Bardeen (blue curve), Hayward (red curve), Simpson-Visser (green curve), and D'Ambrosio-Rovelli (purple curve) regular black holes. In this case, the comparison baseline is the emission spectrum of the Schwarzschild black hole (black curve): the reason is that in this regime ($\vert \xi \vert \gg 1$) the dynamical range of the resulting spectra is much larger than in the conformal case, and the corresponding minimally coupled spectra are nearly indistinguishable from each other, and from the Schwarzschild spectrum, across the plotted range.