Quantum Tunneling of Primordial Black Holes to White Holes: Rates, Constraints, and Implications for Fast Radio Bursts

Abstract

We calculate the present-day and cosmological volumetric rate of primordial black hole (PBH) quantum tunneling events to white holes, incorporating the competition between Hawking evaporation and tunneling, cosmological depletion, realistic mass-dependent abundance constraints, extended mass functions, and the alternative memory-burden scenario. The burst rate is maximized along a narrow ridge in the mass--tunneling-parameter plane where the effective PBH lifetime is comparable to the age of the Universe. Within the canonical Planck-star range of tunneling timescales, FRB-level rates arise only in two highly restricted regions: a low-mass window near the evaporation boundary, and a narrow sequential window where evaporation precedes tunneling; broadening the PBH mass function does not qualitatively alter this conclusion. We further assess observational constraints from FRB repetition statistics, radio spectral properties, prompt and diffuse gamma-ray limits, host-galaxy demographics, and gravitational-wave signatures. All current observations are consistent with a subdominant white-hole contribution to the FRB population but strongly disfavor a dominant origin. The rate calculation does not generically support FRB-level event densities, and any viable FRB interpretation requires narrow, fine-tuned, and strongly assumption-dependent corners of parameter space.

Figures (3)

• Figure 1: Fluence of Planck-star burst emission in the radio (blue/light blue) and gamma-ray (red/orange) channels as a function of distance, for two representative PBH masses: $M = 10^{23}$ kg (solid lines) and $M = 10^{14}$ kg (dashed lines, teal and orange). Shaded bands indicate the uncertainty in the predicted fluence arising from the range of emission efficiencies $\epsilon_{\rm radio} \in [10^{-7},\,10^{-5}]$ and $\epsilon_{\gamma} \in [10^{-5},\,10^{-3}]$, with fiducial values $\epsilon_{\rm radio} = 10^{-6}$ and $\epsilon_{\gamma} = 10^{-4}$ shown as the central lines. Dotted horizontal lines mark the approximate sensitivity thresholds of CHIME/ASKAP ($\mathcal{F}_{\rm min} \simeq 10^{-17}$ erg cm$^{-2}$, radio) and Fermi GBM ($\mathcal{F}_{\rm min} \simeq 10^{-8}$ erg cm$^{-2}$, gamma-ray). The upper axis gives the distance in Gpc. For $M = 10^{23}$ kg the radio fluence remains above the CHIME/ASKAP threshold out to $\sim 1$ Gpc, while the gamma-ray counterpart becomes undetectable beyond $\sim 100$--$300$ Mpc. For $M = 10^{14}$ kg both channels fall below current instrument thresholds at all but sub-kiloparsec distances, illustrating the steep mass dependence of detectability ($\mathcal{F} \propto \epsilon M c^{2} / (4\pi D^{2}$)).
• Figure 2: Present-day volumetric burst rate $\log_{10}R_{\rm PBH}$ [Gpc$^{-3}$ yr$^{-1}$] in the $(M,\alpha)$ plane, evaluated at $z=0$ with $f_{\rm PBH}=1$. The color scale runs from $-2$ (dark purple) to $+6$ (bright yellow). The diagonal ridge of maximal rate traces the locus $\tau_{\rm eff}(M)\simeq t_0$ (Eq. \ref{['eq:ridge']}); to its lower-left the population is exponentially depleted ($\tau_{\rm eff}\ll t_0$), while to its upper-right the instantaneous decay probability is tiny ($\tau_{\rm eff}\gg t_0$). The solid green band marks the FRB-compatible range $10^1\le R_{\rm PBH}\le 10^4\,\mathrm{Gpc^{-3}\,yr^{-1}}$. The upper-left hatched region is excluded because the predicted burst rate would exceed the observed FRB volumetric rate ($R_{\rm PBH}>10^5\,\mathrm{Gpc^{-3}\,yr^{-1}}$), violating the observational upper bound. The horizontal red band indicates the canonical Planck-star range $10^{-2}\lesssim\alpha\lesssim 1$; within this strip the rate is exponentially suppressed for all masses compatible with evaporation constraints. The vertical blue band marks the PBH mass window $M\sim 10^{24}$--$10^{26}$ kg for which the Schwarzschild radius sets the emission wavelength in the GHz band, $\nu_{\rm radio}\sim c/R_S$ (Section \ref{['sec:obs_spectrum']}). The left-boundary hatched strip marks masses already fully depleted by Hawking evaporation ($\tau_{\rm evap}<t_0$). The star marks the unique point where three independent requirements are simultaneously satisfied: the tunneling parameter lies in the canonical Planck-star range ($10^{-2}\lesssim\alpha\lesssim 1$, red band), the PBH mass is consistent with GHz emission from the Schwarzschild scale (vertical blue band, up to an order-unity factor $\kappa$), and the predicted burst rate falls within the observed FRB volumetric rate (green band). The coincidence of these three conditions at a single point in parameter space is a non-trivial consistency check of the Planck-star--FRB hypothesis, though as discussed in the text the rate is in practice far below FRB levels once depletion and abundance constraints are applied. All contours assume a monochromatic mass function; extended distributions are discussed in Section \ref{['sec:rate_massfunction']}.
• Figure 3: Same as Fig. \ref{['fig:RateContour']} but with the PBH dark-matter fraction set to its mass-dependent observational upper bound $f_{\rm PBH,max}(M)$ (inset, grey curve), obtained by interpolating current microlensing, CMB spectral-distortion, and Hawking-evaporation constraints. The color scale and all overlays are identical to Fig. \ref{['fig:RateContour']}, including the hatched upper-left region where $R_{\rm PBH}$ would exceed the observed FRB rate. Abundance constraints enforce $f_{\rm PBH,max}\ll 1$ at $M\lesssim 10^{17}$--$10^{20}$ kg (down to $\sim 10^{-8}$ in the most constrained windows), shifting the entire color scale downward by the same factor and compressing the diagonal ridge to far lower rates. The FRB-compatible band consequently retreats to a narrow wedge confined to very low masses ($M\lesssim 10^{11\text{--}12}$ kg) and very large tunneling parameters ($\alpha\gtrsim 10^{18}$), while the rate-excess exclusion zone shrinks in proportion. The star marks the same triple-coincidence point as in Fig. \ref{['fig:RateContour']}; once $f_{\rm PBH,max}(M)$ is applied, the predicted rate at that point falls well below the FRB-compatible band, illustrating how abundance constraints break the apparent consistency of the canonical scenario. Once realistic abundance limits are imposed, no broad region of the $(M,\alpha)$ plane produces cosmologically significant burst rates; any FRB-level scenario requires PBHs to saturate their abundance bound precisely in the narrow mass window near $10^{11}$--$10^{12}$ kg.