In general, PBHs that form with significant spin or charge have lower surface gravity than PBHs that form with the same mass but vanishing spin and charge. The lower surface gravity yields a lower Hawking temperature. However, Page demonstrated that an initially rotating black hole will preferentially emit particles with spin aligned with the black hole’s angular momentum, causing it to “spin down” to within 1% of the emission rate of a same-mass Schwarzschild black hole before half of its energy is radiated away. The exploding PBHs we consider have present-day masses $M\le {10}^8\mathrm{g}$ and must have formed with initial mass $M_i\sim 5\times 10^{14}\mathrm{g}$. Thus, these PBHs have radiated approximately all of their formation mass and Page’s statement should apply, leading us to assume that present-day exploding PBHs have no spin. He further demonstrated that an initially maximally rotating PBH will have a lifetime that differs from that of its Schwarzschild counterpart only by a factor of 2.32 [29], so we neglect initial spin here. Meanwhile, PBHs discharge any initial charge even more efficiently, leaving stochastic charge fluctuations of the order of $\mathcal{O}(q)\ll \sqrt{G}M$ [44, 45], so we likewise neglect initial charge for the PBHs.

If elementary particles beyond those of the SM exist, they would also be included among the primary emission spectra during PBH Hawking evaporation. On the one hand, increasing the total number of degrees of freedom emitted from a PBH of a given mass would reduce the number of high-energy neutrinos emitted during primary emission for small-mass PBHs near the end of their lifetimes. On the other hand, depending on whether the new particles couple to neutrinos, their presence could increase the number of high-energy neutrinos emitted via secondary emission. Such effects remain strongly model dependent. See, e.g., Refs. [54, 55, 56, 57, 58]. As a concrete example, adding a scalar field with $m\ge 100\mathrm{GeV}$ would increase the Page factor $f(M)$ for $M\lesssim {10}^{11}\mathrm{g}$. Adding an additional heavy degree of freedom would not modify the primary neutrino spectrum Eq. (1), because this depends only on the number of neutrino degrees of freedom. The increase in the Page factor results in an increase in the overall mass-loss rate in Eq. (2) and, thus, a decrease in the integrated total neutrino emission rate in Eq. (5). We find that for this case of a single spin-0 dark degree of freedom with $m=100\mathrm{GeV}$, the value of $N_{\nu }$ decreases by $\sim 2\%$ for $Q_{\min }\gtrsim 100\mathrm{GeV}$.

The neutrinos propagate radially outward with ballistic motion because they are uncharged—and, therefore, are not influenced by galactic magnetic fields—and the cross section for interaction with the interstellar medium is negligible. The thickness of the “shell” of propagating neutrinos is set by the PBH lifetime (see column 4 in Table 1), and, in the cases of interest, ${\tau }_0\lesssim \mathcal{O}({10}^{-5}\mathrm{s})$. We, therefore, can assume that all emitted neutrinos from one explosion would strike a detector at the same time.

We calculate ${\mathrm{\Phi }}_b$ and ${\sigma }_{{\mathrm{\Phi }}_b}$ using standard error propagation, given the values for $E_{\text{break}}$, ${\gamma }_2$, and ${\mathrm{\Phi }}_{\nu }^{\mathrm{IC}}(100\mathrm{TeV})$ reported in Ref. [69]. Next, we note that, when extrapolating the flux to energies $E_{\nu }>E_{\text{break}}$, the relevant quantities obey ${\sigma }_{\mathrm{\Phi }(E_{\nu })}/{\mathrm{\Phi }}_{\nu }(E_{\nu })\ge {\sigma }_{{\mathrm{\Phi }}_b}/{\mathrm{\Phi }}_b$. Since the variance is dominated by the uncertainty in $E_{\text{break}}$, we make the conservative estimate and adopt ${\sigma }_{\mathrm{\Phi }(E_{\nu })}/\mathrm{\Phi }(E_{\nu })={\sigma }_{{\mathrm{\Phi }}_b}/{\mathrm{\Phi }}_b$.

The Page factor $f(M)$ is $\mathcal{O}(1)$ for $M\ge {10}^{15}\mathrm{g}$; it reaches a maximum of $\sim 15$ for $M\le {10}^{11}\mathrm{g}$. $f(M)$ varies significantly with mass beginning around $M={10}^{15}\mathrm{g}$. This means that, for PBHs that form with $M_i\gtrsim M_{*}$, $f(M)$ is nearly constant throughout the present age of the Universe $t_0$. For further discussion, see Ref. [79].

To relate ${\varphi }_{\mathrm{GCC}}(M_i)$ to the mass distribution ${\psi }_{\mathrm{GCC}}(M_i)$ considered in Refs. [24, 83], we use the fact that $n_{\mathrm{PBH}}\simeq {\rho }_{\mathrm{PBH}}/\overline{M}$ for these distributions and ensure that ${\varphi }_{\mathrm{GCC}}(M_i)$ is properly normalized.