Dark radiation from Kerr primordial black holes: the role of superradiance

Abstract

Light primordial black holes (PBHs) that fully evaporate before Big Bang Nucleosynthesis (BBN) produce dark radiation (DR) via Hawking radiation of gravitons, contributing to the effective number of relativistic species $ΔN_{\rm eff}$. If the particle spectrum contains a beyond-the-Standard-Model (BSM) boson with Compton wavelength comparable to the black hole (BH) gravitational radius, superradiant instability extracts angular momentum from the BH into a bosonic cloud, whose gravitational wave (GW) emission contributes an additional source of DR. By simultaneously evolving the BH mass and spin, superradiant mode occupation numbers, comoving entropy and cosmological energy densities in an expanding early-universe background, we find that superradiance generically suppresses $ΔN_{\rm eff}$: by extracting angular momentum before Hawking radiation can convert it into gravitons, superradiance starves the dominant dark-radiation channel. The GWs emitted by the superradiant cloud can partially compensate this loss, but only when the superradiant and BH evaporation timescales are comparable; otherwise the cloud GWs are emitted too early and diluted by cosmological expansion. The results imply that existing $ΔN_{\rm eff}$ bounds on PBH mass and spin derived without superradiance must be revisited if BSM bosons are present in the particle spectrum.

Figures (4)

• Figure 1: Cosmological evolution of the comoving energy densities $\varrho \equiv \rho\,a^3$ (top), the BH spin parameter $a_\ast$ (middle), and the GW-to-SM energy density ratio $\rho_{\rm GW}/\rho_{\rm SM}$ (bottom) as a function of the scale factor $a$, for a benchmark with $M_{\rm ini} = 10^4\,{\rm g}$, $a_{*,\rm ini} = 0.999$, $\beta = 6.40 \times 10^{-8}$, $\alpha_{\rm ini} = 0.10$ ($\mu = 2.7 \times 10^{9}\,{\rm GeV}$, scalar $s = 0$). Left: Hawking-radiation-only baseline. The SM radiation (red) and PBH matter (black) components cross near $a \sim 10^7$, producing a brief PBH-dominated epoch that ends with evaporation near $a \sim 10^{10}$. The spin remains near-extremal throughout most of the evolution, dropping only at the very end of evaporation. The Hawking graviton energy density (GW HR, orange) peaks near evaporation. Right: Full calculation including superradiance. The superradiant scalar cloud (BSM SR, purple) grows rapidly before evaporation, becoming the third-largest energy component. The middle subpanel shows the characteristic staircase spin-down: $a_\ast$ drops sharply from $a_{*,\rm ini}$ to $a_{\ast c}^{|211\rangle}$ as the superradiant mode extracts angular momentum, well before Hawking evaporation sets in. The associated GW emission (GW SR, green) is one to two orders of magnitude below the cloud energy, since only a fraction of the cloud radiates gravitationally. The accelerated spin-down suppresses the Hawking graviton channel, and the total $\Delta N_{\rm eff}$ is reduced relative to the left panel despite the new superradiant cloud GW contribution.
• Figure 2: Same format as fig. \ref{['fig:benchmark']} but for $M_{\rm ini} = 10^8\,{\rm g}$, $\beta = 6.40 \times 10^{-12}$, $\alpha_{\rm ini} = 0.10$ ($\mu = 2.7 \times 10^{5}\,{\rm GeV}$), with $a_{*,\rm ini} = 0.999$ and superradiance active in both panels. Left: Scalar boson ($s = 0$). The two-step staircase in $a_\ast$ (middle subpanel) directly traces the sequential mode dynamics: the $|211\rangle$ mode grows first (first bump in BSM SR near $a \sim 10^7$), spinning down $a_\ast$ to $a_{\ast c}^{|211\rangle}$, and the $|322\rangle$ mode subsequently takes over (second bump near $a \sim 10^{11}$), reducing $a_\ast$ further to $a_{\ast c}^{|322\rangle}$. Each mode produces a corresponding feature in GW SR (green), but the combined $\Delta N_{\rm eff}^{\rm SR} = 5.0 \times 10^{-4}$ is far below the Hawking channel. Right: Vector boson ($s = 1$). The dominant $|1011\rangle$ mode initiates superradiance much earlier ($a \sim 10^3$) due to its faster growth rate, as reflected by the earlier spin-down in the middle subpanel. The GWs emitted at this early epoch undergo far greater redshift before freeze-out, reducing the SR contribution to $\Delta N_{\rm eff}^{\rm SR} = 1.3 \times 10^{-6}$.
• Figure 3: $\Delta N_{\rm eff}$ as a function of $M_{\rm ini}$ for fixed gravitational coupling $\alpha_{\rm ini} = 0.1$ (left) and $\alpha_{\rm ini} = 0.3$ (right), with $\beta > \beta_c$ so that a PBH-dominated epoch precedes evaporation. Blue curves show the Hawking-radiation-only baselines; orange curves include superradiance. Solid and dashed lines correspond to the two initial spins indicated in each legend. In each panel, the lower spin coincides with the critical spin $a_{\ast c}$ of the $|211\rangle$ mode [eq. \ref{['eq:a_crit']}]: $a_{*,\rm ini} = 0.384$ for $\alpha = 0.1$ (left) and $a_{*,\rm ini} = 0.882$ for $\alpha = 0.3$ (right), at which the superradiant growth rate vanishes. The gray line marks the projected $2\sigma$ sensitivity of CMB-HD ($\Delta N_{\rm eff} \approx 0.027$).
• Figure 4: Iso-$\Delta N_{\rm eff}$ contours in the $(M_{\rm ini},\,a_{*,\rm ini})$ plane for $\alpha_{\rm ini} = 0.1$ (left) and $\alpha_{\rm ini} = 0.3$ (right). Blue curves show the Hawking-radiation-only contours at $\Delta N_{\rm eff} = 0.005$ (dotted), $0.01$ (dashed), and $0.027 \approx$ CMB-HD $2\sigma$ sensitivity (solid). Red curves show the corresponding contours with scalar ($s=0$) superradiance included, at $\Delta N_{\rm eff} = 0.005$ (dotted) and $0.01$ (dashed). The green dashed curve shows $\Delta N_{\rm eff} = 0.005$ for vector ($s=1$) superradiance. The region above each contour corresponds to larger $\Delta N_{\rm eff}$. No CMB-HD contour exists for either superradiant scenario.