Mechanism of quasistabilization of primordial black holes

It is argued that primordial black holes with initial masses satisfying $M<{10}^{15}\mathrm{g}$, instead of having explode, might currently be in a quasistable phase contributing to a tiny fraction of the measured dark matter. This statement is based on a computation of black hole evaporation in which energy conservation is taken into account that shows that the backreaction to Hawking radiation favors the quasistabilization of the black hole. The result is specifically shown for general spherically symmetric quantum black holes described by an effective metric independently of the specific framework from which it is derived. The quintessential primordial black hole is fully analyzed as an example.

Figure 1

Given a fixed value for $\sigma$ a schematic representation of the different possibilities in the plane $\{\varphi ,\omega \}$ are shown. Specifically, the darkest region correspond to the values of the exponents that allow black hole explosions in the thermal approach. The grey region close to it is the region where the thermal approach predicts total evaporation without explosion. On the other hand, the striped region (partially overlapping the previous regions) represents the region where the energy conservation approach predicts the absence of total evaporation.

Figure 2

The luminosity of a black hole as a function of its mass (taking into account backscattering) in Planck’s units. Energy conservation has been imposed in order to draw the luminosity $L_{\mathrm{EC}}$ (solid line), while the luminosity $L_{\mathrm{Stand}}$ (dashed line) neglects energy conservation. As can be seen, by considering energy conservation the luminosity reaches a maximum for a non-null mass. If energy conservation is neglected, the luminosity diverges as the black hole approaches its total evaporation.

Figure 3

To the left, we compare the evolution of a black hole mass without taking into account energy conservation $M_{\mathrm{Stand}}(u)$ (dashed line) and taking it into account $M_{\mathrm{EC}}(u)$ (solid line) in both cases starting with the same initial mass of 2 Planck masses at $u=0$. The formation of a slowly evaporating remnant, if energy conservation is taken into account, contrasts with the sudden total evaporation in the “thermal”approximation. To the right, we have depicted the evaporating remnant in the broader range $0\le u\le {10}^6$.

Figure 4

A comparison of the effective spectral distributions of the radiation as seen from infinity (taking into account backscattering). Specifically, we have plotted it for three different BH masses as a function of the energy $E$ in Planck’s units. Energy conservation has been imposed in order to draw the solid lines ($⟨n(E){⟩}_{{\mathrm{Eff}}_{\mathrm{EC}}}$), while the dashed lines have been drawn neglecting energy conservation ($⟨n(E){⟩}_{{\mathrm{Eff}}_{\mathrm{Stand}}}$). For masses bigger that the Planck mass the curves are nearly identical (see, for instance, the first diagram with a mass corresponding to 10 Planck masses). The differences are only considerable if the mass of the black hole is around or below the Planck mass (second and third diagrams, respectively).