Hawking Evaporation and Light Primordial Black Holes

A sufficiently light primordial black hole loses mass by Hawking emission on a cosmological timescale. Its temperature rises as its mass falls, new particle species become available, and the emitted primaries decay or hadronize into observable secondaries. The robust analytic result is the scaling $T_{\rm H}\propto M^{-1}$ and $\tau\propto M^3$ for a neutral, nonrotating hole; precise energy injection requires greybody factors, particle content, and population evolution.

Helpful background. Einstein Equations supplies the Schwarzschild geometry; Relativistic Matter and Compact Objects supplies energy injection into cosmic media; Black-Hole Dynamics and Collapse supplies horizon evolution; Relativistic Cosmology supplies expansion and thermal epochs; and Probability and Statistical Inference supplies population averaging and constraint interpretation.

Hawking temperature and the mass scale

For a four-dimensional Schwarzschild black hole of physical mass $M$, normalized to the static time at infinity, the surface gravity is

$$\kappa=\frac{1}{4GM}.$$

With $c=\hbar=k_{\mathrm B}=1$, the Hawking temperature is

$$T_{\rm H}=\frac{\kappa}{2\pi}=\frac{1}{8\pi GM}.$$

Restoring constants in one representative observable gives

$$k_{\mathrm B}T_{\rm H} =\frac{\hbar c^3}{8\pi GM}.$$

The inverse-mass dependence is the key cosmological ordering: lighter PBHs are hotter and reach particle rest-mass thresholds earlier in their evolution. This calculation assumes a hole much larger than the Planck scale, a slowly changing mass, negligible ambient absorption, and an approximately isolated asymptotically flat near zone embedded in a much larger cosmological spacetime.

Greybody-weighted particle emission

The Hawking occupation factor is thermal, while the spectrum reaching the asymptotic near-zone observer is filtered by the curvature potential. For a neutral, nonrotating hole, the number spectrum of species $i$ with spin $s_i$ can be written

$$\frac{d^2N_i}{dt\,dE} =\frac{g_i}{2\pi} \sum_{\ell,m} \frac{\Gamma_{i\ell m}(E,M)} {\exp(E/T_{\rm H})-(-1)^{2s_i}},$$

where $g_i$ counts internal states and $\Gamma_{i\ell m}$ is the dimensionless greybody transmission probability. Massive emission is kinematically restricted to $E\ge m_i$ and thermally suppressed until $T_{\rm H}$ approaches that scale. Rotating or charged holes add angular-velocity or electric-potential terms to the exponent and require simultaneous evolution of spin or charge.

The time and energy in this expression use the same asymptotic normalization as $T_{\rm H}$. For spherical symmetry, a sum over $m$ may be replaced by its angular degeneracy, but that factor must then be removed from any separately tabulated greybody function. Likewise $g_i$ must not count polarizations or other internal states already included in the tabulation.

Primary quarks and gauge bosons do not arrive at a detector unchanged. Their fragmentation and decay produce secondary photons, electrons, neutrinos, hadrons, and nuclei. Auffinger 2023, § 3.1–§ 3.4 gives the emission formula, greybody treatment, mass/spin evolution, and numerical recipes. A pure blackbody curve therefore cannot be used as a precision photon or neutrino prediction.

Mass loss and the cubic lifetime

Integrating $E$ times the spectrum over energy gives the luminosity. Package all accessible species and greybody factors into a positive dimensionless function $\alpha(M)$:

$$\frac{dM}{dt}=-\frac{\alpha(M)}{G^2M^2}.$$

If $\alpha$ is approximately constant over the mass interval of interest, direct integration yields

$$M(t)^3=M_i^3-\frac{3\alpha}{G^2}(t-t_i), \qquad \tau\simeq\frac{G^2M_i^3}{3\alpha}.$$

Restoring dimensions, the lifetime scale is $\tau\propto G^2M_i^3/(\hbar c^4)$. The $M^3$ scaling is reliable; the coefficient changes when particle thresholds open, greybody factors change, spin evolves, or physics beyond the assumed particle spectrum is introduced. The adiabatic approximation also fails near the Planck regime, where this semiclassical calculation does not decide between complete disappearance, a remnant, or another endpoint.

A useful limiting check comes from the Stefan–Boltzmann estimate. Horizon area scales as $A\propto G^2M^2$ and $T_{\rm H}^4\propto G^{-4}M^{-4}$, so $P\sim AT^4\propto G^{-2}M^{-2}$. It reproduces the mass-loss and lifetime powers but not the greybody-weighted coefficient.

When a particle threshold changes $\alpha(M)$, the controlled replacement for the constant-coefficient lifetime is

$$t(M)-t_i =G^2\int_M^{M_i}\frac{\widetilde M^2}{\alpha(\widetilde M)}\,d\widetilde M.$$

The lower limit must remain within the semiclassical domain. A newly accessible species increases the luminosity and shortens the remaining lifetime; it does not change the already elapsed history retroactively.

Cosmological energy injection

For a physical mass distribution $dn_{\rm PBH}/dM$, the injected primary power per physical volume is

$$\dot Q(t)= \int dM\,\frac{dn_{\rm PBH}}{dM}(M,t) \sum_i\int_{m_i}^{\infty}dE\;E\, \frac{d^2N_i}{dt\,dE}.$$

Cosmological predictions must evolve the mass distribution because evaporation shifts every initial mass to a smaller current mass. They must then propagate secondaries through the relevant plasma. Energy deposited before or during nucleosynthesis, around recombination, or in the later intergalactic medium has different observables and transfer efficiencies. Auffinger 2023, §§ 6–11 reviews the BBN, CMB, background-radiation, cosmic-ray, and extended-mass-function applications, including their particle-physics dependence.

For example, let $n_M=dn_{\rm PBH}/dM$ be the physical number density per unit mass. Before an endpoint is reached, with deterministic evaporation and no new formation or mergers, number conservation gives

$$\partial_t n_M+3Hn_M+\partial_M(\dot M\,n_M)=0.$$

The first transport term dilutes the number density through expansion; the second moves objects between mass bins. Solving along the mass-loss characteristics gives $n_M(M,t)=(a_i/a)^3n_{M,i}(M_i)\lvert dM_i/dM\rvert$. An absorbing low-mass boundary removes objects that reach the prescribed endpoint. Injection is consequently different from deposition: the latter also requires propagation and interaction probabilities for the emitted particles.

Check the scaling before computing a spectrum

Two neutral Schwarzschild PBHs have initial masses $M$ and $2M$, and assume the same constant $\alpha$. Compare their initial temperatures, luminosities, and lifetimes.

Solution

Since $T_{\rm H}\propto M^{-1}$, the heavier hole begins at half the temperature. Since $P=-dM/dt\propto M^{-2}$, its initial luminosity is one quarter as large. Since $\tau\propto M^3$, it lives eight times longer. Thresholds in $\alpha(M)$ modify these ratios quantitatively, but any numerical implementation should recover them when $\alpha$ is held fixed.

Scope and related topics

This page uses the semiclassical emission law to determine PBH lifetimes and cosmological injection. The quantum-state derivation belongs to Hawking Radiation from Gravitational Collapse, detailed transmission physics to Greybody Factors and Asymptotic Flux, and adiabatic backreaction limits to Luminosity and Mass-Loss Laws. The unknown final endpoint and model-dependent observational exclusions are not derived here.

References

• Auffinger, Jérémy. 2023. “Primordial Black Hole Constraints with Hawking Radiation—A Review.” Progress in Particle and Nuclear Physics 131: 104040. DOI.