The 1979 paper by Toussaint, Treiman, Wilczek, and Zee (Phys. Rev. D 19, 1036) established a parametric framework in which the cosmic matter-antimatter asymmetry, quantified by the baryon-to-photon ratio nB/nγn_B/n_\gamma, could arise dynamically from nonequilibrium gravitational particle production — in particular Hawking radiation from primordial black holes — provided microscopic interactions violate baryon number together with C and T. That paper proved that no net baryon-number flux is possible if baryon number is exactly conserved, and explicitly deferred a quantitative program: computing the CP-violating asymmetry parameter ϵ\epsilon in concrete particle-physics models, detailed kinetic calculations of abundances through decoupling and thermalization, the production and initial abundance of heavy particles at M≳1016M\gtrsim10^{16} GeV, the equilibrium-versus-nonequilibrium question for specified microphysics, and the primordial black hole abundance and mass spectrum as an undetermined cosmological input. The methodological components of that program have since been largely absorbed by the subsequent baryogenesis literature and, for the black-hole channel, by an active modern line that has now built the fully coupled kinetic treatment; what remains open is the undetermined early-universe primordial black hole population and whether black-hole-induced baryogenesis can account for the observed asymmetry nB/nγ≈6×10−10n_B/n_\gamma\approx6\times10^{-10} within observationally allowed scenarios.

Background

The universe contains matter (baryons) with essentially no antimatter, and no plausible astrophysical process can separate the two on cosmological scales. The asymmetry is quantified by the baryon-to-photon ratio nB/nγn_B/n_\gamma (the excess of baryons over antibaryons per cosmic background photon), observed to be ≈6×10−10\approx6\times10^{-10} and quoted as ∼10−9\sim10^{-9} at the time of the source paper. A dynamical origin for this number requires three ingredients that are now standard: microscopic violation of baryon number BB (the approximate conserved charge carried by quarks), violation of the discrete symmetries C (particle-antiparticle conjugation) and CP or T (time reversal), and a departure from thermal equilibrium — because in thermal equilibrium the Boltzmann distribution and the CPT theorem (which guarantees equal masses for particles and antiparticles) force equal baryon and antibaryon abundances. Grand unified theories (GUTs), which unify the strong and electroweak interactions at energies of order 101510^{15}-101610^{16} GeV, generically contain the first two ingredients through baryon-number-violating processes of the kind that would mediate proton decay; 'superunified' theories would tie these interactions to gravity themselves.

Hawking radiation — the approximately thermal quantum radiation emitted by black holes, with temperature inversely proportional to the black hole mass — provides a naturally nonequilibrium particle-production process. Primordial black holes (PBHs) are black holes hypothesized to form in the early universe from the collapse of rare overdense regions, with possible masses far below stellar masses; sufficiently light ones evaporate within the age of the universe. The emission is modified by greybody factors, the energy- and spin-dependent transmission coefficients of the curved spacetime outside the horizon, which become central when the emitted particles themselves carry the quantum numbers being tracked.

The source paper (Toussaint, Treiman, Wilczek, and Zee, 1979) proved two structural results within this setting. First, if baryon number is microscopically conserved, the net baryon-number flux to infinity in Hawking radiation is exactly zero — although the radiation can be asymmetric species by species (for example, more Λ\Lambda than Λˉ\bar\Lambda baryons), in which case a black hole preferentially absorbs the complementary species and drives a surrounding baryon-symmetric medium into an asymmetric state; a thermodynamic (Maxwell-demon) argument supports the same conclusion. Second, a net baryon-number flux becomes possible if and only if the microscopic interactions violate baryon number together with C and T — through asymmetric gravitational transmission of radiated particles, or through the asymmetric decay of long-lived emitted particles — and the same conclusions carry over to particle production by the time-dependent gravitational field of the expanding universe. The paper also corrected the first grand-unified baryogenesis proposal (Yoshimura, 1978) by showing that collisions of effectively massless particles in equilibrium generate no asymmetry: nonequilibrium is essential. It then compared the two nonequilibrium engines. Heavy particles fall out of chemical equilibrium once their interaction rates drop below the Hubble expansion rate; their subsequent CP-asymmetric decays, with fractional asymmetry ϵ\epsilon per decay, generate nB/nγn_B/n_\gamma of order ϵ\epsilon times a nonequilibrium dilution factor. Black-hole radiation is a competing engine that is insensitive to the ambient temperature but can only emit particles lighter than the black hole's Hawking temperature, so very heavy particles appear only in the final evaporation stages.

The paper explicitly left the quantitative program open: no concrete microphysical model was specified, so ϵ\epsilon was not computed; the kinetic treatment was crude and parametric (the authors announced a fluctuation-dissipation-based calculation for later work); the PBH abundance and mass spectrum were unknown cosmological inputs, with a speculative possibility of a black-hole-dominated phase or of a 'bounce' cosmology in which the expanding universe emerged from a previous collapsing phase; the initial abundance of very heavy particles (M≳1016M\gtrsim10^{16} GeV) was problematic to estimate because such species never reach thermal equilibrium; and whether baryon-violating interactions were in or out of chemical equilibrium in specified models required dedicated rate calculations. Later developments added a washout effect absent from the 1979 framework: electroweak sphalerons — thermal transitions between topologically distinct gauge-field configurations in the Standard Model — violate the combination B+LB+L of baryon number and lepton number LL (the corresponding charge carried by leptons such as electrons and neutrinos) at high temperature and can erase a generated asymmetry unless it is stored in the orthogonal combination B−LB-L or produced late. The modern literature also developed leptogenesis (baryogenesis via heavy-neutrino decays), which supplies, in a different channel, exactly the quantitative machinery — Boltzmann kinetic networks, ϵ\epsilon computations, washout analysis — that the source paper's program called for.

Problem Statement

In the parametric framework of Toussaint, Treiman, Wilczek, and Zee (1979), gravitationally driven nonequilibrium particle production — in particular Hawking radiation from primordial black holes, supplemented by microscopic baryon-number-, C-, and T-violating interactions — can generate a net baryon asymmetry from an initially symmetric universe. The problem is to convert that framework into a quantitative theory of the observed baryon-to-photon ratio. The microphysical and kinetic components of that program — the asymmetry parameter ϵ\epsilon, the coupled abundance kinetics, and the equilibrium-versus-nonequilibrium rates — are established in the mature treatments in the literature; what remains open is whether there is an early-universe primordial black hole abundance and mass spectrum, within the observationally allowed range (Carr et al., Ref. 12), for which black-hole-induced baryogenesis yields the observed nB/nγ≈6×10−10n_B/n_\gamma\approx6\times10^{-10}. The source paper left that program open on several fronts: no concrete microphysical model was specified, so the asymmetry parameter ϵ\epsilon was not computed for the relevant decay and emission channels; the kinetic treatment was parametric, short of the coupled evolution of an evaporating PBH population with all particle abundances, including greybody factors, entropy dilution from evaporation, decoupling, thermalization, and washout; the cosmological inputs — the early-universe PBH abundance and mass spectrum (and whether a PBH-dominated or bounce phase occurred) and the production and initial abundance of heavy particles with M≳1016M\gtrsim10^{16} GeV — were undetermined; and whether the baryon-violating interactions of a specified model are in or out of chemical equilibrium in the relevant epochs required dedicated rate calculations. The microphysical and kinetic machinery now exists in mature form, including fully coupled numerical treatments of an evaporating PBH population that, for assumed PBH mass spectra, reproduce the observed asymmetry within viable parameter regions; the open frontier is the cosmological input — the early-universe PBH abundance and mass spectrum remain undetermined, and the existing treatments reproduce the asymmetry only for mass spectra posited as input rather than determined from observation.

References

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