Primordial Black Holes with QCD Color Charge

We describe a realistic mechanism whereby black holes with significant QCD color charge could have formed during the early universe. Primordial black holes (PBHs) could make up a significant fraction of the dark matter if they formed well before the QCD confinement transition. Such PBHs would form by absorbing unconfined quarks and gluons, and hence could acquire a net color charge. We estimate the number of PBHs per Hubble volume with near-extremal color charge for various scenarios, and discuss possible phenomenological implications.

The PBH mass at the peak of the mass distribution, M (t c ), is proportional to the mass enclosed within a Hubble volume M H (t c ) at the time of the PBHs' formation, t c [13][14][15][16].This establishes a relationship between the typical PBH mass and the time of collapse.A combination of theoretical and observational bounds leaves a window 10 17 g ≤ M (t c ) ≤ 10 22 g within which PBHs could constitute all of dark matter today [5][6][7][8][9], which in turn constrains the time of collapse to 10 −21 s ≤ t c ≤ 10 −16 s.
At these very early times, the plasma filling the universe had a temperature 10 5 GeV ≤ T (t c ) ≤ 10 7 GeV, exponentially greater than the QCD confinement scale Λ QCD = 0.17 GeV.At such high temperatures, the quarks and gluons in the plasma were unconfined [17][18][19].Therefore PBHs of relevance to DM necessarily formed by absorbing large collections of quarks and gluons from the quark-gluon plasma (QGP), which were not confined within color-neutral hadronic states.
Non-Abelian dynamics among the unconfined quarks and gluons yield a nontrivial distribution of QCD color charge within the QGP [20][21][22].In particular, collective modes of soft gluons, with momenta k soft ∼ g s T , where g s is the dimensionless gauge coupling strength, can produce spatial regions of nonvanishing net color charge, whose typical size is set by the Debye screening length λ D (T ) ∼ 1/(g s T ) [23,24].
The mechanism by which PBHs form, known as "critical collapse," indicates that some PBHs would form with arbitrarily small masses, M ≪ M [25][26][27][28][29][30][31].The longlived PBHs with mass M would constitute DM today, whereas those with M ≪ M would have already evaporated.Nonetheless, these small PBHs would have formed from collapse regions of size ∼ λ D , on scales for which the QGP can have a nontrivial color-charge distribution.Such PBHs would form with net QCD color charge; a subset of these, moreover, would form with extremal enclosed charge, Q = √ G M .On the other hand, DM candidates (with mass ∼ M ) would be color neutral.
Although vacuum solutions of color-charged black holes have been known in the literature for some time [32,33], such simple scenarios have ignored how these objects could have formed.In contrast, we focus on realistic, well-motivated mechanisms by which a population of charged black holes-including near-extremal onescould have formed amid a nontrivial medium in our actual universe.
The abundance of near-extremal PBHs depends sensitively upon the ratio M/ M , which scales with the temperature of the plasma at the time of PBH formation, and hence falls over time as the plasma cools.Although rare and short-lived on cosmological timescales, these nearextremal charged PBHs would constitute an entirely new state of matter, with enclosed QCD charge O(10 13 g s ), unlike the multi-parton states following relativistic heavy ion collisions, which briefly involve O(10 2 ) unconfined charges [34].Moreover, as discussed below, these unusual PBHs could have phenomenological implications [35].
PBHs and Critical Collapse.We restrict attention to PBHs that form from collapse of primordial overdensities amplified during inflation, which remains the most thoroughly analyzed and empirically constrained scenario [4-9, 11, 12].One of the most striking results from decades of studies by the numerical-relativity community is that the mass of the resulting black hole depends upon a oneparameter family of initial data and a single universal critical exponent, in close analogy to phase transitions in statistical physics [25][26][27][28][29][30].In particular, the black hole mass M at the time of collapse t c obeys the relation where M H (t c ) is the mass contained within a Hubble volume at t c , C(r) = 2GδM (r)/r is the compaction as a function of areal radius r [36,37], C is the compaction averaged over a Hubble radius, C c ≃ 0.4 is the threshold arXiv:2310.16877v2[hep-ph] 9 May 2024 for black hole formation [38], and κ is an O(1) dimensionless constant whose value depends on the spatial profile of C(r) and the averaging procedure [9,30,[39][40][41][42][43][44][45].The universal scaling exponent ν depends on the equation of state of the fluid that undergoes collapse; numerical studies show ν = 0.36 for a radiation fluid [26,27,30,46].Early numerical studies only considered spherically symmetric initial conditions, though later studies identified critical-collapse behavior even when relaxing spherical symmetry [29].
Researchers have distinguished "Type I" versus "Type II" forms of critical collapse.In Type I cases, a mass gap appears, setting a smallest (nonzero) value of M , above which M scales as in Eq. (1).In Type II cases, the system remains scale-free and self-similar, with no mass gap, and self-consistent black hole solutions exist even for M → 0 [28,29].Simulations have shown that several cases of cosmological interest are Type II, including collapse within a perfect fluid in the ultrarelativistic regime, as well as in a pure SU(N ) gauge field [30,[47][48][49].
Critical collapse yields a mass distribution with γ ≃ 0.2.Given the probability distribution function P ( C) for C, as we will see below, the mass distribution ψ(M, t c ) features a power-law tail for masses M ≪ M [13][14][15][16].Thus, for Type II collapse, some PBHs will form with arbitrarily small M [50].The radii of the PBHs of interest are so small that Bondi accretion remains negligible over relevant time-scales [51][52][53] (see Supplemental Materials [54]).
Another consequence of the scale-free, self-similar behavior of Type II collapse is that for initial data near the critical value, C ≃ C c , any dimensionful quantity related to the resulting black hole must scale with the same universal exponent ν as the mass [28,29].For such solutions, the initial stages of collapse begin with a region of radius r ∼ 1/H, but as time evolves, there is a net outflow of matter.The critical scaling ensures that the mass enclosed in the collapse region r(t) scales as M enc (r(t))/r(t) = constant, as observed in numerical simulations [14,26].(Note that this is different from estimates outside the context of critical collapse, for which one would expect M enc ∼ r 3 .)This is why black holes that result from an initial compaction C ≃ C c have masses M ≪ M ∼ M H .At the time of collapse t c , black hole formation is triggered and the mass outflow ceases; we call the final collapse radius r c ≡ r(t c ). Physically, all the matter contained within the volume of radius r c becomes contained within the black hole, M enc (r c ) = M .
Given the bound 10 17 g ≤ M (t c ) ≤ 10 22 g within which PBHs could constitute all of dark matter [5][6][7][8][9], and the relationship in Eq. ( 2), the collapse times are constrained to 10 −21 s ≤ t c ≤ 10 −16 s.These times are exponentially earlier than the QCD confinement transition, which occurred at t QCD ≃ 10 −5 s, when the temper-ature of the plasma T became comparable to the QCD scale Λ QCD = 0.17 GeV.As noted above, if PBHs constitute a significant fraction of all DM, they must have formed amid a hot QGP.
QGP and Debye Screening.At temperatures T ≫ Λ QCD , the fluid filling the universe was a hot plasma dominated by unconfined quarks and gluons.Within this hot QGP, color-charged particles such as soft gluons with momenta k soft ∼ g s T undergo (non-Abelian) Debye screening, with screening length λ D (T ) ∼ 1/(g s T ).(For reviews, see Refs.[19-22, 55, 56].)This holds even within an expanding Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime, given the large hierarchy of scales λ D ≪ H −1 .(For example, at t c = 10 −21 s, when the plasma temperature was T ≃ 10 7 GeV, one has λ D ∼ 10 −23 m and H −1 ∼ 10 −12 m [53].)This implies that at some distance r from a charged particle, the effective charge falls as exp[−r/λ D ].
To quantify these effects, we construct an effective field theory for soft gluon modes, whose occupation numbers dominate the plasma.We decompose the full Yang-Mills gauge field Āc µ into hard (a c µ ) and soft (A c µ ) modes, with typical momenta k hard ∼ T and k soft ∼ g s T , respectively, where c = 1, . . ., 8 is the color index.The fields A c µ have large occupation numbers per mode, so they behave as effectively classical fields.The dynamics of these soft modes are governed on length-scales λ ≥ 1/k soft by an effective action [53] which results from integrating out the hard modes.We follow the background-field method, which allows us to choose the gauge symmetry to act only on a c µ .As described in detail in Ref. [20] (see also Ref. [53]), one may then choose a gauge (for example, by including a gaugefixing term such as ∇ i a i a − g s f abc A b i a i c in the full theory) such that no additional gauge-fixing terms or ghosts are required in S eff .Here is the field strength for soft modes, with ∇ µ the spacetime covariant derivative and f abc the SU(3) structure constants; j a µ (x) is the current induced by (non-Abelian) self-interactions with the hard modes (including highmomentum quarks); and L fluid represents contributions to the spacetime evolution from constituents other than the soft modes.
The induced current j µ a (x) depends on the deviation from an equilibrium distribution function for the highenergy charged particles [20,21,53].For spacetimes for which one can set g 0i = 0, and in the approximately in terms of the Debye mass , where N c = 3 is the number of colors, N f is the number of effectively massless quarks, and T is the Tol-man temperature [53].The Debye length is given by λ D ≡ 1/m D .Comparing Eqs. ( 3) and ( 4) reveals that the chromoelectric component A a 0 acquires an effective mass, while the chromomagnetic components A a i remain massless.The equations of motion that follow from In an FLRW background, a closed-form analytic solution for the components A a µ (x) for a point charge may be found [53].We may further identify a quasi-local chromoelectric charge based on the form of the gauge-invariant quantity , where E a i = F a 0i is the chromoelectric field.This yields [53] where the charge at the origin, , is an integer multiple of the unit charge g s , and λ D = λ D (T ).
Given Debye screening, the QGP will be color neutral on long length scales r ≫ λ D , but can have a nontrivial distribution of color charge across shorter length scales r ∼ λ D .In particular, there can exist regions with net color charge, whose spatial extent is set by λ D (T ) [23,24].We may estimate the color charge inside one such region in terms of the number of soft gluons enclosed, with each contributing a unit charge g s in the same direction in color space.These soft gluons will represent a fraction F of all the particles contained within a particular net-color region (including hard gluons and other particle species), given by where α s ≡ g 2 s /(4π) and g * is the number of effectively massless degrees of freedom; at the energy scales of interest, T > m t = 173 GeV, g * = 106.75 for the Standard Model.Eq. ( 6) follows from evaluating the number density n cc soft of soft gluons within a single spatial region of size λ D by truncating the momentum integral over the distribution function at k soft .
The total color charge in a spatial volume depends on the number of distinct net-color regions enclosed.As a conservative estimate, we assume that only one such region located at the center contributes to Q a 0 inside a spherical volume of plasma of radius r c ≥ λ D , while the rest of the plasma in the volume screens Q a 0 .Moreover, we approximate Q a 0 as a point charge, which again underestimates the net charge by overestimating the effect of screening.We may exploit the gauge symmetry to assign a specific color (e.g., a = 1) to the soft gluons within the central net-charge region; within that central region, we approximate the number density for soft gluons of all other charges (a = 2, . . ., 8) to vanish.Then the net color charge Q(r c , T ) in a volume of plasma of radius r c may be approximated as where N cc is the total number of particles in the central region of net color charge.
PBHs with Significant QCD Charge.In order to quantify the net enclosed charge Q(r c , T c ), we estimate N cc for a PBH that forms with mass M at temperature T c as using the fact that in a radiation bath in thermal equilibrium the average energy per particle is ∼ T c , and the number of distinct net-color regions absorbed by the PBH scales as (r c /λ D ) 3 [57].From Eqs. ( 6)-( 8), the net QCD charge contained within a PBH of mass M that forms at time t c is then where we have defined the dimensionless ratio R c ≡ r c /λ D .The net enclosed charge grows inversely with T c , since the average energy per particle falls with T c , so a larger number of charge-carrying particles must be absorbed to form a PBH of mass M at lower temperatures.
On the other hand, as expected, the net charge falls rapidly for R c ≫ 1, given screening within the medium prior to PBH collapse.Based on exact black hole solutions of the Einstein-Yang-Mills equations in vacuum, which are analogous to Reissner-Nordström black holes with electromagnetic charge, an extremal color-charged PBH satisfies Q extr = √ G M extr [32,33].For simplicity, we adopt this value for the maximal charge Q extr even in the presence of plasma external to the PBH.Using Eq. ( 9), we then find that an extremal PBH forms with (dimensionless) collapse radius R extr ≡ r extr c /λ D given by The radius R extr is uniquely determined by the temperature of the plasma at the time of collapse, T c .As noted above, the collapse radius r c and the mass M of a resulting PBH scale similarly, [14,26].Recent numerical simulations have found that the scaling of Eq. ( 1) remains accurate (deviating by less than 15%) even near the peak of the mass distribution, for initial conditions far from the critical value [43].We may therefore compare r c and M for small PBHs with those at the peak of the mass distribution, rc and M .Given Eq. ( 2), we find rc ≃ 1/(2H(t c )), consistent with Carr's original analytic estimate [13].For M < M , the critical scaling r c ∼ GM then yields upon making use of λ D ≃ 1/(g s T c ) and H 2 (t c ) = (π 2 g * /90) T 4 c /M 2 pl to relate the Hubble parameter to the fluid temperature at the time of collapse.PBHs that form with R c ∼ O(1) will be closer to the peak-mass M at higher temperatures (and hence at earlier times) than those that form at lower temperatures.
We may combine Eqs. ( 10) and ( 11) to estimate the abundance of near-extremal PBHs as a function of the collapse time t c .As described in the Supplemental Materials [54], the mass distribution then becomes where m ≡ M/[κM H (t c )], and P ( C) is the probability distribution function for the (spatially averaged) compaction C. As noted above, PBHs with nearextremal charge will form with M ≪ M , or m ≪ 1.
In that limit, we may integrate dn PBH (M, t c )/dM = ρ PBH ψ(M, t c )/M to obtain the number of PBHs per Hubble volume at time t c with masses up to some value M ≪ M .This yields As discussed in the Supplemental Material [54], non-Gaussian features modify P ( C) near the peak M but remain negligible in the limit m ≪ 1.In that limit, P ( C) is well approximated by the usual Gaussian Press-Schechter form, with variance σ 2 in the range of interest 10 −2 ≤ σ 2 ≤ 10 −1 [54].By combining Eqs.(10) and (11), we may evaluate Eq. ( 13) for the case of extremal PBHs, for M → M extr .
At later times t > t c , the total number of PBHs in a Hubble sphere grows as [a(t)/a(t c )] 3 , due to the expansion of the universe.Since a(t) ∝ t 1/2 during radiationdominated expansion, the total number of PBHs within a Hubble sphere at time t is N PBH (up to M , t) = N PBH (up to M , t c )(t/t c ) 3/2 .As shown in Fig. 1, we expect N extr (t QCD ) ≃ 1 extremal PBHs per Hubble volume at the time of the QCD confinement transition, t QCD = 10 −5 s, for a population of PBHs that forms amid a plasma of temperature T c ≃ 5 × 10 7 GeV.
The plasma temperature T c ≃ 5 × 10 7 GeV is remarkably close to the window within which PBHs could account for all of dark matter today, 10 5 GeV ≤ T c ≤ 10 7 GeV.In particular, M (T c = 5 × 10 7 GeV) = 7 × 10 15 g, for which the expected lifetime due to Hawking evaporation t evap ≃ 10 22 s exponentially exceeds the age of our observable universe.Meanwhile, an extremal black hole that forms at T c = 5 × 10 7 GeV has a mass M extr = 2 × 10 7 g and charge Q = M extr /M pl ∼ O(10 13 g s ).
The evaporation lifetime of an uncharged (Schwarzschild) black hole of the same mass is t evap ≃ 10 −4 s > t QCD .Given that the Hawking temperature T H → 0 as a black hole approaches extremality, we expect t evap ≫ t QCD for near-extremal PBHs that form amid a plasma at T c = 5 × 10 7 GeV.
In general, a PBH should form with significant net color charge if r c ∼ λ D ∼ 1/(g s T ).Because of the special scaling features of self-similar critical-collapse solutions, we further expect M extr ∼ r c ; whereas the peak-mass M ∼ 1/H(t c ) ∼ 1/T 2 c .Hence the ratio M extr / M ∼ T c falls over time, as the plasma cools.The number of extremal PBHs scales as (M extr / M ) 1/ν , as per Eq. ( 13).For T c ≪ M pl , this is a small number raised to a large power, confirming the pattern shown in Fig. 1.
Finally we note that our effective field theory for soft gluons remains self-consistent for λ D (T ) ≤ r extr c (T ) ≪ H −1 (T ).For a hot plasma filled with Standard Model particles, λ D (T ) ≃ 10 −31 m (10 15 GeV/T ) and H −1 (T ) ≃ 10 −29 m (10 15 GeV/T ) 2 , indicating that λ D ≪ H −1 for all T ≤ 10 15 GeV, i.e., below the limit on the postinflation reheating temperature set by the present bound on the primordial tensor-to-scalar ratio [58].In addition, Eq. ( 10) confirms that λ D ≤ r extr c for all T ≤ 10 16 GeV.Discussion.Primordial black holes remain a tantalizing candidate with which to address the long-standing mystery of dark matter.Present-day constraints on such a scenario require that the relevant population of PBHs be produced at very early times, exponentially earlier than the time of the QCD confinement transition at t QCD = 10 −5 s.Combining the well-studied phenomenon of critical collapse-which can produce PBHs of arbitrarily small masses-with the nontrivial charge distribution of soft gluons within a hot quark-gluon plasma suggests that some PBHs with net QCD color charge are likely to form at times t c < t QCD .The total number of such charged PBHs per Hubble volume depends sensitively on details like their time of formation.
The evolution of a near-extremal PBH after its formation comprises two periods: net absorption (of mass and compensating charges) starting at t c , followed by Hawking emission.The change occurs at a time t equal when the temperature of the surrounding plasma matches the Hawking temperature of the PBH, as the former decreases with time and the latter increases due to the absorption of compensating charges.(The increase in mass accumulated during ∆t = t equal − t c remains an exponentially small fraction of the original PBH mass, and therefore does not lower the Hawking temperature appreciably [51][52][53][54].) Neither process is likely to completely discharge a near-extremal PBH.First, the PBH's charge is strongly screened in the medium [53], reducing the probability of the PBH preferentially absorbing compensating charges.Second, local temperature gradients in the plasma near the PBH [53] reduce the number of particles absorbed for a given accreted mass ∆M .(See the Supplemental Materials [54] for details on the exponentially inefficient accretion for the PBHs of interest.)We therefore expect a lower bound on the discharge time-scale to be set by the Hawking evaporation time, which, as noted above, satisfies t evap > t QCD for the PBHs of interest.An additional complication that requires further research is how a PBH that is charged under both the fundamental and adjoint representations of SU(3) could discharge following t QCD .
Assuming that charged PBHs immersed within the QGP remain stable at least until t QCD = 10 −5 s, the question of what happens following t QCD becomes critical [59].After the QCD confinement transition, the medium ceases to screen the PBH's enclosed charge (λ D → ∞), and it becomes energetically (very) costly for the PBH to maintain its charge.
A likely scenario following t QCD is that the gravitational potential energy of the PBH would induce a cloud of color-charged (virtual) particles from the vacuum, forming a gray-body penumbra of radius R gb .We estimate R gb by comparing the local particle acceleration to the QCD scale, GM/R 2 gb ∼ Λ QCD , or where r PBH ∼ GM is the radius of the outer trapping surface of the black hole and ℓ QCD ∼ 10 −15 m is the length-scale associated with Λ QCD = 0.17 GeV.Thereafter, the PBH and its cloud would behave as a colorneutral "hadron."(See also Ref. [60].)For PBHs that form near the window 10 −21 s ≤ t c ≤ 10 −16 s relevant for dark matter, such hadrons would have radii about four orders of magnitude smaller than the radius of a proton, but masses more than 30 orders of magnitude greater than the mass of a proton.The rapid transition around t QCD might produce gravitational waves.The "dressing" of a PBH with a charge-canceling cloud could induce local density gradients that would source scalar metric perturbations, which in turn would induce tensor perturbations at second order [61]; the frequency of such a signal would likely peak near f ∼ 1/R gb ∼ 10 27 Hz, well beyond sensitivities of projected detectors.However, if some of these objects persisted well beyond t QCD , they might yield observable effects.For example, it is possible that they could disrupt the thermal equilibrium distribution of protons and neutrons around the onset of big-bang nucleosynthesis.(See also Refs.[62][63][64].)For extremal PBHs that form amid a plasma at T c ∼ 5 × 10 7 GeV, there would exist ∼ 10 7 extremal PBHs (plus clouds) per Hubble volume at the onset of nucleosynthesis.Any such signature would enable a first-of-its-kind probe of the small-mass tail of the PBH distribution.
Beyond these potentially observable consequences, the scenario described here could have implications for the no-hair theorem and cosmic censorship.Whether the presence of a screening medium impacts the non-Abelian hair previously found for QCD-charged black holes in vacuum [32,33,[65][66][67][68][69][70]] requires further study.Other studies, meanwhile, have identified situationssuch as charged black holes in an Einstein-Maxwell holographic model embedded in asymptotically anti-de Sitter spacetime-in which the black holes can evolve into naked singularities as the temperature of the state is decreased to T = 0 [71][72][73][74].In our universe, Debye screening becomes less effective as the temperature of the QGP cools, increasing the PBH's effective charge seen by particles in the plasma.Whether these changes in the medium could push a QCD-charged PBH to become post-extremal remains the subject of further research.