Primordial black holes and the origin of the matter–antimatter asymmetry

Abstract

We review here a new scenario of hot spot electroweak baryogenesis where the local energy released in the gravitational collapse to form primordial black holes (PBHs) at the quark-hadron (QCD) epoch drives over-the-barrier sphaleron transitions in a far from equilibrium environment with just the standard model CP violation. Baryons are efficiently produced in relativistic collisions around the black holes and soon redistribute to the rest of the universe, generating the observed matter–antimatter asymmetry well before primordial nucleosynthesis. Therefore, in this scenario there is a common origin of both the dark matter to baryon ratio and the photon to baryon ratio. Moreover, the sudden drop in radiation pressure of relativistic matter at H⁰/W^±/Z⁰ decoupling, the QCD transition and e⁺e⁻ annihilation enhances the probability of PBH formation, inducing a multi-modal broad mass distribution with characteristic peaks at 10⁻⁶, 1, 30 and 10⁶ M_⊙, rapidly falling at smaller and larger masses, which may explain the LIGO–Virgo black hole mergers as well as the OGLE-GAIA microlensing events, while constituting all of the cold dark matter today. We predict the future detection of binary black hole (BBH) mergers in LIGO with masses between 1 and 5 M_⊙, as well as above 80 M_⊙, with very large mass ratios. Next generation gravitational wave and microlensing experiments will be able to test this scenario thoroughly.

This article is part of a discussion meeting issue ‘Topological avatars of new physics’.

1. Introduction

Primordial black holes (PBHs) in the solar-mass range have attracted a lot of attention since the LIGO/Virgo detection of gravitational waves from coalescing black holes [1,2]. The observed merger rate is compatible with what would be expected if PBHs constitute an appreciable fraction, and possibly all of the cold dark matter (CDM). Moreover, the LIGO/Virgo observations seem to favour mergers with low effective spins, as expected for PBHs, but are hard to explain with models of stellar evolution. An extended mass function with a peak in the range 1–10 M_⊙ could explain LIGO/Virgo observations and would only be constrained by lensing probes (microlensing and supernovae lensing) which are strongly debated and subject to large uncertainties.

Given the revival of interest in PBHs, one must explain why they have the mass and density required for explaining the LIGO/Virgo events, and why these values are comparable to the mass and density of stars. One approach is to choose an inflationary scenario which produces a peak in the power spectrum of curvature fluctuations at the required scale [3]. The required amplitude of the inhomogeneities must be much larger than that observed on cosmological scales but not too large, so this requires fine tuning of both the scale and amplitude.

An alternative approach is to assume the power spectrum is smooth (i.e. featureless), but that there is a sudden change in the plasma pressure at a particular cosmological epoch, allowing PBHs to form more easily. This enhanced gravitational collapse occurs because the critical density fluctuation required for PBH formation (δ_c) decreases when the equation-of-state parameter (w≡p/ρc²) is reduced, and since the PBH collapse fraction scales as $\exp [-{\delta }_{\mathrm{c}}^2/2{\delta }_{\mathrm{rms}}^2]$ for Gaussian fluctuations [4], this can have a strong effect on the fraction of CDM in PBHs. This is especially important for the quark-hadron (QCD) transition at ∼10⁻⁵ s, where lattice-gauge-theory calculations indicate that the sound-speed decreases by around 30% [5–8].

We have exploited this feature in refs. [9–11], pointing out that PBHs formed at the QCD transition would naturally have the Chandrasekhar mass (1.4 M_⊙), this also being the characteristic mass of main-sequence stars. Moreover, we argue that PBH formation should generate a hot outgoing shower of relativistic protons, in which electroweak baryogenesis occurs very efficiently and produces baryons with similar density to the PBHs, as well as a local baryon-to-photon ratio of order unity. After the baryons become distributed throughout space, this naturally produces a global baryon-to-photon ratio of order the PBH collapse fraction (∼10⁻⁹) if PBHs provide all of the dark matter. Their abundance and mass distribution have intrigued both cosmologists and particle physicists alike [12,13]. If they were formed in the early universe from the gravitational collapse of a large curvature fluctuation upon reentry during the radiation era, their mass can be estimated as

$$M_{\mathrm{PBH}}\equiv \gamma \hspace{0.17em}{M}_{\mathrm{hor}}(x)=\gamma \hspace{0.17em}\frac{3\sqrt{5}}{4{\pi }^{3/2}}\hspace{0.17em}\frac{M_P^3}{m_p^2}\hspace{0.17em}\frac{x^2}{g_{\ast }^{1/2}(x)}=0.56\hspace{0.17em}{M}_{\odot }\hspace{0.17em}\frac{\gamma \hspace{0.17em}{x}^2}{g_{\ast }^{1/2}(x)},$$ 1.1

where γ≲1 characterizes the efficiency of gravitational collapse, x≡m_p/T and g_*(x) is the number of degrees of freedom at that temperature. For this PBH mass to be of order the Chandrasekkar mass,

$$M_{\mathrm{Ch}}=\sqrt{\frac{3\pi }{4}}\hspace{0.17em}\frac{\omega }{{\mu }^2}\hspace{0.17em}\frac{M_P^3}{m_p^2}\sim 1.4\hspace{0.17em}{M}_{\odot },$$ 1.2

with ω = 2.018 and μ being the number of free electrons per nuclei, one needs PBH formation to occur at x∼5, or approximately T∼Λ_QCD∼200 MeV. At this temperature, quarks and gluons form the first baryons (protons and neutrons) and mesons (pions). The number of relativistic degrees of freedom drops abruptly and the speed of sound has a dip towards zero, exponentially enhancing the probability of collapse of any large curvature fluctuation that may happen to enter the horizon at that time [3]. Only very few domains will collapse to form PBHs of mass M_PBH, out of the high-density radiation within the horizon. These PBHs constitute a collisionless non-relativistic component whose overall density dilutes more slowly than the surrounding radiation until they dominate the expansion of the universe at matter–radiation equality.

The sudden gravitational collapse of the mass within the horizon at the QCD epoch releases large amounts of entropy and energy per particle, producing a shock wave at an effective temperature well above that of the surrounding plasma. Such a high-density and high-energy hot spot is an ideal place for generating over-the-barrier electroweak sphaleron transitions responsible for Higgs windings around the EW vacuum or, through the chiral anomaly, baryon number generation [14]. The strongly out-of-equilibrium conditions and the CP violation induced by the CKM matrix of the standard model (SM) are enough to produce locally a large density of baryons. Those very few hot pockets where the Baryon Asymmetry of the Universe (BAU) is generated are widely separated while, in the rest of the universe, baryons and antibaryons annihilate. Baryon number density is then radiated away from those hot pockets, by diffusion through the relativistic plasma at the speed of sound, $c_{\mathrm{s}}=c/\sqrt{3}$, until they are homogeneously distributed within a fraction of a second, well before primordial nucleosynthesis.

All the ingredients for efficient electroweak baryogenesis occur naturally around those rare and very localized hot spots associated with primordial black hole collapse during the QCD epoch. The relative energy density of matter to radiation at any time is given by

$$\frac{{\mathrm{\Omega }}_{\mathrm{M}}}{{\mathrm{\Omega }}_{\mathrm{R}}}=\frac{{\mathrm{\Omega }}_{\mathrm{B}}+{\mathrm{\Omega }}_{\mathrm{DM}}}{{\mathrm{\Omega }}_{\mathrm{R}}}\simeq \frac{(1+\chi )\hspace{0.17em}\eta \hspace{0.17em}x}{1.4\hspace{0.17em}{g}_{\ast }(x)}\simeq \frac{1700}{g_{\ast }(z)}\hspace{0.17em}\frac{1+\chi }{1+z},$$ 1.3

where χ = Ω_DM/Ω_B is the relative abundance of DM to baryons, η = n_B/n_γ = 6 × 10⁻¹⁰ is the observed BAU and we have used T = T₀(1 + z) to write x = 4 × 10¹²/(1 + z). At matter–radiation equality, 1 + z_eq = 3300, we have g_* = 3.36 and thus χ∼5.5. However, at PBH formation, the fraction of domains that collapse to form PBH is

$$\beta \equiv \frac{{\mathrm{\Omega }}_{\mathrm{PBH}}}{{\mathrm{\Omega }}_{\mathrm{R}}}=\hspace{0.17em}{f}_{\mathrm{PBH}}\hspace{0.17em}\frac{\chi \hspace{0.17em}{\mathrm{\Omega }}_{\mathrm{B}}}{{\mathrm{\Omega }}_{\mathrm{R}}}\simeq \hspace{0.17em}{f}_{\mathrm{PBH}}\hspace{0.17em}\frac{\chi \hspace{0.17em}\eta \hspace{0.17em}x}{1.4\hspace{0.17em}{g}_{\ast }(x)},$$ 1.4

where f_PBH = Ω_PBH/Ω_DM is the PBH component of DM. Therefore, for PBH formation at the QCD epoch, i.e. for x∼5, and only then, we have that β∼η∼10⁻⁹, if PBH constitute all of the DM. This is an extraordinary coincidence that seems to support the idea that baryogenesis is somehow linked with PBH formation and that the reason why the observed BAU is so small is due to the extreme rarity of the number of Hubble domains that collapse to form PBH, which is itself responsible for a late matter domination after equality.

2. The quark-hadron transition

In order for PBHs to form at the QCD epoch one needs large curvature fluctuation to enter the horizon at the right time for relativistic particles to collapse gravitationally to form a black hole. We could envision some specific dynamics during inflation (e.g. a late plateau like in critical Higgs inflation [15,16]) to produce a broad power spectrum of curvature fluctuations with an amplitude several orders of magnitude larger than those measured at the CMB. Quantum diffusion during those few e-folds that the inflaton (Higgs) field is slowly evolving induces large non-Gaussian tails on the curvature distribution, enhancing the probability of gravitational collapse [17]. But here is where the dynamics during the QCD epoch are again relevant. During the QCD transition the speed of sound drops abruptly by 30% (i.e. c²_s≃0.2 c² instead of c²/3) due to the creation of non-relativistic protons and neutrons out of quarks and gluons. This means that the radiation pressure, which prevented the collapse of mild inhomogeneities, suddenly drops, lowering the critical curvature ζ_c needed for gravitational collapse [18]. Since the probability of the formation of black holes is exponentially sensitive to ζ_c, PBHs that could not have formed before are now allowed with a non-negligible probability. Note that we do not need all Hubble domains to collapse to form PBH, just a billionth of all domains, so this sudden drop in the equation of state of the fluid during the QCD transition could be the trigger for PBH formation precisely at that time, which would then explain why PBH have masses precisely of the order of Chandrasekhar (i.e. solar) mass and not much smaller or much larger. Moreover, depending on the shape of the primordial power spectrum of curvature fluctuations, the mass distribution of PBH is a concrete prediction of this scenario. The largest abundance is associated with the mass within the horizon at T∼Λ_QCD, which is of order a solar mass, when protons and baryons become non-relativistic, as explained above. Then there is a small plateau associated with the mass at T∼m_π, when pions themselves become non-relativistic, which changes slightly the speed of sound, and thus the radiation pressure of the plasma. This mass corresponds to M∼30 M_⊙, which may explain why LIGO finds such an abundance of BH at that mass. At later times, the relativistic degrees of freedom again dominate the expansion of the universe and radiation pressure will prevent collapse, so the PBH mass distribution quickly drops at larger masses, evading all the present constraints. Fortunately, LIGO–Virgo and Kagra will soon measure this mass spectrum rather accurately in the next observing runs and may be able to discover the ‘proton’ peak at low masses together with the ‘pion’ plateau at tens of solar masses.

The gravitational collapse of a ball of radiation the size of the horizon at the QCD epoch onto a black hole of a solar mass is an extremely violent process, where a very dense plasma of relativistic particles acquire energies thousands of times their mass out of the gravitational potential energy of the collapse. Energy and momentum conservation implies that the particles within the horizon that have not fallen into the black hole will be driven out as a shock wave towards the surrounding plasma around the black hole, similar to the shock that blasts the outer layers of a star when it explodes as a supernova, except that here the surrounding plasma is extremely dense and thus ultra-high-energy interactions occur very frequently. An analogous situation can be realized in heavy ion collisions at CERN, although boosted to much larger energies and densities [19]. Such a dense and hot shock wave will induce high-energy processes that can only be realized in the very early universe. In particular, the effective temperature of this ‘little hot big bang’ is above that of electroweak sphaleron transitions and thus it is possible to induce locally windings of the Higgs around the EW vacuum. Such topological configurations are equivalent, through the chiral anomaly, to the creation of baryon number. Since the surrounding plasma (initially beyond the Hubble domain that collapsed to form the BH) is at much lower temperatures, the hot expanding plasma soon quenches and no further sphaleron transitions can wash out the locally produced BAU. Moreover, since the effective CP violation in the SM is strongly temperature dependent, at those quenching temperatures the amount of CPV arising from the CKM matrix [20] is more than enough to generate the observed BAU. Finally, it is the far from equilibrium evolution of the shock wave that allows for baryogenesis to proceed without washout. Thus, all of the Sakharov conditions [21] for baryogenesis are satisfied. Furthermore, the impulse of the shock wave will distribute the baryons from their localized origin around the collapsed PBH to the rest of the universe.

3. Electroweak baryogenesis at the QCD epoch

Let us estimate the energy available for the process of hot spot electroweak baryogenesis (HSEWB) to work as described above. The conservation of energy gives the change in kinetic energy due to the collapse of matter within the horizon down to the Schwarzschild radius of the PBH, R_S = 2GM_PBH/c² = γ d_H,

$$\mathrm{\Delta }K\simeq \left(\frac{1}{\gamma }-1\right)M_{\mathrm{hor}}=\frac{1-\gamma }{{\gamma }^2}\hspace{0.17em}{M}_{\mathrm{PBH}}.$$ 3.1

Note that the smaller γ is, i.e. the less efficient the process of gravitational collapse, the more compact is the resulting PBH, and thus the larger the kinetic energy of the outgoing particles. We can now estimate the energy per particle E₀ acquired by protons in the expanding shell. In ref. [18], the process of gravitational collapse was studied in detail and shown to be explosive, with a shell of relativistic matter expanding out of the collapsing region. The most massive particles at that time are protons. Their number density, between the QCD transition and proton freeze out (i.e. for 20 < T < 200 MeV), is that of a relativistic species, n_p(x) = 1.59 × 10⁴⁰ x^−3/2 e^−x cm⁻³, and therefore

$$E_0=\frac{\mathrm{\Delta }K}{n_p\hspace{0.17em}\mathrm{\Delta }V}\simeq 10\hspace{0.17em}{g}_{\ast }(x)\hspace{0.17em}{x}^{-5/2}\hspace{0.17em}{\mathrm{e}}^x\hspace{0.17em}\mathrm{GeV},$$ 3.2

where we have used γ = 0.6 as a conservative estimate. The energy per proton E₀ is larger for lower efficiencies by a factor 1/(γ + γ² + γ³). At the same time, the density of the relativistic plasma surrounding the collapse horizon is huge at that time, n_gas(x) = 1.13 × 10⁴⁰ g_*(x) x⁻³ cm⁻³, so it behaves like a wall for the relativistic protons that escape the PBH collapse. For example, for those PBH formed at x∼6, with PBH mass M_PBH∼0.8M_⊙, the energy released ΔK = (3/2)Nk_BT gives an effective temperature k_BT_eff = (2/3)E₀≃3 TeV, which is well above the sphaleron barrier [22]. At that effective temperature, the sphaleron rate per unit volume is given by

$${\mathrm{\Gamma }}_{\mathrm{sph}}\sim {\alpha }_{\mathrm{W}}^4\hspace{0.17em}{T}_{\mathrm{eff}}^4.$$ 3.3

The baryon asymmetry induced by the ultrarelativistic partons heating up the surrounding plasma can then be estimated as [14]

$$\frac{n_{\mathrm{B}}}{s}\simeq \frac{n_{\mathrm{parton}}}{s}\times {\mathrm{\Gamma }}_{\mathrm{sph}}(T_{\mathrm{eff}})V\hspace{0.17em}\mathrm{\Delta }t\times {\delta }_{\mathrm{CP}},$$ 3.4

where n_parton is the number density of highly energetic partons (here the protons and antiprotons), Δt∼2 × 10⁻⁵ s (200 MeV T)⁻² is the duration of the sphaleron process and the SM CP violation parameter is given by [22]

$${\delta }_{\mathrm{CP}}(T)=3\times {10}^{-5}{\left(\frac{20.4\hspace{0.17em}\mathrm{GeV}}{T}\right)}^{12}=7.3\times {10}^{10}\hspace{0.17em}{x}^{12}.$$ 3.5

The entropy density, s = (2π²/45) g_*S T³_th, in equation (3.4) is that of the thermalized plasma surrounding the PBH, at temperatures T_th∼70 MeV≪T_eff, quenching the sphaleron transitions and preventing baryon washout.

The production of baryons is thus very efficient for x ≳5, giving rise to n_B∼n_γ or η^local∼1. This maximal BAU is then radiated away at relativistic speeds from the localized hot spots to the rest of the universe. For a fraction β∼10⁻⁹ of Hubble domains that collapse to PBH providing all of the DM (f_PBH = 1), the distance between hot spots, where the BAU has been generated, is given by d∼β^−1/3 d_H(t_QCD)∼10⁻² light-seconds. Moving at the speed of light, the ultrarelativistic baryons move through the plasma until they uniformly distribute the original BAU to the rest of the universe well before primordial nucleosynthesis (t_BBN∼1 − 180 s), thus diluting the BAU and explaining the small baryon-to-photon ratio η∼β∼10⁻⁹ (figure 1).

Figure 1.

Qualitative representation of the three steps in our scenario. (A) Gravitational collapse to a PBH of the curvature fluctuation at horizon re-entry. (B) Sphaleron transitions in the hot spot around the PBH, producing $\eta =\mathcal{O}(1)$ locally through EW baryogenesis. (C) Propagation of baryons to the rest of the Universe through jets, resulting in the observed BAU with η∼β∼10⁻⁹. (Online version in colour.)

The DM-to-baryon ratio, χ∼5, can also be explained in this scenario: most of the PBHs are formed during or after the sudden drop of the sound speed during the QCD transition, when the parton energies are high enough to produce a strong baryon asymmetry. χ is thus given by the ratio of the black hole mass and the ejected mass, which is χ ≈ γ/(1 − γ) ≈ 5 if γ ≈ 0.8. Lower values of γ could nevertheless be accommodated if the temperature below which protons acquire enough energy to drive the baryon-producing sphaleron transitions is reduced, T≲100 MeV, so that only the massive PBHs formed at a later time contribute to the BAU.

4. The origin of the large curvature fluctuations

The softening of the equation of state during the QCD transition boosts the formation of stellar-mass black holes but does not alleviate the need for large curvature fluctuations. One could rely on quantum diffusion during a specific period of inflationary dynamics to account for large non-Gaussian tails of the curvature distribution, but we would still need to understand why then and not somewhere else. We propose that during the QCD epoch a light stochastic spectator field (different from the inflaton [23]) starts to evolve and induces an extra burst of curvature fluctuations providing the necessary critical amplitude for black hole collapse. The spectator field is a curvaton field whose quantum fluctuations during inflation permeate all of space, while its energy density is subdominant, both during inflation and during the subsequent radiation era after reheating. This field remains frozen during radiation (m≪H) until its potential energy density (at the top of its potential) starts to dominate, only in a few patches, the total density of the universe. At this point in time, and in those pockets, the spectator field triggers a second brief period of inflation within those regions, generating local nonlinear curvature fluctuations which reenter the horizon and collapse to form PBHs. In the rest of the universe, curvature fluctuations are statistically Gaussian, unaffected by the dynamics of the spectator field, and do not form PBH.

A natural candidate for the light spectator field is the QCD axion. Its existence is well motivated, providing a robust solution to the strong CP problem, by naturally relaxing the θ parameter to zero [24]. We will assume that the associated Peccei–Quinn symmetry is spontaneously broken before inflation ends.

The axion potential at a temperature T below a few GeV is given by

$$V(a)=m_a^{\mathrm{eff}}{(T)}^2\hspace{0.17em}{f}_a^2\hspace{0.17em}\left[1+\cos \left(\frac{a}{f_a}\right)\right],$$ 4.1

where

$$m_a^{\mathrm{eff}}(T)=m_a\hspace{0.17em}{\left(\frac{T}{T_c}\right)}^{-7/2}\mathrm{for}\hspace{0.17em}T\gtrsim T_c\simeq 100\hspace{0.17em}\mathrm{MeV},$$ 4.2

and otherwise constant and equal to the zero-temperature mass m_a. For the QCD axion, there is a relation between mass and decay constant, m_a f_a≃m_π f_π≃(75 MeV)². Therefore, those patches in which the axion remains at the top of its potential (a≪f_a) will dominate the energy density of the universe at temperatures below

$$T_c={\left(\frac{60\hspace{0.17em}{m}_a^2\hspace{0.17em}{f}_a^2}{{\pi }^2{g}_{\ast }}\right)}^{1/4}\simeq 80\hspace{0.17em}\mathrm{MeV}.$$ 4.3

In those patches, the axion will start rolling down the hill from the rms value generated during inflation, a_ini≃H_inf/2π≪f_a, and inflating the universe until slow-roll ends at $a_{\mathrm{end}}\simeq 2\sqrt{2}\hspace{0.17em}{f}_a^2/{\overline{M}}_P\ll f_a$, where ${\overline{M}}_P=M_P/\sqrt{8\pi }=2.4\times {10}^{18}$ GeV is the reduced Planck mass. This second inflationary period lasts slightly more than one e-fold,

$$\mathrm{\Delta }N\simeq \frac{f_a^2}{{\overline{M}}_P^2}\hspace{0.17em}\ln \left(\frac{a_{\mathrm{end}}^2}{a_{\mathrm{ini}}^2}\right)\lesssim 1.5\mathrm{for}\hspace{0.17em}\hspace{0.17em}{f}_a\lesssim 0.1{\overline{M}}_P.$$ 4.4

During this short burst of inflation, the axion (curvaton) field generates an order one curvature fluctuation $\zeta \simeq {⟨\delta a^2⟩}^{1/2}/{\overline{M}}_P\sqrt{{ϵ}_a}\sim 1$, locally boosting the probability of collapse,

$$P_{\mathrm{PBH}}\simeq C(⟨a⟩)\hspace{0.17em}\frac{\mathrm{\Delta }a}{{⟨\delta a^2⟩}^{1/2}}\sim {10}^{-9},$$ 4.5

where C(〈a〉) depends on the mean field value of the axion field in our Hubble patch [9].

The ingredient that makes this scenario so appealing is that the axion mass is so strongly temperature dependent, that for most of the evolution around the QCD transition at Λ_QCD, the axion is massless and the axion energy density is irrelevant. Only for those domains that are at the top of the axion potential, stochastically determined by inflation, can there be an extra e-fold of inflation that drives order one curvature fluctuations to collapse to form black holes. Thus, PBH production is stochastically predictive and does not depend on the dynamics during inflation, but only on the axion decay constant. Moreover, since the axion mass turns on abruptly, at temperatures above T_c there can be no PBH production, and therefore the PBH mass distribution is naturally cut off at low masses M_PBH < γ M_hor(x_c)≃2M_⊙. In the near future, LIGO–Virgo sensitivity to low BH masses will be able to determine the mass spectrum of coalescing black holes and we could then compare with the predictions of this scenario.

5. Extending the scenario to the thermal history of the universe

Here, we point out an interesting consequence of the above scenario, by extending it beyond the QCD scale. As the background temperature decreases from 100 GeV to 1 MeV, corresponding to the rest masses of the top quark, the Higgs, the W and Z bosons, the proton, the pion and the electron, there are four periods at which the sound speed c_s exhibits sudden dips. The proton dip is the biggest (∼30%) but the others (5 – 10%) may also be relevant due to the exponential dependence of gravitational collapse on the critical curvature fluctuation. These c_s-dips produce distinctive features in the PBH mass function at four mass scales in the range 10⁻⁶ – 10⁶ M_⊙.

An important feature of this scenario is that it predicts the form of the PBH mass distribution precisely. We show that the expected form not only satisfies all the current astrophysical and cosmological constraints but also allows the PBHs to explain numerous observational conundra: (1) microlensing events towards the Galactic bulge generated by planet-mass objects with 1% of the CDM, well above most expectations for free-floating planets; (2) microlensing of quasars, including ones that are so misaligned with the lensing galaxy that the probability of lensing by a star is very low; (3) the unexpected high number of microlensing events towards the Galactic bulge by dark objects in the mass gap between 2 and 5 M_⊙ [25], where stellar evolution models fail to form black holes [26]; (4) unexplained correlations in the source-subtracted X-ray and cosmic infrared background fluctuations [27]; (5) the non-observation of ultra-faint dwarf galaxies (UFDGs) below the critical radius of dynamical heating by PBHs [28]; (6) the mass, spins and coalescence rates for the black holes found by LIGO/Virgo [2]; (7) the still unexplained relationship between the mass of a galaxy and its central intermediate mass black holes (IMBH) or supermassive black holes (SMBH).

Reheating at the end of inflation fills the Universe with radiation. In the absence of extensions beyond the SM of particle physics (e.g. right-handed neutrinos), the Universe remains dominated by relativistic particles with an energy density decreasing as the fourth power of the temperature as the Universe expands. The number of relativistic degrees of freedom remains constant (g_* = 106.75) until around 200 GeV, when the temperature of the Universe falls to the mass thresholds of SM particles.

As shown in figure 2a, the first particle to become non-relativistic is the top quark at T≃m_t = 172 GeV, followed by the Higgs boson at 125 GeV, and the Z and W bosons at 92 and 81 GeV, respectively. These particles become non-relativistic at nearly the same time and this induces a significant drop in the number of relativistic degrees of freedom down to g_* = 86.75. There are further changes at the b and c quark and τ-lepton thresholds, but these are too small to appear in figure 2. Therefore, g_* remains approximately constant until the QCD transition at around 200 MeV, when protons, neutrons and pions condense out of the free light quarks and gluons. The number of relativistic degrees of freedom then falls abruptly to g_* = 17.25. A little later the pions become non-relativistic and then the muons, giving g_* = 10.75. Thereafter, g_* remains constant until e⁺e⁻ annihilation and neutrino decoupling at around 1 MeV, when it drops to g_* = 3.36.

Figure 2.

Relativistic degrees of freedom g_* (a) and equation-of-state parameter w (b), both as a function of temperature T (in MeV). The grey vertical lines correspond to the masses of the electron, pion, proton/neutron, W,  Z bosons and top quark, respectively. The grey dashed horizontal lines indicate values of g_* = 100 and w = 1/3, respectively. (Online version in colour.)

Whenever the number of relativistic degrees of freedom suddenly drops, it changes the effective equation of state parameter w. As shown in figure 2b, there are thus four periods in the thermal history of the Universe when w decreases. After each of these, w resumes its relativistic value of 1/3 but each sudden drop modifies the probability of gravitational collapse of any large curvature fluctuations present at that time. We will see below how these changes in w result in the production of PBHs with different masses and dark-matter fractions.

(a). Primordial black-hole formation

There are a plethora of mechanisms for PBH formation. All of them require the generation of large overdensities, specified by the density contrast, δ≡δρ/ρ, usually assumed to be of inflationary origin [3]. When overdensities re-enter the Hubble horizon, they collapse if they are larger than some threshold δ_c, which generally depends on the equation of state and density profile. For the most studied radiation-dominated and spherically symmetric case, one has δ_c ≈ 0.45 (see e.g. [18]). However, there are other (non-inflationary) scenarios for PBH formation, where the inhomogeneities arise from first-order phase transitions [7,29], bubble collisions [30–32] and the collapse of cosmic strings [33–35], necklaces [36,37], domain walls [38–40] or non-standard vacua [41]. The latter are particularly interesting as they provide a natural scenario for multi-modal mass functions.

The threshold δ_c is a function of the equation-of-state parameter w(T), which is shown in figure 2b, so the thermal history of the Universe can induce pronounced features in the PBH mass function even for a uniform power spectrum. This is because if the PBHs form from Gaussian inhomogeneities with root-mean-square amplitude δ_rms, then the fraction of horizon patches undergoing collapse to PBHs when the temperature of the Universe is T should be [4]

$$\beta (M)\approx \frac{1}{2}\mathrm{erfc}\left[\frac{{\delta }_{\mathrm{c}}(w[T(M)])}{\sqrt{2}\hspace{0.17em}{\delta }_{\mathrm{rms}}(M)}\right],$$ 5.1

where ‘erfc’ is the complementary error function and we have written the temperature in terms of PBH mass, $T\simeq 200\hspace{0.17em}\mathrm{MeV}\hspace{0.17em}\sqrt{M_{\odot }/M\hspace{0.17em}}$. This shows that β(M) is exponentially sensitive to w(M). Throughout this work, we use the numerical results for δ_c from ref. [18]. The present CDM fraction for PBHs of mass M is

$$f_{\mathrm{PBH}}(M)\equiv \frac{1}{{\rho }_{\mathrm{CDM}}}\frac{\mathrm{d}{\rho }_{\mathrm{PBH}}(M)}{\mathrm{d}\ln M}\approx 2.4\hspace{0.17em}\beta (M)\hspace{0.17em}\sqrt{\frac{M_{\mathrm{eq}}}{M}},$$ 5.2

where M_eq = 2.8 × 10¹⁷ M_⊙ is the horizon mass at matter–radiation equality, ρ_CDM is the CDM density and the numerical factor is 2(1 + Ω_CDM/Ω_B) with Ω_CDM = 0.245 and Ω_B = 0.0456 being the CDM and baryon density parameters [42].

There are many inflationary models and these predict a variety of shapes for δ_H(M). Some of them, including single-field models like Higgs inflation [15,16], or two-field models like hybrid inflation [43], produce an extended plateau or dome-like feature in the power spectrum. Instead of focussing on any specific scenario, we will assume here a quasi-scale-invariant spectrum,

$${\delta }_{\mathrm{rms}}(M)=A{\left(\frac{M}{M_{\ast }}\right)}^{(1-n_{\mathrm{s}})/4},$$ 5.3

where the spectral index n_s and amplitude A are treated as free phenomenological parameters. This could also represent any spectrum with a broad peak, such as might be generically produced by a second phase of slow-roll inflation. The amplitude has been chosen to give A = 0.1487 for n_s = 0.97 and M < M_* = 10⁶ M_⊙, in order to get an integrated abundance f^tot_PBH = 1. The ratio between the PBH mass and the horizon mass at re-entry is denoted by γ and we assume γ = 0.8 as a benchmark value, following refs. [9–11]. The resulting mass function is represented in figure 3. It exhibits a dominant peak at M≃2 M_⊙ and three additional bumps at 10⁻⁵ M_⊙, 30 M_⊙ and 10⁶ M_⊙, corresponding to transitions in the number of relativistic degrees of freedom predicted by the known thermal history of the Universe.

Figure 3.

The mass spectrum of PBHs with spectral index n_s = 0.96 (green, dotted), 0.97 (blue, solid), 0.98 (red, dashed). The grey vertical lines corresponds to the EW and QCD phase transitions and e⁺e⁻ annihilation. (Online version in colour.)

6. Constraints on the PBH mass function

In this section, we discuss whether the PBH mass functions shown in figure 3, all of which assume f^tot_PBH = 1, and are compatible with the numerous observational constraints on f_PBH(M). There is an overproduction of light PBHs for n_s ≳0.98 and of heavy ones for n_s≲0.95, but the mass distribution for n_s≃0.97 can provide 100% of the dark matter without violating any current reliable constraints. However, this requires justification because it is sometimes argued that this possibility is already excluded.

In order of increasing mass, the various PBH constraints come from the extragalactic gamma-ray background, neutron star and white dwarf abundances in globular clusters, microlensing surveys, dynamical effects (such as the heating of UFDGs and their stellar clusters and the disruption of wide binaries), radio and X-ray point source counts, and CMB anisotropies generated by PBH accretion. The limits are summarized in ref. [12] and numerous other papers. However, most of these constraints assume a monochromatic PBH mass function (i.e. one with width ΔM∼M). In the present scenario, we predict an extended mass function and cannot simply compare this with the monochromatic constraints.

In order to assess the situation, we adopt the approach advocated in refs. [44–49]. Assuming that the mass distribution scales linearly with f^tot_PBH, each probe sets an upper limit

$$f_{\mathrm{PBH}}^{\max }={\left(\int \frac{\mathrm{d}M}{M}\frac{f_{\mathrm{PBH}}(M)}{C_{\mathrm{mono}}(M)}\right)}^{-1},$$ 6.1

where C_mono(M) is the limit for a monochromatic function of mass M. We have calculated the value of f^max_PBH associated with each probe for n_s = 0.97 but with different astrophysical assumptions. All the probes except one allow f^tot_PBH = 1, including the EROS/MACHO microlensing limits with realistic assumptions. The original ones (no clustering, fixed circular velocities, isothermal DM halo profile) are simplistic, especially when one takes into account that a significant fraction of PBHs could be regrouped into dense clusters [50] (behaving as a single massive lens) or into haloes larger than ∼10³ M_⊙ with a very low probability of being along the line of sight. The one exception is the CMB limit of ref. [51]. However, this is only in tension with our model for M ≳1000 M_⊙ and the steady-state assumption could break down for such large masses. Furthermore, such heavy black holes should have seeded compact haloes before matter–radiation equality, whose impact on the accretion and CMB limits is uncertain. Additional work is clearly needed to derive more secure accurate constraints on the PBH distribution but—given all the current theoretical and astrophysical uncertainties—a distribution with f^tot_PBH = 1 is plausible.

7. Unresolved enigmas of observational cosmology

Besides passing the current observational constraints on the form of the CDM, the PBH mass function with n_s≃0.97 predicted from the known thermal history of the Universe provides a unified explanation for several other puzzling conundra. We discuss these in order of increasing PBH mass.

(a). Planetary-mass microlenses

Very recently, Niikura et al. have reported two interesting microlensing results. The first [52] comes from observations of M31 using the Subaru telescope, which include one possible detection and place strong constraints on PBHs in the mass range 10⁻¹⁰ and 10⁻⁶ M_⊙. The second [53] uses data from the five-year OGLE survey of 2622 microlensing events in the Galactic bulge [54] and has revealed six ultra-short ones attributable to planetary-mass objects between 10⁻⁶ and 10⁻⁴ M_⊙. These would contribute to about 1% of the CDM, which is more than expected for free-floating planets [55]. The latter corresponds to the first bump in our predicted PBH mass function and the abundance, when integrated over the mass range probed by OGLE, coinciding with our best-fit model with n_s≃0.97.

(b). Quasar microlensing

The detection of 24 microlensed quasars [56] suggests that up to 25% of galactic halos could be in PBHs with mass between 0.05 and 0.45 M_⊙. These events could also be explained by intervening stars, but in several cases the stellar region of the lensing galaxy is not aligned with the quasar, which suggests a population of subsolar halo objects with f_PBH > 0.01. For a PBH mass function with n_s = 0.97, one expects f_PBH≃0.05 in this mass range. This is also consistent with claimed detections of microlensing of stars in M31 by halo objects with M between 0.5 and 1 M_⊙ and f_PBH between 15 and 30% [57]. Note that Hawkins has claimed for many years that quasar microlensing data suggest that the dark matter could comprise PBHs [58,59].

(c). OGLE/GAIA excess of dark lenses in the galactic bulge

OGLE has detected around 60 long-duration microlensing events, of which around 20 have GAIA parallax measurements of distances of a few kpc, which break the mass–distance degeneracy of microlensing and allow the determination of masses in the few solar mass range, which imply that they are probably black holes, since stars at those distances would be visible by OGLE [25]. The event distribution from the posterior likelihoods of their masses peaks between 0.8 and 5 M_⊙, overlapping the gap from 2 to 5 M_⊙ in which black holes are not expected to form as the endpoint of stellar evolution [26]. On the other hand, this is consistent with the main peak in the PBH mass distribution if 0.6≲γ≲1 (figure 4).

Figure 4.

The mass distribution of OGLE-GAIA microlensing events (red) from ref. [25]. The grey vertical lines correspond to the mode of the distribution for each microlensing event, and clearly concentrate around the peak of the predicted PBH mass function for γ = 0.8. (Online version in colour.)

(d). Cosmic infrared/X-ray backgrounds

As shown by Kashlinsky et al. [60] and Kashlinsky [61], the unexpected spatial coherence in the fluctuations of the source-subtracted cosmic infrared and soft X-ray backgrounds suggest an overabundance of high-redshift haloes. These could form from the Poisson fluctuations in the PBH number density if solar-mass PBHs comprise a significant fraction of the CDM. In these haloes, a few stars can form and emit infrared radiation, while PBHs can emit X-rays due to accretion. It is challenging to find other scenarios that naturally produce such features.

(e). Ultra-faint dwarf galaxies

For the PBH mass distribution shown in figure 3, the critical radius below which CDM-dominated UFDGs would be dynamically unstable is r_c∼10 – 20 parsecs (depending on the mass of a possible central SMBH). The non-detection of galaxies smaller than this critical radius, despite their magnitude being above the detection limit, suggests compact halo objects in the solar-mass range. Moreover, rapid accretion in the densest PBH halos could explain the extreme UFDG mass-to-light ratios observed [28].

(f). Mass, spin and merger rates for LIGO/Virgo black holes

The effective spin of a black-hole merger is defined by

$${\chi }_{\mathrm{eff}}\equiv \frac{m_1\hspace{0.17em}{\chi }_1\cos ({\theta }_{L{S}_1})+m_2\hspace{0.17em}{\chi }_2\cos ({\theta }_{L{S}_2})}{m_1+m_2},$$ 7.1

where m_i and χ_i = S_i/G m²_i are the black-hole masses and spins, and θ_LSi is the angle between the spin S_i and the orbital angular momentum L. Most of the observed coalesced black holes have effective spins compatible with zero [62]. Although the statistical significance of this result is still low [63], this goes against a stellar binary origin [64,65] but is a prediction of the PBH scenario [13] (figure 5). Whether the binaries are formed early or late, the expected rate of PBH mergers is comparable to that observed [62] if PBHs account for a significant fraction of the CDM [50,66,67]. With our mass distribution, PBHs in the range 10 – 100 M_⊙ have f_PBH(M)∼0.01 even if f^tot_PBH = 1. We have computed the likelihood distribution of merger events with PBHs of masses m₁ and m₂ for n_s = 0.97 using the method of ref. [28] and taking into account the limited sensitivity of the detector at the low frequencies corresponding to large chirp masses. We find that the LIGO/Virgo events are localized in the most likely region. With the expected number of events in the O3 run, LIGO/Virgo should be able to detect mergers with a low mass ratio, q≡m₂/m₁≲0.2, or PBHs larger than 100 M_⊙, which are two distinctive features of our scenario. Finally, we have computed the expected PBH merger rate in the solar-mass range, after normalizing it to the observed rate in the large-mass range, τ = 50 yr⁻¹ Gpc⁻³. We obtain τ ≈ 10³ yr⁻¹ Gpc⁻³ for PBH between 1 and 5 M_⊙, which is below the rate inferred for neutron star mergers but within range of the next LIGO/Virgo runs. Our scenario could therefore be probed by searching for BH mergers in the mass gap, or below the Chandrasekhar mass, that could be distinguished from neutron stars mergers due to the maximum chirp frequency, the well-known f_ISCO≃4400 M_⊙/M_tot, or alternatively if there is no detection of an electromagnetic counterpart. The predicted binary black hole (BBH) merger rate for LIGO/Virgo is shown in figure 6.

Figure 5.

The distributions of effective spin χ_eff and final BH spin af for the 10 GWTC-1 events. The vertical lines, at χ_eff = 0 and a_f = 0.68, respectively, correspond to BH with no initial spin, as predicted for primordial BH. (Online version in colour.)

Figure 6.

The predicted rate of events in LIGO/Virgo for the multi-modal mass distribution expected in this scenario. Most of the GWTC-1 events concentrate in the expected high merger rate peak at $m_A~m_B~50M_{\odot }$, but there is a second peak around $m_A~70M_{\odot }$ with larger mass ratios, $m_B~5M_{\odot }$. It is expected that future LIGO–Virgo runswill also find BBH mergers in that region of masses. (Online version in colour.)

(g). Intermediate mass black holes and supermassive black holes

Given our mass distribution, we have calculated the number of intermediate-mass and supermassive PBHs for each 10¹² M_⊙ halo. Interestingly, we obtain about one 10⁸ M_⊙ PBH per halo and 10 times as many smaller ones, possibly seeding the formation of a comparable number of dwarf satellites and faint CDM haloes. Assuming a standard Press–Schechter halo mass function [68],

$$\frac{\mathrm{d}{n}_{\mathrm{h}}}{\mathrm{d}\ln M_{\mathrm{h}}}\approx \frac{{\rho }_{\mathrm{m}}}{\sqrt{\pi \hspace{0.17em}}\hspace{0.17em}{M}_{\mathrm{h}}}\hspace{0.17em}{\mathrm{e}}^{-M_{\mathrm{h}}/M_{\ast }},$$ 7.2

where ρ_m is the mean cosmological matter density (including both dark matter and baryons) and M_* ≈ 10¹⁴ M_⊙ is the cut-off halo mass. For a given M_h, one can thus identify the corresponding PBH mass that has the same number density

$$\frac{\mathrm{d}{n}_{\mathrm{PBH}}}{\mathrm{d}\ln M}\approx \frac{{\rho }_{\mathrm{m}}\hspace{0.17em}{f}_{\mathrm{PBH}}}{M}.$$ 7.3

This gives a relation M_h ≈ M_PBH/f_PBH, corresponding to roughly one IMBH/SMBH per halo of mass 10³ M_PBH for our distribution, which is in agreement with observations. Furthermore, our n_s = 0.97 mass distribution predicts the observed relation between the central black hole and the halo mass [69], as shown in figure 7, but only if f^tot_PBH≃1. A lower (larger) value of the spectral index would imply too many (few) IMBH/SMBHs. It is remarkable that one can reproduce this unexplained relation with such simple hypothesis, and without invoking super-Eddington accretion. Accretion should nevertheless enhance the mass of heavier SMBHs, which could make the model in even more agreement with observations.

Figure 7.

Expected relation between PBH mass and initial host spheroidal mass for n_s = 0.97 (top, blue) and n_s = 0.96 (bottom, red). Figure adapted from ref. [69]. The larger PBHs tend to accrete more mass than smaller ones, via accretion disks, and therefore it is expected that the predicted curves be shifted upwards, in agreement with observations. (Online version in colour.)

8. Conclusion

The early universe can be used as a probe of fundamental physics at much higher energies than those explored in the present universe. A classic example is the prediction of the observed helium to hydrogen abundance carried out in the 1940s of the last century, when applying nuclear reaction rates to the very hot, very dense and rapidly expanding universe. Now we know that stars cannot build up the observed He abundance through nuclear reactions in their interior, and thus have to be primordial. The same may also happen with the BAU. We can extrapolate the fundamental interactions of the SM of particle physics to the dense and hot early universe and see whether there are the necessary conditions for the BAU to develop. In the scenario that we have proposed, the BAU is generated at the violent process of PBH formation during the QCD transition, triggered by the sudden drop in the radiation pressure, in the presence of large amplitude curvature fluctuations. Baryon number violation is driven by out-of-equilibrium sphaleron processes that are immediately quenched by the surrounding plasma in the expanding universe, preventing baryon wash-out, while the only CP violation needed is that of the CKM phases of the SM. Moreover, the same small fraction of domains that act as hot spots for the efficient production of baryons is responsible for the present low value of the BAU and the dominance of PBH over other forms of matter, while at the same time explaining why baryons and dark matter have similar densities today.

It is interesting how this novel scenario resolves two of the more acute problems of cosmology, the origin of the baryon asymmetry and the nature of dark matter, in one go. Rather than relying on new particle physics interactions at high energy to generate the baryon asymmetry simultaneously on all locations in the universe, via out of equilibrium decay or first-order phase transitions, this scenario suggests it occurs only locally, on just a few rare domains, during the violent gravitational collapse associated with the formation of PBHs, and is later radiated (diffused) to the rest of the universe. The connection between the rareness of those domains, responsible for a late matter domination (thus leaving enough time for the subsequent stellar evolution and structure formation), and the low baryon-to-photon ratio is a completely new way of approaching the problem. Dark matter (in the form of PBH) and baryons are then linked together, explaining their order-one relative ratio. This new scenario of EW baryogenesis at the QCD epoch via PBH works even if the fraction of domains that collapses to form PBH is less than what is needed to generate all of the DM, i.e. if f_PBH < 1, but the nice relation that one obtains between DM and baryons, Ω_DM/Ω_B∼5 if f_PBH = 1, could not be explained in that case.

If LIGO–Virgo interferometers map out in the next few years the mass distribution of coalescing black holes and turn out to be like that of figure 4, then we may conclude that the QCD epoch played a crucial role in the evolution of the Universe, generating at the same time the matter–antimatter asymmetry and the dark matter, setting the stage for primordial nucleosynthesis, stellar evolution and structure formation in the Universe.

Data accessibility

This article has no additional data.

Competing interests

I declare I have no competing interests.

Funding

The author acknowledges support from the Research Project no. FPA2015-68048-03-3P [MINECO-FEDER] and the Centro de Excelencia Severo Ochoa Program no. SEV-2016-0597.