1. Introduction
The supermassive black holes (SMBHs) with a mass $\sim { \mathcal O }({10}^{6}-{10}^{9}){M}_{\odot }$ are ubiquitous in the center of galaxies [1–3], where M⊙ is the solar mass. They are also considered as the central engines of active galactic nuclei (AGN) and quasars [4, 5]. However, the origin of SMBHs is not clear. They may be formed from the stellar black holes (BHs) with a mass $\sim { \mathcal O }(10){M}_{\odot }$ through accretion and mergers [6, 7]. Nevertheless, it is difficult to account for the SMBHs at the high redshift z ∼ 7 [8–10] since the stellar BHs have no time to grow into SMBHs. Another scenario is to consider primordial black holes (PBHs) with a mass $\sim { \mathcal O }({10}^{4}-{10}^{5}){M}_{\odot }$ as the seeds of SMBHs [11, 12]. In this case, the PBHs could subsequently grow up to $\sim { \mathcal O }({10}^{9}){M}_{\odot }$ due to an efficient accretion of matter on the massive seeds and mergings [13, 14]. Additionally, the SMBHs may also originate from the collapse of self-interaction dark matter (DM) [15–18]. See e.g. [19, 20] for reviews of the SMBHs formation.
The PBHs have recently gathered significant interest as the candidates of DM [21–33]. They can be formed due to the collapse of large density fluctuations produced during inflation [34–38], the cosmological phase transitions [39–41], and the collapse of topological defects [42–48] and false vacuum bubbles [49–52], etc. Another scenario is considering the PBHs formation from the quantum chromodynamics (QCD) axion bubbles in the early Universe, leading to the minimum PBH mass $\sim { \mathcal O }({10}^{4}){M}_{\odot }$ [53]. This scenario is similar to the PBHs formation from the baryon bubbles in the Affleck–Dine baryogenesis [54, 55]. The QCD axion bubbles can be generated due to an explicit PQ symmetry breaking after inflation with the multiple vacua, and are formed when the QCD axion starts to oscillate during the QCD phase transition. The axion acquires a light mass after inflation, then starts to oscillate when this mass is comparable to the Hubble parameter and settles down into different potential minima, depending on its initial position. Since the explicit PQ symmetry is supposed to be temporarily broken, the axion potential will disappear before the QCD phase transition. Therefore, the misalignment mechanism is still valid to calculate the final QCD axion abundance. However, the initial misalignment angle will be split into different values by the multiple vacua. During the QCD phase transition, the QCD axions start to oscillate with these different initial angles, in which the high density axion bubbles can be produced with large initial values. An interesting phenomenon of the QCD axion bubbles is the PBHs formation, which can be formed when the axions dominate the radiation in the bubbles.
In this paper, we focus our attention on the PBHs triggered by QCD axion bubbles as the seeds of SMBHs. The axion acquires a light mass due to an explicit PQ symmetry breaking after inflation with a large axion decay constant ${f}_{a}\,\sim { \mathcal O }({10}^{16})\mathrm{GeV}$, and then settles down into two potential minima, ${\phi }_{\min }^{0}$ and ${\phi }_{\min }^{1}$. We consider a general case in which the QCD axion bubbles are formed with the bubble effective angle θeff ∈ (0, π]. The ${\phi }_{\min }^{0}$ accounts for the DM abundance with the initial misalignment angle ∼0, while the ${\phi }_{\min }^{1}$ forms the high energy density QCD axion bubbles with the angle ∼θeff. The PBHs triggered by QCD axion bubbles are formed after the QCD phase transition, which leads to the minimum PBH mass $\sim { \mathcal O }({10}^{4}-{10}^{7}){M}_{\odot }$ for ${f}_{a}\sim { \mathcal O }({10}^{16})\,\mathrm{GeV}$. Compared with the critical value of the $\sim { \mathcal O }({10}^{9}){M}_{\odot }$ SMBHs seeds, the PBH at this mass region may account for the seeds of SMBH.
The paper is structured as follows. In section 2 , we introduce the QCD axion bubbles scenario. In section 3 , we investigate the PBHs triggered by QCD axion bubbles as the seeds of SMBHs. Finally, the conclusion is given in section 4 .
2. QCD axion bubbles with θeff
In this section, we introduce the QCD axion bubbles scenario [53]. The effective potential of the QCD axion is given by
$\begin{eqnarray}{V}_{\mathrm{QCD}}(\phi )={m}_{a}^{2}(T){f}_{a}^{2}\left[1-\cos \left(\displaystyle \frac{\phi }{{f}_{a}}\right)\right],\end{eqnarray}$
with the axion field φ, the axion decay constant fa, the axion angle θ = φ/fa, and the temperature-dependent axion mass $\begin{eqnarray}{m}_{a}(T)\simeq \left\{\begin{array}{ll}{m}_{a,0}{\left(\tfrac{T}{{T}_{\mathrm{QCD}}}\right)}^{-4.08}, & T\geqslant {T}_{\mathrm{QCD}}\\ {m}_{a,0}, & T\lt {T}_{\mathrm{QCD}}\end{array}\right.\end{eqnarray}$
where ma,0 ≃ 5.70(7)μeV(1012 GeV/fa) is the zero-temperature axion mass [56], T is the cosmic temperature, and TQCD ≃ 150 MeV.During the QCD phase transition, the QCD axion starts to oscillate when its mass ma(T) is comparable to the Hubble parameter H(T) at the oscillation temperature Ta. The axion energy density at present can be described by
$\begin{eqnarray}{\rho }_{a}({T}_{0})=\displaystyle \frac{{m}_{a,0}{m}_{a}({T}_{a})s({T}_{0})}{2s({T}_{a})}{f}_{a}^{2}\left\langle {\theta }_{i}^{2}f({\theta }_{i})\right\rangle \chi ,\end{eqnarray}$
with the entropy density s(T), the present cosmic microwave background (CMB) temperature T0, the initial misalignment angle θi, the numerical factor χ ≃ 1.44, and the anharmonic factor f(θi) is given by [57] $\begin{eqnarray}f({\theta }_{i})\simeq {\left[\mathrm{ln}\left(\displaystyle \frac{e}{1-{\theta }_{i}^{2}/{\pi }^{2}}\right)\right]}^{1.16}.\end{eqnarray}$
Then, we can derive the present QCD axion abundance $\begin{eqnarray}\begin{array}{rcl}{{\rm{\Omega }}}_{a}{h}^{2} & \simeq & 0.14{\left(\displaystyle \frac{{g}_{* }({T}_{a})}{61.75}\right)}^{-0.42}\left(\displaystyle \frac{{g}_{* s}({T}_{0})}{3.94}\right)\\ & & \times \,{\left(\displaystyle \frac{{f}_{a}}{{10}^{12}\,\mathrm{GeV}}\right)}^{1.16}\left\langle {\theta }_{i}^{2}f({\theta }_{i})\right\rangle ,\end{array}\end{eqnarray}$
where g*(T) and g*s(T) are the numbers of effective degrees of freedom of the energy density and the entropy density, respectively, and h ≃ 0.68 is the reduced Hubble constant. In order to account for the observed cold DM abundance, the initial misalignment angle should be ${\theta }_{i}\sim { \mathcal O }(1)$ for the scale ${f}_{a}\sim { \mathcal O }({10}^{12})\,\mathrm{GeV}$.The QCD axion bubbles can be generated due to an explicit PQ symmetry breaking after inflation. There are many mechanisms for this explicit PQ symmetry breaking in the early Universe [58–68]. The axion field acquires the effective potential, which can be approximated as7 ) is a small initial misalignment angle8 ), we show TB as a function of the effective angle θeff in figure 1. Two typical values of the scales fa = 1 × 1016 GeV (red) and 1 × 1017 GeV (blue) are selected for comparisons. We find that the temperature ${T}_{B}\,\sim { \mathcal O }(0.1-1)\mathrm{MeV}$ decreases slowly as the effective angle θeff decreases, and varies most rapidly when the angle approaches a small value.
$\begin{eqnarray}{V}_{{/}\!\!\!\!{\mathrm{PQ}}}(\phi )\simeq \displaystyle \frac{1}{2}{m}_{a}^{2}(\phi ){\left(\phi -{\phi }_{\min }^{n}\right)}^{2},\end{eqnarray}$
where ma(φ) is the axion effective mass, ${\phi }_{\min }^{n}$ is the potential minimum, and n is an integer. The explicit PQ symmetry is assumed to be temporarily broken with a large scale ${f}_{a}\sim { \mathcal O }({10}^{16})\,\mathrm{GeV}$. The axion acquires quantum fluctuations and settles down into the minimum ${\phi }_{\min }^{n}$. The QCD axion bubbles are generated when the conventional axion potential VQCD(φ) arises during the QCD phase transition. Therefore, one of the minimum ${\phi }_{\min }^{0}$ of ${V}_{{/}\!\!\!\!{\mathrm{PQ}}}(\phi )$ should be near the minimum of VQCD(φ), ensuring the correct cold DM abundance with the effective initial angle θi,0. In addition, if the initial value is greater than a critical value φcrit, the axion will be stabilized at the another minimum ${\phi }_{\min }^{1}$ with θi,1. Here, we suppose that the QCD axion bubbles are produced at ${\phi }_{\min }^{1}$, i. e. , θi,1 > θi,0, we define the axion bubble effective angle θeff ∈ (0, π]. Then, the effective initial misalignment angle are given by $\begin{eqnarray}{\theta }_{i,n}=\left\{\begin{array}{ll}0-{\theta }_{i}, & n=0\\ {\theta }_{\mathrm{eff}}-{\theta }_{i}, & n=1\end{array}\right.\end{eqnarray}$
corresponding to the potential minima ${\phi }_{\min }^{0}$ and ${\phi }_{\min }^{1}$, respectively. Note that θi in equation ( $\begin{eqnarray}\begin{array}{rcl}{\theta }_{i} & \simeq & 4.29\times {10}^{-3}{\left(\displaystyle \frac{{g}_{* }({T}_{a})}{61.75}\right)}^{0.21}{\left(\displaystyle \frac{{g}_{* s}({T}_{0})}{3.94}\right)}^{-1/2}\\ & & \times \,{\left(\displaystyle \frac{{f}_{a}}{{10}^{16}\,\mathrm{GeV}}\right)}^{-0.58}.\end{array}\end{eqnarray}$
When the axions dominate the radiation in the bubbles, the cosmic background temperature is defined as TB [53]. The axion energy density at TB is equal to the radiation energy density, ρa(TB) = ρR(TB), where ρR is the radiation energy density. Since the axion energy density inside the bubbles at TB can be described by $\begin{eqnarray}\begin{array}{rcl}{\rho }_{a}({T}_{B}) & = & \displaystyle \frac{{m}_{a,0}{m}_{a}({T}_{a})s({T}_{B})}{2s({T}_{a})}\\ & & \times \,{f}_{a}^{2}\left\langle {\left({\theta }_{\mathrm{eff}}-{\theta }_{i}\right)}^{2}f({\theta }_{\mathrm{eff}}-{\theta }_{i})\right\rangle \chi ,\end{array}\end{eqnarray}$
then we can derive the temperature TB $\begin{eqnarray}\begin{array}{rcl}{T}_{B} & \simeq & 2.13\times {10}^{-2}\,\mathrm{MeV}\\ & & \times \,{\left(\displaystyle \frac{{g}_{* }({T}_{a})}{61.75}\right)}^{-0.42}{\left(\displaystyle \frac{{f}_{a}}{{10}^{16}\,\mathrm{GeV}}\right)}^{1.16}\\ & & \times \,\left\langle {\left({\theta }_{\mathrm{eff}}-{\theta }_{i}\right)}^{2}f({\theta }_{\mathrm{eff}}-{\theta }_{i})\right\rangle \chi .\end{array}\end{eqnarray}$
Using equation (
Figure 1. The temperature TB as a function of the axion bubble effective angle θeff. The red and blue lines represent the scales fa = 1 × 1016 GeV and 1 × 1017 GeV, respectively. |
The QCD axion bubbles abundance is related to the inflationary fluctuations. The volume fraction of the bubbles can be described by [53]
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{d}}\beta }{\mathrm{dln}k}\simeq \displaystyle \frac{{\phi }_{\mathrm{crit}}-{\phi }_{i}}{2}P(\mathrm{ln}(k/{k}_{* }),{\phi }_{\mathrm{crit}}),\end{array}\end{eqnarray}$
where φcrit is the critical value, k is the wave number, k* = 0.002 Mpc−1, and $P(\mathrm{ln}(k/{k}_{* }),\phi )$ is the probability density function $\begin{eqnarray}\begin{array}{rcl}P(\mathrm{ln}(k/{k}_{* }),\phi ) & = & \displaystyle \frac{1}{\sqrt{2\pi }\sigma (\mathrm{ln}(k/{k}_{* }))}\\ & & \times \,\exp \left(-\displaystyle \frac{{\left(\phi -{\phi }_{i}\right)}^{2}}{2{\sigma }^{2}(\mathrm{ln}(k/{k}_{* }))}\right),\end{array}\end{eqnarray}$
where $\sigma (\mathrm{ln}(k/{k}_{* }))={H}_{\inf }\sqrt{\mathrm{ln}(k/{k}_{* })}/(2\pi )$ is the variance, $P(\mathrm{ln}(k/{k}_{* }),\phi )=\delta (\phi -{\phi }_{i})$ is the initial condition, and ${H}_{\inf }$ is the Hubble parameter during inflation. Here we take ${\phi }_{\mathrm{crit}}-{\phi }_{i}=4.5{H}_{\inf }$ as a benchmark.3. SMBHs from the QCD axion bubbles
The astronomical observations indicate that SMBHs are ubiquitous in the center of galaxies with a mass ∼109M⊙ at the redshift z ∼ 7 (∼0.76 Gyr) [8–10]. However, the origin of such BHs is still a mystery. The mass of an accreting BH with the time t is given by [12]
$\begin{eqnarray}M(t)={M}_{0}\exp \left(\displaystyle \frac{1-{\epsilon }_{r}}{{\epsilon }_{r}}\displaystyle \frac{t}{{t}_{E}}\right),\end{eqnarray}$
where M0 is the seed BH mass, εr ≃ 0.1 is the radiative efficiency, and tE ≃ 0.45 Gyr. In this case, a seed BH with the mass ${M}_{0}\sim { \mathcal O }({10}^{2}-{10}^{5}){M}_{\odot }$ would take at least ∼0.5 Gyr to grow up to a ∼109M⊙ SMBH [13]. Therefore, the SMBHs must either have the heavy seeds or have a primordial origin.Here, we investigate the PBHs as the seeds of SMBHs, which are produced from the QCD axion bubbles after the QCD phase transition. The initial PBH mass at the formation temperature Tf is [69]10 ) and (14 ), we can derive the minimum PBH mass in the axion bubbles scenario
$\begin{eqnarray}{M}_{\mathrm{PBH}}=\displaystyle \frac{4\pi }{3}\displaystyle \frac{\gamma {\rho }_{R}({T}_{f})}{{H}^{3}({T}_{f})},\end{eqnarray}$
where γ ≃ 0.2 is the gravitational collapse factor [36], and H(Tf) is the Hubble parameter at Tf. Considering the PBHs triggered by QCD axion bubbles, they will be produced when the axions dominate the radiation in the bubbles (Tf ≲ TB), and the bubble size is larger than the horizon size. Using equations ( $\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{M}_{\mathrm{PBH}}^{\min }}{{M}_{\odot }} & \simeq & 6.58\times {10}^{7}\left(\displaystyle \frac{\gamma }{0.2}\right){\left(\displaystyle \frac{{g}_{* }({T}_{a})}{61.75}\right)}^{0.84}\\ & & \times \,{\left(\displaystyle \frac{{g}_{* }({T}_{f})}{10.75}\right)}^{-1/2}{\left(\displaystyle \frac{{f}_{a}}{{10}^{16}\,\mathrm{GeV}}\right)}^{-2.33}\\ & & \times \,{\left(\left\langle {\left({\theta }_{\mathrm{eff}}-{\theta }_{i}\right)}^{2}f({\theta }_{\mathrm{eff}}-{\theta }_{i})\right\rangle \chi \right)}^{-2}.\end{array}\end{eqnarray}$
We show ${M}_{\mathrm{PBH}}^{\min }$ as a function of the bubble effective angle θeff in figure 2. The red and blue lines represent fa = 1 × 1016 GeV and 1 × 1017 GeV, respectively. For fa = 1 × 1016 GeV, we find the minimum PBH mass is roughly at $\sim { \mathcal O }({10}^{4}-{10}^{7}){M}_{\odot }$, which is shown with the gray shadow region. This mass region would be smaller for fa = 1 × 1017 GeV. Considering the PBHs as the seeds of SMBHs, the numerical simulations show that PBHs with the mass $\sim { \mathcal O }({10}^{4}-{10}^{5}){M}_{\odot }$ could subsequently grow up to the $\sim { \mathcal O }({10}^{9}){M}_{\odot }$ SMBHs [70]. In this case, the PBH mass larger than a critical value Mc is considered as the seeds of the SMBHs $\begin{eqnarray}{M}_{c}=\left({10}^{4}-{10}^{5}\right){M}_{\odot }.\end{eqnarray}$
Therefore, the seed PBH mass in the QCD axion bubbles scenario is $\begin{eqnarray}{M}_{s}=\mathrm{Max}\left[{M}_{\mathrm{PBH}}^{\min },{M}_{c}\right].\end{eqnarray}$
Figure 2. The minimum PBH mass ${M}_{\mathrm{PBH}}^{\min }$ (in the solar mass M⊙) as a function of the angle θeff. The red and blue lines represent fa = 1 × 1016 GeV and 1 × 1017 GeV, respectively. The gray shadow region represents the PBH mass region $\sim { \mathcal O }({10}^{4}-{10}^{7}){M}_{\odot }$. |
Then, we discuss the PBH abundance. The energy density of the PBH at the temperature TB is given by11 ), we show the distributions of $\mathrm{log}({f}_{\mathrm{PBH}})$ in the parameters $\{\mathrm{log}({M}_{\mathrm{PBH}}/{M}_{\odot }),{\theta }_{\mathrm{eff}}\}$ plane in figure 3. The left and right panels correspond to the scales fa = 1 × 1016 GeV and 1 × 1017 GeV, respectively. For the same θeff, the value of fPBH decreases as the PBH mass MPBH increases. Note that the blank regions in the plots represent the limits set by ${M}_{\mathrm{PBH}}^{\min }$. For ${f}_{a}\sim { \mathcal O }({10}^{16})\,\mathrm{GeV}$, the PBHs at the mass region $\sim { \mathcal O }({10}^{4}-{10}^{7}){M}_{\odot }$ may account for the seeds of SMBHs. While for ${f}_{a}\sim { \mathcal O }({10}^{17})\,\mathrm{GeV}$, we note that the PBH fractional abundance at the mass region $\sim { \mathcal O }(10-{10}^{4}){M}_{\odot }$ is strongly constrained by the CMB anisotropies measured by Planck [71].
$\begin{eqnarray}{\rho }_{\mathrm{PBH}}({T}_{B})=\displaystyle \frac{3}{4}{T}_{B}s({T}_{B})\displaystyle \frac{{\rm{d}}\beta }{\mathrm{dln}k}.\end{eqnarray}$
The fractional abundance of the PBH at present can be described by $\begin{eqnarray}{f}_{\mathrm{PBH}}=\displaystyle \frac{3}{4}{T}_{B}s({T}_{0})\displaystyle \frac{{\rm{d}}\beta }{\mathrm{dln}k}\displaystyle \frac{1}{{\rho }_{c}}\displaystyle \frac{1}{{{\rm{\Omega }}}_{\mathrm{DM}}},\end{eqnarray}$
where ΩDM ≃ 0.268 is the total cold DM abundance, and ρc is the critical energy density. Using equation (
Figure 3. The distributions of the PBH fractional abundance $\mathrm{log}({f}_{\mathrm{PBH}})$ in the $\{\mathrm{log}({M}_{\mathrm{PBH}}/{M}_{\odot }),{\theta }_{\mathrm{eff}}\}$ plane for fa = 1 × 1016 GeV (left) and 1 × 1017 GeV (right). The shadow regions represent the values of $\mathrm{log}({f}_{\mathrm{PBH}})$. The blank regions represent the limits set by ${M}_{\mathrm{PBH}}^{\min }$. |
Finally, we comment on the SMBHs formation from the baryon bubbles in the Affleck–Dine baryogenesis [72]. The SMBH formation in the QCD axion bubbles scenario has some similarities with that mechanism. They considered the high density baryon bubbles formation from the modified Affleck–Dine mechanism with Hubble induced mass and the finite temperature effect. The PBHs will be formed when the baryon bubbles enter the horizon with the sufficiently large densities. They also show that the PBHs produced by that mechanism may have a reasonable mass as the seeds of SMBHs. In addition, we note that a recent discussion about the SMBHs formation from the QCD axion in [73], they considered the attractive self-interaction of QCD axion DM to form the SMBHs.
4. Conclusion
In summary, we have investigated SMBH formation from the QCD axion bubbles. The PBHs generated from the QCD axion bubbles are considered as the seeds of SMBHs. The QCD axion bubbles can be generated due to an explicit PQ symmetry breaking after inflation. In this work, we do not discuss the specific mechanism of the explicit PQ symmetry breaking. We consider a general case in which the QCD axion bubbles are formed with the bubble effective angle θeff. The PBHs triggered by axion bubbles will be formed when the axions dominate the radiation in the bubbles, which leads to a minimum PBH mass ${M}_{\mathrm{PBH}}^{\min }$. We show the distributions of the temperature TB with different θeff for the scales fa = 1 × 1016 GeV and 1 × 1017 GeV, and also the resulting minimum PBH mass. In order to explain the SMBHs in the galactic centers with the mass ∼109M⊙ at z ∼ 7 with the PBH scenario, the PBH mass should be larger than a critical value $\sim { \mathcal O }({10}^{4}-{10}^{5}){M}_{\odot }$. The PBH fractional abundance distributions in the $\{\mathrm{log}({M}_{\mathrm{PBH}}/{M}_{\odot }),{\theta }_{\mathrm{eff}}\}$ plane are also shown, which is strongly limited by ${M}_{\mathrm{PBH}}^{\min }$. We find that the PBHs triggered by QCD axion bubbles with the mass $\sim { \mathcal O }({10}^{4}-{10}^{7}){M}_{\odot }$ for the scale ${f}_{a}\sim { \mathcal O }({10}^{16})\mathrm{GeV}$ may account for the seeds of SMBHs. The SMBHs formation in the QCD axion bubbles scenario has some similarities with that in the baryon bubbles scenario from the modified Affleck–Dine baryogenesis.