IPMU 170069
False Vacuum Decay
[.5em] Catalyzed by Black Holes
Kyohei Mukaida and Masaki Yamada
Kavli IPMU (WPI), UTIAS,
The University of Tokyo, Kashiwa, Chiba 2778583, Japan
Institute of Cosmology, Department of Physics and Astronomy,
Tufts University, Medford, MA 02155, USA
False vacuum states are metastable in quantum field theories, and true vacuum bubbles can be nucleated due to the quantum tunneling effect. It was recently suggested that an evaporating black hole (BH) can be a catalyst of bubble nucleations and dramatically shortens the lifetime of the false vacuum. In particular, in the context of the Standard Model valid up to a certain energy scale, even a single evaporating BH may spoil the successful cosmology by inducing the decay of our electroweak vacuum. In this paper, we reinterpret catalyzed vacuum decay by BHs, using an effective action for a thinwall bubble around a BH to clarify the meaning of bounce solutions. We calculate bounce solutions in the limit of a flat spacetime and in the limit of negligible backreaction to the metric, where it is much easier to understand the physical meaning, and compare these results with the full calculations done in the literature. As a result, we give a physical interpretation of the enhancement factor: it is nothing but the probability of producing states with a finite energy. This makes it clear that all the other states such as plasma should also be generated through the same mechanism, and calls for finitedensity corrections to the tunneling rate which tend to stabilize the false vacuum. We also clarify that the dominant process is always consistent with the periodicity indicated by the BH Hawking temperature after summing over all possible remnant BH masses or bubble energies, although the periodicity of each bounce solution as a function of a remnant BH can be completely different from the inverse temperature of the system as mentioned in the previous literature.
Contents
1 Introduction and Summary
The discovery of the Higgs boson has established the Standard Model (SM) [1, 2]. For the current center value of the SM parameters, especially the measured top and Higgs masses [3, 4, 5, 6, 7], the Higgs potential develops a lower energy state than the electroweak vacuum at around the intermediate scale well below the Planck [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. This fact implies that the quantum tunneling might lead to a disastrous decay of our vacuum [21, 22]. Fortunately, it is known that, for the current favored value of SM parameters, its lifetime far exceeds the present age of the Universe, and thus our electroweak vacuum is believed to be metastable in the context of SM valid up to a very high energy scale.
Though this argument guarantees the safety of our vacuum in the present Universe, it does not mean that our metastable vacuum can survive throughout the history of the Universe. Thus, this scenario could in principle contradict various cosmological phenomena which can drive the vacuum decay. Many studies have been performed from this viewpoint. For instance, in the early Universe, it is believed that the Universe was filled with thermal plasma composed of SM particles. Since the Higgs interacts with SM particles including the Higgs itself, the thermal fluctuations might activate the decay of our vacuum [23, 8, 11, 12] while these relativistic particles tend to stabilize the Higgs at the same time. It has been shown that this effect does not spoil our Universe for the best fit values of SM parameters [14, 15, 16]. If we further go back through the history of the Universe, we may encounter the phase of inflation and the subsequent (p)reheating. Since light fields with masses smaller than the Hubble parameter acquire fluctuations proportional to the Hawking temperature during inflation [24, 25, 26, 27, 28, 29], the Higgs might overcome the potential barrier via the HawkingMoss instanton [30], which may be interpreted as the thermal hopping owing to the Hawking temperature. This observation puts a severe bound on the Hubble parameter of inflation: with being the instability scale [14, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Although one might think that this constraint can be ameliorated by introducing a small coupling between the inflaton and the Higgs field, recent studies reveal that the (p)reheating stage after inflation could drive the catastrophic decay because the very interaction activates Higgs fluctuations due to the oscillating inflaton [45, 46, 47, 48, 49, 50].
Recently, it was pointed out that a black hole (BH) can be a nucleation site just like a boiling stone in a superheated liquid system, and the vacuum transition rate can be dramatically enhanced (or the potential barrier becomes effectively smaller) around the BH [51, 52, 53] (see Refs. [54, 55, 56, 57, 58, 59] for earlier work). The result of their calculation is independent of the periodicity of the Wickrotated time coordinate, so that they insist that the result can be applied to an arbitrary low temperature system. In particular, the enhancement gets more significant for a smaller BH, and in an extreme case, the Higgs can classically overcome the potential barrier, such as the thermal hopping and the HawkingMoss transition. Applying this result to the Higgs field, they concluded that even a single small BH that evaporates within the current age of the Universe leads to the disaster of our vacuum [60].^{1}^{1}1 See also Refs. [61, 62] for related work. And thus, there should not be such a small BH in our observable Universe. Although such a small BH may not be formed in the usual scenario of cosmology, their conclusion puts a stringent constraint on some cases, such as the formation of primordial BHs in the early Universe [63].^{2}^{2}2 The bubble nucleation process is also important in the context of the multiverse, where bubbles continuously nucleate and observers may live in the baby universes [64, 65, 66, 67, 68, 69, 70, 71]. The enhancement effect of the nucleation rate is applied in Ref. [72] to generate baby universes around BHs.
In this paper, we reinterpret the earlier results derived via a Wickrotated Euclidean field theory in Refs. [51, 52, 53] by invoking an effective action for a thinwall bubble that can also describe the vacuum transition in scalar field theories [55, 56, 73]. We start with the bubble nucleation at a finite temperature in the flat spacetime, and recall that the final bubble nucleation rate can be factorized into the probability of producing states with a finite energy times a tunneling rate of a finite energy. This is also true if we have a BH. By extending the same procedure, we reformulate the bubble nucleation rate around a BH in the case where the backreaction of the bubble on the spacetime can be neglected.^{3}^{3}3 This situation is practically important for realistic applications, for instance, to study the metastable Higgs vacuum [52].
We compare the bubble nucleation rate computed in this way with the full gravitational one in the limit of negligible backreaction. As a result, we clarify the meaning of the enhancement factor, that is, a probability of producing states with a finite energy , which is a Boltzmann factor . It is hard to imagine that a BH only activates bubbles since quantum field theory has many other degrees of freedom to be excited. Hence, we expect all the states with a finite energy should also be generated by the same mechanism. This argument clarifies the need of finitedensity corrections to the bubble nucleation rate regardless of its origin, namely whether or not the Universe is filled by the plasma of the BH Hawking temperature, though the size of corrections depends on it.^{4}^{4}4 See also Sec. 4.
We also confirm that the periodicity of each bounce solution as a function of is not necessarily related to the temperature of the system. However, after summing over all the possible transitions as a function of , we find that the dominant process is always consistent with the periodicity indicated by the temperature of the system. This observation also holds if we have a BH. Although, one still cannot determine the question raised in Ref. [63], whether or not the thermal plasma fills the whole Universe, by only looking at the periodicity of bounce solutions, our procedure indicates that the heart of the problem is free from a BH. The problem is whether or not a finitevolume heat reservoir can emit bubbles whose size is much larger than the size of the reservoir. We leave this issue as an open question.
The following is the summary of our results of this paper:

In a flat spacetime with a finitetemperature plasma, we have shown and , where is the exponent of the quantum tunneling rate, is the energy of the bubble, and is the periodicity of the bubble solution. Since the exponent of the Boltzmann factor satisfies , the inequality implies that the dominant process is given either by or , where is the sphaleron energy.

In the Schwarzschild–de Sitter spacetime, we eventually find and , where is the Hawking temperature associated with the remnant BH and and are the exponents of the quantum tunneling rate coming from the boundary of BH and the other contributions, respectively. The difference of the BH mass before and after the transition, , is equal to the bubble energy by the conservation of energy. We also show that coincides with in the limit where the bubble radius is much larger than the BH radius once we identify the temperature as the Hawking temperature. In particular, in that limit.

In the fixedbackground Schwarzschild–de Sitter spacetime with finitetemperature effects, we again obtain . The behavior of the second derivative is similar to the above full calculation. In the case that the effect of the change of the metric by the bubble is negligible, the nucleation rate coincides with the one derived by the above full calculation only if we identify the temperature of the system as the Hawking temperature of the BH. This observation clarifies that the enhancement factor is nothing but the probability of generating states with a finite energy, which is the Boltzmann factor with a BH Hawking temperature.
This paper is organized as follows. In Sec. 2, we first review the calculation of the tunneling rate for a thinwall bubble in a scalar field theory. We show that the transition rate is dominated either by a vacuum transition without an excited energy or by a sphaleron transition in this system. Next, we take into account gravity and consider the vacuum transition in the Schwarzschildde Sitter spacetime in Sec. 3. In particular, we calculate the bubble energy dependence of transition rate and show that its behavior is similar to the one in a finitetemperature system in a flat spacetime. We also use the effective action for the thinwall bubble and show that the same nucleation rate in the literature can be derived by the thermal activation of the BH Hawking temperature in a certain limit. Section 4 is devoted to the conclusion and discussion, where we briefly explain the physics behind our result and discuss the possibility that the cost of such thermal plasma may significantly reduce the bubble nucleation rate.
2 Transition without gravity
In this section, we review the calculation of the transition rate from a false vacuum to a true vacuum in quantum field theory without gravity, i.e., in the limit of , where is the Newton constant and ( is the Planck scale. We take gravity into account in Sec. 3.
2.1 Tunneling from a false vacuum
The action is given by
(2.1) 
where is a potential for the scalar field, which has a false vacuum at and the true vacuum at .
The lifetime of the false vacuum can be calculated from the path integral as follows:
(2.2)  
(2.3)  
(2.4)  
(2.5)  
(2.6) 
with , where the path integral is performed under the boundary conditions of and . The subscript “bubble" in means that it is a bubble configuration with a certain radius as we specify below. The denominator in the first line comes from the normalization of the initial and final states and gives the factor in the last line, where the subscript “M” indicates this action is defined in Minkowski spacetime.
Now we take the imaginary time and rewrite Eq. (2.5) in terms of the Euclidean path integral. In the saddle point approximation in Euclidean theory, the path integral is approximated by , which is calculated from a bounce solution to the classical Euclidean equation of motion. The action is minimized by a solution where bounces only once. In addition to the single bounce solution, there is an infinite number of solutions where bounces many times, which may be summed in the dilute gas approximation. Then we obtain
(2.7) 
where is a prefactor that is not important for our discussion. Thus we obtain
(2.8)  
(2.9) 
where the Euclidean action is given by
(2.10) 
and the normalization factor is given by
(2.11) 
Let us emphasize that the action is calculated from the bounce solution under the boundary condition of at , and the tunneling process corresponds to a transition from to , i.e., a transition from the metastable ground state to the state at the other side of the potential barrier . Although we mention here the contribution from the perturbation , it does not usually contribute to the exponential factor.^{5}^{5}5 If the system couples with light degrees of freedom, the prefactors could be significant [74, 75, 76, 77, 78]. In other words, we have to be careful what the “treelevel” action is in computing the bounce. Throughout this paper, we do not consider this issue further, and simply assume that we somehow know the treelevel action that is appropriate to compute the bounce.
The bounce solution obeys the Euclidean equation of motion that is given by the variational principle of in terms of :
(2.12)  
(2.13) 
where is the Laplacian. In quantum field theory, the degrees of freedom is infinity because of the spacial dependence of the field. In many cases, however, we can use some symmetries to reduce the degrees of freedom to unity.
The Euclidean action has an O(4) symmetry in quantum field theory, so let us first focus on O(4) symmetric solutions. In the thinwall approximation, the scalar field configuration is approximated by the following O(4) symmetric configuration:
(2.14) 
where () is the radial coordinate of the Euclidean spacetime and is the radius of the bubble. Note that this is an instanton solution where the time variable runs from to and the configuration is nontrivial in a small interval .
The number of degrees of freedom is reduced to be unity by the O(4) symmetry, so that we can consider a onedimensional system with the variable . Plugging the O(4) symmetric thinwall configuration back into the action, we can express the action as a function of the bubble radius ,
(2.15) 
where the first term comes from the contribution inside the bubble, while the second term comes from the surface of the bubble. We define the energy density difference as and the surface energy density of bubble as
(2.16) 
The variable obeys a constraint that originates from the Euclidean equation of motion. The same equation can be derived from the variational principle of in terms of the variable . Then we find the following results:
(2.17)  
(2.18) 
We find
(2.19) 
It is instructive to consider the same theory with an O(3) spherical symmetric assumption, which can be generalized to calculate transition rates in a finite temperature. In the thinwall approximation, the worldsheet metric is written as (see e.g., Ref. [73])
(2.20) 
The action for the domain wall is given by the worldsheet area in addition to the difference of potential energy inside and outside bubble, so that we obtain
(2.21) 
where may be regarded as the gamma factor of the domain wall. The first term is the surface term, which contains the kinetic energy of bubble, and the second term comes from the contribution inside the bubble. The conserved energy can be derived from
(2.22) 
where the Lagrangian can be read from the above action. This is rewritten as
(2.23)  
(2.24) 
In the second equality, we use the fact that the initial energy is zero. Taking the imaginary time , this can be rewritten as
(2.25) 
The bounce solution is given by
(2.26) 
so that we obtain the value of the Euclidean bounce action as
(2.27) 
Noting that , these results are consistent with the above results using the O(4) symmetric assumption.
2.2 Tunneling with a finite energy
Now we can consider a transition from an excited state around the false vacuum by using the O(3) approximation and the thinwall approximation. As one can see from Eq. (2.23) and Fig. 1, a state with a finite energy allows an O(3) symmetric bubble with , whose radius is or . The amplitude of the transition from a bubble with at to another one with at , which is going to expand, is obtained from
(2.28) 
The transition rate is then given by
(2.29)  
(2.30)  
(2.31) 
where the last path integral is performed under the boundary conditions such that at , at , and at . By using the saddlepoint approximation, the path integral is replaced by the dominant contribution where is the bounce solution obeying the Euclidean equation of motion with the boundary condition of . The amplitude is normalized by . As a result, the transition rate can be expressed as
(2.32) 
where comes from the normalization. Note again that is obtained from the Euclidean equation of motion from to while is an O(3) symmetric bubble with a fixed radius .
The bounce with a finite energy satisfies
(2.33) 
where is the initial energy. There are two solutions with a vanishing wall velocity , and , which are obtained from as can be seen from Fig. 1. Note here that the initial energy should be small enough to have these two solutions. The critical energy, above which we do not have solutions to Eq. (2.23), is obtained from the condition . The critical solution is
(2.34)  
(2.35) 
Here, the subscript “sp” indicates that this is nothing but the sphaleron, as we will see in the next Sec. 2.3. When we regard as a position variable of a particle in a onedimensional system, the constraint Eq. (2.33) can be rewritten as the following conservation law of “energy":
(2.36)  
(2.37) 
Plugging the solutions into Eq. (2.21), we get the bubble nucleation rate for ,
(2.38)  
(2.39) 
where we use . Once we regard the factor as the effective mass of the bubble, the result is similar to that in the onedimensional quantum mechanical system.
Note again that the above transition means that a bubble with a radius , which is not , tunnels into the one with a radius . Hence, we need to specify the way to excite the initial state to the bubble with the radius . If such bubbles are continuously produced and collapse in the initial state with a finite probability, the vacuum decay rate may be expressed as the probability of creating bubbles with times the probability of tunneling from to , namely . Here, note that we first assume the field configuration as Eq. (2.14) and reduce the number of degrees of freedom to unity. Since there are infinite degrees of freedom for the scalar field, it is generally difficult to give the energy so that all the energy is converted to such a macroscopic configuration, that is, the initial bubble with a radius . The thermal state is an example that we have such an excited initial condition naturally. In this case, all degrees of freedom have a typical energy of order with the Boltzmann weight, and hence the probability of creating the initial bubble is nothing but and is nonzero.
2.3 Tunneling with a thermal energy
Now we can consider a transition in the scalar field theory in a thermal background with a temperature of . The transition rate can be calculated by the integral of the Boltzmann factor times quantum tunneling rate (see e.g., Ref. [79]):
(2.40)  
(2.41)  
(2.42) 
where is the classical transition rate. Note that, for , the bubble nucleation rate is unity, , where is the sphaleron energy as explained below.
Let us evaluate the transition rate approximately. As discussed in the case of onedimensional quantum mechanics, the question traces back to the behavior of . If for , one may evaluate the integral via the steepest descent method by expanding the exponent as follows:
(2.43) 
If one finds the solution to
(2.44)  
(2.45)  
(2.46) 
where is the time periodicity of the bounce solution, the integral may be approximated by the Gaussian. In this case the first integral is dominated by the energy :
(2.47)  
(2.48) 
where we use . This equation indicates that the transition rate can be decomposed into two parts as Eq. (2.47): the Boltzmann factor and the quantum tunneling rate. On the one hand, the quantum tunneling rate tells one that the bubble with , which is not equal to , tunnels to . On the other hand, the Boltzmann factor represents the probability that the bubble with energy reaches the point of . Therefore, in total, Eq. (2.48) gives the transition rate where the bubble goes from to via thermal excitation and then to via quantum tunneling. If there is no solution to Eq. (2.46), the integral is dominated by the boundary between Eq. (2.41) and Eq. (2.42), which ends up .
However, at least for the thermal transition in quantum field theory under the thinwall approximation with the O symmetry, the second derivative of the bounce action with respect to its energy is always . Let us first confirm this property. Figure 2 shows the bounce and its first derivative, , as a function of the energy . One can see that is an increasing function with respect to , and thus . Therefore, the saddle point is not a minimum of the exponent in Eq. (2.41). Rather, the integral is dominated by its edges, or . As a result, the transition rate can be expressed as
(2.49)  
(2.50)  
(2.51) 
Recalling the sphaleron energy and the bounce action , we can go further. It is clear that the sphaleron transition dominates the decay process when the temperature is large enough to satisfy . The threshold temperature is given by
(2.52)  
(2.53) 
Thus we conclude that the transition rate is summarized as
for ,  (2.54)  
for .  (2.55) 
Before closing this section, we would like to explain the relation between the above results and the wellknown method of putting the theory on (i.e., periodic Wickrotated time ) and evaluating the imaginary part of the free energy [23]. In this case, the geometry forces all the configurations to be periodic, including the bounce. The quantum one, Eq. (2.54), corresponds to the dominant periodic instanton. Interestingly, though there exist other branches of periodic instantons with a finite energy , the vacuum one, , dominates for the thinwall approximation under the O symmetry. Note here that, since is much larger than the radius of the vacuum bubble as can be seen from Eq. (2.53), the vacuum bubble may be embedded in for . The classical one, Eq. (2.55), is obtained by the dimensional reduction of , which is a good approximation if the energy scale of the bubble is much smaller than the temperature. It is clear that the static solution can be embedded in the periodic spacetime.
3 Transition in the Schwarzschild–de Sitter spacetime
Gravity changes the spacetime in accordance with finiteenergy objects. In the case of our interest, not only the BH but the bubble could be the origin of such distortions. Roughly speaking, in the context of the vacuum decay, the change of the spacetime affects the cost to nucleate the bubble. One can guess that gravity should modify the vacuum decay rate. Therefore, in this section, we switch on gravity and investigate its effect on the vacuum decay. The action is given by [80]
(3.1)  
(3.2)  
(3.3) 
where is the determinant of the metric and is the Ricci scalar. We take the Planck unit, () (i.e., the Newton constant is taken to be ), unless otherwise stated. The curvature scalar contains terms with second derivatives, which can be removed by integration by parts and write the action only by first derivatives so that we can use the path integral approach in the gravitational theory. As a result, the boundary term arises, which is the integral of the trace of the second fundamental form of the boundary [81]:
(3.4) 
where is the determinant of the threedimensional metric on the boundary surface, is the volume of the boundary, and is the unit normal. Note that the region of boundary depends on the metric, and hence the boundary term is determined only after we specify the metric.
In particular, we consider a bubble nucleation in (anti)de Sitter spacetime with a BH, which is described by the Schwarzschildde Sitter metric,
(3.5)  
(3.6) 
where is a BH mass and is a vacuum energy. The areas of the boundary, , at the surface of the BH () and at the cosmological horizon () are given by and in this metric, respectively, where is the inverse of the apparent horizon length related to the vacuum energy as . This metric respects only an O(3) symmetry. We use the O(3) symmetric assumption to find bounce solutions in the following analysis.
3.1 Bubble nucleation via tunneling
Here we illustrate how to evaluate the bounce action in the presence of a BH with gravitational backreaction following Refs. [51, 52, 53, 60]. Since we employ a thinwall approximation, we may use the Euclidean metric defined separately in the outer and inner regions of the bubble:
(3.7)  
(3.8) 
where is the initial BH mass and is the remnant BH mass after the bubble nucleation. The zeros of define the horizon of each patch. We have at most two zeros for each : the BH horizon and the de Sitter horizon for respectively. The natural periodicities to eliminate the conical deficits at each horizon are the following: at and at .
We now specify the setup of our interest. In the following, we focus on the case where we initially have a BH with . Also, we assume that the vacuum energy of the scalar field changes from to due to this transition. On the other hand, the remnant BH mass is taken to be an arbitrary parameter. We do not impose that the Hawking temperature of the initial black hole should coincide with that inferred from the periodicity of the bubble solution. This mismatch will cause a conical deficit at the BH horizon, and thus we have to cope with it appropriately as done in Ref. [51].
Before evaluating the bounce action, we briefly illustrate how to obtain the wall trajectory of the nucleated bubble with fixed , , and . The wall trajectory is parametrized by the proper time of a comoving observer of the wall,
(3.9) 
where the dot denotes the derivative with respect to the proper time . The Israel junction condition yields
(3.10) 
where is the tension of the bubble and
(3.11) 
We can explicitly rewrite it as
(3.12) 
The junction condition implies that the wall velocity has to satisfy the following conservation law of “energy”:
(3.13)  
(3.14) 
where and , and
(3.15) 
where and ().^{6}^{6}6 Note that the here is different from the gamma factor of the domain wall, , defined below. Once we fix the all the parameters and , we can obtain the wall trajectory as a function of the proper time, , in principle. In our setup, we have fixed and and hence we have a family of solutions as a function of the remnant BH mass, .
Now we are in a position to evaluate the Euclidean action by the solution to Eq. (3.13). The gravitational Euclidean action is given by
(3.16) 
where and represent the regions inside and outside of the bubble, respectively, and represents the boundary induced by the bubble. Note again that accounts for the boundaries at the horizons. The Einstein equation and the Israel junction condition imply that the action can be rewritten as
(3.17) 
In the thinwall approximation, the bubble Euclidean action is given by
(3.18)  
(3.19) 
If there is no de Sitter horizon, we have to drop the last term.
The bubble nucleation rate around the BH is calculated from
(3.20) 
where is the total Euclidean action and is the action without the bubble. If the gravitational action changes, the boundary term may also change. Assuming that the bubble is not as large as the de Sitter horizon, one can see that the boundary from the BH horizon only contributes to the difference. The explicit form of is given by
(3.21)  
(3.22)  
(3.23) 
where we use the Israel junction . The third term in is related to the contribution of the conical deficit. The term accounts for the change of the BH entropy which comes from the area of the boundary at the horizon. We regard this result as a function of for later convenience. In summary, the bubble nucleation rate from the initial state, (a BH of , vacuum energy of ), to the final state, (a BH of , vacuum energy ), is given by
(3.24) 
We emphasize that the initial mass of the BH , the initial and final energy densities , and the surface energy density of the bubble are determined by the initial conditions and the potential of the scalar field, while the remnant BH mass after the bubble nucleation is not yet determined in the above calculation. In the sense of the path integral approach, we should sum over all the nucleation rates in terms of the variable . Thus, we need to find a minimal value of the action in terms of ,
(3.25) 
As has been pointed out in Ref. [51], it is possible that the remnant BH mass is larger than the initial BH mass though it usually gives subdominant contributions. Here we comment on the lower bound on the integral Eq. (3.25). If is sufficiently large, there is a lower bound of , below which the “potential" is always larger than zero and there is no solution to Eq. (3.13). At the critical point, and the solution is static. See also the discussion below.
Let us take the variation of the bounce action with respect to so as to approximate the integral of Eq. (3.25). The boundary term gives
(3.26) 
By numerically solving the equation of motion and calculating the transition rate (See Figs. 3 and 4), we also find that the variation of the other terms satisfies
(3.27) 
Combining these results, we obtain
(3.28) 
The above result Eq. (3.28) is similar to the one obtained in the previous sections [see Eq. (2.46)]. In particular, if we require , we find the relation , which may imply that the transition is due to the thermal effect with Hawking temperature. However, the transition rate is not minimized at the saddle point unless as we discussed in the previous section. In fact, we numerically check that this condition is not always satisfied. We show an example in Fig. 3, where we assume , , and .^{7}^{7}7 Note that (). These parameters lead to and . Note that the dependence can be trivially factorized in our results such that , , and though we take as a reference value to plot them. In the left panel, monotonically decreases as increases and it is minimized at the maximal value of , which corresponds to the static solution. Note again that, if the seed BH mass, , is sufficiently heavy, there exists a maximal value of , above which we do not have solutions to Eq. (3.13) and at which the solution becomes static. In the right panel, increases as increases for moderately large , which implies that there is no saddle point at least in that range of .