We propose a new mechanism of primordial black hole formation via an aborted phase transition during the early matterdominated stage of reheating after inflation. In reheating induced by the decay of a pressureless fluid dominating the Universe at the end of inflation dubbed as reheaton the temperature of the radiation bath typically increases reaching a maximum temperature Trm max and then decreases. We consider a firstorder phase transition induced by the increase of the temperature that is aborted as Trm max is higher than the critical temperature but not sufficiently high for the bubble nucleation rate to overcome the expansion of the Universe. Although bubbles never fully occupy the space some may be nucleated and expand until the temperature once again decreases to the critical temperature. We argue that these bubbles shrink and disappear as the temperature drops further leaving behind macroscopic spherical regions with positive density perturbations. These perturbed regions accrete the surrounding matter reheatons and eventually collapse into primordial black holes whose mass continues to grow until the onset of radiation domination. We estimate the abundance of these primordial black holes in terms of the bubble nucleation rate at Trm max and demonstrate that the abundance can be significantly large from a phenomenological perspective.
['Astrophysics - Cosmology and Nongalactic Astrophysics', 'General Relativity and Quantum Cosmology', 'High Energy Physics - Phenomenology']
Primordial black holes from an aborted phase transition
{'Primordial black holes from an aborted phase transition': "Wen-Yuan Ai, 1, ∗ Lucien Heurtier, 1, † and Tae Hyun Jung 2, ‡  \n1 Theoretical Particle Physics and Cosmology,  \nKing's College London, Strand, London WC2R 2LS, UK  \n2 Particle Theory and Cosmology Group, Center for Theoretical Physics of the Universe,  \nInstitute for Basic Science (IBS), Daejeon, 34126, Korea  \nWe propose a new mechanism of primordial black hole formation via an aborted phase transition during the early matter-dominated stage of reheating after inflation. In reheating, induced by the decay of a pressureless fluid dominating the Universe at the end of inflation, dubbed as reheaton, the temperature of the radiation bath typically increases, reaching a maximum temperature T max , and then decreases. We consider a first-order phase transition induced by the increase of the temperature that is aborted as T max is higher than the critical temperature but not sufficiently high for the bubble nucleation rate to overcome the expansion of the Universe. Although bubbles never fully occupy the space, some may be nucleated and expand until the temperature once again decreases to the critical temperature. We argue that these bubbles shrink and disappear as the temperature drops further, leaving behind macroscopic spherical regions with positive density perturbations. These perturbed regions accrete the surrounding matter (reheatons) and eventually collapse into primordial black holes whose mass continues to grow until the onset of radiation domination. We estimate the abundance of these primordial black holes in terms of the bubble nucleation rate at T max , and demonstrate that the abundance can be significantly large from a phenomenological perspective.  \nIntroduction -Primordial black holes (PBHs) are black holes that form in the early Universe in a non-stellar way (see Ref. [1] for a recent review). Their possible existence throughout cosmic history has rich phenomenological implications [1-3] and a broad mass range of PBHs are compelling candidates for the dark-matter component of the Universe [4-6] that might be on the verge of being probed using solar ephemerides precision measurements [7, 8]. Moreover, PBHs could also explain a variety of conundrums, including the recently observed microlensing signal candidates, the correlations in the cosmic infrared and X-ray backgrounds, and the origin of the supermassive black holes in galactic nuclei at high redshift [9]. Moreover, it is possible that the LIGO/Virgo black hole mergers [10, 11] has a primordial origin [12].  \nSo far, most of the PBH formation mechanisms involved the gravitational collapse of large curvature perturbations generated during inflation. To generate such large curvature perturbations, the inflation model is required to have peculiar features, e.g., an inflection point or a plateau in a small field range of the potential [5, 1322], a potential hill [23-26], multiple phases of inflation or hybrid inflation [27-38], a non-canonical kinetic term [39, 40], multifield inflation [41, 42], light spectator fields [43-46], and other possibilities (e.g. [47-49]). In addition, PBH formation has been considered in connection with preheating after inflation [50-53] although their formation in this context was recently questioned [54].  \nLong after the idea was suggested in Refs. [55, 56], recent works reconsidered that PBHs may also be formed from a first-order phase transition (FOPT) [57-62]. This  \nidea was then further investigated in Refs. [63-77]. This possibility is particularly exciting, as FOPTs are naturally present in many particle physics models and have far-reaching phenomenological consequences, such as the emission of a stochastic gravitational wave background.  \nIn this Letter , we propose a new PBH formation mechanism in which an FOPT occurs while the Universe's temperature increases during reheating after inflation. This FOPT is thus a heating phase transition [78-80] rather than a cooling phase transition that occurs as the temperature decreases in the early Universe. The special ingredient of our scenario is an abortion of the FOPT assuming that the maximal temperature reached in reheating is higher than the critical temperature but lower than the temperature that guarantees the phase transition to complete. In the following, we introduce the specifics of the aborted phase transition, explain how PBH can form in this setup, and relate the PBH mass and abundance to the dynamics of the perturbative reheating and the phase transition sector considered.  \nReheating sector -Before going into the details, let us be clear in our setup. When inflation ends, we consider the Universe to be filled with a pressureless fluid slowly decaying into particles that quickly get thermalized, producing a relativistic plasma. We refer to this decaying matter component as the reheaton , χ . As χ decays, the radiation sector's temperature first increases, reaching the maximal temperature T max , and decreases as the Universe expands. The temperature evolution in terms of the scale factor a can be described by [81]  \nT ( a ) = c 1 T max [ ( c 2 a a max ) -3 2 -( c 2 a a max ) -4 ] 1 4 , (1)  \nwhere a max is the scale factor at T = T max , c 1 = 2 6 / 5 (27 1 / 5 5) -1 / 4 ≈ 1 . 30 and c 2 = 2 6 / 5 3 -2 / 5 ≈ 1 . 48.  \nFIG. 1. An example of the temperature dependence of scalar potential.  \n<!-- image -->  \nDenoting by Γ χ the decay width of the reheaton, matter domination lasts until the plasma temperature reaches the reheating temperature T RH ∼ √ Γ χ M Pl , with M Pl = 2 . 4 × 10 18 GeV being the reduced Planck mass, below which radiation domination starts. In general, there is no direct relation between T RH and T max , as the value of the latter depends on the time at which the reheating starts, so it is natural to assume that there is a large hierarchy between them.  \nAborted phase transition -Now, let us consider a real scalar field ϕ which breaks a symmetry 1 , spontaneously, by getting a nonzero vacuum expectation value. Assuming that the scalar sector undergoes an FOPT along the temperature change, one can define three characteristic temperatures that play an important role: the critical temperature, T c at which two local minima are degenerate, the spinodal (binodal) temperature T 1 ( T 2 ) above (below) which the potential barrier disappears (see Fig. 1 for the schematic description of thermal effective potential V T ( ϕ ) at each temperature). During inflation, the temperature is zero, and ϕ is stabilized in the symmetrybreaking vacuum assuming that the inflation scale is not too large compared to the curvature scale of the potential.  \nWhile the thermal bath is heated, the scalar potential V ( ϕ ) receives thermal corrections and there can be two types of phase transitions in general. During the change of T = 0 → T max > T c , the symmetry-restoring vacuum becomes more stable compared to the symmetrybreaking vacuum, and the phase transition occurs. This  \nphase transition is called symmetry-restoring , or heating phase transition (see, e.g. Ref. [78-80], for related discussions in various contexts). In previous studies, it is assumed that the heating phase transition is completed and that the Universe settles down in the symmetryrestoring phase. Then, as the temperature drops back, the symmetry-breaking vacuum becomes more stable again, and the symmetry-breaking (or cooling) phase transition starts at the bubble nucleation temperature.  \nOn the contrary, in this Letter , we assume that T max is greater than the critical temperature T c , but not large enough to make the bubble nucleation rate catch up with the spacetime expansion. Thus, the phase transition is aborted at T max by the temperature's turning around. Bubbles can still be formed, but since they never collide with each other, they just expand during T > T c and shrink back when T < T c . We argue that these bubbles eventually lead to PBH formation and that the abundance of such PBHs can be significant.  \nFate of bubbles in the aborted phase transition -Initially, the bubble grows since the free energy density difference, ∆ V T ≡ V T ( ϕ b ) -V T ( ϕ s ), is positive, where ϕ b and ϕ s denote the symmetry-breaking and restoring extrema of the thermal effective potential, respectively. Once the wall starts expanding, the perturbed plasma would backreact to the wall, creating a backreaction force P back ( v w ), which has a dependence on the wall velocity. In general, a terminal velocity exists and should be reached after a short acceleration period, determined by ∆ V T = P back ( v w ) [79, 80].  \nAs the temperature changes, ∆ V T also changes, so the wall velocity adiabatically follows the terminal velocity at each temperature; it reaches a maximal value at T max and decreases as T decreases. At T = T c , since the terminal velocity becomes v w = 0 by definition, this is the moment when the bubble stops expanding and has the largest comoving radius, which we denote as r c, 2 . The subscript c, 2 will be used to indicate quantities estimated at the critical temperature reached for the second time throughout this Letter . The critical temperature was reached for the first time during the temperature-increasing process, T = 0 → T max , for which we use the labeling of c, 1.  \nWe can estimate r c, 2 as r c, 2 = ∫ t c, 2 t nuc d t ' v w ( t ' ) /a ( t ' ) ∼ ¯ v ( η c, 2 -η nuc ) where η is the conformal time defined via d t = a d η , ¯ v denotes the averaged wall velocity in bubble expansion, and the subscript nuc indicates quantities estimated at the time when this bubble is nucleated. In a matter-dominated universe, we have H ( a ) ∝ a -3 / 2 and thus  \nr c, 2 ∼ ¯ v ( η c, 2 -η nuc ) = 2 ¯ v a c, 2 H c, 2 [ 1 -( a nuc a c, 2 ) 1 2 ] . (2)  \nThis shows that the comoving radius at t c, 2 is of the order of the comoving Hubble radius r H = 1 / ( aH ).  \nAfterwards, at T < T c , the net pressure ∆ V T becomes negative, and the bubble starts shrinking. The bubble wall velocity stays following its terminal (negative) velocity, which induces fluid motion of the plasma in this  \nFIG. 2. A schematic chronology of our PBH formation scenario. A symmetry-restoring bubble nucleates at around a max and expands with the bubble wall indicated by the blue line. At a c, 2 , the bubble wall stops expanding, turns around, and shrinks until it completely disappears at a zero . This leaves a spherical overdense region of macroscopic size (dashed blue line). This region accretes surrounding matter (reheaton), and the accretion collapses into a PBH via the post-collapse accretion mechanism at a BH . The PBH mass grows until the radiation domination starts at a RH .  \n<!-- image -->  \nregion. This shrinking occurs slightly below T c , while the vacuum energy difference is comparable to the pressure of the radiation plasma. Therefore, the bubble wall does not run away, as shown in more detail in the supplementary material, Section S2. In the absence of any runaway during both its expansion and contraction phases, the energy budget of the bubble wall's kinetic motion is negligible. To understand what happens in this region, we can thus focus on the balance between vacuum and thermal energy, where the latter should be understood to include the fluid's bulk motion. During bubble expansion, i.e. when T > T c , the thermal energy is first transferred into vacuum energy which redshifts slower than the radiation plasma with cosmic expansion. Later on, the bubble's contraction at T < T c converts vacuum energy back to thermal energy. Therefore, the energy density of the region perturbed by the wall's motion is greater than the unperturbed region far away from the nucleation site.  \nBecause the Universe is still matter-dominated as we assume T max ≳ T c ≫ T RH , the presence of such an overdensity easily leads to PBH formation via the postcollapse accretion mechanism [82-84] (see also Refs. [53, 84-96] for PBH formation during matter domination in a variety of different aspects). After the bubble completely disappears, the overdense region, with a macroscopic size as large as the comoving radius r c, 2 defined in Eq. (2), creates a gravitational potential and triggers an accretion of reheaton into this region. The accretion of the reheaton (which is pressureless in our setup) finally leads to the whole region collapsing into a black hole with an initial mass of order 10 -2 M H [82] where M H = 4 πM 2 Pl /H is the Hubble mass for a given background expansion rate H . As shown in Ref. [82], after it forms, the black hole quickly increases in mass by absorbing the surrounding matter. Once the PBH mass reaches about one Hubble mass, the rapid accretion is expected to be slowed down, and the mass simply follows the scaling of one  \nHubble mass M BH ∼ M H ∝ a 3 / 2 . This mass-growing process ends when radiation domination starts. Eventually, the final PBH mass is simply determined by the value of the Hubble mass at the time of the reheating, which we evaluate by considering that there is a matterradiation equality at T RH , giving  \nM PBH ∼ 3 . 5 × 10 -12 M ⊙ α ( 10 5 GeV T RH ) 2 ( 100 g ∗ ( T RH ) ) 1 / 2 , (3)  \nwhere g ∗ ( T RH ) is the number of effective relativistic degrees of freedom present in the plasma at the reheating time, and α ≲ 1 is an efficiency factor, which we take to be O (0 . 1) for simplicity. As can be seen from Eq. (3), M PBH is insensitive to the phase transition properties and solely determined by the value of T RH once they are formed. The distribution of PBHs formed from an aborted phase transition is thus expected to be monochromatic.  \nNote that the time scale of the PBH formation and mass-growing process is around the Hubble time scale [82, 84] at the time when the density perturbation is generated. Therefore, as long as T max ≳ T c ≫ T RH , there is enough time for the PBH to form and grow. The chronology of our PBH formation scenario is schematically summarized in Fig. 2, where one can also check our notations for important events.  \nPBHabundance -The PBH relic abundance can be estimated by counting the expected number of symmetryrestoring bubble nucleations during the aborted phase transition. It is thus sensitive to the bubble nucleation rate per unit volume, Γ( T ) ∼ T 4 e -S 3 /T where S 3 is the minimal energy of the scalar configuration to make a thermal escape from the local minimum, which can be obtained by the three-dimensional Euclidean action of the O (3) bounce solution [97-99]. Since the phase transition is aborted, Γ is maximized at the moment where  \nT = T max , and most of the symmetry-restoring bubbles are nucleated around this time.  \nTo be specific, let us consider a sufficiently large comoving total volume V . The number of nucleated bubbles at time t nuc , corresponding to a nuc , is given by  \nd N PBH ( a nuc ) = d a nuc a nuc H ( a nuc ) × V a 3 nuc Γ( T ( a nuc )) . (4)  \nIntegrating it from t c, 1 to t and dividing the result by V a ( t ) 3 gives the integrated number density at t  \nn PBH ( t ) = ( a max a ( t ) ) 3 ∫ a ( t ) a c, 1 ( a nuc a max ) 2 Γ( T ( a nuc ))d a nuc a max H ( a nuc ) . (5)  \nFrom this, one can obtain the PBH dark matter fraction f PBH = n PBH ( t today ) M PBH / ( ρ c Ω DM ).  \nHere, we proceed with a rough estimation using a model-independent approach with the following approximations. First of all, we take the Taylor expansion of S 3 /T around T max in the log scale;  \nS 3 T ≃ S 3 T ∣ ∣ ∣ ∣ T = T max -ˆ β max ln ( T T max ) , (6)  \nwhere we define the rapidity parameter ˆ β max as  \nˆ β max ≡ -d( S 3 /T ) dln T ∣ ∣ ∣ ∣ T = T max . (7)  \nThen we can approximate Γ( T ) as Γ( T ) ≃ Γ( T max )( T/T max ) ˆ β max +4 . In the Supplementary Material, we evaluate S 3 /T and ˆ β max in the case of the so-called Abelian Higgs model and obtain ˆ β max around 10 4 -10 6 . We use this value as a benchmark in what follows. In addition, using Eq. (1) to evaluate T ( a ), we obtain in the limit a ≈ a max  \nT ( a ) ≃ T max exp [ -3 4 ( a a max -1 ) 2 ] . (8)  \nAlthough (7) and (8) are only valid around T max , we checked numerically that they lead to a good approximation for f PBH as long as ˆ β max > 50 because the largest contribution to f PBH comes from Γ( T max ).  \nThen, Eq. (5) can be approximated as  \nn PBH ( t RH ) ≃ √ 4 π 3 ˆ β max ( a max a RH ) 3 Γ( T max ) H max , (9)  \nfor ˆ β max > 50. Assuming that the PBH yield is unchanged after reheating temperature, we obtain f PBH as  \nf PBH = M PBH n PBH /s ρ DM /s  \n∼ 1 α ( T RH 10 5 GeV ) ( ( Γ( T max ) H 4 max ) 10 -16 ) ( 10 5 ˆ β max ) 1 2 ( a RH /a max 10 2 ) 3 2 , (10)  \nFIG. 3. Fraction of the dark matter relic density that is composed of PBHs, as a function of the PBH mass or equivalently the reheating temperature, using the benchmark values α = 0 . 1, a RH /a max = 10 2 and ˆ β max = 10 5 . Shaded areas correspond to regions of the parameter space excluded by BBN, CMB anisotropies, cosmic-ray detection, microlensing, gravitational wave detection and accretion (as reported in [3]).  \n<!-- image -->  \n/circledot  \nwhere the observed dark matter relic abundance is taken to be ρ DM /s ≃ 0 . 4 eV [100].  \nIn Fig. 3, we depict Γ( T max ) /H 4 max required to give a sizable f PBH for different M PBH (or T RH ) for α = 0 . 1, a RH /a max = 10 2 and ˆ β max = 10 5 , taking g ∗ ( T ) to be the Standard Model value [101]. We also show relevant constraints coming from the null observation of PBH evaporation signal (cyan), lensing by PBHs (purple), gravitational waves (blue), and accretion (green), taken from Ref. [3]. The dotted line on the right edge represents the lower bound of T RH ≳ 5 MeV (and thus an upper bound of M PBH ) coming from the big bang nucleosynthesis [102107] while the one on the left edge depicts the critical PBH mass M ⋆ ≃ 5 × 10 14 gram below which PBHs evaporate completely before the present [108, 109]. For masses smaller than M ⋆ , we also indicate constraints from BBN (pink) and CMB anisotropies (orange) on evaporating PBHs [3]. As one can see from this figure, a broad range of values for Γ( T max ) /H 4 max lead to an abundance of PBHs that is of phenomenological interest, including PBHs that could constitute the whole dark matter of our Universe.  \nSummary and Discussion -In this Letter , we have proposed a new PBH formation mechanism in aborted phase transition during reheating. A symmetry-restoring bubble is nucleated and expands during T > T c , and it shrinks back as the temperature drops below T c . This generates a macroscopic size of over-density perturbation with a spherical symmetry, which eventually collapses  \ninto a PBH via the post-collapse accretion mechanism during matter domination. The mass of PBHs formed in this process grows quickly by absorbing the surrounding matter, and its final mass is determined by T RH as given in Eq. (3). We estimate the PBH abundance (10) in terms of the bubble nucleation rate around T max parametrized by the effective rapidity parameter ˆ β max at T max , and show that it can be sizable in the aspect of phenomenology.  \nOur findings rely on the post-collapse accretion mechanism [82-84], in which a small overdensity accretes the matter present in the Hubble patch during a matterdominated era, leading to the formation of a black hole. 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Laurent, and J. van de Vis, JCAP 07 , 002 (2023), arXiv:2303.10171 [astro-ph.CO].", 'S1. AN EXAMPLE MODEL TO EVALUATE THE PHASE TRANSITION RAPIDITY PARAMETER': 'In this section, we consider a benchmark model and obtain ˆ β max . The model we consider is a simple Abelian Higgs model where a complex scalar field Φ is charged under a U (1) gauge interaction with a charge unity. We further assume that the theory is classically scale invariant, so the tree-level potential is given by  \nV (Φ) = λ | Φ | 4 , (S1)  \nwhere λ is the self-quartic coupling. The spontaneous symmetry breaking is radiatively generated as originally shown in Ref. [110].  \nTo include the loop effects conveniently, we take the RG scale µ = µ ∗ which is defined by λ ( µ ∗ ) = 0. The existence of such µ ∗ is guaranteed by the positive beta function of λ coming from the gauge boson loop. Denoting ϕ for the radial degree of Φ, the one-loop effective potential can be written as  \nV ( ϕ ) = δλ 4 ϕ 4 + 1 4 β λ ϕ 4 log ϕ µ ∗ , (S2)  \nwhere δλ = 3 g 4 16 π 2 (log g 2 -5 / 6) and β λ = 6 g 4 / 16 π 2 with g being the gauge coupling. This potential is minimized at v ϕ = e 1 / 6 µ ∗ /g and the potential energy difference is given by ∆ V 0 = 3 e 2 / 3 128 π 2 µ 4 ∗ .  \nWe include the thermal correction coming from the gauge boson loop,  \n̸  \n∆ V T =0 = 3 T 4 2 π 2 J B ( m 2 V /T 2 ) , (S3)  \nwith the field-dependent gauge boson mass m V = gϕ and the J B function given by  \nJ B ( y 2 ) = ∫ ∞ 0 d xx 2 log [ 1 -e -√ x 2 + y 2 ] . (S4)  \nNote that the scalar-loop contribution vanishes due to our choice of RG scale, λ ( µ ∗ ) = 0.  \nIn this specific setup, we find two important model properties. First, T c is independent of the size of gauge coupling g since the zero-temperature potential energy difference is given by ∆ V 0 = 3 e 2 / 3 128 π 2 µ 4 ∗ independently of g . Second, the binodal temperature T 1 (where the potential barrier disappears) is also g -independent. This is because the effective field range of the thermal correction ϕ eff , T has the same coupling dependence with v ϕ ∼ µ ∗ /g ; ϕ eff , T can be estimated by m V ( ϕ eff , T ) ∼ T , so ϕ eff , T ∼ T/g . We numerically find that T c ≃ 0 . 37 µ ∗ and T 1 ≃ 0 . 44 µ ∗ . For our PBH formation scenario, T max must be between T 1 and T c , which is not impossible although it requires tuning (note that there are already multiple coincidences of time scales in the standard cosmology).  \nThese properties can be changed by including additional fields. For instance, if we include a Dirac fermion ψ that couples to ϕ via a Yukawa interaction, T c decreases while T 1 does not change much. This can increase the ratio of T 1 and T c , reducing the required level of tuning T max .  \nFor a temperature between T c and T 1 , we obtain the bounce action by using the CosmoTransitions [111]. The result of S 3 /T is given in the left panel of Fig. S4. Then, we obtain the rapidity parameter as shown in the right panel of Fig. S4, which shows that 10 4 ≲ ˆ β max ≲ 10 6 .', 'S2. DYNAMICS OF A SYMMETRY-RESTORING BUBBLE': 'In this section, we show that the bubble wall typically reaches a terminal velocity, i.e., has a non-runaway behavior, in both the expansion and contraction stages. Bubble wall dynamics is a highly complicated subject, requiring one to solve the Boltzmann equations for the particle distribution functions (which are integro-differential equations), the background scalar equation of motion, and the fluid equations for the hydrodynamics [112-116]. To determine whether or not a bubble wall runs away, i.e. accelerates all the way until colliding with another bubble, friction in the γ w → ∞ limit is usually compared to the vacuum energy difference [117] (although this may not always be valid [118]). Below, we also do a similar analysis, using the simple Bodeker-Moore criterion [117].  \nFIG. S4. Our numerical result of S 3 /T and ˆ β = -d( S 3 /T ) / dln T for the Abelian Higgs model.  \n<!-- image -->  \n0.36  \n0.38  \n0.40  \n0.42  \n0.44  \nT  \n/  \nμ  \n*', 'A. Bubble expansion ( T > T c )': '̸  \nLet us first consider bubble expansion. Note that, for T > T c , the vacuum energy inside the bubble is greater than outside, i.e. ∆ V 0 < 0, so the vacuum energy always gives a negative pressure that tries to contract the bubble. On the other hand, the thermal pressure difference ∆ V T =0 is positive, and this is the driving force of the bubble expansion. When a bubble is formed at T > T c , the bubble wall gets accelerated since the net pressure is positive (by the definition of T c ). When the bubble wall velocity is nonzero, the thermal driving force is reduced. This can be seen from the fact that, in the example of 1-to-1 transmission processes, the momentum transfer in the wall-rest frame decreases as the fluid velocity increases; ∆ p z = √ p 2 z +∆ m 2 -p z ∼ ∆ m 2 / 2 p z where z is the direction of the bubble wall propagation, p z is the momentum of a particle coming toward the bubble wall from outside, and ∆ m 2 > 0 is the mass-squared difference. Therefore, as velocity increases, the thermal driving force decreases until it reaches the equilibrium with the vacuum energy pressure.  \nAs pointed out in Ref. [80], the thermal driving force has a nonzero asymptotic value in v w →∞ limit, which we also call Bodeker-Moore thermal force  \nP BM = ∑ i C i g i c i ∆ m 2 i T 2 24 , (S5)  \nwhere c i = 1(1 / 2) for bosons (fermions). Here g i is the number of internal degrees of freedom of species i that couple with the scalar ϕ , and ∆ m 2 i is the difference of the squared-mass in broken and symmetric phases. C i is approximately given by  \nC i T 2 24 ≈      T 2 24 if m out i ≪ T , 1 2 m out i ( m out i T 2 π ) 3 / 2 e -m out i /T if m out i ≫ T , (S6)  \nwith m out i being the mass outside of the wall, i.e. in the broken phase.  \nIf m out i are larger than the temperature, which is the case for our model considered in the last section, P BM would be suppressed because the number density of those heavy particles is Boltzmann-suppressed. This means that the asymptotic value of the driving force is small, ensuring the existence of equilibrium with ∆ V 0 at some velocity.  \nWe note that P BM is the force caused only by the 1 → 1 processes. There can be additional forces caused by particle-production processes [119-125], i.e., when a particle splits into two or more particles when it transits across the wall. These next-to-leading-order forces may behave as true friction as in a cooling phase transition [79]. We also note that hydrodynamic effects can induce a barrier of the frictional pressure at the Jouguet velocity [118, 126-128]. All these factors would just make our conclusion more solid.', 'B. Bubble contraction ( T < T c )': 'For T < T c , the dynamics of the bubble wall can be understood in the usual way although our bubble is still symmetry-restoring and contracts. Actually, the contraction process under consideration can be likened to the contraction of a false-vacuum bubble (sometimes referred to as a false-vacuum island) in a cooling and symmetry-breaking  \nT  \ng  \no  \nl  \nd  \n/  \n)  \nT  \n/  \n3  \nS  \n(  \nd  \n-  \n=  \nβ  \n7  \n10  \n6  \n10  \n5  \n10  \n4  \n10  \n1000  \nc  \nT  \n=  \nT  \ng  \n0.4  \n0.6  \n0.7  \n0.85  \n1  \nT  \n=  \nT  \nFOPT. During contraction, the vacuum energy difference accelerates the bubble wall velocity while the thermal effect acts as friction. In this case, when the wall velocity increases, the friction increases and has an asymptotic value of P BM [117]. Thus, if | ∆ V 0 | < P BM , there exists a terminal velocity where the friction and ∆ V 0 make an equilibrium.  \nBefore proceeding, note that the temperature range in our process is all around T c . As shown in the previous section, T max /T c < T 1 /T c cannot be large in the model-building aspect, and therefore, the temperature when the bubble shrinks and disappears, which we denote T zero , should be also close to T c .  \nNow let us again consider the largeγ w limit. In bubble contraction, the driving force is  \nP driving = | ∆ V 0 | , (S7)  \nwhile the Bodeker-Moore thermal force is [117]  \nP friction = P BM = ∑ i g i c i ∆ m 2 T 2 24 ∼ g ⋆,ϕ ∆ m 2 T 2 24 , (S8)  \nis a true friction where g ⋆,ϕ is the effective degrees of freedom that couple to ϕ .  \nOn the other hand, we have the relation of ∆ V 0 ≃ g ⋆,ϕ π 2 90 T 4 c which is smaller than P BM for ∆ m 2 > T 2 . Therefore, we conclude that the bubble wall still does not run away even without taking into account friction from 1-to-2 or 1-to-many processes and hydrodynamic obstruction [118].'}