Supercool subtleties of cosmological phase transitions

With bubble nucleation and growth specified, we can determine the transition progress through the false vacuum fraction, $P_{f}$. The volume occupied by bubbles is difficult to determine without performing a stochastic field-fluid simulation (see e.g. Ref. [124]) and measuring the fraction of simulation volume encompassed by bubbles. Instead, we can proceed analytically to model the false vacuum fraction as [125, 126, 91, 114]⁵⁵5This model for phase transitions has in fact been known in the field of materials science since the 1930s, where it is referred to as the JMAK equation [127, 128, 129, 130, 131]. Importantly, there is a wealth of literature detailing the assumptions and caveats hidden within the derivation of Refs. [125, 126]. We review this literature in an upcoming paper [132]. For now, we refer the reader to Refs. [133, 134] for a discussion of the correctness and assumptions of the JMAK equation.

$$P_{f}(t)=\exp(-\mathcal{V}_{t}^{\text{ext}}(t)),$$

(2.4)

where $\mathcal{V}_{t}^{\text{ext}}$ is the (fractional) extended volume of true vacuum bubbles, given by

$$\mathcal{V}_{t}^{\text{ext}}(t)=\frac{4\pi}{3}\int_{t_{0}}^{t}\!dt^{\prime}\,\Gamma(t^{\prime})\frac{a^{3}(t^{\prime})}{a^{3}(t)}R^{3}(t^{\prime},t).$$

(2.5)

Here, $t_{0}$ is the time when the transition first becomes possible — taken to be when the Universe cools to the critical temperature. As space expands, the number density of previously nucleated bubbles is reduced by the cubed ratio of scale factors. Note that $\mathcal{V}_{t}^{\text{ext}}$ double-counts overlapping regions of bubbles and includes phantom bubbles which nucleate within pre-existing bubbles. The extended volume is not equivalent to the true vacuum fraction $P_{t}=1-P_{f}$ except in the early stages of a transition where bubbles nucleate and grow in isolation. To calculate the true vacuum fraction directly, the nucleation rate would need to be multiplied by $P_{f}$ and the volume of individual bubbles would need to account for deformation due to impingement. It is considerably more convenient to instead exponentiate the extended volume. Various alternative derivations for the false vacuum fraction exist in the literature (see e.g. Ref. [135] for a review), which also lead to (2.4) and (2.5). For example, a recent derivation in cosmological literature [91] avoids the explicit nucleation of phantom bubbles but again leads to these results. One may interpret $\mathcal{V}_{t}^{\text{ext}}$ as the fraction of the volume of all real and phantom bubbles over the total volume in which the phase transition occurs (i.e. the entire Universe).⁶⁶6Because the nucleation rate (2.1) is the number of bubbles nucleated per volume per time, the volume in $\Gamma$ can cancel with the total transition volume in $\mathcal{V}_{t}^{\text{ext}}$. We use this in arriving at (2.5). We emphasise that one cannot simply replace $\mathcal{V}_{t}^{\text{ext}}$ in (2.4) with the true vacuum fraction, or multiply $\Gamma$ by $P_{f}$ in (2.5).

The false vacuum fraction is then

$$P_{f}(t)=\exp(-\frac{4\pi}{3}v_{w}^{3}\int_{t_{0}}^{t}\!dt^{\prime}\,\Gamma(t^{\prime})\!\left(\int_{t^{\prime}}^{t}\!dt^{\prime\prime}\,\frac{a(t^{\prime})}{a(t^{\prime\prime})}\right)^{\!\!3}),$$

(2.6)

where we have used the assumptions in (2.3) for bubble growth. These assumptions are readily justified in this work. We consider strongly supercooled transitions, in which case the terminal wall velocity $v_{w}\rightarrow 1$ is expected for the entire time window where nucleation is significant.⁷⁷7Some models may still permit $v_{w}<1$ despite strong supercooling. Our discussion in Section 3 considers arbitrary bubble wall velocities. We note that for bubbles that expand as deflagrations or hybrids, reheating can affect the evolution of the transition. This effect is discussed in Appendix D. The pressure difference between phases is given time to increase, permitting rapid acceleration of the bubble walls. With a low nucleation rate, bubbles can grow considerably before colliding, allowing ultra-relativistic velocities to be attained. In this case the initial radius of a bubble becomes a negligible contribution to its evolution.

The nucleation rate in (2.1) is expressed as a function of temperature. It will be convenient to also express (2.6) in terms of temperature. We can use the time-temperature relation:

$$\derivative{T}{t}=-TH(T),$$

(2.7)

where $H$ is the Hubble parameter defined below. This relation is an approximation based on the MIT bag equation of state [136], but it holds well in the models we consider. Further details are available in Appendix B. Assuming flat space, with vanishing Gaussian curvature in the FLRW metric, the Hubble parameter can be determined from Friedmann’s first equation as

$$H(T)=\frac{\dot{a}(T)}{a(T)}=\sqrt{\frac{8\pi G}{3}\rho_{H}(T)},$$

(2.8)

where $G=6.7088\!\times\!10^{-39}\,\text{GeV}^{-2}$ is Newton’s gravitational constant [137].

With the relation (2.7) between temperature and time, we can now express (2.6) in terms of temperature

$$P_{f}(T)=\exp(-\frac{4\pi}{3}v_{w}^{3}\int_{T}^{T_{c}}\!dT^{\prime}\,\frac{\Gamma(T^{\prime})}{T^{\prime}H(T^{\prime})}\!\left(\int_{T}^{T^{\prime}}\!dT^{\prime\prime}\frac{1}{T^{\prime\prime}H(T^{\prime\prime})}\frac{a(T^{\prime})}{a(T^{\prime\prime})}\right)^{\!\!3}\,),$$

(2.9)

where we take the initial temperature of the transition to be the critical temperature, $T_{c}$. Adiabatic expansion ensures

$$\frac{a(T_{2})}{a(T_{1})}=\frac{T_{1}}{T_{2}},$$

(2.10)

as shown in Appendix C, so we can simplify (2.9) as

$$P_{f}(T)=\exp(-\frac{4\pi}{3}v_{w}^{3}\int_{T}^{T_{c}}\!dT^{\prime}\,\frac{\Gamma(T^{\prime})}{T^{\prime 4}H(T^{\prime})}\!\left(\int_{T}^{T^{\prime}}\!dT^{\prime\prime}\frac{1}{H(T^{\prime\prime})}\right)^{\!\!3}).$$

(2.11)

It now remains to determine the energy density $\rho_{H}$. For the purposes of cosmological phase transitions, we can consider the Lagrangian density

$$\mathcal{L}(\boldsymbol{\phi},T)=\frac{1}{2}\partial_{\mu}\boldsymbol{\phi}\partial^{\mu}\boldsymbol{\phi}-V(\boldsymbol{\phi},T)$$

(2.12)

with scalar potential $V(\boldsymbol{\phi},T)$.⁸⁸8We use the symbol $V$ for the potential and $\mathcal{V}$ for the volume. All fields that do not gain a vacuum expectation during the transition can be ignored, leaving a set of scalar fields $\boldsymbol{\phi}$ on which the dynamics of the phase transition depend. The minima of this potential correspond to vacua, or phases, while the potential itself is the free energy density; $\mathcal{F}(\boldsymbol{\phi},T)=V(\boldsymbol{\phi},T)$. The energy density for an arbitrary field configuration is

$$\rho(\boldsymbol{\phi},T)=T\partialderivative{p}{T}-p(\boldsymbol{\phi},T),$$

(2.13)

by the first law of thermodynamics, with pressure $p=-\mathcal{F}$. In terms of the potential, the energy density is

$$\rho(\boldsymbol{\phi},T)=V(\boldsymbol{\phi},T)-T\partialderivative{V}{T}.$$

(2.14)

During the phase transition we take the energy density $\rho_{H}$ in Friedmann’s equation (2.8) to be the difference in energy density between the false vacuum and the zero-temperature ground state of the potential

$$\rho_{H}(T)=\rho(\boldsymbol{\phi}_{f}(T),T)-\rho(\boldsymbol{\phi}_{\text{gs}}(0),0),$$

(2.15)

which expands to,

$$\rho_{H}(T)=V(\boldsymbol{\phi}_{f}(T),T)-T\partialderivative{V(\boldsymbol{\phi}_{f}(T),T)}{T}-V(\boldsymbol{\phi}_{\text{gs}}(0),0).$$

(2.16)

Here, $\boldsymbol{\phi}_{f}$ and $\boldsymbol{\phi}_{\text{gs}}$ are the field configurations corresponding to the false vacuum and the ground state, respectively. In general this ground state may not be the vacuum that describes our current Universe, which would leave a non-zero cosmological constant $\rho(\boldsymbol{\phi}_{\text{cur}}(0),0)-\rho(\boldsymbol{\phi}_{\text{gs}}(0),0)$. Conservation of energy is an implicit assumption in this form of the energy density of the Universe. As the transition progresses, regions of the Universe will no longer be in the false vacuum, so using only $\rho(\boldsymbol{\phi}_{f},T)$ instead of a combination of $\rho(\boldsymbol{\phi}_{f},T)$ and $\rho(\boldsymbol{\phi}_{t},T)$ (the subscript $t$ denoting the true vacuum) would appear erroneous. However, energy liberated by expanding true vacuum bubbles reheats the surrounding plasma. Following Ref. [57], we thus assume energy conservation outside of adiabatic cooling.

It will be convenient to separate out the parts of the potential that scale as radiation (i.e. $\propto T^{4}$), which yields the radiation energy density

$$\rho_{R}(T)=\frac{\pi^{2}}{30}g_{*}T^{4}=\frac{T^{4}}{\xi_{g}^{2}}.$$

(2.17)

Here $g_{*}$ is the number of relativistic degrees of freedom in the plasma, with a value of $106.75$ at the electroweak scale in the Standard Model, and we introduced $\xi_{g}^{2}$ as a shorthand notation for $30/(\pi^{2}g_{*})$. The remaining terms in the energy density can be considered to be the ‘vacuum contribution’, given by

$$\rho_{V}(T)=\rho_{H}(T)-\rho_{R}(T).$$

(2.18)

Typically, $\rho_{V}$ freezes to a constant value at low temperatures (see Figure 1, or e.g. Figure 33 in Ref. [62]) and is subdominant at high temperatures (e.g. towards the critical temperature). Then, a reasonable approximation is

	$\displaystyle\rho_{H}(T)$	$\displaystyle\approx\rho_{R}(T)+\rho_{V}(0)$
		$\displaystyle=\rho_{R}(T)+V(\boldsymbol{\phi}_{f}(0),0)-V(\boldsymbol{\phi}_{\text{gs}}(0),0).$		(2.19)

One can separate periods of the transition where the energy density was radiation- and vacuum-dominated by defining $T_{\text{eq}}$ such that

$$\rho_{V}(T_{\text{eq}})=\rho_{R}(T_{\text{eq}}).$$

(2.20)

Then we can roughly assume radiation domination when $T>T_{\text{eq}}$ and vacuum domination when $T<T_{\text{eq}}$. We will make use of this concept of distinct eras in our analytic discussion in Section 3. Note that (2.19) further assumes $\boldsymbol{\phi}_{f}$ still exists at $T=0$, which is not always true. One could instead use $\rho_{V}(T_{*})$ instead of $\rho_{V}(0)$, where $T_{*}$ is the temperature at which the phase $\boldsymbol{\phi}_{f}$ disappears.

We argue that a correct description of the expansion rate during a phase transition would ideally reflect a difference in expansion rate inside bubbles, outside bubbles, and near bubble walls. In a thin-wall approximation, one could neglect the latter region, leaving a difference in expansion rate in the two phases. The applicability of such a treatment to the averaged, analytic measure of nucleation and transition progress considered here is not immediately obvious. We reserve an exploration of this marriage of approaches for future work. In line with the averaged analysis for the transition progress, we use (2.16) with the assumption that, due to reheating, it captures the average energy density of the Universe. We use (2.16) in our numerical results and (2.19) in our analytic discussion.

It is useful to define several temperatures corresponding to milestones in the phase transition. The nucleation, percolation and completion temperatures, among others, are often introduced to capture the moment of their respective milestones. Here we discuss these temperatures and their corresponding events.

Significant nucleation does not occur immediately after the critical temperature, because the action diverges at $T_{c}$. The number of bubbles nucleated within a Hubble volume (of order $H^{-3}$) at temperature $T$ is [138, 99]

$$N(T)=\int_{T_{c}}^{T}\!dT^{\prime}\,\frac{\Gamma(T^{\prime})P_{f}(T^{\prime})}{T^{\prime}H^{4}(T^{\prime})},$$

(2.21)

where (2.7) has been used to transform the time integral. A factor of $4\pi/3$ could be included in (2.21) to consider a spherical Hubble volume, though such a change would usually have little impact due to how rapidly $N$ changes with temperature. We omit this factor to connect with the broader literature. Still, (2.21) differs from the common description of the number of nucleated bubbles [61],

$$N^{\text{ext}}(T)=\int_{T_{c}}^{T}\!dT^{\prime}\,\frac{\Gamma(T^{\prime})}{T^{\prime}H^{4}(T^{\prime})},$$

(2.22)

in that (2.21) does not count phantom bubbles (i.e. those nucleated inside other bubbles). Phantom bubbles are important for calculating the extended volume (2.5), but should not be counted here because phantom bubbles cannot nucleate. We have labelled the number of bubbles in (2.22) with the superscript “ext” to signify the counting of all bubbles in the extended true vacuum volume.

The nucleation temperature, $T_{n}$, is defined as the temperature when on average one bubble has nucleated in a Hubble volume,

$$N(T_{n})=1,$$

(2.23)

an event we will refer to as “unit nucleation” for convenience.⁹⁹9This event is usually referred to as nucleation in the literature. Here we want to differentiate it from the generic nucleation of a bubble. The usual definition is

$$N^{\text{ext}}(T_{n})=1.$$

(2.24)

Although approximate, (2.24) will yield a virtually identical nucleation temperature to (2.23). An exception is when the transition is strongly supercooled, with very few bubbles such that $P_{f}$ differs from unity appreciably before the nucleation temperature is reached. In Section 5.1 we will demonstrate cases where the extracted nucleation temperatures from (2.23) and (2.24) differ noticeably.

The nucleation temperature is often used in studies of phase transitions as a characteristic or reference temperature for important processes, such as gravitational wave production, baryogenesis or topological defect formation. It is conveniently simple to determine provided one has a method of computing the action. There are a few approximations used in the literature for the nucleation temperature that further simplify its determination, as discussed in Section 5.1. The nucleation temperature is often thought to represent the start of a transition. For a fast transition, the nucleation temperature may lie close to both the critical and completion temperatures. An alternative initial temperature is when non-negligible false vacuum conversion has occurred; for instance, Ref. [58] use $P_{f}(T_{I})=0.99$. For a strongly supercooled transition, the nucleation temperature may be significantly higher than the completion temperature (or not exist at all, as we will show), and is no longer a suitable reference temperature. This brings us to the next milestone temperature of interest.

The percolation temperature, $T_{p}$, is defined by

$$P_{f}(T_{p})\approx 0.71.$$

(2.25)

This condition is based on results from percolation threshold studies (see e.g. Refs. [139, 140, 141]). Percolation is defined as having an infinite cluster of — in our case — connected bubbles [142, 143]. For a finite-sized transforming medium, percolation is similarly defined as having a cluster of connected bubbles with size of the order of the medium (as depicted in Figure 2).¹⁰¹⁰10In other contexts such as a fluid filtering through a porous medium, the percolation threshold is the maximum connectivity (of the solid regions) in the medium such that the fluid can pass from one side of the medium to the other (i.e. there is a path through the entire medium for the fluid to flow). In the present context, the percolation threshold is the minimum fraction of the Universe converted to the true vacuum such that there is a connected cluster of bubbles that span the Universe. Percolation coincides with a time when bubbles are colliding (by definition, and as evidenced by Figure 2), marking an important time for gravitational wave production. While the ability to determine the ideal reference temperature for gravitational wave production is currently lacking, the percolation temperature is recommended over the nucleation temperature [47, 62, 99].

The completion (or final) temperature, $T_{f}$, is the temperature at which the transition completes. Ideally, this would be when $P_{f}(T_{f})=0$. However, (2.11) only predicts zero false vacuum fraction asymptotically because (2.4) is a probabilistic treatment of bubble nucleation and growth. Instead, we can define the completion temperature as

$$P_{f}(T_{f})=\varepsilon,$$

(2.26)

where $\varepsilon\ll 1$. We consider $\varepsilon=0.01$ in this work. Such a definition ensures that the transition is highly likely to have completed by the time the Universe cools to $T_{f}$. We remark that $T_{f}$ varies very weakly with $\varepsilon<0.01$ except in very strongly supercooled transitions where $P_{f}$ approaches a non-zero value as $T\rightarrow 0$. Remembering that we do not consider zero-temperature quantum tunnelling in this work, if a transition has $P_{f}(0)>\varepsilon$ due to thermal fluctuations and finite-temperature quantum tunnelling, we label it as incomplete.

Another criterion that should be satisfied for a successful transition is that the average physical volume of the false vacuum,

$$\mathcal{V}_{\text{phys}}(t)=a^{3}(t)P_{f}(t),$$

(2.27)

eventually decreases with time [144].¹¹¹¹11More precisely, (2.27) is the average proper volume of the false vacuum contained inside a unit comoving volume, hence the conversion factor $a^{3}$. That is,

$$\derivative{t}\!\left.\left(a^{3}(t)P_{f}(t)\right)\right|_{t>t_{*}}<0,$$

(2.28)

where $t_{*}$ is the time at which the physical volume of the false vacuum stops increasing. This leads to the condition

$$3H(t)-\derivative{\mathcal{V}_{t}^{\text{ext}}}{t}<0$$

(2.29)

for some $t>t_{*}$, where we have used the definition of the Hubble parameter given in (2.8). Or in terms of temperature, using (2.7),

$$3+T\derivative{\mathcal{V}_{t}^{\text{ext}}}{T}<0$$

(2.30)

for some $T<T_{*}$, where $T_{*}$ corresponds to $t_{*}$. Ref. [61] notes that (2.30) may not be satisfied when $P_{f}(T)\approx 0.71$, potentially spoiling percolation. For transitions where (2.30) is only satisfied for temperatures below $T_{p}$, it is suggested that percolation (and by extension completion) is questionable. To understand this situation, consider two nearby but non-overlapping bubbles. If the growth of the bubbles towards each other is slower than the expansion of the false vacuum between the bubbles, then they will never meet. This situation is possible even as the fraction of space in the false vacuum along the line connecting the centres of the bubbles approaches zero.

To be confident in the success of a transition and accuracy of the percolation condition (2.25), ideally we would have (2.30) satisfied at both $T_{p}$ and $T_{f}$. A weaker constraint is that there is a temperature for which $\mathcal{V}_{\text{phys}}$ is decreasing, without regard for the magnitude or duration of this decrease [61]. We demonstrate in Figure 3 and more generally in Section 3.6 that this constraint is insufficient for the strongly supercooled transitions we consider in this study. Other simple constraints can be considered in lieu of integrating the change in $\mathcal{V}_{\text{phys}}$ over time. One such constraint is that $\mathcal{V}_{\text{phys}}(T_{f})<\mathcal{V}_{\text{phys}}(T_{p})$, which maps to the condition $T_{f}\gtrsim 0.24T_{p}$. Another possible constraint is that $\mathcal{V}_{\text{phys}}$ is decreasing at $T_{f}$. The (perhaps model-dependent) hierarchy of these constraints is presented in Section 3.6.

Another relevant temperature for gravitational wave predictions is the reheating temperature, $T_{\text{reh}}$. As the energy liberated by vacuum conversion thermalises, the plasma reheats. This can have important consequences on the progress of the phase transition if bubbles expand as deflagrations or hybrids (see e.g. Refs. [145, 74]), and still affects the gravitational wave signal even for detonations. For instance, the frequency spectrum of gravitational waves is red-shifted due to cooling from $T_{\text{reh}}$ to the present day vacuum temperature, rather than cooling from $T_{f}$ or $T_{p}$ [57, 60]. One can estimate the temperature of the plasma after the phase transition completes on the grounds of energy conservation [57], by solving

$$\rho(\boldsymbol{\phi}_{f}(T_{f}),T_{f})=\rho(\boldsymbol{\phi}_{t}(T_{\text{reh}}),T_{\text{reh}}),$$

(2.31)

where the solution lies in the range $T_{f}<T_{\text{reh}}<T_{c}$. The reheating temperature can also be evaluated at the onset of percolation by replacing $T_{f}$ with $T_{p}$ in (2.31). The reheating temperature is not directly relevant to this study but will be reported in numerical scans in Section 5.

With all temperatures of interest defined, the remainder of this paper is dedicated to comparing these temperatures to each other and determining their relevance. We will argue that the nucleation temperature based on its standard definition (2.24) is not a fundamental quantity, and that its only use is as a proxy for the more appropriate and well-motivated percolation temperature. A thorough analysis of a phase transition can determine the percolation temperature, rendering the nucleation temperature obsolete, with the slight caveat that the percolation temperature is currently more computationally expensive to determine. In this work, we provide scenarios where the typical use of the nucleation temperature is unsuitable and would lead to incorrect predictions of the dynamics and completion of the transition.

Unit nucleation (with its corresponding nucleation temperature $T_{n}$) (2.23) is when an average of one bubble appears per Hubble volume. This may occur very soon after the transition first becomes possible in a fast (or weakly supercooled) transition. In a strongly supercooled transition, this may occur when a significant fraction of the Universe has already been converted to the true vacuum. The few bubbles have sufficient time to expand to sizes that are not achievable in a fast transition. For a discussion of the use of the nucleation temperature, see the text following (2.24).

The difference between the nucleation and critical temperatures is an indication of the degree of supercooling, or how long the Universe remained in the metastable phase below the critical temperature. A supercooling parameter can be defined as

$$\delta_{sc,n}=\frac{T_{c}-T_{n}}{T_{c}}.$$

(3.1)

This measure of supercooling has two important limitations. Obviously, unit nucleation must occur for this parameter to be defined. Additionally, given a value for $\delta_{sc,n}$, one still has no insight as to the progress of the transition.

Percolation (with its corresponding percolation temperature $T_{p}$) (2.25) is when there is a cluster of connected bubbles that spans across the transforming medium. For spherical bubbles, this occurs when approximately $30\%$ of the Universe has been converted to the true vacuum. The event of percolation aligns with when a significant proportion of bubble collisions occur, suggesting its relevance for gravitational wave generation [23, 24, 146, 147]. The percolation temperature can deviate significantly from the nucleation temperature in strongly supercooled transitions, as we will show in Section 5. Thus, another supercooling parameter can be defined as

$$\delta_{sc,p}=\frac{T_{c}-T_{p}}{T_{c}}.$$

(3.2)

Similarly to $\delta_{sc,n}$, this has the limitation that it is only defined if percolation occurs. However, given a value $\delta_{sc,p}$, one can determine the approximate temperature window for which the false vacuum fraction deviates significantly from unity.

In this paper we highlight that unit nucleation and percolation are independent events. The occurrence of one does not guarantee the occurrence of the other. Satisfying (2.23) suggests a significant number of bubbles have been nucleated, which naturally assists percolation. However, we will show two interesting scenarios that defy expectation:

1.     Scenario 1:

   Unit nucleation without percolation.

2.     Scenario 2:

   Percolation without unit nucleation.

These scenarios may also be expressed in terms of completion rather than percolation. Scenario 1 is what one might expect when a transition is tuned to be more strongly supercooled. Bubbles nucleate, but not enough for them to percolate before the Universe cools well below the scale of the transition (e.g. the electroweak scale). Slow bubble growth also aids in realising this scenario. Scenario 2 is less obvious. This requires very few bubbles to nucleate, and for those bubbles to grow sufficiently fast to eventually coalesce and encompass the entire Universe. This naturally requires bubbles to expand for a long time and at a significant fraction of the speed of light. A high temperature phase transition aids in realising this scenario, as well as a large and long-lasting potential barrier to suppress bubble nucleation.

The absence of unit nucleation is often used as a guarantee that the transition cannot be successful. We argue that its definition is somewhat misleading. Nucleating less than one bubble per Hubble volume does not preclude the possibility of those bubbles reaching a Hubble volume in which no bubble nucleated. Unit nucleation represents an obvious early stage of a fast transition where the number of bubbles per Hubble volume can be many orders of magnitude above unity. However, unit nucleation fails to represent a consistent milestone for supercooled transitions. This is demonstrated in Section 5.

Eventually, we wish to demonstrate the existence of Scenarios 1 and 2 in a concrete model. First we consider an analytic treatment that will provide insight into the conditions necessary for these scenarios to be realised. Later, we will verify the accuracy of this analytic treatment by cross-checking with a full numerical transition analysis in the models described in Section 4.

Scenario 1 requires

$$N(0)\geq 1~{}~{}\mathrm{and}~{}~{}P_{f}(0)>0.71,$$

(3.3)

while Scenario 2 requires

$$N(0)<1~{}~{}\mathrm{and}~{}~{}P_{f}(0)\leq 0.71.$$

(3.4)

Here $N(0)$ is the average number of bubbles nucleated per Hubble volume, and $P_{f}$ is the fraction of the Universe remaining in the false vacuum. Both of these quantities are evaluated at zero temperature. However, our analysis could be generalised to a transition with a low temperature cutoff.

Our aim is to find threshold values of some parameter for these scenarios. We will use the bubble wall velocity, $v_{w}$, as this deciding parameter. Consider Scenario 1 (unit nucleation without percolation), and assume unit nucleation occurs while varying the extended volume of true vacuum bubbles. On one side of this threshold value for $v_{w}$, we have unit nucleation without percolation, and on the other side we have unit nucleation with percolation. Whether a transition has a bubble wall velocity above or below the threshold value is more important than how far from the threshold value the transition is. This is because we merely want to demonstrate the existence of the scenarios and understand how and when they arise.

Direct integration of (2.11) and (2.21) is not feasible analytically unless many approximations are made. This subsection is dedicated to stating and justifying our assumptions to assist in the comparison of $N(0)$ and $P_{f}(0)$, and in the next subsection we detail the method of doing so. In fact, we take $N(0)$ as input and compare $P_{f}(0)$ to this while avoiding direct integration of (2.11). While the following assumptions are convenient for an analytic investigation, we later demonstrate numerically through benchmarks — without these assumptions — that Scenarios 1 and 2 can occur. The expected action curve, $S(T)$, is shown in Figure 4 with a minimum at $T_{S_{\text{min}}}$, and informs the first two assumptions. This shape of the action curve is typical of supercooled transitions, see e.g. Refs. [58, 60, 59].

1.     Assumption 1:

   The action is minimised at temperature $T_{S_{\text{min}}}$, where $S(T_{S_{\text{min}}})$ and $T_{S_{\text{min}}}$ are both positive. The action is approximately quadratic close to its minimum. Because the nucleation depends exponentially on the action, the nucleation rate is approximately Gaussian, centred near $T_{S_{\text{min}}}$. We define $T_{\Gamma}$ as the temperature that maximises the nucleation rate, about which the Gaussian nucleation rate is centred. Although we expect $T_{\Gamma}\approx T_{S_{\text{min}}}$, we note that $\left.\derivative{\Gamma}{T}\right|_{T_{S_{\text{min}}}}=\frac{4\Gamma(T_{S_{\text{min}}})}{T_{S_{\text{min}}}}\neq 0$.¹²¹²12If reheating occurs during the transition, then we should instead compute $\derivative{\Gamma}{t}$ to find the minimum of $\Gamma(T(t))$, which would occur near the time that minimises $S(T(t))$ (see e.g. Ref. [74]).

2.     Assumption 2:

   At zero temperature, the number of nucleated bubbles is not affected significantly by whether or not phantom bubbles are counted; that is, $N(0)\approx N^{\text{ext}}(0)$. If we consider Scenario 1, then percolation does not occur so $P_{f}\approx 1$. If we consider Scenario 2, then very few bubbles nucleate and yet percolation occurs. This suggests that bubbles nucleate early and nucleation is later suppressed while prolonged bubble growth continues. In this case, the bubbles nucleate when $P_{f}\approx 1$. Thus, we generically expect Assumption 2 to hold in these two scenarios for the models we consider.

3.     Assumption 3:

   The energy density (2.16) can be decomposed as a radiation component and a vacuum component. The vacuum component can be approximated as $\rho_{V}(T)\approx V(\boldsymbol{\phi}_{f}(0),0)-V(\boldsymbol{\phi}_{\text{gs}}(0),0)$, as in (2.19). This approximation is reasonable for transitions where the vacuum component away from zero temperature is dominated by the radiation component given by (2.17). As discussed below (2.19), this decomposition can be generalised to transitions where the false vacuum phase does not persist at zero temperature. We use (2.16) rather than (2.19) for the energy density in our numerical simulations.

In Assumption 1 we defined $T_{\Gamma}$ as the temperature that maximises the nucleation rate. For transitions that complete after cooling below $T_{\Gamma}$ (i.e. $T_{f}<T_{\Gamma}$), we naturally find $T_{\Gamma}\approx T_{S_{\text{min}}}$, where $T_{S_{\text{min}}}$ is the temperature that minimises the action. The maximum difference between $T_{S_{\text{min}}}$ and $T_{\Gamma}$ in the benchmarks listed in Section 4 is 0.7%. Hence, we may use these temperatures interchangeably in the models we consider, which will be important for Section 5.

As a reminder, we wish to simultaneously compare $N(0)$ to 1 and $P_{f}(0)$ to 0.71, which are the number of nucleated bubbles and the false vacuum fraction at zero temperature, respectively. In the following, we decompose $P_{f}$ in such a way that eventually permits direct comparison of $N(0)$ and $P_{f}(0)$. Using Assumption 2, we can approximate (2.21) as

$$N(T)\approx\int_{T}^{T_{c}}\!dT^{\prime}\,\frac{\Gamma(T)}{T^{\prime}H^{4}(T^{\prime})}=\int_{T}^{T_{c}}\!dT^{\prime}\,X(T^{\prime}),$$

(3.5)

where we have defined

$$X(T)=\frac{\Gamma(T)}{TH^{4}(T)}.$$

(3.6)

To assist in comparing the number of nucleated bubbles to the true vacuum volume, we also want to similarly rewrite $P_{f}$, so that both $N$ and $P_{f}$ involve integrals of $X$. Introducing

$$Y(T^{\prime},T)=\frac{v_{w}}{T^{\prime}}\int_{T}^{T^{\prime}}\!dT^{\prime\prime}\,\frac{H(T^{\prime})}{H(T^{\prime\prime})},$$

(3.7)

we can decompose $P_{f}$ as

$$P_{f}(T)=\exp(-\frac{4\pi}{3}\int_{T}^{T^{\prime}}\!dT^{\prime}\,X(T^{\prime})Y^{3}(T^{\prime},T)).$$

(3.8)

One can recognise that

$$Y(T_{\Gamma},T)=\frac{r(T_{\Gamma},T)}{r_{H}(T_{\Gamma})},$$

(3.9)

where

$$r(T^{\prime},T)=v_{w}\int_{T}^{T^{\prime}}\!dT^{\prime\prime}\frac{1}{a(T^{\prime\prime})T^{\prime\prime}H(T^{\prime\prime})}$$

(3.10)

is the comoving radius of a bubble nucleated at $T_{\Gamma}$ that has grown until $T$, and

$$r_{H}(T)=\frac{1}{a(T)H(T)}$$

(3.11)

is the comoving Hubble radius. Then $Y$ can be interpreted as the ratio of radii of the most common bubbles to the Hubble volume, in comoving coordinates.¹³¹³13We use comoving radius because the proper radius of bubbles is ill-defined at zero temperature, since $a(T)\propto T^{-1}$. The remainder of this subsection is dedicated to an analytic comparison of the number of bubbles nucleated and the true vacuum volume, aided by the use of well-motivated assumptions and approximations.

The conditions for Scenario 1, unit nucleation without percolation, and Scenario 2, percolation without unit nucleation, are given in (3.3) and (3.4), respectively. The exclusivity between the conditions for these two scenarios means that once we have found conditions for one scenario, it will be easy to find the conditions for the other. The conditions for Scenario 1 can be expressed as

$$\int_{0}^{T_{c}}\!dT\,X(T)\geq 1\;\;\;~{}\mathrm{and}~{}\;\;\;\exp(-\frac{4\pi}{3}\int_{0}^{T_{c}}\!dT\,X(T)Y^{3}(T,0))>0.71.$$

(3.12)

It will be convenient to work with the extended volume of true vacuum bubbles, $\mathcal{V}_{t}^{\text{ext}}$, which appears inside the exponent of $P_{f}$ (see (2.4)), rather than $P_{f}$ directly. Let $f_{i}=P_{f}(T_{i})$ denote the false vacuum fraction for some milestone temperature $T_{i}$. We have $f_{p}=0.71$ for percolation and $f_{f}=0.01$ for completion, according to our definitions in Section 2.2. We can express the condition $P_{f}(0)>0.71$ as

$$\int_{0}^{T_{c}}\!dT\,X(T)Y^{3}(T,0)<-\frac{3\ln(f_{p})}{4\pi}=c_{p}^{3},$$

(3.13)

where we have defined

$$c_{i}=\left(-\frac{3\ln(f_{i})}{4\pi}\right)^{\!\!\frac{1}{3}}.$$

(3.14)

The factor of $3/(4\pi)$ would not be present if a spherical Hubble volume was used in $N$ (see (2.21)).

Multiplying the right-hand side of (3.13) by a factor of one and using Assumption 2, we have

$$\int_{0}^{T_{c}}\!dT\,X(T)Y^{3}(T,0)<\frac{c_{p}^{3}}{N(0)}\int_{0}^{T_{c}}\!dT\,X(T).$$

(3.15)

We take $N(0)$ as input in the following analysis. The number of bubbles nucleated throughout the transition requires knowledge of the action as a function of temperature, which we cannot hope to capture in a model-independent way. In fact, our analysis will be completely model-independent, except that the model must admit a phase transition at roughly the electroweak scale, and this transition must satisfy the assumptions detailed in Section 3.2.

Now consider percolation more generally. The inequality (3.15) becomes an equation in the general case, which may be written as

$$\int_{0}^{T_{c}}\!dT\,X(T)A(T)=Z,$$

(3.16)

where $Z$ is a real number that can take either sign or even vanish, and

$$A(T)=Y^{3}(T,0)-\frac{c_{p}^{3}}{N(0)}.$$

(3.17)

The sign of $Z$ indicates whether percolation occurs: $Z<0$, $Z=0$ and $Z>0$ are respectively where percolation does not occur, the threshold of percolation ($P_{f}(0)=0.71$), and where percolation does occur. That is, $\operatorname{sign}(Z)=\operatorname{sign}(0.71-P_{f}(0))$. To classify whether a transition satisfies the conditions of Scenario 1, Scenario 2, or neither, requires only the determination of $\operatorname{sign}(Z)$, along with $N(0)$ which we take as input. However, the integral in (3.16) still cannot be evaluated analytically. We can make one further simplification to avoid the integration in (3.16) altogether.

The form of $Y(T,0)$ is shown in Figure 5 for the benchmark M2-BP1 (defined in Section 4.2). This form holds in general and can be understood as follows. During vacuum domination ($T\ll T_{\text{eq}}$), a constant Hubble parameter in (3.7) yields a constant: $Y(T,0)\approx v_{w}$. During radiation domination ($T\gg T_{\text{eq}}$), instead $dY(T,0)/dT$ is approximately constant, which can be seen by using $H(T)\approx T^{2}/\xi_{g}$ in (3.7). Importantly, $Y(T,0)$ monotonically increases with temperature from zero temperature to the critical temperature, and consequently so does $A(T)$. Because $X(T)$ is strictly positive, we have $\operatorname{sign}(X(T)A(T))=\operatorname{sign}(A(T))$. Assumption 1 implies that $X(T)$ is an even function (specifically Gaussian) about $T_{\Gamma}$. Then we have $Z=0$ if $A(T)$ is odd about $T_{\Gamma}$. Further, we have $\operatorname{sign}(Z)=\operatorname{sign}(c)$ if $A(T)-c$ is odd about $T_{\Gamma}$. Because we only need the sign of $Z$ (and not the magnitude) to determine whether percolation occurs, we can reduce the problem of checking (3.15) to simply computing $\operatorname{sign}(c)=\operatorname{sign}(A(T_{\Gamma}))$, provided $A(T)-c$ is odd about $T_{\Gamma}$. This property of $A(T)$ can be seen by the fact that it scales as a constant for low temperatures $T\ll T_{\text{eq}}$, and becomes most strongly temperature-dependent at high temperatures $T\gg T_{\text{eq}}$, scaling as $T^{3}$. Due to the exponential suppression away from $T_{\Gamma}$ by $X(T)$, any deviation from the odd function we desire is washed out. This leaves an approximately odd function about $T_{\Gamma}$, provided the Gaussian nucleation rate is not broadly peaked.

We numerically confirm that evaluating $\operatorname{sign}(A(T_{\Gamma}))$ is a reasonably accurate way to predict whether percolation occurs (with comparison to numerical results shown later in Figures 6 and 7). We expect this to hold in general, unless the energy density of the Universe scales differently with temperature than assumed for the models considered here. Another caveat is that alternate nucleation scenarios can be conceived where there is a prolonged window of significant nucleation, in which case suppression of the nucleation rate away from the maximum is not strong. In either case, the same general principles for realising Scenarios 1 and 2 should still apply, however, the analytic treatment used here would need to be significantly altered.

The percolation condition resulting from the simplification $\operatorname{sign}(Z)=\operatorname{sign}(A(T_{\Gamma}))$ is that percolation occurs if

$$Y(T_{\Gamma},0)>\frac{c_{p}}{N^{\frac{1}{3}}(0)}.$$

(3.18)

Then, checking (3.18) and whether $N(0)$ is larger or smaller than unity is sufficient to determine whether Scenarios 1 and 2 occur.

In Section 3.3 we showed that the conditions for unit nucleation without percolation reduce to simply finding when

$$Y(T_{\Gamma},0)<\frac{c_{p}}{N^{\frac{1}{3}}(0)},$$

(3.19)

with $N(0)\geq 1$ taken as input. Because $Y$ is an integral of Hubble parameters, it remains to evaluate the Hubble parameter. Using Assumption 3, we can approximate the Hubble parameter by considering a constant vacuum energy density: $\rho_{V}(T)\approx\rho_{0}$. The Hubble parameter is then

$$H(T)\approx\sqrt{\frac{8\pi G}{3}\left(\frac{T^{4}}{\xi_{g}^{2}}+\rho_{0}\right)}.$$

(3.20)

where $\rho_{0}=V(\boldsymbol{\phi}_{f}(0),0)-V(\boldsymbol{\phi}_{\text{gs}}(0),0)$. We define $T_{\text{eq}}$ as the temperature for which the radiation and vacuum energy densities are equal;

$$\rho_{R}(T_{\text{eq}})=\rho_{V}(T_{\text{eq}}).$$

(3.21)

Under the approximation of constant vacuum energy density, we have

$$T_{\text{eq}}^{2}=\xi_{g}\sqrt{\rho_{0}}.$$

(3.22)

Then we can express (3.20) as

$$H(T)\approx\sqrt{\frac{8\pi G}{3}\rho_{0}}\sqrt{\left(\frac{T_{\Gamma}}{T_{\text{eq}}}\right)^{\!\!4}+1}.$$

(3.23)

This approximation for the Hubble parameter permits an analytic result for the integral of the ratio of Hubble parameters in (3.7). The condition for Scenario 1 can be expressed as a bound on the bubble wall velocity:

$$v_{w}<\frac{c_{p}}{N^{\frac{1}{3}}(0)}\Bigg{[}\sqrt{\left(\frac{T_{\Gamma}}{T_{\text{eq}}}\right)^{\!\!4}+1}\,\,{}_{2}F_{1}\!\left(\frac{1}{4},\frac{1}{2};\frac{5}{4};-\!\left(\frac{T_{\Gamma}}{T_{\text{eq}}}\right)^{\!\!4}\right)\Bigg{]}^{-1},$$

(3.24)

where ${}_{2}F_{1}$ is a hypergeometric function. Interestingly, the bubble wall velocity required for percolation depends only on two quantities: the number of bubbles nucleated during the transition, $N(0)$; and the temperature at which bubble nucleation is maximised, $T_{\Gamma}$, relative to the temperature when the radiation and vacuum energy densities are equal, $T_{\text{eq}}$.

The result for the bubble wall velocity bound (3.24) is presented in Figure 6. To have unit nucleation without percolation (i.e. Scenario 1), we must have $N(0)\geq 1$ in addition to satisfying the bound on the bubble wall velocity in (3.24). This scenario is realised in the shaded region of Figure 6, where $N(0)=1$ was chosen. The choice of $N(0)$ scales the vertical axis; a value of $N(0)$ above this threshold of unity would permit percolation for lower bubble wall velocities. Intuitively, the results suggest that percolation does not occur if the bubbles expand sufficiently slowly. In a phase transition with slow bubble walls, the transition will complete via nucleation of many small bubbles. Or, if the nucleation rate is suppressed as we consider here, the bubbles will not percolate and the transition will not complete. The reciprocal factor in brackets in (3.24) has an upper bound of unity as $T_{\Gamma}/T_{\text{eq}}\rightarrow 0$. Then for $N(0)<c_{p}^{3}$, even $v_{w}=1$ is not predicted to yield percolation for $T_{\Gamma}\ll T_{\text{eq}}$. This corresponds to the average number of nucleated bubbles (throughout the entire transition) per Hubble volume being very low; specifically $N(0)\lesssim 0.08$, which is an order of magnitude below unit nucleation. Nevertheless, percolation can still occur with even fewer bubbles if $T_{\Gamma}\gg T_{\text{eq}}$. For $N(0)\gg 1$, we expect percolation to occur unless bubbles grow very slowly during adiabatic expansion. Although we take $v_{w}$ to be a free parameter, bubbles that expand as deflagrations or hybrids cause reheating in the false vacuum whereas detonations cause reheating in the true vacuum. This is an important distinction [145, 58, 74] that we do not capture in this study, with our analysis being most suitable for detonations.¹⁴¹⁴14Modelling of the false vacuum fraction evolution in the presence of reheating from deflagrations, as well as the effects of reheating on the gravitational wave signal, was recently performed in Ref. [114]. Reheating from bubbles expanding as deflagrations or hybrids could significantly delay percolation and may even prevent percolation for strongly supercooled transitions.¹⁵¹⁵15However, with sufficiently strong supercooling, deflagrations may not be a stable hydrodynamic solution [148, 149]. The instability is expected to accelerate the bubble walls, perhaps somewhat mitigating the hindrance to percolation from reheating. The realisation and analysis of deflagrations in the presence of strong supercooling may be an interesting avenue for future investigation.

We now discuss the implications of the obtained bubble wall velocity bounds. In Figure 6 we see that percolation is hardest to achieve when bubbles are predominantly nucleated during vacuum domination ($T_{\Gamma}\ll T_{\text{eq}}$); in that a larger bubble wall velocity is required for percolation compared to cases where bubbles nucleate when the radiation energy density is still significant ($T_{\Gamma}\gtrsim T_{\text{eq}}$). This result is expected because the bubbles that nucleate during radiation domination have both the radiation- and vacuum-dominated eras during which to grow, whereas bubbles that nucleate during the vacuum-dominated era can only grow during vacuum domination. Thus, the ratio $T_{\Gamma}/T_{\text{eq}}$ indicates how long bubbles can grow for. For Scenario 1, unit nucleation without percolation, we must have $N(0)\geq 1$, otherwise unit nucleation does not occur. Then the largest bubble wall velocity for which percolation does not occur is $v_{w}=c_{p}\approx 0.43$, according to the bound for $v_{w}$ with $T_{\Gamma}\ll T_{\text{eq}}$ and $N(0)=1$. Thus, percolation is guaranteed to occur if unit nucleation occurs, provided the bubbles expand as detonations or hybrids, which have $v_{w}>c_{s}\simeq 0.58$ for a perfect fluid in the MIT bag equation of state. Strictly speaking, Scenario 1 lies outside the regime of validity of our analysis because we neglect reheating, and therefore only model detonations correctly. However, in Appendix D we discuss how reheating affects Scenarios 1 and 2 by hindering percolation, and we reason that reheating should only strengthen the case for the existence of Scenario 1 for deflagrations. Additionally, reheating may allow Scenario 1 to be realised for bubbles growing as hybrids. Although we generically expect detonations for strong supercooling, hybrids may be possible if friction from the plasma is sufficient. We then predict that Scenario 1 is possible for strongly supercooled transitions (with $N(0)\approx 1$) that proceed with bubbles growing as deflagrations or hybrids. Conversely, we predict that unit nucleation implies percolation for transitions that are not strongly supercooled or where bubbles grow as detonations.

Formulating the condition for Scenario 1 as a bound on the bubble wall velocity is quite intuitive. Percolation occurs once a large enough fraction of the Universe is contained in true vacuum bubbles. A low bubble wall velocity results in bubbles expanding slower, delaying and perhaps precluding the onset of percolation. One can further connect this to the bubble radius using (3.9) in (3.19) to give the condition

$$\frac{r(T_{\Gamma},0)}{r_{H}(T_{\Gamma})}<\frac{c_{p}}{N^{\frac{1}{3}}(0)}$$

(3.25)

for Scenario 1, again with $N(0)\geq 1$. The left-hand side is the bubble’s comoving radius as a fraction of the comoving Hubble radius at the time of nucleation. For percolation to occur, this fractional radius $\mathcal{R}(T)=r(T_{\Gamma},T)/r_{H}(T_{\Gamma})$ must to be at least $c_{p}/N^{\frac{1}{3}}(0)$ as $T\rightarrow 0$. When considering a spherical Hubble volume (i.e. including the factor of $4\pi/3$ in (2.21)) and using $f_{p}=0.71$, we have the constraint $\mathcal{R}(0)\gtrsim 0.70/N^{\frac{1}{3}}(0)$ for percolation to occur. Replacing $c_{p}$ with $c_{f}$ in (3.25) and using $f_{f}=0.01$, we have the constraint $\mathcal{R}(0)\gtrsim 1.7/N^{\frac{1}{3}}(0)$ for completion to occur.

The analysis of Scenario 2 follows from that of Scenario 1. The condition for Scenario 2 equivalent to (3.19) is given in (3.18), and we must have $N(0)<1$. Fortunately, we can reuse all results from Scenario 1 simply by reversing the inequalities. Scenario 2 occurs in the non-shaded region above the curve in Figure 6.¹⁶¹⁶16This second usage of Figure 6 is possible because we consider $N(0)=1$; the threshold of unit nucleation. Reducing $N(0)$ infinitesimally in Figure 6 will not affect the partitions in the plot, but will then be considering the case of no unit nucleation. Even with a low number density of nucleated bubbles, percolation and completion are still possible, provided the bubbles expand quickly and have enough time to expand (that is, they nucleate at a high enough temperature relative to $T_{\text{eq}}$). Percolation is more difficult to achieve if bubbles predominantly nucleate during the vacuum-dominated era ($T_{\Gamma}\ll T_{\text{eq}}$) because the bubbles have less time to grow. As long as $N(0)>c_{p}^{3}\approx 0.08$, percolation without unit nucleation remains possible even if $T_{\Gamma}\ll T_{\text{eq}}$, provided the bubbles grow as detonations. Superluminal bubble wall velocities are required for percolation if $N(0)<c_{p}^{3}$ and bubbles nucleate during the vacuum-dominated era. If $N(0)\sim 1$ and bubbles grow as deflagrations or hybrids, reheating in front of the bubble walls may spoil percolation (as discussed in Appendix D). For instance, the bubble wall velocity bound when bubbles nucleate predominantly in the radiation-dominated era ($T_{\Gamma}\gtrsim T_{\text{eq}}$) suggests that bubble wall velocities even in the range $v_{w}\sim$ 0.1– 0.2 could yield percolation. However, percolation is hindered by the suppression of bubble nucleation and growth due to reheating in deflagrations and hybrids, so larger bubble wall velocities than predicted are required for percolation. Thus, the obtained bubble wall velocity bounds are necessary but not sufficient conditions for Scenario 2 outside of the detonation regime. Besides, a constant bubble wall velocity throughout the transition is not a reasonable approximation when reheating occurs [145, 74]. Quantitative predictions for Scenarios 1 and 2 in the case of reheating are beyond the scope of this study.

The obtained bubble wall velocity bounds provide no insight of the time ordering of unit nucleation and percolation events. A natural extension to Scenario 2 is percolation before unit nucleation. A similar analytic treatment to that used for Scenarios 1 and 2 could be applied for this scenario, with some difficulty and much tedium. We argue that such a scenario is largely unimportant, with this assessment following from our findings in Section 5 that the event of unit nucleation is an arbitrary milestone. We note that Ref. [62] demonstrate strongly supercooled cases where $T_{n}$ can become close to $T_{p}$, and also cases where $T_{n}<T_{\text{eq}}$. Ref. [59] consider cases where $T_{p}<T_{\text{eq}}<T_{n}$,¹⁷¹⁷17Although Ref. [59] stated the ordering of event times, we have neglected reheating, so the ordering of temperatures is the reverse ordering of times. and mention that scale-invariant models can have $T_{p}<T_{n}<T_{\text{eq}}$. We find both cases in our benchmarks (stated in Section 4), but, like Refs. [59, 62], we do not find any benchmarks where $0<T_{n}<T_{p}$.

The bubble wall velocity bound obtained for Scenarios 1 and 2 in Sections 3.4 and 3.5 can also be applied to unit nucleation without completion and completion without unit nucleation, respectively, simply by replacing $c_{p}$ with $c_{f}$ (see (3.14)). The results are shown in Figure 7. We see that completion is no longer possible for $T_{\Gamma}\lesssim T_{\text{eq}}/2$ at the threshold of unit nucleation, even with $v_{w}=1$. Additionally, completion is not possible (even neglecting reheating) for deflagrations at the threshold of unit nucleation, unless $T_{\Gamma}\gtrsim 1.38T_{\text{eq}}$. However, one may naïvely expect that percolation can still occur in some of the cases where completion does not. We will demonstrate that this is not necessarily true.

Even if the fraction of the Universe in the false vacuum decreases to some completion threshold $P_{f}(T_{f})=\epsilon\ll 1$ and beyond, finite pockets of the false vacuum can persist if the rate of false vacuum conversion does not exceed the rate of expansion of the false vacuum. At the end of Section 2.2 we mentioned five completion criteria (CC):

1.     CC1  

   The false vacuum fraction becomes sufficiently small.

2.     CC2  

   The physical volume of the false vacuum, $\mathcal{V}_{\text{phys}}$, decreases for some temperature $T>T_{f}$.

3.     CC3  

   $\mathcal{V}_{\text{phys}}$ decreases on average from percolation to completion: $\mathcal{V}_{\text{phys}}(T_{f})<\mathcal{V}_{\text{phys}}(T_{p})$. This constraint conveniently maps to the condition $T_{f}\gtrsim 0.24T_{p}$.

4.     CC4  

   $\mathcal{V}_{\text{phys}}$ is decreasing at $T_{f}$: $\left.\displaystyle\derivative{\mathcal{V}_{\text{phys}}}{t}\right|_{T_{f}}<0$.

5.     CC5  

   $\mathcal{V}_{\text{phys}}$ is decreasing at $T_{p}$: $\left.\displaystyle\derivative{\mathcal{V}_{\text{phys}}}{t}\right|_{T_{p}}<0$.

In this study we do not determine simple, model-independent analytic conditions for these constraints equivalent to the conditions found in Sections 3.4 and 3.5 for Scenarios 1 and 2. Instead we numerically find the bubble wall velocity, $v_{w}$, for which each of these constraints are satisfied. We do this for each of the benchmarks stated in Section 4, which we use in Figures 6 and 7. We consistently find that the value of $v_{w}$ for which CC2, CC3, CC4 and CC5 are satisfied in our benchmarks are respectively 10-15%, 18-21%, 21-23% and 70-123% above the value of $v_{w}$ required for completion (CC1), where the ranges show the full variation in results amongst our benchmarks. This holds for each benchmark, with the required bubble wall velocity being progressively larger in the order of CC1-5. A significant variation between benchmarks is only observed for CC5. We tentatively propose the more general bound (analogous to (3.18)) for a successful transition,

$$Y^{3}(T_{\Gamma},0)\geq\frac{\alpha_{\text{CC}i}c_{f}^{3}}{N(0)},$$

(3.26)

with $\alpha_{\text{CC}1}=1$, $\alpha_{\text{CC}2}=1.15$, $\alpha_{\text{CC}3}=1.21$, $\alpha_{\text{CC}4}=1.23$ and $\alpha_{\text{CC}5}=2.23$. Recalling the situation depicted in Figure 3, we recommend the use of CC5, or at least CC4, for a conservative estimate of transition completion. If one wishes to rule out the possibility of transition completion, then failure to satisfy CC1 is sufficient. In general, one should be cautious if any of the success criteria are not satisfied.

Although these results suggest that a transition completes (i.e. $T_{f}>0$) if CC5 is satisfied, it remains to be proven. We have provided analytic arguments that support the claim that a transition completes if and only if (3.26) is satisfied, using $\alpha_{\text{CC}1}$. No such analytic arguments have been made for the physical volume of the false vacuum and we are not aware of any existing proof that $\derivative{t}\mathcal{V}_{\text{phys}}(T_{p})<0\implies P_{f}(0)\rightarrow 0$.

To further investigate the ordering of $\alpha_{\text{CC}i}$ seen in our benchmarks, we consider a simple transition model that allows for analytic investigation of $\mathcal{V}_{\text{phys}}$ and $P_{f}$. We refer to this as the simultaneous nucleation model. We assume instantaneous nucleation of all bubbles at some temperature $T_{*}$ such that $\Gamma(T)=n_{B}\delta(T-T_{*})$, with $n_{B}$ being the number density of bubbles. We also assume vacuum domination throughout the transition, with the further approximation $H(T)=H_{*}$. Here, $T_{*}$ and $H_{*}$ are both constant. With these simplifications, the extended volume in (2.11) is

$$\mathcal{V}_{t}^{\text{ext}}(T<T_{*})=\frac{4\pi}{3}b^{3}\!\left(1-\frac{T}{T_{*}}\right),$$

(3.27)

where we have defined

$$b=v_{w}\!\left(\frac{n_{B}}{T_{*}H_{*}^{4}}\right)^{\!\!\frac{1}{3}}=v_{w}N^{\frac{1}{3}}(0).$$

(3.28)

The last equality in (3.28) is found using (2.21), with $N(0)$ being the number of bubbles per Hubble volume nucleated throughout the entire transition. The percolation and final temperatures are defined through $P_{f}(T_{i})=f_{i}$, or equivalently $\mathcal{V}_{t}^{\text{ext}}(T_{i})=-\ln(f_{i})$. In this simple model, the percolation and final temperatures are given by

$$T_{i}=T_{*}\!\left(1-\frac{c_{i}}{b}\right),$$

(3.29)

where $c_{i}$ is as defined in (3.14). Additionally, we can determine whether the physical volume of the false vacuum, $\mathcal{V}_{\text{phys}}$, is decreasing with time. This condition can be checked by finding $d\mathcal{V}_{t}^{\text{ext}}/dT$, as in (2.30). This is simple in the simultaneous nucleation model under consideration, because

$$\derivative{\mathcal{V}_{t}^{\text{ext}}}{T}=-\frac{4\pi b^{3}}{T_{*}}\!\left(1-\frac{T}{T_{*}}\right)^{\!\!2}.$$

(3.30)

Then (2.30) becomes

$$3-4\pi b^{3}\frac{T_{i}}{T_{*}}\!\left(1-\frac{T_{i}}{T_{*}}\right)^{\!\!2}<0.$$

(3.31)

We can use (3.29) in (3.31) to check whether $\mathcal{V}_{\text{phys}}$ is decreasing at the percolation or completion temperature. This yields the constraint

$$b>c_{i}+\frac{3}{4\pi c_{i}^{2}}.$$

(3.32)

It is possible to determine analytic bounds on $b=v_{w}N^{\frac{1}{3}}(0)$ for all completion criteria in the simultaneous nucleation model, using (3.29)–(3.32). The bounds are

1.     CC1  

   $T_{f}>0$:   $b>c_{f}\approx 1.032$.

2.     CC2  

   $\displaystyle\derivative{\mathcal{V}_{\text{phys}}}{t}<0$ for some $T_{f}<T(t)<T_{*}$:   $\displaystyle b>\left(\frac{81}{16\pi}\right)^{\!\!\frac{1}{3}}\approx 1.172$.

3.     CC3  

   $\mathcal{V}_{\text{phys}}(T_{f})<\mathcal{V}_{\text{phys}}(T_{f})$:   $\displaystyle b>\left[\left(\frac{f_{p}}{f_{f}}\right)^{\!\!\frac{1}{3}}c_{f}-c_{p}\right]\left[\left(\frac{f_{p}}{f_{f}}\right)^{\!\!\frac{1}{3}}-1\right]^{-1}\approx 1.223$.

4.     CC4  

   $\displaystyle\left.\derivative{\mathcal{V}_{\text{phys}}}{t}\right|_{T_{f}}<0$:   $\displaystyle b>c_{f}+\frac{3}{4\pi c_{f}^{2}}\approx 1.256$.

5.     CC5  

   $\displaystyle\left.\derivative{\mathcal{V}_{\text{phys}}}{t}\right|_{T_{p}}<0$:   $\displaystyle b>c_{p}+\frac{3}{4\pi c_{p}^{2}}\approx 1.701$.

In determining the bound for CC2, one can recognise that $T_{i}/T_{*}=1/3$ minimises (3.31). The condition for percolation is $b>c_{p}$. Thus, the condition for unit nucleation without percolation is $v_{w}<c_{p}/N^{\frac{1}{3}}(0)$ and $N(0)\geq 1$, just as in the vacuum-dominated limit ($T_{\Gamma}\ll T_{\text{eq}}$) of the model-independent analysis in Section 3.4.