SCIENCE CHINA Physics, Mechanics & Astronomy, Volume 61 , Issue 8 : 080411(2018) https://doi.org/10.1007/s11433-018-9216-7

## Energy budget of cosmological first-order phase transition in FLRW background

• AcceptedMar 29, 2018
• PublishedApr 23, 2018
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### Abstract

We study the hydrodynamics of bubble expansion in cosmological first-order phase transition in the Friedmann-Lema^ıtre-Robertson-Walker (FLRW) background with probe limit. Different from previous studies for fast first-order phase transition in flat background, we find that, for slow first-order phase transition in FLRW background with a given peculiar velocity of the bubble wall, the efficiency factor of energy transfer into bulk motion of thermal fluid is significantly reduced, thus decreasing the previously-thought dominated contribution from sound wave to the stochastic gravitational-wave background.

### Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11690022, 11435006, 11447601, and 11647601), the Strategic Priority Research Program of China Academy Sciences (Grant No. XDB23030100), the Peng Huanwu Innovation Research Center for Theoretical Physics (Grant No. 11747601), and the Key Research Program of Frontier Sciences of China Academy Sciences. We acknowledge the use of HPC Cluster of ITP-CAS. SJW would like to thank the invitation, support and warm hospitality from Mark Hindmarsh, Kari Rummukainen and David J. Weir during his visit at Helsinki institute of physics, Helsinki university, Finland. SJW wants to thank Thomas Konstandin for his valuable observation so that the application range of our conclusion in the second version of this manuscript has shrunk down to the slow first-order phase transition. SJW also wants to thank Huai-Ke Guo, Run-Qiu Yang for helpful discussions.

### Supplement

Appendix

Bubble wall velocity

In the previous study of macroscopic hydrodynamics, we have assumed that the bubble wall expansion has reached the stationary state in bubble center frame with a presupposed bubble wall peculiar velocity. In this appendix, we outline the usually-adopted model-independent approach for estimating the bubble wall velocity by taking into account two new features: one is that, the bubble wall may never runaway in bubble center frame according to the recent claim in ref. [53] due to extra friction from transition radiation; the other one is that, the background spacetime has experienced the Hubble expansion, therefore it is necessary to check the form of the Boltzmann equation and the EOM of the scalar-fluid system, which turns out to be unchanged with appropriate redefinition in bubble center frame.

Boltzmann equation

In the bubble center frame with comoving coordinates $\mathrm{d}s^2=a(\bar{t}+\bar{t}_n)^2(-\mathrm{d}\bar{t}^2+\delta_{ij}\mathrm{d}\bar{x}^i\mathrm{d}\bar{x}^j)$, the corresponding 4-momentum is defined by $p^\mu=\mathrm{d}\bar{x}^\mu/\mathrm{d}\lambda$. For particle of mass $m$, the wordline parameter $\lambda$ is chose as $\tau/m$, where the proper time $\tau$ is defined by $-\mathrm{d}\tau^2=\mathrm{d}s^2$. Then the components of 4-momentum $p^\mu=m\bar{\gamma}(1,\bar{v}^i)/a$ is written by the Lorentz factor $\bar{\gamma}=1/\sqrt{1-\bar{v}^2}$ of the norm $\bar{v}^2=\delta_{ij}\bar{v}^i\bar{v}^j$ of peculiar 3-velocity $\bar{v}^i=\mathrm{d}\bar{x}^i/\mathrm{d}\bar{t}$. Then the norm of the 3-momentum $\mathbf{p}^2=g_{ij}p^ip^j=m^2\bar{\gamma}^2\bar{v}^2$ is equal to the norm $\mathbf{\bar{p}}^2=\delta_{ij}\bar{p}^i\bar{p}^j$ of barred 3-momentum defined by $\bar{p}_i=p_i/a=ap^i=\bar{p}^i=m\bar{\gamma}\bar{v}^i$. The on-shell relation is then $p^2=g_{\mu\nu}p^\mu~p^\nu=-E^2+\mathbf{p}^2=-m^2$ with $E^2=-g_{00}(p^0)^2$ and $\mathbf{p}^2=g_{ij}p^ip^j=\delta_{ij}\bar{p}^i\bar{p}^j=\mathbf{\bar{p}}^2$. Conventionally, the unbarred 3-momentum is referred as the physical peculiar momentum, and the barred 3-momentum is referred as the comoving peculiar momentum, which can be raised up and down as if they are in the Euclidean space. The physical meaning of the comoving peculiar velocity is that they are actually the relative velocity with respect to the comoving frame (a frame comoving with Hubble expansion, not bubble expansion). The distribution function $f(\bar{x}^\mu,p_\mu)$ in phase space evolves by the Boltzmann equation: \begin{align}\frac{\mathrm{d}}{\mathrm{d}\lambda}f \equiv\left(\frac{\mathrm{d}\bar{x}^\mu}{\mathrm{d}\lambda}\frac{\partial}{\partial \bar{x}^\mu}+\frac{\mathrm{d}p_\mu}{\mathrm{d}\lambda}\frac{\partial}{\partial p_\mu}\right)f =C[f], \tag{153} \end{align} where $C[f]$ is the usual collision term, and the directional covariant derivatives are of form: \begin{align}\frac{\mathrm{d}\bar{x}^\mu}{\mathrm{d}\lambda}&=\frac{\mathrm{d}\bar{x}^\mu}{\mathrm{d}\lambda}=p^\mu; \tag{154} \\ \frac{\mathrm{d}p_\mu}{\mathrm{d}\lambda}&=\frac{\mathrm{d}p_\mu}{\mathrm{d}\lambda}-\Gamma^\sigma_{\mu\nu}p_\sigma p^\nu\equiv mF_\mu, \tag{155} \end{align} The external force can be defined by the geodesic equation: \begin{align}\frac{\mathrm{d}p^\mu}{\mathrm{d}\lambda}+\Gamma^\mu_{\nu\sigma}p^\nu p^\sigma\equiv mF^\mu, \tag{156} \end{align} which has a special form $F_\mu=-\partial_\mu~m$ resulted from the directional covariant derivative of on-shell relation for spatial-dependent effective mass term. Therefore, the form of Boltzmann equation is unchanged as in the flat background, namely \begin{align} \left(p^\mu\frac{\partial}{\partial x^\mu}+mF_\mu\frac{\partial}{\partial p_\mu}\right)f=C[f]. \tag{157} \end{align}

Equation-of-motion

In the curved spacetime, the energy-momentum tensor for the thermal fluid is of form \begin{align}T^{\mu\nu}_f &=\sum\limits_{i=\mathrm{B},\mathrm{F}}g_i\int\frac{\mathrm{d}p_1\mathrm{d}p_2\mathrm{d}p_3}{(2{\pi})^3\sqrt{-g}p^0}p^\mu p^\nu f_i; \tag{158} \\ &=\sum\limits_{i=\mathrm{B},\mathrm{F}}g_i\int\frac{\mathrm{d}\bar{p}_1\mathrm{d}\bar{p}_2\mathrm{d}\bar{p}_3}{(2{\pi})^3E_i}p^\mu p^\nu f_i, \tag{159} \end{align} where $\sqrt{-g}=\sqrt{-g_{00}}a^3$, $\sqrt{-g_{00}}p^0=E_i=\sqrt{\mathbf{\bar{p}}^2+m_i^2}$, and $p_i/a=\bar{p}_i$ have been used. The above form is covariant by noting that \begin{align} \int\frac{\mathrm{d}^3\mathbf{p}}{\sqrt{-g}2p^0}&=\int\frac{\mathrm{d}^3\mathbf{\bar{p}}}{2E} \\ &=\int\mathrm{d}^3\mathbf{\bar{p}}\int_{-\infty}^\infty\mathrm{d}E\delta(E^2-\mathbf{\bar{p}}^2-m^2)\theta(E). \tag{160} \end{align} Multiplying both side of eq. (157) by eq. (160) with extra multiplier $p_\nu$, and then summing over the particle of species $i$ gives rise to \begin{align}\sum\limits_ig_i\int\frac{\mathrm{d}^3\mathbf{\bar{p}}}{(2{\pi})^3E_i}p^\mu p_\nu\partial_\mu f_i &=\nabla_\mu T_{f\nu}^\mu=-\nabla_\mu T_{\phi\nu}^\mu \tag{161} \\ &=-\nabla_\nu\phi\left(\nabla_\mu\nabla^\mu\phi-\frac{\partial V_0}{\partial\phi}\right), \tag{162} \\ \sum\limits_ig_i\!\int\!\!\frac{\mathrm{d}^3\mathbf{\bar{p}}}{(2{\pi})^3E_i}mF_\mu p_\nu\frac{\partial}{\partial p_\mu}f_i &=\!\nabla_\nu\phi\sum\limits_ig_i\frac{\mathrm{d}m_i^2}{\mathrm{d}\phi}\!\int\!\!\frac{\mathrm{d}^3\mathbf{\bar{p}}}{(2{\pi})^32E_i}f_i, \tag{163} \end{align} where total conservation law $\nabla_\mu(T_f^{\mu\nu}+T_\phi^{\mu\nu})=0$ is used in the first line, and integration by part with $F_\mu=-\partial_\mu~m$ is used in the second line. The collision term simply vanishes upon above manipulations if collisions of particles happen at some points connected with geodesic equation. Therefore, the EOM of scalar-fluid system is obtained as: \begin{align} -\nabla_\mu\nabla^\mu\phi+\frac{\partial V_0}{\partial\phi}+\sum\limits_i g_i\frac{\mathrm{d}m_i^2}{\mathrm{d}\phi}\int\frac{\mathrm{d}^3\mathbf{\bar{p}}}{(2{\pi})^32E_i}f_i=0. \tag{164} \end{align} Splitting the distribution function into equilibrium and non-equilibrium parts $f_i=f_i^\mathrm{eq}+\delta~f_i$ with $f_i^\mathrm{eq}=\exp(-E_i/T)/(1\mp\exp(-E_i/T))$, one found that the derivative of the finite temperature part of effective potential, \begin{align}\frac{\partial V_T}{\partial\phi}&=\frac{\partial}{\partial\phi}\sum\limits_i\pm g_iT\int\frac{\mathrm{d}^3\mathbf{\bar{p}}}{(2{\pi})^3}(1\mp\exp(-E_i/T)) \tag{165} \\ &=\sum\limits_i\pm g_iT\int\frac{\mathrm{d}^3\mathbf{\bar{p}}}{(2{\pi})^3}\frac{\mp f_i^\mathrm{eq}}{T}\left(-\frac{\mathrm{d}E_i}{\mathrm{d}\phi}\right) \tag{166} \\ &=\sum\limits_ig_i\frac{\mathrm{d}m_i^2}{\mathrm{d}\phi}\int\frac{\mathrm{d}^3\mathbf{\bar{p}}}{(2{\pi})^32E_i}f_i^\mathrm{eq}, \tag{167} \end{align} is exactly the equilibrium part of the third term in the left hand side of eq. (164), therefore the EOM of scalar-fluid system reads \begin{align} -\nabla_\mu\nabla^\mu\phi+\frac{\partial\mathcal{F}}{\partial\phi}-\mathcal{K}(\phi)=0, \tag{168} \end{align} where the driving term comes from the free energy density, and the friction term comes from the departure from equilibrium, \begin{align}\mathcal{K}(\phi)=-\sum\limits_i g_i\frac{\mathrm{d}m_i^2}{\mathrm{d}\phi}\int\frac{\mathrm{d}^3\mathbf{\bar{p}}}{(2{\pi})^32E_i}\delta f_i. \tag{169} \end{align} Due to a recent finding in ref. [53], the friction term should contain a Lorentz factor that grows without bound for an accelerating bubble wall, thus leading to an eventual balance between driving force and friction force. Therefore the bubble wall cannot runaway in this case. Such phenomenological parametrization of friction term: \begin{align}\mathcal{K}(\phi)=T_N\widetilde{\eta}u^\mu\partial_\mu\phi \tag{170} \end{align} has already been proposed in refs. [47,49] before a modified parametrization of friction term [50,51] for runaway behavior [52].

Expansion equation

To write down the explicit form of the EOM eq. (168), it is usually conventional and convenient to work under planar limit where the bubble wall moves along $z$ direction. With use of the comoving coordinate system, the bubble center frame (comoving with Hubble expansion) is presented by $(\bar{t},\bar{z})$, and the bubble wall frame (comoving with bubble expansion) is presented by $(\bar{t}',\bar{z}')$. The comoving position of the bubble wall is thus presented by $\bar{z}_w(\bar{t})$ with comoving peculiar velocity $\bar{v}_w(\bar{t})$ 5)( and its corresponding Lorentz factor $\bar{\gamma}_w(\bar{t})$. In the bubble wall frame, the scalar profile depends only on $\bar{z}'$ through $\phi(\bar{z}')$ with suitable boundary conditions $\phi(\bar{z}'=-\infty)=\phi_-$, $\phi(\bar{z}'=0)=\phi_-/2$, $\phi(\bar{z}'=+\infty)=\phi_+$. When written in the bubble center frame, the scalar profile depends both on the $\bar{t}$ and $\bar{z}$ by $\phi(\bar{t},\bar{z})=\phi(\bar{\gamma}_w(\bar{t})[\bar{z}-\bar{z}_w(\bar{t})])\equiv\phi(\bar{z}')$ through a local Lorentz transformation \begin{align}\bar{t}'&=\bar{\gamma}_w(\bar{t})[\bar{t}-\bar{v}_w(\bar{t})\bar{z}]; \tag{171} \\ \bar{z}'&=\bar{\gamma}_w(\bar{t})[\bar{z}-\bar{z}_w(\bar{t})]. \tag{172} \end{align} The time derivatives of scalar profile in bubble center frame are computed directly as: \begin{align}\frac{\partial}{\partial\bar{t}}\phi(\bar{t},\bar{z})&=\phi'(\bar{z}')[\dot{\bar{\gamma}}_w(\bar{z}-\bar{z}_w)-\bar{\gamma}_w\dot{\bar{z}}_w]; \tag{173} \\ \frac{\partial^2}{\partial\bar{t}^2}\phi(\bar{t},\bar{z}) &=\phi''(\bar{z}')[\dot{\bar{\gamma}}_w(\bar{z}-\bar{z}_w)-\bar{\gamma}_w\dot{\bar{z}}_w]^2 \\ & +\phi'(\bar{z}')[\ddot{\bar{\gamma}}_w(\bar{z}-\bar{z}_w)-2\dot{\bar{\gamma}}_w\dot{\bar{z}}_w-\bar{\gamma}_w\ddot{\bar{z}}_w]. \tag{174} \end{align} Therefore, the first term in EOM 168 can be worked out in bubble center frame as: \begin{align}\nabla_\mu\nabla^\mu\phi&=\partial_\mu\partial^\mu\phi+\Gamma^\mu_{\mu\nu}\partial^\nu\phi \\ &=\frac{1}{a(\bar{t}+\bar{t}_n)^2}\left(\frac{\partial^2}{\partial\bar{z}^2}-\frac{\partial^2}{\partial\bar{t}^2}\right)\phi(\bar{t},\bar{z}) \tag{175} \\ &=\frac{1}{a^2}\left(\bar{\gamma}_w^2\phi''(\bar{z}') -\phi''(\bar{z}')[\dot{\bar{\gamma}}_w(\bar{z}-\bar{z}_w)-\bar{\gamma}_w\dot{\bar{z}}_w]^2\right. \\ & \left.-\phi'(\bar{z}')[\ddot{\bar{\gamma}}_w(\bar{z}-\bar{z}_w)-2\dot{\bar{\gamma}}_w\dot{\bar{z}}_w-\bar{\gamma}_w\ddot{\bar{z}}_w]\right). \tag{176} \end{align} To evaluate above expression, one can introduce the surface tension as: \begin{align}\sigma\equiv\int_{-\infty}^\infty\mathrm{d}\bar{z}'\phi'(\bar{z}')^2, \tag{177} \end{align} then the mean value of some quantity $F(\bar{z}')$ cross the bubble wall can be defined by \begin{align}\langle F\rangle\equiv\frac{1}{\sigma}\int_{-\infty}^\infty\mathrm{d}\bar{z}'\phi'(\bar{z}')^2F(\bar{z}'). \tag{178} \end{align} If $F$ is an odd function across the bubble wall, then its mean value $\langle~F\rangle$ should be zero. As an example, the bubble wall position can be defined in this way by \begin{align}\int\mathrm{d}\bar{z}(\partial_{\bar{z}}\phi)^2(\bar{z}-\bar{z}_w) &=\int\mathrm{d}\bar{z}'\frac{\mathrm{d}\bar{z}}{\mathrm{d}\bar{z}'}\phi'(\bar{z}')^2\bar{\gamma}_w^2(\bar{z}-\bar{z}_w) \tag{179} \\ &=\int_{-\infty}^\infty\mathrm{d}\bar{z}'\phi'(\bar{z}')^2\bar{z}' \tag{180} \\ &=\langle\bar{z}'\rangle\sigma=0. \tag{181} \end{align} Note that the boundary conditions imply \begin{align}\int_{-\infty}^\infty\mathrm{d}\bar{z}'\phi'(\bar{z}')\phi''(\bar{z}')=\int_{-\infty}^\infty\mathrm{d}\phi\,\phi''=\phi'|_{-\infty}^\infty=0. \tag{182} \end{align} Therefore, one can evaluate the first term in EOM 168 multiplied by $\phi'(\bar{z}')$ and integrated across the bubble wall as: \begin{align}&\int_{-\infty}^\infty\mathrm{d}\bar{z}'\phi'(\bar{z}')\nabla_\mu\nabla^\mu\phi \\ &=\frac{1}{a(\bar{t}+\bar{t}_n)^2}\int_{-\infty}^\infty\mathrm{d}\bar{z}'\phi'(\bar{z}')^2[2\dot{\bar{\gamma}}_w\dot{\bar{z}}_w+\bar{\gamma}_w\ddot{\bar{z}}_w] \tag{183} \\ &\equiv\frac{\sigma}{a^2}\bar{\gamma}^3_w(1+\dot{\bar{z}}^2_w)\ddot{\bar{z}}_w. \tag{184} \end{align} For stationary expansion of the bubble wall in the bubble center frame, this term is simply zero.

Next, the second term in EOM eq. (168), when multiplied by $\phi'(\bar{z}')$ and integrated across the bubble wall, gives rise to the driving force, \begin{align}F_\mathrm{dr} &\equiv\int_{-\infty}^\infty\mathrm{d}\bar{z}'\phi'(\bar{z}')\frac{\partial\mathcal{F}}{\partial\phi}(\phi(\bar{z}'),T(\bar{z}')) \tag{185} \\ &=\int_{-\infty}^\infty\mathrm{d}\bar{z}'\left(\frac{\mathrm{d}\mathcal{F}}{\mathrm{d}\bar{z}'}-\frac{\partial\mathcal{F}}{\partial T}T'(\bar{z}')\right) \tag{186} \\ &=\mathcal{F}|_-^+-\int_{T_-}^{T_+}\mathrm{d}T^2\frac{\partial\mathcal{F}}{\partial T^2} \tag{187} \\ &\simeq\epsilon|_-^+-\left\langle\frac{\partial\mathcal{F}}{\partial T^2}\right\rangle(T_+^2-T_-^2) \tag{188} \\ &=a_+T_+^4\alpha_+-\frac13(a_+-a_-)T_+^2T_-^2 \tag{189} \\ &=a_+T_+^4\left[\alpha_+-\frac13\left(1-\frac{a_-}{a_+}\right)\frac{T_-^2}{T_+^2}\right], \tag{190} \end{align} where in the forth line the integral is approximated by its average value across the wall, \begin{align}\left\langle\frac{\partial\mathcal{F}}{\partial T^2}\right\rangle \equiv\frac12\left(\frac{\partial\mathcal{F}_+}{\partial T_+^2}+\frac{\partial\mathcal{F}_-}{\partial T_-^2}\right), \tag{191} \end{align} and in the last line the ratio of temperatures across the wall can be inferred from eq. (52), \begin{align}\frac{w_-}{w_+}=\frac{a_-T_-^4}{a_+T_+^4}=\frac{\bar{v}_+\bar{\gamma}_+^2}{\bar{v}_-\bar{\gamma}_-}\Rightarrow\frac{T_-^2}{T_+^2} =\sqrt{\frac{a_+}{a_-}\frac{\bar{v}_+}{\bar{v}_-}\frac{\bar{\gamma}_+^2}{\bar{\gamma}_-^2}}. \tag{192} \end{align} The last term in EOM eq. (168), when multiplied by $\phi'(\bar{z}')$ and integrated across the bubble wall, gives rise to the friction force, \begin{align}F_\mathrm{fr} &\equiv\int_{-\infty}^\infty\mathrm{d}\bar{z}'\phi'(\bar{z}')T_N\widetilde{\eta}u^\mu\partial_\mu\phi \tag{193} \\ &=\int_{-\infty}^\infty\mathrm{d}\bar{z}'\phi'(\bar{z}')T_N\widetilde{\eta}\frac{\bar{\gamma}(\bar{v})}{a(\bar{t}+\bar{t}_n)} \\ & \times\left(\phi'(\bar{z}')[\dot{\bar{\gamma}}_w(\bar{z}-\bar{z}_w)-\bar{\gamma}_w\dot{\bar{z}}_w]+\bar{v}\phi'(\bar{z}')\bar{\gamma}_w\right) \tag{194} \\ &=\frac{T_N\widetilde{\eta}}{a}\int_{-\infty}^\infty\mathrm{d}\bar{z}'\phi'(\bar{z}')^2 (\bar{\gamma}\bar{v}\bar{\gamma}_w-\bar{\gamma}\bar{\gamma}_w\bar{v}_w) \tag{195} \\ &=\frac{\sigma}{a}T_N\widetilde{\eta}\left(\bar{\gamma}_w\langle\bar{\gamma}\bar{v}\rangle-\bar{\gamma}_w\bar{v}_w\langle\bar{\gamma}\rangle\right). \tag{196} \end{align} If one introduces $\eta$ to simply parameterize the friction term as $\eta~a_NT_N^4\bar{\gamma}_w\langle\bar{\gamma}\bar{v}\rangle$, then the peculiar wall velocity of a stationary bubble expansion can be obtained from: $F_\mathrm{dr}=F_\mathrm{fr}$, namely \begin{align}\alpha_+-\frac13\left(1-\frac{a_-}{a_+}\right)\frac{T_-^2}{T_+^2}=\eta\frac{\alpha_+}{\alpha_N}\bar{\gamma}_w\langle\bar{\gamma}\bar{v}\rangle, \tag{197} \end{align} which can be readily solved for given $\alpha_N(\alpha_+)$ and $\eta$. In practice, $\alpha_N(\alpha_+)$ and $\bar{v}_w$ are input into above equation to see if the outcome of $\eta$ could match the estimation from the microphysics of specific model.

It should not be confused with the fluid velocity $\bar{v}(\bar{\xi}_w)$ at the bubble wall in the previous sections.

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• Figure 1

(Color online) Illustration for the junction condition at the bubble wall. The blue shaded region is in true vacuum and the red shade region is in false vacuum. For a local region at the bubble wall, the junction condition can be derived at an interface $\Sigma$ inside some volume $\mathcal{V}$ enclosed by surface $\mathcal{S}$.

• Figure 2

(Color online) The wall frame peculiar velocity $\bar{v}_-$ and $\bar{v}_+$ of bulk fluid in the back ($x$-axis) and front ($y$-axis) of the bubble wall for given strength factor $\alpha_+$. The blue shaded region is detonation mode, and the red dashed region is deflagration mode. The green shaded region is forbidden so that only the weak detonation and weak deflagration are allowed. The strong deflagration will be decay into the Jouguet deflagration, which is the hybrid mode with $\bar{v}_-=c_s^-$.

• Figure 3

(Color online) The similarity solutions $v(\xi)$ and $\bar{v}(\bar{\xi})$ without input matching conditions to the EOM of bulk fluid motion in bubble center frame for fast (a) and slow (b) first-order phase transitions in flat (a) and FLRW (b) background. In flat background, the grey shaded region above $v=\xi$ is forbidden, and the red shaded region is deflagration mode while blue shaded region is detonation mode, which are separated by the shockwave front defined by $\mu(\xi,v)\xi=c_s^2$ and rarefaction front defined by $\mu(\xi,v)=c_s$. The same classification is also shown for FLRW background in comoving coordinate system, except that the solution curves will not be ended at the same improper node point $(\bar{\xi},\bar{v})=(c_s,0)$ as in the flat background any more.

• Figure 4

(Color online) The velocity profiles $v(\xi)$ (left column) and $\bar{v}(\bar{\xi})$ (right column) of the EOM with input junction conditions from detonation (first line), deflagration (second and third lines) and hybrid (last line) modes in fast (left column) and slow (right column) first-order phase transitions for a given strength factor $\alpha_+$ and a bubble wall velocity $\xi_w$ (left column) and $\bar{\xi}_w$ (right column). The corresponding enthalpy profiles are also presented as small panels. The main difference of velocity profiles is that they are more narrow in the slow first-order phase transition than in the fast first-order phase transition.

• Figure 5

(Color online) The velocity profiles are shown in first line for $\alpha_+=0.1$ and different bubble wall velocities. In second line, $\alpha_+$ as a function of the bubble wall velocity are plotted for $\alpha_N=0.01,~0.03,~0.1,~0.3,~1,~3$. The velocity profiles are shown in third line for $\alpha_N=0.1$ and different bubble wall velocities. In last line, the maximal flow velocity in bubble center frame as a function of the bubble wall velocity is shown for $\alpha_N=0.01,~0.03,~0.1,~0.3,~1,~3$. All panels in left and right columns are solved for fast and slow first-order phase transitions in flat and FLRW backgrounds, respectively.

• Figure 6

(Color online) The fluid peculiar velocities in the bubble wall frame with respect to the bubble wall peculiar velocity for given $\alpha_N=0.1$. The fluid peculiar velocities $\bar{v}_+$ and $\bar{v}_-$ just in the front (red solid line) and back (blue solid line) of the bubble wall are shown along with the rarefaction wave (blue shaded region) and compression shockwave (green shade region) proceeded with shock front peculiar velocity $\bar{\xi}_{sh}$ (green solid line).

• Figure 7

(Color online) The efficiency factors with respect to the bubble wall (peculiar) velocity for given asymptotic strength factors for fast (top) and slow (middle) first-order phase transitions and their ratio (bottom).

• Figure 8

(Color online) The numerical fitting of efficiency factors $\kappa_A$, $\kappa_B$, $\kappa_C$ and $\kappa_D$ are shown in (a)-(d) if the bubble wall velocity is non-relativistic, acoustic, Jouguet and ultra-relativistic, respectively. The slop of efficiency factor at the continuous transition from deflagration region to hybrid region is fitted in (e). It is worth noting that the slop of efficiency factor at the transition from hybrid region to detonation region is not continuous. Interpolating these fitting formulas at the boundaries of deflagration, hybrid and detonation regions, one could find the fitting formulas over the whole parameter space of $(\bar{\xi}_w,\alpha_N)$.

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