WallGo investigates: Theoretical uncertainties in the bubble wall velocity

Abstract

We examine theoretical uncertainties in state-of-the-art calculations of the bubble wall velocity during first-order cosmological phase transitions. By utilising the software WallGo for two extensions of the Standard Model, we find several $O(1)$ uncertainties arising from the number of particles taken out of equilibrium, the logarithmically and power enhanced collision integrals, the treatment of thermal masses, the nucleation temperature, the $\tanh$ ansatz, and the perturbative order of the effective potential. However, we show that the linearisation of the Boltzmann equations is generally a good approximation with much smaller associated errors. We further clarify the limitations of the quasiparticle approximation in regions with negative mass squared. This study provides a detailed uncertainty budget and highlights where future efforts should be directed to improve the reliability of wall velocity and hence gravitational wave predictions.

Figures (11)

• Figure 1: Linearisation criteria across the xSM scan, computed by WallGo and stored in the WallGoResults object. Left: linearizationCriterion1, based on the ratio $\mathcal{P}[\delta f_1]/\mathcal{P}[f_{\rm eq}]$. Right: linearizationCriterion2, based on $\mathcal{P}[\delta f_2]/\mathcal{P}[f_{\tt WallGo}]$, vanishing in both the LTE and ballistic limits.
• Figure 2: The wall velocity for BM:xSM2 (left) and BM:IDM1 (right) for different choices of out-of-equilibrium particles, as indicated in the legend. Labels denote: $t$ for top quark, $g$ for gluon, $W$ for $W$ boson $b$ for bottom quark, and $A$ for the $A$ and $H^\pm$ scalars in the IDM. In the right panel, "QCD" and "QCD, Weak" indicate whether only QCD or both QCD and weak interactions were included in the collision terms. For the IDM, the bands reflect the uncertainty in $v_w$ arising from the choice of VEV used in the collision terms $v_{\rm coll} \to [0.5,1.0] \times v_{\rm n}$; see eq. \ref{['eq:VEV:col:factor']}.
• Figure 3: Contributing $2 \to 2$ scattering processes for $t_{\hbox{\tiny\rm{L}}} b_{\hbox{\tiny\rm{L}}} \to t_{\hbox{\tiny\rm{L}}} b_{\hbox{\tiny\rm{L}}}$ in the SM. Zig-zag lines denote $\mathrm{SU}(2)$ vector bosons ($Z$, $W^{\pm}$), curly lines represent gluons, and directed lines correspond to quarks ($t_{\hbox{\tiny\rm{L}}}$, $b_{\hbox{\tiny\rm{L}}}$). While there is only a single $t$-channel LL gluonic contribution Arnold:2003zc, the weak sector receives power-enhanced (P) contributions from the $u$-channel and NLL contributions from the $s$-channel diagrams.
• Figure 4: Estimate of the collision integral in the limit where $\mathcal{C} \propto 1/\delta f$, computed in the Standard Model with a Higgs mass of $m_{\hbox{\tiny\rm{$H$}}} = 34$ GeV, and only out-of-equilibrium, weakly interacting, left-handed top quarks. Left: power-enhanced terms. Right: logarithmically enhanced terms. See the main text for details on the plotted quantities.
• Figure 5: Bubble wall velocity $v_w$ as a function of collisionMultiplier. Left: BM:xSM1 with the top quark, $t$, out of equilibrium and varying $\lambda_{hs}$. Right: BM:IDM1 for $m_{\hbox{\tiny\rm{$A$}}} = 265$ GeV and different out-of-equilibrium particle sets, as indicated in the legend. The gray band indicates the range ${\tt collisionMultiplier} \in [10^{-1/2}, 10^{1/2}]$.