## Author(s)

Garcia Garcia, Isabel, Petrossian-Byrne, Rudin## Abstract

We show that interactions between axion-like particles (ALPs) and co-dimension one defects, such as phase-transition bubble walls and solitonic domain walls, can lead to important changes in the evolution of both walls and ALPs. The leading effect arises from the change in the ALP decay constant across the interface, which naturally follows from shift-symmetric interactions with the corresponding order parameter. Specifically, we show that for thin walls moving relativistically, an ALP background -- such as e.g. axion dark matter -- gives rise to a frictional force on the interface that is proportional to $\gamma^2$, with $\gamma$ the Lorentz factor of the wall, and that this effect is present in both the oscillating and frozen axion regimes. We explore the broader consequences of this effect for bubble and domain walls in the early universe, and show that this source of friction can be present even in the absent of a conventional medium such as radiation or matter. Possible implications include modifications to the dynamics of bubble and domain walls and their corresponding gravitational wave signatures, as well as the generation of a dark radiation component of ALPs in the form of ultra-relativistic `axion shells' with Lorentz factor $\gamma_\text{shell} \simeq 2\gamma^2 \gg 1$ that may remain relativistic until the present day.

## Figures

In general, the decay constant of an axion can change across a domain or bubble wall formed in the early universe as a result of spontaneous symmetry breaking. The change takes place smoothly over some distance $L$ corresponding to the wall thickness. Typically, a change in decay constant also leads to a change in the mass of the canonically normalized axion field, as indicated in the figure. WLOG, we take the decay constant to vary from $f^2$ to the right of the wall to $\tilde f^2 = f^2 + \Delta f^2$ to the left of the wall.

Colored contours correspond to values of $r_{\text{dm}, *}^{1/2} \left( \frac{\Delta f^2}{f^2} \right)$ for which the bubble walls of a cosmological phase transition reach an equilibrium velocity, as a function of the temperature of the thermal plasma at which the transition takes place. The contours are obtained by requiring that $\gamma_\eq \leq \min \left\{ (mL)^{-1}, (H_* R_n)^{-1} \right\}$. Each color corresponds to a different ALP mass, as indicated. For illustration, we take $\alpha = 0.1$, $R_n = L$, and $L=T_*^{-1}$.

The various colored contours are obtained as described in the caption of Fig.~\ref{fig:plots}. Each color corresponds to a different ALP mass and decay constant, as indicated, such that the correct dark matter relic density is obtained via misalignment with $\theta_i \sim 1$. For illustration, we take $\alpha = 0.1$, $R_n = L = v^{-1}$ with $v = 10^{-2} f$.

Dark matter axions kicked by a phase transition in the early universe can potentially survive till today as relativistic relics, though this is highly sensitive to many parameters. We show here contours of the present day energy of the relic axions for two example scenarios with $\alpha = 0.1$ (which saturates the limit $\Delta N_{\rm eff}\lesssim 0.3$), and the ALP DM field is fixed by the misalignment mechanism with $\theta_i=1$ and $\Omega_{\rm ALP} \approx 0.25$. The red line ($H=m$) separates regimes when the axion is frozen (top) and oscillating (bottom) during the PT. The two shaded regions denote points where equilibrium due to axion friction is inconsistent (bottom), or too late (top). \textbf{Left:} `Thermal transition' -- all scales $v,L,R_n$ are set by $T_*$. The $PQ$ radial mode is taken relatively light, $m_\rho = 10^7\GeV$, and $\eta = 0.1$. The black dot corresponds to the benchmark point explored in text. In the white region reflected axions lose their kinetic energy just by redshift, and are a tiny fraction of the dark matter today. \textbf{Right:} `Cold transition', all scales are set by the UV $f$, though the transition occurs much later by tunnelling. $\eta = 0.01$.

We study the relative importance of axion dark matter friction to that of SM transition radiation, in theories where the electroweak PT is first order and super-cooled, here as a function of axion decay constant $f$ and the effective scale $m_\rho/\sqrt{\eta}$ suppressing the axion-Higgs portal coupling. For this example, we take a vacuum energy difference $\Delta V = (100 \GeV)^4$ and a transition temperature $T_*=150 \MeV$ close to the QCD catalysis scale. The axion mass is chosen by Eq.(\ref{eq:massFromf}) with $\theta_i =1$, which gives the right present day DM in the `frozen' regime ($m \ll H_{\rm ew}$ during the PT) to the right, and a much smaller abundance in the `oscillating' regime ($m \gg H_{\rm ew}$) to the left, as explained in the text. We highlight the boundary between the two regimes around $f\sim 2\cdot 10^{13}\GeV$, where the exact solution starts to quickly oscillate, and we appropriately replace it by an average there onwards. $\gamma_{\rm a, eq}$ and $\gamma_{\rm SM, eq}$ are equilibrium Lorentz factors of the bubble expansion, if the axion and SM processes were the only source of friction pressure respectively. For this example, the axion field dominantes only in the small dark blue region of negative contours above $f\gtrsim 10^{13}\GeV$. The ratio $\gamma_{\rm a, eq}/\gamma_{\rm SM, eq}$ is directly linked to the energy budget at the end of the PT according to Eq.\eqref{eq:EnergyBudget}. Taking $L,R_n = v_{\rm ew}^{-1}$, the two shaded regions rule out points where equilibrium is inconsistent (left), or is too late (right). For $L, R_n = T_{*}^{-1}$, these constraints move to the inner dashed lines.

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