We study the spectrum of gravitational waves produced by a first order phase transition in a hidden sector that is colder than the visible sector. In this scenario, bubbles of the hidden sector vacuum can be nucleated through either thermal fluctuations or quantum tunnelling. If a cold hidden sector undergoes a thermally induced transition, the amplitude of the gravitational wave signal produced will be suppressed and its peak frequency shifted compared to if the hidden and visible sector temperatures were equal. This could lead to signals in a frequency range that would otherwise be ruled out by constraints from big bang nucleosynthesis. Alternatively, a sufficiently cold hidden sector could fail to undergo a thermal transition and subsequently transition through the nucleation of bubbles by quantum tunnelling. In this case the bubble walls might accelerate with completely negligible friction. The resulting gravitational wave spectrum has a characteristic frequency dependence, which may allow such cold hidden sectors to be distinguished from models in which the hidden and visible sector temperatures are similar. We compare our results to the sensitivity of the future gravitational wave experimental programme.

Article funded by SCOAP3

$, so the resulting hidden sector gauge boson masses are $m_A = \frac{1}{2} g \left<\phi\right>$. We consider theories with $g^2 \gg \lambda$ and further assume that the tree level mass squared in eq.~\eqref{eq:phipot} satisfies $\left| m^2 \right| \ll \left<\phi\right>^2$ (this will be seen to be consistent). In this part of parameter space the 1-loop Coleman-Weinberg potential is comparable to the tree level potential~\cite{Coleman-ml-1973jx}. We make this choice because the combination of the tree and 1-loop potentials can lead to a first order phase transition but also to induce a barrier in the potential at zero temperature between a meta-stable vacuum and the true vacuum. Neither of these features are possible if the tree level potential eq.~\eqref{eq:phipot} dominates. In the regime we consider it is convenient to write the mass squared parameter in eq.~\eqref{eq:phipot} in terms of a dimensionless parameter $\tilde{m}^2$ and a renormalisation group (RG) scale $w$ \beq m^2 \equiv \tilde{m}^2 \frac{9 g^4}{1024 \pi^2} w^2 ~. \eeq We choose the RG scale to coincide with the VEV $\left<\phi\right> = w$ (in parts of parameter space in which a symmetry breaking vacuum exists). It is straightforward to evaluate the 1-loop correction to $\phi$'s potential. Given the assumption of a small quartic coupling we can neglect loops of $\phi$ itself, and the result comes only from the hidden sector gauge bosons. After adding appropriate counterterms, the total zero temperature potential is \beq \label{eq:v0} V_0\left(\phi\right) = \frac{9 g^4}{1024 \pi^2} \left[ \frac{1}{2} \tilde{m}^2 w^2 \phi^2 + \phi^4 \left( \log\left(\frac{\phi^2}{w^2}\right)- \frac{(\left(2+\tilde{m}^2 \right)}{4} \right) \right]~. \eeq As usual the quartic coupling has been replaced with the renormalisation scale $w$ and renormalisation conditions by dimensional transmutation.\footnote{The contribution to the zero temperature potential from $\phi$ itself is of the form $V = \frac{1}{64 \pi^2} \left( V''(\phi) \right)^2 \log(\frac{V''\left(\phi\right)}{w^2})$. Since $V'' \sim 9 g^4 w^2/(64 \pi^2)$ our analysis is consistent for $g\sim 1$.} If $\tilde{m}^2\leq 0$ the point $\left<\phi\right> =0$ is unstable at zero temperature, if $0<\tilde{m}^2<2$ it is a metastable minimum, and if $\tilde{m}^2>2$ this is the true vacuum. If $0<\tilde{m}^2<2$ the difference in energy density between the true vacuum at $\left<\phi\right> \neq 0$ and the metastable vacuum at $\left<\phi\right> =0$ at zero temperature is \beq \label{eq:deltaV} \rho_{\rm vac} = \frac{9 g^4}{1024 \pi^2} \frac{\left(2- \tilde{m}^2 \right)}{4} w^4 \,, \eeq and the mass of $\phi$ in the symmetry breaking vacuum is \beq \label{eq:mphi2} m_{\phi}^2=\frac{9}{512\pi^2} \left( 4-\tilde{m}^2\right) g^4 w^2 ~. \eeq Phase transitions in the early universe depend on $\phi$'s potential at finite temperature. The simplest estimate of this is obtained by combining the zero temperature potential, eq.~\eqref{eq:v0}, with the naive one loop finite temperature correction $V_T$~\cite{Kapusta-ml-2006pm} \beq V\left(\phi\right)= V_0\left(\phi\right) + V_T\left(\phi\right)~. \eeq The contribution to $V_T\left(\phi\right)$ from the hidden sector gauge bosons is \beq \label{eq:finiteTpot} V_T\left(\phi\right) \supset \frac{n_i T^4}{2\pi^2} \int_0^{\infty} q^2 \log \left(1- \exp\left(-\sqrt{q^2+m_A^2\left(\phi\right)/T^2} \right) \right)~dq \,, \eeq where $n_i=9$ and $m_A\left(\phi\right) = \frac{1}{2} g \phi$. The one loop correction from $\phi$ loops has a similar form and at temperatures around the time of the phase transition is subleading to eq.~\eqref{eq:finiteTpot}, as is the case with the zero temperature potential.\footnote{This can be seen directly by expanding the integral analogous to that in eq.~\eqref{eq:finiteTpot}.} Although it demonstrates the existence of a phase transition, the simple one-loop thermal potential is known to lead to significant inaccuracies in many models and can even lead to incorrect predictions of the order of a transition. Different approaches have been proposed to capture the effects missed by eq.~\eqref{eq:finiteTpot} (a recent discussion can be found in~\cite{Curtin-ml-2016urg}). In our present work we are interested in phenomenological possibilities rather than precise demarcation of the parameter space. It is therefore sufficient to only improve eq.~\eqref{eq:finiteTpot} by resumming an infinite set of daisy diagrams. This fixes the most severe shortcoming of eq.~\eqref{eq:finiteTpot} by removing IR divergences that would otherwise spoil the perturbative loop expansion~\cite{Carrington-ml-1991hz,Fendley-ml-1987ef}. In practice the daisy resummation can be performed by simply replacing the masses in eq.~\eqref{eq:finiteTpot} \beq \label{eq:daisy} m_i^2\left(\phi\right) \rightarrow m_i^2\left(\phi\right) + \Pi_i\,, \eeq where $\Pi_i$ is the finite temperature self energy of the species $i$. At leading order in $T^2$ the longitudinal components of the hidden sector gauge bosons have $\Pi_{{\rm long}} = \frac{11}{6} g^2 T^2$ and the transverse gauge bosons have no dependence at this order, and $\Pi_{\phi} = \frac{1}{2} \left(g^2 + \lambda \right)T^2$~\cite{Espinosa-ml-1995se}. \begin{figure}[t]\centering \includegraphics[height=4.8cm]{pot1.pdf}\hspace{0.1cm} ]]>

2m_h$), but for our purposes it is enough to note that its decay rate once EW symmetry is broken, assuming $m_{\phi}\gg m_{h}$, is \beq \Gamma_{\phi} \simeq \frac{\lambda_p^2 w^2 }{32 \pi m_{\phi}} \,, \eeq which leads to a lifetime \beq \label{eq:gammahp} \Gamma^{-1}_{\phi} \simeq \left(\frac{10^{-12}}{\lambda_p}\right)^2 \left(\frac{100\,\GeV}{w} \right) {\rm s} ~. \eeq In some parts of parameter space eqs.~\eqref{eq:gammaferm} and~\eqref{eq:gammahp} allow $\phi$ to decay before BBN for values of $\lambda_p$ that do not destroy the temperature hierarchy between the hidden and visible sectors. However, this is not possible if $\phi$ is relatively light. In such cases, the simplest option to obtain a viable model is to introduce light hidden sector fermions, with mass $m_{\psi}$, that $\phi$ can decay to. We assume that the coupling of $\phi$ to these states is sufficiently large that it decays fairly fast, but that these states only interact with each other weakly and after the phase transition their comoving number density is constant. If $\alpha_{\rm h} \gtrsim 1$ their yield can be estimated similarly to eq.~\eqref{eq:phialpye} and to avoid overclosure of the universe requires \beq \label{eq:mpsilim} m_{\psi} \lesssim \frac{\eV}{\alpha^{3/4}}\,, \eeq while if $\alpha_{\rm h}\lesssim 1$ we need \beq m_{\psi} \lesssim \frac{\eV}{\epsilon^{3}}\,, \eeq similarly to how eq.~\eqref{eq:phirel} was derived. If $\alpha \sim 1$ the relic population of $\psi$ forms a warm dark matter component, which must be subdominant to the main cold dark matter. There can also be a significant relic abundance of the hidden sector gauge bosons. Given our assumption about the hierarchy of masses in the hidden sector these have an annihilation channel to $\phi$ (with a cross section that is proportional to $g^2/w^2$), so their relic abundance is set by freeze-out. If the hidden sector is at approximately the same temperature as the visible sector, the gauge boson relic abundance is the same as has been studied in the literature. In this case there are large regions of parameter space that have either an under-abundance of gauge boson dark matter, or the full required abundance (provided that $w \lesssim 10^5 \,\GeV$ due to the usual unitarity bound)~\cite{Hambye-ml-2008bq}. If the hidden sector is cold relative to the visible sector, this will affect the gauge boson relic abundance. The Hubble parameter will be larger when the hidden sector temperature drops below $m_A$, owing to the energy density in the visible sector, so freeze-out will happen at a slightly higher hidden sector temperature than would otherwise be the case. On the other hand, the final dark matter yield will be dramatically decreased since the entropy of the universe is higher, which typically more than compensates the previous effect. An upper bound on the gauge boson yield can be obtained similarly to that of $\phi$ in eq.~\eqref{eq:phirel} and~\eqref{eq:phialpye}. This corresponds to assuming that gauge boson annihilations become inefficiently immediately after the phase transition. Since we consider parts of parameter space in which the gauge bosons masses are similar to the mass of $\phi$, the resulting bounds on $\alpha$ and $\epsilon$ are similar to eqs.~\eqref{eq:epconstPHI} and~\eqref{eq:alphaRelicLim}. We therefore conclude that if $\alpha$ and $\epsilon$ are such that a population of stable $\phi$ particles is cosmologically safe, the hidden sector gauge bosons will also not overclose the universe. Models in which $\phi$ can decay to the visible sector via a Higgs portal operator are only possible with not too different hidden and visible sector temperatures (otherwise the temperature hierarchy would be erased), so there are large regions of parameter space such that the gauge boson relic abundance is viable. Finally in models that require light hidden sector fermions for $\phi$ to decay to, these states can break the hidden sector custodial symmetry, and therefore allow the gauge bosons to also decay. ]]>

=0$) is initially favoured. As before the temperature of the visible and hidden sectors are allowed to differ by a ratio $\epsilon$, and we assume there is no energy transfer between the sectors.\footnote{When presenting results we quote $\epsilon$ at the time of the hidden sector phase transition.} In figure~\ref{fig:contours} we plot contours of the minimum values of $S_4$ and $S_3/T_{\rm h}$ as a function of the dimensionless parameters of the hidden sector, for models such that a barrier persists at zero temperature. The value of $w$ does not affect these results, since it is the only relevant scale in the calculation. We consider relatively large gauge couplings $1\gtrsim g \gtrsim 2.5$. Towards the upper end of this range the accuracy of our perturbative calculations may be compromised, however since $g<\sqrt{4\pi}$ we do not expect the qualitative dynamics to change significantly. Therefore, despite this source of potential numerical imprecision, we regard our hidden sector as a useful toy model to explore the phenomenological possibilities that can arise more generally. The minimum value of $S_3/T_{\rm h}$ is smaller than that of $S_4$ over all of the parameter space in figure~\ref{fig:contours}. Further, $S_3/T_{\rm h}$ is also smaller than $S_4$ at hidden sector temperatures $T_{\rm h} \sim w$ (as is the case for the points in parameter space shown in figure~\ref{fig:act}). These features are not surprising. As discussed in~\cite{Espinosa-ml-2008kw}, in the thin wall approximation the actions scale as \beq \label{eq:s3scal} S_{3}/T_{\rm h} \sim \frac{w}{T_{\rm h}} \left(\frac{w^4}{\Delta V}\right)^2 \,, \eeq and \beq \label{eq:s4scal} S_4 \sim \left(\frac{w^4}{\Delta V} \right)^3 \,, \eeq where $\Delta V$ is the difference in energy density between the two vacua. Even though the thin wall approximation often does not give precise numerical results, the feature that if $S_3/T_{\rm h} \gg 1$ then $S_4 \gg S_3/T_{\rm h}$ is typical across many classes of models (although it would be interesting to find theories for which it does not hold). As a result, if a first order phase transition happens at a temperature $T_{\rm h} \sim w$, this will be through nucleation of bubbles by thermal fluctuations (including in the case that no barrier remains at zero temperature). \begin{figure}[t]\centering \includegraphics[height=7cm]{s3p.pdf}\hspace{0.5cm} ]]>

0$. Given that $\gamma_{\rm w}$ reaches extremely large values in runaway transitions, $\sim 10^{14}$, such a contribution could prevent runaway walls even if it has an extremely suppressed coefficient, e.g.\ from multiple loop factors.\footnote{A $\propto \log \gamma_{\rm w}$ dependence might not be enough to stop the bubble walls accelerating before the transition completes.} This is a major source of uncertainty on the dynamics of the bubble walls in such models, and it is an important topic to resolve in the future. ]]>

n_{\rm v} < H(T_{\rm v}) \,, \eeq where $\left< \sigma v \right>$ is the thermally averaged cross section between the visible and hidden sectors and $n_{\rm v}$ is the number density of the relevant states in the visible sector.\footnote{A hidden sector that is hotter than the visible sector is less interesting. The universe must be dominated by the visible sector before BBN, so in this case the energy density in the hidden sector must be transferred to the visible sector before this time. There are regions of parameter space in which this is possible. However the gravitational wave signal that would arise is similar in shape and amplitude to if the two sectors were in thermal equilibrium throughout.} For relatively small temperature differences the condition in eq.~\eqref{eq:GammaI} is approximately sufficient to maintain a hierarchy. However, for large temperature ratios a stronger condition is required. The hidden sector temperature must not be increased by energy transfer from the visible sector, even if this energy is not enough for the two sectors to reach the same temperature. The resulting constraint can be estimated by demanding that the energy density transferred per Hubble time to the hidden sector is smaller than that already present in the hidden sector, all times prior to the phase transition. This corresponds to \beq \label{eq:cond} \frac{n_{\rm v}^2 \left<\sigma v\right> T_{\rm v}}{H^4} \lesssim \frac{T_{\rm h}^4}{H^3} \,, \eeq where we have dropped factors of order 1. If a hidden sector is initially colder than this, but the portal coupling is not large enough for full thermalisation, it would be partially heated up until eq.~\eqref{eq:cond} is satisfied. The hidden sector that we consider can couple to the visible sector in multiple different ways, and we focus on a simple Higgs portal operator as an example. This interaction takes the form \beq \label{eq:hpop2} \mathcal{L} \supset -\frac{1}{2}\lambda_p \left|\Phi\right|^2 \left|H \right|^2 \,, \eeq where $H$ is the SM Higgs doublet. Although $\lambda_p =0$ is technically natural, it is not unreasonable to suppose a non-zero value might be present, for example due to heavy states that couple to both sectors. If the hidden sector phase transition happens when the visible sector temperature is above the scale of the electroweak (EW) phase transition, the cross section between the two sectors at the relevant times is \beq \left<\sigma v\right> \sim \frac{\lambda_p^2}{32\pi T_{\rm v}^2} \,, \eeq and eq.~\eqref{eq:cond} becomes \beq \label{eq:cond2} \lambda_p \lesssim 10^{-8} \epsilon^{3/2} \left( \frac{w}{\GeV} \right)^{1/2}~. \eeq When $\epsilon=1$ this matches the previously known condition for the two sectors to not thermalise at temperatures above the EW scale $\lambda\lesssim 10^{-7}$~\cite{Bento-ml-2001yk}. As expected, smaller values of the portal couplings are required to maintain a large temperature hierarchy between the two sectors, and tiny portal couplings are needed if $\epsilon \ll 1$. The condition on $\lambda_p$ is different if the hidden sector phase transition happens when the visible sector temperature is below the EW scale. In this case the dominant energy transfer from the visible sector happens immediately after the EW phase transition, since at this point the decay channel $h \rightarrow \phi\phi$ is open and there is still a thermal population of the SM Higgs (which will later be exponentially suppressed).\footnote{Since we assume that the hidden sector is colder than the visible sector this is automatically kinematically allowed.} The resulting energy transfer can be approximated from the Higgs branching fraction to the hidden sector~\cite{Bento-ml-2001yk}, which leads to the constraint to maintain a temperature hierarchy \beq \label{eq:cond1a} \lambda_p \lesssim 10^{-10} \epsilon^2 ~. \eeq This again matches the condition for thermalisation when $\epsilon =1$, and it is much stronger for small $\epsilon$. Such tiny portal couplings are far beyond the reach of any direct experimental searches.\footnote{A temperature difference could be maintained with larger values of the portal coupling if the reheating temperature of the universe was below the EW scale. In this case energy is only transferred to the hidden sector through scattering of light SM fermions via an off-shell Higgs, and the rate that this occurs at is strongly suppressed. Portal couplings that are large enough to have observational consequences might be possible in this case, although we do not investigate it further.} ]]>

w$, and for the moment we also assume that the hidden and visible sectors are at the same temperature. The visible sector reheat temperature must be $\gtrsim 5\,\MeV$ to preserve the successful predictions of BBN. This constrains the inflaton decay rate $\Gamma_{\rm inf}$ to \beq \MeV \lesssim \left( \Gamma_{\rm inf} M_{\rm Pl} \right)^{1/2} \,, \eeq which means that the maximum temperature after inflation $T_{\rm max}$ (which is larger than the reheating temperature) will be~\cite{Giudice-ml-2000ex} \beq \begin{aligned} T_{\rm max} &\simeq \left(H_I M_{\rm Pl}^2 \Gamma_{\rm inf} \right)^{1/4} \\ &\gtrsim 3000 \,\GeV \left(\frac{H_I}{100\,\GeV} \right)^{1/4} ~. \end{aligned} \eeq Therefore for hidden sectors at a scale $w\lesssim \TeV$ the universe automatically reaches a temperature at which the hidden sector symmetry is restored if $H_I \gtrsim w$. Subsequently, as the temperature drops a thermal transition will happen in preference to a tunnelling transition, as in the previously considered case.\footnote{A minor caveat to this argument is that the relation between the Hubble parameter and temperature is altered during reheating, because the universe is matter dominated. This makes it slightly less likely that a transition completes through thermal fluctuations. In practice the difference between the two actions, e.g.\ from eqs.~\eqref{eq:s3scal} and~\eqref{eq:s4scal}, is sufficiently large that this does not lead to a tunnelling transition in any of the parameter space of our model.} For hidden sectors at higher scales it is possible that its symmetry is restored during inflation and the sector subsequently undergoes a tunnelling transition, despite being at the same temperature as the visible sector. This requires $T_{\rm max} < w$, and that the tunnelling transition happens before $H$ drops below $\sim w^2/M_{\rm Pl}$, when the universe would reenter an inflationary phase. The conflicting pressures of reheating above the scale of BBN and having $T_{\rm max} < w$ mean that tunnelling transitions only happen in a small region of parameter space. Additionally, the entropy injection by the inflaton decays after the phase transition dilute the present day gravitational wave signal in this case. These issues are avoided if the hidden sector is cold relative to the visible sector. The Hubble scale during inflation can be sufficiently high that the hidden sector symmetry is restored, $H >w$. However, if the inflaton decays dominantly to the visible sector the hidden sector temperature might never get close to $w$, despite the visible sector being reheated above the scale of BBN. In this case a thermal transition will not take place, but provided $S_4$ is sufficiently large a tunnelling transition can happen before the hidden sector vacuum energy density dominates the universe. The condition that a tunnelling transition completes is $\Gamma_4 \gtrsim \sqrt{\rho_{\rm vac}}/\left(3 M_{\rm Pl} \right)$, as before. The parts of parameter space that satisfy this condition are those in figure~\ref{fig:typeofT} right that undergo a transition (including parts shown as going through a thermal transition in the previous context of that plot). The visible sector temperature at the time of the transition is again given by eq.~\eqref{eq:TvisTun} (assuming, for simplicity, that the transition happens after reheating completes i.e.\ when the Hubble parameter is $\lesssim H\left(T_{\rm RH}\right)$). ]]>