Hearing without seeing: gravitational waves from hot and cold hidden sectors

TL;DR

This work investigates gravitational waves from first-order phase transitions in hidden sectors that are colder than the visible sector. Using a concrete SU(2) gauge theory with a dark Higgs, it shows how thermal transitions compete with tunnelling in cold sectors, and how bubble-wall friction and energy transfer shape the GW spectrum. The authors derive how cosmological constraints bound the temperature hierarchy and energy release, and they provide spectral fits for bubble collisions and sound waves to assess detectability in future experiments. A key finding is that cold hidden sectors can emit GWs in frequency ranges not allowed by BBN constraints, offering a possible indirect probe of otherwise inaccessible hidden sectors, though observable signals occupy limited regions of parameter space. Overall, the paper clarifies how the nature of the hidden sector temperature relative to the visible sector imprints on GW signals and informs prospects for experimental discovery.

Abstract

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.

Figures (10)

• Figure 1: The scalar potential $V\left(\phi\right)$ of the hidden sector that we consider, at different points in parameter space (left and right), at zero temperature (solid) and increasing temperature (dashed, dotted). The model plotted in the left panel has a barrier between the two vacua that remains at zero temperature, and the barrier disappears at zero temperature in the model in the right panel.
• Figure 2: Examples of the dependence of the critical bubble nucleation actions $S_3/T_{\rm h}$ and $S_4$ on the hidden sector temperature $T_{\rm h}$. The parameters $g$ and $\tilde{m}^2$ define the model via Eq. \ref{['eq:v0']}. The left panel corresponds to a hidden sector for which the energy barrier between the two minima vanishes at zero temperature, so both $S_3/T_{\rm h}$ and $S_4$ go to zero. The right panel corresponds to a model in which a barrier remains at zero temperature, so $S_4$ asymptotes to a constant and $S_3/T_{\rm h} \rightarrow \infty$.
• Figure 3: Contours of the minimum values of $S_3/T_{\rm h}$ (left) and $S_4$ (right) as a function of the parameters of the hidden sector (defined in Section \ref{['sec:model']}).
• Figure 4: The bubble nucleation rate as a function of the hidden sector temperature for a model with $g=2$ and $\tilde{m}^2=0.04$ (as in the left panel of Figure \ref{['fig:act']}) and $w=1~{\, {\rm GeV}}$. Results are shown for a hidden sector at the same temperature as the visible sector ($\epsilon=1$, left) and for a hidden sector that is much colder ($\epsilon=10^{-8}$, right). The Hubble parameter in the two cases is also plotted, assuming that the transition occurs prior to the hidden sector false vacuum energy dominating the energy density of the universe (dashed black), and assuming the phase transition does not complete prior to this (dotted black). If $\epsilon=1$ (left panel) the hidden sector false vacuum energy begins to dominate at $T_{\rm v}= T_{\rm h} \simeq 0.2 \rm {\, {\rm GeV}}$ when the two Hubble curves diverge. A transition happens when $H(t)$ is first smaller than one of $\Gamma_3^{1/4}$ or $\Gamma_4^{1/4}$. Therefore the warm hidden sector (left) undergoes a thermally nucleated transition, while the cold hidden sector (right) misses a thermal transition, but subsequently goes through a tunnelling transition.
• Figure 5: The type of phase transition that occurs (yellow: thermal, blue: tunnelling) over the hidden sector parameter space for $w=1~{\, {\rm GeV}}$, when the hidden sector is at the same temperature as the visible sector (left) and when the hidden sector is much colder with $\epsilon =10^{-8}$ (right). The contours give the values $\alpha$ for the transitions. In the white region no phase transition takes place. For cold hidden sectors there are models that cannot lead to an acceptable cosmological history, as described in the text.