Observational prospects for gravitational waves from hidden or dark chiral phase transitions

We study the gravitational-wave (GW) signature of first-order chiral phase transitions ($\chi \mathrm{PTs}$) in strongly interacting hidden or dark sectors. We do so using several effective models in order to reliably capture the relevant nonperturbative dynamics. This approach allows us to explicitly calculate key quantities characterizing the $\chi \mathrm{PT}$ without having to resort to rough estimates. Most importantly, we find that the transition’s inverse duration $\beta$ normalized to the Hubble parameter $H$ is at least 2 orders of magnitude larger than typically assumed in comparable scenarios, namely, $\beta /H\gtrsim \mathcal{O}(10^4)$. The obtained GW spectra then suggest that signals from hidden $\chi \mathrm{PTs}$ occurring at around 100 MeV might be in reach of LISA, while DECIGO and BBO may detect a stochastic GW background associated with transitions between roughly 1 GeV and 10 TeV. Signatures of transitions at higher temperatures are found to be outside the range of any currently proposed experiment. Even though predictions from different effective models are qualitatively similar, we find that they may vary considerably from a quantitative point of view, which highlights the need for true first-principle calculations such as lattice simulations.

Figure 1

Temperature-dependent global minimum of the effective potential ${\overline{\sigma }}_{\min }$ as predicted by the employed low-energy effective models for benchmark point A of Table 1. The discontinuity at some critical temperature $T_c$ is characteristic of a first-order phase transition.

Figure 2

Action functional $S_3/T$ evaluated at the bounce solution as a function of the dimensionless ratio $T/T_c$ for benchmark point A of Table 1. For each of the considered effective models, we show the results of our explicit calculations (black circles) alongside the function $b(1-T/T_c{)}^{-\gamma }$ fitted to the aforementioned data points (solid lines). For comparison, we also plot interpolating cubic splines (dashed lines).

Figure 3

Gravitational-wave spectra as predicted for the benchmark points of Table 1 together with the power-law integrated sensitivity curve [112] for the LISA experiment assuming 5 years of data taking and a threshold signal-to-noise ratio of 5. The strain noise power spectral density for LISA was adopted from Ref. [113]. The displayed signal bands were computed from Eq. (40) for a fixed $g_{\star }=47.5$ by varying the bubble-wall velocity between $v_b=0.75$ and $v_b=1$. The dashed curves show the corresponding spectra obtained from Eqs. (44) and (47) for $v_b=1$ and the same $g_{\star }$ as before.

Figure 4

Inverse duration of the hidden chiral phase transition $\beta$ normalized to the Hubble parameter $H$ as a function of the transition’s critical temperature for benchmark point A of Table 1. The shown bands illustrate the dependence of the result on the effective number of relativistic d.o.f. $g_{\star }$, which range from $g_{\star }=1$ (upper edge) to $g_{\star }=100$ (lower edge).

Figure 5

Gravitational-wave spectra as predicted for the benchmark points of Table 1, but rescaled according to Eq. (51) with $\xi =100$ to obtain transitions at $T_n=\mathcal{O}(10\mathrm{GeV})$. The spectra are shown together with the power-law integrated sensitivity curves [112] for various space-borne GW experiments assuming 5 years of data taking and a threshold signal-to-noise ratio of 5. The strain noise power spectral densities for B-DECIGO, FP-DECIGO, and BBO were adopted from Refs. [113, 114, 115], respectively. The displayed signal bands were computed from Eq. (40) for a fixed $g_{\star }=47.5$ by varying the bubble-wall velocity between $v_b=0.75$ and $v_b=1$. The dashed curves show the corresponding spectra obtained from Eqs. (44) and (47) for $v_b=1$ and the same $g_{\star }$ as before.

Figure 6

Gravitational-wave spectra as predicted for the benchmark points of Table 1, but rescaled according to Eq. (51) with $\xi =10^3$ to obtain transitions at $T_n=\mathcal{O}(100\mathrm{GeV})$. The spectra are shown together with the power-law integrated sensitivity curves for various space-borne GW experiments (see the caption of Fig. 5 for details). The displayed signal bands were computed from Eq. (40) for a fixed $g_{\star }=47.5$ by varying the bubble-wall velocity between $v_b=0.75$ and $v_b=1$. The dashed curves show the corresponding spectra obtained from Eqs. (44) and (47) for $v_b=1$ and the same $g_{\star }$ as before.

Figure 7

Gravitational-wave spectra as predicted for the benchmark points of Table 1, but rescaled according to Eq. (51) with $\xi =10^5$ to obtain transitions at $T_n=\mathcal{O}(10\mathrm{TeV})$. The spectra are shown together with the power-law integrated sensitivity curves for various space-borne GW experiments (see the caption of Fig. 5 for details). The displayed signal bands were computed from Eq. (40) for a fixed $g_{\star }=47.5$ by varying the bubble-wall velocity between $v_b=0.75$ and $v_b=1$. The dashed curves show the corresponding spectra obtained from Eqs. (44) and (47) for $v_b=1$ and the same $g_{\star }$ as before. Note that even though the signals lie in the frequency range of ground-based observatories like LIGO, KAGRA, or the Einstein Telescope, the sensitivities of those experiments are insufficient for a detection; see, e.g., Ref. [116].

Figure 8

Signal-to-noise ratio as a function of the transition’s critical temperature for benchmark point A of Table 1. Calculations in which any potential suppression of the GW amplitude due to a shortened sound-wave period in very fast transitions is ignored (left) are compared to those where such a suppression is incorporated via the heuristic arguments leading to Eqs. (44) and (47) (right).

Figure 9

Possible topologies of two-loop diagrams contributing to $V_2$ [graphs (a) and (b)], as well as the associated one-loop contributions to the scalar self-energy [graphs (c) and (d)]. All dashed lines correspond to full, field-dependent mesonic propagators ${\overline{G}}_i$, with $i$ being either $\sigma$, $a$, ${\eta }^{\prime }$, or $\pi$. Vertices are obtained from the shifted LSM tree-level potential $V_{\mathrm{tree}}^{\mathrm{LSM}}(\overline{\mathrm{\Phi }}+\mathrm{\Phi })$, where $V_{\mathrm{tree}}^{\mathrm{LSM}}$ and the collective meson field $\mathrm{\Phi }$ are given in Eqs. (23) and (24), respectively, while $\overline{\mathrm{\Phi }}≔\overline{\sigma }\mathbb{1}/\sqrt{6}$. In the Hartree-Fock approximation used in the present paper, only diagrams of the double-bubble topology (a) are taken into account in calculating $V_2$, so that only tadpole graphs (c) contribute to the self-energy.