Gravitational Wave Echoes of the First Order Phase Transition in a Kination-Induced Big Bang

Abstract

Gravitational waves (GWs) produced during first-order phase transitions (FOPTs) in the early universe provide a powerful probe of nonstandard cosmological histories. We study GW production from a FOPT ending a kination-dominated epoch in the Kination-Induced Big Bang scenario, in which a period of kination domination terminates through a phase transition that reheats the universe into radiation domination. A rolling scalar field drives the kination epoch. In the specific model we consider, its derivative coupling to a second scalar (tunneling field) dynamically traps the latter in a false vacuum, with the phase transition triggered as the kination field slows due to Hubble friction. We compute the resulting stochastic GW background from bubble nucleation and collisions, presenting analytic estimates and numerical results for the peak amplitude and frequency. In all cases we find an upper bound $Ω_{\rm GW} h^2\lesssim 2\times10^{-7}$ from the bubble percolation condition. In the case where the false vacuum energy dominates at the transition (yet the kination field drives the FOPT), we find $Ω_{\rm GW}h^2\gtrsim 10^{-12}$. We further find that the Hubble scale during the phase transition across a broad set of model parameters is bounded by $\mathscr{O}(10^{-13})M^2/M_{\rm Pl}\lesssim H_* \lesssim \mathscr{O}(0.1)M^2/M_{\rm Pl}$, where $M$ is the mass-scale controlling the strength of the interaction between the kination and tunneling fields. The predicted signal spans frequencies from nHz to MHz, allowing the model to explain the signal reported by Pulsar Timing Array experiments and to be constrained or probed by interferometers such as LISA, Advanced LIGO, Cosmic Explorer, and BBO. Interestingly, a FOPT can occur even if the bare tunneling potential has a single minimum, as metastability is generated dynamically by the coupling between the tunneling and the kination field.

Figures (18)

• Figure 1: The potential in a kinetically induced first order phase transition as a function of the tunneling field amplitude and the kination field velocity $V=V(\chi,\dot\phi)$. The kination field $\phi$ initially has a large velocity $\dot\phi$. The tunneling field $\chi$ is then trapped in a metastable minimum through its coupling to the kination field ${\cal L}_{\rm {int}} = -\chi^2(\partial\phi)^2/M^2$. While the kination field slows down through Hubble friction as $\dot\phi\propto a^{-3}$ a second deeper minimum occurs along the $\chi$-direction. Once the barrier between the two minima becomes sufficiently small $\chi$ tunnels into the true minimum and kination ends through a first order phase transition.
• Figure 2: The exact Lambert $W$ function is shown in red. The small-argument approximation $W(x)\simeq x$ is shown in black dot-dashed. The full large-argument approximation $W(x)\simeq \ln (x) \left(1-\frac{\ln (\ln (x))}{1+ \ln (x)}\right)$ is shown in blue-dashed. Successively more drastic approximations are shown in green-dashed $W(x)\simeq \ln (x) \left(1-\frac{\ln (\ln (x))}{\ln (x)}\right)$ and then the brown-solid $W(x)\simeq \ln(x)$. Our parameters lie in the $x\gg 1$ regime and we will thus use the successively more drastic approximations given by the green-dashed and brown lines in Eqs. \ref{['eq:phidotcAB']} and \ref{['eq:phidotfinalsimplification']} respectively.
• Figure 3: Three panels, clockwise from the top left: The determination of $\dot\phi_c$ through the intersection of the numerically derived curves for $\Gamma$ (blue) and $\beta^4/8\pi$ (red) for $(M,\mu, m_\chi, \lambda) = (10^{10}{\rm {MeV}}, 0.01 \lambda^2 M, 0.1\mu, 0.01)$, $(10^{10}{\rm {MeV}}, 0.01 \lambda^2 M, 2\mu, 0.01)$ and $(10^{10}{\rm {MeV}}, 0.01 \lambda^2 M, 0.1\mu, 0.1)$. The vertical gray lines show our analytic approximation for $\dot{\phi_c}$. One can see that our analytic approximation can be used as a good starting point for finding the exact value of $\dot\phi_c$, which is necessary for numerically computing the peak frequency and amplitude of GW spectra (see Section \ref{['sec:parameterscan']}). The bottom left panel shows the ratio $\dot\phi_c^2 / M^2\mu^2$ as a function of $\mu$. The approximate value of Eq. \ref{['eq:phidotsimplest']}, which is derived using our analytic approximations for $\lambda, m_\chi/\mu\ll 1$, is shown in black. The numerical value in the same regime with $\lambda=0.01$ and $m_\chi=0.1\mu$ is shown in blue. The other colors refer to cases not compatible with the limits of our approximate solution in the black line. The red curve has $\lambda=0.01$ but $m_\chi=\mu$ and the green curve has $\lambda=0.01$ and $m_\chi=2\mu$. The orange and brown curves have $\lambda=0.1, 0.3$ respectively, while keeping $m_\chi=0.1\mu$. In all cases, $\dot\phi_c^2 / M^2\mu^2 = O(1)$.
• Figure 4: Various quantities of interest as a function of the ratio $m_\chi/\mu$. For all panels $M=10^{10}$ MeV, solid curves correspond to $\mu/M=0.1\lambda^2$, $\lambda=0.01$. Dashed curves correspond to $\mu/M=0.01\lambda^2$, $\lambda=0.01$, dotted curves correspond to $\mu/M=0.1\lambda^2$, $\lambda=0.1$ and dot-dashed curves correspond to $\mu/M=0.01\lambda^2$, $\lambda=0.1$. Left: The inverse duration of the phase transition $\beta/H$ (blue) and $6S_4$ (red). In the limit $m_\chi / \mu\ll 1$ we expect $\beta/H = 6S_4$, which is verified by the overlapping blue and red curves in that limit. The horizontal green curve corresponds to the analytic prediction for $\beta/H$ in the limits $\lambda \ll 1$ and $m_\chi/\mu\ll 1$ (the regime of the analytic calculations). The horizontal black line shows the lower bound $\beta/H=3$ required for successful bubble percolation. Middle: The effective mass (red) and the rescaled critical velocity (blue). The black curve shows $m_\chi^2/\mu^2$ and the horizontal green line corresponds to the approximate expression of $2 \dot\phi_c^2/M^2\mu^2$ per Eq. \ref{['eq:phidotsimplest']}. In the limit of $m_\chi / \mu\ll 1$, $m_{\chi,\rm{eff}}^2 \simeq 2\dot\phi^2_c/M^2$, as can be seen by the fact that the black curve is subdominant and the red and blue curves overlap in that limit. Right: The energy ratio $\alpha$ (blue), along with the analytical approximation of Eq. \ref{['eq:alphacalc']} (red). The horizontal green line shows $\alpha=1$.
• Figure 5: GW production for the case $\alpha>1$ (where the FOPT triggered by the kination field takes place during a brief period of vacuum domination subsequent to the kination domination): Upper panel: The peak amplitude and frequency of the stochastic GW spectrum in our model of a kinetically-induced FOPT, along with the sensitivity curves for various experiments shown as black curves. The upper horizontal gray line corresponds to the absolute maximum value of $\Omega_{GW}^{peak}h^2 \simeq 2\times 10^{-7}$, which is reached at the threshold of bubble percolation $\beta/H_*=3$. For all colored curves shown we fix $\lambda=0.01$ and $r=\mu/(\lambda^2 M)=0.1$, leading to $\alpha>1$ and thus no suppression of GW production. The green curve corresponds to our analytic estimate of Eq. \ref{['eq:omegafapprox']}, which is a function solely of $M$, obtained in the limit $m_\chi / \mu \ll 1$ for fixed $\lambda, r$. The red curves correspond (bottom to top) to $m_\chi/\mu =0,0.5,1,1.4,1.8,2,2.2,2.4,2.6$ with the curve $m_\chi/\mu=0.5$ being dashed and all others being solid. For each red curve, $M$ grows from left to right in the range $1\le M/{\rm {GeV}}\le 10^{12}$ (although some reach the percolation threshold, and correspondingly the horizontal gray line, before reaching $M= 10^{12}\,{\rm {GeV}}$). The blue curves correspond (left to right) to a fixed value of $M/{\rm {GeV}}= 1,10^2,10^4,10^6,10^8,10^{10},10^{12}$ with $m_\chi/\mu$ growing from $0$ at the bottom until each curve reaches the percolation threshold. Lower panels: The quantity $\beta/H_*$ (where $\beta^{-1}$ is the duration of the phase transition) as a function of $M$ (left) and $m_\chi/\mu$ (right) for the same parameters as in the upper panel. The horizontal gray line shows the percolation threshold at $\beta/H_* = 3$. In the lower left panel, different colors corresponds to different values of $m_\chi/\mu$ ranging from $m_\chi/\mu=0$ (top, dark red) to $m_\chi/\mu=2.6$ (bottom, yellow). The dashed curve corresponds to $m_\chi/\mu=0.5$ and we see that it is almost indistinguishable from $m_\chi/\mu=0$. In the lower right panel, different shades of blue correspond to different values of $M\in [1,10^{12}]\, {\rm {GeV}}$, the same range used in the upper panel. In the lower right panel, we see that $m_\chi/\mu$ could only exceed $\sim 3$ for values of the EFT cutoff $M < 1$GeV, too low to accommodate BBN after the phase transition.