Hubble-Scale Tachyonic Shocks from Low-Scale Inflation -- A New Gravitational-Wave Window on Inflation

TL;DR

The paper investigates how low-scale, single-field inflation can produce rapid tachyonic instabilities at the end of inflation, driving nonlinear preheating on Hubble scales and generating gravitational waves. Through a hilltop/inflection potential $V(\phi)=V_0-\lambda\phi^{2n}$ and CMB normalization via $A_S$, the authors show an explosive growth of inflaton fluctuations before coherent oscillations and the formation of relativistic, bubble-like shocks. These shocks efficiently convert vacuum energy into gradient energy and radiate gravitational waves with a peak frequency set by the inflationary Hubble parameter, offering a novel observational window into MeV–EeV inflation scales. The mechanism remains robust for general potentials and could imply accompanying phenomena such as domain walls, baryogenesis, and dark-matter production, motivating searches for GW signals across current and future detectors.

Abstract

Current bounds on the tensor-to-scalar ratio imply that the energy scale of inflation may lie below the grand-unified scale. In this paper, we show that in a broad class of single-field inflation models with sufficiently small energy scales, an extremely efficient tachyonic instability develops at the end of inflation. This instability rapidly drives the system into a nonlinear regime before coherent oscillations can be established, leading to a first-order phase-transition--like phenomenon without tunneling or barrier crossing. The resulting ultra-relativistic shock fronts surrounding the bubble interiors expand to near the Hubble scale, corresponding to the most strongly enhanced tachyonic modes, and collide with one another, producing energetic inflaton particles and gravitational waves. As a result, the post-inflationary dynamics can differ significantly from the conventional high-scale inflationary scenario. Interestingly, inflation at MeV--EeV energy scales can be probed via gravitational-wave observations, including pulsar timing arrays, ground-based detectors, and future space-based experiments. Recent limits from the LIGO--KAGRA--Virgo collaboration already constrain EeV-scale inflation, while pulsar timing array results may be interpreted as evidence for gravitational waves generated by GeV-scale inflation. We also briefly discuss further implications of the resulting tachyonic shocks.

Figures (6)

• Figure 1: $\delta\phi_{\rm end}/\bar{\phi}_{\rm end}$ with $\kappa=1$ as a function of the inflationary energy scale $V_0^{1/4}$. The red curves correspond to $n=3/2,2,\ldots,10$ from left to right. In the blue shaded region, the linear perturbation is violated during the rolling of $\bar{\phi}$ toward $\bar{\phi}_{\rm end}$, and a tachyonic shock occurs. Eq.(\ref{['eq:efold']}) and $A_S=2\times 10^{-9}$ are used.
• Figure 2: Two-dimensional lattice simulation for the case $n=3/2$, corresponding to an inflection-point inflation model. The upper-left panels labeled ① and ② show the time evolution of the scalar field and the energy components, respectively. The broad purple arrow and the narrow red arrow indicate the benchmark time evolution at which we present the reduced power spectra of $\dot{\phi}$ and snapshots of the gradient energy, labeled ③ and ④, respectively. In panel ③, the purple solid lines correspond to the broad purple arrow, while the red dashed lines correspond to the narrow red arrow. See Appendix \ref{['app:1']} for details of the numerical simulations. Here $m_\phi = (3/2)\sqrt{V_0}/v$ denotes the vacuum mass.
• Figure 3: Same as \ref{['fig:lattice']} but with $n=2$, i.e., a hilltop inflation. Here $m_\phi =4/\sqrt{3} \sqrt{V_0}/v$ is the vacuum mass.
• Figure 4: The evolution of a spherically symmetric vacuum bubble. The upper panel shows the field profile $\phi(r)$, while the lower panel displays the gradient energy density on spherical shells, $2\pi r^2 (\nabla\phi)^2$. We set the initial field value outside the bubble to $\overline{\phi}_{i,\mathrm{out}} = v/200$ and the initial bubble radius to $r_{i,b} = 10/m_\phi$ with the potential of Eq.(\ref{['eq:benchmarks']}) of $n=3/2$. The color changes from purple to red as time progresses. The time slices are sampled as $\delta t = 2/m_\phi$ with $m_\phi=(3/2) \sqrt{V_0}/v$.
• Figure 5: The gravitational wave prediction from single field low scale inflation by assuming the instantaneous reheating. $\epsilon_{\rm GW}=0.1$ and $\epsilon_{\rm GW}=0.001$ are shown by solid lines and dashed lines, respectively. Also shown are limits/preference from the PTA results as well as the limit set by LVK collaboration LIGOScientific:2025bgj, and future reaches of SKA Janssen:2014dkaWeltman:2018zrl, LISA LISA:2017pwj, DECIGO Kawamura:2011zz, ET Punturo:2010zzSathyaprakash:2012jk, CE Evans:2021gyd, and LIGO-O5 TheLIGOScientific:2014jea. One can see the whole region satisfying Eq.(\ref{['eq:nonlinear']}), can be covered, and the 1EeV is already constrained by the LVK collaboration while 2GeV is favored by the PTA result NANOGrav:2023gorAntoniadis:2023ottReardon:2023gzhXu:2023wog.