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

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.

• The paper establishes that sufficiently low-scale single-field inflation develops nonlinear tachyonic fluctuations before coherent oscillations, generating relativistic bubbles and shock fronts when the inflationary scale falls below roughly 10^12 GeV.
• Simulations show bubble-wall collisions produce a gravitational-wave spectrum rising as k^3 below its peak and falling as k^-2 above it, with an estimated peak abundance near 10^-7 and frequencies spanning PTA to interferometer bands.
• The results open observational tests of MeV–EeV inflation through gravitational waves and enhanced spectral running, while requiring improved simulations and first-principles calculations to refine the predicted signal.

Overview

This paper identifies a generic feature of low-scale single-field inflation: when the inflationary energy scale is sufficiently small, tachyonic growth of inflaton fluctuations becomes nonlinear before coherent homogeneous oscillations of the condensate can develop. The authors term the resulting phenomenon a tachyonic shock: a first-order phase-transition-like dynamics—relativistically expanding bubble-like structures, shock fronts, and bubble collisions—that occurs without tunneling or barrier crossing. Because the instability is seeded by modes at the Hubble scale rather than by microscopic inflaton masses, the characteristic frequency of the emitted gravitational waves (GWs) is $k_{\rm GW} \sim H_{\rm inf}$, placing the signal in the observational band of pulsar timing arrays (PTAs), ground-based interferometers, and future space-based detectors. This opens a GW window onto inflation at MeV–EeV energy scales, regimes in which primordial tensor modes are unobservable in the CMB (2602.15825).

Setup and motivation

The analysis assumes potentials near the slow-roll region of the form

$V \approx V_0 - \lambda\,\phi^{2n},$

covering hilltop ($n = 2, 3, \dots$) and inflection-point ($n = 3/2, 5/2, \dots$) models; such forms arise from axionic constructions, non-minimal gravity couplings, supersymmetry, and heavy QCD axion inflation. CMB normalization fixes $\lambda V_0^{n-2}$ for given $V_0$ and e-fold number $N$, and the end of inflation occurs when $-\eta \approx 1$, i.e., when $3H_{\rm inf}^2 \approx -V''(\bar\phi_{\rm inf})$. A central point is that with the observed scalar amplitude $A_S \approx 2\times10^{-9}$, small $V \approx V_0 - \lambda\,\phi^{2n},$0 forces $V \approx V_0 - \lambda\,\phi^{2n},$1 to be extremely tiny during inflation while the potential curvature must change rapidly near the end—a combination that makes strong tachyonic amplification inevitable at low scales.

The paper also notes that small $V \approx V_0 - \lambda\,\phi^{2n},$2 generically enhances the running $V \approx V_0 - \lambda\,\phi^{2n},$3 to levels probed by SPHEREx and DESI/SKA combinations, an independent consistency test of low-scale scenarios.

Stage 1: explosive tachyonic instability

In linear theory, modes with $V \approx V_0 - \lambda\,\phi^{2n},$4 grow as $V \approx V_0 - \lambda\,\phi^{2n},$5, outpacing the background rolling $V \approx V_0 - \lambda\,\phi^{2n},$6. Combining this scaling with the vacuum-mode amplitude yields the estimate

$V \approx V_0 - \lambda\,\phi^{2n},$7

which is infrared dominated relative to the scale-invariant spectrum. The key quantitative result is that the ratio $V \approx V_0 - \lambda\,\phi^{2n},$8 grows as $V \approx V_0 - \lambda\,\phi^{2n},$9 decreases, so for any $n = 2, 3, \dots$0 nonlinearity ($n = 2, 3, \dots$1) is reached before higher-order terms stabilize the potential provided

$n = 2, 3, \dots$2

with the threshold below roughly $n = 2, 3, \dots$3 GeV for any finite $n = 2, 3, \dots$4. Notably, the effective timescale governing the instability, $n = 2, 3, \dots$5, is independent of $n = 2, 3, \dots$6 and takes values $n = 2, 3, \dots$7.

Stages 2 and 3: bubbles, shocks, collisions

Once fluctuations reach the background amplitude, regions with locally larger field values roll first, seeding expanding bubbles separated by order $n = 2, 3, \dots$8. Because the enhanced modes have wavelength $n = 2, 3, \dots$9, gradient energy cannot stall the expansion; energy conservation gives wall Lorentz factors $n = 3/2, 5/2, \dots$0, producing Lorentz-contracted shock fronts that continue extracting latent energy even after the exterior begins oscillating. Two-dimensional lattice simulations (using modified {\tt CosmoLattice}) for benchmark $n = 3/2, 5/2, \dots$1 and $n = 3/2, 5/2, \dots$2 potentials confirm the power-spectrum scaling, gradient-energy domination in the walls, and eventual collision. In integer-$n = 3/2, 5/2, \dots$3 hilltop cases, collisions populate the second vacuum, forming sub-horizon domain walls that collapse rapidly due to population bias—an implication for topological-defect production if the inflaton breaks a symmetry $n = 3/2, 5/2, \dots$4.

Gravitational-wave signal and observational prospects

Since about 10% of the total energy resides in relativistic walls colliding over a duration $n = 3/2, 5/2, \dots$5, the peak abundance is estimated as

$n = 3/2, 5/2, \dots$6

A three-dimensional $n = 3/2, 5/2, \dots$7 lattice simulation confirms the spectral shape: $n = 3/2, 5/2, \dots$8 growth below the peak (causality) and $n = 3/2, 5/2, \dots$9 falloff above it, matching the relativistic-wall limit of known first-order phase-transition spectra. The resulting coverage is striking:

Energy scale	Status
$\lambda V_0^{n-2}$0 GeV	Already constrained by LIGO–KAGRA–Virgo (for $\lambda V_0^{n-2}$1)
$\lambda V_0^{n-2}$2 GeV	Favored region consistent with PTA stochastic background
$\lambda V_0^{n-2}$3 MeV	Excluded by BBN (insufficient reheating temperature)

Thus PTA data admit interpretation as GWs from GeV-scale inflation—relevant to trans-Planckian censorship conjecture motivations—and future SKA, LISA, DECIGO, ET, and CE observations could cover essentially the entire parameter space where tachyonic shocks occur. Combined with the enhanced running prediction, a detected signal would constitute a distinctive signature of single-field low-scale inflation.

Generality and related phenomena

The authors argue the result extends beyond single-term potentials: for Taylor-expanded $\lambda V_0^{n-2}$4, successive dominance of higher powers multiplies the fluctuation enhancement factors, so the single-term analysis is conservative unless fine-tuned cancellations are imposed. Additional consequences discussed include Baumkuchen domain walls in ALP hilltop inflation, dark matter production and baryogenesis from ultra-relativistic bubble expansion, asymmetric thermalization with boosted fragmentation momenta altering standard reheating assumptions, electroweak symmetry restoration via induced Higgs walls enabling sphaleron-driven baryogenesis below the weak scale, and possible primordial black hole formation from overdense Hubble patches.

Limitations and open questions

Several caveats bear directly on the quantitative claims. The lattice simulations artificially enhance initial fluctuations by factors $\lambda V_0^{n-2}$5–$\lambda V_0^{n-2}$6 because resolving the true hierarchy ($\lambda V_0^{n-2}$7, requiring impractically large lattices) is impossible; the authors justify this via the well-understood linear tachyonic stage but it remains a numerical compromise. The gravitational-wave abundance rests on analogy with first-order phase transitions parametrized by an efficiency factor $\lambda V_0^{n-2}$8; a detailed spectrum calculation is explicitly deferred to future work. The condition for the tachyonic shock does not directly apply to waterfall-field hybrid-inflation variants, and exceptions arise for potentials with special symmetry structure (e.g., ALP hilltop, where domain walls rather than simple bubble dilution dominate).

Conclusion

The paper establishes that sufficiently low-scale single-field inflation generically ends not in coherent oscillations but in a Hubble-scale tachyonic shock: nonlinear fluctuation growth seeds relativistic bubbles whose collisions produce energetic particles and gravitational waves peaking at $\lambda V_0^{n-2}$9. This connects inflationary dynamics across the MeV–EeV range to current and upcoming GW observables, with LVK bounds already excluding part of the parameter space and PTA hints interpretable as GeV-scale inflation. Confirmation requires more detailed numerical treatment of the full dynamical hierarchy and a first-principles computation of the GW spectrum.