Probing Planck-Scale Physics with High-Frequency Gravitational Waves

Abstract

We develop a framework for testing quantum gravity through the stochastic gravitational-wave background produced by evaporating near-Planck-mass primordial black holes. Because gravitons free-stream from the emission region without rescattering, they preserve a direct spectral record of the black-hole temperature--mass relation $T(M)$, a relation that is erased for all other Hawking-radiated species by rapid thermalization. We translate six representative phenomenological beyond-semiclassical frameworks (the generalized uncertainty principle, loop quantum gravity, noncommutative geometry, asymptotic safety, string/Hagedorn physics, and tunneling backreaction) into distinct $T(M)$ parametrizations and compute the resulting gravitational wave spectra numerically. Modifications that suppress $T(M)$ shift the spectral peak by up to ten decades in frequency, in some cases into the sensitivity bands of next-generation interferometers or resonant-cavity detectors, while models imposing a hard evaporation cutoff produce distinctive peak morphologies that discriminate between quantum-gravity scenarios. We further discuss the impact of different choices for post-inflationary conditions in the very early universe. We find that the relative spectral displacement between the standard Hawking prediction and any modified model is cosmology-independent, hence spectral shape rather than absolute peak frequency provides the cleanest probe of Planck-scale physics.

Figures (6)

• Figure 1: Representative temperature--mass relations $T(M)$ for the modified black-hole evaporation models considered in this work, shown on logarithmic axes. All curves asymptote to the standard Hawking behavior $T\propto M^{-1}$ at large mass while exhibiting qualitatively distinct behavior near the Planck scale: plateau/Hagedorn-like saturation, cooling (post-peak decrease), GUP-, LQG-, and noncommutative-geometry--inspired forms with minimum masses, and tunneling/backreaction corrections.
• Figure 2: Same as Fig. \ref{['fig:TM_models_log']} on linear axes, highlighting the relative ordering of peak temperatures, the presence or absence of maxima, and the suppression or termination of evaporation near a minimum mass.
• Figure 3: Illustrative evaporation histories $M(t)$ for initial mass $M_0=10^3\,M_{\rm Pl}$ in several modified $T(M)$ scenarios. The figure shows how modified models can (i) delay the late-stage evolution, (ii) produce a prolonged low-mass tail, or (iii) effectively arrest evaporation at a finite remnant mass. Time is shown in the dimensionless units of Appendix \ref{['sec:app']}.
• Figure 4: Gravitational wave energy density spectra $\Omega_{\mathrm{GW}}(f_0)\,h^2$ from primordial black hole evaporation for six modified Hawking temperature models, computed for $M_0 = 10\,M_{\mathrm{Pl}}$. Each panel shows the standard Hawking result (black solid) alongside three representative parameter choices for: Plateau suppression (top left, $M_c = 3,\,3000,\,3\times10^{6}$), Cooling (top right, $M_c = 5,\,100,\,3000$), LQG/Planck-star (middle left, $M_{\min} = 0.3,\,3,\,9.9$), Non-commutative geometry (middle right, $M_{\min} = 1,\,8,\,9.99$), Tunneling corrections (bottom left, $\gamma = 3,\,30,\,99$), and Hagedorn/string-inspired (bottom right, $M_{H} = 10,\,10^{4},\,10^{8}$). All spectra are normalized to $\Delta N_{\mathrm{eff}} = 0.3$; the corresponding BBN/$N_{\mathrm{eff}}$ constraint band is shown in light blue with the central value indicated by an orange dotted line. Also shown for reference are approximate power-law sensitivity curves for the Einstein Telescope (ET, dark gray dashed) and Cosmic Explorer (CE, light gray dashed), and two resonant-cavity sensitivity projections: an optimistic scenario with $\nu_{\mathrm{ref}} = 1\,\mathrm{GHz}$ (purple dotted) and an ultra-optimistic scenario with $\nu_{\mathrm{ref}} = 50\,\mathrm{MHz}$ (purple dot-dashed). Increasing the model-specific cutoff parameter generically shifts the spectral peak to lower frequencies and suppresses high-frequency emission relative to the standard Hawking case.
• Figure 5: Gravitational wave energy density spectra $\Omega_{\mathrm{GW}}(f_0)\,h^2$ from primordial black hole evaporation for six modified Hawking temperature models, shown for two initial masses: $M_0 = 10\,M_{\mathrm{Pl}}$ (top) and $M_0 = 300\,M_{\mathrm{Pl}}$ (bottom). In each panel the standard Hawking result is shown as a black solid curve, with the following modified-temperature models overlaid: Tunneling corrections ($\gamma = 99$, blue dashed), Plateau suppression ($M_c = 3\times10^{4}$, orange dot-dashed), Cooling ($M_c = 3000$, green dotted), LQG/Planck-star ($M_{\min} = 9.9$, red dense-dashed), Non-commutative geometry ($M_{\min} = 9.99$, purple dash-dot-dot), and Hagedorn/string-inspired ($M_H = 10^{8}$, brown dense-dotted). All spectra are normalized to $\Delta N_{\mathrm{eff}} = 0.3$; the BBN/$N_{\mathrm{eff}}$ constraint band is shown in light blue with the central value ($\Delta N_{\mathrm{eff}} = 0.30$) indicated by an orange dotted line. Gray dashed curves show approximate power-law sensitivity envelopes for the Einstein Telescope (ET, dark gray) and Cosmic Explorer (CE, light gray); purple dotted and dot-dashed curves show resonant-cavity sensitivity projections for optimistic ($\nu_{\mathrm{ref}} = 1\,\mathrm{GHz}$) and ultra-optimistic ($\nu_{\mathrm{ref}} = 50\,\mathrm{MHz}$) configurations, respectively. Increasing $M_0$ shifts all spectral peaks to lower frequencies, while the relative displacement between the standard Hawking peak and those of the modified models reflects the characteristic mass scale introduced by each scenario.