Trans-Planckian Oscillations in Quantum Gravity

• Trans-Planckian oscillations are oscillatory features in correlators and spectra that emerge when probing quantum fields at or beyond the Planck scale.
• They arise from mechanisms such as UV-completion modifications, nonadiabatic mode evolution, and nonperturbative effects in both gravitational and cosmological settings.
• These oscillations signal the breakdown of local field theory at short distances, offering constraints on UV completions and insights into quantum spacetime structure.

Trans-Planckian oscillations refer to characteristic features—usually in the form of oscillatory behavior—in correlators, spectra, or response functions arising when quantum fields, gravity, or effective descriptions are probed at or beyond the fundamental “Planck scale” (ℓₚ, mₚ, or a length/energy scale set by quantum gravity or a dynamical cutoff). These oscillations are a recurrent motif in diverse approaches to quantum gravity, early-universe cosmology, and high-energy gravitational scattering, and are intimately linked to the breakdown of standard local field theory at trans-Planckian distances. Their occurrence or absence depends sensitively on the structure of the theory—in particular on whether degrees of freedom above the Planck scale are allowed as genuine propagating quantum modes, and how quantum gravity "cloaks" or transmutes UV (ultraviolet) physics into IR (infrared) behavior.

1. Trans-Planckian Oscillations: General Mechanisms and Theoretical Context

Trans-Planckian oscillations arise generically in settings where the effective theory is deformed, UV-completed, or extended so that the short-distance structure is fundamentally altered. The sources of oscillations fall broadly into three categories:

• Spectral modifications due to UV-completion: Attempts to introduce additional massive poles or nonlocal features in the graviton propagator can produce oscillatory corrections in coordinate-space correlators (for example, via inverse Fourier transforms of complex spectral densities). However, in healthy (ghost-free) gravitational theories, such poles are rendered non-propagating above $m \gg m_P$ due to black hole (BH) formation, and their effect is mapped into the IR as classical black holes, leading to exponential suppression rather than observable oscillations (1006.0984).
• Initial state modifications and mode matching: In cosmological settings, if the initial state is set by an $\alpha$-vacuum or otherwise imposed at a finite conformal time (e.g., when the physical wavenumber $k/a$ equals an explicit cutoff $\Lambda$), matching conditions induce k-dependent phase shifts in the mode functions of quantum fluctuations. This generates oscillatory "correction factors" in the power spectra, visible as ripples or features in momentum space (Cheraghchi et al., 2023, 1211.6753). The amplitude and detectability of these oscillations are controlled by the ratio $H/\Lambda$ (inflationary Hubble scale to cutoff).
• Nonperturbative instanton-like effects in quantum gravity toy models: In $T\bar{T}$-deformed conformal field theories, which serve as soluble models of Planckian physics, nonperturbative completions of correlation functions manifest pronounced trans-Planckian oscillations as one probes lengths below the built-in scale $\ell_P$ (Hirano et al., 22 Jul 2025). These oscillations result from the interference between perturbative and instanton contributions, marking the breakdown of local QFT at $|x| \lesssim \ell_P$.

2. Trans-Planckian Oscillations in Gravity and the UV/IR Correspondence

The analysis of gravitational scattering amplitudes reveals that, in a ghost-free theory flowing to Einstein gravity in the IR, extensions with extra massive states (e.g., higher-spin fields or massive scalars) modify the graviton propagator with potentially oscillatory features above the Planck scale. The general spectral representation (with positive spectral densities $\rho_2(s) \geq 0$) ensures that the gravitational coupling $\alpha_{\rm grav}(p)/\alpha_{\rm Ein}(p)$ is a non-decreasing function of momentum $p^2$, i.e., gravity cannot become weaker in any ghost-free extension (1006.0984):

$$\frac{\alpha_{\rm grav}(p)}{\alpha_{\rm Ein}(p)} = 1 + p^2 \int_0^\infty ds \frac{\rho_2(s)}{p^2 + s}$$

Any new pole at $p^2 = -m^2$ (with $m \gg m_P$) that could induce oscillatory corrections in the position-space potential is physically mapped to a classical black hole state: attempts to resolve distances $L \ll L_P$ require energies sufficient for black hole formation, and the would-be quantum degree of freedom is replaced by a classical object with exponentially suppressed contribution in physical amplitudes ($\sim e^{-m/m_P}$). Thus, manifest "trans-Planckian oscillations" in scattering amplitudes are absent or IR-mapped in any ghost-free, self-complete theory of gravity (1006.0984).

The UV/IR correspondence is codified in the observation that high-energy (short-distance) corrections cannot produce observable quantum oscillatory effects; instead, all such putative modifications are either washed out or encoded in exponentially suppressed contributions from classical BH states.

3. Trans-Planckian Oscillations in Cosmology: Initial State Sensitivity and Power Spectrum Features

In the context of cosmological inflation, the trans-Planckian problem is that observable modes today may have originated with physical wavelengths smaller than the Planck length at the beginning of inflation (1211.6753). Unknown UV physics could leave observable oscillatory imprints in the primordial power spectrum.

Two distinct mechanisms yield oscillatory corrections:

• Modified initial conditions (e.g., $\alpha$-vacua, Danielsson prescription): If the initial state for each mode is imposed not in the infinite past but at a finite conformal time $\tau_i$ determined by $k = a(\tau_i) \Lambda$, the resulting Bogoliubov coefficients $\alpha_k$, $\beta_k$ acquire $k$-dependent phases. The power spectrum receives a correction:

$$\mathcal{P}^\alpha(k) = \mathcal{P}(k) \bigg[1 - (1 - 2\nu^R(\tau_i)) \frac{\sin(2k\tau_i)}{2k\tau_i}\bigg]$$

where $\nu^R$ encodes slow-roll or model-dependent parameters (Cheraghchi et al., 2023). The $(\sin(2k\tau_i)/(2k\tau_i))$ term generates oscillatory features with frequency set by the cutoff scale, and amplitude suppressed by $H/\Lambda$.

• Modified dispersion relations and nonadiabaticity: If the mode evolution is nonadiabatic when physical wavenumbers are trans-Planckian due to e.g., modifications of the dispersion relation, excited initial states (nonzero $\beta_k$) produce oscillatory modulations in the power spectrum, often parametrizable as:

$$\Delta P_\zeta(k) \propto \cos\bigg(2\frac{\Lambda}{H}(1 + \cdots) + \phi \bigg)$$

where the corrections are "log-periodic" in $k$ and correspond to rapid but small-amplitude oscillations (sometimes called "ripples") (1211.6753). Observation bounds (CMB measurements) constrain the amplitude and frequency of such features.

4. Trans-Planckian Oscillations in Effective Theories and Quantum Gravity Toy Models

$T\bar{T}$-deformed two-dimensional CFTs provide an explicit nonperturbative setting in which trans-Planckian oscillations are realized and calculated (Hirano et al., 22 Jul 2025). Introducing a deformation parameter $\mu$ and a corresponding minimal length $\ell_P = \sqrt{|\mu|}$, nonperturbative correlators for primary operators of dimension $\Delta$ exhibit the following structure for $|x_{12}|$ close to $\ell_P$:

$$\langle \mathcal{O}_\Delta(x_1)\mathcal{O}_\Delta(x_2)\rangle \sim \frac{\Gamma(1-\Delta)}{|x_{12}|^{2\Delta}} Z^{-\Delta} e^{-(1/Z)\,{}_1F_1(1-\Delta; 1; 1/Z)}$$

with $$Z = -\operatorname{sgn}\mu \frac{4\ell_P^2}{\pi|x_{12}|^2}\log(|x_{12}|/\ell_P)$$. Expanding near $|x_{12}| \gtrsim \ell_P$, the interference between perturbative and nonperturbative sectors produces rapid oscillations in the correlator. These are interpreted as "trans-Planckian oscillations," a direct result of coupling the QFT to random geometry (i.e., quantum gravity). At even smaller distances ($|x_{12}| \ll \ell_P$), this oscillatory behavior transitions to exponential suppression of correlations due to geometric randomness—the local structure of spacetime is effectively erased, and the correlator scaling changes from a power law to

$|x|^2 \rightarrow \ell_P^2 \log(\ell_P/|x|)$

demonstrating a fundamentally modified effective notion of distance.

This behavior serves as a controlled illustration of general expectations for quantum spacetime—namely, that trans-Planckian oscillations mark the onset of nonlocal, nonclassical behavior, and that further into the UV, locality and traditional correlator-based diagnostics break down due to underlying geometric "randomness" or quantum gravitational fluctuations.

5. Cosmological and Black Hole Contexts: Observability and Physical Interpretation

Oscillatory corrections arising from trans-Planckian physics are subject to potential observational signatures but are often exponentially suppressed or phase-randomized. In inflationary cosmology, trans-Planckian oscillations in the power spectrum are constrained by backreaction and data to be of modest amplitude (proportional to $H/\Lambda$) (1211.6753). In quantum gravity regimes, as in black hole interiors, discrete, oscillator-like quantum spectra replace the classical continuous picture, and the fundamental oscillation quantizes previously singular variables (Sanchez, 7 Apr 2024). These quantum oscillations unify the continuous gravitational and quantum pictures, eliminating singularities and resolving physical quantities into regular, level-structured forms.

In dynamical scenarios—such as Hawking radiation near black hole horizons or gravitational collapse—trans-Planckian redshifts and the exponential growth of certain higher-derivative contributions in scattering amplitudes signal the breakdown of effective field theory and the necessity of Planckian (UV) completion (Ho et al., 2021).

A table summarizing key sources and outcomes of trans-Planckian oscillations:

Physical Regime	Mechanism Producing Oscillations	Physical Outcome / Observability
Ghost-free quantum gravity	Extra propagator poles, spectral density	Oscillations IR-mapped, cloaked by black holes
Cosmology: modified initial data	$\alpha$-vacua, mode matching at cutoff	Oscillatory power spectrum, amplitude $\sim H/\Lambda$
$T\bar{T}$-deformed CFTs	Nonperturbative instanton interference	Explicit oscillatory correlators, transition to randomness
Loop quantum cosmology	Modified dispersion, nonadiabatic mode evolution	Spectral modulations sensitive to model / $e$-folds

6. Theoretical Implications and Constraints on UV Completion

The existence or suppression of trans-Planckian oscillations acts as a theoretical diagnostic for the physical content of proposed UV completions:

• In gravity, the absence of observable, excited quantum oscillations above the Planck scale in a ghost-free setting supports Einstein gravity's self-completeness and argues against a naïvely Wilsonian UV completion (1006.0984).
• In effective cosmological models, significant trans-Planckian oscillations in power spectra are often limited by backreaction or Adiabaticity, and models featuring minimally invasive corrections (e.g., small $|x| \sim \ell_P$) are best aligned with data (1211.6753).
• Nonperturbative models demonstrate that quantum gravitational effects induce oscillations as the classical geometric picture breaks down, but these oscillations self-erase at even smaller distances, paralleling expected behaviors such as dimensional reduction and nonlocality (Hirano et al., 22 Jul 2025).

7. Summary

Trans-Planckian oscillations are robust markers of the profound changes that occur in physical theories as they are probed at distances or energies at or beyond the Planck scale. In ghost-free quantum gravity, any would-be oscillatory quantum effect is either prohibited or mapped into the IR as classical physics, consistent with UV self-completeness. In cosmology and quantum gravity models, explicit oscillatory features occur when new physics (in initial conditions, dispersion relations, or nonperturbative completions) alters the propagation or statistics of field modes, but these effects are typically suppressed or masked in observable quantities either by exponential factors or geometric randomness. The paper of trans-Planckian oscillations remains central in constraining and interpreting the microscopic structure of quantum spacetime, delineating the boundary between effective field theory and the emergent, fundamentally quantum nature of Planckian physics.