Why the Trans-Planckian Censorship Conjecture Matters

Inflation stretches quantum fluctuations from microscopic origins to astronomical scales. In conventional models, the modes that seed cosmic microwave background anisotropies began with wavelengths far smaller than the Planck length. Such trans-Planckian wavelengths lie outside the domain where known physics is reliable, raising the unsettling possibility that inflationary predictions depend on unknown ultraviolet completions. To address this concern, Bedroya and Vafa proposed the Trans-Planckian Censorship Conjecture (TCC). It asserts that no mode that was once shorter than the Planck length should ever exit the Hubble horizon. Applied to inflation, the conjecture imposes an upper bound on the number of e-folds for a given Hubble rate: e^N < M_Pl/H. Equivalently, N < ln(M_Pl/H). This simple inequality has deep consequences for model building, restricting both the energy scale and the duration of inflation.

The bound arises by considering a mode with physical wavenumber k/a. During inflation the scale factor grows exponentially, so modes that start at the Planck scale would be stretched beyond the Hubble radius after N e-folds if N is too large. The TCC forbids this scenario, effectively censoring trans-Planckian modes from ever becoming classical. If inflation obeys the conjecture, then for H about 10¹³ GeV the maximum allowed e-folds is around 35, far less than the canonical 60 required to solve the flatness and horizon problems. Reconciling the TCC with successful cosmology may require lower inflation scales, multiple stages of reheating, or alternative mechanisms for generating density perturbations.

This calculator implements the inequality directly. The user supplies the Hubble scale during inflation, expressed in gigaelectronvolts, and a target number of e-folds. The script computes N_max = ln(M_Pl/H) using the reduced Planck mass M_Pl ≈ 2.435×10¹⁸ GeV. It then reports whether the chosen N violates the conjecture. For additional insight, the tool also outputs the characteristic energy scale of inflation, V^1/4 = (3^1/4√{H M_Pl} ), which sets the scale of primordial gravitational waves. If the bound is violated, the result is flagged to emphasize the tension.

While the TCC is a conjecture rather than a proven theorem, it is motivated by attempts to embed inflation in a consistent theory of quantum gravity. Swampland criteria emerging from string theory suggest that effective field theories with too much parametric freedom may not admit ultraviolet completions. The TCC can be viewed as one such criterion, forbidding excessively long periods of accelerated expansion. Its implications reach beyond cosmology, affecting proposals for dark energy and late-time acceleration. Any epoch of accelerated expansion must respect the censorship condition or else risk relying on trans-Planckian physics.

The following table illustrates the bound for several Hubble scales:

H (GeV)	Nmax
1e13	34.6
1e10	41.7
1e7	48.8

As H decreases, the allowed number of e-folds increases modestly. Achieving the traditional N ≈ 60 requires H below roughly 10³ GeV, corresponding to an inflationary energy scale far below the reach of current high-energy theories. Critics of the TCC argue that such low scales are incompatible with generating sufficient perturbations or with reheating the universe efficiently. Proponents counter that alternative mechanisms or additional epochs of expansion might reconcile the constraints. The debate remains lively, reflecting deeper questions about the nature of quantum gravity.

Ultimately, the TCC shines a spotlight on our ignorance of Planck-scale physics. It challenges cosmologists to construct models that remain self-consistent all the way back to the highest energies, or else to accept that conventional inflation may need revision. Whether the conjecture holds in a full theory of quantum gravity is an open question. Nevertheless, exploring its consequences fosters healthy scrutiny of cherished assumptions and encourages the search for robust predictions that survive beyond the reach of perturbative field theory.