Trans-Planckian Censorship Conjecture (TCC)

• TCC is a quantum gravity-inspired conjecture that restricts cosmic inflation by preventing sub-Planckian quantum fluctuations from becoming classical.
• It imposes universal bounds on e-folds and the Hubble parameter, impacting inflationary models, dark energy, and the swampland criteria.
• The conjecture establishes a UV/IR interplay that constrains effective field theories and shapes our understanding of early and late-time cosmic acceleration.

The Trans-Planckian Censorship Conjecture (TCC) is a quantum-gravity-motivated restriction on the cosmological evolution of the universe, formulated to prevent the horizon crossing—and subsequent classicalization—of fluctuation modes whose physical wavelength was ever smaller than the Planck length. Originating in the context of the "swampland" program, the TCC provides a universal bound on the amount of accelerated expansion allowed in any effective field theory (EFT) compatible with quantum gravity. This principle has profound implications for inflationary model-building, the structure of the cosmological landscape, late-time acceleration, and the interpretation of standard swampland criteria.

1. Mathematical Formulation and Physical Origin

The TCC asserts that during any period of accelerated cosmic expansion, no fluctuation mode with initial proper wavelength below the Planck length ($\ell_{\rm Pl}$) should ever be redshifted beyond the Hubble horizon and become an effectively classical perturbation. Formally, for a phase lasting from $t_i$ to $t_f$,

$$\lambda_{\rm phys}(t_i) \leq \ell_{\rm Pl} \;\Longrightarrow\; \lambda_{\rm phys}(t_f) < H^{-1}(t_f)$$

Given that $\lambda_{\rm phys}(t) = a(t)\, \lambda_{\rm com}$, the requirement is

$$\ell_{\rm Pl} \frac{a(t_f)}{a(t_i)} < H^{-1}(t_f) \;\Longrightarrow\; \frac{a(t_f)}{a(t_i)} < \frac{1}{\ell_{\rm Pl} H(t_f)} = \frac{M_{\rm Pl}}{H(t_f)}$$

Consequently, for quasi-de Sitter expansion with constant $H$, this yields

$$N \equiv \ln\frac{a(t_f)}{a(t_i)} < \ln \frac{M_{\rm Pl}}{H}$$

The TCC thus links the ultraviolet (Planckian) and infrared (horizon or Hubble scale) cutoffs of the EFT in a manner not enforced by general relativity or semiclassical quantum field theory, positing a new form of UV/IR interplay characteristic of quantum gravity (Schneider, 2022).

2. Physical Motivation and Scope

The TCC can be seen as a momentum-space analog of cosmic censorship, preventing the observable imprint of physics that lies outside the domain of validity of low-energy EFT. It responds to the "trans-Planckian problem" in inflation, whereby modes now stretched to cosmological scales would have originated at sub-Planckian lengths, beyond the predictive power of any known EFT description (Schneider, 2022, Brandenberger, 2021).

The conjecture thereby forbids any period of inflation (or late-time acceleration) long enough to render Planck-scale quantum fluctuations observable as cosmological structure. In operational terms, it imposes that the number of e-folds $N$ of any phase of cosmic acceleration must satisfy $N < \ln(M_{\rm Pl}/H)$.

3. Swampland Universality and TCC Variants

Unlike the standard swampland conjectures—such as the de Sitter or distance conjectures—which are formulated as universal restrictions on the space of EFT Lagrangians or scalar potentials $V(\phi)$, the TCC is fundamentally scenario-based, applying to realized cosmological histories. Saito, Shirai, and Yamazaki (Saito et al., 2019) systematically analyzed how one might attempt to elevate the TCC to a genuine swampland criterion:

• Universal TCC (UTCC): Demands all classically observable solutions of an EFT satisfy the TCC. This implementation enforces strict universality but is phenomenologically too strong, excluding any EFT with even a single TCC-violating trajectory—including eternal inflation and any potential with a positive maximum.
• Classical TCC (CTCC): Restricts only the classical (tree-level) solutions to satisfy the TCC. This version eliminates scenarios with quantum backreaction, but in doing so it discards quantum fluctuations essential for structure formation, conflicting with basic quantum mechanics.
• Observational TCC (OTCC): Requires that TCC is valid in our observed universe, allowing for potentially TCC-violating patches elsewhere. This version leads to phenomenological bounds on inflation, reheating, and tensor fluctuations, but cannot exclude other metastable vacua or restrict lifetimes of positive-energy vacua.
• Probabilistic TCC (PTCC): Allows for rare TCC-violating histories, constrained to occur only with negligible probability. This introduces a cosmological measure problem, relying on specific and possibly ad hoc definitions of probability in cosmology.

These distinctions highlight a deep conceptual tension: the TCC as traditionally formulated is a restriction on solutions (cosmological data), whereas swampland universality demands a restriction at the level of EFTs or potentials ($V(\phi)$) (Saito et al., 2019).

4. Phenomenological Implications and Bounds

Under even the minimal (OTCC) version, the TCC leads to stringent model-independent constraints for early-universe cosmology:

• The Hubble parameter during inflation is bounded as $H_{\rm inf} \lesssim 10^{-9}\ M_{\rm Pl} \sim 10^9$ GeV in models with instant reheating, or $H_{\rm inf} \lesssim 10$–$10^3$ GeV with extended reheating or early matter domination (Torabian, 2019, Dhuria et al., 2019).
• The maximum allowed number of e-folds is $N_{\rm inf} < \ln(M_{\rm Pl}/H_{\rm inf})$.
• Predicted tensor-to-scalar ratios are minuscule, $r \lesssim 10^{-30}$ for single-stage inflation (Bedroya et al., 2019, Sanna et al., 2021).
• Late-time acceleration must be transient: TCC forbids eternally accelerating de Sitter (imposing a constraint $w(a) > -1/3$ in the asymptotic future), independently ruling out simple $\Lambda$CDM and certain modified gravity models (Li et al., 10 Apr 2025).

For dark energy parameterizations (CPL, BA, JBP, EXP, LOG), the TCC restricts the allowed equation-of-state space to "quintom-B" regions where $w$ crosses $-1$ from below to above and asymptotes to $w > -1/3$ (Li et al., 10 Apr 2025).

5. Consequences for Inflationary Model Building and Field Potentials

The TCC places severe restrictions on inflationary models:

• High-scale and large-field inflation is ruled out by forbidding sufficient e-folds at high $H$; only extremely low-scale or multi-phased ("weak TCC") inflationary scenarios can evade these bounds.
• In single-field models, plateaux must end rapidly, and small-field, ultra-flat—but not too extended—potentials are preferred. Significant negative running of the scalar spectral index is generically expected (Kadota et al., 2019).
• Constraints on the axion decay constant for $N$-axion models are sharpened to $f\sqrt{N} \lesssim 0.6M_{\rm Pl}$, limiting the viability of axion-driven inflation. Axion quintessence remains TCC-compatible only with carefully tuned initial conditions (Shlivko, 2023).
• For non-minimal inflation, the effective Planck mass at the beginning of inflation substantially relaxes the TCC constraint, allowing high-scale inflation if the effective Planck mass is large during the initial phase (Guleryuz, 2021).

6. Connections to the Refined TCC and Quantum Gravity Principles

A refined TCC has been proposed (Cai et al., 2019), derived from the strong Weak Gravity Conjecture (WGC) and entropy bounds. It replaces the simple bound $e^N < M_{\rm Pl}/H$ with a weaker condition involving the average slow-roll parameter,

$$e^{N_\epsilon/\mathcal O(1)} \lesssim \mathcal O(1)\,\frac{M_{\rm Pl}}{H}$$

where $N_\epsilon = N/\langle\epsilon_H^{-1/2}\rangle$. This relaxes the fine-tuning problem of slow-roll inflation and restores phenomenological viability for models with higher $H$, but at the expense of universality.

The status of TCC as a true swampland conjecture thus remains open. Unlike the de Sitter conjecture, which restricts the scalar potential across all field space, the TCC resists embedding as a theory-space constraint without additional dynamical ingredients, selection rules, or probabilistic interpretations. Its ultimate foundation may require a more profound quantum-gravity principle (Saito et al., 2019, Schneider, 2022).

7. Geometric and Quantum-Cosmological Consequences

Mathematically, the TCC, when expressed as an integrability bound on the Hubble rate $\int H\,dt < \ln(M_{\rm Pl}/H_f)$, provides necessary conditions for the geodesic completeness of FRW spacetimes—prohibiting finite-time blowups and ensuring cosmological spacetimes are both past and future geodesically complete under standard hyperbolicity conditions (Cotsakis et al., 2022). In quantum cosmological path integrals, implementing the TCC as a complex boundary term in the action exponentially suppresses those histories that would violate the conjecture, yielding a quantum-modified upper bound on the Hubble rate at the end of inflation and negligible corrections to the primordial power spectrum (Mondal, 2022).

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References

• (Saito et al., 2019) Saito, Shirai & Yamazaki, "Is Trans-Planckian Censorship a Swampland Conjecture?"
• (Cotsakis et al., 2022) Cotsakis & Miritzist, "Trans-Planckian censorship and spacetime singularities"
• (Schneider, 2022) Butterfield, "A (strictly) contemporary perspective on trans-Planckian censorship"
• (Bedroya et al., 2019) Bedroya & Vafa, "Trans-Planckian Censorship and the Swampland"
• (Brandenberger, 2021) Brandenberger, "Trans-Planckian Censorship Conjecture and Early Universe Cosmology"
• (Shlivko, 2023) Rudelius, "Trans-Planckian censorship constraints on properties and cosmological applications of axion-like fields"
• (Cai et al., 2019) Cai & Wang, "A refined trans-Planckian censorship conjecture"
• (Li et al., 10 Apr 2025) Anagnostopoulos et al., "Quantum Gravity Meets DESI: Dynamical Dark Energy in Light of the Trans-Planckian Censorship Conjecture"