Trans-Planckian Censorship Bound

What is the Trans-Planckian Censorship Conjecture?

The Trans-Planckian Censorship Conjecture (TCC) is a proposal from the "Swampland" program in string theory. It suggests that in any consistent theory of quantum gravity, sub-Planckian quantum fluctuations should never be stretched by cosmic expansion to become larger than the Hubble horizon. If they did, they would "freeze out" and become classical cosmological perturbations, meaning our macroscopic universe would be directly determined by physics we don't understand (trans-Planckian modes).

The Mathematical Bound

To satisfy the TCC, the duration of inflation (measured in e-folds, $N$) must be limited by the energy scale of inflation ($H$):

Where $M_{\mathrm{Pl}}\approx 2.435\times {10}^{18}\text{GeV}$ is the reduced Planck mass.

Implications for Cosmology

This bound creates a tension for standard inflationary models. To solve the horizon and flatness problems, we typically need $N\approx 60$. However, the TCC implies:

• High Scale Inflation ($H\approx {10}^{13}\text{GeV}$): Allows only $N\approx 12$, which is insufficient for standard cosmology without modification.
• Low Scale Inflation: To allow $N=60$, $H$ must be extremely small ($<{10}^9\text{GeV}$), which implies a vanishingly small tensor-to-scalar ratio ($r$).

Tensor-to-Scalar Ratio ($r$)

The calculator estimates the tensor-to-scalar ratio, a key observable for upcoming CMB experiments (like LiteBIRD or CMB-S4), using the slow-roll approximation $r\approx 16ϵ$. In many models, $r$ is related to the scale of inflation by:

If the TCC holds, detecting primordial gravitational waves ($r>0.001$) would rule out standard slow-roll inflation, forcing us to consider alternative early universe scenarios.

Swampland vs. Landscape

String theory suggests a vast "Landscape" of possible vacua. The "Swampland" consists of effective field theories that look consistent but cannot be completed into a full theory of quantum gravity. The TCC is a "Swampland Conjecture"—a proposed rule to distinguish the good theories from the bad. While not proven, it guides theoretical physicists in searching for viable models of our universe.