Trans-Planckian Censorship Bound

Overview: What this calculator does

This calculator evaluates the Trans-Planckian Censorship Conjecture (TCC) bound for simple inflationary scenarios. Given an inflationary Hubble scale (in GeV) and a desired number of e-folds , it checks whether the scenario is compatible with the TCC and provides an indicative tensor-to-scalar ratio under slow-roll assumptions.

Use it to:

• Estimate the maximum allowed e-folds for a given inflationary scale.
• See whether your chosen violates the TCC bound.
• Get an order-of-magnitude estimate of the tensor-to-scalar ratio implied by the scale of inflation.

Inputs, variables, and typical ranges

The form above takes three main inputs. The presets in the Inflation Scenario menu simply suggest typical values for and ; you can override them with custom numbers.

Inflation Scenario

A qualitative label for the inflationary scale:

• GUT Scale Inflation (High Energy): , close to typical Grand Unified Theory scales.
• Starobinsky-like (Intermediate): lower scale, often tuned to give .
• Low Scale Inflation (TeV): scenarios with down near the TeV scale or below.

Inflationary Hubble scale (GeV)

The Hubble parameter during inflation, in giga–electron-volts. The allowed range in the form (roughly to ) is wide enough to cover most speculative models. A commonly studied value is , corresponding to high-scale inflation.

Desired e-folds

The total number of e-folds of inflation you want your model to achieve. Standard cosmological arguments typically require to solve the horizon and flatness problems, depending on reheating details.

The calculator uses the reduced Planck mass as a fixed reference scale.

Definition of the Trans-Planckian Censorship Conjecture

The Trans-Planckian Censorship Conjecture is a proposal emerging from the swampland program in string theory and quantum gravity. Intuitively, it states that in any consistent theory of quantum gravity, modes that start at sub-Planckian wavelengths should never be stretched by cosmic expansion to super-horizon scales where they freeze out as classical perturbations.

Otherwise, the large-scale structure of our Universe would be directly controlled by physics at trans-Planckian energies, beyond the regime where effective field theory is trustworthy. The TCC is thus a proposed consistency condition that rules out some effective field theories, placing them in the swampland rather than the allowed landscape of viable low-energy descriptions.

Mathematical form of the TCC bound

For a simple, approximately de Sitter phase of inflation with Hubble scale , the TCC leads to an upper bound on the number of e-folds of accelerated expansion. A commonly used form of the bound is:

Taking the natural logarithm of both sides gives a simple expression for the maximum allowed number of e-folds:

This is the core relation implemented by the calculator: for a given Hubble scale , it computes and compares it to your chosen . If your desired e-folds exceed , the scenario violates the TCC under the assumptions listed below.

Connection to the tensor-to-scalar ratio

In slow-roll single-field inflation, the tensor-to-scalar ratio , measuring primordial gravitational waves, is related to the inflationary energy scale. A widely used approximate relation is

where is the inflaton potential energy density. For quasi-de Sitter inflation, and are related via the Friedmann equation, and one can derive an approximate mapping between and . The calculator uses a simple slow-roll prescription (through ) to provide an order-of-magnitude estimate of compatible with your chosen .

If the TCC bound forces inflation to occur at very low , the implied becomes extremely small, often far below the sensitivity of near-future CMB polarization experiments. This is why the TCC has strong implications for the observability of primordial gravitational waves.

Worked example: high-scale vs low-scale inflation

Consider first a high-scale inflationary scenario with , typical of GUT-scale models. Using the TCC bound,

This allows only about 12 e-folds of inflation. If you enter and into the calculator, it will report a violation of the TCC bound, since .

Now consider a low-scale inflationary scenario with fixed, as is standard for solving cosmological problems. Reversing the bound,

For , this gives

Such a low Hubble scale typically corresponds to very small values of , often below . In this regime, a positive detection of primordial tensors (say ) would be hard to reconcile with TCC-compatible slow-roll inflation.

Interpreting the calculator results

After you enter and and click Check Bound, you can use the outputs as follows:

• TCC consistency: If the reported is larger than your desired , then a simple TCC implementation does not immediately rule out your scenario. If , the TCC suggests that such a long phase of inflation at that scale is inconsistent.
• Maximum allowed e-folds: Treat as a “target” that your model should not exceed. For model building, it can guide the combination of , , and reheating history you explore.
• Tensor-to-scalar ratio : The displayed is only an indicative value. If the corresponding is above the sensitivity of upcoming CMB missions, a non-detection could put pressure on that class of models; conversely, a detection of in conflict with the TCC-compatible range would challenge the conjecture or the underlying assumptions.

Comparison of typical inflationary scenarios

The table below summarizes, at a rough order-of-magnitude level, how different inflationary scales map onto TCC bounds and expected . Values are illustrative and may differ from the precise numbers used in your calculation.

Assumptions, limitations, and scope

The implementation here is intentionally minimal and should be interpreted as a phenomenological tool, not a full numerical cosmology solver. Key assumptions include:

• Single-field, slow-roll inflation: The mapping between , , and assumes a standard slow-roll regime with a single canonical inflaton field.
• Approximately constant : The bound is derived for a quasi-de Sitter phase with nearly constant Hubble parameter during the e-folds being counted.
• Reduced Planck mass convention: We use . Different conventions (e.g., unreduced Planck mass) change numerical prefactors but not the qualitative conclusion.
• Simplified reheating treatment: The calculator does not model reheating or post-inflationary entropy production explicitly; the required to solve cosmological problems can depend on these details.

The calculator is not reliable for:

• Non-standard cosmologies (e.g., bouncing or cyclic models).
• Models with strongly varying or multiple distinct phases of acceleration.
• Scenarios with modified gravity or non-canonical kinetic terms where standard slow-roll relations break down.

Swampland context and further reading

The TCC fits into a broader set of swampland conjectures that attempt to characterize which low-energy effective field theories can arise from a consistent theory of quantum gravity, such as string theory. Related ideas include the Swampland Distance Conjecture, the de Sitter conjecture, and bounds on scalar field excursions during inflation.

For a deeper dive, you may wish to consult the original and follow-up literature on the TCC and swampland bounds, as well as observational reviews on inflation and primordial gravitational waves. This calculator is intended as a quick exploratory tool that complements more detailed analytical and numerical studies.