Trans-Planckian Censorship Bound

Inputs, variables, and typical ranges

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

Inflation Scenario

A qualitative label for the inflationary scale:

• GUT Scale Inflation (High Energy): $H\sim {10}^{13}\text{ GeV}$, close to typical Grand Unified Theory scales.
• Starobinsky-like (Intermediate): lower scale, often tuned to give $r\lesssim 0.01$.
• Low Scale Inflation (TeV): scenarios with $H$ down near the TeV scale or below.

Inflationary Hubble scale $H$ (GeV)

The Hubble parameter during inflation, in giga–electron-volts. The allowed range in the form (roughly ${10}^{-5}$ to ${10}^{19}\text{ GeV}$) is wide enough to cover most speculative models. A commonly studied value is $H\approx {10}^{13}\text{ GeV}$, corresponding to high-scale inflation.

Desired e-folds $N$

The total number of e-folds of inflation you want your model to achieve. Standard cosmological arguments typically require $N\approx 50-60$ to solve the horizon and flatness problems, depending on reheating details.

The calculator uses the reduced Planck mass $M_{\text{Pl}}\approx 2.435\times {10}^{18}\text{ GeV}$ as a fixed reference scale.

Formula: Mathematical form of the TCC bound

For a simple, approximately de Sitter phase of inflation with Hubble scale $H$, the TCC leads to an upper bound on the number of e-folds $N$ 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:

Formula: N_max ≃ ln ⁡ (M_Pl / H).

$N_{max}\simeq ln\left(\frac{M_{Pl}}{H}\right).$

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

Connection to the tensor-to-scalar ratio $r$

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

Formula: V^1/4 ≈ 10^16 GeV(r / 0.01) 1 / 4,

$V^{\frac{1}{4}}\approx {10}^{16}\text{GeV}(\frac{r}{0.01}){}^{\frac{1}{4}},$

where $V$ is the inflaton potential energy density. For quasi-de Sitter inflation, $H$ and $V$ are related via the Friedmann equation, and one can derive an approximate mapping between $H$ and $r$. The calculator uses a simple slow-roll prescription (through $r\approx 16\epsilon$) to provide an order-of-magnitude estimate of $r$ compatible with your chosen $H$.

If the TCC bound forces inflation to occur at very low $H$, the implied $r$ 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 $H\approx {10}^{13}\text{ GeV}$, typical of GUT-scale models. Using the TCC bound,

Formula: N max ≃ ⁡ ln ⁡ ((2.4 × 10^18 GeV) / (10^13 GeV)) = ⁡ ln ⁡ (2.4 × 10^5) ≈ 12.

$N{\max }_{}\simeq \ln \left(\frac{2.4\times {10}^{18} \text{GeV}}{{10}^{13} \text{GeV}}\right)=\ln (2.4\times {10}^5)\approx 12.$

This allows only about 12 e-folds of inflation. If you enter $H={10}^{13}\text{GeV}$ and $N=60$ into the calculator, it will report a violation of the TCC bound, since $60\gg 12$.

Now consider a low-scale inflationary scenario with $N=60$ fixed, as is standard for solving cosmological problems. Reversing the bound,

Formula: e^N ≼ M_Pl / H ⇒ H ≼ M_Pl e^-N.

$e^N\preccurlyeq \frac{M_{Pl}}{H} \Rightarrow H\preccurlyeq M_{Pl}{e}^{-N}.$

For $N=60$, this gives

Formula: H ≲ 2.4 × 10^18 ⁢ GeV ⁢ e^−60 ∼ 10^9 ⁢ GeV (or smaller, depending on conventions).

$H\lesssim 2.4\times {10}^{18}\text{GeV}{e}^{-60}\sim {10}^9\text{GeV (or smaller, depending on conventions)}.$

Such a low Hubble scale typically corresponds to very small values of $r$, often below ${10}^{-4}$. In this regime, a positive detection of primordial tensors (say $r\gtrsim {10}^{-3}$) would be hard to reconcile with TCC-compatible slow-roll inflation.

Interpreting the calculator results

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

• TCC consistency: If the reported $N_{\max }$ is larger than your desired $N$, 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 $N_{\max }$ as a “target” that your model should not exceed. For model building, it can guide the combination of $H$, $N$, and reheating history you explore.
• Tensor-to-scalar ratio $r$: The displayed $r$ is only an indicative value. If the corresponding $r$ is above the sensitivity of upcoming CMB missions, a non-detection could put pressure on that class of models; conversely, a detection of $r$ 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 $r$. 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 $H$, $V$, and $r$ assumes a standard slow-roll regime with a single canonical inflaton field.
• Approximately constant $H$: The bound $N_{\max }\approx \ln \left(\frac{M_{\mathrm{Pl}}}{H}\right)$ 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 $M_{\text{Pl}}\approx 2.435\times {10}^{18}\text{ GeV}$. 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 $N$ 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 $H$ or multiple distinct phases of acceleration.
• Scenarios with modified gravity or non-canonical kinetic terms where standard slow-roll relations break down.

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.

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.