Inflation Slow-Roll Parameter Calculator

Slow-Roll Dynamics for Monomial Inflation

This calculator is built around a specific inflationary toy model: a single scalar field with a monomial potential. In that setting, the potential is simple enough to make the slow-roll expressions readable, but still rich enough to show how the steepness of the potential affects the duration of inflation, the scalar tilt, and the tensor signal. The formulas here are therefore best read as a compact bridge between textbook slow-roll theory and the observables that appear in cosmic microwave background constraints.

Introduction to Inflation Slow-Roll Parameters

This calculator focuses on monomial inflationary potentials of the form $V\left(\phi \right)=\lambda {\phi }^n$. In plain language, that means the potential energy grows as a power of the inflaton field. By choosing the power n, a field value $\phi$, and a number of e-folds $N$, you can estimate the first slow-roll parameter $\epsilon$, the second slow-roll parameter $\eta$, the scalar spectral index $n_s$, and the tensor-to-scalar ratio $r$. These are the quantities most often compared with cosmic microwave background observations because they capture both the shape of the primordial spectrum and the size of primordial gravitational-wave modes.

In canonical single-field inflation the first slow-roll parameter is defined as $\epsilon =\frac{M^2}{2}\left(\frac{V\text{'}}{V}\right)$. It measures how steep the potential is relative to its height. Inflation requires $\epsilon \ll 1$; once $\epsilon$ approaches unity, accelerated expansion ends. The second slow-roll parameter $\eta$ tracks the curvature of the potential and helps determine how the primordial power spectrum tilts away from exact scale invariance.

Observables such as the scalar spectral index $n_s\approx 1-6\epsilon +2\eta$ and the tensor-to-scalar ratio $r\approx 16\epsilon$ are then obtained at leading order. Because modern observations constrain both $n_s$ and $r$, even a simple calculator like this one is useful for building intuition about which inflationary potentials are favored and which combinations of n and φ are pushed into tension with data.

How to Use This Inflation Slow-Roll Calculator

Use the form below to probe a monomial slow-roll model with your own values. The calculator reads the potential power, the field value, and the e-fold count, then evaluates the leading-order slow-roll quantities with the same relationships used later in the reference table. The live result shown under the button comes from the field value $\phi$ you enter, while $N$ is used for the helper estimate and the table entries.

Here is what each input means for this specific model. The potential power n sets the exponent in $V\; \propto \; \phi ⁿ$, so larger values generally make the potential steeper at large field values. The field value φ is entered in reduced Planck mass units, which is why the calculator treats it as a dimensionless number. The e-folds N field does not replace the direct $\phi$ calculation; instead it supports the table and the auxiliary estimate that map a target e-fold count to an approximate field value.

To see a result, enter numeric values and press Compute Parameters. The output reports $\epsilon$, $\eta$, $n_s$, and $r$, followed by a quick comparison with the page's screening threshold. If any entry is not numeric, if n is negative, if $\phi$ is not positive, or if $N$ is smaller than 1, the calculator shows an error message rather than a misleading result.

For intuition, change one parameter at a time. At fixed n, increasing $\phi$ usually lowers both $\epsilon$ and $\eta$ because each one falls with ${\phi }^2$ in the denominator. At fixed $\phi$, raising n tends to make the potential steeper and pushes $r$ upward. Watching those shifts is often more useful than focusing on a single output line, especially when you are trying to compare a few monomial shapes rather than fit data exactly.

Formula for Inflation Slow-Roll Parameters

For a monomial potential, the derivative terms collapse into compact slow-roll relations that the calculator evaluates directly from your inputs. In this page, the main button output is based on the field value you provide, while the table uses the same formulas after first estimating $\phi$ from $N$.

Formula: ε = n^2 / (2 φ^2)

$\epsilon =\frac{n^2}{2{\phi }^2}$

Formula: η = (n(n − 1)) / φ^2

$\eta =\frac{n(n-1)}{{\phi }^2}$

Formula: n_s = 1 − 6 ε + 2 η

$n_s=1-6\epsilon +2\eta$

Formula: r = 16 ε

$r=16\epsilon$

The helper relation below is used for the table and is the reason the $N$-based examples differ from the direct-$\phi$ output. In the slow-roll approximation, the number of e-folds between a field value and the end of inflation is approximately $N\approx \frac{{\phi }^2-{\phi }_{\mathrm{end}}^2}{2n}$. The script uses the approximation $\phi \approx \sqrt{2nN+\frac{n^2}{2}}$ when filling the example table. That means the table and the button answer related questions: the table starts from $N$, while the main calculator uses the field value you enter manually.

This distinction matters for interpretation. If you want the result to correspond to a specific e-fold count in the same approximation used by the table, choose a field value consistent with that relation. If instead you are exploring a field point directly, the calculator still gives the correct leading-order slow-roll quantities for that chosen $\phi$ and n.

Worked Example: Quadratic Inflation at φ = 5

This worked example follows the calculator's actual formulas for a quadratic monomial potential with n = 2 and a field value $\phi =5$ in reduced Planck units. The first slow-roll parameter becomes $\epsilon =\frac{4}{2\times 25}=0.08$. The second parameter is $\eta =\frac{2}{\times }$, which is $0.08$ as well. Then the scalar spectral index is $n_s=1-6\times 0.08+2\times 0.08=0.68$, and the tensor-to-scalar ratio is $r=16\times 0.08=1.28$.

That result is easy to follow numerically, which is exactly why it is useful as a check on the live calculator. It also shows why interpretation matters. A result like $r=1.28$ is far above the page's quick comparison threshold, so the calculator flags it as inconsistent with the simple observational screen. The point of the example is not that quadratic inflation is permanently ruled out in every context, but that a specific choice of n and $\phi$ can produce a tensor signal that is much too large for a quick slow-roll sanity check.

As another check, look at the prefilled table below. For n = 2 and $N=60$, the helper function first estimates the field value from the e-fold relation and then computes $n_s$ and $r$. Those values are generated with the same leading-order relations as the main calculator, but they start from $N$ rather than from a manually entered $\phi$. Comparing the manual form result with the table result is a good way to see how a field-value input and an e-fold-based estimate can lead to very different numerical outputs.

Reading the Inflation Slow-Roll Outputs

When you read the output, start with $\epsilon$ and $\eta$. In a monomial inflation model, these are the basic indicators of whether the slow-roll approximation is self-consistent. Values much smaller than one generally support the approximation, while values approaching one suggest inflation is ending or the approximation is becoming unreliable. Next, inspect $n_s$. A value slightly below one corresponds to the observed red tilt of the scalar power spectrum. Finally, check $r$, which measures the relative strength of primordial tensor modes. Larger $r$ means a stronger gravitational-wave signal and usually a higher inflationary energy scale.

The page includes a small comparison against a commonly quoted observational upper bound on $r$. That message is useful as a quick screen, but it should not be treated as a full likelihood analysis. Real cosmological constraints depend on the exact dataset, confidence level, reheating assumptions, and whether the model is being compared at a fixed pivot scale. The calculator is best understood as a first-pass estimator rather than a substitute for a full parameter-inference pipeline.

Monomial Reference Table and Trends

To illustrate the interplay between parameters, the table below lists slow-roll predictions for several choices of the power n and e-fold number $N$. The results show a familiar trend for monomial inflation: larger n generally increases the tensor-to-scalar ratio, while larger $N$ tends to reduce $\epsilon$ and therefore lower $r$. These examples are generated automatically by the existing script and provide a quick benchmark for comparison with your own inputs.

The slow-roll formalism also helps estimate the duration of reheating, the period after inflation when the inflaton decays into standard particles. Different reheating histories effectively change the mapping between observable scales and the number of e-folds, thereby altering predictions for $n_s$ and $r$. While this calculator fixes $N$ as an input and uses it mainly in the helper function, more advanced analyses treat it as a derived quantity that depends on the thermalization temperature and the equation of state during reheating.

In addition to scalar and tensor spectra, monomial slow-roll parameters also influence higher-order statistics such as the bispectrum. In simple single-field models, non-Gaussianity is usually suppressed and often scales schematically like . That is one reason the slow-roll framework remains so central: the same small parameters that govern the background evolution also shape many observable signatures.

Limitations and Assumptions in the Slow-Roll Approximation

This calculator is intentionally simple, and its limitations are important. First, it assumes a canonical single-field inflation model with a smooth monomial potential. It does not handle multifield dynamics, non-canonical kinetic terms, sharp features, turns in field space, or potentials with plateaus and inflection points where higher-order corrections may matter. Second, the displayed formulas are leading-order slow-roll expressions. They are excellent for intuition and rough estimates, but precision work may require next-order corrections and a more careful treatment of the horizon-crossing conditions.

Third, the page mixes two related but distinct ways of specifying the inflationary state. The form lets you enter $\phi$ directly, while the helper function used for the table estimates $\phi$ from $N$. That is not a bug in the script, but it does mean you should be clear about which quantity you are treating as fundamental in your own use. If you want a self-consistent monomial model at a chosen e-fold count, use the e-fold relation to guide your field choice. If you simply want to inspect the slow-roll parameters at a particular field value, the form output is the relevant result.

Finally, observational interpretation should be cautious. The page compares $r$ with a single threshold, but real cosmological constraints involve full datasets and model assumptions. A result that looks acceptable here is not automatically a viable inflation model, and a result that fails the quick comparison may still motivate theoretical study in a broader context. The calculator is best used for learning, screening, and building intuition before moving on to detailed numerical or statistical analysis.

By providing an interactive tool, this page bridges the gap between textbook formulas and hands-on exploration. Students can vary n, $\phi$, and $N$ to see how the resulting values of $n_s$ and $r$ change, building intuition about which monomial models survive observational scrutiny. Researchers may also use it as a quick estimator when sketching out new ideas. Even in an era of precision cosmology, simple slow-roll relations remain one of the clearest ways to connect an inflaton potential to observable consequences.