Lyth Bound Field Excursion Calculator: Inflationary Δφ from r and N

Introduction: Lyth-bound field ranges in inflationary cosmology

The Lyth bound connects the size of primordial tensor modes to the distance the inflaton must travel during inflation. Entering a tensor-to-scalar ratio r and a number of e-folds N into this calculator turns that pair into the corresponding minimal field excursion Δφ and shows whether the excursion stays below or rises above one reduced Planck mass. That makes the tool useful for a fast check on whether a proposed inflationary model behaves like a small-field or large-field scenario before you move on to a more detailed analysis.

In the standard slow-roll picture, the tensor-to-scalar ratio is related to the first slow-roll parameter ε by r = 16ε. Over N e-folds, the inflaton's field range is then approximated by

Combining those relations gives the familiar Lyth-bound form, written here in reduced Planck units with M_Pl = 1:

This calculator uses the equality version Δφ = (M_Pl/√8) √r N so that the displayed output reflects the minimal excursion implied by the chosen inputs. Because the square root of r appears alongside a factor of N, the result reacts strongly to the duration of inflation as well as to the tensor amplitude. Even a modest change in either input can move the answer from comfortably sub-Planckian to clearly super-Planckian, which is why the page also labels the result for quick interpretation.

The longer discussion below explains how the bound arises from the inflaton trajectory and why it matters when you compare model classes. In practical terms, the calculator is a compact way to see how a candidate value of r translates into a field-range requirement, without having to redo the algebra each time you test a different e-fold count. That is especially helpful when you are scanning between optimistic and conservative assumptions and want to know which parameter is doing most of the work.

The table below shows representative Lyth-bound calculations for a few choices of r and N, letting you see how the result and the sub-Planckian or super-Planckian label move together:

These examples illustrate the basic scaling built into the formula: larger tensor ratios enlarge the field excursion, and longer inflationary stretches enlarge it again. The calculator's classification output follows the same threshold used in the table, so the text result gives you an immediate check on whether the computed excursion crosses one reduced Planck mass.

There are also interpretive details to keep in mind when you use the result in model building. The inferred excursion depends on how you associate N with observable scales, so changes in the reheating history can shift the comparison even when r is unchanged. The clean formula shown here also assumes the usual monotonic slow-roll intuition, which is why it should be treated as a transparent baseline rather than a complete description of every inflationary trajectory. If a model introduces scale dependence, bends in field space, or unusual kinetic terms, the output is still informative, but it is no longer the whole story.

From a theory standpoint, the Lyth bound is often used as a quick filter for whether an inflationary potential needs extra structure to remain under control across a wide field range. A large excursion does not automatically invalidate a model, but it does raise questions about symmetry protection, effective-field-theory reliability, and how the underlying potential is stabilized over the range implied by the chosen r and N. That is the kind of question this calculator is meant to surface early, before you commit to a deeper calculation or a full numerical study.

To conclude the introduction, the calculator gives you a direct bridge from the observational side of inflation to the model-building side. If you are comparing competing scenarios, it helps to know not only whether r is large or small, but how that choice translates into field distance once you also choose the inflationary duration. In that sense, the page is less about a single number and more about the relationship between tensors, e-folds, and the geometry of the inflaton path.

Saving Lyth-bound field excursion estimates

When you are comparing inflationary scenarios, the copy button lets you keep a record of each computed Δφ value alongside the r and N inputs that produced it. That makes it easier to line up several candidate models and see whether they differ mainly because of the tensor amplitude, the number of e-folds, or both. For notebook work, talks, or model scans, having the result in a copyable text form saves you from re-entering the same assumptions over and over again.

It also helps when you want to show a collaborator how sensitive the field-range estimate is to a particular assumption. A copied result keeps the calculator's interpretation attached to the input pair that generated it, so you can compare nearby scenarios without losing track of which choice of r and N belongs to which Δφ.

How to use this Lyth bound field excursion calculator

1. Enter Tensor-to-Scalar Ratio r as a plain numerical value for the tensor amplitude you want to test.
2. Enter Number of e-folds N for the inflationary interval you want the Lyth bound to cover.
3. Run the calculation, then change one input at a time to see how the field excursion responds across nearby inflationary scenarios.

Formula: the Lyth-bound estimate from r and N

The displayed estimate follows directly from the Lyth-bound expression shown above, with the reduced Planck mass carried through the calculation in the JavaScript. Once r and N are supplied, the page evaluates Δφ and also checks the ratio Δφ / M_Pl so the answer can be labeled as sub-Planckian or super-Planckian. That keeps the output tied to the same conventions used in the explanation and lets you compare scenarios without translating the result by hand.

Worked example: compare two Lyth-bound scenarios

Use this section as a qualitative guide for reading the calculator, not as a canned story that replaces the inputs you enter yourself. If you hold N fixed and raise r, the field excursion grows like the square root of the tensor ratio; if you hold r fixed and increase N, the excursion grows linearly. Those two scalings are the main reason the calculator is useful for quick checks: they show at a glance whether the tensor amplitude or the inflationary duration is driving the field-range requirement.

When you compare two candidate models, start by changing only one input so you can see which assumption matters most. A smaller r may keep the excursion modest even when N is fairly large, while a longer inflationary period can push the estimate upward even if r remains subdued. The point of a worked comparison here is not to memorize a single number, but to develop an intuition for how quickly the Lyth bound responds when you move through the parameter space one step at a time.

Limitations and assumptions of the Lyth bound estimate

This calculator is a compact planning estimate built around the standard slow-roll Lyth bound, not a full inflationary solver that follows every detail of a particular model. It captures the simple relationship between r, N, and Δφ, but it does not attempt to fold in every reheating history, scale dependence, multi-field turn, or non-canonical kinetic effect that can complicate the interpretation of a real scenario. For that reason, the output is best treated as a clear baseline for discussion rather than as the last word on a model's viability.

The result is only as reliable as the conventions behind the inputs. If r comes from a forecast, a constraint, or a paper using a different notation, make sure the number you enter matches the same convention assumed by the formula on this page, and that the e-fold count refers to the same inflationary interval. The calculator can show you how the bound scales, but it cannot repair a mismatch between source data, model assumptions, or the way those assumptions were defined.

It is also worth remembering that the Lyth bound is most transparent when the inflaton rolls monotonically in a simple single-field picture. If your model has features, bends in field space, or other departures from that picture, the displayed Δφ still provides a useful reference point, but it should be checked against the specific dynamics of the model rather than read as a universal verdict. In short, the page is designed to help you think clearly about the size of the field excursion implied by a chosen r and N, while leaving the deeper theoretical judgment to the model-specific analysis that follows.