Inflation Slow-Roll Calculator

Inputs

Tip: You may work in any consistent unit system. If you use GeV, set M_pl in GeV and keep derivatives consistent with your field units.

Formula used

This calculator uses the potential slow-roll definitions (reduced Planck mass form):

• ε_V = (M_pl^2 / 2) · (V′/V)^2
• η_V = M_pl^2 · (V″/V)
• ξ_V^2 = M_pl^4 · (V′·V‴ / V^2) (requires V‴)

Leading-order observable relations:

• n_s ≈ 1 − 6ε_V + 2η_V
• r ≈ 16ε_V
• α_s ≈ 16ε_Vη_V − 24ε_V^2 − 2ξ_V^2 (requires V‴)

If V > 0, it also computes H ≈ √(V / (3 M_pl^2)) and V^{1/4} as an energy-scale proxy.

How to use this calculator

1. Choose a Planck mass option (standard or custom).
2. Enter V, V′, and V″ at your chosen field value.
3. Optionally enter V‴ to compute ξ_V^2 and running α_s.
4. Press Calculate to show results above the form.
5. Use Download CSV for spreadsheets, or Download PDF for printing.

Example data table

Example values below use a custom M_pl = 1 (Planck units) to keep numbers readable.

Inflation slow-roll article

1) Why slow-roll matters

Slow-roll inflation models a scalar field phi moving down a potential V(phi) with enough friction from cosmic expansion to keep the motion gradual. When the roll is slow, the expansion is close to exponential and the primordial fluctuation spectrum can be predicted from a few local derivatives of the potential.

2) What the inputs represent

Enter V, Vp, and Vpp evaluated at the same phi. These three numbers fix the leading slow-roll parameters and therefore the tilt and tensor strength. If you also enter Vppp, the calculator estimates xi_V^2 and the running of the spectral index.

3) Epsilon_V and the end of inflation

Epsilon_V measures the steepness of the potential slope. Small epsilon_V (for example 1e-4 to 1e-2 in many scenarios) supports prolonged inflation. If epsilon_V approaches 1, the slow-roll approximation fails and inflation is expected to end near that field value.

4) Eta_V and spectral tilt control

Eta_V captures the local curvature of V(phi). It influences the scalar tilt through ns about 1 - 6*epsilon_V + 2*eta_V. Even with tiny epsilon_V, a modest negative eta_V can produce a red tilt (ns less than 1), which is commonly favored by observations.

5) From epsilon_V to r

The tensor-to-scalar ratio is reported using r about 16*epsilon_V. This makes r a direct slope diagnostic. As a quick check, epsilon_V = 0.002 implies r about 0.032, while epsilon_V = 5e-4 implies r about 0.008.

6) Running alpha_s and higher derivatives

The running alpha_s is typically small in smooth potentials, often around the 1e-3 level or below. A nonzero Vppp can increase xi_V^2 and therefore alpha_s, which may indicate features or rapid curvature changes. Treat alpha_s as exploratory if Vppp is uncertain.

7) Energy scale, H, and the amplitude check

When V is positive, the tool estimates the Hubble scale H and an energy proxy V^(1/4). It also predicts the scalar amplitude As via As = V / (24*pi^2*Mpl^4*epsilon_V). With a target As near 2.1e-9, you can infer the V needed to match your chosen epsilon_V.

8) Practical workflow with this calculator

First tune eta_V to land ns in your desired band, then tune epsilon_V to control r. Next use the target As option to normalize the overall energy scale. Finally, if you are exploring running, vary Vppp and watch how alpha_s changes. For model building, many analyses consider roughly 50 to 60 e-folds before the end of inflation.

FAQs

1) What does slow-roll mean in this calculator?

It means epsilon_V and |eta_V| are much smaller than one, so the field evolves gradually and the leading-order relations for ns, r, and alpha_s remain a good approximation.

2) Do I have to use GeV units?

No. Use any consistent units. If you work in Planck units, choose Custom Planck mass and set Mpl = 1, then keep V and its derivatives consistent with your field normalization.

3) Why do some outputs require V > 0?

H, V^(1/4), and As rely on a positive inflationary energy density. If V is zero or negative, those derived quantities are not meaningful in this slow-roll potential framework.

4) What if epsilon_V is greater than or equal to 1?

That usually signals slow-roll has broken down and inflation is ending. The tool still computes values, but the reported ns, r, and As formulas may not be reliable beyond the slow-roll regime.

5) What should I enter for the target As?

A commonly used reference is about 2.1e-9 at a standard pivot scale. Use it as a normalization check, and adjust if your analysis uses a different convention.

6) Why is alpha_s sometimes blank?

Running requires xi_V^2, which depends on Vppp. If you do not enter Vppp, the calculator cannot estimate xi_V^2 consistently, so it leaves alpha_s unreported.

7) Is the e-fold estimate N exact?

No. It is a rough estimate using N about (V/Vp)*DeltaPhi/Mpl^2. A full treatment integrates (V/Vp) over phi and depends on the end-of-inflation condition.