Inflaton potentialV(φ) / V(φ₀)
The shallow shoulder is the quasi-inflection region: the field loses speed there even though the change in the potential itself is visually subtle.
Interactive lab · SIGWAY
From inflation to gravitational waves
Start from a single-field inflationary potential or draw the primordial curvature spectrum directly, then explore the structures that can seed scalar-induced gravitational waves.
SIGWAY kernel lab
Paint a primordial spectrum
Draw the binned curvature spectrum and see how pairs of scalar modes source the induced gravitational-wave signal.
Loading coefficient grid…
Input𝒫ζ(k)
Drag across the plot to paint
ResponseΩGWh²(f)
LISA · 4 yr PLI · radiation domination
The induced spectrum is quadratic in the curvature perturbations: every output frequency receives contributions from pairs of primordial modes, so changing one bin generally reshapes a broad range of the gravitational-wave spectrum. PTA and LIGO views use the radiation-era kernel’s scale invariance to shift both frequency grids together. LISA and LIGO show power-law-integrated sensitivity curves; the PTA reference is the NANOGrav 15-year free-spectrum posterior rather than a forecast sensitivity.
Single-field inflation lab
From an inflaton potential to 𝒫ζ(k)
Follow the background through a near-inflection point and compare the full Mukhanov–Sasaki solution with the usual horizon-crossing slow-roll estimate.
USR interval—
Minimum εH—
Peak frequency—
Peak 𝒫ζ—
BackgroundεH(N)
The shaded interval uses ηH < −3 as a conservative marker of ultra-slow-roll evolution.
Mukhanov–Sasaki modes𝒫ζ(f)
Full Mukhanov–Sasaki solutionHorizon-crossing slow-roll estimate
What is being calculated?
The inflation playground solves the background Klein–Gordon and Friedmann equations and then evolves each scalar mode from Bunch–Davies initial conditions to super-horizon freeze-out with SIGWAY’s Mukhanov–Sasaki solver. The quasi-inflection potential follows the single-field model introduced here.
Ultra-slow-roll spectra are unusually sensitive to small potential changes. The comparison curve deliberately shows why a horizon-crossing slow-roll estimate is unsafe here: slow-roll violation can change the result qualitatively, and perturbations can continue evolving after horizon exit. The stored solutions are linear Mukhanov–Sasaki results; stochastic effects and primordial non-Gaussianity are outside this illustration.
The spectrum painter starts one step later. Scalar perturbations generate gravitational waves at second order when they re-enter the horizon. Because each output frequency receives contributions from pairs of curvature modes, a narrow primordial feature can produce a broader, structured response. The 50-bin view is standard; 100 bins allow finer features. Detector overlays provide the observational scale for LISA, PTAs, and LIGO.