From quantum ripples to cosmic texture
Start from a tiny quantum jitter during inflation and follow it out of the horizon, into a frozen seed and onto the Planck constraints. Then recompute a spectrum yourself and read how researchers search for features and non-Gaussian signals.
How a Quantum Ripple Becomes Cosmic Structure
In the first instant, space stretched so fast that quantum jitters froze in place. Follow one ripple from empty space to the seeds of galaxies.
Why a Stretched Ripple Stops Changing After Horizon Exit
Compare a ripple’s fixed comoving wavenumber with the growing aH, find when it exits the horizon, and show why its curvature then stays constant.
Deriving the Mode Equation Behind a Computed Spectrum
Write the inflaton background in e-folds, derive the curvature mode equation, start it in the Bunch–Davies vacuum and check it with the Wronskian.
What the Tilt, Tensors and Squeezed Triangles Measure
Derive the spectral tilt and tensor ratio, test two potentials against Planck and PDG limits, and see why single-field squeezed non-Gaussianity is tiny.
Recompute a Primordial Spectrum and Test the Solver
Run a small Python reference, check it against exact power-law spectra, recompute five fixed models and compare their tilt and tensor ratio with Planck.
Reading Planck’s Searches for Inflation Features and Non-Gaussianity
A guided reading of Planck 2018 X and IX and the PDG 2025 review: step templates, fit improvement versus evidence, bispectrum templates and reheating.
Recompute the spectra yourself
Download the Python reference, benchmarks, inputs and expected outputs (23973 B)
NumPy and SciPy integrate the background and every mode from its vacuum state. The expected files were made with Python 3.9.6, NumPy 2.0.2 and SciPy 1.13.1. The ZIP contains no browser Engine and no paper text.
SHA-256 and what the files are
01825a57661f03e95e517178320bf1bd20f1bf050aae74318a37deb51a799a77
The five inputs are the fixed models behind the Lab presets. The expected outputs are our own model calculations, not observations. A fresh extraction reproduced them exactly on the same software versions.
Sources and calculation scope
- Planck Collaboration, Planck 2018 results. X. Constraints on inflation, arXiv:1807.06211v2
Full v2 PDF (2 August 2019) read for Sections 4.2 and 7.1–7.3: equations 47–48, the step template in equations 57–62, Table 11 priors, Table 12 step results and Figure 28. The headline spectral index and amplitude quoted in this course come from its parameter tables.
- Planck Collaboration, Planck 2018 results. IX. Constraints on primordial non-Gaussianity, arXiv:1905.05697v1
Full v1 PDF read for Sections 2.1–2.2 (equations 1–3, written for the Bardeen potential rather than the curvature perturbation), Table 5 final SMICA T+E values after lensing-ISW subtraction, and Table 10 feature-model peak statistics with the look-elsewhere correction.
- Ellis, Vennin & Wands, Review 23: Inflation, Review of Particle Physics (PDG 2025, revised August 2025)
Full review read. Used: Section 23.3.4 equations 23.55–23.57 and its sentence on the ACT combination, Section 23.6 on reheating and the number of e-folds after the pivot exits, and Section 23.7.3.2 equations 23.101–23.103 for the sign convention of the local non-Gaussian amplitude and its single-field squeezed-limit value.
- Adams, Cresswell & Easther, Inflationary perturbations from a potential with a step (2001), astro-ph/0102236v2
Full v2 read. Equation 1 defines the tanh step we use; equations 6–19 set out the mode equation in e-fold time and its initial data. Our code is written independently and puts the step on the Starobinsky plateau and on a quadratic potential, not only on the paper’s quadratic case.
- Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models (2003), astro-ph/0210603v5
Full v5 read. Equation 1.2 gives the squeezed-limit argument. In that paper the tilt symbol means the deviation from scale invariance, which modern papers write as the spectral index minus one.
- Creminelli & Zaldarriaga, Single field consistency relation for the 3-point function (2004), astro-ph/0407059v2
Full v2 read. Equation 1 states the consistency relation for any single-field model. Equation 5 defines the local non-Gaussian amplitude with the opposite sign to Planck and PDG; we use the PDG sign and do not use or rewrite equation 6.
- Stewart & Lyth, A more accurate analytic calculation of the spectrum of cosmological perturbations produced during inflation (1993), gr-qc/9302019v1
Full v1 read, with printed pages 4–6 checked as images. Equations 31, 41 and 43 give the exact power-law-inflation spectra that our solver must reproduce.
- CLASS explanatory.ini, lesgourg/class_public commit c5d93d01990116162b408b789b62ec5669fc4ea4
Primordial-spectrum options read: the pivot wavenumber setting, the inflation_V numerical mode and the external_Pk route for handing CLASS your own table. We did not run CLASS for this course.
- Original background and mode calculations with an independent Python reference
Our own calculations: the Lab integrator and a separate NumPy/SciPy reference agree to better than three parts in ten thousand on all five presets. The ZIP holds the Python reference, closed-form benchmarks, five inputs and expected outputs. No Engine code and no paper text is included.