---
title: "Why a Stretched Ripple Stops Changing After Horizon Exit"
description: "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."
canonical_url: "https://sandboxphysics.com/stories/inflation-perturbations/why-a-stretched-ripple-stops-changing/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/inflation-perturbations/why-a-stretched-ripple-stops-changing/"
last_modified: "2026-11-12"
---

[From quantum ripples to cosmic texture](https://sandboxphysics.com/stories/inflation-perturbations/)

# Why a Stretched Ripple Stops Changing After Horizon Exit

Sandbox Physics · 2026-11-12

This chapter turns the freezing picture into three checkable statements. Inflation is the epoch when the comoving Hubble radius shrinks. A mode exits when its wavenumber equals $aH$. After exit, the curvature ripple of a single-field model stays constant.

## Before you begin

Exponentials, logarithms and one derivative. Units have {{c=1}}, and a prime means a derivative with respect to e-folds {{N=\\ln a}}.

## Two lengths to compare

Label points by comoving coordinates, which expand with space. A ripple then keeps a fixed comoving wavenumber $k$, while its physical wavelength $2\pi a/k$ grows with the scale factor $a$.

The Hubble length $1/H$ is the distance at which expansion carries two points apart at light speed. Points much closer than that can still exchange signals within one expansion time. Dividing by $a$ gives the comoving Hubble radius $1/(aH)$, the yardstick for a comoving ripple.

So the whole question is a comparison between $k$ and $aH$. A mode is inside the horizon when $k\gg aH$ and outside when $k\ll aH$.

$$
\lambda_{\mathrm{phys}}=\frac{2\pi a}{k},\qquad \frac{k}{aH}\gtrless 1
$$

The physical wavelength grows with $a$. Whether a mode is inside or outside the horizon depends only on the ratio $k/(aH)$.

Reading: Ellis, Vennin & Wands, Review 23: Inflation, Review of Particle Physics (PDG 2025, revised August 2025) · Adams, Cresswell & Easther, Inflationary perturbations from a potential with a step (2001), astro-ph/0102236v2

## Inflation means a shrinking comoving Hubble radius

Because $aH=\dot a$, the comoving Hubble radius shrinks exactly when $\ddot a>0$. Accelerating expansion and a shrinking $1/(aH)$ are the same statement.

In e-fold time it is cleaner. With the Hubble slow-roll parameter $\epsilon=-\dot H/H^2$, the growth rate of $aH$ is $1-\epsilon$. Inflation is $\epsilon<1$, and we define its end as the moment $\epsilon=1$. On the Lab’s plateau, $\epsilon$ is only about $2\times10^{-4}$ when the pivot scale exits.

$$
\frac{d}{dt}\left(\frac{1}{aH}\right)=-\frac{\ddot a}{\dot a^{2}},\qquad \frac{d\ln(aH)}{dN}=1-\epsilon
$$

The first form ties shrinking to acceleration; the second gives the rate in e-folds.

Reading: Ellis, Vennin & Wands, Review 23: Inflation, Review of Particle Physics (PDG 2025, revised August 2025) · Planck Collaboration, Planck 2018 results. X. Constraints on inflation, arXiv:1807.06211v2

## Counting e-folds to horizon exit

A mode exits when $k=aH$. Two modes therefore exit $\int(1-\epsilon)\,dN=\ln(k_2/k_1)$ apart. When $\epsilon$ is tiny, the gap in e-folds is simply the logarithm of the wavenumber ratio.

Worked example: the Lab tracks $0.002$, $0.01$ and $0.05\,\mathrm{Mpc}^{-1}$. The predicted gaps are $\ln 5\approx1.61$ and $\ln 25\approx3.22$ e-folds. In the computed figure the dotted exit lines sit near $58.2$, $56.6$ and $55.0$ e-folds before the end, which matches.

Those numbers need one convention to tie the model to real distances. We require that $k_*=0.05\,\mathrm{Mpc}^{-1}$ exits when $N_*$ e-folds of inflation remain. The default is $N_*=55$. Changing $N_*$ slides every observed scale along the potential.

![Three computed curvature modes shrink inside the horizon and level off after they exit, next to the growing comoving Hubble wavenumber](https://sandboxphysics.com/_astro/freezing.Da1MGV-H.svg)

Our calculation for the smooth $R^2$ plateau with $N_*=55$. Left or top: the size $|\mathcal{R}_k|$ of three complex mode functions, divided by its final value. Dotted lines mark $k=aH$. Right or bottom: $aH$ in $\mathrm{Mpc}^{-1}$ rises while each fixed wavenumber stays flat; a mode exits where the lines cross. Model output, not data. [Open the full figure](https://sandboxphysics.com/_astro/freezing.Da1MGV-H.svg)

Reading: Original background and mode calculations with an independent Python reference · Planck Collaboration, Planck 2018 results. X. Constraints on inflation, arXiv:1807.06211v2

## Inside the horizon: a wave whose size falls

Deep inside the horizon, gravity barely matters on the scale of one wavelength. The field combination $v=z\mathcal{R}$ with $z=a\phi'$ then behaves like a free wave with fixed amplitude $1/\sqrt{2k}$.

Since $z$ grows almost like $a$, the curvature ripple $|\mathcal{R}_k|$ falls like $1/a$, by a factor $e$ per e-fold. The figure starts each mode at $k/aH=100$; over the next $4.6$ e-folds its size falls by about $100$, as $e^{4.6}\approx100$ predicts.

Reading: Adams, Cresswell & Easther, Inflationary perturbations from a potential with a step (2001), astro-ph/0102236v2 · Original background and mode calculations with an independent Python reference

## Outside the horizon: why the ripple stays constant

The mode equation of the next chapter can be written in the conserved-flux form below. Once $k/aH\ll1$, the right-hand side vanishes, so $z^2aH\,\mathcal{R}'$ is constant.

Then $\mathcal{R}'\propto1/(z^2aH)$. With $z^2=2a^2\epsilon$ and slowly varying $H$ and $\epsilon$, this falls like $e^{-3N}$. One solution is constant; the other dies out within a few e-folds. That constant is the frozen value.

This conclusion relies on a single field. PDG 2025 notes that with several fields, non-adiabatic fluctuations can keep changing the curvature perturbation outside the horizon.

$$
\frac{d}{dN}\left(z^{2}aH\,\mathcal{R}'\right)=-\left(\frac{k}{aH}\right)^{2}z^{2}aH\,\mathcal{R},\qquad z=a\phi'
$$

Reduced Planck mass $M_{\mathrm{pl}}=1$. Outside the horizon the right side is negligible, and the flux on the left stays constant.

Reading: Ellis, Vennin & Wands, Review 23: Inflation, Review of Particle Physics (PDG 2025, revised August 2025) · Adams, Cresswell & Easther, Inflationary perturbations from a potential with a step (2001), astro-ph/0102236v2

## What freezing does not mean

Freezing fixes each mode’s amplitude, not a predictable value. The frozen ripple is a random draw from the quantum state, so theory predicts its variance, the power spectrum $\mathcal{P}_{\mathcal{R}}(k)$. The Lab’s map is one such random draw.

After inflation, modes re-enter the growing horizon and start evolving again, becoming the acoustic patterns of the microwave background and the seeds of structure. Following that later evolution takes transfer functions, the next stage beyond this chapter’s model.

Reading: Ellis, Vennin & Wands, Review 23: Inflation, Review of Particle Physics (PDG 2025, revised August 2025) · Planck Collaboration, Planck 2018 results. X. Constraints on inflation, arXiv:1807.06211v2

## Exercises and explained answers

1. Keep $\epsilon=0.01$ constant. How many e-folds separate the exits of two modes whose wavenumbers differ by a factor of $25$?

   Read the explanation

   The exit condition gives $(1-\epsilon)\,\Delta N=\ln25$, so $\Delta N=3.219/0.99\approx3.25$ e-folds. The small $\epsilon$ stretches the gap by only one per cent.

2. The integrator starts each mode at $k/aH=100$ and reads it out at $k/aH=10^{-3}$. How many e-folds does one mode take on the plateau?

   Read the explanation

   The ratio changes by $10^5$, so $\Delta N\approx\ln10^5/(1-\epsilon)\approx11.5$ e-folds. Most of the cost lies in the first part, where the mode still oscillates.

3. Outside the horizon, by what factor does the decaying part $\mathcal{R}'$ shrink over two e-folds if $H$ and $\epsilon$ stay fixed?

   Read the explanation

   $\mathcal{R}'\propto e^{-3N}$, so the factor is $e^{-6}\approx2.5\times10^{-3}$. After a few e-folds only the constant solution is left.

## Try the prediction

### [Read horizon-exit times from the Lab](https://sandboxphysics.com/labs/inflation-mode-freezing/)

1. Pause the Lab and drag the time slider to $58.2$. On the potential panel, the dashed line for the longest tracked mode meets the ball.
2. Drag on to $55$. The pivot mode exits there, and its curve in the mode-freezing chart flattens within about two e-folds.
3. Move the e-folds-after-pivot slider from $55$ to $46$ and watch every exit line slide along the potential.

[Previous: How a Quantum Ripple Becomes Cosmic Structure](https://sandboxphysics.com/stories/inflation-perturbations/how-a-quantum-ripple-becomes-cosmic-structure/) [Next: Deriving the Mode Equation Behind a Computed Spectrum](https://sandboxphysics.com/stories/inflation-perturbations/slow-roll-and-the-mode-equation/)

## Sources and calculation scope

1. [Planck Collaboration, Planck 2018 results. X. Constraints on inflation, arXiv:1807.06211v2](https://arxiv.org/abs/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.

2. [Ellis, Vennin & Wands, Review 23: Inflation, Review of Particle Physics (PDG 2025, revised August 2025)](https://pdg.lbl.gov/2025/reviews/rpp2025-rev-inflation.pdf)

   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.

3. [Adams, Cresswell & Easther, Inflationary perturbations from a potential with a step (2001), astro-ph/0102236v2](https://arxiv.org/abs/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.

4. [Original background and mode calculations with an independent Python reference](https://sandboxphysics.com/stories/inflation-perturbations/materials/inflation-spectrum-recompute-v1.zip)

   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.

## Keep reading

From quantum ripples to cosmic texture · Reading path

### Horizon exit, the mode equation and spectral constraints

1. [Article **How a Quantum Ripple Becomes Cosmic Structure**](https://sandboxphysics.com/stories/inflation-perturbations/how-a-quantum-ripple-becomes-cosmic-structure/)
2. [You are here **Why a Stretched Ripple Stops Changing After Horizon Exit**](https://sandboxphysics.com/stories/inflation-perturbations/why-a-stretched-ripple-stops-changing/)
3. [Article **Deriving the Mode Equation Behind a Computed Spectrum**](https://sandboxphysics.com/stories/inflation-perturbations/slow-roll-and-the-mode-equation/)
4. [Article **What the Tilt, Tensors and Squeezed Triangles Measure**](https://sandboxphysics.com/stories/inflation-perturbations/what-tilt-tensors-and-squeezed-triangles-measure/)
5. [Article **Recompute a Primordial Spectrum and Test the Solver**](https://sandboxphysics.com/stories/inflation-perturbations/recompute-a-primordial-spectrum/)
6. [Article **Reading Planck’s Searches for Inflation Features and Non-Gaussianity**](https://sandboxphysics.com/stories/inflation-perturbations/read-feature-and-non-gaussianity-searches/)