Inflation: Quantum Ripples to Cosmic Texture

How does a tiny early ripple become today’s large structure? During inflation, space stretched so fast that quantum jitters in the field were pulled beyond the reach of any signal. Once a ripple is larger than that horizon it stops oscillating and keeps its size. Later, gravity grows these frozen ripples into the hot and cold spots of the microwave background and into galaxies.

Drag a step on the inflaton potential and watch quantum ripples freeze as they leave the horizon. Compare the spectrum with Planck limits.

Spectral tilt at the pivot scale $0.965$

Tensor-to-scalar ratio $3.55\times10^{-3}$

Largest ripple from the step $33.3\%$

Largest squeezed non-Gaussian amplitude $-0.651$

Each mode freezes after it leaves the horizon

$k=0.002\,\mathrm{Mpc^{-1}}$
$k=0.01\,\mathrm{Mpc^{-1}}$
$k=0.05\,\mathrm{Mpc^{-1}}$

Dotted lines mark horizon exit; the white line is the current time.

Primordial curvature spectrum

mode-by-mode computation
same potential without the step
ideal slow-roll reference $H^2/(8\pi^2\epsilon)$
Planck 2018 power-law fit, 68%

Squeezed limit of the bispectrum

$f_{\rm NL}^{\rm sq}(k)=\tfrac{5}{12}\,[1-n_s(k)]$
Planck 2018 local template, 68% $-0.9\pm5.1$

Only this limit is computed. The Planck band constrains a constant-amplitude template.

Model assumptions and sources

Single canonical scalar field, reduced Planck units, time measured in e-folds $N=\ln a$ . Each mode starts in the de Sitter Bunch–Davies vacuum at $k/aH=100$ and is read at $k/aH=10^{-3}$ .
The overall height of the potential is re-fixed so the smooth model gives $\ln(10^{10}A_s)=3.044$ at $k_*=0.05\,\mathrm{Mpc^{-1}}$ ; $k_*$ equals $aH$ when $N_*$ e-folds remain.
The step uses the tanh form of Adams, Cresswell and Easther (2001). Its height and the sharp preset are illustrative, not fits; Planck 2018 found no preference for features.
Only the squeezed bispectrum limit is shown. Sharp features also produce equilateral and oscillating bispectra that this Lab does not compute.
The map draws independent Gaussian Fourier amplitudes with the computed variance on a periodic 6.3 Gpc patch, from $k=10^{-3}$ to the grid limit. It is a picture of the two-point function, not a simulated sky.

Independent checks: the exact power-law spectrum, the de Sitter mode function, the canonical Wronskian, step halving and a separate Python solver.

A ripple stops changing once it outgrows the horizon

BackgroundDuring inflation a nearly constant energy density makes space grow by a factor e in each e-fold, while the Hubble length barely changes. A quantum ripple of fixed comoving size is first much smaller than the Hubble length and oscillates. A few e-folds later it is much larger and its curvature amplitude stays fixed.

Why it mattersThose frozen amplitudes are the seeds measured in the microwave background. The slope of the potential when each scale left the horizon sets how strong the seed is, so the measured spectrum constrains the shape of the potential.

Focus question: Why does each ripple remember only the moment it left the horizon?
One-sentence intuition: Outside the horizon the curvature mode has a constant solution, so the spectrum maps the potential near each exit time.

Core mathematical model

Background in e-folds

H^2=\frac{V}{3-\epsilon},\qquad \epsilon=\frac{\phi'^2}{2},\qquad \phi''=-(3-\epsilon)\,\phi'-\frac{V_{,\phi}}{H^2}

Primes are derivatives with respect to the e-fold number $N$ , in reduced Planck units. Inflation ends when $\epsilon=1$ .

One Fourier mode of the curvature

\mathcal R_k''+\left(3-\epsilon+\frac{2\phi''}{\phi'}\right)\mathcal R_k'+\left(\frac{k}{aH}\right)^{2}\mathcal R_k=0

Inside the horizon the last term dominates and the mode oscillates with a decaying amplitude. Outside it the last term is negligible and a constant solution remains.

What the Lab plots

\mathcal P_{\mathcal R}(k)=\frac{k^3}{2\pi^2}\,\lvert\mathcal R_k\rvert^2,\qquad n_s-1=\frac{d\ln\mathcal P_{\mathcal R}}{d\ln k},\qquad f_{\rm NL}^{\rm sq}=\frac{5}{12}\,(1-n_s)

The spectrum is read after the mode has frozen. The last relation is the single-field squeezed limit in the convention $\zeta=\zeta_g+\tfrac35 f_{\rm NL}\zeta_g^2$ .

Common difficulties

The horizon is not a wall

Typical misconceptionA mode stops at the horizon because something blocks it.

Better mental modelNothing blocks it. The oscillation term shrinks as space stretches until the pressure gradient can no longer push the ripple back.

A step changes many scales

Typical misconceptionA step in the potential should change one wavenumber only.

Better mental modelA sudden change in the field speed disturbs modes still inside the horizon, so it leaves ringing on smaller scales as well.

Run the experiment

01
Watch one mode freeze
Let the animation run and follow the left chart below the stage.
What to observe: Each curve falls while its mode oscillates inside the horizon, then turns flat after its dotted exit line.
02
Drag the step
Drag the gold handle up and sideways on the potential.
What to observe: A taller step makes larger ripples in the spectrum; moving it changes which scales ring.
03
Compare with Planck
Choose Steep slope, strong gravitational waves, then Shorter reheating history.
What to observe: The steep model gives a tensor ratio well above current limits; fewer e-folds after the pivot lowers the tilt.

Inflation: Quantum Ripples to Cosmic Texture

A ripple stops changing once it outgrows the horizon

Core mathematical model

Background in e-folds

One Fourier mode of the curvature

What the Lab plots

Common difficulties

The horizon is not a wall

A step changes many scales

Run the experiment

Watch one mode freeze

Drag the step

Compare with Planck

Related experiments and articles

Articles

Horizon exit, the mode equation and spectral constraints