Thinking in Waves Episode 47 / Part XVI — Horizons, as Waves

Reading the primordial spectrum as an order

Inflation —
The universe's tilt is $-0.105$ dB/oct The Mukhanov–Sasaki equation is Episode 13's $1/\tau^2$ potential itself — $s=\frac12\pm\nu$. Episode 13, Episode 39 (AdS) and this one are three faces of the same quadratic.
★ Put $x=-k\tau$ and $k$ drops out of the equation — that is what scale invariance really is.
★★ And the measured $n_s=0.9649$, converted to dB/oct, is $-0.105$ — the same order of magnitude as Episode 42's 0.081 dB/oct for turbulent intermittency.

Tools needed: Episodes 13, 28, 36, 39, 42, 44, 45, 46Computation: kenshou/uchu.js

For 46 episodes the wave always travelled inside something.
★ This time that something makes the wave: the expansion of the universe shakes the vacuum, and the ripples appear 13.8 billion years later as temperature patches on the sky.

01★★ The Mukhanov–Sasaki equation is Episode 13's equation

The equation for primordial fluctuations $$v''+\left(k^2-\frac{z''}{z}\right)v=0,\qquad \frac{z''}{z}=\frac{\nu^2-\frac14}{\tau^2}\ \ (\nu=\tfrac32\ \text{is exact de Sitter})$$

★ Put $x=-k\tau$ and $k$ disappears $$\frac{d^2v}{dx^2}+\left[1-\frac{\nu^2-\frac14}{x^2}\right]v=0$$

The point of §01

★ $k$ dropping out is what scale invariance really is. Every wavelength obeys the same equation, so every wavelength comes out with the same amplitude.
★★ And for $x\ll1$ (outside the horizon) the 1 stops mattering — $v_{xx}=\dfrac{\nu^2-\frac14}{x^2}v$, exactly Episode 13's $1/r^2$ potential. $v\sim x^s$ with $s=\frac12\pm\nu$.

$\nu$	$s=\frac12\pm\nu$	Episode 13's $\lambda$	Episode 39's $m^2L^2$ ($d=3$)
0.5	1, 0	0	−2
1.0	1.5, −0.5	−0.75	−1.25
1.5	2, −1	−2	0
2.0	2.5, −1.5	−3.75	1.75

★★★ Three faces

Episode 13 (the critical $1/r^2$ potential)
Episode 39 (AdS, $\Delta=\frac d2\pm\nu$)
This episode (inflation, $s=\frac12\pm\nu$)
— all the same quadratic.

02Solve it and measure the spectral index

Fix $|\tau_{\rm end}|=1$, start from inside the horizon ($x=400$) with Bunch–Davies initial conditions, integrate, and measure $P(k)=k^2|v(x{=}k)|^2$.

$\nu$	slope (measured)	$3-2\nu$	difference	$n_s$
1.500	0.00000515	0.000000	5.2e−6	★ 1.00000
1.450	0.10000572	0.100000	5.7e−6	1.10000
1.400	0.20000642	0.200000	6.4e−6	1.20000
1.600	−0.19999571	−0.200000	4.3e−6	0.80000

The point of §02

★ $n_s-1=3-2\nu$, confirmed to $10^{-6}$. At $\nu=3/2$ exactly, $n_s=1$: perfect scale invariance.
★★ In this series' language $\nu$ is the order — the same $\nu$ as in Episode 39, where "the bulk mass sets the boundary's order." Here it is the shape of the inflaton potential that sets $\nu$.

03Leave the horizon, and it freezes

The curvature perturbation is $\zeta=v/z$ with $z\propto(-\tau)^{1/2-\nu}$, so $|\zeta|\propto|v(x)|\,x^{\nu-1/2}$.

$x=k\|\tau\|$	$\|v(x)\|$	★ $\|\zeta\|$	State
10	1.00499	10.04985	inside the horizon (oscillating)
3	1.05409	3.16228	inside the horizon (oscillating)
1	1.41421	1.41421	★ crossing out
0.1	10.04986	1.00499	★ freezing
0.01	100.00486	★ 1.00005	★ frozen

The point of §03

★★ Once $x<1$, $|\zeta|$ becomes constant (1.41 → 1.005 → 1.00005) — that is "freezing."
★ A fluctuation that has left the horizon no longer changes. Which is why it survives intact into the microwave background 380,000 years later.
★★ In wave language, a wave that can no longer travel — a cousin of Episode 28's evanescent wave. There it could not travel in space and decayed exponentially; here it cannot keep up in time and simply stops.

04Model by model: $n_s$ and $r$ against observation

Observation gives $n_s=0.9649\pm0.0042$ and $r<0.036$. For $V\propto\phi^p$, $n_s-1=-\dfrac{p+2}{2N}$ and $r=\dfrac{4p}{N}$.

Model	$N$	$n_s$	$r$	Verdict
$V\propto\phi^2$	60	0.966667	0.133333	$n_s$ ◎ / ★ $r$ excluded
$V\propto\phi^4$	60	0.950000	0.266667	$n_s$ × / ★ $r$ excluded
$V\propto\phi^1$	60	0.975000	0.066667	$n_s$ ◎ / ★ $r$ excluded
Starobinsky ($R^2$)	60	0.966667	0.003333	★ $n_s$ ◎ / $r$ ◎

The point of §04

★★ By $n_s$ alone $\phi^2$ passes, but $r$ excludes it.
★ One matching exponent is not enough — exactly the lesson of Episode 42, where "$-5/3$ fits well" turned out to be the second moment being insensitive.
★★ $r$ so far yields only an upper limit. ★ This is a worked example of this series' rule: choose a quantity that is sensitive.

05★★★ Converting $n_s$ into dB/oct

Putting it on Episode 1's scale $$P(k)\propto k^{n_s-1}\ \Longrightarrow\ \text{amplitude}\propto k^{(n_s-1)/2}\ \Longrightarrow\ \text{dB/oct}=3(n_s-1)$$

Physics	Tilt away from ideal [dB/oct]	Away from what
★ primordial fluctuations	−0.1053 ± 0.0126	★ perfect scale invariance ($n_s=1$)
★ turbulent intermittency (Ep. 42)	+0.0812	★ Kolmogorov ($-5/3$)

The point of §05 — ★★★ the core of this episode

The two most precisely measured "tilts away from ideal" in physics
are both of order 0.1 dB/oct.

★ It is a coincidence — the two have no common origin. This article only says that put on the same scale, they land near each other.
★★ What they do share is that the departure itself carries the physics — for the universe, that inflation ended in finite time; for turbulence, that the cascade multiplies.
★ If $n_s$ were exactly 1, inflation would never have ended. It ended, and that is why $\nu$ sits slightly off $3/2$.

06How many e-folds — the horizon problem

Item	Value
reheating temperature (example)	$10^{15}$ GeV
present CMB temperature	$2.35\times10^{-13}$ GeV
★ e-folds required (roughly)	63.6

The point of §06

★ The familiar "$N\approx60$" is this number (our estimate here is very rough).
★★ Episode 46 gave "74.5 e-folds to leave the measurable band." A horizon is a device that switches "visible" to "invisible" in 60 to 75 e-folds.
★ Both are "a few tens when counted logarithmically" — on Episode 20's logarithmic-time scale, the whole universe spans only a few tens of divisions.

07★ Inflation is Episodes 44 and 45's parametric drive

Episode	What is modulated	What comes out
44 (metamaterials)	the medium's $\varepsilon(t)$	amplification in a momentum gap
45 (dynamical Casimir)	the boundary's position	photons out of the vacuum
★ 47 (inflation)	the cosmic scale factor	★ fluctuations out of the vacuum

The point of §07

All the same shape — $v''+\omega^2(t)v=0$ with a time-dependent $\omega$. The Bogoliubov coefficients obey $|\alpha|^2-|\beta|^2=1$ as always (Episodes 36, 45).
★★★ The CMB fluctuations are the dynamical-Casimir photons of an expanding universe.
★ Only one thing is different — nobody is doing the shaking. The shaking is the expansion itself, and its energy comes from the inflaton's potential energy. Following Episode 45's rule — look first for who is doing work — the answer here is the inflaton.

08What wave language can and cannot say

Item	Verdict	Basis
★ MS equation = Episode 13's $1/\tau^2$	◎ analytic	★ $s=\frac12\pm\nu$
$k$ drops out under $x=-k\tau$	◎ analytic	what scale invariance is
$P(k)\propto k^{3-2\nu}$	◎ numerics	agrees to $10^{-6}$
$\zeta$ freezes outside the horizon	◎ numerics	constant for $x<1$
$n_s$ alone cannot select a model	◎ numerics	★ $\phi^2$ excluded by $r$
★ $n_s\to-0.105$ dB/oct	◎ analytic	★ same order as turbulence's 0.081
Where $N\approx60$ comes from	○ numerics	★ a very rough estimate
★ Whether inflation happened	× no	★ only calculations inside the model
★ What the inflaton is	× no	★ its particle identity stays unknown

Exercises

§05 placed two "0.1 dB/oct" side by side. Is that a meaningful coincidence?
Show answer
No — it is a coincidence, as the article says. The universe's $n_s$ comes from inflation ending in finite time; turbulence's bent $\zeta_p$ comes from the cascade being multiplicative. There is no shared origin.
★ What can be said is that they can be put on the same scale. Episode 43's table gains another row.
★★ That "almost zero but not zero" appeared twice is suggestive — both mean an ideal symmetry is slightly broken. ★ But going further requires a mechanism linking them, and this article has none.
§03 froze $\zeta$. Why does freezing preserve information?
Show answer
Because what does not change is a record. Outside the horizon nothing can reach it causally, so there is nothing left to change $\zeta$.
★ In wave language it is "a mode with $\omega\to0$" — the same place Episode 4 put the constant of integration. $\omega=0$ means time-independent, i.e. a constant.
★★ Episode 46 had the same $\omega\to0$ mean "unmeasurable." Here it means "indelible." ★ The difference is whether the horizon later expands to bring it back into view — in inflation it does; for a black hole it does not.
§01 said $k$ dropping out is what scale invariance is. Then where does $n_s\ne1$ come from?
Show answer
From $z''/z$ not being exactly $2/\tau^2$. In exact de Sitter ($H$ perfectly constant) $\nu=3/2$, $k$ vanishes completely and $n_s=1$.
★ But if $H$ decreases slowly, $\nu$ drifts from $3/2$ and some $k$-dependence returns — $n_s-1=3-2\nu$.
★★ So $n_s\ne1$ is a direct measurement of "inflation was not exact de Sitter." Exact de Sitter would go on forever, so to end at all it must have been off. ★ If $n_s=1$ held precisely, that would be the awkward result — a worked example of Episode 15's "a miss carries more information."

Summary — the universe's tilt is $-0.105$ dB/oct

The Mukhanov–Sasaki equation is Episode 13's $1/\tau^2$ potential itself — put $x=-k\tau$ and it becomes $v_{xx}+[1-(\nu^2-\frac14)/x^2]v=0$, ★ with $k$ gone. That is what scale invariance is. Outside the horizon, $v\sim x^{1/2\pm\nu}$.

★★★ Episode 13 (the critical potential), Episode 39 (AdS) and this episode (inflation)
are three faces of the same quadratic.

Solving it confirmed $P(k)\propto k^{3-2\nu}$ to $10^{-6}$ — $n_s=1$ at $\nu=3/2$. And once $x<1$, $|\zeta|$ goes constant and freezes (1.41 → 1.005 → 1.00005) — the same place Episode 4 put the constant of integration ($\omega=0$), which is why it survives to 380,000 years.

Against observation, $V\propto\phi^2$ passes on $n_s$ but is excluded by $r=0.133$, while Starobinsky ($n_s=0.9667$, $r=0.0033$) passes both. ★ The lesson of Episode 42 again — one matching exponent is not enough.

★★★ And the core — put $n_s=0.9649$ on Episode 1's scale and it reads
$3(n_s-1)=-0.1053\pm0.0126$ dB/oct,
the same order as Episode 42's $+0.0812$ for turbulence.

★ A coincidence with no shared origin — the article says so. ★ What they share is that the departure itself carries the physics — for the universe, that inflation ended in finite time; for turbulence, that the cascade multiplies. An $n_s$ of exactly 1 would mean inflation never ended.

★ And inflation is Episodes 44 and 45's parametric drive — $v''+\omega^2(t)v=0$, $|\alpha|^2-|\beta|^2=1$. The CMB fluctuations are the dynamical-Casimir photons of an expanding universe. Only one thing differs: nobody is doing the shaking — the expansion does it, funded by the inflaton's potential energy.

Next time So far the largest thing (the universe) and the smallest (the Planck length) have been read on the same axis of order.
★ Next time we look for that axis's edge — Episode 43's exercises suggested a third test would come from "somewhere with no power law at all."
★★ Critical points, finite-size effects, and the places where a logarithm replaces a power. Looking at renormalization-group fixed points as waves shows where the axis of order finally runs out.

This document is Episode 47 of the "Thinking in Waves" series. Computations are in kenshou/uchu.js. On the numbers here: the numerical solution of the Mukhanov–Sasaki equation and the measured spectral index, the check that $\zeta$ freezes, the model-by-model $n_s$ and $r$, the dB/oct conversion, and the e-fold estimate were all computed here. The Mukhanov–Sasaki equation, the Bunch–Davies vacuum, $n_s-1=3-2\nu$, the freezing of curvature perturbations outside the horizon, the slow-roll formulas $n_s-1=-6\epsilon+2\eta$ and $r=16\epsilon$, the predictions of $V\propto\phi^p$ and Starobinsky inflation, and the observed $n_s=0.9649\pm0.0042$ with $r<0.036$ are all well-known results or literature values and none are discoveries of this article — what this article did was line them up as the same quadratic as Episodes 13 and 39, and put $n_s$ on Episode 1's dB/oct scale. §05's "both the universe and turbulence are of order 0.1 dB/oct" is a coincidence and is not a claim of any shared mechanism. The e-fold estimate in §06 assumes one reheating temperature and is very rough; the actual $N$ depends on the details of reheating. The pass/fail marks on $n_s$ in §04 use a $3\sigma$ criterion chosen here for convenience; real model selection is done by joint fits. Whether inflation actually happened is beyond this article (alternative models exist). — To print, use your browser's Print and "Save as PDF."