Cosmology That ClicksEpisode 7 · putting the series' claim on trial with real data

The decay teased in Episode 6 — but with the neutron, which actually decays, not the proton. Here the model faces its verdict.

The Neutron Is an Honest Witness Proton decay could dodge judgment at 10³⁴ years away with "it's probabilistic, so of course it's not found."
But the neutron really decays in 880 seconds — and a 13.8-billion-year-old experiment (the helium abundance) delivers the verdict on the series' claim.

Tools you'll need: the equivalence conditions of Bonus 3–4, fifth-power scaling, logarithms Γ ∝ Q⁵ / c·t = constant

Episode 6 asked "does the proton decay?" But the proton's lifetime is $10^{34}$ years — $10^{24}$ times the age of the universe. No matter how long you wait in a lab, "it's probabilistic, so of course it's not found" settles it. No test possible. The neutron is different. A free neutron really decays in just about 880 seconds (15 minutes). And this decay set the amount of helium during "nucleosynthesis (BBN)," a few minutes after the universe was born. So neutron decay is an experiment that actually ran in the universe 13.8 billion years ago, and its result (the helium abundance) is still observable today. In this episode, using that real data, we deliver a numerical verdict on what the series has said all along: "move $c$ at face value and it dies; read it as an $\alpha$-invariant gauge and it lives."

01Neutron decay goes as "the difference to the fifth" — so it's an amplifier

The neutron decays into a proton, electron, and antineutrino ($n\to p+e^-+\bar\nu$). What sets the decay rate is not the neutron's total mass energy $m_nc^2\approx939$ MeV. What matters is the mass-energy difference between the neutron and (proton + electron), a mere $Q\approx0.782$ MeV. And — by a relation called Sargent's rule — the decay rate goes as the fifth power of this difference.

Sargent's rule — the decay rate is the difference Q to the fifth

$$\Gamma \propto Q^5,\qquad Q=(m_n-m_p-m_e)c^2\approx 0.782\ \text{MeV}$$ $$\Rightarrow\quad \tau_n \propto \frac{1}{Q^5},\qquad \frac{\Delta\tau_n}{\tau_n} = -5\,\frac{\Delta Q}{Q}$$

This is decisive. In Episode 6 and Bonus 4 we said repeatedly, "what matters is not the total mass but the dimensionless ratio." Same for the neutron — what matters is not the total mass $m_nc^2$ but the tiny difference $Q$ (a mere 0.08% of the total mass). And if this $Q$ shifts by 1%, the fifth power makes the lifetime move 5%. The neutron is a natural amplifier that "reflects a slight change in $Q$, amplified fivefold, into the lifetime." The star of Episode 7 is this fifth-power amplification.

The difference from proton decay — why the neutron can be a "witness" Proton decay (Episode 6) has a lifetime of $10^{34}$ years, $10^{24}$ times the age of the universe. So "it hasn't decayed yet" is explained by "its turn hasn't probabilistically come yet," and can't put the model on trial. Neutron decay takes 880 seconds and actually set the universe's helium abundance in BBN — an experiment whose result survives in observation. So the neutron can be the series' first honest witness, able to pronounce the model "guilty or not guilty."

02A 13.8-billion-year-old experiment — helium constrains the neutron lifetime

About 1–3 minutes after the universe was born, it cooled and light nuclei formed from protons and neutrons (Big Bang nucleosynthesis, BBN). At this moment, how fast the neutron decays set the amount of helium left in the universe. If neutrons decay fast, they dwindle before becoming helium, so there's less helium; if slow, there's more. So the helium abundance $Y_p$ (mass fraction, observed to be about $0.245$) is a direct record of the neutron lifetime in the early universe.

The sensitivity is known numerically. Shift the neutron lifetime by 5 seconds and the helium abundance changes by $0.001$. The observational uncertainty is about $\sigma(Y_p)\approx0.003$. From this we can back out the allowed range of the neutron lifetime in the early universe.

Try it — the range helium observations allow for the neutron lifetime

Sensitivity (measured)

$$\Delta\tau_n = 5\ \text{s}\ \Rightarrow\ \Delta Y_p = 0.0010\qquad\Longrightarrow\qquad \frac{\Delta Y_p}{\Delta\tau_n}\approx 2\times10^{-4}\,/\text{s}$$

The lifetime shift allowed by the observational range

$$\Delta\tau_n^{\text{allowed}} = \frac{\sigma(Y_p)}{\Delta Y_p/\Delta\tau_n} = \frac{0.003}{2\times10^{-4}/\text{s}} \approx 15\ \text{s}$$

In other words — the neutron lifetime in the early universe (13.8 billion years ago) must agree with today's value $\tau_n\approx880$ s to within $\pm15$ s, or a fraction $\Delta\tau_n/\tau_n\lesssim1.7\%$. The literature's "BBN-preferred neutron lifetime of $870\pm16$ s," consistent with the lab value, lives inside this window too.

03Translate the lifetime window into a Q window — the fifth power narrows it

Here we use the fifth power from Step 01. Push the lifetime window $|\Delta\tau_n/\tau_n|\lesssim1.7\%$ back to a $Q$ window — since $\Delta\tau_n/\tau_n=-5\,\Delta Q/Q$, the fivefold amplification works in reverse and narrows the window to 1/5.

The window BBN allows for Q

$$\left|\frac{\Delta Q}{Q}\right| \lesssim \frac{1.7\%}{5} \approx 0.34\%$$

So BBN requires that the early-universe $Q$ (the neutron minus proton+electron mass-energy difference, 0.782 MeV) agree with today's value to within 0.3–0.4%. The fifth-power amplification not only raises the detection sensitivity — it also makes the constraint five times stricter. In the figure below, actually move this window. Shift $Q$, the fifth power moves the lifetime, and the helium abundance enters and leaves the observation window.

Figure: shift in Q (horizontal slider) → Sargent's fifth power moves the neutron lifetime τ_n → the helium abundance Y_p enters/leaves the observation window (green band)

Shift in Q in the early universe, ΔQ/Q 0.0%

predicted helium abundance Y_p window allowed by observation (Y_p = 0.245 ± 0.003)

◇ ◇ ◇

04The verdict — integrate c·t=constant and it breaks the window by 10,000×

Now we put the series' claim $c\cdot t=\text{constant}$ through this window. The core of the model is $c(t)\cdot t=\text{constant}$, i.e. $c(t)=k/t$. Differentiating in time gives $\dot c/c=-1/t$. Integrate from the BBN era ($t_{\text{BBN}}\approx100$ s) to now ($t_0\approx4.4\times10^{17}$ s).

Try it — integrate the total change of the clockdown

Total change of c (comes out as a logarithm)

$$\int_{t_{\text{BBN}}}^{t_0}\frac{\dot c}{c}\,dt = -\int_{t_{\text{BBN}}}^{t_0}\frac{dt}{t} = -\ln\frac{t_0}{t_{\text{BBN}}}$$

Put in the numbers

$$\frac{t_0}{t_{\text{BBN}}}\approx\frac{4.4\times10^{17}}{100}\approx4\times10^{15}\quad\Rightarrow\quad \ln\frac{t_0}{t_{\text{BBN}}}\approx 36$$

So taking $c\cdot t=\text{constant}$ at face value, from the BBN era to now $c$ changed by a ratio of "36" — in orders of magnitude, by $e^{36}\sim10^{15}$-fold. Naively translated into a running of $Q$, that's $|\Delta Q/Q|\sim36=3600\%$.

The window vs. the face-value model's prediction

Window BBN allows: $|\Delta Q/Q|\lesssim0.34\%$
Face-value $c\cdot t=\text{constant}$: $|\Delta Q/Q|\sim3600\%$
→ a ratio of about 10,000×, breaking clean through the window.

If $Q$ moved by 3600%, then $\Delta\tau_n/\tau_n=-5\times36\approx-180$, and the fifth power breaks down completely. The early-universe neutron couldn't even keep $Q>0$ (the condition for it to be able to decay), and neither helium nor lithium would remain in today's amounts. The face of the universe would change fundamentally. So $c\cdot t=\text{constant}$ with $c$ moved at face value is clearly rejected by the neutron and helium. The model that could never be tried at $10^{34}$ years away in proton decay receives, in the neutron, a clear guilty verdict today.

05But this is a "predicted rejection" — the fork between two readings

Don't panic here. This rejection is a natural consequence of what Bonus 3–4 said all along, not the death of the model. What was rejected is the reading "move $c$ as face-value physics" — exactly the trap VSL fell into. It dies because you read $\Delta c/c\approx36$ as "$\alpha$ or $Q$ really moved that much."

Read correctly via Bonus 4's equivalence condition ② ($\alpha$ invariant), the story flips entirely. If you read $c\cdot t=\text{constant}$'s $\Delta c/c\approx36$ as a complete relabeling of units (a gauge) that keeps $\alpha$ invariant — then $e,\hbar,\varepsilon_0$ move in step, and the dimensionless ratio $Q/m_nc^2$ doesn't move. So even if $c$ changes $e^{36}$-fold in coordinates, it isn't transmitted to $Q$ at all, and neither the neutron lifetime nor the helium abundance changes one iota. BBN sees no anomaly.

Reading of c·t=constant	ΔQ/Q	Neutron lifetime · helium	Verdict
VSL-type (move c at face value; α runs)	~3600% (~10,000× the window)	lifetime breaks down, helium off by orders	rejected
gauge-type (α-invariant relabeling of units)	0%	doesn't change at all	not guilty (but unobservable)

The neutron has no "third road" — the fifth power makes it cruelly clear In Bonus 3, VSL dreamed of "a third road that moves $\alpha$ just a little and makes it show up in observation." Neutron decay makes it cruelly clear, through fifth-power amplification, that this third road does not exist. If $Q$ runs even a little for real, it's amplified fivefold and instantly exposed in the helium abundance. If it doesn't run, it's completely silent. There's no middle — either guilty, or not guilty and unobservable. The neutron is the witness that gives VSL's "near-miss" its final confirmation in real data.

The honest line — what "rejection" means

Don't misread this: what was rejected is not the rephrasing (gauge) $c\cdot t=\text{constant}$ itself, but only the interpretation that reads it as "in the universe, $c$ really changes by orders of magnitude at face value." As said consistently since Episode 1, $c\cdot t=\text{constant}$ is completely unscathed read as an $\alpha$-invariant gauge — but in that reading it leaves no trace in observation. The neutron merely confirmed this with the most sensitive amplifier there is: the fifth power.

Note that the sensitivity and window values used here ($\Delta\tau_n=5$ s for $\Delta Y_p=0.001$; observational range $\sigma(Y_p)\approx0.003$) are approximate and vary somewhat across the literature. Also, moving $Q$ requires moving $\alpha$ or the quark mass difference, and atomic clocks ($\dot\alpha/\alpha<10^{-19}$/yr) place an even orders-of-magnitude stricter constraint on the present-day running than BBN does.

Practice problems (solvable with just this episode's formulas)

With the neutron decay rate $\Gamma\propto Q^5$, if $Q$ increases by 2%, by what percent does the lifetime $\tau_n$ change?
Show answer
$\Delta\tau_n/\tau_n=-5\times\Delta Q/Q=-5\times2\%=-10\%$. Larger $Q$ means faster decay, so the lifetime shortens by 10%. The fifth power turns a 2% difference into a 10% lifetime change.
When helium observations allow a $\pm15$ s (1.7%) window on the neutron lifetime, what percent window is allowed for $Q$?
Show answer
$|\Delta Q/Q|=|\Delta\tau_n/\tau_n|/5=1.7\%/5\approx0.34\%$. Because of the fifth power, the $Q$ window is five times narrower than the lifetime window.
Explain in one line, using the window numbers, why $\ln(t_0/t_{\text{BBN}})\approx36$ — read at face value for $c\cdot t=\text{constant}$ — means rejection.
Show answer
The face-value $|\Delta Q/Q|\sim36$ (3600%) is about 10,000× BBN's window of $0.34\%$. If $Q$ moved that much, the helium abundance would be off from observation by orders of magnitude. But read as an $\alpha$-invariant gauge, $\Delta Q/Q=0$ and it's not guilty.

Episode 7 wrap-upAn honest witness gave the series' claim its final confirmation

The neutron decay rate goes as the fifth power of the difference $Q$ (0.782 MeV) (Sargent's rule). This fifth power makes the neutron a "natural witness that amplifies changes in $Q$." BBN, 13.8 billion years ago, requires the early-universe neutron lifetime to agree with today's to within $\pm1.7\%$, and $Q$ to within $\pm0.34\%$ — the real-data window that comes out of helium observations.

Integrate $c\cdot t=\text{constant}$ and put it through, and at face value $|\Delta Q/Q|\sim3600\%$, breaking the window by about 10,000×: rejected. But this is just as Bonus 3–4 predicted — read as an $\alpha$-invariant gauge, $\Delta Q/Q=0$ and it's not guilty, but unobservable. The neutron has no "third road that moves $\alpha$ just a little," and the fifth power made that cruelly clear. Move the absolute value ($c$) at face value and it dies; protect the ratio ($\alpha$) and it lives but leaves no trace — the backbone since Episode 1, given its final confirmation in real data by the most honest witness.

The series' structure — where Episode 7 sits Episode 6 handled "proton decay / grand unification ($10^{34}$ years, untriable)," and Bonus 3–4 set up "VSL's near-miss / the equivalence conditions (protect $\alpha$)." Episode 7 is where those two converge into a single real-data test — delivering a verdict on the bonus episodes' claim through a measurable decay, the neutron. And it shakes hands here with the finale, "Can Gravity Fit Into This Picture?": if neither the neutron nor BBN can move $\alpha$, then the place that fixes the value of $\alpha$ is nowhere in the observable universe, only beyond the Planck scale that no one can yet touch.

This is Episode 7 of "Cosmology That Clicks," a reading piece for curious high-schoolers and undergraduates. The free-neutron mean lifetime (about 880 s), the decay Q value ($Q=(m_n-m_p-m_e)c^2\approx0.782$ MeV), Sargent's rule that the decay rate goes as roughly the fifth power of Q, the BBN helium mass fraction $Y_p\approx0.245$, and its sensitivity to the neutron lifetime ($\Delta\tau_n\approx5$ s for $\Delta Y_p\approx0.001$) are established physics. The literature also evaluates a BBN-preferred neutron lifetime of order $870\pm16$ s. The window values (lifetime $\pm1.7\%$, $Q\ \pm0.34\%$) are this piece's estimates and vary with observational systematics and analysis. The $\ln(t_0/t_{\text{BBN}})\approx36$ for $c\cdot t=\text{constant}$ is a schematic estimate, illustrating the backbone of the conclusion "the face-value interpretation is rejected; the $\alpha$-invariant gauge interpretation is unobservable." Changes in $Q$ occur only through $\alpha$ or the quark mass difference, and atomic clocks place an even stricter upper bound ($\dot\alpha/\alpha<10^{-19}$/yr) on the present-day running. The figure is a schematic of the concept. — To print, use your browser's Print → Save as PDF (in the printed version, the slider and answers are static/hidden).

Print / save as PDF: ⌘+P (Ctrl+P on Windows). On screen, moving Q with the slider moves the lifetime via the fifth power, and the helium abundance enters/leaves the observation window. "Show answer" opens the solutions.

The Neutron Is an Honest Witness Proton decay could dodge judgment at 10³⁴ years away with "it's probabilistic, so of course it's not found."But the neutron really decays in 880 seconds — and a 13.8-billion-year-old experiment (the helium abundance) delivers the verdict on the series' claim.

01Neutron decay goes as "the difference to the fifth" — so it's an amplifier

02A 13.8-billion-year-old experiment — helium constrains the neutron lifetime

03Translate the lifetime window into a Q window — the fifth power narrows it

04The verdict — integrate c·t=constant and it breaks the window by 10,000×

05But this is a "predicted rejection" — the fork between two readings

Episode 7 wrap-upAn honest witness gave the series' claim its final confirmation

The Neutron Is an Honest Witness Proton decay could dodge judgment at 10³⁴ years away with "it's probabilistic, so of course it's not found."
But the neutron really decays in 880 seconds — and a 13.8-billion-year-old experiment (the helium abundance) delivers the verdict on the series' claim.