c·t = CONST, THAT CLICKS EPISODE 30 ／ Part IV — looking at the side that measures
Finding a place that amplifies beats improving precision
Measuring varying
constants for real
Atomic clocks, the Oklo natural reactor, quasar absorption lines — three different physics, one skeleton.
And we know \(\alpha\) to 26 bits over only 0.1% of the logarithmic history.
Episodes 28 and 29 handled two theories of the "constants vary" kind. Today we look at the side that actually measures — atomic clocks, the Oklo natural reactor, quasar absorption lines. Three completely different pieces of physics, and yet the same single line of skeleton. And at the end, one number: we know \(\alpha\) to 26 bits over only 0.1% of the universe's logarithmic history.
01Three ways of measuring
| Method | \(\alpha\) of when | Bound on \(|\Delta\alpha/\alpha|\) | Bits pinned |
|---|---|---|---|
| Atomic clocks (Yb⁺ E3 vs Sr) | today (a rate) | \(1.4\times10^{-8}\) | 26.1 bit |
| Oklo natural reactor | 1.8 Gyr ago | \(1.1\times10^{-8}\) | 26.4 bit |
| Quasar absorption lines | \(z\sim2\) (10.5 Gyr ago) | \(1.0\times10^{-5}\) | 16.6 bit |
| CMB | \(z=1100\) | \(4.0\times10^{-3}\) | 8.0 bit |
| Nucleosynthesis | \(t=1\) s | \(1.0\times10^{-2}\) | 6.6 bit |
The laboratory value of \(\alpha\) itself is far more precise — \(\alpha^{-1}=137.035999177(21)\), a relative precision of \(1.6\times10^{-10}\), 32.5 bits. But that is "the value today", not evidence that it has not moved.
02The heart — all three are built from the same line
\(K\) is the amplification — the factor translating a tiny change in \(\alpha\) into a large change in the observable
| Method | Amplification \(K\) | Precision | What does the amplifying |
|---|---|---|---|
| Atomic clocks | 7 | \(10^{-18}\) | the difference in \(\alpha\)-sensitivity between two transitions |
| Oklo | \(10^{7}\) | \(2\times10^{-2}\) | 97.3 meV as a difference of MeV-scale quantities |
| Quasars | 0.3 | \(3\times10^{-6}\) | statistics from bundling many lines |
The thing this episode most wants to say
Oklo is strong not because the measurement is precise.
Its precision is 2% — \(10^{16}\) times coarser than an atomic clock — and it yields the same bound because the amplification is \(10^7\).
Finding a place that amplifies beats improving precision.
03Where does Oklo's amplification come from?
1.8 billion years ago at Oklo in Gabon, a natural uranium deposit spontaneously sustained a fission chain reaction. The ashes remain. The key is a neutron capture resonance in \(^{149}\mathrm{Sm}\).
the resonance energy \(E_r=97.3\) meV appears as a difference of MeV-scale quantities
$$E_r\ \sim\ (\text{nuclear binding energy})-(\text{Coulomb energy})\ \sim\ 10^6\ \mathrm{eV}-10^6\ \mathrm{eV}$$moving \(\alpha\) shifts only the Coulomb term, so
$$\frac{\Delta E_r}{E_r}\ \sim\ \frac{10^6\ \mathrm{eV}}{0.0973\ \mathrm{eV}}\times\frac{\Delta\alpha}{\alpha}\ \simeq\ 10^{7}\times\frac{\Delta\alpha}{\alpha}$$This is exactly the inverse of the situation when Episode 19 measured coincidences. There, "large numbers cancelling to something small" was the surprise. Here that smallness is being used as the amplifier in a measuring instrument. A cancellation of orders can be a mystery or a tool.
04Where on the logarithmic axis does the data sit?
Episode 2 counted the whole history of the universe as 140.24 logarithmic steps. Place the five measurements on that ruler.
| Method | Step | Bits pinned |
|---|---|---|
| Nucleosynthesis | 99.63 | 6.6 |
| CMB | 129.74 | 8.0 |
| Quasar absorption lines | 138.81 | 16.6 |
| Oklo | 140.10 | 26.4 |
| Atomic clocks | 140.24 | 26.1 |
Conclusion of §04
Data exist from step 99.63 to 140.24 — 29% of the whole history.
For the remaining 71% there is not one measurement of \(\alpha\).
And precision above 20 bits covers steps 140.10 to 140.24 — 0.1% of the history.
Figure: logarithmic step across (the whole history is 140.24), bits pinned about \(\alpha\) up. Grey is where no data exist. Move the slider through epochs to read how much we know — all of it is piled at the right edge.
05The gaps
| Gap | Steps | Length | Why nothing can be measured |
|---|---|---|---|
| Nucleosynthesis → CMB | 99.6 → 129.7 | 30.1 steps | plasma; light does not get through |
| CMB → quasars | 129.7 → 138.8 | 9.1 steps | the dark ages; nothing luminous yet |
The data live on three islands — nucleosynthesis, the CMB, and \(z\lesssim4\) onwards. Between them there is only theoretical interpolation. Episode 28 noted that a phase-transition VSL escapes exclusion by hiding before nucleosynthesis; in fact there are another 30 steps of hiding place between nucleosynthesis and the CMB.
06The reveal — measurement lives where this series' tool cannot reach
| Quantity | What it is | Weight |
|---|---|---|
| Amplification \(K\) | a ratio of ratios | \(0\) |
| \(\Delta\alpha/\alpha\) | the fractional change of a dimensionless quantity | \(0\) |
| Bits pinned | the log of a ratio | \(0\) |
Conclusion of §06
Measurement of constants sits entirely in the zero column of Episode 16's map of weights.
── A place this series' tool cannot touch at all. Which is exactly why it can be the referee.
Episode 13 measured the tool's limit: "a conformal transformation touches only size". Today is the flip side — the referee sits entirely outside the tool. That is how VSL could be judged in Episode 28, MOND in Episode 29, and \(c\cdot t=\)const itself in Episode 3. Judgement is possible because the yardstick does not move.
07An aside — observation beats anthropics
| What breaks if \(\alpha\) moves | Required \(|\Delta\alpha/\alpha|\) |
|---|---|
| the 7.65 MeV carbon resonance (triple-alpha) disappears | \(4\times10^{-2}\) |
| the \(^4\)He yield departs from observation | \(1\times10^{-2}\) |
| the timing of recombination shifts | \(4\times10^{-3}\) |
| the actual observational bound | \(1\times10^{-8}\) |
The argument that "life could not exist unless \(\alpha\) had this value" (anthropics) demands at most 4%. The actual observational bound is \(10^{-8}\) — six orders tighter. That \(\alpha\) has not moved is confirmed far beyond anything anthropics can explain.
① The amplification \(K\) is an order-of-magnitude marker. Oklo's \(10^7\) is estimated from "the resonance energy is a difference of MeV-scale quantities"; the exact value depends on nuclear structure calculations and the literature spans \(10^7\)–\(10^8\). The atomic clocks' \(\Delta K\simeq7\) depends on which transitions are compared.
② The Oklo analysis carries nuclear-physics assumptions. The reactor temperature, the neutron spectrum, and the treatment of simultaneous variation in constants other than \(\alpha\) (such as \(m_q/\Lambda_{\rm QCD}\)) move the bound by factors of a few — \(10^{-8}\) is a representative value.
③ Quasar absorption lines remain disputed (same caveat as Episode 28 ②). Webb and collaborators claim a significant variation, with Keck and VLT disagreeing in sign. The \(10^{-5}\) here is a conservative bound, not a single measurement.
④ "Bits pinned" is this document's quantity, defined as \(-\log_2|\Delta\alpha/\alpha|\). It reads as "how many leading binary digits of \(\alpha\) have not moved", and is a different quantity from Episode 19's surprise in bits (which is relative to a prior range). Do not conflate them.
⑤ §04's "29% of the whole history" is a fraction measured in logarithmic steps. Measured in ordinary time, data exist for 99.99999...% of it — only in a logarithmic measure does "71% blank" appear. This is not a matter of which is correct but of what one chooses to treat as evenly spaced (as in Episodes 2 and 20).
⑥ §07's \(4\times10^{-2}\) is a representative estimate for the triple-alpha resonance. The tolerable tuning depends on the method of calculation, with the literature spanning 0.5%–4%. The conclusion — observation is orders tighter — is unaffected.
Exercises (solvable with this episode's formulas alone)
- Write the form common to all three measurements.
Show the answer
(change in the observable) \(=K\times\Delta\alpha/\alpha\), hence \(|\Delta\alpha/\alpha|<\) (precision) \(/K\). \(K\) is the amplification, and the larger it is, the stronger the bound from the same precision. - Why is Oklo's amplification \(10^7\)?
Show the answer
Because the \(^{149}\mathrm{Sm}\) resonance at 97.3 meV appears as a difference of MeV-scale quantities. Moving \(\alpha\) shifts only the Coulomb term, so \(\Delta E_r/E_r\sim(10^6\,\mathrm{eV}/0.0973\,\mathrm{eV})\times\Delta\alpha/\alpha\simeq10^7\times\Delta\alpha/\alpha\). A cancellation of orders becomes the amplifier of an instrument. - Oklo's precision is a coarse 2%. Why does it still match an atomic clock?
Show the answer
Because the amplification is \(10^7\): \(2\times10^{-2}/10^7=2\times10^{-9}\). An example of finding a place that amplifies beating improving precision. - Over what fraction of the universe's logarithmic history does \(\alpha\) data exist?
Show the answer
From nucleosynthesis (step 99.63) to today (140.24), \((140.24-99.63)/140.24=\) 29%. For the remaining 71% there is no data at all, and precision above 20 bits covers only 0.1%. - (Harder) Why can measurements of constants serve as the referee for this series' judgements?
Show the answer
Because the amplification, \(\Delta\alpha/\alpha\) and the bits pinned are all dimensionless — weight 0, sitting in the zero column of Episode 16's map, where this series' tool (the conformal transformation) cannot reach. A yardstick that cannot be moved is what allows VSL (Episode 28), MOND (Episode 29) and \(c\cdot t=\)const itself (Episode 3) to be judged.
Summary — whoever finds the amplifier wins
Three ways of measuring — atomic clocks (today, 26.1 bits), the Oklo natural reactor (1.8 Gyr ago, 26.4 bits), quasar absorption lines (10.5 Gyr ago, 16.6 bits). Three different pieces of physics, one skeleton: (change in the observable) \(=K\times\Delta\alpha/\alpha\).
The heart is the amplification \(K\). Oklo's precision is 2%, \(10^{16}\) times coarser than an atomic clock, and it gives the same bound — because the amplification is \(10^7\). The \(^{149}\mathrm{Sm}\) resonance at 97.3 meV appears as a difference of MeV-scale quantities, so a tiny change in \(\alpha\) shows up by orders. A cancellation of orders can be a mystery or a tool — what Episode 19 measured as "surprise" is here an amplifier.
Placed on the logarithmic axis, the landscape changes. Data exist over 29% of the 140.24 steps, and precision above 20 bits over 0.1%. They sit on three islands, with a 30-step gap between nucleosynthesis and the CMB. We have confirmed \(\alpha\)'s constancy through a much narrower window than one imagines.
And the reveal — amplification, \(\Delta\alpha/\alpha\) and bits pinned are all weight 0. Measurement of constants sits entirely where this series' tool cannot reach. The flip side of Episode 13's "a conformal transformation touches only size" — judgement is possible because the referee does not move. As an aside, anthropics demands 4% while observation delivers \(10^{-8}\), six orders tighter.
This document is Episode 30 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. The atomic clock bound \(|\dot\alpha/\alpha|<1.0(1.1)\times10^{-18}\)/yr is Lange et al. (2021, PRL 126, 011102); the Oklo constraint follows a line of analyses since Shlyakhter (1976); the many-multiplet method for quasar absorption lines is due to Webb, Murphy, Flambaum and collaborators. CODATA 2022's \(\alpha^{-1}=137.035999177(21)\) is standard. The amplification \(K\) is an order-of-magnitude marker: Oklo's \(10^7\) is estimated from the \(^{149}\mathrm{Sm}\) 97.3 meV resonance being a difference of MeV-scale quantities, and the exact value depends on nuclear structure calculations, spanning \(10^7\)–\(10^8\) in the literature. The Oklo analysis assumes a reactor temperature, a neutron spectrum, and a treatment of simultaneous variation in constants other than \(\alpha\), any of which moves the bound by factors of a few. Quasar absorption lines remain disputed: Webb and collaborators claim a significant variation, with Keck and VLT disagreeing in sign — the \(10^{-5}\) used here is a conservative bound, not a single measurement. "Bits pinned", defined as \(-\log_2|\Delta\alpha/\alpha|\), is this document's quantity and is different from Episode 19's surprise in bits (which is relative to a prior range). §04's "29% of the whole history" is measured in logarithmic steps; in ordinary time it would be essentially 100% — this is a choice about what to treat as evenly spaced (Episodes 2 and 20). §07's \(4\times10^{-2}\) is a representative estimate for the triple-alpha resonance, with the literature spanning 0.5%–4%; the conclusion is unaffected. The 140.24 logarithmic steps are \(\ln(t_0/t_P)\) (Episode 2). Linear expansion (\(c\cdot t=\)const, \(R_h=ct\)) is a minority model under examination. The academic standard remains the \(\Lambda\)CDM model including inflation. ── To make a PDF, use your browser's Print dialogue (sliders freeze and answers are hidden in the print version).