c·t = CONST, THAT CLICKS EPISODE 32 ／ Part IV — what a notation gains by becoming a theory
This is what Episode 4's picture looks like implemented as a field theory
Wetterich's
cosmon
The scalar field that grows the masses plays the role of dark energy itself.
It pays 10.7 bits and buys back up to 408.
When Episode 4 counted "delete everything deletable and one mass is left", it named only the source of that picture — Wetterich, the person who actually wrote "a universe without expansion" as a field theory. Now we go inside. Where Episode 4's picture was notation, the cosmon is a dynamical theory implementing that notation — so it has predictions. One question today: what does a notation gain when it becomes a theory?
01Episode 4's picture versus the cosmon
| Episode 4's picture | The cosmon | |
|---|---|---|
| What moves | one mass, \(\tilde m=a(t)m\) | one mass, \(m\propto\chi\) |
| \(a(t)\) or \(\chi(t)\) is | a function put in by hand | determined by field equations |
| Parameters | 0 | 2, in the potential \(V(\chi)\) |
| Predictions | 0 | \(w(z)\), structure formation |
Conclusion of §01
Same picture, and yet whether \(m(t)\) is chosen or determined makes them entirely different things.
── Episode 4 is notation; the cosmon is a theory.
In Wetterich's model a scalar field \(\chi\) (the cosmon) sets every particle mass, \(m\propto\chi\). And the same \(\chi\) also plays the role of dark energy. The field that grows the masses and the field that accelerates the universe are one field.
02What it pays — the price of parameters
an exponential potential \(V(\chi)=M^4e^{-\alpha\chi/M}\) has two
$$2\times\tfrac12\log_2(1701)=2\times5.37=10.7\ \text{bits}$$Episode 4's picture had zero, so 10.7 bits have been paid. What is bought?
03What it buys — the cosmological constant's tuning
The thing this episode most wants to say
Setting that value by hand means setting 408 bits.
Quintessence claims it comes out automatically from an attractor —
paying 10.7 bits to buy back up to 408.
That is why dynamical dark energy is attractive despite adding parameters. Episode 5 counted a parameter at 5.37 bits — if 408 bits get explained, two of them are cheap. (Not all of it is bought back; §07 does the honest accounting.)
04Why it does not die like VSL in Episode 28
Structurally this is the most interesting part. The cosmon is also a "constants vary" theory, and yet its fate differs from Episode 28's VSL.
the cosmon sets every mass by the same \(\chi\), so
$$\frac{m_p}{m_e},\quad \frac{m_n}{m_p},\quad \alpha\quad \text{are all exactly invariant}$$| Dimensionless ratio | Observational bound | Bits pinned | Cosmon's prediction |
|---|---|---|---|
| \(\alpha\) | \(1.4\times10^{-8}\) | 26.1 | exactly 0 |
| \(m_p/m_e\) | \(1.4\times10^{-7}\) | 22.8 | exactly 0 |
| \(m_n/m_p\) | \(1\times10^{-2}\) | 6.6 | exactly 0 |
Conclusion of §04
Same "constants vary", and life or death turns on whether the ratios are protected.
VSL moved \(\alpha\) and collided head-on with 26 bits of constraint (Episode 28).
The cosmon protects the ratios, so measurements of constants catch nothing at all.
This is not an accident but a design principle. Episode 3: "no claim without a comparison". Episode 9: "atoms shrink, and everything to compare against shrinks with them". The cosmon builds that structure into the theory — one field sets every mass, so the ratios cannot move.
05So what does judge it?
If measurements of constants cannot distinguish it, judgement moves elsewhere — to the equation of state of dark energy.
with a prior range \(w\in[-2,0]\)
$$\text{bits pinned}=-\log_2\frac{0.06}{2}=5.1\ \text{bits}$$Figure: the arena of the equation of state \(w\). A cosmological constant is the single point \(w=-1\); quintessence has width. Move the potential's slope \(\alpha\) and \(w\) drifts in and out of the observational band — this is the arena the notation gained by becoming a theory.
DESI (2024) reports hints favouring models with time-varying \(w\) — an ongoing story, which this document does not adjudicate. But being on an arena where judgement is possible at all is the decisive difference from Episode 4's picture.
06Does the singularity go away?
The most striking of Wetterich's claims is "the universe had no beginning". It concerns the same place Episode 4's table counted as \(\tilde R=0\).
Episode 6 wrote that "nothing has actually vanished on changing pictures — the geometric singularity came back the moment a dimensionless ratio was formed". The same caution applies here. Though in the cosmon picture masses too go to zero in the past, so massive particles approach masslessness — whether "the singularity is removed" depends on how that limit is handled. Contested.
07The honest ledger — what is not bought back
| Content | |
|---|---|
| Bought back | the magnitude of \(\rho_\Lambda\) (up to 408 bits, if the attractor explains it) |
| Not bought ① | the potential's scale \(M\) still has to be set |
| Not bought ② | the "why now" problem (Episode 12) remains — an attractor does not explain today's coincidence |
| Not bought ③ | the strength of \(\chi\)'s coupling to matter is a new freedom |
So only part of the 408 bits is actually recovered — the order shrinks but does not go to zero. Episode 12 counted that "neither the cosmological constant problem nor the why-now problem moves when you swap pictures". The cosmon is not a swap of pictures but a theory, so it bites on the first. Not on the second.
08The surgery — name against content
| Ep. | Theory | What the name points at | Content | Surgery already done? |
|---|---|---|---|---|
| 28 | VSL | (A) a change of units | (B) \(\alpha\) varies | no |
| 31 | CCC | both (A) and (B) | (B) Hawking points | yes |
| 32 | the cosmon | (A) "a universe without expansion" | (B) a dynamical scalar field | yes, inside the paper |
The paper's title, "A universe without expansion", speaks from the (A) side. But Wetterich himself states explicitly that the two pictures are equivalent by a Weyl transformation — so the confusion of Episode 28's VSL does not arise. The title is provocative; the surgery is done inside the paper. The same type as Episode 31's CCC.
09The reveal — what a notation gains by becoming a theory
| What is gained | Change | Meaning in the ledger |
|---|---|---|
| Parameters | 0 → 2 | pay 10.7 bits |
| Predictions | 0 → \(w(z)\) | enter a 5.1-bit arena |
| Explanation | none → the magnitude of \(\rho_\Lambda\) | buy back up to 408 bits |
| Falsifiability | none → yes | judged by departures from \(w=-1\) |
Conclusion of §09
In Episode 25's terms —
a notation only shortens \(L(\text{law})\); a theory pays \(L(\text{parameters})\) to reduce \(L(\text{residual})\).
Different arenas. And as Episode 25 showed, only the latter can be used for judgement.
① The cosmon model here summarises a body of work by Wetterich (quintessence, 1988; "A Universe without expansion", 2013, among others). This document does not fix on one specific model but treats only the structure "one scalar field sets every mass" — real models come in many variants.
② "Two parameters" assumes an exponential potential. Real models involve more complicated potentials or couplings of \(\chi\) to dark matter (growing neutrino quintessence and the like), adding parameters — 10.7 bits is a lower-side estimate.
③ "Buying back 408 bits" is a maximum. Tracker quintessence eases the fine-tuning of the magnitude of \(\rho_\Lambda\), but per §07 the potential's scale and the "why now" problem remain. The order shrinks; it does not go to zero. The 408 bits also uses the \(-\log_2\) reading of Episode 30's "bits pinned", which is strictly a different quantity from Episode 5's MDL currency (same caveat as Episode 30 ④) — they are placed side by side to convey magnitude.
④ \(w=-1.03\pm0.03\) is a representative combination (Planck + SNe + BAO). The number moves with the dataset and the model assumed (constant \(w\) or \(w_0w_a\)). DESI (2024) reports hints favouring dynamical dark energy, but the assessment of significance is ongoing and is not adjudicated here.
⑤ Whether "the singularity is removed" is contested. Keeping curvature invariants finite and having geodesic completeness are different claims — the former is demonstrable, the latter depends on the treatment of massive particles (same caveat as Episode 6 ③).
⑥ This document neither supports nor refutes the cosmon model. It only counts, in the ledger, what was gained relative to Episode 4's picture (a notation).
Exercises (solvable with this episode's formulas alone)
- What is the essential difference between Episode 4's picture and the cosmon?
Show the answer
Whether \(m(t)\) is chosen or determined. Episode 4 put \(a(t)\) in by hand, so zero predictions; the cosmon has field equations determining \(\chi(t)\), so it has predictions. The same picture is a different thing as notation and as theory. - Find the bits the cosmon pays and the maximum it can buy back.
Show the answer
Pays: 2 parameters × \(\tfrac12\log_2(1701)=5.37\) = 10.7 bits. Buys at most: \(-\log_2(1.13\times10^{-123})=\) 408 bits (the tuning of \(\rho_\Lambda/M_{\rm Pl}^4\)). Two parameters are cheap at that price. - Why does the cosmon not die like Episode 28's VSL?
Show the answer
Because every mass is set by the same \(\chi\), so mass ratios are fixed. It catches nothing from the constraints on \(\alpha\) (26.1 bits), \(m_p/m_e\) (22.8) or \(m_n/m_p\) (6.6). VSL moved \(\alpha\) and collided head-on — life or death turns on whether the ratios are protected. - So what does judge the cosmon?
Show the answer
The dark energy equation of state \(w\). Observation gives \(w=-1.03\pm0.03\), which with a prior of \([-2,0]\) pins 5.1 bits. The difference between a cosmological constant (exactly \(w=-1\)) and moving quintessence shows up here. - (Harder) In Episode 25's terms, what does a notation gain by becoming a theory?
Show the answer
A notation only shortens \(L(\text{law})\); a theory pays \(L(\text{parameters})\) to reduce \(L(\text{residual})\). As Episode 25 showed, \(L(\text{law})\) depends on the description language and cannot be used for judgement, leaving only the other two — so a notation is not on the arena of judgement and a theory is.
Summary — becoming a theory puts you on the arena
Episode 4's picture and the cosmon are the same picture of "one mass moving". The difference is one point — whether \(m(t)\) is put in by hand or determined by field equations. In Wetterich's model a scalar field \(\chi\) sets every particle mass, and the same \(\chi\) plays the role of dark energy.
In the ledger, it pays 2 parameters = 10.7 bits and buys the tuning of \(\rho_\Lambda/M_{\rm Pl}^4=1.13\times10^{-123}\), which set by hand would be 408 bits — two parameters are cheap if an attractor explains that. Not all of it is recovered, though (the potential's scale, the "why now" problem, the coupling strength remain).
Structurally the most interesting part was why it does not die like Episode 28's VSL. Because every mass is set by the same \(\chi\), the mass ratios are fixed — \(\alpha\) (26.1 bits) and \(m_p/m_e\) (22.8 bits) are exactly invariant and catch nothing from measurements of constants. VSL moved \(\alpha\) and collided head-on. Same "constants vary", and life or death turns on whether the ratios are protected — the structure of Episodes 3 and 9, built in as a design principle.
So judgement moves to the dark energy equation of state (\(w=-1.03\pm0.03\), 5.1 bits). And the reveal, in Episode 25's terms: a notation only shortens \(L(\text{law})\); a theory pays \(L(\text{parameters})\) to reduce \(L(\text{residual})\). Different arenas — and only the latter can be used for judgement.
This document is Episode 32 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. Cosmon/quintessence is due to Wetterich (1988, Nucl. Phys. B302, 668), and the growing-mass picture of cosmology to Wetterich (2013, Phys. Dark Univ. 2, 184, "A Universe without expansion"). This document does not fix on one specific model but treats only the structure "one scalar field sets every mass" — real models come in many variants. "Two parameters" assumes an exponential potential \(V=M^4e^{-\alpha\chi/M}\); more complicated potentials or couplings of \(\chi\) to dark matter (growing neutrino quintessence) add parameters, so 10.7 bits is a lower-side estimate. "Buying back 408 bits" is a maximum: tracker quintessence eases the tuning of the magnitude of \(\rho_\Lambda\) but leaves the potential's scale and the "why now" problem — the order shrinks but does not go to zero. The 408 bits uses the \(-\log_2\) reading of Episode 30's "bits pinned", strictly a different quantity from Episode 5's MDL currency (same caveat as Episode 30 ④). \(w=-1.03\pm0.03\) is a representative Planck + SNe + BAO value and moves with the dataset and the model assumed — DESI (2024) reports hints favouring dynamical dark energy, but the significance is still being assessed and is not adjudicated here. The \(w(z)\) in the figure is a schematic of exponential-tracker behaviour, not a numerical solution of a specific model. The bounds on dimensionless ratios from atomic clocks, molecular clocks and nucleosynthesis are as in Episode 30. Whether "the singularity is removed" is contested: keeping curvature invariants finite and having geodesic completeness are different claims (Episode 6 ③). This document neither supports nor refutes the cosmon model; it counts in the ledger what was gained relative to Episode 4's picture. 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).