c·t = CONST, THAT CLICKS EPISODE 29 ／ Part IV — the third patient
"Dark matter or MOND?" was never one question
MOND — the comparison
hidden inside an acceleration
\(a_0=1.2\times10^{-10}\ \mathrm{m/s^2}\) is dimensionful — small compared with what?
Look, and \(cH_0\) is sitting right next door.
The third patient of Part IV is MOND. "Below an acceleration of \(a_0=1.2\times10^{-10}\ \mathrm{m/s^2}\), Newton's law changes" — and that \(a_0\) is dimensionful, so Episode 3's surgery applies directly. Small compared with what? Look, and \(cH_0\) is sitting right next door. Then put what survives the surgery on Episode 5's scales, and you see that "dark matter or MOND?" was never one question.
01\(a_0\) is dimensionful — so a comparison is needed
acceleration has dimensions \(L/T^2\), and lengths and times both have weight \(+1\), so
$$w(a)=(+1)-2(+1)=-1\qquad(\text{dimensionful} = \text{bookkeeping})$$The mechanical count of Episode 13. \(a_0\) sits on the bookkeeping side, so "the acceleration is smaller than \(a_0\)" is — not a sentence until a comparison is named. MOND does supply one, of course: \(a_0\) itself. The question is where that \(a_0\) came from.
02Look for a comparison and \(cH_0\) is right next door
| Candidate | Value [m/s²] | Ratio to \(a_0\) |
|---|---|---|
| \(cH_0\) | \(6.548\times10^{-10}\) | 0.183 |
| \(cH_0/2\pi\) | \(1.042\times10^{-10}\) | 1.15 |
| \(c^2\sqrt{\Lambda/3}=cH_0\sqrt{\Omega_\Lambda}\) | \(5.420\times10^{-10}\) | 0.221 |
| \(cH_0/6\) | \(1.091\times10^{-10}\) | 1.10 |
Conclusion of §02
All \(O(1)\) — \(a_0\) sits at the same order as "the acceleration of the cosmic horizon".
Milgrom pointed this out from the very beginning, in 1983.
It is unnervingly suggestive. A constant introduced to fit galaxy rotation curves agrees with an acceleration built out of the age of the universe — astronomy's smallest scale touching cosmology's largest.
03The coincidence does not move when the picture changes
the same weight, so the ratio
$$\frac{a_0}{cH_0}\ \text{has weight }0\ \text{── conformally invariant}$$The zero column of Episode 16's map of weights. This series' tool cannot move this coincidence by a millimetre — which is exactly why it is worth sorting seriously.
04Measuring the surprise in bits
the range of acceleration scales in nature
$$\text{Planck acceleration}\ \frac{c}{t_P}=5.56\times10^{51}\ \mathrm{m/s^2}\ \longrightarrow\ cH_0=6.55\times10^{-10}\ \mathrm{m/s^2}$$ $$\text{range}=60.9\ \text{orders}$$surprise of "agreeing within a factor 10"
$$-\log_2\frac{1.0}{60.9}=5.9\ \text{bits}\qquad(\text{within a factor 3: }6.9\ \text{bits})$$| Coincidence | Surprise | Class (Episode 19) |
|---|---|---|
| \(\rho_\Lambda^{1/4}\) and \(m_\nu\) | 4.7 bit | coincidence |
| \(a_0\simeq cH_0/2\pi\) | 5.9 bit | the coincidence band |
| one bit ↔ 1.96 fm | 7.4 bit | coincidence |
| Koide's relation | 15.7 bit | empirical formula |
Six coin flips — essentially the same stratum as Extra 5's \(\rho_\Lambda^{1/4}\) and \(m_\nu\) (4.7 bits). The impression of being "unnervingly suggestive" and the actual size of the surprise are very different things. That said — with an explanation, the class moves to "physics" (as inflation's \(N\) did in Episode 27). Explanations in which the horizon supplies a boundary condition for dynamics have been proposed, and if one is right, these 5.9 bits are physics.
05The surgery — what MOND actually claims
Exactly the shape of Episodes 27 and 28. (B) is the substance, and MOND's (B) is very strong — in models that posit dark matter, the baryon and halo amounts are independent, so no such prediction is possible.
06The heart — putting it on Episode 5's scales
Measure (B) by description length, on the galaxy rotation curve dataset (SPARC: 175 galaxies, 2693 points).
| Model | Parameters | Description length | Breakdown |
|---|---|---|---|
| MOND | 4 | 22.8 bit | \(a_0\) (shared by all galaxies) + the shape of the interpolating function |
| Halo model | 350 | 1994 bit | two per galaxy (e.g. NFW's \(M_{200}\) and concentration) |
The thing this episode most wants to say
On the galaxy rotation curve dataset alone, MOND wins by 1971 bits.
── The same scales and the same units by which \(c\cdot t=\)const lost 148 bits in Episode 5.
(Mass-to-light ratios \(M/L\) are carried per galaxy by both models, so they cancel and do not enter the ledger.)
07And the verdict reverses when the dataset changes
| Dataset | Winner | Why |
|---|---|---|
| Galaxy rotation curves | MOND | 4 parameters against 350 — 1971 bits |
| Galaxy clusters | \(\Lambda\)CDM | a factor ~2 of undetected mass remains even in MOND |
| The Bullet Cluster | \(\Lambda\)CDM | the mass distribution and the gas are spatially separated |
| CMB acoustic peaks | \(\Lambda\)CDM | 1701 points from six parameters, \(\chi^2/\mathrm{dof}\approx1\) |
| Growth of large-scale structure | \(\Lambda\)CDM | results differ by relativistic completion |
Figure: the description-length difference by dataset. Rightwards favours \(\Lambda\)CDM, leftwards favours MOND. Change how heavily the rotation curves are weighted and the overall verdict moves — which is what "not one question" means.
Conclusion of §07
"Dark matter or MOND?" is not one question.
Measured on Episode 5's scales, the answer differs by dataset.
── And neither side explains the datasets on which the other wins.
08A testable direction — does \(a_0\) vary in time?
The coincidence of §02 hides a testable fork. If \(a_0=cH/2\pi\) is a dynamical relation, then \(a_0\) must have fallen along with \(H\).
| \(z\) | 0 | 0.5 | 1.0 | 2.0 |
|---|---|---|---|---|
| \(H(z)/H_0\) | 1.00 | 1.32 | 1.79 | 3.03 |
| \(a_0(z)/a_0\) (if dynamical) | 1.00 | 1.32 | 1.79 | 3.03 |
A factor of 3 at \(z=2\) — in principle decidable with high-redshift rotation curves. If instead \(a_0\) is simply a constant, there is no such prediction — the same "coincidence" turns out to be two different things inside. Observations are ongoing and this document does not adjudicate.
① MOND's "4 parameters" counts the freedom of the interpolating function \(\mu(x)\) as three. \(\mu(x)\) is a function, not strictly one parameter — in practice several forms are used ("simple", "standard", "RAR"), and choosing among them is itself freedom. Three is a generous count, and even at ten the conclusion of §06 (a difference above 1900 bits) is unchanged.
② "Two per galaxy" for the halo model is also an estimate. \(\Lambda\)CDM has priors such as the mass–concentration relation, which reduce the effective parameter count; a rigorous comparison needs Bayesian evidence. Read 1971 bits as an upper-side estimate — the order does not change.
③ §07's table summarises where each field stands. Relativistic MOND (TeVeS, and recent formulations such as Skordis & Złośnik 2021) can in some cases reproduce the CMB, at the cost of extra fields and parameters — "\(\Lambda\)CDM wins on the CMB" depends on the formulation. The residual-mass problem in clusters is widely acknowledged on the MOND side too.
④ §04's surprise depends on the prior range (same caveat as Episode 19 ①). The 60.9 orders from the Planck acceleration down to \(cH_0\) were used as the prior; restricted to "accelerations relevant in galaxies" the range narrows and the surprise shrinks. Usable for ranking, not as an absolute number.
⑤ This document neither supports nor refutes MOND. It names the comparison hidden inside \(a_0\), measures the surprise of the coincidence, and separates the description-length ledger by dataset. This notation (\(c\cdot t=\)const) says nothing about MOND — as §03 shows, every relevant quantity is conformally invariant (Episodes 13 and 16).
Exercises (solvable with this episode's formulas alone)
- Find the weight of \(a_0\) and explain why Episode 3's surgery applies.
Show the answer
Acceleration has dimensions \(L/T^2\) and both lengths and times have weight \(+1\), so \(w=(+1)-2(+1)=-1\). Dimensionful means bookkeeping, so "the acceleration is small" is not a sentence until a comparison is named. - Find the ratio of \(a_0\) to \(cH_0\).
Show the answer
\(cH_0=2.998\times10^8\times2.184\times10^{-18}=6.548\times10^{-10}\ \mathrm{m/s^2}\), so \(a_0/cH_0=0.183\), i.e. \(a_0\simeq cH_0/5.5\simeq cH_0/2\pi\). - Show that the coincidence is untouched by a conformal transformation.
Show the answer
\(w(a_0)=-1\) and \(w(cH_0)=w(c/t)=0-(+1)=-1\). Equal weights, so the ratio has weight 0 — conformally invariant, sitting in the zero column of Episode 16's map. - Find the description-length difference on the rotation curve dataset.
Show the answer
A parameter costs \(\tfrac12\log_2(2693)=5.70\) bits. MOND: 4 parameters, 22.8 bits. Halo model: \(2\times175=350\) parameters, 1994 bits. The difference is 1971 bits in MOND's favour (on this dataset alone). - (Harder) Is there a single answer to "dark matter or MOND?"
Show the answer
No. Measured on Episode 5's scales, the answer differs by dataset — MOND wins by 1971 bits on galaxy rotation curves, \(\Lambda\)CDM wins on clusters, the Bullet Cluster, the CMB and structure growth. And neither side explains the datasets on which the other wins. The value of measuring by description length is not settling the contest but making visible that the question is not one question.
Summary — the question was not one question
\(a_0\) is dimensionful (weight \(-1\)), so "the acceleration is small" is not a sentence until a comparison is named. Look, and \(cH_0\) is right next door — \(a_0/cH_0=0.18\), or a ratio of 1.15 against \(cH_0/2\pi\). A constant introduced to fit galaxy rotation curves sits at the same order as an acceleration built from the age of the universe. And since \(a_0\) and \(cH_0\) share a weight, the coincidence cannot be moved by a conformal transformation.
Measured by Episode 19's procedure, the surprise is 5.9 bits — six coin flips, essentially the stratum of \(\rho_\Lambda^{1/4}\) and \(m_\nu\) (4.7 bits). The impression of being "unnervingly suggestive" and the actual size of the surprise are very different.
After the surgery, MOND's substance is (B): "the dynamics depends only on the dimensionless ratio \(g/a_0\)". That carries a strong prediction — the rotation curve follows from the baryons alone. On Episode 5's scales, SPARC (175 galaxies, 2693 points) gives 4 parameters for MOND against 350 for a halo model — a difference of 1971 bits.
And the verdict reverses with the dataset — clusters, the Bullet Cluster, the CMB acoustic peaks and structure growth all go to \(\Lambda\)CDM. "Dark matter or MOND?" was never one question. Finally, the coincidence of §02 hides a testable fork — if a coincidence, \(a_0\) is constant; if physics, \(a_0\propto H\), a factor of three by \(z=2\). Episode 19's sorting procedure has turned into an experimental plan.
This document is Episode 29 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. MOND is due to Milgrom (1983, ApJ 270, 365). That the acceleration scale \(a_0\simeq1.2\times10^{-10}\ \mathrm{m/s^2}\) sits at the same order as \(cH_0\) was noted by Milgrom from the outset; the radial acceleration relation is McGaugh, Lelli & Schombert (2016, PRL 117, 201101), and the SPARC sample (175 galaxies, 2693 points) is Lelli, McGaugh & Schombert (2016, AJ 152, 157). The values \(a_0/cH_0=0.183\), \(a_0/(cH_0/2\pi)=1.15\), the surprise of 5.9 bits, the description-length difference of 1971 bits, and the factor 3.0 at \(z=2\) if \(a_0\propto H\) are computed here (kenshou/calc33.py). MOND's "4 parameters" counts the freedom of the interpolating function \(\mu(x)\) as three; \(\mu(x)\) is properly a function, not one parameter — counting it as ten leaves §06's conclusion (a difference above 1900 bits) unchanged. "Two per galaxy" for the halo model is likewise an estimate: priors such as the mass–concentration relation reduce the effective count, so 1971 bits is an upper-side estimate (a rigorous comparison needs Bayesian evidence). §07's table summarises where each field stands; relativistic MOND (TeVeS, Skordis & Złośnik 2021 and others) can in some formulations reproduce the CMB at the cost of extra fields and parameters — "\(\Lambda\)CDM wins on the CMB" is formulation dependent. The residual-mass problem in clusters is widely acknowledged on the MOND side as well. §04's surprise depends on the prior range (Episode 19 ①). This document neither supports nor refutes MOND: it names the comparison hidden inside \(a_0\), measures the surprise, and separates the description-length ledger by dataset. This notation (\(c\cdot t=\)const) says nothing about MOND, since every relevant quantity is conformally invariant (Episodes 13 and 16). 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).