Ten episodes of saying "these are equivalent". So what happens if you run it at face value?
This series has said, consistently: "space stretches", "light slows down" and "atoms shrink" are three ways of telling the same physics. A correct claim. But as the honest-line box of Episode 5 said, whether the expansion law \(a\propto t\) is itself correct is an entirely separate question. Today we go to receive the verdict on that. The courtroom is Big Bang nucleosynthesis. And to say the outcome first — at face value, it loses. Moreover, defend it in any of the three pictures and the verdict is identical.
Within the first few minutes of the universe, protons and neutrons bound together to make helium. The amount is well measured: a mass fraction of \(Y_p\simeq0.245\) — about a quarter of the matter in the universe is helium. Standard cosmology accounts for this beautifully.
And nucleosynthesis is extraordinarily sensitive to how the universe expands, because it comes down to a race.
The weak interaction (shuttling neutrons ⇄ protons, at rate \(\Gamma\))
vs
the expansion of the universe (pulling things apart and stopping the reactions, at rate \(H\))
While \(\Gamma\) is winning, the ratio of neutrons to protons takes its equilibrium value, set by the temperature. The instant expansion wins (\(\Gamma=H\)) the ratio freezes out, and every remaining neutron ends up in helium — the "bits freezing" of Episode 10 of the previous series.
When freeze-out happens is decided by a contest between the temperature dependences of \(\Gamma\) and \(H\).
The weak interaction rate (the same in either universe)
$$\Gamma\;\propto\;T^5$$Standard cosmology (radiation dominated, \(a\propto t^{1/2}\))
\(T\propto t^{-1/2}\) and \(H\propto1/t\), so \(H\propto T^2\). The freeze-out condition gives
$$T^5\sim T^2\;\Longrightarrow\;T_f\sim0.8\ \mathrm{MeV}$$\(c\cdot t=\text{const}\) (\(a\propto t\))
\(T\propto1/a\propto1/t\) and \(H=1/t\), so \(H\propto T\). The freeze-out condition becomes
$$T^5\sim T\;\Longrightarrow\;\text{freeze-out does not happen until far lower temperature}$$Drop the exponent of the temperature from 2 to 1 and the contest reverses. Expansion is too feeble and the weak interaction keeps winning — estimates in the literature put the freeze-out temperature roughly two orders of magnitude lower than standard.
This is the fatal blow. The equilibrium neutron-to-proton ratio, written with the mass difference \(Q=1.293\) MeV, is
$$\frac{n}{p}=e^{-Q/k_BT}$$An exponential. The lower the temperature, the more exponentially the neutrons are depleted. Delay the freeze-out and you are made to sit through all of that decline.
| Freeze-out \(T_f\) | \(n/p\) | Helium \(Y_p\) | Verdict |
|---|---|---|---|
| 0.8 MeV (standard) | 0.199 | 0.33 → 0.25* | agrees with the observed 0.245 |
| 0.5 MeV | 0.075 | 0.14 | not enough |
| 0.3 MeV | 0.013 | 0.027 | nowhere near enough |
| 0.1 MeV | \(2.4\times10^{-6}\) | \(4.8\times10^{-6}\) | no helium can be made |
| 0.05 MeV | \(5.9\times10^{-12}\) | \(1.2\times10^{-11}\) | annihilated |
* In the standard case some neutrons β-decay after freeze-out, taking \(n/p\) from 1/6 to 1/7, giving \(Y_p=2(1/7)/(1+1/7)=0.25\).
Lower the freeze-out temperature by a mere factor of 8 and helium drops by five orders of magnitude. That is what happens when \(c\cdot t=\text{const}\) is run at face value. Lewis, Barnes and Kaushik (2016) did the actual computation and concluded that the helium mass fraction comes out of order \(10^{-3}\) against the observed 0.245. The title of their paper is "Primordial nucleosynthesis in the \(R_h=ct\) cosmology: pouring cold water on the simmering Universe".
Drag the slider left. Going from 0.8 to 0.3 alone sends the curve off a cliff; at 0.1 it is pinned to the floor of the plot. Exponentials show no mercy.
Here the series' work pays off. The verdict does not change with the picture you speak in.
| Picture | What it says is happening |
|---|---|
| ① Space stretches | The temperature falls as \(1/t\). It falls too slowly for freeze-out to arrive in time. |
| ② Light slows down | The same (only the time coordinate has changed — Episode 3). |
| ③ Masses grow | The temperature is constant. The mass difference \(Q\) grows as \(\propto t\), too slowly, so by the time \(Q/k_BT\) exceeds 1 the weak interaction has used the neutrons up. |
Version ③ is especially satisfying. As Episode 3 showed, in the conformally transformed picture the universe does not cool; masses grow instead. What decides whether helium is made is, in either picture, the dimensionless ratio \(Q/k_BT\). Whether the numerator grows or the denominator falls is all that differs; the value of the ratio is the same — so the verdict is the same too.
\(Q/k_BT\) is dimensionless → it does not move with the gauge → physics
Hence no picture can secure an acquittal.
Separate from nucleosynthesis, there is a more fundamental objection. Mitra (2014) argued that the \(R_h=ct\) universe can be written in static form by a coordinate transformation, which in the flat case implies zero density — a vacuum — making it, like the Milne universe, "a vacuum in disguise".
The verdict in Episode 7 of the previous series read "rejected at face value; acquitted in an \(\alpha\)-invariant gauge but unobservable". Restated in the language of conformal transformations, it becomes: the equivalence is a consequence of relativity (\(c_B\cdot a=\text{const}\)); what fails is the expansion law substituted into it (\(a\propto t\)). And what today makes plain is that this series built the tool for not confusing those two.
The equivalence derived from relativity is, as bonus episode ④ of "Cosmology That Clicks" showed, \(c_B\cdot a=\text{const}\) (together with \(\alpha\) invariance). Constant when multiplied by \(a\) — not by \(t\). That relation holds whatever the form of \(a(t)\), and applies unchanged to a \(\Lambda\)CDM universe. The equivalence itself is therefore not the kind of claim observation can reject.
So where does \(c\cdot t=\text{const}\) come from? It is \(c_B\cdot a=\text{const}\) with \(a\propto t\) substituted in. Indeed \(c_B\cdot t=c_0\,t/a\), which is constant only when \(a\propto t\). In the actual universe (log-averaged \(a\propto t^{0.51}\)) one gets \(c_B\cdot t\propto t^{0.49}\), growing by a factor of \(10^{29.7}\) between the Planck time and today.
So two different things were travelling under one name — the equivalence (\(c_B\cdot a=\text{const}\); a consequence of relativity; always true) and the expansion law (\(a\propto t\); decided by observation). This bonus episode rejects only the latter. The way of speaking — "light slows down" — is left entirely untouched.
The estimate of \(T_f\) in section 02 is an order-of-magnitude argument. Comparing \(\Gamma\propto T^5\) with \(H\) is the standard way to estimate a freeze-out temperature, but obtaining an accurate \(Y_p\) requires solving the whole reaction network numerically (which is what Lewis et al. 2016 did). The table in section 03 is likewise a guide, computed from the equilibrium value at freeze-out, and does not include the details of subsequent neutron decay and nuclear reactions.
There are also counter-arguments on the \(R_h=ct\) side. Melia and collaborators maintain that model-independent distance measurements favour \(R_h=ct\). The "rejection" here refers to extrapolating the model at face value into the early universe (the nucleosynthesis era) and is a separate matter from the debate over fits to low-redshift observations. Do not close any of these disputes after hearing only one side.
Nucleosynthesis is extraordinarily sensitive to the expansion law. The neutron-to-proton ratio freezes out from the race between the weak interaction \(\Gamma\propto T^5\) and expansion \(H\), and that sets the helium abundance. In the standard case (\(H\propto T^2\)) freeze-out occurs at \(T_f\simeq0.8\) MeV and \(Y_p=0.25\). But with \(a\propto t\) one has \(H\propto T\), delaying freeze-out by about two orders of magnitude in temperature. Since \(n/p=e^{-Q/k_BT_f}\) is exponential, the neutrons are used up in the meantime and almost no helium is produced (Lewis et al. 2016).
And crucially, the verdict is the same however you defend it. What decides matters is the dimensionless ratio \(Q/k_BT\), which does not move with the gauge — the finale's decision procedure applies unchanged. The conformal transformation gave \(c\cdot t=\text{const}\) three ways of speaking, but adding ways of speaking does not earn a model a single extra point. Building the tool that keeps those two apart was the greatest achievement of these ten episodes.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen the slider changes the freeze-out temperature and shows the helium vanishing. "Show answer" opens the solutions.