\(c\cdot t=\text{const}\) came out of the idea that the universe is a computer with finite resources. Was that idea right?
The auxiliary line \(c\cdot t=\text{const}\) originally came from an information-theoretic intuition: the universe is a computer with finite resources, so the bigger it gets, the slower the computation per unit place must run. Let us now turn the decision procedure built in Episode 10 on that motivation itself. The conclusion, up front — the motivation is remarkably accurate and the implementation misses its target. Exactly the shape of the diagnosis passed on VSL in bonus episode 3 of the previous series. And chasing the miss leads to a place where gravity, quantum theory, information and time meet at a single point.
Seen as a computer, the most basic resource of the universe is "how far can I get a signal to". The quantity for that is the comoving Hubble radius.
It is "the range you can reach causally right now", measured in comoving coordinates. Look for the expansion law that holds it fixed and the answer is unique.
For \(a\propto t^p\), \(r_H\propto t^{1-p}\). It comes to a dead stop only at \(p=1\).
| Expansion law | \(r_H\) | What happens to the computer |
|---|---|---|
| \(p<1\) (radiation, matter) | grows | New comoving regions come into view — memory appears that was never in contact (the horizon problem) |
| \(p=1\) (\(c\cdot t=\)const) | constant | Nothing enters, nothing leaves. The address space does not change |
| \(p>1\) (Λ, inflation) | shrinks | Regions already in contact become unreachable — information is lost |
So \(c\cdot t=\text{const}\) is the unique expansion law for which a finite-resource computer's address space does not move. The motivation was accurate. It is the fourth face of Episode 5's "triple boundary at \(w=-1/3\)".
Memory (bits on the horizon)
$$N=\frac{\pi R_H^2}{\ell_P^2\ln 2}=2.96\times10^{122}\ \text{bit}$$Total operations (integrating the Margolus–Levitin bound \(2E/\pi\hbar\))
$$\Omega=2.1\times10^{121}\ \text{ops}\qquad(\text{agreeing with Lloyd's }10^{120}\text{, 2002})$$Clock (times light has crossed the horizon)
$$\ln\frac{t_0}{t_P}=\mathbf{140}\ \text{times}$$Against \(10^{122}\) bits of memory, the clock has ticked 140 times. A machine wildly rich in memory and poor in cycles.
And this computer runs exactly at the thermodynamic limit.
The ratio is 1.0000. This is the identity \(E=T_HS\) (it holds in any FLRW), but it reads as: the universe runs the Landauer limit with zero margin. Episode 10 of the previous series — "\(k_BT\ln2\) to erase one bit" — balances on a cosmic scale.
If the motivation is accurate, it should turn into a model. Take Episode 3's "masses grow" not as a gauge statement but as physics.
The hypothesis (this alone)
$$\frac{m}{M_{\rm Pl}}\propto a$$Then
$$\rho_m=n\,m\propto a^{-3}\cdot a=a^{-2}$$Which is exactly the \(\rho\propto a^{-2}\) that \(w_{\rm tot}=-1/3\) demands. "Masses grow in proportion to \(a\)" alone gives \(w=-1/3\) with no tuning. And the problem from bonus ① — a negative dark-energy density once matter is included — disappears at the same time.
This is coupled quintessence, and the parameters are not free.
Integrate numerically and this solution turns out to be an attractor — perturb \(\rho_m\) or \(\dot\sigma\) by \(\pm50\%\) and it comes back to \(Ht\to1\), \(\Omega_m\to0.30\). For the first time, "why is \(w\) exactly \(-1/3\)?" has an answer: because the solution attracts. Up to here it is good news for the model.
There is exactly one thing this picture cannot save: radiation.
The strength with which the dilaton exchanges information with matter is proportional to the trace of the stress–energy tensor, \(T^\mu{}_\mu\) (Episode 5). But —
The dilaton can rescue mass, but it cannot rescue light.
\(\rho_r\propto a^{-4}\) grows faster going back than \(\rho_{\rm tot}\propto a^{-2}\), so radiation must eventually take over the total. The moment is
$$\sqrt{\Omega_r}=9.6\times10^{-3}\qquad\Longrightarrow\qquad \text{radiation dominates for }z>103$$Recombination (\(z=1100\)) lies entirely inside that. \(c\cdot t=\text{const}\) cannot hold before recombination. That leaves two branches.
Only at knob position 0 is the line perfectly horizontal — the address space frozen. Add even a trace of radiation and the line bends on the left, squeezing the flat part into the right-hand edge. At the real value:
| \(c\cdot t=\)const (no radiation) | with radiation (branch B) | ΛCDM | |
|---|---|---|---|
| Growth of address space (volume) | 1 | \(1.2\times10^{89}\) | \(\sim10^{88}\) |
| Causally disconnected patches | 1 | \(1.2\times10^{4}\) | \(9.6\times10^{3}\) |
| Range where \(a\propto t\) holds | all of it | 6.4% of logarithmic history | ── |
The horizon problem returns, at essentially ΛCDM's severity. The universe as a computer cannot manage its own radiation — and radiation is exactly what dominates the early universe. A design that freezes the address space fails to operate in the era where it is most needed.
Let us measure the universe-as-computer once more.
| Entropy \(S/k_B\) | Fraction of capacity | |
|---|---|---|
| Horizon capacity (maximum) | \(2.05\times10^{122}\) | 1 |
| Total entropy today | \(3.1\times10^{104}\) | \(1.5\times10^{-18}\) |
| CMB photons (the early state) | \(2.0\times10^{88}\) | \(1.0\times10^{-34}\) |
Today's total entropy is dominated by supermassive black holes (Egan & Lineweaver 2010). Even so, only \(10^{-18}\) of the memory is filled.
Turn that around — the universe is still 99.9999999999999998% conformally flat. Which is why FLRW works, and why this series' tool works.
This is the deepest point of the episode. In Episode 7 we counted "10 components of a four-dimensional metric = 9 light cones + 1 ruler marking". That "9" has a name — the Weyl tensor \(C_{\mu\nu\alpha\beta}\), a quantity that a conformal transformation does not move at all.
| The conformal-factor side (1 component) | The Weyl-tensor side (9 components) |
|---|---|
| moves under a conformal transformation | does not move (\(C\) is conformally invariant) |
| gauge, bookkeeping (Ep.4: net zero) | physics |
| ghost = wrong-sign kinetic term (Ep.9) | the propagating gravitational-wave degrees of freedom |
| carries no gravitational entropy | is gravitational entropy (Penrose) |
| carries \(a(t)\) | carries structure, clumping, black holes |
| initial condition is free | \(C=0\) initially (Weyl curvature hypothesis) |
| carries no arrow of time | ★ the arrow of time lives only here |
The previous series' motto — units are bookkeeping, dimensionless is physics — is realised as the internal structure of the gravitational field itself. And the arrow of time lives only on the physics side.
At last we can say why no prediction came out.
\(c\cdot t=\text{const}\) is a condition on \(a(t)\) = a condition on the conformal factor = a condition on the bookkeeping.
Constrain the bookkeeping and no prediction follows.
To have predictions you must constrain the Weyl side — causal structure — light sheets.
And a "finite resource" principle of exactly that shape already exists.
It is fully conformally invariant — light sheets (null surfaces) do not move under a conformal transformation (Episode 7), and \(A/\ell_P^2\) is dimensionless (even in the Weyl frame, \(A\to A/a^2\) and \(\ell_P^2\to\ell_P^2/a^2\) leave it alone). A gauge-independent finite-resource condition — and therefore one shaped so that it can predict.
The motivation was right. Writing the universe as a computer with finite resources is a legitimate idea. It is just that — what should have been constrained was not \(a(t)\), but the information on light sheets.
The numbers for branch B in section 04 (patch count \(1.2\times10^4\), acoustic-peak position) are estimates from the simplified hybrid model \(H^2=H_0^2[\Omega_ra^{-4}+(1-\Omega_r)a^{-2}]\), not a Boltzmann-code calculation. Read them as order-of-magnitude. While writing this article the author estimated the acoustic peak with radiation left out and got the order of magnitude wrong — the very theme of this episode, that the component you drop is the one that matters most, tripped up the writer.
In the table of section 06, "ghost = gravitational entropy upside down" is continuous with the Gross–Perry–Yaffe negative mode of Euclidean Schwarzschild (the negative specific heat of black holes), but recent work shows that mixing between the transverse-traceless and trace modes makes a naive identification delicate. No claim beyond "they share the same sign" is made here. The definition of gravitational entropy itself (how to count \(C\)) is also not settled.
Seen information-theoretically, \(c\cdot t=\text{const}\) has a clear identity — the unique expansion law for which the comoving Hubble radius is constant. The computer's address space neither grows nor shrinks. It is exactly the watershed between the horizon problem (memory appears that was never contacted) and information loss (contacted memory becomes unreachable). Today's spec sheet: \(10^{122}\) bits of memory, \(10^{121}\) operations, and a clock that has ticked 140 times — with the energy per bit sitting exactly on the Landauer limit.
But build the model by taking "masses grow \(\propto a\)" as physics (\(\lambda\) and \(\beta\) fixed by \(\Omega_m\) alone, and the solution an attractor) and radiation alone cannot be saved — it is conformally invariant and cannot couple to the dilaton (Episode 7). Radiation dominates for \(z>103\), the one selling point evaporates, and the horizon problem returns at ΛCDM's level. The deeper reason is that the gravitational field itself splits into the conformal factor (bookkeeping, ghost, no arrow of time) and the Weyl tensor (physics, gravitational entropy, the arrow of time). \(c\cdot t=\text{const}\) was constraining the bookkeeping side. What should have been constrained is the information on light sheets.
Print / PDF: ⌘+P (Ctrl+P on Windows). On screen the slider changes the amount of radiation and shows the constancy of the address space breaking down. "Show answer" opens the solutions.