c·t = CONST, THAT CLICKS EPISODE 20 ／ The door the previous series left open
We constrained it, and no prediction came out — the reason is the interesting part
Actually constraining
the light sheets
Impose Bousso's bound as a constraint: what gets excluded?
Today's margin is 33 orders. The only place it bites was the Planck era.
Extra 2 of the previous series ended on this: "if you write the universe as a computer with finite resources, what should be constrained is not \(a(t)\) but the information on the light sheets. Nobody has done that calculation." Extra 3 applied Bousso's covariant entropy bound to the apparent horizon, derived \(s\le3H/4\ell_P^2\), and got as far as showing that saturating it identically gives \(a\propto t^{1/3}\). Today we go further — imposed as a constraint rather than a saturation, does a prediction come out? The answer first: no. And why not is the interesting part.
01Following the bound through time
with \(s\propto a^{-3}\) and \(H\propto1/t\)
$$f\ \propto\ t^{\,1-3p}\qquad(a\propto t^p)$$Take the standard thermal history (radiation \(p=1/2\) → matter \(p=2/3\)), set \(f\approx1\) at the Planck era, and stack up to today.
| Epoch | \(t\) | \(f\) | Margin |
|---|---|---|---|
| Planck era | \(5.4\times10^{-44}\) s | \(\approx1\) | 0 orders (saturated) |
| Electroweak | \(10^{-11}\) s | \(7.3\times10^{-17}\) | 16 orders |
| Nucleosynthesis | 1 s | \(2.3\times10^{-22}\) | 22 orders |
| Recombination | 380,000 yr | \(2.4\times10^{-29}\) | 29 orders |
| Today | 13.8 Gyr | \(6.7\times10^{-34}\) | 33 orders |
Conclusion of §01
The bound saturates exactly at the Planck era, and the margin opens steadily from there.
Today's margin is 33 orders. ── The only epoch actually constrained is the Planck era.
02Saturating at the Planck era was an identity
"The bound saturates exactly at the Planck era" looks meaningful. Let us measure it with Episode 19's tool.
at the Planck era \(T\sim M_{\rm Pl}\), hence
$$s\sim T^3\sim\frac{1}{\ell_P^3},\qquad H\sim\frac{1}{t_P}\sim\frac{c}{\ell_P}$$substituting
$$f=\frac{s}{3H/4\ell_P^2}\sim\frac{1/\ell_P^3}{(1/\ell_P)/\ell_P^2}=1$$Conclusion of §02
\(O(1)\) from dimensional analysis alone. An [identity] in Episode 19's classification — surprise 0 bits.
"The bound saturates at the Planck era" was a check, not a discovery.
After Dirac's large numbers (Episode 7) and the Landauer limit (Episode 10), this is the third "looks meaningful, is an identity". The procedure built last episode went straight to work.
03The heart — as a constraint, far too weak
So what does imposing this bound exclude? For each expansion law, find when \(f=1\) is reached going backwards.
| \(p\) | Expansion law | Exponent \(1-3p\) | When \(f=1\) | Verdict |
|---|---|---|---|---|
| 1/3 | stiff (\(w=1\)) | 0 | ── (\(f\) constant) | the only law that can stay saturated |
| 1/2 | radiation | \(-0.5\) | \(2\times10^{-49}\) s | before the Planck time — never violated |
| 2/3 | matter | \(-1\) | \(2.9\times10^{-16}\) s | formally violated (really radiation then) |
| 1 | \(c\cdot t=\)const | \(-2\) | \(11\) s | violated (the verdict of Extra 3) |
Here is the heart. The bound excludes only expansion laws that reach \(f>1\) before the Planck era. The real universe is radiation dominated there, so it passes safely.
The thing this episode most wants to say
Constraining the information on light sheets yields no prediction.
With 33 orders of margin today, essentially every standard expansion law passes.
In Episode 19's language — this constraint carries almost zero information.
Figure: the history of the occupancy \(f=s/(3H/4\ell_P^2)\). The vermilion ceiling is \(f=1\) (the bound). The standard thermal history touches the ceiling at the Planck era and descends 33 orders from there. The slider changes the expansion law: for \(p>1/2\) the curve punches through the ceiling going backwards.
04The universe that stays saturated is stiff
So "the universe that always maximises its packing efficiency" is stiff (kination) — not ours. Here is the previous series' table of resources again.
Which resource you fix splits the answer three ways — and the observed universe is none of them (it changes over, radiation → matter → \(\Lambda\)). The very idea of "fixing one resource" is not enough to determine the shape of the universe — that is this episode's answer to Extra 2's question.
05So is the door closed?
Honestly: half closed.
| What Extra 2 hoped | What we found |
|---|---|
| Constraining the light-sheet information yields a prediction | it does not (33 orders of margin, almost zero information) |
| That is a better form than constraining \(a(t)\) | correct as a form (conformally invariant, gauge independent) |
| Where the bound has meaning | the Planck era only — and saturation there is an identity |
Extra 2's "what should be constrained is the light-sheet information" was right as a form — Bousso's bound is conformally invariant, gauge independent, and thus has the shape of something that can be judged in Episode 3's sense. But applied, the only place it bites is the Planck era.
① \(s\le3H/4\ell_P^2\) is Bousso's covariant entropy bound applied to one particular surface, the apparent horizon. The bound is a claim about arbitrary surfaces, so other surfaces could give stronger conditions — what is shown here is "at least on this surface the constraint is weak" (same caveat as §8 of Extra 3 in the previous series).
② Comoving conservation of entropy is assumed. With entropy production (reheating) the past \(s\) would have been smaller and the margin wider still (strengthening the conclusion).
③ "\(f\approx1\) at the Planck era" is a one-loop estimate. The coefficient moves with the choice of \(g_*\) (Extra 3 estimated \(T\le2.84M_{\rm Pl}/\sqrt{g_*}\) for standard cosmology). §02 claims that \(f\) comes out \(O(1)\) from dimensional analysis, not that the coefficient is exactly 1.
④ The \(f=1\) times in §03 assume a single \(p\) at all epochs. The 11 s for \(c\cdot t=\)const is the same order as Extra 3's 5 s, the difference coming from the value taken for today's \(f\). The real universe changes expansion laws, so tracing back with one \(p\) is a coarse approximation.
⑤ "No prediction" is a conclusion about this form of constraint, not a claim that the holographic principle is powerless — there are many contexts (AdS/CFT among them) where holography is a powerful computational tool. The point here is the single one that it is weak as a constraint determining a cosmological \(a(t)\).
Exercises (solvable with this episode's formulas alone)
- Derive the time dependence of \(f=s/(3H/4\ell_P^2)\) from \(a\propto t^p\).
Show the answer
\(s\propto a^{-3}\propto t^{-3p}\) and \(H\propto1/t\), so \(f\propto t^{-3p}/t^{-1}=t^{1-3p}\). Only at \(p=1/3\) does the exponent vanish. - Why is \(f\approx1\) at the Planck era, and is it surprising?
Show the answer
With \(T\sim M_{\rm Pl}\), \(s\sim1/\ell_P^3\) and \(H\sim c/\ell_P\), substitution gives \(f\sim1\). It comes out of dimensional analysis alone, so in Episode 19's classification it is an identity — 0 bits of surprise. A check, not a discovery. - Which expansion law saturates the bound identically?
Show the answer
The one making \(f\propto t^{1-3p}\) constant: \(p=1/3\), i.e. \(a\propto t^{1/3}\) (\(w=1\), stiff/kination). "The universe that always maximises packing efficiency" — but not ours. - Imposed as a constraint, what does this bound exclude?
Show the answer
Only expansion laws reaching \(f>1\) before the Planck era. With 33 orders of margin today, essentially all standard ones pass — almost zero information as a constraint. Though \(c\cdot t=\)const applied at all epochs does violate it at \(t\sim10\) s. - (Harder) Was Extra 2's "constrain the light-sheet information" wrong?
Show the answer
Right in form, insufficient in effect. Bousso's bound is conformally invariant and gauge independent, so it has the right shape to be judged in Episode 3's sense. Applied, though, it bites only at the Planck era — and saturation there is an identity. The conclusion is that "the universe is a computer with finite resources" has meaning only in the quantum gravity regime — not a failure but a determination of range.
Summary — the only place it bites was the Planck era
We went through the door the previous series left open — does constraining light-sheet information yield a prediction? Using Extra 3's \(s\le3H/4\ell_P^2\) and following the occupancy \(f\), it saturates exactly at the Planck era and the margin opens by 33 orders (16 at electroweak, 22 at nucleosynthesis, 33 today).
"Saturating exactly at the Planck era" looks meaningful, but measured with Episode 19's tool it was an identity — \(T\sim M_{\rm Pl}\) gives \(s\sim1/\ell_P^3\) and \(H\sim c/\ell_P\), so \(f\sim1\) from dimensional analysis alone. Zero bits of surprise. The third such case after Dirac's large numbers (Episode 7) and the Landauer limit (Episode 10).
And the heart — it is far too weak as a constraint. With 33 orders of margin, all it excludes are laws that violate before the Planck era, and standard ones all pass. Constraining light-sheet information yields no prediction. Saturating identically gives \(a\propto t^{1/3}\) (stiff), which is not our universe — fix the bit count and you get de Sitter, fix the address space and you get \(a\propto t\), fix the occupancy and you get \(a\propto t^{1/3}\). Which resource you fix splits the answer three ways, and the observed universe is none of them.
So the door is half closed. Extra 2's "constrain the light-sheet information" was right as a form — Bousso's bound is conformally invariant and gauge independent — but it bites only at the Planck era. Not a failure but a determination of range: as Episode 16 measured the limit of the conformal transformation, today measured the limit of the holographic bound. "The universe is a computer with finite resources" has meaning only in the quantum gravity regime. Part V goes there.
This document is Episode 20 of "c·t = const, That Clicks", written for physics-minded high-school and university readers. The covariant entropy bound is due to Bousso (1999). Applying it to the apparent horizon to get \(s\le3H/4\ell_P^2\), and the fact that saturating it identically gives \(a\propto t^{1/3}\) (\(w=1\)), were shown in Extra 3 of the previous series. The occupancy history (\(\approx1\) at the Planck era, \(7.3\times10^{-17}\) at electroweak, \(2.3\times10^{-22}\) at nucleosynthesis, \(2.4\times10^{-29}\) at recombination, \(6.7\times10^{-34}\) today) and the \(f=1\) times per expansion law are computed here (kenshou/calc24.py). \(s\le3H/4\ell_P^2\) applies the bound to one particular surface, the apparent horizon; other surfaces could give stronger conditions (same caveat as §8 of Extra 3). Comoving conservation of entropy is assumed; entropy production would widen the margin further (strengthening the conclusion). "\(f\approx1\) at the Planck era" is a one-loop estimate whose coefficient moves with \(g_*\) — §02 claims \(O(1)\) from dimensional analysis, not a coefficient of exactly 1. The \(f=1\) times in §03 assume a single \(p\) at all epochs; the 11 s for \(c\cdot t=\)const is the same order as Extra 3's 5 s. "No prediction" concerns this form of constraint and is not a claim that holography in general is powerless — AdS/CFT and others make it a powerful computational tool. This document's assessment of Extra 2's proposal (right in form, insufficient in effect) is this series' reading. 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).