Mass That ClicksFinale · Technical Appendix ② / Settling Wall ① (the w problem)

A c·t (conformal-time) cutoff meets the acceleration data head-on

Settling Wall ①: \(w(a)\) RVM ended at \(w_0=-1\). NADE, with a conformal-time (= \(c\cdot t\) system) cutoff, gives \(w_0\neq-1\).
We write that \(w(a)\) down and pit its sign against the evolution DESI hints at — one falsifiable sheet.

Prerequisites: the Appendix (one sheet for a discrete universe), Wall ① (the w problem), the running vacuum's \(w(a)\) Conclusion: \(w_0\approx-0.8\) agrees; the sign of \(w_a\) decides it

Of the appendix's four walls, we settle Wall ① (the \(w\) problem). Earlier, when we solved the running vacuum \(\Lambda(H)\) exactly, we got \(w_0=-1\) exactly, and couldn't reproduce DESI's evolution. This time we use conformal time \(\eta\) (= the causal cutoff of the \(c\cdot t\) system) — new agegraphic dark energy (NADE). It gives \(w_0\neq-1\), turning your \(c\cdot t\) into a prediction that the acceleration data can settle black-and-white. The conclusion up front: the magnitude (\(w_0\approx-0.8\)) agrees with DESI, but the direction of the evolution (the sign of \(w_a\)) collides head-on.

01The model: IR cutoff = conformal time \(\eta\) (causal)

New agegraphic DE (Wei–Cai 2008)
$$\rho_{\rm de}=\frac{3n^2M_{\rm Pl}^2}{\eta^2},\qquad \eta=\int_0^t\frac{dt'}{a}=\int_0^a\frac{da'}{a'^2H}$$

\(n\) = an O(1) single parameter. \(\eta\) = conformal time (the comoving horizon) = the natural clock of the \(c\cdot t\) gauge. Past only = causal (unlike the event horizon, it does not clash with discreteness / causality principles).

02Writing down \(w(a)\)

From \(\Omega_{\rm de}=n^2/(H^2\eta^2)\) and \(d\eta/d\ln a=1/(aH)\), we get \(d\ln\rho_{\rm de}/d\ln a=-2\sqrt{\Omega_{\rm de}}/(na)\). Substituting this into \(w=-1-\frac13 d\ln\rho_{\rm de}/d\ln a\):

NADE's equation of state, and the evolution of Ω_de
$$w_{\rm de}(a)=-1+\frac{2}{3n}\,\frac{\sqrt{\Omega_{\rm de}(a)}}{a}$$ $$\Omega_{\rm de}'=\Omega_{\rm de}(1-\Omega_{\rm de})\Big[\,3-\frac{2}{na}\sqrt{\Omega_{\rm de}}\,\Big]\quad('=d/d\ln a)$$

03Decisive point ① — \(w_0\neq-1\) comes out (breaking RVM)

Today (\(a=1,\ \Omega_{\rm de0}\approx0.7\)):

Today's value $$w_0=-1+\frac{2}{3n}\sqrt{\Omega_{\rm de0}}=-1+\frac{0.558}{n}$$

\(n=2\to w_0=-0.72\), \(\ n=3\to w_0=-0.81\), \(\ n=3.5\to w_0=-0.84\). At \(n\approx3\), \(w_0\approx-0.81\) = right on DESI's central value (\(\approx-0.83\)). It breaks RVM's \(w_0=-1\) degeneracy: a \(c\cdot t\)-type cutoff naturally produces a quintessence-like today — \(\eta\) is still growing today = DE is still evolving today, and this \(c\cdot t\) clock is what does it.

04Decisive point ② — the "direction" of the evolution is opposite to DESI

Trace the full history (\(n=3\)): \(\ w\to-\tfrac23\) (matter era, \(a\to0\)) \(\ \to\ -0.81\) (now) \(\ \to\ -1\) (future). Always \(w>-1\) (quintessence, never phantom), and \(w\) decreases monotonically toward \(-1\) with time — read in CPL, that means \(w_a>0\) (thawing). But DESI has \(w_a<0\) (phantom in the past, crossing \(-1\)). In the figure below, you can see them agree today and split in opposite directions toward the past.

Figure: \(w(a)\). Blue = NADE (\(c\cdot t\) conformal-time cutoff, ODE integrated numerically). Orange = the CPL that DESI hints at (\(w_0{=}-0.83,\ w_a{=}-0.75\)). The two nearly agree "now" (\(a{=}1\)), but run opposite into the past: NADE stays \(w>-1\) (thawing), while DESI crosses \(-1\) into phantom territory. Change \(n\) with the slider.
Move n and NADE's w(a) changes.
NADE (c·t conformal time) DESI hint (CPL) w=−1 (phantom divide)

05The decisive verdict

Item
NADE (c·t cutoff)
DESI (2024) hint
Today's \(w_0\)
\(-1+0.558/n\); at \(n{\approx}3\), −0.81
≈ −0.83 agrees
Evolution \(w_a\)
about +0.1 (thawing)
≈ −0.75 (phantom crossing) opposite sign
Crossing \(w=-1\)
no (always \(w>-1\))
yes

So — the magnitude (\(w_0\approx-0.8\)) is a hit, the direction of the evolution (\(w_a\)) is opposite. If DESI's "phantom crossing" is confirmed, the pure conformal-time cutoff is rejected. Conversely, if the "evolution" turns out to be thawing quintessence (\(w_a>0\)) or a systematic error, NADE survives. Either way, your \(c\cdot t\) has become a falsifiable prediction that the acceleration data settles black-and-white.

The honest line

This is a prediction, not a confirmation. And DESI's evolution hint itself is still provisional (\(\sim2\text{–}4\sigma\), dependent on the dataset and systematics, with the \(-1\) crossing possibly being systematics). So "rejection" comes with the proviso "if DESI's crossing is real." If DESI-DR2 / Euclid in the near future pin down the sign of \(w_a\), the life or death of \(c\cdot t\) + a conformal-time cutoff is decided.

Also, NADE is a one-parameter (\(n\)) model, and \(\Omega_{\rm de0}\) is fixed by integration (not fit freely). \(c\cdot t=\text{constant}\) and conformal time \(\eta\) are close relatives but not identical; here we use NADE as a representative of a "causal, past-only \(ct\)-system cutoff." The local speed of light is invariant.

Questions to check yourself
  1. Why can NADE produce \(w_0\neq-1\), while the running vacuum (RVM) gave \(w_0=-1\)?
    One answer
    RVM has \(\rho_\Lambda\propto H^2\), and at \(a=1\) the square bracket of the effective EoS vanishes exactly, giving \(w_0=-1\). NADE has \(\rho_{\rm de}\propto1/\eta^2\), and because conformal time \(\eta\) is still growing today = DE is still evolving today, \(w_0=-1+\frac{2}{3n}\sqrt{\Omega_{\rm de0}}\neq-1\). The difference is whether there is "a clock that is still ticking."
  2. If DESI confirms phantom crossing (\(w_a<0\)), what happens to NADE?
    One answer
    NADE is always \(w>-1\) (thawing, \(w_a>0\)) and does not cross \(-1\), so it is rejected. As long as you use a conformal-time cutoff on its own, phantom behavior does not appear. Conversely, if the evolution is on the thawing side, it survives. Either way, observation settles it = falsifiable.

Appendix ② summaryWall ① condensed into one falsifiable prediction

NADE, with a conformal-time (\(c\cdot t\) system) cutoff, gives \(w_{\rm de}(a)=-1+\frac{2}{3n}\sqrt{\Omega_{\rm de}}/a\). Today \(w_0=-1+0.558/n\) (\(-0.81\) at \(n{\approx}3\)) = matches DESI's magnitude. But it is always \(w>-1\) (thawing, \(w_a>0\)), opposite in sign to DESI's phantom crossing (\(w_a<0\)). So: if DESI's crossing is real, the pure conformal-time cutoff is rejected; if it's thawing / systematics, it survives — the acceleration data settles it black-and-white.

Wall ① (the \(w\) problem) has condensed from an "uncrossable wall" into "one prediction settled by the sign of \(w_a\)." RVM (\(w_0=-1\)) and NADE (\(w_0\approx-0.8\), \(w_a>0\)) are clearly distinguished and falsified within DESI's \(w(a)\). This is something numerology could never do.

At this point on the road Starting from the diagnosis of numerology \(1/(Cn)^D\) — through lattice gauge theory, emergent Lorentz, induced gravity, the cosmological constant via CKN, and now — we've arrived at a single equation and a single decision point: the conformal-time cutoff predicts \(w_0\approx-0.8\), and collides head-on with DESI in the sign of \(w_a\). Your \(c\cdot t=\text{constant}\), ignored by viXra though it may be, now stands as a concrete, falsifiable prediction to be tested by observations of the accelerating expansion. It may be confirmed, or it may fail — but that is exactly the best place a hypothesis can reach. The near-future data will return the answer.
Thank you, truly, for this long road so far. Next: the day DESI-DR2 announces the sign of \(w_a\).
This document is the Finale · Technical Appendix ② of the "Mass That Clicks" series, a reading piece for physics-loving high-schoolers and undergraduates. For new agegraphic dark energy (NADE, Wei–Cai 2008; IR cutoff = conformal time \(\eta\), \(\rho_{\rm de}=3n^2M_{\rm Pl}^2/\eta^2\)), it is established that the equation of state is \(w_{\rm de}=-1+\frac{2}{3n}\sqrt{\Omega_{\rm de}}/a\), the evolution equation is \(\Omega_{\rm de}'=\Omega_{\rm de}(1-\Omega_{\rm de})[3-\frac{2}{na}\sqrt{\Omega_{\rm de}}]\), the matter-era solution is \(\Omega_{\rm de}\simeq n^2a^2/4\) (\(\Rightarrow w\to-2/3\)), and at \(n\approx3\), \(w_0\approx-0.8\) with always \(w>-1\) (non-phantom). That DESI (2024, DR1) preferred \(w_0w_a\)CDM over \(\Lambda\)CDM and gave \(w_0\approx-0.83,\ w_a\approx-0.75\) (dataset-dependent, \(\sim2\text{–}4\sigma\), a provisional hint including the possibility of systematic error), and that this implies phantom crossing, are also as reported. The figure compares \(w(a)\) obtained by numerically integrating the above ODE from the initial condition \(\Omega_{\rm de}=n^2a^2/4\) against CPL \(w=w_0+w_a(1-a)\). This piece does not claim the correctness of any particular model; it lays out a falsifiable prediction alongside the current state of observation. \(c\cdot t=\text{constant}\) is a restatement in coordinates and units; the local speed of light is invariant. — To print, use your browser's "Print" and "Save as PDF" (in the print version the slider and answers are frozen and hidden).

Print / save as PDF: Ctrl+P (⌘+P on Mac). On screen, change n with the slider to see how NADE's w(a) splits from the DESI hint. Click "One answer" to reveal each solution.