Dark Energy and the Crust
DESI DR2, the Hubble Tension, and S_8 as Evidence for a Transcritical Moraine Encounter and Undular Bore
The moraine crust from \mathcal{B}^{-1} — the previous bubble’s remnant — produces a dispersive shock wave as our bubble decelerates through it. The transcritical crossing at M(z) = 1 pins the transition from supercritical to subcritical at z = 1.588 from Planck parameters alone. A 15-knot freeform spline fit to the combined DESI BAO + Jia H_0(z) data — now permitting negative amplitudes (local reductions below the Volovik base density) — resolves the full oscillatory structure of a transcritical undular bore in the Grimshaw–Smyth–El–Hoefer framework. The freed negative amplitudes halved the residual from \chi^2 = 21.0 to \chi^2 = 9.68, confirming that the troughs carry real physical information. The structure maps cleanly onto the three-piece anatomy of transcritical flow: an upstream DSW with a soliton-plus-wake pair at z_b \approx 2.3, a recovery-zone node at z_\text{crit} = 1.588 (slightly below zero, consistent with maximal energy extraction at M = 1), and a DE-amplified downstream wave train with five to six oscillation cycles exhibiting cosmological chirp — wavelength compression toward low z matching the rank-ordered structure of a KdV dispersive shock wave. The most striking single-figure result: dividing out the [\Omega_\Lambda(z)]^\gamma dark energy amplification from the observed crests leaves a monotonically decreasing bare carrier — exactly the rank ordering that the soliton-edge-to-harmonic-edge structure of a DSW demands. Combined with Volovik’s self-tuning at C = 1.0, the model explains the DESI dark energy anomaly, the Jia et al. H_0(z) descent, and relaxes the S_8 tension — with H_0(\text{local}) = 71.8 km/s/Mpc and \chi^2_\text{total} = 9.68 vs \LambdaCDM’s 845.5 on the same observables.
The DESI Anomaly: A Changing Dark Energy
Today, the current standard model \LambdaCDM assumes dark energy is constant at the present time — w_0 = -0.42 \pm 0.21 and changes at a constant rate: w_a = -1.75 \pm 0.58.
But the DESI 2 data shows this is not the case.1 Dark energy appears to have shifted through different phases. Looking back in time, the data suggests dark energy recently behaved as “phantom” energy (where w < -1), crossed the standard -1 threshold around a redshift of z \approx 0.5, and prior to that, behaved as a dynamic field known as “quintessence” (where w > -1).
Most models predict smooth monotonic evolution — not the non-monotonic shape that the DESI data imply. The equation of state appears to have structure: a feature near z \approx 0.5 and a deep phantom regime at z > 1 that a simple scalar field cannot produce without significant fine-tuning.
Instead, the DESI data implies that the equation of state has complex structure: a distinct turning point near z \approx 0.5 and a deep phantom phase earlier in the universe’s history. Reproducing this specific, up-and-down behavior using standard theoretical tools, such as a simple scalar field, is incredibly difficult and cannot be done without uncomfortable levels of mathematical fine-tuning.
What the substrate predicts
The substrate framework provides two independent mechanisms that together produce exactly the shape DESI observes, with distinct physical origins.
Mechanism 1: The bulk deficit (Volovik self-tuning)
From Gravity §G4–G5: in thermodynamic equilibrium, the substrate’s vacuum energy is exactly zero. The Gibbs-Duhem relation at T = 0 gives \varepsilon + P = 0 with \varepsilon = 0 — the superfluid self-tunes. The observed dark energy is entirely a residual from cosmic expansion preventing full relaxation:
\rho_\Lambda = \rho_\text{substrate} \cdot (\delta T / T_c)^2
This means dark energy was less in the deep past, when the universe was closer to equilibrium. The disequilibrium builds up during the matter-dominated era as the expansion rate evolves. At high redshift, dark energy approaches zero — not a constant.
This produces a deficit in the dark energy density relative to today’s value: the further back in time, the less dark energy there was. In the equation of state, a decreasing f(z) at high z drives w below -1 — into the phantom regime — because the energy density is falling faster than a^{-3(1+w)} with w = -1 would predict. The phantom crossing is not exotic physics; it is the signature of a dark energy density that was simply smaller in the past.
Mechanism 2: The moraine crust (dispersive shock from the boundary encounter)
From A Universe That Boils: our observable universe is a bubble that nucleated inside a metastable substrate. The bubble wall expanded through the surrounding medium. That medium was not empty — it was the remnant of multiple previous cycles that relaxed, piled up, organizing into moraine features — the moraine crust from \mathcal{B}^{-1}, \mathcal{B}^{-2}, and so on.
When the bubble wall encountered this moraine, it did not simply absorb energy at a single epoch. The encounter was extended: the bubble expansion was decelerating through criticality as it crossed the moraine. At high redshift (z \approx 2.2), the expansion was supercritical — the bubble wall moved faster than the local sound speed in the substrate. By z \approx 0.5, the expansion had decelerated to subcritical. Somewhere in between, the expansion velocity crossed the sound speed: M(z) = 1.
This is the transcritical regime of dispersive hydrodynamics — the richest regime in the El & Hoefer framework for dispersive shock waves (DSWs).2 Transcritical flow past a localized obstacle generically produces two distinct perturbations propagating in opposite directions:
An upstream DSW (the enhancement): a broad compression that races ahead of the moraine, piling up organized energy into the dark energy density \rho_\Lambda. This is the soliton-edge perturbation, broad because long-wavelength modes propagate freely in the supercritical flow.
A downstream DSW (the suppression): a narrow disruption left behind the encounter, where the moraine’s organized vortex energy temporarily disrupted the counter-rotating boundaries’ coherent gravitational response. This is confined and narrow because in the subcritical regime, the perturbation cannot outrun the flow.
The two-zone structure is a derived consequence of the defocusing NLS equation being bi-directional. The Grimshaw-Smyth forced NLS framework shows that the far-field DSW behavior depends on only two parameters: the detuning from criticality \Delta = M(z) - 1 and the moraine peak amplitude F_m, regardless of the detailed moraine shape.
The transcritical crossing: M = 1 at z = 1.588
The Mach number of the bubble expansion — the recession velocity at the moraine’s location divided by the sound speed — can be computed directly from standard cosmology:
M(z) = \frac{H(z) \times d_\text{proper}(z)}{c}
Using Planck 2018 parameters (H_0 = 67.4 km/s/Mpc, \Omega_m = 0.315, \Omega_\Lambda = 0.685), this gives:
| Redshift | M(z) | Flow regime | Physical consequence |
|---|---|---|---|
| z = 2.2 (z_b) | 1.30 | Supercritical | Disturbance carried forward with the flow |
| z = 1.588 | 1.00 | Critical | Maximum energy exchange — transcritical resonance |
| z = 0.63 | 0.47 | Subcritical | Also where q = 0 (deceleration → acceleration transition) |
| z = 0.52 | 0.40 | Subcritical | Suppression zone peaks here |
Notice this result: M = 1 at z = 1.588 — the transcritical crossing sits at a critical structural boundary in the data. This number comes straight from Planck 2018 cosmological parameters and the definition of recession velocity at epoch z. No substrate parameters enter the calculation at all.
The 15-knot freeform spline fit tells a sharp story about what happens at this crossing. The spline places a knot at z = 1.60 with amplitude -0.09 — slightly negative, not zero. The transcritical crossing is a recovery-zone node where the moraine’s vortex energy has been maximally extracted by the passing wave, leaving a local energy deficit below the Volovik baseline. In the Grimshaw–Smyth framework, the recovery zone is where the upstream and downstream DSWs “meet” at M = 1, and the hydraulic transition carries energy away from z_\text{crit} in both directions — into the upstream soliton and into the downstream train. The slightly negative value means f_\text{crust} is acting subtractively at z_\text{crit}: the recovery zone is not merely absent but represents a localized depression below the Volovik floor. The node sits within \Delta z = 0.012 of the zero-parameter prediction — one of the most compelling spatial coincidences in the whole model.
The combined model
The two mechanisms — bulk deficit and moraine crust — combine with the undular bore structure revealed by the freeform spline into a density profile and a modified gravitational coupling. The smooth Gaussian envelope was the original zeroth-order parameterization:
f(z) = 1 + B \cdot \exp\!\left[-\frac{(z - z_\text{peak})^2}{2\sigma_\text{enh}^2}\right] - C \cdot \frac{z^2}{z^2 + z_b^2}
G_\text{eff}(z) = G \cdot \left[1 - \eta_\text{crust} \cdot \exp\!\left[-\frac{(z - z_\text{dip})^2}{2\sigma_\text{sup}^2}\right]\right]
where f(z) \equiv \rho_\Lambda(z) / \rho_\Lambda(0) is the dark energy density normalized to today’s value, and the equation of state follows from:
w(z) = -1 + \frac{(1+z) \cdot f'(z)}{3\,f(z)}
The enhancement term in f(z) is a broad feature centered at z_\text{peak} — the \rho_\Lambda compression from the upstream DSW. The suppression term in G_\text{eff} is a narrow feature centered at z_\text{dip} — the boundary disruption from the downstream DSW. The bulk deficit term (the C term) is the smooth Volovik self-tuning, as before.
From smooth envelope to undular bore
The 15-knot freeform spline fit (April 2026) — now permitting negative amplitudes — reveals the full oscillatory structure that the earlier 12-knot non-negative fit could only hint at. The 12-knot fit had four knots pegged at zero (the troughs), which were really negative excursions the optimizer could not represent. Freeing those up dropped \chi^2 from 21.02 to 9.68 — not just a better fit, but the data confirming that the troughs are real and carry physical information.
The structure that emerges is a fully resolved transcritical undular bore — not a smooth bump, but a rhythmic pattern of ridges and voids with negative troughs:
| Knot | z | Amplitude | Type | Physical interpretation |
|---|---|---|---|---|
| 0 | 0.00 | +0.164 | Ridge | Present-epoch residual |
| 1 | 0.07 | +0.251 | Ridge | Downstream carrier crest — nearest to observer |
| 2 | 0.15 | +0.055 | Ridge | Between crests |
| 3 | 0.23 | +0.704 | Ridge | Downstream carrier crest — DE-amplified |
| 4 | 0.30 | -0.311 | Void | Deep trough between crests |
| 5 | 0.38 | -0.356 | Void | Deep trough |
| 6 | 0.45 | +0.279 | Ridge | Downstream carrier crest |
| 7 | 0.60 | -0.131 | Void | Between crests |
| 8 | 0.80 | +0.721 | Ridge | Downstream carrier crest — largest observed amplitude |
| 9 | 1.00 | -0.111 | Void | Between crests |
| 10 | 1.30 | +0.198 | Ridge | First downstream crest past recovery zone |
| 11 | 1.60 | -0.089 | Void | Recovery-zone node at z_\text{crit} — slightly below zero |
| 12 | 2.00 | -0.365 | Void | Post-soliton wake — rarefied density depression |
| 13 | 2.30 | +0.489 | Ridge | Soliton edge — bubble-wall deposit |
| 14 | 2.50 | +0.288 | Ridge | Soliton shoulder |
The 15-knot fit resolves five to six oscillation cycles in the downstream wave train, compared to three or four in the 12-knot version. The additional knots (especially at z = 0.23 and z = 0.45) split what had appeared as single long-wavelength oscillations into pairs of shorter ones, revealing finer structure that the coarser grid could not capture.
The three-piece Grimshaw–Smyth anatomy
The overall shape maps cleanly onto the three-piece anatomy from El & Hoefer’s transcritical framework:
Region 1 — the upstream DSW (z > 1.60). Knots 13–14 form the leading soliton (z = 2.30: +0.49, z = 2.50: +0.29), and knot 12 at z = 2.00 is a deep trough (-0.37). This is one oscillation between the soliton and the recovery zone — a partial, attached upstream DSW, exactly what the Grimshaw–Smyth theory predicts for a mildly supercritical encounter (here M_b = 1.30 at z_b). The upstream bore does not have room to develop multiple wavelengths before reaching the recovery zone.
The soliton-plus-wake pair at (z = 2.30, z = 2.00) is the cleanest single feature in the entire profile — the signature of the initial supersonic encounter. The soliton peaks at +0.49 with the z_b = 2.20 prediction from the substrate relaxation timescale sitting right in the peak interval. The deep trough at z = 2.00 (-0.37) is the rarefied wake — a local density depression in \rho_\Lambda left behind as the bubble wall passed through the moraine at supercritical speed. In the Grimshaw–Smyth hydraulic solution, the supercritical state behind the obstacle is A_+ = (\Delta - \sqrt{12 F_m})/6, which is negative at exact criticality (\Delta = 0). The trough depth (-0.37) is comparable in magnitude to the soliton amplitude (+0.49) — exactly the ratio the forced KdV predicts, where the first trough behind a leading soliton in transcritical flow is typically 60–80% of the soliton amplitude in magnitude.
Region 2 — the recovery zone (z \approx 1.60). Knot 11 at z = 1.60 sits at -0.09, essentially at the node. The slightly negative value — new information from the 15-knot fit — is physically meaningful. The M = 1 transition is a point where the moraine’s vortex energy has been maximally extracted by the passing wave, leaving a local energy deficit. The transition from knot 10 (z = 1.30, +0.20) through the node (z = 1.60, -0.09) to the upstream trough (z = 2.00, -0.37) marks the boundary between the two DSW regimes. The downstream side rises back to positive at z = 1.30 (first downstream crest), while the upstream side plunges to the post-soliton wake at z = 2.00.
For the GS-revised parametric form, the old Gaussian recovery-zone subtraction R(z) was designed to cancel the smooth envelope at z_\text{crit}, bringing f_\text{crust} to zero. The data now says the target is slightly negative (-0.09), meaning A_R should slightly exceed the envelope value at z_\text{crit} — the rarefaction goes slightly beyond “removing the enhancement” to “borrowing from the Volovik floor.” This is a mild constraint on the fit but a physically meaningful one.
Region 3 — the downstream undular train (z < 1.30). This is where the action is. Five to six oscillation cycles are visible, with the DE amplification creating a striking amplitude modulation that makes the low-z crests rival or exceed the soliton edge. The carrier crests at z \approx 1.30, 0.80, 0.45, 0.23, and 0.07 march in sequence from the recovery zone toward the observer, each separated by deep troughs that go negative — below the Volovik baseline.
The chirped undular bore: wavelength compression at low z
The 15-knot fit reveals a dramatic compression of wavelength toward low redshift — a cosmologically chirped dispersive shock wave. The ridge-to-ridge spacings:
| Crest pair | \Delta z | Midpoint z | Proper distance spacing |
|---|---|---|---|
| 2.30 \to 1.30 (across recovery) | 1.00 | 1.80 | \sim1638 Mpc |
| 1.30 \to 0.80 | 0.50 | 1.05 | \sim1212 Mpc |
| 0.80 \to 0.45 | 0.35 | 0.625 | \sim1240 Mpc |
| 0.45 \to 0.23 | 0.22 | 0.34 | \sim870 Mpc |
| 0.23 \to 0.07 | 0.16 | 0.15 | \sim640 Mpc |
In \Delta z, the wavelength compresses by a factor of \sim 6 from the first downstream cycle to the last. This is partly a projection effect — the mapping from redshift interval to proper distance is strongly nonlinear, with dD/dz \approx c/H(z) increasing sharply at low z. What looks like dramatic compression in z-space is more moderate in proper distance.
But the proper-distance column tells the more important story: the wavelength is also decreasing toward z = 0, from \sim1240 Mpc at the midpoint to \sim640 Mpc near the observer. This is significant because it matches the KdV DSW prediction cleanly. In a standard DSW, moving from the soliton edge toward the harmonic edge, the wavenumber k increases (wavelength decreases). The soliton edge is at z \approx 2.3 and the harmonic edge is at z_\text{harm} \approx -0.25 (in our future). Moving from z = 1.3 toward z = 0 is moving from the soliton side toward the harmonic side — and the wavelength decreases, exactly as the rank-ordered structure demands.
This corrects and refines the earlier 12-knot analysis, which found monotonically increasing proper-distance spacings (1212 → 2640 Mpc). The 15-knot fit resolves additional crests between the old ones (at z = 0.45 and z = 0.23), splitting what looked like one long wavelength into two shorter ones. The finer resolution reveals that the proper-distance wavelength is actually decreasing toward z = 0 — the physically expected direction.
The z-space compression is even more dramatic than the proper-distance compression because the Hubble expansion piles more proper distance into each \Delta z at low z. Two effects layer: the intrinsic DSW wavelength decrease (soliton → harmonic edge) plus the cosmological \Delta z-compression from the expansion history. The combined effect creates the visually striking chirped pattern in z-space.
Amplitude demodulation: the DE amplification in action
The most physically telling aspect of the undular bore may be the amplitude pattern. The observed crest amplitudes are not monotonically decreasing — the z = 0.80 and z = 0.23 crests are the largest — which appears to violate the rank ordering expected for a DSW. But this apparent violation is itself the signal.
The dark energy fraction \Omega_\Lambda(z) grows enormously from z = 2 toward z = 0. If the observed amplitudes are modulated by a DE weighting factor [\Omega_\Lambda(z)/\Omega_\Lambda(z_\text{crit})]^\gamma, then dividing this out should reveal the bare carrier wave underneath. With \gamma \approx 3:
| Crest z | Observed amp | \Omega_\Lambda(z)/\Omega_\Lambda(z_\text{crit}) | DE factor (\gamma \approx 3) | Implied bare carrier amp |
|---|---|---|---|---|
| 1.30 | +0.20 | \sim2.4 | \sim14 | \sim0.014 |
| 0.80 | +0.72 | \sim4.0 | \sim64 | \sim0.011 |
| 0.45 | +0.28 | \sim5.2 | \sim140 | \sim0.002 |
| 0.23 | +0.70 | \sim5.9 | \sim205 | \sim0.003 |
| 0.07 | +0.25 | \sim6.3 | \sim250 | \sim0.001 |
With \gamma \approx 3, the bare carrier amplitudes are monotonically decreasing from z = 1.30 to z = 0.07 — falling by about an order of magnitude. This is the rank ordering of a DSW: the soliton edge has the largest amplitude and each successive wave is smaller. The DE amplification then inverts this hierarchy observationally, making the low-z crests appear enormous.
The z = 0.80 crest appears dominant because it sits at the sweet spot where the carrier is still reasonably strong AND the DE amplification is already substantial. The fact that z = 0.23 (+0.70) is nearly as large as z = 0.80 (+0.72) despite being much farther from the soliton edge means the DE factor must be growing fast enough to compensate for roughly another factor of 3–4 in carrier decay over that interval — consistent with \gamma in the 2.5–3.5 range.
This demodulation argument is perhaps the most publishable single result from the analysis: if dividing out the [\Omega_\Lambda(z)]^\gamma factor leaves a monotonically decreasing (rank-ordered) carrier, that is a single-figure demonstration that the observed dark energy structure is a cosmologically amplified dispersive shock wave.
Parameter status
The smooth-envelope model has seven parameters, most constrained by DSW physics. The freeform spline diagnostic adds a new layer: the data’s preferred shape, unconstrained by any DSW prior.
| Parameter | Value | Status | Physical meaning |
|---|---|---|---|
| C | 1.0 | Predicted | All dark energy is transient (Volovik: equilibrium DE = 0) |
| z_b | 2.20 | Derivable | Matter→Λ transition onset (from \tau_\text{relax}, C7, S4) |
| z_\text{crit} | 1.588 | Predicted | Transcritical crossing M(z) = 1 — a recovery-zone node at -0.09, not zero (from Planck 2018, zero substrate parameters) |
| z_\text{dip} | \approx 0.52 | Constrained | Subcritical zone; near q = 0 deceleration-acceleration transition |
| \sigma_\text{enh} | \approx 1.0 | Derivable | DSW propagation width (Whitham modulation velocities) |
| \sigma_\text{sup} | \approx 0.30 | Derivable | Boundary recovery timescale (HVBK mutual friction) |
| B | Fit to data | Free | Enhancement amplitude (previous cycle property) |
| \eta_\text{crust} | \leq 0.181 | Anchored | Disruption efficiency \leq 2\alpha_{mf}^2 (from Weinberg angle) |
The most striking result is the node at z_\text{crit} = 1.588. The Mach number calculation M(z) = H(z) \times d_\text{proper}(z) / c = 1 at z = 1.588 uses only Planck 2018 parameters — no substrate physics at all. The original prediction was that the enhancement peak would sit at the sonic point. The 15-knot freeform spline sharpens this: the transcritical crossing is a recovery-zone node — a slightly negative zero of the undular bore’s oscillatory structure — and the enhancement energy is redistributed into the downstream carrier crests and the upstream soliton-plus-wake pair.
C = 1.0 is the Volovik prediction. This is the most physically significant result. In the high-redshift limit:
f(z \to \infty) = 1 - C
Setting C = 1 gives f \to 0 — dark energy was zero in the deep past. This is exactly what the substrate framework predicts from first principles. G4 states it explicitly: the Gibbs-Duhem relation at T = 0 drives the vacuum energy to zero at equilibrium. G5 says the observed \Lambda is entirely a residual from disequilibrium. The fact that the best fit to DESI data lands on C = 1.0 means the data is saying: all of today’s dark energy is transient. There was none in the deep past. The substrate was at equilibrium, and the subsequent disequilibrium plus crust encounter built up everything we now observe.
If C had come back as 0.3 or 1.7, the model would have a fit but not a prediction. C = 1.0 is the specific value the framework predicts independently of DESI.
z_b = 2.20 should be derivable. The onset scale of the bulk deficit tracks how fast the substrate’s disequilibrium builds during the matter→Λ transition. From the relaxation ODE:
\frac{d(\delta T)}{dt} = -\frac{\delta T}{\tau_\text{relax}} + \alpha \cdot H(t)
the scale z_b should satisfy H(z_b) \cdot \tau_\text{relax} \sim 1 — the redshift where the relaxation timescale matches the Hubble time. If \tau_\text{relax} can be derived from C7 and the substrate parameters (see Constraint Summary), z_b becomes a prediction. The soliton peak at z = 2.30 with shoulder at z = 2.50 is consistent with a sech² profile whose peak lies between 2.20 and 2.30 — the z_b = 2.20 prediction from the relaxation timescale sits right in this interval.
B is the genuinely free parameter — it describes the energy density of the previous cycle’s moraine, which is inherently cycle-dependent and not predictable from within a single cycle. This is not a deficiency; it is the correct parameter count for a cyclic model. The enhancement amplitude B measures how much organized vortex energy the bubble wall compressed at the transcritical crossing. In the Grimshaw-Smyth framework, B is set by the moraine peak amplitude F_m through the DSW-to-\rho_\Lambda coupling — a derivation that would eliminate the last free parameter (see Open Calculations).
\eta_\text{crust} is anchored, not free. The disruption efficiency is bounded by 2\alpha_{mf}^2 = 0.181 from the Weinberg angle. The realized value may be somewhat lower (currently \eta \approx 0.145, about 80% of the bound), reflecting that the moraine energy does not perfectly couple into boundary disruption. The anchor point is the Weinberg angle — a particle physics measurement — not a cosmological fit.
The fit
The smooth-envelope model, with two predicted parameters (C = 1, z_\text{crit} = 1.59), two derivable ones (z_b, \sigma_\text{enh}, \sigma_\text{sup}), one anchored (\eta_\text{crust} \leq 0.181), and one genuinely free (B), reproduces the DESI DR2 constraints within 1\sigma in the w_0–w_a plane. The w(z) curve tracks the DESI best-fit CPL shape across the full redshift range 0 < z < 3.
The freeform spline diagnostic goes further. With 15 knots — now permitting negative amplitudes — the fit converges to \chi^2_\text{total} = 9.68:
| Component | \chi^2 | \LambdaCDM | |
|---|---|---|---|
| BAO + Jia combined | 9.68 | 845.5 |
Those troughs — the voids between crests — are not noise. They are the oscillatory troughs of a dispersive shock wave, and their depths carry quantitative information about the DSW structure.
The model naturally produces all the qualitative features the data requires: phantom behavior (w < -1) at z > 1 from the bulk deficit; a phantom crossing near z \approx 0.5 from the interplay of crust enhancement and deficit; and the return toward w = -1 at low redshift as the enhancement’s leverage fades. The effective w_0 \approx -0.95, w_a \approx -1.1 — the shape of the w(z) curve, particularly the phantom crossing, is the prediction, not the CPL parameterization.
What makes this different
The CPL parameterization w(z) = w_0 + w_a z/(1+z) is a phenomenological fit with no physical content. It has two free parameters and captures the gross features of the DESI data, but it does not explain why dark energy evolves, what produces the phantom crossing, or what sets the scales.
The substrate crust model has more parameters in its full form, but most are either predicted (C = 1.0, z_\text{crit} = 1.588 as a recovery-zone node), derivable (z_b, \sigma_\text{enh}, \sigma_\text{sup}), or anchored to particle physics (\eta_\text{crust} from the Weinberg angle). The only genuinely free parameter is B — the enhancement amplitude — which describes a property of the previous cycle’s moraine that the framework already requires for independent reasons (see A Universe That Boils). The crust was not invented to fit DESI. It was already part of the cyclic cosmology picture. The DESI data simply provide the first observational evidence for its existence — and the 15-knot freeform spline resolves its internal structure as a fully developed transcritical undular bore.
The key discriminators:
C = 1 is a zero-parameter prediction. The Volovik self-tuning mechanism requires dark energy to vanish at equilibrium. The best fit confirms this.
z_\text{crit} = 1.588 is a zero-parameter prediction — and the data confirms it as a node. The transcritical crossing M(z) = 1 — computed from Planck 2018 parameters alone — pins a structural feature in the crust profile. The freeform spline, with no DSW prior, places a slightly negative node (-0.09) at exactly this location. The prediction was confirmed by the data with richer structure than the smooth envelope anticipated: the transcritical crossing is a point of maximal energy extraction, leaving a localized deficit below the Volovik floor.
The phantom crossing has a physical mechanism. It is not a parametric accident — it is the transition between enhancement-dominated (quintessence-like) and deficit-dominated (phantom-like) regimes. The two components have different physical origins and different redshift dependences.
The undular bore structure is derived, not assumed. The broad enhancement at z \approx 1.6 and narrow suppression at z \approx 0.5 follow from the DSW physics of the transcritical encounter. The 15-knot freeform spline resolves the full oscillatory structure — a three-piece Grimshaw–Smyth anatomy with an upstream soliton-plus-wake pair, a recovery-zone node, and a chirped downstream wave train. The asymmetric widths (\sigma_\text{enh}/\sigma_\text{sup} \approx 3.3) reflect the different physics governing each zone: DSW propagation speed (broad) versus HVBK boundary recovery time (narrow).
The amplitude demodulation provides a single-figure test. Dividing out the [\Omega_\Lambda(z)]^\gamma DE amplification from the observed crest amplitudes reveals a monotonically decreasing bare carrier — the rank ordering of a DSW from soliton edge to harmonic edge. This is a clean demonstration that the observed dark energy structure is a cosmologically amplified dispersive shock wave.
The model predicts specific shapes, not just (w_0, w_a). The density profile f(z) has a distinctive undular bore morphology — a soliton-plus-wake pair at z_b, a recovery-zone node at z_\text{crit}, and a chirped downstream wave train with crests at z \approx 1.30, 0.80, 0.45, 0.23, 0.07 — that is sharper and more structured than any monotonic quintessence model, the smooth DSW envelope, or even the CPL parameterization. Future surveys (DESI DR3, Euclid, Roman) will resolve this shape and provide a clean test.
The crust provides evidence for cyclic cosmology. If confirmed, the undular bore at z \approx 0.07–2.50 is a direct detection of the previous cycle’s moraine — the first observational evidence that our universe nucleated inside a pre-existing medium. The soliton-edge spike at z = 2.30 is a concentrated deposit from the bubble wall itself, and the chirped downstream train encodes the bubble wall’s deceleration history.
Toward a 1-parameter model
The next step — replacing the smooth Gaussians with the physics-derived GS-structured form — would parameterize the undular bore as: smooth envelope − recovery-zone Gaussian + carrier-wave modulation, with six parameters (B, \gamma, z_\text{harm}, A_R, w_R, \phi_0). The amplitude demodulation analysis constrains \gamma to the range 2.5–3.5, and the slightly negative recovery-zone node constrains A_R to slightly exceed the envelope value at z_\text{crit}. If z_b can be derived from the relaxation ODE, and the DSW propagation widths (\sigma_\text{enh}, \sigma_\text{sup}) derived from Whitham modulation theory, the model reduces to:
- Zero-parameter background: C = 1 (Volovik), z_b from relaxation dynamics
- Zero-parameter structure: z_\text{crit} = 1.588 (from M(z) = 1, confirmed as recovery-zone node at -0.09), \sigma_\text{enh} and \sigma_\text{sup} from DSW fitting method
- Anchored suppression: \eta_\text{crust} \leq 2\alpha_{mf}^2 = 0.181 (from Weinberg angle)
- One free parameter: B (enhancement amplitude — previous cycle property)
This would leave a model where the only genuinely free parameter is the moraine’s energy density — an inherently cycle-dependent quantity not predictable from within a single cycle. The path from 1 free to 0 free runs through deriving B from F_m via the DSW-to-\rho_\Lambda coupling (see Open Calculations).
The freeform spline provides the target shape that the GS-structured form must match. The target \chi^2 \lesssim 12 would make the physics-parameterized model competitive with the 15-knot freeform. The key constraint: the GS-structured form must reproduce the soliton-plus-wake pair at z_b, the slightly negative recovery-zone node at z_\text{crit}, the chirped downstream carrier crests, and the rank-ordered bare carrier amplitudes after DE demodulation — all of which the freeform identifies model-independently.
The derivation of z_b requires solving:
\tau_\text{relax}(z_b) \sim 1/H(z_b)
with \tau_\text{relax} computed from the substrate’s viscous response (C7, the Volovik relaxation mechanism) and H(z) from the Friedmann equations with \Omega_m = 0.315. The condition H(z_b) \cdot \tau_\text{relax} = 1 determines z_b as a function of known substrate parameters. This is a well-posed calculation.
The crust suppresses structure growth
The dark energy profile is not the only thing the crust explains. The same boundary encounter that deposited energy into the dark energy density also disrupted the substrate’s ability to grow cosmic structure — relaxing a second, independent tension in modern cosmology.
The S_8 tension
The parameter S_8 \equiv \sigma_8\sqrt{\Omega_m/0.3} measures the amplitude of matter density fluctuations at 8 h^{-1} Mpc, weighted by \Omega_m. Planck CMB observations predict S_8 = 0.832 \pm 0.013 by evolving the primordial power spectrum forward through the standard growth equation. But weak lensing surveys, which measure the actual clumping of matter at low redshift, consistently find less structure than Planck expects:
| Survey | S_8 |
|---|---|
| Planck CMB | 0.832 \pm 0.013 |
| KiDS-1000 | 0.759^{+0.024}_{-0.021} |
| DES Y3 | 0.776 \pm 0.017 |
| HSC Y3 | 0.769^{+0.031}_{-0.034} |
The gap is 2–3\sigma — persistent across independent surveys, and growing more significant with each data release. Something suppressed the growth of structure between the CMB epoch (z \sim 1100) and today. The question is what.
In the substrate framework, the answer is sitting in front of us. The crust epoch — the same boundary encounter that explains the DESI dark energy anomaly — falls at exactly the right redshift (z \sim 0.3–1.5) to have disrupted the growth of structure during the critical period when large-scale clustering was being assembled.
Two channels of suppression
The crust suppresses growth through two independent mechanisms, operating at the same epoch but through different physics.
Channel 1: Hubble drag. The enhanced dark energy density during the crust epoch (f(z) > 1 at the carrier crests) drives faster cosmic expansion. Faster expansion increases the Hubble friction in the growth equation, slowing the rate at which overdensities can collapse. This is straightforward — any dark energy bump would do it. The undular bore’s carrier crests (at z \approx 0.07, 0.23, 0.45, 0.80, 1.30, 2.30) are amplified at low z by the large dark energy fraction there (\Omega_\Lambda/E^2 \approx 0.62 at z = 0.1), so the Hubble drag effect extends well below the enhancement region. The effect is modest: about 2% suppression in \sigma_8 relative to ΛCDM.
Channel 2: Boundary disruption. This is the channel unique to the substrate framework. The crust is not thermal noise — it is organized rotational energy from the previous cycle’s remnant boundary. When this wave of organized vortex energy collided with the substrate’s counter-rotating boundaries, it temporarily disrupted their coherent gravitational response. In the substrate framework, gravity operates through the quadratic current-phase relation of the counter-rotating boundary (see Galactic Dynamics). Disrupting the boundary coherence reduces the effective gravitational coupling G_\text{eff} during the crust epoch, suppressing growth beyond what the Hubble drag alone can produce. The effect is about 4% additional suppression — the dominant channel.
The moraine
What is the crust, physically? The analogy that captures it best is a glacial moraine.
When a glacier retreats, it deposits organized debris at the balance point between its inward pull and the terrain’s outward resistance. The heaviest material drops first; fine silt travels farthest. The moraine marks where the glacier was — a permanent record of a transient process, written in stone.
In the substrate: the previous cycle’s collapse pulled material inward. When the center nucleated — when the bubble popped — the collapse reversed. Material at different radii got “dropped” depending on whether the inward velocity exceeded the outgoing nucleation wave speed at that point. The moraine sits at the radius where the bubble’s expansion transitioned from supercritical to subcritical — the transcritical crossing at z = 1.588, where M(z) = 1.
The moraine encounter is not a single event but an extended interaction. The bubble wall entered the moraine’s outer edge at z \approx 2.2 (supercritical, M = 1.30), crossed through criticality at z \approx 1.588, and exited the inner edge at z \approx 0.5 (subcritical, M = 0.40). The DSW physics produces a broad upstream compression (the \rho_\Lambda enhancement, which races ahead) and a narrow downstream disruption (the G_\text{eff} suppression, which stays localized near the encounter exit). This is the same “beach” analogy: a wave washing up on shore piles up energy ahead (broad shoaling), then leaves a narrow wash zone behind as it recedes.
And like a moraine, the crust has internal structure — it is built from organized rotational energy, not thermal noise. The dc1 vortices from \mathcal{B}^{-1} that constitute the moraine were spinning at whatever the local v_\text{rot,outer} was at the time of the previous bubble’s relaxation — a fossil rotation rate from an earlier, higher-density epoch. That is why it can disrupt the substrate’s gravitational coherence, and why the disruption has a specific, calculable efficiency.
The moraine ripples: an undular bore in the data
The 15-knot freeform spline fit reveals the full oscillatory structure of the moraine — not a smooth lump, but a resolved transcritical undular bore with deep troughs that extend below the Volovik baseline. The spline, unconstrained by any DSW prior, converges to a pattern that maps cleanly onto the Grimshaw–Smyth–El–Hoefer framework for transcritical dispersive shock waves.
Ridges (moraine compression crests):
| z | Amplitude | Physical interpretation |
|---|---|---|
| 0.07 | +0.251 | Downstream carrier crest — nearest observer, DE-amplified |
| 0.23 | +0.704 | Downstream carrier crest — second-largest observed amplitude |
| 0.45 | +0.279 | Downstream carrier crest |
| 0.80 | +0.721 | Downstream carrier crest — largest observed amplitude |
| 1.30 | +0.198 | First downstream crest past recovery zone |
| 2.30 | +0.489 | Soliton edge — bubble-wall deposit |
Voids (moraine troughs — now resolved as negative excursions):
| z | Amplitude | Physical interpretation |
|---|---|---|
| 0.30 | -0.311 | Deep trough between downstream crests |
| 0.38 | -0.356 | Deep trough |
| 0.60 | -0.131 | Between crests |
| 1.00 | -0.111 | Between carrier crests |
| 1.60 | -0.089 | Recovery-zone node at z_\text{crit} — below baseline |
| 2.00 | -0.365 | Post-soliton rarefied wake |
The pattern has a rhythm — the alternating crests and troughs of a dispersive shock wave, cosmologically chirped by the expansion history and amplified at low redshift by the growing dark energy fraction. The 15-knot fit resolves this rhythm into five to six complete oscillation cycles downstream of the recovery zone, compared to the three or four that the 12-knot non-negative fit could capture.
What the ripples tell us about the physics
The 15-knot freeform spline provides three clean results that the earlier analysis missed:
Result 1: The soliton-plus-wake pair is the cleanest feature. The soliton peaks at z = 2.30 (+0.49) with a shoulder at z = 2.50 (+0.29), and the deep trough at z = 2.00 (-0.37) is the rarefied wake. The amplitude ratio (trough magnitude / soliton amplitude \approx 0.75) falls within the 60–80% range predicted by the forced KdV equation for the first trough behind a leading soliton in transcritical flow. This pair is the direct signature of the initial supersonic encounter — the bubble wall passing through the moraine at M = 1.30.
Result 2: The recovery zone is a deficit, not a zero. The 12-knot fit pegged z_\text{crit} at zero. The 15-knot fit resolves it at -0.09 — a slight deficit below the Volovik floor. This means the M = 1 transition extracted energy so thoroughly from the moraine at this location that it left a local depression in \rho_\Lambda. The GS framework predicts this: the hydraulic transition at M = 1 carries energy away in both directions, and the recovery zone is the energy-depleted center of that redistribution.
Result 3: The downstream bore is a chirped, DE-amplified DSW. The five to six cycles of the downstream train show both wavelength compression (toward the harmonic edge at low z) and amplitude modulation (from the growing \Omega_\Lambda fraction). Dividing out the DE amplification reveals a monotonically decreasing bare carrier — the rank-ordered structure of a standard DSW. This is the single most diagnostic feature: a smooth dark energy perturbation would not produce rank-ordered carrier amplitudes after demodulation.
The disruption efficiency: zero new parameters
The efficiency of the disruption is the result that ties everything together. The crust energy couples into the substrate’s gravitational response through the same HVBK mutual friction interface (\alpha_{mf}) that governs every boundary interaction in the framework — from the electroweak sector to the quantum potential. The disruption is a two-step process:
Crust energy couples into the counter-rotating boundary through mutual friction. Efficiency: \alpha_{mf}.
The coupled energy disrupts the boundary’s coherent gravitational response — the quadratic current-phase relation that produces the MOND field equation. Efficiency: \alpha_{mf} again.
Each step loses most of the energy to thermalization; only the fraction \alpha_{mf} passes through. The combined disruption is multiplicative — both steps must succeed — giving \alpha_{mf}^2. The factor of 2 comes from HVBK theory: at the substrate’s operating point (\alpha_{mf} = 0.3, intermediate between the low-temperature limit and the lambda point), both dissipative and reactive components of the mutual friction force contribute comparably.3
The result:
\eta_\text{crust} = 2\alpha_{mf}^2 = 2\left(\frac{\sin^2\theta_W}{1 - \sin^2\theta_W}\right)^2 = 0.181
This is 18% — only a fifth of the crust energy at peak actually disrupts the gravitational coherence. The rest passes through or thermalizes without affecting the quadratic current-phase relation. This is physically reasonable. The counter-rotating boundaries are robust topological structures, not fragile assemblies. It takes a precisely coupled perturbation to disrupt their phase coherence, and even then, most of the energy misses.
The modified gravitational coupling during the crust epoch is:
G_\text{eff}(z) = G \cdot \left[1 - \eta_\text{crust} \cdot \exp\!\left(-\frac{(z - z_\text{dip})^2}{2\sigma_\text{sup}^2}\right)\right]
with z_\text{dip} \approx 0.52 and \sigma_\text{sup} \approx 0.30 — a narrow suppression zone centered in the subcritical regime, where the downstream DSW disrupted boundary coherence most intensely. The profile decays as the counter-rotating layers re-cohere through mutual friction relaxation.
The chain that produces this number starts in a particle collider and ends in the large-scale distribution of galaxies:
\sin^2\theta_W = 0.2312 \;\xrightarrow{\text{C8}}\; \alpha_{mf} = 0.3008 \;\xrightarrow{2\alpha_{mf}^2}\; \eta_\text{crust} = 0.181 \;\xrightarrow{f(z),\, G_\text{eff}}\; S_8 = 0.817
The Weinberg angle — measured in e^+e^- collisions — determines how much the previous cycle’s moraine suppressed galaxy formation in this one. Zero new parameters.
The \eta_\text{crust} = 2\alpha_{mf}^2 scaling is the natural HVBK answer for a two-step mutual friction process, but the argument is currently heuristic — not derived from a full turbulence calculation of the crust-boundary interaction. The growth calculation uses the linearized growth equation with a modified G_\text{eff}; the MOND field equation’s nonlinearity makes the full perturbation theory more subtle (see WIP-20). The current S_8 = 0.817 relaxes the tension with weak lensing surveys from \sim 3\sigma to \sim 1–2\sigma but does not fully resolve it and must be recomputed with the 15-knot undular-bore f(z) profile. The most likely sources of additional suppression are: (a) the undular-bore profile structure, which now places six carrier crests at different redshifts than the smooth envelope; (b) the nonlinear MOND growth calculation; and (c) full Friedmann self-consistency with both f(z) and G_\text{eff}(z).
The Jia H_0(z) descent
Here is the summary info from the observations from the Jia et al. (2025) DESI DR2 binned reconstruction of H_0(z).4
| Bin | z_\text{mid} | H_0 (km/s/Mpc) |
|---|---|---|
| 1 | 0.1 | 72.20 \pm 0.19 |
| 2 | 0.3 | 71.62 \pm 0.36 |
| 3 | 0.5 | 69.78 \pm 0.51 |
| 4 | 0.7 | 68.13 \pm 0.67 |
| 5 | 2.5 | 67.23 \pm 0.84 |
This is a clean, monotonic descent from \sim 72 to \sim 67 km/s/Mpc — exactly what the moraine crust model predicts, and for a specific physical reason.
The \rho_\Lambda enhancement is distributed across the undular bore’s carrier crests (z \approx 0.07 to 2.30), far from the low-redshift bins where it has the most leverage. But dark energy is a fraction of the total energy density, and that fraction varies enormously with redshift. At z = 0.1, the dark energy fraction \Omega_\Lambda / E^2 \approx 0.62 — dark energy dominates. At z = 2.5, it is negligible. So the crust enhancement, though broadly centered at z \approx 0.8–1.6, has its maximum leverage on H_0 at low z, where dark energy is a large fraction of the total.
The result: the crust enhancement naturally produces a descending H_0(z) because the enhancement’s leverage decreases monotonically with redshift as matter comes to dominate. At z = 0.1, the effective boost is \sim 4\% in H, which is exactly the 4.8 km/s/Mpc elevation. At z = 2.5, the boost is negligible, and H_0 returns to the Planck value. The smooth, monotonic descent is a prediction, not a fit.
The Jia data’s monotonic descent is more natural for the DSW model than the non-monotonic shape seen in earlier GP-regression reconstructions (e.g., Wu et al. 2025).
The freeform spline fit greatly inproves the Jia fit achieving \chi^2_\text{Jia} = 4.49 across 5 bins with H_0(\text{local}) = 71.80 km/s/Mpc — a substantial improvement. The per-bin residuals are:
| z | Observed | \sigma | Predicted | d/\sigma |
|---|---|---|---|---|
| 0.10 | 72.20 | 0.19 | 72.20 | -0.00 |
| 0.30 | 71.62 | 0.36 | 71.06 | -1.55 |
| 0.50 | 69.78 | 0.51 | 69.49 | -0.56 |
| 0.70 | 68.13 | 0.67 | 68.89 | +1.13 |
| 2.50 | 67.23 | 0.84 | 67.82 | +0.70 |
The residual at z = 0.3 (-1.55\sigma) is the largest, and it sits precisely at one of the undular bore’s voids — the freeform spline places a deep trough there (-0.31). The smooth-envelope model had trouble at this redshift because it predicted a monotonically declining f(z) with no structure at z = 0.3. The undular bore’s rhythmic crest-trough pattern naturally introduces the fine structure that the Jia data hints at. Improved binning in the z = 0.3–0.7 range from DESI DR3 would directly test whether this trough is real — a sharp discriminant between the smooth and structured models.
The biggest remaining tension is a direct conflict between one DESI 2 observation and the middle Jia bin at z \approx 0.5. The 15-knot spline shows this is right where the structure is most complex: knots 4–6 span z = 0.30 to 0.45 with a sharp void-to-ridge transition (-0.31 at z = 0.30, -0.36 at z = 0.38, +0.28 at z = 0.45). A BAO measurement centered at z = 0.5 would be averaging over this steep gradient, and the effective redshift of the measurement could shift the comparison value significantly. Additional knot coverage near z = 0.5 may help, but the fundamental issue is that the spline is trying to represent a rapid oscillation near the Nyquist limit of the current knot spacing.
The connection to the DESI dark energy anomaly is direct: Jia’s Equation (19) shows that if w(z) evolves as the data suggests, then H_0(z) naturally descends. The substrate framework goes one step further — it explains why w(z) evolves (the moraine crust undular bore encounter) and predicts the structural features from M(z) = 1 at z = 1.588.
DESI reference data
The fit uses the DESI DR2 + CMB combined constraints:5
| Quantity | Value |
|---|---|
| w_0 | -0.42 \pm 0.21 |
| w_a | -1.75 \pm 0.58 |
| Correlation \rho_{w_0, w_a} | -0.85 |
| \Omega_m | 0.315 |
The error ellipse in the w_0–w_a plane is rotated by \sim 17° from the w_0 axis, with semi-axes (0.160, 0.920) and (0.262, 1.508) for the 1\sigma and 2\sigma contours respectively.
Open calculations
Derive z_b from \tau_\text{relax}. Compute \tau_\text{relax} from C7 and the substrate viscous parameters, then solve H(z_b) \cdot \tau_\text{relax} = 1 for z_b. If z_b \approx 2.2 falls out, the background is fully determined with zero free parameters. The soliton peak at z = 2.30 with shoulder at z = 2.50 is consistent with a sech² profile centered between 2.20 and 2.30.
GS-structured refit of the undular bore. The 15-knot freeform spline provides the target shape; the next step is to parameterize the undular bore with physics-derived variables: smooth envelope − recovery-zone Gaussian + carrier-wave modulation, with six parameters (B, \gamma, z_\text{harm}, A_R, w_R, \phi_0). The amplitude demodulation constrains \gamma \approx 2.5–3.5, and the slightly negative recovery-zone node constrains A_R to slightly exceed the envelope at z_\text{crit}. The target is \chi^2_\text{total} \lesssim 12 to beat the freeform on AIC. The GS-structured form must reproduce the soliton-plus-wake pair at z_b, the negative node at z_\text{crit}, the chirped downstream carrier crests, and the rank-ordered bare carrier after DE demodulation.
GP-dispersion wavelength check. The 15-knot freeform spline provides six ridge positions (z = 0.07, 0.23, 0.45, 0.80, 1.30, 2.30) with ridge-to-ridge spacings in proper distance ranging from $$640 Mpc to $$1638 Mpc — decreasing toward low z, consistent with the wavenumber increase expected from soliton edge to harmonic edge. Computing the predicted local wavenumber k(z) from the substrate’s GP dispersion relation at the local Mach number and comparing to these five measured spacings is the bridge from microscopic coherence length (\xi \sim 100\;\mum) to Mpc-scale moraine ripples. If the predicted k(z) profile matches the observed wavelength compression, that is a publishable headline result.
DSW-to-observable coupling: B from F_m. The enhancement amplitude B is the genuinely free parameter. It is set by F_m — the peak amplitude of the moraine forcing in the Grimshaw-Smyth transcritical framework — through the coupling between DSW wave amplitude and \rho_\Lambda compression. At criticality, the Grimshaw-Smyth amplitudes are |A_\pm| = \sqrt{F_m/3}. The asymmetry between enhancement amplitude (B) and suppression efficiency (\eta) does NOT come from A_-/A_+ splitting (these are equal at \Delta = 0) but from different coupling channels: direct compression for \rho_\Lambda versus \alpha_{mf}^2-filtered disruption for G_\text{eff}. Deriving the \rho_\Lambda coupling would eliminate the last free parameter.
HVBK recovery timescale. The suppression width (\sigma_\text{sup} \approx 0.30) is a boundary recovery timescale set by HVBK mutual friction dynamics, probably \tau \sim 1/(\alpha_{mf}\,\omega_0). This is a separate calculation from the DSW propagation widths. The ratio \sigma_\text{enh}/\sigma_\text{sup} \approx 3.3 is the ratio of the DSW propagation timescale to the HVBK relaxation timescale — a meaningful physical prediction requiring both calculations.
Predict f(z) shape for future surveys. The density profile f(z) now has a distinctive chirped undular bore morphology — soliton-plus-wake pair at z_b, recovery-zone node at z_\text{crit}, downstream carrier crests at z \approx 1.30, 0.80, 0.45, 0.23, 0.07 with wavelength compression toward the harmonic edge — that differs sharply from CPL, any polynomial dark energy model, or even the smooth DSW envelope. Euclid and Roman will measure w(z) at higher resolution — the model predicts specific oscillatory deviations from CPL that are testable. The amplitude demodulation figure — showing rank-ordered bare carrier amplitudes after dividing out [\Omega_\Lambda(z)]^\gamma — would be the single most diagnostic test: it distinguishes a cosmologically amplified DSW from any smooth dark energy model.
Check consistency with CMB. The high-redshift behavior f(z) \to 0 for C = 1 affects the integrated Sachs-Wolfe effect and the late-time ISW signal. Verify that C = 1 is consistent with Planck CMB constraints independently of the BAO data.
Connect to a_0(z) evolution. The MOND scale a_0(z) = a_0(0)(1+z)^{3/2} from Galactic Dynamics is affected by the crust — the G_\text{eff} suppression at z \approx 0.5 predicts a \sim 7–9\% dip in the MOND acceleration scale at that epoch. This is a cross-domain prediction connecting galactic dynamics to cosmology through the crust.
Full nonlinear MOND growth calculation (WIP-20). The linearized growth calculation with a modified G_\text{eff} is a placeholder. The MOND field equation is nonlinear (|\nabla\Phi|\nabla\Phi), and the time-dependent MOND scale a_0(z) = a_0(0)(1+z)^{3/2} means that structure-forming regions in the MOND regime today may have been deeper in the MOND regime during the crust epoch. A full calculation requires solving the cosmological perturbation equations with the nonlinear Poisson equation, time-dependent a_0(z), and self-consistent H(z).
Verify the 2\alpha_{mf}^2 scaling from first principles (WIP-20). The heuristic argument (two-step coupling through mutual friction, factor of 2 from HVBK dissipative + reactive) should be confirmed by a full HVBK turbulence calculation of the crust-boundary interaction. The key question is whether the reactive coefficient \alpha_{mf}' at the substrate’s operating point truly contributes comparably to the dissipative coefficient \alpha_{mf}.
Full Friedmann self-consistency (WIP-18, WIP-21). Solve the Friedmann equations with \rho_\Lambda(z) = \rho_\Lambda(0) \cdot f(z) and G_\text{eff}(z) simultaneously, and verify consistency with DESI BAO distance measurements. The 15-knot undular-bore profiles for both f(z) and G_\text{eff}(z) should be used, with the M(z) = 1 prediction anchoring the recovery-zone node at z_\text{crit}.
Refine the Jia H_0(z) fit with undular bore. The freeform spline achieves \chi^2_\text{Jia} = 4.49/5 bins with H_0(\text{local}) = 71.80. The remaining -1.55\sigma residual at z = 0.3 is the tightest constraint. The DESI/Jia tension at z \approx 0.5 — where the spline structure is most complex — requires careful treatment of the effective redshift for BAO measurements averaging over a steep gradient. Improved binning in the DESI DR3 data at z = 0.3–0.7 would directly test the rhythmic crest-trough pattern predicted by the downstream carrier wave.
DSW propagation widths from Whitham modulation theory. The Gaussian profiles for the enhancement (\sigma_\text{enh} \approx 1.0) and suppression (\sigma_\text{sup} \approx 0.30) are placeholders. The DSW fitting method (El & Hoefer, Sec. 4.1) provides the edge speeds and amplitudes for the substrate’s pressure law P \propto \rho^{5/3} without solving the full modulation equations. With the 15-knot undular bore structure now fully resolved — including the chirped wavelength compression and the rank-ordered bare carrier — the Whitham calculation should target the carrier wavelength profile \lambda(z) and the soliton-plus-wake pair shape, not just the smooth envelope.
Footnotes
DESI Collaboration, “DESI 2024 VI: Cosmological Constraints from Baryon Acoustic Oscillations,” arXiv:2404.03002, 2024. Updated in DR2, 2025.↩︎
El, G.A. & Hoefer, M.A., “Dispersive shock waves and modulation theory,” Physica D 333, 11–65, 2016.↩︎
In the HVBK formalism, the mutual friction force has dissipative (\alpha_{mf}) and reactive (\alpha_{mf}') components. At the substrate’s operating temperature, both channels are active, giving a total efficiency 2\alpha_{mf}^2. See Hall & Vinen (1956), Bekarevich & Khalatnikov (1961).↩︎
Jia et al. (2025), ApJL 994 L22, Table 2. Using DESI+PP calibration.↩︎
DESI Collaboration (2025), DR2 combined analysis.↩︎