The Substrate’s Equation of State
Why a stiff superfluid still expands, nucleosynthesizes, and clusters like cold dark matter
How can the substrate be both dark matter with, n_1 m_1 = \rho_\text{DM}, and at the same time behave like a vacuum. It’s one condensate playing both roles (Substrate Particles). And the substrate is stiff: P = \rho c^2, sound speed c_s = c, the barotropic relation that hands Gravity its exact Painlevé–Gullstrand metric and hands the linearized theory its factor of 4. This means a w = 1 stiff fluid redshifts as a^{-6}, dominates the early universe, wrecks Big-Bang nucleosynthesis and the microwave background, and — with a sound speed at c giving a horizon-scale Jeans length — cannot cluster into halos at all. Most of all how does it explain the cosmological background.
The “P” above comes from three different quantities, none is a w = 1 background.
Three pressures, not one
The single symbol P has been standing for three physically distinct pressures:
| What it is | Governs | EOS parameter | Speed | |
|---|---|---|---|---|
| 1. Background \bar P | the coarse-grained, volume-averaged pressure the bulk condensate presents to the Einstein equations | Friedmann H(z), BBN, CMB peak positions | w_\text{bg} = \bar P/\bar\rho c^2 \approx 0 | c_{s,\text{bg}} \approx 0 |
| 2. Acoustic-metric \delta P/\delta\rho | the slope of the EOS — how a ripple’s pressure responds to its density, the analog-gravity “light cone” | local GR tests, the factor of 4, the PG metric | \delta P/\delta\rho = c^2 (stiff) | c_s = c |
| 3. Clustering | the medium’s response to over-density: instability, not restoring pressure | halo seeding, JWST-early structure | tachyonic (\omega^2 < 0) above n_\text{tr} | growth, \lambda \ll horizon |
The curious observation of the a^{-6} redshift comes from the collapse of rows 1 and 2 into a single “w = 1.” They are not the same number. A stiff fluid dilutes as a^{-6} only when its background value \bar P equals its energy density \bar\rho c^2 — that is, when w_\text{bg} = +1, a genuinely relativistic degenerate state. The substrate’s stiffness comes from row 2, the slope \delta P/\delta\rho = c^2, not in the value \bar P. And the value is pinned near zero.
The background: Gibbs–Duhem pins \bar P \to 0
Why is \bar P \approx 0 rather than \bar\rho c^2? Because the substrate self-tunes to the marginal point. In complete thermodynamic equilibrium a superfluid vacuum obeys the Volovik/Gibbs–Duhem identity (Gravity § The Cosmological Constant Problem)
\varepsilon + P = 0,
and at the marginal density n_\text{tr} the two are not merely equal-and-opposite but individually driven to zero: any excess pressure changes the density, which changes the chemical potential, which drives a flow that relaxes the pressure away. The substrate does not sit at P = \rho c^2 as a background state; it self-organizes to the density where the coarse-grained pressure vanishes — the same \mu \to 0 marginal point that makes light massless (Substrate Particles § The Marginal Point).
What survives at that point is the slope, not the value. This is exactly the property that needs the logarithmic equation of state: where the sound speed is density-independent giving \rho\,F'(\rho) = \text{const}, and the solution is the logarithm, whose slope \delta P/\delta\rho = c^2 holds everywhere while its value \bar P passes through zero at n_\text{tr}. A linear stiff fluid P = \rho c^2 cannot do this — its value and slope are locked together. The log EOS is precisely the equation of state that decouples them: stiff to ripples, pressureless to Friedmann.
Concretely, the coarse-grained stress-energy the substrate presents to cosmology is
\bar\rho\,c^2 = \underbrace{n_1 m_1 c^2}_{\text{rest mass} \;=\; \rho_\text{DM}c^2,\; w=0} \;+\; \underbrace{u(\rho)}_{\text{internal (log) energy} \;=\; \rho_\Lambda c^2,\; w\approx-1},
the two terms diluting as a^{-3} and \approx const respectively. There is no a^{-6} term because there is no w = +1 term. The substrate presents to Friedmann as cold dark matter plus a cosmological-constant-scale residual — which is the entire observed dark sector, and nothing else.
The microphysics underneath comes from Chavanis’s logotropic model, a unified dark fluid with a single component whose rest-mass energy plays dark matter and whose internal energy plays dark energy under a logarithmic EOS P = A\ln(\rho/\rho_P), whose background is indistinguishable from \LambdaCDM up to the present (DESI § the logotropic comparison). The substrate is a logotrope with a condensate underneath it. Chavanis showed such a fluid passes BBN and the CMB.
The 0.776c is spin, not wind
With the sound speed near c how do vortex cores with rotational velocity spinning at 0.776\,c avoid kinetic pressure? The reason is that the 0.776\,c is the internal circulation, not the bulk peculiar velocity.
The effective quantum orbits at v_\text{rot,inner} = c\sqrt{2\alpha_{mf}} = 0.776\,c inside a 150 fm vortex core (Substrate Particles § Inner Scale). That circulation is topologically bound — it is the very thing the framework calls mass: “mass is leaking rotational kinetic energy” (Mass as Rotational Energy). In the coarse-grained stress-energy it contributes to the energy density \rho c^2, exactly where rest mass belongs, and not to a bulk translational pressure. The bulk condensate is at rest: \langle \mathbf{v}_\text{bulk}\rangle = 0, its rest-frame density is \rho_\text{DM}.
A gas of fast-spinning tops, each pinned in place, is pressureless dust no matter how fast the tops spin — the spin is rest energy, the wind is zero. That is the substrate. The kinetic pressure that would source w = 1 requires a bulk velocity dispersion \langle v_\text{bulk}^2\rangle, and the dc1 sea is a zero-temperature condensate (\lambda_{dB} \gg n_1^{-1/3}, a BEC — Substrate Particles): its bulk velocity dispersion is the tiny condensate quantum-pressure term, not a thermal \sim c spread. Cold, in the precise cosmological sense.
Clustering: an instability replaces the Jeans length
And how does a sound speed c_s = c give a Jeans length \lambda_J \sim c_s/\sqrt{G\bar\rho} \sim c/H \sim the horizon, so the medium is pressure-supported on every sub-horizon scale and can never collapse into halos.
This assumes clustering is a contest between self-gravity and a restoring pressure — the standard Jeans picture, where a large c_s wins and forbids collapse. In the log substrate that contest does not decide structure formation, because above the marginal density the medium has no restoring pressure at all — it has a growing mode. The Bogoliubov spectrum factorizes,
\epsilon(p) = \sqrt{\left(\tfrac{p^2}{2m_1} - \epsilon_1\right)\left(\tfrac{p^2}{2m_1} - \epsilon_2\right)}, \qquad \epsilon_2 = \epsilon_1 + 2\beta^{-1},
and for n_0 > n_\text{tr} the low-momentum branch turns tachyonic — \epsilon^2 < 0, \omega imaginary, exponential growth (Substrate Particles § The Marginal Point; Early Structure Formation § The Deeper Engine). The over-dense uniform medium is already unstable on a band of scales and must break into self-bound clumps; the breakup of the over-density is structure formation, driven by the condensate’s own instability, not waiting on gravity to overcome a horizon-scale pressure.
The scale where this matters most is the unstable band that runs from the healing length \xi \sim 100\;\mum up to a longest unstable wavelength set by how far the background sits above n_\text{tr} — and at the epoch of the first galaxies the mean density sat \sim 10^3 (at recombination \sim 10^9) times above marginal (Early Structure Formation), opening the unstable band across an enormous range of sub-horizon scales. So the clustering length is not the horizon; it is microphysically small and everything larger than it grows. The stiff phonon at c sets the light cone, an analog-metric statement about how ripples and radiation propagate — it never set the clustering length, and mistaking one for the other is the whole of the Jeans objection.
Today the mean density has relaxed down onto n_\text{tr} (the late-time attractor — which is why we see one sharp c). There the tachyonic band closes and the coarse-grained hydrodynamic sound speed — the adiabatic c_{s,\text{bg}} a cosmologist would insert for a clustering component — is the gentle logotrope value c_{s,\text{bg}}^2 = A/\rho \ll c^2, not c. The luminal c is the Bogoliubov phonon (the photon’s cousin), a different mode of the same condensate from the hydrodynamic clustering mode. One medium, two sound speeds: c for the analog metric, \approx 0 for gravitational clustering — exactly the pair cold dark matter needs.
The factor of 4
The factor of 4 remains a local, self-gravitation, not the factor for the background. The linearized theory’s \rho_\text{eff} = \rho + 3P/c^2 = 4\rho uses the acoustic-metric EOS of row 2 — the substrate’s response to a mass embedded in it: the factor \rho_\text{eff} = 4\rho applies to the substrate’s own self-gravitation (P = \rho c^2); matter sources embedded in the substrate have their own equations of state — radiation \rho_\text{eff} = 2\rho, dust \rho_\text{eff} = \rho (Factor of 4). Rows 1 and 2 are that same kinematics/dynamics split (BLV 2005) read at cosmological scale: the acoustic metric (row 2, stiff) gives the local light cone and the factor of 4; the coarse-grained source (row 1, cold) gives the expansion history. The substrate can be stiff to a black hole’s infalling ripple and cold to the Hubble flow at the same time for the same reason it is stiff in slope and soft in value — the log EOS decouples them.
What BBN and the CMB actually see
With the background established as cold (w_\text{bg} \approx 0) rest-mass-dominated dark matter plus a \rho_\Lambda-scale residual, the two classic early-universe tests are one sentence each.
Big-Bang nucleosynthesis. The substrate contributes as a non-relativistic condensate, not as radiation: it is a zero-temperature BEC, so it adds no relativistic degrees of freedom, \Delta N_\text{eff} \approx 0, and the expansion rate at T \sim 1 MeV is the standard radiation-dominated one. There is no a^{-6} component to overwhelm radiation at early times — that term does not exist in row 1 — so the light-element abundances are untouched.
CMB acoustic peaks. The peaks require a component with w \approx 0, c_s \approx 0 to seed the potential wells the baryon–photon fluid oscillates in. The substrate is exactly that: rest-mass-dominated, pressureless, and (being \sim 10^9 above marginal at z = 1100) aggressively clustering on sub-horizon scales. The internal log-energy / dark-energy piece is \sim \rho_\Lambda/\rho_\text{DM}(z{=}1100) \sim 10^{-9} of the density then, negligible for the acoustic physics. So the peak positions and heights are set by the standard baryon–photon physics in standard CDM wells.
What is derived here is the sign-and-scale statement: the substrate presents to Friedmann as w_\text{bg} \approx 0 CDM plus a \rho_\Lambda residual, with no w = 1 background and hence no a^{-6} catastrophe, and it clusters by intrinsic instability rather than despite a horizon-scale Jeans length. What is not yet computed is the precision CMB C_\ell spectrum with the substrate’s modified (instability-driven) growth rate substituted for linear CDM growth — the existing open item (Early Structure Formation, Open Calculation 6) — and the growth rate of the tachyonic band against cosmological expansion (Open Calculation 5). The claim of this chapter is the weaker, load-bearing one: the equation of state is not contradictory, and the numbers it presents to cosmology are cold, not stiff.
Summary: the numbers the substrate hands to Friedmann
- Background gravitating EOS: w_\text{bg} \approx 0 (rest-mass, clustering component, exact for the dust part) plus a \rho_\Lambda-scale internal-energy piece with w \approx -1. No w = +1 term, no a^{-6}. The dark sector is this one fluid.
- Background sound speed (clustering): c_{s,\text{bg}} \approx 0 (adiabatic logotrope value \ll c; tachyonic — a growing mode — when over-dense). This is what builds halos and CMB wells.
- Acoustic-metric EOS (perturbation slope): \delta P/\delta\rho = c^2, c_s = c for the Bogoliubov phonon. This is the “stiff P = \rho c^2” — the analog-gravity light cone that gives the PG metric and the factor of 4. It is a statement about ripples, not the expansion history.
- The 0.776c is internal vortex circulation (rest-mass energy), not bulk peculiar velocity, and does not enter the coarse-grained stress-energy as kinetic pressure.
The apparent contradiction dissolves the moment the three P’s are held apart: the substrate is stiff in the slope of its equation of state and soft in the value, and the logarithmic EOS is precisely the equation of state that makes those two independent. It is luminal to a photon, cold to the Hubble flow, and unstable to a halo — all at once, and without tuning.