Early Structure Formation

Why JWST’s ‘Impossible’ Galaxies Are Expected

The Puzzle

The James Webb Space Telescope’s first deep surveys revealed something that standard cosmology did not predict: galaxies at redshifts z > 10 that are too massive, too luminous, and too numerous for \LambdaCDM structure formation models.1 In a universe where gravity works the same way at all epochs and dark matter halos grow by hierarchical merging, there simply isn’t enough time in the first 400 million years to assemble 10^9\,M_\odot galaxies with high star-formation rates and blue UV colors.2

Maisie’s Galaxy — spectroscopically confirmed at z = 11.4, seen just 390 million years after the Big Bang — exemplifies the problem.3 With stellar mass \sim 10^{8.5}\,M_\odot and vigorous star formation (\log\,\text{sSFR} \sim -8.2\;\text{yr}^{-1}), it formed its stars over a period of only 30–120 million years. Standard models struggle to explain how this happened so fast.

The substrate framework resolves this naturally. It predicts that gravity was more effective at high redshift — not because G changed, but because the MOND acceleration scale a_0 was larger in the denser early substrate. No new parameters, no modifications. It falls directly from the boundary parity mechanism derived in Galactic Dynamics.

The Core Prediction

From Galactic Dynamics, the MOND acceleration scale is set by the substrate density:

a_0 = c\sqrt{G\,\rho_\text{DM}}

The substrate density dilutes with cosmic expansion like any pressureless component:

\rho_\text{DM}(z) = \rho_\text{DM}(0)\,(1+z)^3

Therefore:

\boxed{a_0(z) = a_0(0)\,(1+z)^{3/2}}

This is not a tuning or an assumption — it is an algebraic consequence of combining two established results. The boundary parity mechanism that produces the quadratic current-phase relation does not change with redshift; only the substrate density changes.

At key epochs:

Epoch Redshift z a_0(z)/a_0(0) a_0(z) [m/s²] Age of universe
Today 0 1.0 1.2 \times 10^{-10} 13.8 Gyr
Low-z surveys 0.5 1.84 2.2 \times 10^{-10} 8.6 Gyr
DESI/Euclid reach 1.0 2.83 3.4 \times 10^{-10} 5.9 Gyr
Cluster-scale transition 2.0 5.20 6.2 \times 10^{-10} 3.3 Gyr
JWST galaxies 8.0 27.0 3.2 \times 10^{-9} 0.64 Gyr
Maisie’s Galaxy 11.4 43.7 5.2 \times 10^{-9} 0.39 Gyr
Highest JWST candidates 15 64.0 7.7 \times 10^{-9} 0.27 Gyr

At z = 11.4, the acceleration scale below which the substrate responds coherently was 44 times higher than today.

a₀(z) = a₀(0)·(1+z)3/2 (substrate) Standard MOND (constant a₀) Key epochs

Physical interpretation

The acceleration a_0 marks the boundary between two regimes of gravitational response in the substrate (Galactic Dynamics, GD7):

  • Above a_0 (a \gg a_0): Gravitational perturbations evolve faster than the substrate’s free-fall time t_\text{ff} = 1/\sqrt{G\rho_\text{DM}}. Each boundary scatters the ebbing current independently. Response is incoherent — Newtonian gravity.

  • Below a_0 (a \ll a_0): Perturbations evolve slowly enough for the substrate to respond coherently. The parity-symmetric quadratic CPR produces enhanced gravitational acceleration: g_\text{eff} = \sqrt{a_0 \cdot g_N}, which exceeds g_N by the factor \sqrt{a_0/g_N}.

At high redshift, with a_0 much larger, the coherent regime extends to much higher accelerations. Structures that would be in the Newtonian regime today were deep in the MOND regime during the epoch of first galaxy formation.

Enhanced Gravitational Collapse

The MOND radius shrinks at high redshift

For a baryonic mass M_b, the Newtonian gravitational acceleration at distance r is g_N = GM_b/r^2. The MOND transition occurs at the radius where g_N = a_0:

r_M = \sqrt{\frac{GM_b}{a_0}}

This is the MOND radius — beyond it, gravity is enhanced; within it, gravity is Newtonian. At redshift z:

r_M(z) = \sqrt{\frac{GM_b}{a_0(z)}} = \frac{r_M(0)}{(1+z)^{3/4}}

At z = 11.4: r_M(z) = r_M(0) / 6.6. The MOND enhancement extends 6.6 times closer to the center of a collapsing structure than it does today.

For a protogalactic cloud of M_b = 10^9\,M_\odot:

Epoch r_M r_M (physical)
Today \sqrt{GM_b/a_0(0)} \sim 1.1 kpc
z = 11.4 r_M(0)/6.6 \sim 0.17 kpc

At z = 11.4, the transition radius is only \sim 170 pc — meaning that essentially the entire protogalactic cloud (which extends over several kpc) is in the MOND-enhanced regime. Today, only the outer regions beyond \sim 1 kpc would experience MOND enhancement for the same mass. The early universe was deep in the coherent substrate response across the full spatial extent of forming galaxies.

Gravitational enhancement factor

In the deep-MOND regime, the effective gravitational acceleration is:

g_\text{eff} = \sqrt{a_0 \cdot g_N}

The enhancement factor over Newtonian gravity is:

\frac{g_\text{eff}}{g_N} = \sqrt{\frac{a_0}{g_N}}

Consider a protogalactic gas cloud at the epoch of Maisie’s Galaxy (z = 11.4) with baryonic mass M_b \sim 10^9\,M_\odot at radius R \sim 3 kpc (a characteristic scale for a forming protogalaxy):

g_N = \frac{GM_b}{R^2} \approx 1.5 \times 10^{-11}\;\text{m/s}^2

At z = 0: \;a_0/g_N = 8.0, so g_\text{eff}/g_N = 2.8.

At z = 11.4: \;a_0(z)/g_N = 350, so g_\text{eff}/g_N = 18.7.

The same cloud experiences 6.6× stronger gravitational enhancement at z = 11.4 compared to z = 0. This factor — \sqrt{a_0(z)/a_0(0)} = (1+z)^{3/4} — measures the increased efficiency of gravitational collapse in the early universe.

The MOND Jeans mass

The Jeans mass sets the minimum mass for gravitational collapse. In the Newtonian regime:

M_{J,N} \propto \frac{c_s^3}{G^{3/2}\,\rho_b^{1/2}}

In the deep-MOND regime, the enhanced gravity reduces the Jeans mass.4 The effective gravitational acceleration at the Jeans scale is g_\text{eff} = \sqrt{a_0 \cdot g_N}, which yields (for a uniform cloud at the Jeans boundary):

M_{J,\text{MOND}} \propto \frac{c_s^4}{G\,a_0}

The key difference: M_{J,\text{MOND}} depends on a_0, not on G^{3/2}\rho^{1/2}. At redshift z:

\frac{M_{J,\text{MOND}}(z)}{M_{J,\text{MOND}}(0)} = \frac{a_0(0)}{a_0(z)} = \frac{1}{(1+z)^{3/2}}

At z = 11.4:

M_{J,\text{MOND}}(z=11.4) = \frac{M_{J,\text{MOND}}(0)}{43.7}

The minimum mass for gravitational collapse is 44 times smaller at z = 11.4 than today. Many more structures cross the collapse threshold. Structures that would be too small to collapse under MOND-enhanced gravity today could collapse readily in the early universe.

⚠️ The MOND Jeans mass expression assumes the deep-MOND limit and a simplified uniform-density cloud. A full treatment requires the interpolation function \mu(a/a_0) at intermediate accelerations (open calculation from Galactic Dynamics) and the detailed density profile of the collapsing perturbation.

Collapse timescale

The free-fall time in the deep-MOND regime is:

t_\text{ff,MOND} \sim \frac{1}{(a_0 \cdot G\,\rho_b)^{1/4}}

This is shorter than the Newtonian free-fall time t_\text{ff,N} = 1/\sqrt{G\rho_b} by a factor of (G\rho_b/a_0)^{1/4} — which is less than unity for systems in the MOND regime.

At redshift z, with a_0(z) = a_0(0)(1+z)^{3/2} and \rho_b(z) = \rho_b(0)(1+z)^3:

t_\text{ff,MOND}(z) \propto \left[a_0(z) \cdot G\,\rho_b(z)\right]^{-1/4} \propto (1+z)^{-9/8}

At z = 11.4: t_\text{ff,MOND} = t_\text{ff,MOND}(0)\, / \,(12.4)^{9/8} \approx t_\text{ff,MOND}(0)\, /\, 17.

The collapse timescale is roughly 17 times shorter at z = 11.4 than it would be for the same baryonic cloud parameters today. Combined with the 44× reduced Jeans mass, this means the early universe was dramatically more efficient at forming structures.

⚠️ The (1+z)^{9/8} exponent includes both the a_0 evolution (contributing 3/8) and the baryonic density evolution (contributing 3/4). The Hubble expansion rate H(z) was also higher, partially counteracting the faster collapse. A full perturbation calculation (open calculation, below) must account for both.

Three Effects Working Together

The substrate framework predicts three reinforcing effects on early structure formation:

a₀(z) = a₀(0)·(1+z)³ᐟ² Higher substrate density at early times Lower Jeans mass 44× smaller at z = 11.4 Faster collapse ~17× at z = 11.4 Deeper potentials 6.6× stronger g_eff More structures collapse Assemble mass faster Higher star formation JWST "impossible" galaxies Massive, luminous galaxies at z > 10 Expected in the substrate framework

1. Lower MOND Jeans mass

More structures can collapse because the threshold mass is 44× smaller at z = 11.4. Baryonic gas clouds that would be gravitationally stable today were unstable in the early universe. This directly increases the number density of collapsed structures.

2. Faster collapse

Structures that DO collapse do so on shorter timescales (~17× faster at z = 11.4). This means galaxies can assemble more stellar mass in the limited time available — directly addressing the “too massive, too fast” complaint about JWST galaxies.

3. Deeper potential wells

The enhanced gravitational acceleration g_\text{eff} = \sqrt{a_0 \cdot g_N} creates deeper effective potential wells, leading to higher gas infall rates and higher star-formation efficiency. This explains the high specific star-formation rates observed in early JWST galaxies (\log\,\text{sSFR} \sim -8 yr^{-1}).

These three effects work in the same direction. None requires tuning. All follow from a_0(z) = a_0(0)(1+z)^{3/2}, which itself follows from the boundary parity mechanism operating in a substrate whose density dilutes with expansion.

Comparison with Standard Approaches

Feature \LambdaCDM Standard MOND Substrate framework
a_0 evolution N/A $a_0 = $ constant a_0(z) = a_0(0)(1+z)^{3/2}
Early galaxy abundance Underpredicts JWST by \sim 10\times Not computed (no cosmology) Enhanced (lower Jeans mass)
Mechanism Hierarchical merging in DM halos Modified gravity (fixed) Coherent substrate response (evolving)
Free parameters for this Star-formation efficiency (adjustable) None (but no prediction either) Zero
Predicted a_0 at z = 1 N/A 1.2 \times 10^{-10} 3.4 \times 10^{-10}

\LambdaCDM has responded to the JWST tension by invoking higher star-formation efficiency, reduced feedback, or modified initial mass functions. These are model adjustments, not predictions. Standard MOND has no cosmological framework and cannot address the question. The substrate’s prediction is parameter-free and was derivable from the galactic dynamics sector before JWST launched.

Connection to dark matter fraction observations

At high redshift, the larger a_0(z) extends the MOND enhancement to higher accelerations — meaning the phonon-mediated gravitational boost operates at smaller radii, deeper inside galaxies. Observers measuring rotation curves at z \sim 1–2 should see:

  • The radial acceleration relation (RAR) shifted to higher g_\text{bar} at the transition
  • An apparent increase in “dark matter fraction” at fixed physical radius (stronger enhancement)
  • The Tully-Fisher relation v^4 = a_0(z)\,G\,M_b with systematically higher normalization at high z

Some of these effects may already be present in existing data. Recent studies report that galaxies at z \sim 1–2 have rotation curves consistent with higher effective mass-to-light ratios — qualitatively consistent with larger a_0.

The a_0/(cH_0) Ratio as a Structural Prediction

The relation a_0 = c\sqrt{G\rho_\text{DM}} can be rewritten using the Friedmann equation H_0^2 = (8\pi G/3)\rho_\text{total}:

\frac{a_0}{cH_0} = \sqrt{\frac{3\,\Omega_\text{DM}}{8\pi}} = \sqrt{\frac{3 \times 0.264}{8\pi}} = 0.178

Measured: a_0/(cH_0) = (1.2 \times 10^{-10})/(3 \times 10^8 \times 2.2 \times 10^{-18}) = 0.182.

Match: ~2% — well within observational uncertainties on both a_0 and H_0.

This is sometimes called the “cosmic coincidence” because it relates a galactic-scale quantity (a_0) to a cosmological quantity (H_0). In \LambdaCDM, it has no explanation. In the substrate, it is an algebraic identity: both a_0 and H_0 are determined by \rho_\text{DM} and G, which are the same substrate parameters.

At redshift z, the relation generalizes:

\frac{a_0(z)}{c\,H(z)} = \sqrt{\frac{3\,\Omega_\text{DM}(z)}{8\pi}}

This is a structural prediction that can be tested epoch by epoch. In the matter-dominated era, \Omega_\text{DM}(z) is approximately constant (\sim 0.85 of the total), so a_0/(cH) remains \mathcal{O}(0.1–0.3) for all z \lesssim 3. The “cosmic coincidence” a_0 \approx cH_0/6 is not fine-tuned to the present epoch — it holds at every epoch when matter dominates. At each redshift, the ratio of the MOND scale to the Hubble scale should track the dark matter fraction — not because of fine-tuning, but because both quantities emerge from the same substrate.

Quantitative Estimate: Galaxy Luminosity Function

The abundance of galaxies above a mass threshold M_* is exponentially sensitive to the ratio M_*/M_J:

n(>M_*, z) \propto \exp\!\left(-\frac{\delta_c^2}{2\,\sigma^2(M_*, z)}\right)

where \delta_c \approx 1.686 is the collapse threshold and \sigma^2(M_*, z) is the variance of density fluctuations at mass scale M_*. In \LambdaCDM, \sigma^2 grows via linear perturbation theory. In the substrate, the MOND-enhanced gravity boosts the effective growth rate.

A rough estimate of the enhancement: in the deep-MOND regime, the effective G is replaced by G_\text{eff} = G\sqrt{a_0/g_N}. The perturbation growth rate scales as \sqrt{G_\text{eff}\,\rho}, which gives:

\frac{\sigma_\text{MOND}(z)}{\sigma_\text{CDM}(z)} \sim \left(\frac{a_0(z)}{g_N}\right)^{1/4}

Since the exponential depends on 1/\sigma^2, even a modest increase in \sigma leads to a large increase in abundance. For the marginal objects that are at the edge of \LambdaCDM predictions (which is exactly where the JWST tension lives), a 25–50% increase in \sigma can boost the predicted abundance by factors of 10–100 — easily resolving the tension.

Status: Scaling estimate

This is a scaling estimate. A proper calculation requires numerical integration of the perturbation equations with the MOND interpolation function and the full thermal history. This is the most important open calculation for this section.

Observational Tests

Test 1: RAR evolution with redshift (DESI/Euclid/SKA)

Prediction: The radial acceleration relation shifts with redshift. At z = 1, the transition acceleration g_\dagger should be \sim 2.8\times higher than today (3.4 \times 10^{-10} m/s² vs 1.2 \times 10^{-10} m/s²).

Method: Measure rotation curves of disk galaxies at z \sim 0.5–2 with spatially resolved spectroscopy. Plot g_\text{obs} vs g_\text{bar}. The transition point should shift systematically to higher g_\text{bar} with increasing z.

Distinguishing power: Standard MOND predicts no evolution of the RAR. \LambdaCDM predicts no universal RAR at all. The substrate predicts a specific, quantitative evolution with (1+z)^{3/2}.

Timeline: DESI spectroscopic galaxy survey (ongoing); Euclid weak lensing + spectroscopy (data taking); SKA HI surveys (next decade).

z = 0 (today) z = 1 (DESI/Euclid) z = 3 g_obs = g_bar (Newton)

Test 2: Tully-Fisher normalization evolution

Prediction: The baryonic Tully-Fisher relation v^4 = a_0(z)\,G\,M_b has a normalization that scales as (1+z)^{3/2}. At fixed baryonic mass, the flat rotation velocity should be higher at high redshift:

v_\text{flat}(z) = v_\text{flat}(0) \times (1+z)^{3/8}

At z = 1: v_\text{flat} should be \sim 30\% higher than for the same-mass galaxy today.

Method: Compare Tully-Fisher relations at different redshift bins. Molecular gas observations (ALMA, NOEMA) provide M_b; spatially resolved kinematics (JWST NIRSpec IFU, VLT/MUSE) provide v_\text{flat}.

Test 3: Galaxy stellar mass function at z > 8

Prediction: The substrate predicts more massive galaxies at z > 8 than \LambdaCDM, with the excess growing with redshift. The MOND Jeans mass scales as (1+z)^{-3/2}, so the number of collapsed structures above any mass threshold increases dramatically.

Quantitative target: At z \sim 10, the substrate predicts \sim 10–100\times more galaxies above M_* = 10^9\,M_\odot than the \LambdaCDM halo mass function permits (pending full perturbation calculation). This is the range of the observed JWST excess.

Method: Continued JWST deep field surveys with spectroscopic confirmation. Galaxy UV luminosity functions at z > 10.

Test 4: Void-galaxy environmental dependence

Prediction: Galaxies in void environments experience a slightly different a_0 (lower local \rho_\text{DM}) and should show a slightly shifted RAR compared to cluster galaxies at the same redshift. The offset is proportional to the local density contrast: \delta a_0/a_0 = \delta\rho_\text{DM}/(2\rho_\text{DM}).

Method: Compare rotation curves of void galaxies vs filament galaxies at matched baryonic mass and redshift.

Test 5: Cosmic dawn 21 cm signal

Prediction: The enhanced gravitational collapse at z > 15 should produce earlier and more vigorous star formation, affecting the 21 cm signal from the cosmic dawn. The substrate predicts a stronger absorption trough at higher redshift than \LambdaCDM models.

Method: HERA, SKA-Low, lunar far-side radio telescopes.

Open Calculations

Required to sharpen the prediction

  1. Full MOND perturbation theory with evolving a_0(z). Integrate the coupled baryon-substrate perturbation equations through the matter-dominated era with a_0 \propto (1+z)^{3/2}. This will produce a quantitative prediction for the galaxy stellar mass function at each redshift, directly comparable to JWST data. Priority: high.

  2. MOND interpolation function at high z. The interpolation between Newtonian and MOND regimes is \mu(a/a_0). At high z with larger a_0, more systems are in the interpolation zone. The specific form of \mu matters for precision predictions. From Galactic Dynamics: McGaugh’s Bose-Einstein empirical form suggests quantum statistics of boundary transmission — this should be derivable from the HVBK equations.

  3. Nonlinear collapse in the MOND regime. The Jeans analysis is linear. Actual galaxy formation involves nonlinear collapse, fragmentation, and feedback. Adapting existing MOND N-body codes (e.g., POrtable Remeshing for Particle Methods) to include evolving a_0(z) would provide direct comparison to observed galaxy demographics.

  4. Baryon fraction evolution. In the substrate framework, “dark matter” is the substrate itself, not a separate particle species. The baryon-to-total-mass ratio within collapsed structures depends on the phonon-mediated force profile. Computing this profile at different a_0(z) values would predict how the inferred dark matter fraction evolves with redshift.

Required for broader cosmological consistency

  1. CMB power spectrum with evolving a_0. The CMB was emitted at z \approx 1100, where a_0(z) \approx 4.3 \times 10^4 \times a_0(0). At this epoch, essentially ALL baryonic perturbations are in the deep-MOND regime. This would significantly affect the acoustic oscillation pattern — but the substrate framework already derives the correct CMB spectrum through the standard baryon-photon physics (the MOND-like enhancement applies to the gravitational sector, not the photon-baryon coupling). A careful calculation must verify that the modified gravitational growth rate is consistent with the observed CMB C_l spectrum.

  2. Reconciliation with cluster phenomenology. Galaxy clusters (v_\text{disp} > v_L \approx 750 km/s) are in the incoherent (Newtonian) phase. At high redshift, the Landau critical velocity v_L = v_\text{rot,outer} = \omega_0\xi does not evolve (it depends on substrate parameters that are constant), so the galaxy/cluster separation should be the same at all redshifts. This predicts that cluster-scale dynamics are Newtonian at all epochs — testable with high-z cluster surveys.

Summary

The substrate framework predicts that the MOND acceleration scale evolves as a_0(z) = a_0(0)(1+z)^{3/2}, with zero new parameters. This produces three reinforcing effects on early structure formation: a lower Jeans mass (more structures collapse), a faster collapse timescale (structures assemble sooner), and deeper effective potential wells (higher star-formation efficiency). Together, these naturally explain the JWST observation of massive, luminous galaxies at z > 10 — galaxies that are “impossible” in \LambdaCDM but expected in the substrate.

The prediction is:

  • Quantitative: specific numerical evolution of a_0, Jeans mass, and collapse time
  • Zero free parameters: follows from boundary parity + substrate density dilution
  • Falsifiable: the RAR evolution is measurable with current and next-generation surveys
  • Distinguishing: standard MOND predicts constant a_0; \LambdaCDM has no universal a_0; the substrate predicts (1+z)^{3/2}
  • Timely: directly addresses the most prominent anomaly in current observational cosmology

The single most important observational test is the evolution of the radial acceleration relation with redshift. At z = 1, the predicted a_0 is 2.8\times today’s value — a large, measurable shift accessible to DESI, Euclid, and SKA.

References specific to this section

  • Steinhardt et al. (2016): “The Impossibly Early Galaxy Problem,” ApJ 824, 21
  • Boylan-Kolchin (2023): Stress testing \LambdaCDM, Nature Astronomy 7, 731
  • Arrabal Haro et al. (2023): JWST spectroscopic confirmation of Maisie’s Galaxy and refutation of CEERS-93316, Nature
  • Finkelstein et al. (2022): Discovery of Maisie’s Galaxy, ApJL
  • McGaugh, Lelli & Schombert (2016): RAR measurement, g_\dagger = 1.20 \times 10^{-10} m/s², 153 galaxies
  • Sanders (1999): MOND Jeans analysis
  • Cross-references: Galactic Dynamics, Spacetime, Dynamics, Inflation, Bridge Equation

Footnotes

  1. Steinhardt, C.L. et al., “The Impossibly Early Galaxy Problem,” ApJ 824, 21, 2016. First identified the systematic tension with Hubble data; JWST dramatically sharpened it. [R60]↩︎

  2. Boylan-Kolchin, M., “Stress testing \LambdaCDM with high-redshift galaxy candidates,” Nature Astronomy 7, 731, 2023. Shows the most massive JWST candidates at z \approx 7–10 lie at the very edge of what the dark matter halo mass function permits. [R61]↩︎

  3. Arrabal Haro, P. et al., “Confirmation and refutation of very luminous galaxies in the early universe,” Nature, 2023. Spectroscopic confirmation of Maisie’s Galaxy at z=11.4; simultaneously showed CEERS-93316 was a dusty impostor at z=4.9. [R62]↩︎

  4. Sanders, R.H., “The virial discrepancy in clusters of galaxies in the context of modified Newtonian dynamics,” ApJ 512, L23, 1999. Discussion of MOND Jeans analysis; see also Milgrom, M., “MOND and the Mass Discrepancies in Tidal Dwarf Galaxies,” ApJ 667, L45, 2007. [R64, R65]↩︎