---
source_pdf: Late_Time_Potential_Shallowing_and_Low_Acceleration_Hints.pdf
title: "Late-Time Potential Shallowing and Low-Acceleration"
site: https://densityfielddynamics.com/
author: Gary Alcock
framework: Density Field Dynamics (DFD)
format_note: "Markdown extracted from the source PDF for clean AI ingestion. No editorial changes; mathematical typography in the PDF is authoritative."
---

Late-Time Potential Shallowing and Low-Acceleration
Hints:
A Minimal Scalar-Refractive Interpretation with
Laboratory Falsifiability
Gary Alcock
October 1, 2025
Abstract
Several recent measurements continue to stress General Relativity (GR) in the
late-time universe. First, a model-independent, direct measurement of the Weyl gravitational potential from DES Year 3 weak-lensing × clustering finds the lowest-redshift
bins are 2–3σ shallower than ΛCDM+GR expectations. Second, DESI DR2 BAO—in
combination with supernovae and a CMB distance prior—exhibit dataset-dependent
preference for dynamical dark energy over a pure cosmological constant. Third, independent, late-time determinations of H0 (time-delay cosmography; JWST-Cepheid
cross-checks of the local distance ladder) keep the Hubble tension alive as a robust crossmethod discrepancy. In parallel, Gaia wide-binary tests at accelerations ≲ 10−10 m s−2
remain active and contested. We show that a minimal scalar refractive framework—in
which photons see an optical index n = eψ , matter accelerates as a = (c2 /2)∇ψ, and ψ
obeys a quasilinear Poisson equation with a low-acceleration crossover—naturally yields
(i) time-weakening lensing potentials as the mean density dilutes and (ii) MOND-like
phenomenology in the deep-field regime, while (iii) remaining indistinguishable from
GR in Solar-System PPN tests and (iv) offering a decisive, laboratory falsifier via clock
redshift comparisons between solid-state cavities and atomic transitions. We emphasise these observations as motivations, not proofs; the laboratory discriminator carries
the ultimate burden of evidence.

1

Introduction

GR remains extraordinarily successful in high-gradient and Solar-System regimes. At late
times and low accelerations, however, several independent datasets continue to show mild
but persistent tensions with ΛCDM+GR. Most notable are: (i) the DES Y3 direct Weylpotential measurement showing shallower low-z wells; (ii) DESI DR2 BAO combinations
indicating a dataset-dependent preference for w(z) ̸= −1; (iii) the durability of the H0 split
across methods (distance ladder with JWST cross-checks, time-delay cosmography). At the
same time, wide-binary tests of gravity at a ∼ 10−10 m s−2 remain contested and under active
1

refinement. We ask a restricted, operational question: can a minimal scalar refractive picture
capture the qualitative directions of these anomalies while staying fully compliant with PPN
constraints and yielding an unambiguous, lab-grade falsifier?

2

Minimal scalar-refractive framework

We consider a single scalar field ψ(x) defining an optical medium
c
n(x) = eψ(x) ,
c1 (x) = = c e−ψ ,
n

(1)

with the weak-field matter response
c2
a = ∇ψ ≡ −∇Φ,
2

c2
Φ ≡ − ψ,
2

(2)

and a quasilinear field equation with a single crossover function µ:




8πG
∇· µ(|∇ψ|/a⋆ ) ∇ψ = − 2 ρ − ρ̄ .
c

(3)

Here a⋆ sets the low-acceleration crossover. The normalisation is chosen so that in highgradient regimes (µ → 1) one recovers the Newtonian potential and all 1PN optical tests
of GR (light deflection, Shapiro delay) exactly. In the deep-field regime, µ(x) ∼ x yields
|∇ψ| ∝ 1/r and asymptotically flat rotation curves, i.e. MOND-like phenomenology, without
adding dark matter explicitly. This construction is minimal : a single scalar with a single
interpolation µ.

Action principle, coupling, and PPN limit
To address physical mechanism and avoid ad hoc postulation, consider the action
 4



Z


√
c
|∇ψ|
2
4
a⋆ H
− ψ (ρ − ρ̄) + SSM e−ψ Aµ , Ψmatter .
S = d x −g
16πG
a⋆

(4)

Here H is a dimensionless function and SSM denotes the Standard-Model sector with photons
coupled through the optical metric (phase velocity vphase = c e−ψ ) while massive fields follow
the weak-field acceleration law above. Varying (4) with respect to ψ yields


 
8πG
1 dH
|∇ψ|
∇· µ
∇ψ = − 2 (ρ − ρ̄),
µ(y) ≡
.
(5)
a⋆
c
y dy
Thus the interpolation µ is generated by a single scalar functional H; the limits µ → 1 (high
gradient) and µ ∼ y (deep field) follow from H being quadratic for y ≫ 1 and ∝ y 2 /2 for
y ≪ 1, respectively. PPN sketch. Expanding (4) around a static, weak-field source with
gµν = ηµν + δgµν and ψ ≪ 1, one finds to O(v 2 /c2 ) that g00 = −1 + 2Φ/c2 + O(c−4 ) and
2
gij = δij (1 + 2Φ/c2 ) + O(c−4 ) with Φ = − c2 ψ sourced by (3). Hence light deflection and
Shapiro delay correspond to γ = 1, and the quadratic response of H in the high-gradient
limit yields β = 1 at 1PN order; preferred-frame/non-conservative PPN parameters vanish
at leading order.
2

Units and normalization of µ
Because ψ is dimensionless, |∇ψ| has units of inverse length. It is convenient to write the
argument of µ in terms of the acceleration a ≡ (c2 /2) |∇ψ|:
x ≡

a
|∇ψ|
=
.
2
(2a⋆ /c )
a⋆

With this choice, the interpolation µ(x) is a function of a/a⋆ as in standard MOND-like
notation, while Eq. (3) retains the form given.

Interpolation µ(x) and the scale a⋆
Representative choices that capture both regimes are
µsimple (x) =

x
,
1+x

x
µstandard (x) = √
.
1 + x2

(6)

Both satisfy µ → 1 for x ≫ 1 and µ ∼ x for x ≪ 1. The scale a⋆ is not a fine-tuned
constant but encodes the transition from linear (Newton/GR) response to the deep-field
regime; phenomenologically, a⋆ ∼ 10−10 m s−2 brackets the galactic crossover and is precisely
where wide-binary tests are probing.

3

Late-time potential shallowing (DES)

GR+Λ anticipates nearly constant late-time gravitational potentials on large scales; departures are typically ascribed to evolving dark energy or modified growth functions. DES Y3
report a direct, model-independent estimate of the Weyl potential in four redshift bins using
combined galaxy-galaxy lensing and clustering; the two lowest-z bins are measured ∼ 2σ
and ∼ 2.8σ below ΛCDM expectations. In Eq. (3), the source of ψ tracks (ρ − ρ̄). As the
universe dilutes, the line-of-sight mean approaches ρ̄(t) and the typical ψ-gradient weakens,
leading generically to shallower lensing potentials at late times:
∆ρ
∆Φ
∼
Φ
ρ

⇒

late-time shallowing as ρ ↓ .

(7)

Quantitatively, the DES low-z deficit corresponds to a fractional reduction at the O(10%)
level (consistent with a 2–3σ deviation when mapped to the fiducial covariance), which
is the expected order from modest dilution of the large-scale ψ-gradient without invoking
exotic microphysics. This qualitative trend matches the DES finding and requires no exotic
dark-energy microphysics beyond the effective refractive response of the cosmic medium.

FRW implementation
Write ψ(x, a) = ψ̄(a) + δψ(x, a) and ρ = ρ̄(a) [1 + δ(x, a)] in a spatially flat FRW background
with scale factor a. In comoving coordinates, ∇2phys = a−2 ∇2 . Assuming µ is slowly varying
3

|∇ψ| (schematic)
ΛCDM (approx. const. potential)
dilution ⇒ weaker ∇ψ

scale factor a
Figure 1: Schematic comparison: the scalar-refractive picture generically weakens the lineof-sight ψ-gradient with cosmic dilution, producing shallower late-time lensing potentials
than a strictly constant-potential baseline.
on the large scales of interest, one obtains at linear order and in the quasistatic regime
(k ≫ aH):
8πG
2
δΦ ≡ − c2 δψ.
(8)
µ(x̄) ∇2 δψ ≃ − 2 a2 ρ̄(a) δ(x, a),
c
Hence
a2 ρ̄(a) D(a)
,
(9)
δΦk (a) ∝
µ(x̄(a)) k 2
with D(a) the linear growth factor. In GR (µ = 1) this reduces to the familiar result:
δΦ roughly constant in matter domination and decaying once dark energy dominates. In
the scalar-refractive picture, any secular drift of µ(x̄(a)) due to the slow evolution of the
background |∇ψ| produces an additional, controlled decay factor. Toy parametrization.
Taking µ−1 (x̄(a)) = 1 + ϵ0 [a/at ]p with (ϵ0 , p) ∼ (0.1, 1) and at ∼ 0.7 yields a ∼10% reduction
in δΦ between z ≈ 0.6 and z ≈ 0.2, consistent in order-of-magnitude with DES. We present
this as a toy µ–evolution model; a full Boltzmann treatment is left for future work.

4

Dynamical late-time background (DESI DR2, cautiously)

DESI DR2 BAO, when combined with SNe and a CMB distance prior, shows a datasetdependent preference for dynamical dark energy w(z) over Λ. We treat this not as proof of
new physics but as convergent motivation: late-time geometry appears flexible Renough that
a refractive description—in which optical path-lengths are effectively Dopt = 1c eψ ds—can
account for mild departures from a rigid-Λ background
without compromising early-time
R
1
CMB fits. Toy model. For small ψ, Dopt ≈ c (1 + ψ) ds so the inferred distance-redshift
relation acquires a fractional bias ∆D/D ≃ ⟨ψ⟩LOS . Parametrising ⟨ψ⟩LOS (z) by a smooth
function (e.g. a cubic spline anchored at the DESI effective redshifts) induces an effective
w(z) in standard fits without invoking a fluid; small, percent-level ψ biases can mimic mild
dynamical-w preferences in the same redshift range, consistent with the cautious language
used here.
4

a(r) (schematic)

low-a crossover
∼ 10−10 m s−2

Newton/GR

r

Figure 2: Illustrative acceleration profiles: a low-a crossover (dashed) flattens relative to
Newton/GR (solid) near a ∼ 10−10 m s−2 . Wide-binary studies currently disagree over the
presence of such a deviation.

5

Low-acceleration regime (wide binaries; active and
contested)

Gaia wide binaries probe internal accelerations down to a ∼ 10−10 m s−2 . Some analyses
report a ∼20% velocity excess beyond ∼3000 au consistent with MOND-like expectations;
others demonstrate that realistic triple-population modelling and stricter data cuts drive
the signal back toward Newtonian dynamics. Given current disagreement, wide binaries
are best viewed as an active, near-term battleground precisely at the scale where Eq. (3)
transitions (µ ∼ x). For orientation, the µ-crossover radius follows from x = a/a⋆ ≃ 1.
Using a = (c2 /2)|∇ψ| and the point-mass high-gradient solution |∇ψ| = 2GM/(c2 r2 ), one
has a = GM/r2 and x = GM/(a⋆ r2 ). Thus the crossover radius is
r
r× =

GM
≈ 7.1 × 103 au
a⋆



M
M⊙

1/2 

1.2 × 10−10 m s−2
a⋆

1/2
,

(10)

i.e. (3–7) × 103 au for M ∼ (0.2–1)M⊙ , matching the observational dispute range now under
scrutiny. Our point is limited: the direction of the disputed anomaly aligns with the minimal
scalar-refractive crossover.

6

Consistency and counter-evidence

Any alternative must squarely face null tests. A key geometry vs. dynamics test, EG , has
recently been measured with ACT DR6 CMB-lensing × BOSS galaxies and found consistent
with ΛCDM/GR and largely scale-independent within current precision. Weak-lensing S8
results have also evolved: the KiDS-Legacy cosmic-shear analysis is consistent with Planck
ΛCDM. These findings do not contradict the qualitative late-time trends above, but they
emphasise caution: late-time tensions are uneven across probes and evolving with improved
analyses.

5

Quantitative benchmarks and laboratory error budget
Cavity–atom slope (decisive prediction). For two stationary platforms separated by
∆h, the gravitational potential difference is ∆Φ ≃ g ∆h. The scalar-refractive picture yields
a ratio redshift between an evacuated optical cavity (tracking vphase = c e−ψ ) and a co-located
atomic transition:
∆f
∆Φ
= κ 2 ,
f cav/atom
c

κ = 1 (scalar refractive) ,

κ = 0 (GR).

(11)

Derivation of κ = 1. Locally, fcav ∝ vphase /(2L) ∝ e−ψ (with L a proper length stabilized
2
against elastic sag). Thus ∆fcav /fcav = −∆ψ. Using Φ = − c2 ψ, one has ∆ψ = −2 ∆Φ/c2 so
∆fcav /fcav = +2 ∆Φ/c2 . Atomic transitions redshift with proper time, ∆fat /fat = +∆Φ/c2
to leading order. Therefore for the ratio R = fcav /fat across two heights:




∆R
∆f
∆f
∆Φ
∆Φ
=
−
= (2 − 1) 2 = 2 ,
R
f cav
f at
c
c
i.e. κ = 1. With ∆h = 100 m and g ≃ 9.81 m s−2 ,
∆f
g ∆h
≈ 2 ≈ 1.1 × 10−14 per 100 m,
f
c

(12)

providing a clear target for present-day optical metrology. A cross-material (e.g. ULE vs. Si)
and cross-species (e.g. Sr vs. Yb) ratio design isolates the universal geometry-locked slope
from material dispersion or atomic structure.
DES shallowing (order-of-magnitude). Mapping the reported 2–3σ low-z deficit to fractional amplitude implies O(10%) weaker Weyl potential than the Planck-ΛCDM expectation
in those bins, consistent with dilution of ∇ψ along typical lines of sight.
Wide-binary crossover (orientation). For a solar-mass system, a = GM/r2 crosses
∼ 10−10 m s−2 for separations of order (3–7) × 103 au, overlapping the regime where Gaia
analyses disagree.
Scale / Probe
Prediction (scalar refractive)
Status
Solar System (PPN) γ = β = 1; preferred-frame ≈ 0
GR-consistent
DES (low-z Weyl)
∆Φ/Φ = O(10%) shallower
2–3σ low at low z
Galactic rotation
|∇ψ| ∝ 1/r; flat v; TF scaling
Empirical trend
Wide binaries
Crossover near a⋆ ∼ 10−10 m s−2
Active, contested
Lab (100 m)
(∆f /f )cav/atom ≈ 1.1 × 10−14
Near-term falsifier
Table 1: Representative quantitative benchmarks across regimes.

6

7

Laboratory falsifiability (decisive path)

The decisive test is local and composition-resolved. In a verified nondispersive band, a vacuum optical cavity’s resonance frequency scales with the phase velocity vphase = c/n = c e−ψ ,
while co-located atomic transition frequencies track internal energy intervals. Comparing a
cavity to an atomic clock at two different gravitational potentials isolates a ratio redshift: GR
predicts a strict null (both redshift equally), whereas the scalar-refractive picture allows a
small, geometry-locked slope ∝ ∆Φ/c2 . A cross-material, cross-species ratio protocol cleanly
separates material/atomic systematics; the observable is route- and potential-dependent,
not device-dependent. This experiment carries the model’s risk: a strict null at laboratory
sensitivity falsifies the framework.

Embedding and symmetry remark
While the present work stays agnostic about a full high-energy completion, Eq. (4) sketches
a minimal embedding: a single scalar controlling the optical metric seen by photons and
sourcing an effective potential for matter. Deep-field universality arises from the single
interpolation function µ(x); no multiple free functions are introduced. The µ ∼ x behaviour
reflects an emergent scale-free response in the |∇ψ| ≪ a⋆ sector rather than fine-tuning a
specific exponent.

8

Conclusions

We have outlined a minimal scalar-refractive model that: (i) matches Solar-System PPN
constraints; (ii) qualitatively reproduces late-time potential shallowing as the universe dilutes
and a low-a crossover phenomenology at a ∼ 10−10 m s−2 ; (iii) remains decisively falsifiable
via laboratory cavity–atom redshift ratios. We regard current cosmological anomalies as
motivations, not conclusions. If future DESI/LSST-era analyses strengthen dynamical latetime signals while EG and shear constraints continue to tighten, the scalar-refractive picture
will face sharper quantitative tests. Regardless, the laboratory ratio test provides a clean
decision procedure independent of cosmological systematics.

References
DES Weyl potential (model-independent): I. Tutusaus et al., “Measurement of the Weyl
potential evolution from the first three years of Dark Energy Survey data,” Nature Communications
15, 9295 (2024).
DESI DR2 dynamical w (dataset-dependent): S. Adil et al., “Dynamical dark energy in light
of the DESI DR2 BAO,” Nature Astronomy (2025); see also arXiv:2504.06118 (updated 2025) for
a data-combination analysis consistent with the journal version.
H0 status (independent late-time anchors): S. Birrer et al. (TDCOSMO), “Cosmological
constraints from strong lensing time delays,” arXiv:2506.03023 (2025) and references therein; L.
Breuval et al., “Latest updates on the Hubble tension from JWST by the SH0ES team,” AAS
246 (2025), abstract; see also contemporaneous JWST-based coverage confirming HST Cepheid

7

calibrations.
EG gravity test (GR-consistent): L. Wenzl et al., “The Atacama Cosmology Telescope DR6:
gravitational lensing × BOSS EG test,” Phys. Rev. D 111, 043535 (2025).
KiDS-Legacy shear (Planck-consistent): B. Stölzner et al., “KiDS-Legacy: Consistency of
cosmic shear measurements with Planck,” arXiv:2503.19442 (2025).
Wide binaries (active & contested): C. Pittordis & W. Sutherland, “Wide binaries from Gaia
DR3: testing GR vs. MOND with realistic triple modelling,” Open Journal of Astrophysics (2025);
X. Hernandez, “A recent confirmation of the wide binary gravitational anomaly,” MNRAS 537,
2925 (2025).

8