---
source_pdf: ka_and_the_a2_Invariant__with_Strong_Fields_added__v1_1.pdf
title: "ka and the a2 Invariant:"
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."
---

ka and the a2 Invariant:
A Unified Acceleration Scale from Galaxies to Atomic Clocks
Gary Alcock
December 2, 2025

Abstract
Modern gravity phenomenology exhibits at least three apparently unrelated small acceleration
scales: the Milgrom scale a0 organizing galaxy rotation curves, the cosmic acceleration scale
aΛ ∼ cH0 , and the sensitivity of precision clock tests to tiny violations of local position invariance.
Conventional frameworks—ΛCDM with cold dark matter on the one hand, and modified-gravity
models on the other—typically treat these scales as independent or emergent features of very
different sectors: dark halos, dark energy, and laboratory metrology.
Here I show that a broad class of scalar refractive-index theories of gravity admits a single,
universal “acceleration-squared” invariant
a2 ≡ a·a,
linked to the gradient energy of a scalar refractive field ψ via a dimensionless self-coupling ka .
In the weak-field, quasi-static limit the field equation can be written as
∇·a +

ka 2
a = −4πGρ,
c2

with a = −c2 ∇ψ the physical free-fall acceleration and ρ the mass-energy density. The ka a2
term represents genuine gravitational self-interaction in the scalar sector, but in a form that is
far simpler than the tensorial nonlinearity of general relativity.
I then show how this structure naturally generates a single preferred acceleration-squared
√
scale a2⋆ ∝ (c2 /ka ) Gρ that simultaneously: (i) reproduces MOND-like scaling g ≃ a⋆ gN in
galaxies when the ka a2 term dominates the bare Poisson term; (ii) yields a cosmic background
value a2⋆ ∼ c2 H02 in an FRW universe with density of order the critical density; and (iii) enters
directly into species-dependent gravitational redshift anomalies for atomic clocks, via scalar
couplings KA encoding the internal structure of each atomic transition.
The construction here is deliberately minimal: I restrict attention to a conformally flat optical
metric gµν = e2ψ ηµν and a single scalar degree of freedom in the weak-field, quasi-static regime.
In companion DFD work, the same scalar is embedded in a nonperturbative optical metric with
a k-essence-type action W (X) for X = ∂µ ψ ∂ µ ψ, together with a transverse-traceless wave sector
whose propagation matches LIGO/Virgo/KAGRA constraints, and a separate PDE analysis
establishes local well-posedness of the resulting quasilinear system. The present paper should
therefore be read as pinning down the weak-field scalar self-interaction driven by ka and its
empirical consequences across galaxies, cosmology, and clocks; any full DFD completion must
recover this a2 structure while matching the strong-field and gravitational-wave constraints
addressed in the companion work.

1

Contents
1 Introduction

2

2 Scalar refractive-index framework

4

2.1

Refractive index and effective metric . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

2.2

Field equation with self-interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.3

Regime hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

3 The a2 invariant and the scale a⋆

6

3.1

Dimensional analysis and definition of a⋆ . . . . . . . . . . . . . . . . . . . . . . . . .

6

3.2

Connection to MOND-like phenomenology . . . . . . . . . . . . . . . . . . . . . . . .

7

3.3

Cosmic acceleration scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

4 Species-dependent couplings and atomic clocks

9

4.1

Effective coupling coefficients KA . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

4.2

Incorporating the a2 invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

5 Experimental determination of ka

10

5.1 Astrophysical constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10

5.2

Clock-based strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

5.3

Consistency with existing tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

6 Limitations, strong fields, and gravitational waves

11

7 Implications for the DFD program

12

8 Conclusions and outlook

13

1

Introduction

Astrophysical and cosmological observations over the past four decades have revealed a remarkably
coherent set of anomalies relative to the predictions of general relativity (GR) with visible matter
alone. Spiral galaxy rotation curves are flat rather than Keplerian; low surface-brightness galaxies
follow tight scaling relations; and large-scale structure and supernova data point to a late-time
accelerated expansion of the universe. [1–3, 6]
The dominant response has been the ΛCDM paradigm, which retains GR but postulates cold dark
matter and a cosmological constant. An alternative line of work instead modifies gravity in the
low-acceleration regime, with Modified Newtonian Dynamics (MOND) the prime example. [4, 5]
2

MOND introduces a characteristic acceleration a0 ∼ 10−10 m/s2 governing the transition between
Newtonian and deep-MOND behavior in galaxies.
A striking and still poorly understood fact is that a0 is numerically close to the cosmic acceleration
scale aΛ ∼ cH0 inferred from supernovae and the cosmic microwave background. [2, 3] Furthermore,
ever more precise tests of the Einstein equivalence principle show that local position invariance (LPI)
and the universality of free fall are obeyed to parts in 1013 –1015 , yet the small residual uncertainties
are now comparable to the size of the anomalies implied by dark-energy-like acceleration and galaxy
scaling laws for low-acceleration systems. [7–11]
At the same time, scalar and vector-tensor theories of modified gravity have proliferated. [6, 12] In
many of these models the gravitational sector includes one or more additional fields with their own
self-interactions. However, the accelerations a0 and aΛ are usually put in by hand, or emerge from
very different pieces of the theory, and there is no a priori reason why the same scale should play a
role in both galaxy dynamics and cosmic expansion.
Within the broader Density Field Dynamics (DFD) framework, gravity and optics are encoded in
a single scalar refractive field ψ(x) defining an optical metric gµν = e2ψ ηµν . Companion papers
develop the full theory: one constructs a strong-field completion with a nonperturbative optical
metric and a k-essence-type action W (X) for X = ∂µ ψ ∂ µ ψ, together with a transverse-traceless
gravitational-wave sector whose propagation matches current LIGO/Virgo/KAGRA constraints,
while a separate PDE-focused analysis establishes local well-posedness and finite propagation speed
for the resulting quasilinear ψ-equation. The present work does not revisit those constructions;
instead it focuses on a particularly simple and tightly constrained piece of the weak-field sector.

Goal of this paper
The aim of this paper is to isolate and analyze a simple structural feature that appears naturally
in scalar refractive-index theories of gravity and that ties these disparate phenomena together:
a universal acceleration-squared invariant a2 ≡ a · a that enters the field equation through a
dimensionless self-coupling ka .
The key points are:
1. In a scalar refractive-index framework, the metric seen by light and matter is encoded in a
single scalar field ψ(x) that modulates the local refractive index n(x) = eψ(x) .
2. The weak-field, quasi-static limit can be arranged so that the physical free-fall acceleration
field is
a(x) = −c2 ∇ψ(x),
reproducing Newtonian gravity in the appropriate regime.
3. A minimal nonlinear completion introduces a gradient self-coupling term proportional to |∇ψ|2
in the field equation with a dimensionless coefficient ka . In terms of the acceleration this term
becomes proportional to ka a2 /c2 .
4. In spherically symmetric systems with characteristic density ρ, there is a natural accelerationsquared scale
4πGρc2
a2⋆ ∼
,
ka
3

which controls both galaxy-scale dynamics and the background cosmological expansion if one
takes ρ to be of order the mean cosmic density.
5. When matter is described by different species of bound states (e.g. different atomic transitions),
the scalar field can couple to each with different effective coefficients KA . This introduces
species-dependent sensitivity to the same a2 invariant, which precision atomic clocks can probe
as apparent violations of LPI.
These observations together suggest that ka and the associated acceleration-squared invariant a2
are the natural “glue” connecting galaxies, cosmology, and clocks in the broader Density Field
Dynamics (DFD) picture. The present paper focuses exclusively on this structural connection and
the minimal mathematics needed to make it precise, leaving the strong-field and radiation sectors to
the companion work mentioned above.

2

Scalar refractive-index framework

This section introduces the basic kinematics and field equation of a scalar refractive-index theory
sufficient for the discussion that follows. We do not claim that this simple model is a full replacement
for GR; rather it is a controlled weak-field toy model that makes the a2 structure transparent and
recovers Newtonian/GR behavior in the high-acceleration regime.

2.1

Refractive index and effective metric

Consider a scalar field ψ(x) on a background Minkowski spacetime with coordinates (t, x) and metric
ηµν = diag(−1, 1, 1, 1). Define a position-dependent refractive index

and an effective metric

n(x) = eψ(x) ,

(1)

gµν = e2ψ(x) ηµν .

(2)

In the eikonal approximation, light propagation is governed by null geodesics of gµν , and the local
coordinate speed of light is reduced by e−ψ compared to c in the background frame.
For slowly moving massive particles, the nonrelativistic limit of the geodesic equation in static ψ
yields an effective potential
Φeff (x) = c2 ψ(x),
(3)
so that the free-fall acceleration is
a(x) = −∇Φeff (x) = −c2 ∇ψ(x).

(4)

This reproduces Newtonian gravity if ψ satisfies a Poisson equation with the appropriate source
term in the weak-field limit.

4

2.2

Field equation with self-interaction

The simplest purely Newtonian limit would require ψ to satisfy
∇2 ψ =

4πG
ρ,
c2

(5)

so that combining with Eq. (4) one finds ∇ · a = −4πGρ.
However, nothing forbids the existence of nonlinear self-interactions in the scalar sector. The class
of models we consider here are defined by the modified field equation
∇2 ψ − ka |∇ψ|2 =

4πG
ρ,
c2

(6)

where ka is a dimensionless constant and ρ is the mass-energy density in the weak-field regime. We
assume |ka | ∼ O(1) so that the modification becomes important only when the acceleration is small
compared to the characteristic scale a⋆ defined below.
Using Eq. (4), we can rewrite Eq. (6) directly in terms of the physical acceleration field. First note
that
a2
|∇ψ|2 = 4 ,
a2 ≡ a · a.
(7)
c
Moreover,
1
(8)
∇2 ψ = − 2 ∇ · a.
c
Substituting into Eq. (6) gives
1
a2
4πG
− 2 ∇ · a − ka 4 = 2 ρ,
(9)
c
c
c
or, multiplying through by −c2 ,
∇·a+

ka 2
a = −4πGρ.
c2

(10)

Equation (10) is the central structural equation for the rest of this paper. It shows that, in this class
of scalar refractive models, a single invariant combination a2 = a · a appears linearly in the field
equation with coefficient ka /c2 . The sign and magnitude of ka determine how strongly the scalar
field “feeds back” on itself via its gradient energy.
Dimensional consistency check. All three terms in Eq. (10) have dimensions of inverse time
squared:
• [∇ · a] = (m/s2 )/m = s−2 ,
• [ka a2 /c2 ] = (1)(m2 /s4 )/(m2 /s2 ) = s−2 ,
• [4πGρ] = (m3 /kg · s2 )(kg/m3 ) = s−2 .
The equation is therefore dimensionally consistent with ka a pure number.

5

2.3

Regime hierarchy

Equation (10) also makes the hierarchy of regimes transparent. Comparing the divergence term and
the self-interaction term gives three qualitatively distinct behaviors:
Regime

Condition

Behavior

Solar system / high-a

ka 2
a
c2
ka
∇ · a ∼ 2 a2
c
ka
∇ · a ≪ 2 a2
c

Linear (Newtonian / GR limit)

Crossover / galactic
Deep field / low-a

∇·a≫

MOND-like transition
Nonlinear a2 ∝ aN scaling

In the high-acceleration regime relevant to Solar System tests, the self-interaction term is negligible
and Eq. (10) reduces to the usual Newtonian Poisson equation. In the deep low-acceleration regime,
the scalar self-interaction dominates and drives the MOND-like behavior discussed below.

3

The a2 invariant and the scale a⋆

3.1

Dimensional analysis and definition of a⋆

Since ka is dimensionless, the combination ka a2 /c2 has the same dimensions as ∇ · a, namely
inverse time squared (equivalently, acceleration per unit length). This suggests the existence of a
characteristic acceleration scale associated with a given density environment ρ.
To see this, consider a region of approximately uniform density ρ and characteristic size L, such that
∇ · a ∼ a/L. The field equation (10) then implies, schematically,
a
ka
+ 2 a2 ∼ 4πGρ.
L c

(11)

This quadratic relationship between a and ρ admits two limiting regimes:
• If a is large enough that a/L ≫ ka a2 /c2 , we recover the standard Newtonian scaling a ∼ 4πGρL.
• If a is small enough that ka a2 /c2 ≫ a/L, the nonlinear self-interaction term dominates, and
we obtain
ka 2
4πGρc2
2
a
∼
4πGρ
⇒
a
∼
.
(12)
c2
ka
This motivates defining a characteristic acceleration-squared scale
a2⋆ (ρ) ≡

4πGρc2
,
ka

(13)

so that in the deeply nonlinear regime we have
a2 ∼ a2⋆ (ρ),

a ∼ a⋆ (ρ).
6

(14)

Dimensional consistency check.
[a2⋆ ] =

(m3 /kg · s2 )(kg/m3 )(m2 /s2 )
m2
= 4 = [a]2 . ✓
1
s

(15)

Two points are important here:
1. The scale a⋆ depends on the ambient density ρ. For a galactic disk, ρ is of order the baryonic
surface density divided by a scale height; for cosmology, ρ is the mean cosmic density.
2. The dependence is via a2⋆ , not a⋆ itself. This becomes crucial when comparing to phenomenology
√
such as MOND, where the deep-regime scaling is g ∼ a0 gN , i.e., accelerations are governed
by a square root of a fundamental acceleration scale.

3.2

Connection to MOND-like phenomenology

In MOND, the modified Poisson equation reads schematically [4, 5]
   
|g|
∇· µ
g = −4πGρ,
a0

(16)

where g is the gravitational field (acceleration), a0 is the MOND acceleration scale, and µ(x) is an
interpolation function such that µ(x) → 1 for x ≫ 1 and µ(x) → x for x ≪ 1. In the deep-MOND
regime |g| ≪ a0 , one finds


|g|
∇·
g ≈ −4πGρ,
(17)
a0
which in spherical symmetry leads to the scaling relation
g 2 ≈ a0 gN ,

(18)

with gN the Newtonian acceleration.
The structure in Eq. (10) is different but closely related. If we identify a with the gravitational field
g, then our modification takes the form
∇·a+

ka 2
a = −4πGρ.
c2

(19)

In a spherically symmetric configuration sourced by a point mass M , the Newtonian solution satisfies
∇ · aN = −4πGρ and aN (r) = GM/r2 . When the nonlinear term becomes important, the balance
equation becomes roughly
ka 2
GM
a ∼ 4πGρeff ∼ 3 ,
(20)
c2
r
where we have used ρeff ∼ M/(4πr3 /3) for order-of -magnitude purposes. This yields
a2 ∼

c2 GM
.
ka r3

(21)

c2
ka r

(22)

Combining with aN = GM/r2 , we obtain
2

a ∼



7


aN .

If the system has a characteristic radius r ∼ R, then we can define an effective acceleration scale
c2
,
ka R

(23)

a2 ∼ aeff
0 aN .

(24)

aeff
0 ≡
so that

This is formally the same scaling as in deep-MOND, with a0 replaced by an effective aeff
0 set by
ka and the size of the system. In more realistic disk geometries, R is replaced by an appropriate
combination of disk scale lengths and heights, but the structural relationship a2 ∝ aN persists.
Dimensional consistency check.
[aeff
0 ]=

3.3

[c2 ]
m2 /s2
m
=
= 2 = [a]. ✓
[ka ][R]
m
s

(25)

Cosmic acceleration scale

In a homogeneous and isotropic FRW cosmology with scale factor a(t) and Hubble parameter
H = ȧ/a, the Newtonian analogue of the Friedmann equation can be written as
ä
4πG
Λc2
=−
(ρ + 3p/c2 ) +
.
a
3
3
The observed late-time acceleration is characterized by a scale
aΛ ∼ cH0 ,

(26)
(27)

where H0 is the present-day Hubble parameter. [2, 3]
In the scalar refractive-index picture, one can interpret the cosmic expansion as a large-scale
configuration of the scalar field ψ with slowly varying gradient on Hubble scales. The acceleration of
comoving observers relative to the scalar field definition of “free fall” is then governed by an effective
a2 term of the same structural form as in local systems, with ρ replaced by the mean cosmic density
ρ̄ ∼ 3H02 /(8πG).
Plugging this into Eq. (13) gives
a2⋆ (ρ̄) ∼

4πG 2 4πG 3H02 2 3c2 H02
ρ̄c ∼
c =
.
ka
ka 8πG
2ka

Thus the cosmological a⋆ scale is

(28)

r

3
cH0 .
(29)
2ka
For ka of order unity, this is naturally of order cH0 ≈ 7 × 10−10 m/s2 without any additional tuning.
a⋆ (ρ̄) ∼

Dimensional consistency check.
[cH0 ] = (m/s)(s−1 ) = m/s2 = [a]. ✓

(30)

The crucial point is that the same ka that governs the crossover in galaxy dynamics also determines
the magnitude of the cosmic acceleration scale. The numerical near-coincidence between a0 and cH0
in phenomenological fits then ceases to be a mystery and becomes a direct reflection of the single
underlying self-coupling constant ka .
8

4

Species-dependent couplings and atomic clocks

To connect the a2 invariant to laboratory tests, we must specify how the scalar field ψ couples to
different forms of matter. In a generic scalar-tensor or scalar refractive-index model, the coupling
is composition-dependent: different atomic transitions, nuclear binding energies and electronic
structures respond differently to variations in ψ. [7]

4.1

Effective coupling coefficients KA

Let us consider an atomic transition A with frequency νA . In the presence of the scalar field ψ, we
allow for a linearized dependence
δνA
= KA δψ,
(31)
νA
where KA is a dimensionless sensitivity coefficient encoding how the transition energy depends on
the underlying dimensionless constants that are themselves functions of ψ (fine-structure constant,
electron-proton mass ratio, etc.).
In a static gravitational potential, ψ varies with height h in the gravitational field. For small height
differences in a uniform gravitational field a, we have
a
δψ ≈ − 2 δh,
c
using Eq. (4). Thus the fractional frequency shift between two heights separated by ∆h is


a ∆h
∆ν
≈ −KA 2 .
ν A
c

(32)

(33)

Comparing two different species A and B at the same locations yields a fractional ratio shift
∆(νA /νB )
a ∆h
≈ −(KA − KB ) 2 .
νA /νB
c

(34)

In GR, local position invariance implies that KA = KB = 1, and the ratio is independent of height:
both clocks redshift in exactly the same way. [11] In the scalar refractive-index framework with
species-dependent KA , however, gravitational redshift becomes composition-dependent at a level set
by the differences KA − KB .

4.2

Incorporating the a2 invariant

The structure of Eq. (34) already shows that clock comparison experiments are directly sensitive to
the acceleration a. To connect this to the acceleration-squared invariant, recall that the background
field a itself is constrained by the field equation (10):
∇·a+

ka 2
a = −4πGρ.
c2

(35)

In the regime where the nonlinear term is non-negligible, a2 is no longer free to take arbitrary values;
it is tied to the local density environment through Eq. (13).
9

Thus, at leading order, we can write
a2 ≈ a2⋆ (ρ) =
so that
a≈

4πGρc2
,
ka

√
2 πGρ c
√
.
ka

(36)

(37)

Substituting into Eq. (34) gives

√
√
2 πGρ c ∆h
∆(νA /νB )
2 πGρ
∆h.
≈ −(KA − KB ) √
= −(KA − KB ) √
νA /νB
c2
ka
ka c

(38)

Several features are worth emphasizing:
• The magnitude of the effect scales with
structure of the field equation.

p
ρ/ka , not linearly with ρ. This reflects the a2

• Once ka is fixed, Eq. (38) defines a completely predictive relationship between the density
environment, the height separation, and the composition dependence of gravitational redshift.
• Atomic clock networks spanning different height ranges (e.g. on towers, satellites, or deep
underground laboratories) and√using different clock species become a direct probe of ka through
the combination (KA − KB )/ ka . [9, 10]

5

Experimental determination of ka

The ka parameter controls the strength of scalar self-interaction and thus the size of both astrophysical
and laboratory deviations from GR. Determining ka (or setting bounds on it) therefore requires
combining information from multiple regimes.

5.1

Astrophysical constraints

Galaxy rotation curves and their scaling relations can be used to infer an effective acceleration scale
agal
0 in the deep low-acceleration regime. [5] In the scalar self-interaction picture, this effective scale
is related to ka and the characteristic density and size of the galaxy by
agal
0 ∼

c2
.
ka Reff

(39)

−10 m/s2 , this provides one handle on k for
If one adopts a phenomenological value agal
a
0 ≈ 1.2 × 10
typical disk galaxies of known Reff .
p
Cosmological data, on the other hand, constrain the combination a⋆ (ρ̄) ∼ cH0 3/(2ka ) discussed
above. Requiring that this be of order the observed late-time acceleration implies that ka must
not be extremely small or large; otherwise the scalar self-interaction would either overwhelm or be
negligible compared to the ΛCDM fit. [6, 12]

These considerations suggest that ka is plausibly of order unity in natural units, though the precise
value depends on the detailed matching between the simple scalar model considered here and full
observational data.
10

5.2

Clock-based strategies

Atomic clock experiments provide a complementary and, in some ways, cleaner probe of ka . The
basic strategy is:
1. Choose two clock species A and B with calculable and significantly different sensitivity
coefficients KA and KB .
2. Deploy clocks at two or more heights separated by a distance ∆h in a gravitational field with
known density profile ρ(h), such as the Earth’s near-surface environment.
3. Measure the fractional ratio shift ∆(νA /νB )/(νA /νB ) as a function of ∆h and compare to the
GR prediction (which is essentially zero for the ratio).
√
4. Use Eq. (38) to infer or bound the combination (KA − KB )/ ka , and thus ka once the KA
are known or constrained from atomic theory. [7]
Current and near-future optical lattice clock networks, both ground-based and space-based, already
operate at fractional frequency precision better than 10−17 –10−18 . [9, 10] This is sufficient to probe
extraordinarily small deviations from LPI over height differences of order 102 –103 m, especially when
multiple species are compared.

5.3

Consistency with existing tests

Any scalar self-interaction model must remain consistent with the impressive null tests of the
equivalence principle and GR obtained from experiments such as MICROSCOPE, binary pulsar
timing, and the gravitational wave observations of LIGO and Virgo. [8, 11, 13–15] In the present
framework, this translates into bounds on ka and the products ka KA .
The essential point is that the same ka enters all three regimes we have discussed:
• galaxy dynamics (through agal
0 ),
• cosmic acceleration (through a⋆ (ρ̄)),
• clock tests (through the ratio shifts in Eq. (38)).
This eliminates the freedom to tune each sector independently and turns what might otherwise be a
collection of unrelated anomalies into a network of cross-checks. Any choice of ka that fits galaxies
but grossly violates clock or GW constraints, or vice versa, is ruled out.

6

Limitations, strong fields, and gravitational waves

The analysis in this paper is intentionally restricted to the weak–field, quasi–static sector of a scalar
refractive–index theory, where a single scalar field ψ and its gradient determine the effective potential
and test–mass acceleration. In this limit, the relevant invariant is
a2 ≡ a · a,
11

and the self–interaction parameter ka fixes how departures from Newtonian gravity emerge in
low–acceleration environments. All of the results above are derived in this regime: static or slowly
varying configurations, no strong–field horizons, and no explicit radiation sector.
From the broader DFD point of view, this is a controlled truncation rather than a complete
theory. In the full framework, the refractive field ψ fixes an optical metric gµν [ψ], and a separate
transverse–traceless radiation sector can be added consistently, yielding tensor gravitational waves
with the observed polarizations and near–luminal propagation speed. The strong–field structure of
that completion, and its confrontation with LIGO/Virgo events and horizon–scale tests, are treated
in a companion analysis on strong fields and gravitational waves in DFD (work in preparation). A
separate PDE–focused study establishes local well–posedness and finite propagation speed for the
underlying quasilinear ψ–equation in that setting (also in preparation).
The present work deliberately does not re–derive or fit the tensor gravitational–wave sector. In
particular, we do not attempt to:
• compute full inspiral–merger–ringdown waveforms in the ka –deformed theory;
• revisit polarization constraints from LIGO/Virgo beyond the requirement that a viable completion retain a transverse–traceless sector;
• analyse strongly curved, dynamical spacetimes where higher–order invariants or additional
fields may become important.
Within its narrow scope, the contribution of this paper is therefore precise: it isolates a single
acceleration–squared invariant a2 and shows how a scalar self–interaction governed by ka can link
galaxies, cosmology, and clocks in the weak–field, quasi–static regime. Consistency with strong–field
phenomena and gravitational waves is imposed at the level of the underlying DFD completion but
not re–analysed here. A natural next step is a global fit in which the same ka and a2 structure is
confronted simultaneously with rotation curves, cosmological observables, clock experiments, and
the strong–field and GW constraints developed in the companion works.

7

Implications for the DFD program

Within the broader Density Field Dynamics program, the central idea is that a single scalar density
or refractive field controls both the effective metric for light and matter and the stochastic structure
of quantum measurement. Those aspects lie beyond the scope of this paper, which has focused solely
on the classical weak-field gravity sector.
Nevertheless, the emergence of a universal acceleration-squared invariant a2 with self-coupling ka
has several important implications:
1. It provides a simple and robust organizing principle: everywhere the scalar field has a gradient,
there is an associated local scale a⋆ (ρ) set by Eq. (13). Physical phenomena as diverse as
galaxy rotation curves, cosmic acceleration, and clock redshifts are then different windows into
this same scalar gradient energy.
2. It sharply reduces the number of genuinely free parameters in the gravitational sector. Once ka
is fixed (or tightly constrained) by any one class of observations, the others become predictions
rather than independent fits.
12

3. It suggests a natural hierarchy of regimes. High-acceleration systems such as the Solar System
lie firmly in the linear regime ∇ · a ≫ ka a2 /c2 , reproducing GR and Newtonian gravity
to high accuracy. Low-acceleration, low-density environments lie in the nonlinear regime
ka a2 /c2 ≳ ∇ · a, where MOND-like and dark-energy-like phenomena emerge.
4. It provides a clean target for both theoretical and experimental work: the precise determination
of ka and the mapping of where, in density and acceleration space, the transition between
linear and nonlinear regimes occurs.

8

Conclusions and outlook

We have identified and analyzed a simple but powerful structural feature of scalar refractive-index
theories of gravity: a universal acceleration-squared invariant a2 = a·a that appears linearly in the
field equation with a dimensionless self-coupling ka /c2 . This leads naturally to a density-dependent
acceleration scale a⋆ (ρ) that:
• produces MOND-like scaling in galaxies without introducing an arbitrary new constant unrelated to the density environment;
• matches the order of magnitude of the cosmic acceleration scale when ρ is taken to be the
mean cosmic density;
• directly controls composition-dependent gravitational redshift effects for atomic clocks via
species-dependent couplings KA .
The main conceptual achievement is that a single structural parameter ka —together with the invariant
a2 —links three previously disparate acceleration scales: galactic a0 , cosmic aΛ , and laboratory-scale
sensitivities in precision metrology. This closes a loop in the gravitational sector of the Density Field
Dynamics program: once ka is fixed by any one of these regimes, the others are no longer free to
vary independently.
From an experimental perspective, the most promising near-term probes of ka are multi-species
atomic clock networks, which can measure or bound composition-dependent gravitational redshift
at levels far beyond what is accessible to astrophysical observations alone. On longer timescales, a
consistent fit of galaxy dynamics, cosmic expansion, and clock tests within this framework, together
with the strong-field and gravitational-wave constraints developed in the companion DFD work,
would constitute strong evidence for a scalar refractive component to gravity and would dramatically
sharpen the case for DFD as a viable extension or alternative to GR.

Acknowledgements
The author thanks the many experimental teams developing ever more precise optical clocks, without
which the connection between fundamental gravity and metrology would remain purely theoretical.

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References
[1] A. Einstein. Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 49:769–822,
1916.
[2] A. G. Riess et al. Observational evidence from supernovae for an accelerating universe and a
cosmological constant. The Astronomical Journal, 116(3):1009–1038, 1998.
[3] S. Perlmutter et al. Measurements of Ω and Λ from 42 high-redshift supernovae. The Astrophysical Journal, 517(2):565–586, 1999.
[4] M. Milgrom. A modification of the Newtonian dynamics as a possible alternative to the hidden
mass hypothesis. The Astrophysical Journal, 270:365–370, 1983.
[5] B. Famaey and S. S. McGaugh. Modified Newtonian dynamics (MOND): Observational
phenomenology and relativistic extensions. Living Reviews in Relativity, 15:10, 2012.
[6] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis. Modified gravity and cosmology. Physics
Reports, 513(1–3):1–189, 2012.
[7] J.-P. Uzan. Varying constants, gravitation and cosmology. Living Reviews in Relativity, 14:2,
2011.
[8] P. Touboul et al. MICROSCOPE mission: first results of a space test of the equivalence principle.
Physical Review Letters, 119(23):231101, 2017.
[9] P. Delva et al. Test of gravitational redshift with stable clocks in eccentric-orbit satellites.
Physical Review Letters, 121(23):231101, 2018.
[10] E. Dierikx et al. Comparing optical clocks over fiber: A review. Metrologia, 57(3):034004, 2020.
[11] C. M. Will. The confrontation between general relativity and experiment. Living Reviews in
Relativity, 17:4, 2014.
[12] A. Joyce, B. Jain, J. Khoury, and M. Trodden. Beyond the cosmological standard model.
Physics Reports, 568:1–98, 2015.
[13] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration). Observation of
gravitational waves from a binary black hole merger. Physical Review Letters, 116(6):061102,
2016.
[14] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration). GW170817:
Observation of gravitational waves from a binary neutron star inspiral. Physical Review Letters,
119(16):161101, 2017.
[15] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration). Tests of general
relativity with the binary black hole signals from the first and second observing runs. Physical
Review D, 100(10):104036, 2019.

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