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This document is a preprint and working draft intended for community feedback.
It has not been peer-reviewed and has not been formally published.
© 2026 - Donald Airey. All rights reserved.
This work is intended for submission to a peer-reviewed journal.
Public posting of this draft does not constitute prior publication and does not waive the author’s right to publish a revised version.
This document is a preprint and working draft intended for community feedback.
It has not been peer-reviewed and has not been formally published.
© 2026 - Donald Airey. All rights reserved.
This work is intended for submission to a peer-reviewed journal.
Public posting of this draft does not constitute prior publication and does not waive the author’s right to publish a revised version.
A Phenomenological Acceleration Law from Galaxy Dynamics Consistent with Type Ia Supernova Distances
A Phenomenological Acceleration Law from Galaxy Dynamics Consistent with Type Ia Supernova Distances
A Kinematic Origin of Gravity and Cosmic Expansion
Abstract
Abstract
We test whether a single empirically determined acceleration scale can link galaxy dynamics and cosmological distance measurements. A phenomenological acceleration law is calibrated using the baryonic Tully--Fisher relation from the SPARC galaxy sample, yielding a fixed acceleration scale. An analytic, closed-form distance--redshift relation depending on this scale is then evaluated against the Pantheon+SH0ES Type~Ia supernova dataset, with the remaining parameter calibrated on the Cepheid subset and tested on out-of-sample supernovae.
The model yields a substantially lower covariance-weighted $\chi^2$ than a spatially flat $\Lambda$CDM model with Planck 2018 parameters under identical calibration. Information criteria relative to the FLRW model give $\Delta\mathrm{AIC}=1321$ and $\Delta\mathrm{BIC}=1316$, indicating a decisive statistical preference. The distance relation is analytic and differs from the standard FLRW integral form.
These results show that a single acceleration scale inferred from galaxy dynamics can parameterize a closed-form distance--redshift relation that predicts Type~Ia supernova distances significantly more accurately than the FLRW model, without dark matter or dark energy, providing a compact and computationally efficient representation of the observed distance--redshift relation.
Keywords
Keywords
cosmology: observations -- distance scale -- supernovae: general -- galaxies: kinematics and dynamics
Setup
Setup
Environment Cleanup
Environment Cleanup
Typesetting Rules
Typesetting Rules
Introduction
Introduction
Type~Ia supernovae (SNe~Ia) provide one of the most direct observational probes of the cosmic expansion history through the luminosity distance--redshift relation. Large compilations such as Pantheon+~\cite{Brout2022}, building on earlier compilations~\cite{Scolnic2018}, and constructed using standard light-curve fitting techniques such as SALT2~\cite{Guy2007}, enable high-precision tests of cosmological models using covariance-weighted statistical methods. In the standard framework, Type~Ia supernova observations are interpreted within a parameterized expansion history specified by the Hubble constant and the relative contributions of matter, radiation, and dark energy, with spatial flatness often assumed. These parameters determine the redshift dependence of the expansion rate and, through it, the luminosity distance.
H(z)
An alternative approach is to consider constrained analytic forms for the distance--redshift relation and to evaluate their performance directly against the data. In this work, we examine a model in which the distance relation is parameterized by a small set of quantities, one of which is an acceleration scale determined independently from galaxy dynamics. This approach enables a direct test of whether a single empirically determined scale can consistently describe both galactic and cosmological observables.
The acceleration scale is obtained from fits to the baryonic Tully--Fisher relation using the SPARC galaxy sample, yielding a fixed value that is not adjusted in the supernova analysis. The remaining parameter is calibrated using the Cepheid subset of the Pantheon+ sample, and the resulting model is evaluated on the out-of-sample supernovae using the full covariance matrix.
The goal of this paper is to assess whether this constrained analytic relation provides a quantitatively consistent description of the observed SNe~Ia distance moduli across redshift under identical calibration conditions. No assumption is made regarding the underlying gravitational or cosmological theory, and the analysis is restricted to the observables considered here.
Phenomenological Acceleration Law
Phenomenological Acceleration Law
The purpose of this work is to test whether a single empirically determined acceleration scale can link two otherwise distinct observational regimes: circular motion in rotationally supported galaxies and the luminosity distances of Type~Ia supernovae. We define a phenomenological acceleration law for circular orbits, determine the acceleration scale $A$ from the baryonic Tully--Fisher relation, and then hold it fixed in the supernova analysis. No underlying gravitational theory is assumed. This separation is essential: the acceleration scale entering the distance--redshift relation is not fitted to the supernova sample.
Circular Orbits
Circular Orbits
We consider circular motion in an acceleration field consisting of a Newtonian component and a constant inward term . For a test body in a circular orbit of radius about an enclosed baryonic mass (r), the radial acceleration is
A
r
M
b
In[]:=
a
in
GM[r]
2
r
Out[]=
-A-
GM[r]
2
r
The radial acceleration of a circular orbit is:
In[]:=
a
cent
2
v
r
The condition for circular motion then requires that this radial acceleration provide the centripetal acceleration,
In[]:=
a
cent
a
in
Out[]=
Ar+
GM[r]
r
2
v
Solving for the enclosed baryonic mass yields
In[]:=
M[v_,r_]=M[r]/.First[Solve[[r][r],M[r]]]
a
cent
a
in
Out[]=
-
r(Ar-)
2
v
G
Circular - Orbit Solutions
Circular - Orbit Solutions
This equation defines a two-parameter family of circular-orbit solutions relating baryonic mass , orbital radius , and orbital speed .
M
r
v
For fixed orbital speed , the mass function is a concave quadratic in . Differentiating gives
v
M(r)
r
dM
dr
2
v
G
The mass at a given velocity therefore reaches a maximum at
In[]:=
r
max
∂
r
Out[]=
2
v
2A
Substituting this radius into equation (1) yields the maximum baryonic mass compatible with a circular orbit at speed ,
v
In[]:=
M
max
r
max
Out[]=
4
v
4AG
The surface defines a family of circular-orbit solutions, as shown in Figure 1, with the ridge (v) giving the upper envelope of baryonic mass for a given orbital speed. Galaxies dominated by circular rotation are expected to saturate this boundary and follow a characteristic mass--velocity relation.
M(v,r)
M
max
In[]:=
M
max
r
max
Out[]=
4
v
4AG
In[]:=
params=GUnitConvertQuantity["GravitationalConstant"],,AQuantity,;massSurface[v_,r_]:=ConditionalExpressionUnitConvert[M[v,r],"SolarMass"],QuantityMagnitudeUnitConvert-Ar,>0/.params;massScale=;fundamentalPlane=Plot3DUnitConvertmassSurfaceQuantityv,,Quantity[r,"Kiloparsecs"],"SolarMass"massScale,{v,0,100},{r,0,30},AxesLabel"speed (km )","radius (kpc)","mass ",PlotRangeAll,PlotStyle->Directive[GrayLevel[0.5]],MeshStyle->Directive[GrayLevel[0.3]],Lighting->"Neutral";maximumMassCurve=ParametricPlot3Dv,QuantityMagnitudeUnitConvertQuantityv,,"Kiloparsecs",UnitConvertQuantityv,,"SolarMass"massScale/.params,{v,0,100},PlotStyle->{Directive[Red,Thick]},PlotPoints->100,MaxRecursion->4;fundamentalPlaneFigure=Show[{fundamentalPlane,maximumMassCurve},ViewPoint{-2,-3,1},ViewVertical{0,0,1},ImageSize->Large]
3
"Meters"
"Kilograms"
2
"Seconds"
1
11
10
"Meters"
2
"Seconds"
2
v
2
"Meters"
2
"Seconds"
1
Quantity[,"SolarMass"]
9
10
"Kilometers"
"Seconds"
-1
s
9
10
M
☉
r
max
"Kilometers"
"Seconds"
M
max
"Kilometers"
"Seconds"
Out[]=
Figure 1. The baryonic mass surface is shown as a function of orbital speed and orbital radius for an illustrative acceleration scale . The red ridge marks the locus (v) corresponding to the maximum baryonic mass permitted at each orbital speed. Galaxies populating this ridge reproduce the baryonic Tully--Fisher relation.
M(v,r)
v
r
Am
-11
10
-2
s
M
max
Baryonic Tully--Fisher Relation
Baryonic Tully--Fisher Relation
M
max
M∝
4
v
A
This parameter is determined empirically by fitting the circular-orbit envelope to the SPARC galaxy sample of McGaugh, Lelli and Schombert, using the assumptions 0.5, 1.33, zero intrinsic mass-to-light uncertainty, and the quality cut . These rotationally supported disk systems are well suited to this analysis, as their kinematics closely follow circular-orbit behavior.
Υ
★
f
gas
Qual≤3
For each galaxy the rotation curve provides an asymptotic circular velocity , while the baryonic mass is computed as +. The value of is determined by minimizing the statistic between the theoretical envelope and the observed pairs.
v
M
b
M
b
Υ
★
L
3.6
f
gas
M
HI
A
2
χ
(,v)
M
b
In[]:=
tapURL="https://tapvizier.cds.unistra.fr/TAPVizieR/tap/sync";adql="SELECT * FROM \"J/AJ/152/157/table1\"";raw=Import[URLBuild[tapURL,<|"request"->"doQuery","lang"->"adql","format"->"csv","query"->adql|>],"CSV"];header=First[raw];rows=Rest[raw];upsStar=0.5;fracUpsErr=0.0;gasFactor=1.33;maxQual=3;column=AssociationThread[header->Range[Length[header]]];get[row_,key_]:=row[[column[key]]];baryonicMass[row_]:=Module[{l36=N@get[row,"L3_6"],mhi=N@get[row,"MHI"]},upsStar*l36+gasFactor*mhi];massError[row_]:=Module[{l36=N@get[row,"L3_6"],el36=N@get[row,"e_L3_6"],mhi=N@get[row,"MHI"],dist=N@get[row,"Dist"],edist=N@get[row,"e_Dist"],fracDist,eStarPhot,eStarML,eGasDist,eStarDist},fracDist=If[dist>0,edist/dist,0.0];eStarPhot=upsStar*el36;eStarML=fracUpsErr*upsStar*l36;eStarDist=2*fracDist*(upsStar*l36);eGasDist=2*fracDist*(gasFactor*mhi);Sqrt[eStarPhot^2+eStarML^2+eStarDist^2+eGasDist^2]];vflat[row_]:=N@get[row,"Vflat"];vflatError[row_]:=N@get[row,"e_Vflat"];validRowQ[row_]:=Module[{v=vflat[row],ev=vflatError[row],q=get[row,"Qual"]},NumericQ[v]&&NumericQ[ev]&&v>0&&ev>=0&&q<=maxQual];goodRows=Select[rows,validRowQ];btfrData=({baryonicMass[#],vflat[#],massError[#],vflatError[#]}&)/@goodRows;
The best-fit acceleration scale is
Distance-Redshift Relation
Distance-Redshift Relation
We consider a phenomenological analytic form for the comoving distance,
The luminosity distance follows as
This analytic form differs from the standard FLRW integral representation and provides an alternative distance-redshift relation for comparison with observations.
Distance Modulus
Distance Modulus
Type-Ia supernova observations are reported in terms of the distance modulus, which relates the observed luminosity distance to the measured brightness of each supernova,
Kinematic Parameters
Kinematic Parameters
We treat the parameters in the analytic distance--redshift relation as kinematic quantities that set the temporal, velocity, and acceleration scales of the model, without assuming an underlying dynamical theory.
A boundary condition is imposed at the observation epoch by requiring that the parameter combination reproduce the locally measured speed of light,
Table 1 - Kinematic parameters determined from BTFR and Cepheid-calibrated Pantheon+ subset.
Type Ia Supernovae
Type Ia Supernovae
We compare the analytic distance--redshift relation with the Pantheon+SH0ES Type~Ia supernova dataset using covariance-weighted statistics. The resulting Hubble diagram and residuals are shown in Figure 3.
Figure 5. Pantheon+SH0ES distance moduli with PME (red) and spatially flat FLRW (blue) predictions evaluated on the same absolute calibration. Lower panel shows residuals relative to each model.
χ² Comparison
χ² Comparison
However, the difference is structural, not numeric. In FLRW, the expansion history is governed primarily by free density parameters. In PME, the same large-scale behavior is a direct consequence of the manifold kinematics.
Conclusion
Conclusion
Matter does not generate gravitational acceleration, nor does it deform spacetime. Instead, baryonic structure redistributes an ambient background acceleration scale, concentrating flux and producing spatial gradients. In the local bound regime this redistribution yields an inhomogeneous acceleration component obeying a Gauss-type constraint that reproduces the familiar inverse-square behavior associated with Newtonian gravity. The resulting local acceleration,
arises without invoking forces, potentials, or nonlocal interactions.
Within this framework, momentum is not exchanged through forces acting between distant bodies. Momentum change is the local, observable response of mass to a structured acceleration flux. Newton’s second law is retained operationally but reinterpreted: it describes how matter couples to an existing background acceleration scale rather than how bodies exert forces upon one another. Conservation of momentum follows from conservation of acceleration flux under redistribution, replacing action–reaction pairs with flux balance.
This construction resolves a long-standing conceptual tension in classical gravity. Newton’s laws successfully describe motion but provide no physical account of how gravitational influence is transmitted. Acceleration Flux Theory supplies that missing mechanism while preserving the empirical content of Newtonian dynamics. In doing so it also exposes the limitation of Newton’s first law: inertial motion is not fundamental but an approximation that neglects a universal background acceleration. Objects at rest do not remain at rest; they evolve under a persistent kinematic acceleration scale whose effects appear once acceleration is treated as primary.
Gravity therefore emerges not as a force and not as geometry, but as the organized redistribution of a conserved background acceleration scale. The same kinematic structure that governs cosmic expansion also governs local attraction, unifying gravitational phenomena across scales without additional assumptions or dark sectors. Acceleration Flux Theory thus provides a physically intelligible, local, and deterministic foundation for gravity—one that resolves several longstanding conceptual difficulties, including action at a distance, dynamical spacetime curvature, and the separation between cosmic expansion and local gravitational attraction, while remaining fully compatible with observed dynamics.