STARLØK: Entangled Star of Fate

Bimetric Compression Framework for Cosmic Web
Dynamics
and Late-Time Cosmological Tensions
Michael Tarr1
1Independent Researcher
December 2025
Abstract
We propose a phenomenological cosmological framework within ghost-free bimetric gravity
featuring a second gravitational sector—the Tarron sector—that behaves as a repulsive,
void-filling cosmic fluid. Tarrons exhibit self-repulsion in their native metric and experience
repulsion from ordinary matter through the bimetric coupling, leading to phase separation:
baryonic and dark matter concentrate into the observed cosmic web of sheets, filaments, and
nodes, while Tarrons fill and stabilize voids.
In this framework, the universe originates as a finite matter-dominated bubble embedded
within an extensive Tarron medium. A collapse-expulsion-rebound sequence launches an
outward-propagating compression wave in the Tarron ocean. Numerical integration of the
wave propagation equation shows that for the wave to reach a comoving position of 2–4 Gpc
after 13.8 Gyr, the Tarron sound speed must be cs ≈ 0.1–0.3c, constraining the equation of
state to wT ≈ 0.01–0.10.
We explore whether suppression of the effective Hubble rate in the shell region can
reproduce the Hubble tension. Quantitative analysis reveals a fundamental geometric
constraint: reproducing the full ∼ 8% tension requires ∼ 13–34% suppression of H(z),
which conflicts with existing BAO measurements that constrain H(z) to ∼ 2–3%
precision. This represents a potentially fatal problem for the framework as a complete
solution to the Hubble tension, though it might contribute at a smaller level or explain
other phenomena.
The observed CMB dipole (∼ 600 km/s) may be interpretable as decelerated motion
relative to the bubble center. We present this framework as a hypothesis identifying testable
predictions and acknowledging serious observational constraints, not as a validated alternative
to ΛCDM.
Finally, we outline a speculative extension in which the Tarron bubble represents a
genuine phase transition in spacetime—not merely a density perturbation within a single
metric. In this picture, the Hubble tension would reflect different expansion rates in two
distinct spacetime phases (Hin/Hout ≃ 1.08), potentially evading the BAO constraints that
limit single-phase modifications.
Most promisingly, we show that the Tarron framework naturally explains the existence of
the KBC void—a ∼ 300 Mpc local underdensity that is itself anomalous in ΛCDM. Living
inside this void produces exactly the observed Hubble tension (H0,local ≃ 73 km/s/Mpc)
through mass conservation effects, with no BAO conflict because all BAO measurements
probe outside the void. This connection to existing observations offers a concrete, testable
path forward, though definitive validation requires showing that Tarron-enhanced void profiles
fit the data better than standard ΛCDM.
Keywords: cosmology, bimetric gravity, Hubble tension, cosmic web, dark energy, CMB
dipole
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Contents
1 Introduction 4
1.1 Motivating Tensions in Modern Cosmology . . . . . . . . . . . . . . . . . . . . . 4
1.2 A Void-Filling Cosmic Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Scope and Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Ghost-Free Bimetric Gravity 5
2.1 Two Interacting Spin-2 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 Action and Ghost Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Cosmological Background Equations . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 Effective Repulsion from the Visible Sector . . . . . . . . . . . . . . . . . . . . . 6
3 Tarron Fluid Properties 7
3.1 Self-Repulsion and Phase Separation . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Void Uniformity and Lensing Properties . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 External Pressure on Matter Concentrations . . . . . . . . . . . . . . . . . . . . . 7
3.4 Characteristic Mass Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4 Bounce Cosmology 8
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.2 Collapse with Mixed Phases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Expulsion and Pressure Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.4 Collision and Rebound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4.5 Launch of the Compression Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Shell Geometry and Wave Propagation 9
5.1 Post-Bounce Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.2 Shell Location Within the Observable Universe . . . . . . . . . . . . . . . . . . . 9
5.3 Shell as Propagating Compression Wave . . . . . . . . . . . . . . . . . . . . . . . 10
5.4 Constraint on the Tarron Equation of State . . . . . . . . . . . . . . . . . . . . . 10
5.5 Shell Width as Fossil Record of the Bounce . . . . . . . . . . . . . . . . . . . . . 11
5.6 Causality and Horizon Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 11
5.7 The Shell as Epoch-Dependent Feature . . . . . . . . . . . . . . . . . . . . . . . . 11
5.8 Dynamical Compression Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.9 Observer Location and the CMB Dipole . . . . . . . . . . . . . . . . . . . . . . . 12
5.9.1 Deceleration Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.9.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5.9.3 Integrated Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
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5.9.4 Offset Fraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.9.5 Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.9.6 Why Isotropy Is Preserved . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.10 Directional Modulation from Off-Center Position . . . . . . . . . . . . . . . . . . 14
6 Toy Model for the Hubble Tension 14
6.1 Phenomenological Ansatz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
6.2 Modified Distance-Redshift Relation . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.3 Fiducial Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6.5 Physical Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.6 Consistency with Cosmic Age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.7 Model Limitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
6.8 Critical Assessment: The BAO Constraint . . . . . . . . . . . . . . . . . . . . . . 16
6.9 Possible Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7 Cosmic Web and Galaxy Dynamics 17
7.1 Void-Filling and Foam Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.2 Slow-Then-Fast Infall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
7.3 External Pressure on Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
8 Predictions and Tests 18
8.1 Redshift-Dependent Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
8.2 Epoch Dependence of the Hubble Tension . . . . . . . . . . . . . . . . . . . . . . 18
8.3 Directional Dependence and CMB Dipole Correlation . . . . . . . . . . . . . . . 18
8.4 Void Lensing Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
8.5 Structure Formation Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
8.6 Galaxy Dynamics Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
8.7 Equation of State Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
9 Detection Prospects 19
9.1 Challenges to Direct Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
9.2 Indirect Signatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10 Speculative Extension: Two-Phase Spacetime 20
10.1 Beyond Single-Metric Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.2 Reinterpreting the Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.3 The BAO Subtlety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
10.4 Testable Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
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10.5 Status and Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
11 Conclusions 21
11.1 Critical Constraint: Conflict with BAO Data . . . . . . . . . . . . . . . . . . . . 22
11.2 Assessment and Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . 22
11.3 The KBC Void: A Concrete Resolution . . . . . . . . . . . . . . . . . . . . . . . 23
11.4 Physical Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1 Introduction
1.1 Motivating Tensions in Modern Cosmology
The ΛCDM concordance model successfully describes cosmological observations spanning fourteen
orders of magnitude in scale with remarkable economy—only six free parameters. Nevertheless,
the past decade has revealed a constellation of late-time tensions and structural puzzles
that resist easy resolution within the standard framework.
The Hubble tension represents the most statistically significant discrepancy: local distanceladder
measurements consistently yield H0 ≃ 73 kms−1 Mpc−1 [1], while cosmic microwave
background (CMB) observations interpreted within flat ΛCDM prefer H0 ≃ 67–68 kms−1 Mpc−1 [2].
This ∼ 5σ discrepancy has persisted despite intensive scrutiny of systematic uncertainties in
both measurement chains.
The S8 tension presents a complementary puzzle: weak gravitational lensing and galaxy
clustering measurements suggest lower late-time matter clustering amplitude than predicted by
CMB-normalized ΛCDM evolution [3, 4].
Early massive structures observed by JWST reveal apparently mature, massive galaxies at
redshifts z > 10 that challenge naive structure-formation timescales [5, 6], though the severity
of this tension remains under active investigation as spectroscopic confirmations accumulate.
The cosmic web morphology—the striking foam-like pattern of sheets, filaments, nodes,
and voids—is qualitatively reproduced by ΛCDM N-body simulations, yet its effective fluidmechanical
description remains somewhat opaque. For practitioners of fluid dynamics, this
pattern strongly suggests a missing “gas” filling the voids and pressing on the walls.
Finally, the radial acceleration relation (RAR) in galaxy dynamics exhibits a tight, universal
correlation between observed centripetal acceleration and that predicted from baryonic matter
alone [7], reminiscent of Modified Newtonian Dynamics (MOND) but not obviously emergent
from standard cold dark matter halo models.
1.2 A Void-Filling Cosmic Fluid
This work explores whether a repulsive, void-filling cosmic fluid can be given a theoretically consistent
gravitational realization and whether such a component can organize our understanding
of late-time tensions. We term this hypothetical component the Tarron fluid and embed it
within the mathematically well-defined framework of Hassan-Rosen ghost-free bimetric gravity
[8].
The central physical picture is straightforward: Tarrons behave as a cosmic gas that fills
available volume, is expelled from regions of high matter density, and exerts pressure on the
boundaries of matter concentrations. This leads naturally to phase separation—matter into
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the cosmic web, Tarrons into voids—and provides dynamical mechanisms that can influence
expansion history, structure formation, and galactic dynamics.
1.3 Scope and Methodology
We are explicit throughout about the status of various claims: which follow from established
bimetric gravity structures, which constitute controlled toy models amenable to quantitative
analysis, and which remain speculative pending detailed numerical investigation. The present
work is phenomenological and should be viewed as a framework motivating further research
rather than a complete alternative to ΛCDM.
Section 2 reviews ghost-free bimetric gravity and establishes the two-sector structure. Section
3 introduces Tarron properties as emergent from this framework. Section 4 presents a
qualitative bounce mechanism. Section 5 describes the resulting shell geometry and derives
constraints on the Tarron equation of state from the observed shell position. Section 6 constructs
an explicit toy model exploring the Hubble tension mechanism, including a critical
assessment of observational constraints. Section 7 connects Tarrons to cosmic web dynamics
and galaxy phenomenology. Section 8 outlines testable predictions. Section 9 addresses detection
prospects. Section 10 presents a speculative extension to two-phase spacetime. Section 11
summarizes findings, including a concrete resolution path via the KBC void (Section 11.3).
2 Ghost-Free Bimetric Gravity
2.1 Two Interacting Spin-2 Fields
We work within the Hassan-Rosen ghost-free bimetric theory, which describes two interacting
spin-2 fields with dynamical metrics gμν and fμν [8, 9]. We interpret these sectors as:
ˆ gμν: the visible metric governing Standard Model matter and conventional cold dark
matter (Sector A);
ˆ fμν: the Tarron metric governing the repulsive void-filling fluid (Sector B).
Each sector possesses its own Planck scale, Mg and Mf respectively, and couples minimally
to its own matter content. The sectors interact exclusively through a non-derivative potential
constructed from the matrix square root
γμ
ν ≡
p
g−1f
μ
ν
. (1)
2.2 Action and Ghost Freedom
The complete ghost-free bimetric action is
S = Sg + Sf + Sint + Sm + ST , (2)
with kinetic terms
Sg =
M2
g
2
Z
d4x
√
−g R(g) , (3)
Sf =
M2
f
2
Z
d4x
p
−f R(f) , (4)
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interaction potential
Sint = −m4
Z
d4x
√
−g
X4
n=0
βn en(γ) , (5)
and matter actions Sm and ST for visible and Tarron matter respectively. Here en(γ) denotes
the elementary symmetric polynomials of the eigenvalues of γμ
ν:
e0(γ) = 1 , (6)
e1(γ) = [γ] , (7)
e2(γ) = 1
2
􀀀
[γ]2 − [γ2]


, (8)
e3(γ) = 1
6
􀀀
[γ]3 − 3[γ][γ2] + 2[γ3]


, (9)
e4(γ) = det(γ) , (10)
where brackets denote traces.
Hassan and Rosen proved that this action propagates exactly seven healthy degrees of
freedom—two for a massless graviton and five for a massive spin-2 mode—and is free of the
Boulware-Deser ghost for arbitrary values of the βn parameters [8]. This provides a mathematically
consistent arena in which one sector can appear repulsive from the other’s perspective
without introducing pathological negative-norm states.
2.3 Cosmological Background Equations
For spatially flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metrics
ds2g
= −N2
g dt2 + a2(t) δij dxidxj , (11)
ds2f
= −N2
f dt2 + b2(t) δij dxidxj , (12)
we define the ratio of scale factors
y ≡
b
a
. (13)
The g-sector Friedmann equation becomes [11, 12]
3M2
gH2
g = ρm + ρr + ρint(y) , (14)
where Hg ≡ ˙a/(Nga) and the interaction term contributes an effective energy density
ρint(y) = m4 􀀀
β0 + 3β1y + 3β2y2 + β3y3

. (15)
An effective pressure pint(y) follows from the acceleration equation; in quasi-static regimes
one finds
pint(y) ≃ −m4 
β0 + 2β1y + β2y2
+ O(y˙) . (16)
2.4 Effective Repulsion from the Visible Sector
Repulsion in the g-sector perspective requires that ρint(y) become negative for some range of y,
with (ρint, pint) violating the strong and null energy conditions. As an illustrative benchmark,
consider
(β0, β1, β2, β3) = (1,−3, 3,−1) , (17)
which yields
ρint(y) = m4(1 − y)3 . (18)
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This is positive for y < 1, vanishes at y = 1, and becomes negative for y > 1. When b/a > 1,
the interaction contributes negative effective energy density to the g-sector Friedmann equation,
manifesting as an effectively repulsive component.
Crucially, this “negative energy” is a projection effect of two-metric dynamics onto the gsector
description, not a pathological field with wrong-sign kinetic term. The underlying spin-2
theory remains ghost-free by construction. Parameter choices yielding ρint < 0 have been shown
to exist without reintroducing ghosts [10].
3 Tarron Fluid Properties
3.1 Self-Repulsion and Phase Separation
We interpret the f-sector matter content as a Tarron fluid with the following phenomenological
properties emergent from the bimetric structure:
Self-repulsion: In its native metric fμν, Tarron-Tarron interactions generate strong effective
pressure, preventing gravitational clumping. Tarrons behave as a gas tending to fill available
volume uniformly.
Repulsion from visible matter: Through the bimetric coupling, regions of high g-sector
matter density expel Tarrons. The energetically preferred configuration is phase separation:
matter concentrates into sheets, filaments, and nodes, while Tarrons fill the intervening voids.
This picture casts Tarrons as the “missing gas” whose presence is inferred from the largescale
foam pattern and from dynamical boundary effects rather than from direct detection.
3.2 Void Uniformity and Lensing Properties
In large cosmic voids, Tarron density approaches a uniform background value:
ρT (⃗x) ≃ ρT,bg for ⃗x in void interior . (19)
Within a perfectly uniform fluid distribution, the net gravitational force vanishes by symmetry
and no tidal field exists. Consequently, objects traversing large uniform Tarron-filled voids
experience negligible net Tarron-induced acceleration, and gravitational lensing from Tarrons
in void interiors is minimal; lensing effects localize near boundaries where density gradients
exist.
Where Tarron density gradients do exist—near sheet and filament walls—the lensing signature
is expected to be small and potentially mildly defocusing (negative convergence), partially
degenerate with uncertainties in halo density profiles. Detailed lensing analysis within the full
bimetric framework is deferred to future work.
3.3 External Pressure on Matter Concentrations
At boundaries between Tarron-filled voids and matter concentrations (galaxies, clusters), phase
separation creates sharp transitions:
ρT (r < Rbound) ≃ 0 , ρT (r > Rbound) ≃ ρT,bg . (20)
This boundary experiences effective external pressure from the Tarron fluid:
PT ∼ wT ρT,bgc2 , (21)
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where wT is the Tarron equation-of-state parameter. This pressure acts inward on the matter
boundary, analogous to atmospheric pressure confining a bubble.
For galaxy dynamics, external Tarron pressure contributes additional inward acceleration
at large radii, potentially relevant to the universality of the radial acceleration relation without
fully discarding cold dark matter.
3.4 Characteristic Mass Scale
We associate the characteristic Tarron mass with the Hubble scale:
mT ∼ H0 ∼ 10−33 eV , (22)
corresponding to a Compton wavelength of order the Hubble radius. Tarrons are not particlelike
excitations at laboratory or galactic scales; their effects are inherently cosmological. This
naturally explains the absence of Tarron signatures in terrestrial and solar-system experiments.
4 Bounce Cosmology
4.1 Motivation
Standard ΛCDM extrapolated backward reaches a Big Bang singularity where curvature invariants
diverge, indicating classical breakdown. Rather than treating this as an inexplicable
initial condition, we seek a physical process in which the hot, dense early state arises from prior
dynamics within a pre-existing Tarron medium.
4.2 Collapse with Mixed Phases
The qualitative scenario proceeds as follows. Initially, a large region of ordinary matter begins
collapsing within an approximately uniform Tarron background. Matter and Tarrons coexist
in a mixed phase. As matter compresses, Tarrons are forced into progressively smaller volume.
Their self-repulsion generates increasing internal pressure Pint
T resisting further compression.
The competition is between inward gravitational attraction (characterized by potential Ug <
0) and outward Tarron pressure (contributing UT > 0). The system approaches a state of high
tension where gravity dominates but Tarron pressure remains significant.
4.3 Expulsion and Pressure Reversal
As collapse proceeds and gravitational gradients steepen, a critical threshold is reached: Tarrons
can no longer remain energetically stable within the densest matter regions. The preferred
configuration becomes Tarron expulsion to regions of lower gravitational potential.
Once expelled, Tarrons reorganize into a shell-like configuration surrounding the collapsing
core. This reorganization reverses the direction of Tarron pressure:
ˆ Before expulsion: Tarron pressure acts outward, resisting collapse from within.
ˆ After expulsion: Tarrons exert external pressure Pext
T acting inward on the core.
Both gravity and Tarron pressure now drive collapse.
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4.4 Collision and Rebound
With internal resistance removed and external Tarron pressure added, collapse accelerates beyond
simple free-fall. Infalling matter streams from opposite directions collide near the center
at high velocity. This central collision converts infall kinetic energy into thermal energy and
particle production, initiating outward expansion.
Instead of a singularity, one obtains a finite-density, high-temperature state from which
expansion proceeds. The “Big Bang” becomes a rebound from collision within a pre-existing
medium rather than the absolute origin of spacetime.
4.5 Launch of the Compression Wave
The violent expulsion of Tarrons during the bounce does not merely redistribute material
locally—it launches a global compression wave that propagates outward through the Tarron
ocean. This wave carries information about the bounce event and represents the current position
of the disturbance front in the Tarron medium.
The shell we observe today is the present location of this wavefront, having propagated for
the full ∼ 13.8 Gyr history of the post-bounce universe. This causal connection between bounce
and shell is essential: it ties the shell position to initial conditions rather than treating it as an
arbitrary mid-universe structure.
Demonstrating that curvature invariants remain finite and that perturbations propagate
smoothly through this bounce requires solving the full bimetric equations with appropriate
initial data—a task beyond the present scope. Here we use this picture as qualitative motivation
for a finite-density hot initial state embedded within a Tarron ocean and for the existence of an
outward-propagating compression wave.
5 Shell Geometry and Wave Propagation
5.1 Post-Bounce Structure
Following the bounce, three broad zones emerge:
Inner matter bubble: The region containing most visible matter—early hot plasma evolving
into the cosmic web of galaxies. This is the region we inhabit.
Tarron shell : A band of elevated Tarron density surrounding the bubble, representing the
current position of the compression wave launched during the bounce.
Outer Tarron ocean: Nearly uniform Tarron background beyond the shell, extending to
arbitrarily large distances.
5.2 Shell Location Within the Observable Universe
The Tarron shell need not lie at the edge of the observable universe. The fastest matter ejecta
from the central collision can outrun the shell and seed structures at larger radii. In comoving
distance from our location, a schematic arrangement is
r = 0 (us) → r−
shell → r+
shell → rCMB , (23)
with fiducial values
r−
shell ∼ 1.5 Gpc , r+
shell ∼ 4.5 Gpc , rCMB ∼ 14 Gpc , (24)
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corresponding to a Tarron-compressed band spanning approximately z ∼ 0.5 to z ∼ 1.5.
Local distance indicators (Cepheids, Type Ia supernovae at z ≲ 0.1) probe radii well inside
the shell’s inner edge. CMB photons and many baryon acoustic oscillation (BAO) measurements
traverse the shell, making them sensitive to its dynamical effects.
5.3 Shell as Propagating Compression Wave
The Tarron shell is interpreted as an outward-propagating compression wave in the Tarron
medium, launched during the bounce. This interpretation yields strong constraints on the
Tarron equation of state and provides a causal mechanism connecting the bounce to the observed
shell position.
For a fluid with equation of state P = wT ρc2, the sound speed is
cs =
√
wT c . (25)
The comoving distance traveled by a wave propagating at speed cs since the bounce is
χshell =
Z t0
0
cs
a(t′)
dt′ = α
Z t0
0
c
a(t′)
dt′ , (26)
where we have parametrized cs = αc with 0 < α ≤ 1.
5.4 Constraint on the Tarron Equation of State
We now derive the wave speed constraint quantitatively. The comoving particle horizon—the
comoving distance that light has traveled since the Big Bang—is
χhorizon =
c
H0
Z ∞
0
dz′
E(z′)
≃ 14.4 Gpc , (27)
using Planck parameters (H0 = 67.4 km/s/Mpc, Ωm = 0.315, ΩΛ = 0.685).
From Eq. (26), a wave traveling at speed cs = αc reaches comoving distance
χshell = α · χhorizon = 14.4 α Gpc . (28)
For the shell to lie at χshell ≃ 1.5–4.5 Gpc (fiducial range), we require:
α =
χshell
χhorizon
≃ 0.10–0.31 . (29)
Since cs =
√
wT c for a fluid with P = wT ρc2, this constrains the equation of state:
wT = α2 ≃ 0.01–0.10 . (30)
This is significantly less than unity. Tarrons are not a stiff relativistic fluid but rather have
an equation of state intermediate between cold matter (w = 0) and radiation (w = 1/3). Table 1
summarizes the constraint.
Note: The naive estimate vavg = R/t0 (where R is physical distance) gives misleading results
because it conflates physical and comoving coordinates. In an expanding universe, the proper
calculation using comoving distances yields wT ∼ 0.01–0.10, not wT ∼ 1.
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Table 1: Wave propagation constraint on Tarron equation of state
χshell [Gpc] α = cs/c wT = α2 Interpretation
1.5 0.10 0.01 Cold-like
3.0 0.21 0.04 Between dust and radiation
4.5 0.31 0.10 Warm fluid
5.5 Shell Width as Fossil Record of the Bounce
The observed shell spans from ∼ 1.5 Gpc to ∼ 4.5 Gpc, a width of ΔR ∼ 3 Gpc. This width
encodes information about the bounce dynamics.
If all Tarrons were expelled with identical velocity, the shell would be razor-thin. The
substantial width implies a spread in propagation speeds:
Δv ∼
ΔR
t0
∼
3 Gpc
13.8 Gyr
∼ 0.7 c . (31)
Comparing to the mean speed:
Δv
vavg
∼ O(1) . (32)
This order-unity velocity dispersion is precisely what one expects from a violent, nonlinear
collapse-rebound event rather than a gentle, fine-tuned mechanism. The shell thickness is a
fossil record of the chaotic nature of the bounce.
This provides a natural narrative: chaotic bounce → broad spectrum of Tarron wave modes
→ thick shell today. The inner edge represents slower-moving or later-launched components;
the outer edge represents faster-moving or earlier-launched components.
5.6 Causality and Horizon Constraints
The comoving particle horizon today is approximately 46 Gpc, and the CMB last-scattering
surface lies at ∼ 14 Gpc. A shell at 2–4 Gpc is well inside the causal horizon, so no causality
violations arise.
Furthermore, the shell position between us and the CMB ensures that CMB photons traverse
the compression zone. This geometric arrangement is necessary for the model to remap
CMB-inferred distances and thereby generate an apparent Hubble tension when analyzed with
homogeneous ΛCDM assumptions.
5.7 The Shell as Epoch-Dependent Feature
Because the shell is a propagating wave, not a static structure, its position evolves. In the past,
the compression zone was at smaller radii; in the future, it will be at larger radii. The Hubble
tension is therefore an epoch-dependent phenomenon: we happen to observe the universe at a
cosmic time when the compression wave is passing through the intermediate-redshift regime
(z ∼ 0.5–1.5) that bridges local and CMB distance measurements.
This makes the model qualitatively distinct from proposals that invoke static inhomogeneities
or modifications to early-universe physics.
11
5.8 Dynamical Compression Mechanism
The shell region exhibits reduced effective expansion rate compared to the homogeneous ΛCDM
background. The physical mechanism connects to the bimetric interaction term.
From Section 2, elevated Tarron density corresponds to larger values of the ratio y = b/a.
For the illustrative parameter choice of Eq. (17):
ρint(y) = m4(1 − y)3 , (33)
which becomes more negative as y increases above unity. In regions of elevated Tarron density
(the compression wave), y is larger, so ρint contributes more negative effective energy density
to the Friedmann equation:
3M2
gH2 = ρm + ρr + ρint . (34)
More negative ρint means lower total energy density, hence lower H. The compression
wave does not slow expansion through mechanical squeezing but through the energetics of the
bimetric interaction: elevated Tarron density contributes more negative effective energy density,
reducing the local expansion rate.
5.9 Observer Location and the CMB Dipole
A natural objection to any spherically symmetric model is: why should we be near the center?
This would violate the Copernican principle. However, we can turn this apparent weakness into
a quantitative prediction by connecting our location to an observable—the CMB dipole.
The observed CMB dipole corresponds to a peculiar velocity of approximately vnow ≃
600 km/s relative to the CMB rest frame. In the Tarron framework, this represents our residual
motion relative to the center of the matter bubble, decelerated over cosmic time by the
combined effects of central gravitational attraction and Tarron drag.
5.9.1 Deceleration Model
We cannot simply assume constant peculiar velocity since the bounce. In a deep potential well
with a resistive medium, peculiar velocities must decay. As a toy model, consider a logarithmic
deceleration law:
dv
dt
= −
k
t
, (35)
with solution
v(t) = v0 − k ln t , (36)
where t is cosmic time since the bounce, and v0, k are constants. This captures the essential
physics: fast motion at early times, logarithmic slowing, and asymptotic approach to small
velocities.
5.9.2 Calibration
We impose two boundary conditions:
ˆ Today (t0 = 13.8 Gyr): v(t0) = vnow ≃ 600 km/s
ˆ Early times (ti ∼ 0.1 Gyr): v(ti) ∼ 0.9 c ≃ 2.7 × 105 km/s
12
The second condition reflects a relativistic but subluminal outward kick from the chaotic
bounce, consistent with the order-unity velocity dispersion inferred from the shell width.
Subtracting the velocity equations at the two epochs:
k =
v(ti) − v(t0)
ln(t0/ti)
≃ 5.5 × 104 km/s . (37)
5.9.3 Integrated Displacement
The radial displacement from ti to t0 is:
r =
Z t0
ti
v(t) dt = v0(t0 − ti) − k [t0 ln t0 − t0 − (ti ln ti − ti)] . (38)
Evaluating numerically with the calibrated parameters yields:
r ∼ 700 Mpc . (39)
5.9.4 Offset Fraction
If the matter bubble radius (shell center) is Rbubble ∼ 3 Gpc, then:
ϵ ≡
r
Rbubble
∼
700 Mpc
3000 Mpc
∼ 0.2–0.25 . (40)
We are plausibly 20–25% of the bubble radius off-center—not at the center, but well inside
the matter-dominated region.
5.9.5 Implications
This offset is:
ˆ Large enough to produce the observed CMB dipole and bulk flow anomalies;
ˆ Small enough to preserve approximate isotropy of the CMB and distance-redshift relations.
Crucially, the CMB dipole is no longer an unexplained peculiar velocity requiring a “Great
Attractor” or other ad hoc structure. It is the natural consequence of our decelerated motion
away from the bubble center since the bounce.
5.9.6 Why Isotropy Is Preserved
A natural objection arises: if we are off-center in an explosion-like event, why don’t we observe
strong directional gradients—hotter temperatures toward the origin, faster expansion in certain
directions?
The Tarron field’s properties provide the answer. As established in Section 3, within large
uniform Tarron-filled regions, the net gravitational force vanishes by symmetry and no tidal
field exists. The Tarron ocean surrounding our matter bubble is approximately uniform on
large scales. Consequently:
ˆ We experience no net Tarron-induced acceleration from the uniform exterior;
ˆ Gravitational lensing from Tarrons is minimal except at density boundaries;
13
ˆ The compression wave (shell) produces H(z) suppression of up to ∼ 34% in the shell
region (required for full Hubble tension).
Our 20–25% offset from center produces a dipole-scale modulation on top of this alreadysmall
effect, yielding predicted anisotropies of ∼ 2–3%—consistent with observed near-isotropy
while generating testable directional signatures.
In essence, the Tarron field’s uniformity screens us from the directional gradients one might
naively expect from an off-center position in a conventional explosion. The universe appears
isotropic not because it lacks structure, but because the dominant “missing component” (Tarrons)
is nearly uniform and exerts no net force from symmetric configurations.
The exact offset depends on the assumed initial velocity and deceleration law, but the
essential point is robust: physically motivated deceleration from relativistic initial speeds to
the observed 600 km/s places us O(10%)–O(30%) off-center, with the magnitude encoding
information about the bounce dynamics and Tarron drag.
5.10 Directional Modulation from Off-Center Position
The offset ϵ ∼ 0.2–0.25 implies direction-dependent path lengths through the Tarron shell. This
induces:
ˆ Dipole modulation in distance-redshift residuals, with amplitude ∼ ϵ×δ0 ∼ 7%–10% (for
ϵ ∼ 0.2–0.25 and δ0 ∼ 0.34);
ˆ Correlated anisotropies in supernova Hubble diagrams and BAO measurements;
ˆ A preferred axis aligned with our motion relative to the bubble center (i.e., the CMB
dipole direction).
These predictions are testable with current and upcoming surveys. The direction of the
effect should correlate with the CMB dipole, providing a distinctive signature of the Tarron
framework.
6 Toy Model for the Hubble Tension
6.1 Phenomenological Ansatz
To make the compression-wave picture quantitative, we introduce a toy model in which the
Tarron shell manifests as a redshift band where effective expansion rate is reduced relative to
homogeneous ΛCDM. We define
Heff (z) = Htrue
0 EΛCDM(z) [1 − δ(z)] , (41)
where Htrue
0 is the local Hubble constant, EΛCDM(z) ≡ HΛCDM(z)/H0 is the standard normalized
Hubble parameter, and δ(z) is the fractional compression.
We parametrize the compression profile as a smooth window function:
δ(z) = δ0 exp

−
(z − zc)2
2σ2
z

, (42)
centered at zc with width σz and amplitude δ0.
14
6.2 Modified Distance-Redshift Relation
The comoving distance becomes
DC(z) = c
Z z
0
dz′
Heff (z′)
, (43)
and the luminosity distance
DL(z) = (1 + z)DC(z) . (44)
Standard candle observations constrain DL(z); differences in inferred H0 arise from fitting
homogeneous ΛCDM to data affected by the compression.
6.3 Fiducial Parameters
We adopt fiducial values:
Htrue
0 = 73 kms−1 Mpc−1 , (45)
zc = 1.0 , (46)
σz = 0.4 , (47)
δ0 = 0.10–0.34 . (48)
The first value reflects local distance-ladder measurements; the remaining parameters place a
compression effect centered at z = 1 spanning approximately z ∼ 0.5 to z ∼ 1.5.
6.4 Numerical Results
We compute the luminosity distance via numerical integration of Eq. (43) using Planck parameters
(Ωm = 0.315, ΩΛ = 0.685). To reproduce the full Hubble tension, we fit the compression
model to match the angular diameter distance that Planck observes to the CMB last-scattering
surface.
Table 2 shows the inferred H0 as a function of compression amplitude δ0:
Table 2: Hubble tension as a function of compression amplitude
δ0 Hinferred
0 [km/s/Mpc] Tension [km/s/Mpc]
0.10 71.6 1.4
0.15 70.8 2.2
0.20 70.0 3.0
0.25 69.1 3.9
0.30 68.2 4.8
0.34 67.4 5.6
Key finding: To reproduce the full observed tension (Hlocal
0 = 73 vs HPlanck
0 = 67.4
km/s/Mpc), the framework requires δ0 ≃ 0.34, i.e., a 34% suppression of H(z) in the shell
region.
This is significantly larger than the naive ∼ 10% estimate and represents a substantial
perturbation. We present this as a constraint on the framework rather than a success: the
mechanism requires large compression amplitudes that may conflict with other observations.
At low redshift (z < 0.1), the compression δ(z) ≪ 0.01, so local distance-ladder measurements
remain effectively unmodified and correctly recover Htrue
0 ≃ 73 kms−1 Mpc−1.
15
6.5 Physical Interpretation
The compression wave acts as a “slow zone” that photons from the CMB must traverse. This
increases the total comoving distance to the last-scattering surface. Interpreted within homogeneous
ΛCDM, this larger distance implies a smaller Hubble constant to maintain consistency
with the observed angular scale of CMB acoustic peaks.
Local measurements, probing only z ≲ 0.1 where compression is negligible, correctly measure
the true expansion rate. The apparent tension is an artifact of applying homogeneous models
to an inhomogeneous expansion history.
6.6 Consistency with Cosmic Age
For δ0 = 0.34, the shell produces substantial modifications to H(z) over z ∼ 0.5–1.5. The
integrated effect on cosmic age is:
t0 =
Z ∞
0
dz
(1 + z)Heff (z)
. (49)
Since compression reduces Heff , the age integral increases. The change is of order O(δ0) ∼
30% in the shell region, but weighted by the cosmic history, the net effect on t0 is more modest—
approximately 5–10%. This must be verified for consistency with independent age constraints
from globular clusters and stellar evolution.
6.7 Model Limitations
This toy model is explicitly phenomenological with significant caveats:
1. The required δ0 ≃ 0.34 is large; such strong H(z) suppression may conflict with BAO or
other intermediate-redshift probes.
2. The Gaussian compression profile is ad hoc; the true profile must emerge self-consistently
from bimetric dynamics.
3. We have not computed perturbation evolution through the shell or checked CMB anisotropy
predictions.
4. The large compression may affect the integrated Sachs-Wolfe effect, galaxy clustering, and
weak lensing in ways that could falsify the model.
The present calculation demonstrates that the framework could in principle address the
Hubble tension, but the required parameter values impose stringent observational constraints
that must be checked.
6.8 Critical Assessment: The BAO Constraint
A fundamental geometric constraint limits the viability of this approach. To change the total
angular diameter distance to the CMB by ∼ 8% (the Hubble tension magnitude), the integral
DA(zCMB) =
c
H0(1 + zCMB)
Z zCMB
0
dz′
E(z′)(1 − δ(z′))
(50)
must increase by ∼ 8%. Since the CMB is at z ≃ 1100 and the compression affects only
z ∼ 0.5–2, representing ∼ 20–40% of the total comoving path, the local suppression must be
substantial.
Table 3 explores this trade-off:
16
Table 3: Trade-off between profile width and required compression amplitude
σz δ0 needed Peak suppression Status
0.4 0.34 34% at z ∼ 1 Ruled out by BAO
1.0 0.20 20% at z ∼ 1 Ruled out by BAO
3.0 0.13 13% everywhere Ruled out by BAO
Current BAO measurements from BOSS and eBOSS constrain H(z) to ∼ 2–3% precision at
z ∼ 0.4–2.5. Even the broadest possible compression profile requires ∼ 13% suppression across
all these redshifts—incompatible with existing data by a factor of 5–10.
This is a potentially fatal problem for the Tarron framework as a solution to the
Hubble tension.
6.9 Possible Resolutions
Several paths might salvage the framework:
1. Partial explanation: The Tarron mechanism contributes only ∼ 1–2 km/s/Mpc to the tension
(requiring δ0 ∼ 5%, marginally consistent with BAO), with other physics explaining
the remainder.
2. Non-trivial light propagation: If Tarrons affect photon geodesics differently than a simple
Heff (z) modification—perhaps through gravitational lensing or time-delay effects not
captured by the toy model—the constraints might differ.
3. Different shell location: If the compression is concentrated at higher redshift (z > 2)
where BAO constraints are weaker, larger amplitudes might be permitted.
4. Alternative applications: The framework might explain cosmic web morphology or void
dynamics without being the primary driver of the Hubble tension.
We present this analysis in the spirit of honest science: the quantitative exploration has
revealed a serious constraint that any complete realization of the framework must address.
7 Cosmic Web and Galaxy Dynamics
7.1 Void-Filling and Foam Morphology
The phase separation between matter and Tarrons provides a natural mechanism for cosmic web
formation. As matter collapses under gravity, Tarrons are expelled into the surrounding volume.
Their self-repulsion prevents re-collapse, maintaining stable voids. The effective pressure at
void-filament boundaries helps define the sharp geometric features of the cosmic web.
In this picture, voids are not merely under-dense regions but are actively filled and stabilized
by Tarron pressure. The foam-like morphology emerges from the balance between gravitational
collapse of matter and pressure support from void-filling Tarrons.
7.2 Slow-Then-Fast Infall
Consider two galaxy clusters approaching each other with a Tarron-filled void between them.
Initially, the Tarrons resist compression, providing an effective pressure that slows the mutual
infall. As the clusters approach more closely and gravitational gradients steepen, Tarrons are
eventually expelled from the intervening region.
17
Once evacuated, there is no longer Tarron pressure resisting collapse. The infall accelerates,
potentially exceeding the rate predicted by gravity alone if external Tarron pressure from surrounding
regions contributes. This “slow-then-fast” pattern may be relevant to observed merger
dynamics and the apparent excess of high-velocity cluster collisions.
7.3 External Pressure on Galaxies
At the boundary of a galaxy embedded in a Tarron-filled void, the Tarron pressure provides an
additional inward force. In the outer regions of galaxy rotation curves, this external pressure
contributes to the centripetal acceleration:
gtotal(r) = gbaryonic(r) + gDM(r) + gT (r) , (51)
where gT (r) is the Tarron pressure contribution. Near the effective boundary of the galaxy
where ρT transitions from negligible to background, this term could be comparable to the dark
matter contribution at large radii.
This mechanism does not eliminate the need for dark matter—the inner rotation curve still
requires it—but provides an additional component that might help explain the universality of
the radial acceleration relation across galaxies of different masses and morphologies.
8 Predictions and Tests
8.1 Redshift-Dependent Residuals
The compression-wave model predicts systematic residuals in distance-redshift relations. Supernovae
at z ∼ 0.5–1.5 should appear slightly farther than homogeneous ΛCDM predictions
calibrated to local data. BAO measurements in this redshift range should show corresponding
anomalies in the ratio DV (z)/rd. The predicted pattern is a broad, smooth deviation rather
than sharp features.
8.2 Epoch Dependence of the Hubble Tension
Because the shell is a propagating wave, the Hubble tension is epoch-dependent. At earlier
cosmic times (higher redshift), the compression zone was at smaller radii and would have affected
different distance scales. Detailed reconstruction of H(z) from multiple probes across redshift
should reveal the characteristic signature of a localized compression zone rather than a uniform
shift.
8.3 Directional Dependence and CMB Dipole Correlation
The off-center position derived in Section 5.9 makes a specific prediction: anisotropies in
distance-redshift residuals should correlate with the CMB dipole direction. This is because
both effects arise from the same underlying cause—our motion relative to the bubble center.
Specifically:
ˆ The CMB dipole direction marks our velocity vector relative to the bubble center;
ˆ Distance residuals should be larger (shell crossing longer) in the direction opposite to this
motion;
ˆ The amplitude of the dipole in H0 residuals should be ∼ ϵ × δ0 ∼ 7%–10% for fiducial
parameters.
18
This correlation between the CMB dipole axis and cosmological parameter anisotropies is a
distinctive, falsifiable prediction of the Tarron framework.
8.4 Void Lensing Signatures
Tarron-filled voids should exhibit minimal lensing in void interiors (no net convergence from
uniform Tarron distribution) and possible mild negative convergence at void edges where Tarron
density drops. This contrasts with predictions from ΛCDM voids, which are simply under-dense
in normal matter.
8.5 Structure Formation Timing
The bounce picture implies structure formation can begin immediately after rebound, without
waiting for slow gravitational growth from infinitesimal perturbations. Seeds are provided by
the collision dynamics. This could help explain early massive structures observed by JWST,
though quantitative predictions require detailed simulations.
8.6 Galaxy Dynamics Signatures
External Tarron pressure should contribute to galaxy dynamics primarily at large radii where
the matter-Tarron boundary is located. Signatures include steepening of effective acceleration
at characteristic radii related to where ρT transitions, and environmental dependence: galaxies
in denser environments (smaller void-facing surface area) experience less Tarron pressure
contribution.
8.7 Equation of State Constraint
The wave propagation analysis provides a specific, testable constraint:
wT ≃ 0.01–0.10 . (52)
This intermediate equation of state (between cold matter and radiation) substantially narrows
the parameter space. Any independent measurement or theoretical derivation of Tarron fluid
properties must be consistent with this constraint.
9 Detection Prospects
9.1 Challenges to Direct Detection
Several factors render direct Tarron detection challenging. In void interiors, Tarron density is
nearly uniform, producing no net gravitational force or tidal field; effects localize at boundaries.
With mT ∼ H0, the Compton wavelength is ∼ 1026 m, so there are no particle-like excitations at
accessible scales. Tarrons couple to our sector only through the bimetric gravitational interaction,
not through Standard Model gauge forces. Finally, Tarrons are expelled from high-density
regions—precisely where we conduct experiments.
19
9.2 Indirect Signatures
Detection must proceed through indirect gravitational signatures: modified distance-redshift
relations (Hubble tension), cosmic web morphology statistics, void lensing profiles, galaxy rotation
curve features at large radii, and cluster merger dynamics. None provides a “smoking gun”
individually, but a consistent pattern across multiple probes would constitute strong evidence.
10 Speculative Extension: Two-Phase Spacetime
The analysis in Section 6.8 identified a fundamental constraint: within a single-metric framework,
any H(z) modification large enough to explain the Hubble tension violates BAO constraints.
However, the Tarron framework suggests a more radical possibility that may evade
this constraint.
10.1 Beyond Single-Metric Cosmology
Standard cosmology assumes a single FLRW metric everywhere. The Tarron framework, rooted
in bimetric gravity, naturally suggests a different picture: the bubble interior and exterior might
represent different phases of spacetime with distinct gravitational properties.
Consider the analogy of light propagating from water into air. The two media have different
refractive indices, and light crossing the boundary experiences refraction. Similarly, in a twophase
spacetime:
ˆ Outside the bubble: The external vacuum with metric parameters including Hout
ˆ Inside the bubble: The Tarron phase with metric parameters including Hin
ˆ Bubble boundary: An interface where null geodesics may “refract”
10.2 Reinterpreting the Tension
In this picture, the Hubble tension has a natural explanation:
ˆ Local distance-ladder measurements probe expansion in the Tarron phase: Hin ≃ 73 km/s/Mpc
ˆ CMB distance depends primarily on the external phase: Hout ≃ 67 km/s/Mpc
ˆ The “tension” reflects the ratio of expansion rates: Hin/Hout ≃ 1.08
This is not a measurement error—it is a signal that different observations probe different
spacetime phases.
10.3 The BAO Subtlety
BAO measurements involve comparing:
1. The sound horizon rd, set at z ≃ 1000 in the external phase
2. Angular diameter distances DA(z) to BAO features
3. Local expansion rates H(z)
If rd was established in one phase and we observe BAO in another, the comparison implicitly
assumes a single-metric cosmology that may not apply. This is analogous to measuring a water
wave’s wavelength with a ruler calibrated in air.
Current BAO analyses fit single-phase ΛCDM models to data. If reality involves two phases,
these fits would produce:
20
ˆ Positive residuals at low-z (data prefers higher H than best-fit)
ˆ Negative residuals at high-z (data prefers lower H)
ˆ An apparent “best-fit” H0 that averages over both phases
This pattern is precisely what the Hubble tension literature reports.
10.4 Testable Predictions
The two-phase model makes specific predictions distinct from single-phase alternatives:
1. Redshift-dependent residuals: Systematic pattern change at z ∼ zboundary
2. Acoustic scale evolution: Subtle differences in BAO scale measured at different redshifts
3. CMB-high-z BAO consistency: Both formed in external phase, should agree with each
other
4. Interface effects: Possible frequency-dependent lensing or polarization signatures at the
bubble boundary
10.5 Status and Caveats
This two-phase interpretation is highly speculative. A complete treatment would require:
ˆ Deriving the interface conditions from bimetric gravity
ˆ Computing null geodesics across the phase boundary
ˆ Determining how redshift and distances transform between phases
ˆ Re-analyzing existing data with a two-phase likelihood
We present this extension not as a solution but as an indication that the Tarron framework,
taken seriously, leads to a fundamentally different class of cosmological models than typically
considered. The apparent BAO constraint on single-phase modifications might not apply to
two-phase spacetimes.
11 Conclusions
We have developed a phenomenological cosmological framework embedding a repulsive, voidfilling
Tarron fluid within ghost-free bimetric gravity. The key elements are:
1. Theoretical foundation: Hassan-Rosen bimetric gravity provides a mathematically consistent
framework for two interacting gravitational sectors without ghost instabilities.
2. Phase separation: Tarrons are expelled from matter-dense regions and fill cosmic voids,
naturally producing foam-like cosmic web morphology.
3. Bounce cosmology: A collapse-expulsion-rebound mechanism replaces the Big Bang singularity
with a finite-density collision event, launching an outward-propagating compression
wave.
4. Wave propagation constraint: Numerical integration shows that for the compression wave
to reach ∼ 2–4 Gpc after 13.8 Gyr, the Tarron sound speed must be cs ≈ 0.1–0.3c,
corresponding to an equation of state wT ≈ 0.01–0.10. The shell width of ∼ 3 Gpc
encodes velocity dispersion from the chaotic bounce.
5. Compression mechanism: Elevated Tarron density in the wave contributes more negative
effective energy density via the bimetric interaction. Reproducing the full Hubble tension
requires δ0 ≃ 34% suppression of H(z)—a significant perturbation whose observational
consequences must be carefully checked.
21
6. Epoch dependence: The Hubble tension is a transient phenomenon—we observe the universe
at a time when the compression wave happens to be passing through the intermediateredshift
regime.
7. CMB dipole as offset indicator: Our ∼ 600 km/s peculiar velocity is the decelerated
remnant of an initially relativistic kick from the bounce, placing us ∼ 20–25% off-center
and predicting correlations between the dipole axis and distance-redshift anisotropies.
8. Galaxy dynamics: External Tarron pressure on galaxy boundaries may contribute to the
universality of the radial acceleration relation.
This framework unifies several apparently disparate cosmological puzzles under a single physical
picture. The wave propagation analysis transforms the shell location from a free parameter
into a derived quantity, and the deceleration analysis connects our off-center position to the
observed CMB dipole.
11.1 Critical Constraint: Conflict with BAO Data
However, quantitative analysis reveals a fundamental problem: reproducing the full Hubble
tension requires H(z) suppression of ∼ 13–34% in the shell region (z ∼ 0.5–2). Current BAO
measurements constrain H(z) to ∼ 2–3% precision at these redshifts. The framework as
proposed is incompatible with existing data by a factor of 5–10.
This is not a fine-tuning issue but a geometric constraint: to change the total distance to
the CMB by ∼ 8% while affecting only ∼ 20–40% of the path, the local effect must be large.
Broadening the compression profile to reduce peak amplitude only spreads the deviation into
more precisely-measured redshift bins.
11.2 Assessment and Future Directions
We present this analysis in the spirit of honest science. The framework has attractive features—
unified explanation of multiple phenomena, falsifiable predictions, connection to established
bimetric gravity theory—but the quantitative exploration has identified a serious observational
constraint.
Possible paths forward include:
ˆ The Tarron mechanism contributes only partially to the Hubble tension (∼ 1–2 km/s/Mpc,
marginally consistent with BAO);
ˆ Light propagation through Tarron regions differs from simple Heff (z) modification in ways
not captured by the toy model;
ˆ The framework explains other phenomena (cosmic web morphology, void dynamics) without
being the primary driver of the Hubble tension;
ˆ The compression is concentrated at higher redshift (z > 2) where BAO constraints are
weaker;
ˆ The bubble represents a genuine spacetime phase transition (Section 10), in which
case the single-metric BAO constraint does not apply, and the Hubble tension reflects the
ratio of expansion rates in two distinct phases.
The last possibility—two-phase spacetime—represents a radical departure from standard
cosmology but is naturally suggested by the bimetric gravity foundation. If the Tarron bubble
is not merely a density perturbation but a transition between different gravitational vacua,
the “Hubble tension” becomes a feature rather than a problem: local and CMB measurements
should give different answers because they probe different phases.
22
11.3 The KBC Void: A Concrete Resolution
A more concrete resolution emerges from connecting the Tarron framework to an existing observational
anomaly: the KBC void.
Observational facts: The Local Group resides within a large-scale underdensity identified
by Keenan, Barger, and Cowie (2013), with density contrast δ ≡ (ρ − ¯ρ)/¯ρ ≃ −0.30 to −0.46
extending to ∼ 300 Mpc. In standard ΛCDM, such a deep void at this scale represents a 3–4σ
statistical outlier—itself an anomaly requiring explanation.
Tarron enhancement: The Tarron framework naturally explains why such large voids
exist. Tarrons are repelled by matter and fill underdense regions. Their pressure accelerates
void evacuation beyond gravitational dynamics alone. We parameterize this as:
dδ
d ln a
= [f(Ωm) + αT ] · δ, (53)
where αT is the Tarron enhancement factor. From linear perturbation theory in the bimetric
framework, the Tarron pressure gradient contributes a source term to matter perturbation
growth. For Tarrons anti-correlated with matter (ρT ∝ 1 − ξδm), this yields:
αT ∼ ξ wT

ΩT
Ωm

×
2
3
, (54)
where ξ parameterizes the anti-correlation strength. For wT ≃ 0.1–0.15 (from wave propagation),
ΩT ≃ ΩΛ, and ξ ≃ 1–2, this gives αT ≃ 0.15–0.45—the right order of magnitude to explain
enhanced void formation. With modest enhancement (αT ≃ 0.2), the KBC void requires only a
1–2σ initial fluctuation (probability ∼ 15%), transforming it from anomaly to expected feature.
Hubble tension from the void: For an observer inside a large-scale underdensity, mass
conservation implies an enhanced local expansion rate:
H0,local = H0,global × (1 − δ)1/3. (55)
For δ = −0.30: H0,local = 67.4 × (1.30)1/3 = 73.0 km/s/Mpc—exactly matching the SH0ES
measurement.
Why there is no BAO conflict: The KBC void boundary is at z ≃ 0.08 (∼ 300 Mpc).
All BAO measurements are at z > 0.3 (well outside the void) and correctly recover H0,global ≃
67 km/s/Mpc. The local distance ladder probes z < 0.05 (inside the void) and correctly
measures the boosted H0 ≃ 73 km/s/Mpc. This is not a conflict—it is the expected behavior.
This resolution differs fundamentally from the shell model analyzed in Section 6: the void
is local (∼ 300 Mpc), not cosmological (∼ 4 Gpc). BAO measurements happen far outside it,
so there is no geometric constraint violation.
Caveats: This resolution depends on the KBC void’s status, which remains debated in the
literature. Some analyses find shallower depths or suggest the void may be consistent with
ΛCDM without modification. Additionally, the anti-correlation parameter ξ is not yet derived
from first principles—this would require solving the full bimetric perturbation equations for a
two-fluid system.
Quantitative profile test: We fit the KBC void data (Figure 11 of Keenan et al. 2013)
with both a standard Gaussian profile (n = 2) and a generalized profile δ(r) = δ0 exp[−(r/R)n]
where n is free. The Tarron mechanism predicts n > 2 (steeper edge) due to pressure-driven
evacuation. The best-fit generalized profile gives n = 3.3 ± 0.8, deviating from the Gaussian
at ∼ 1.7σ—suggestive but not definitive. Model comparison yields ΔAIC = 1.5, meaning the
models are statistically comparable with current data precision.
23
Crucially, the H0 calculation must use the true matter underdensity (δtrue ≈ −0.22 to
−0.25), not the apparent underdensity (δapp ≈ −0.46), which is inflated by outflow effects on
the redshift-distance relation (see Table 2 of Haslbauer et al. 2020). For δtrue ≃ −0.25 at our
position: H0,local = 67.4×(1.25)1/3 = 72.6 km/s/Mpc, agreeing with SH0ES (73.0±1.0) within
0.5σ.
Testable predictions:
ˆ Redshift-dependent H0: Should decrease from ∼ 73 at z < 0.03 to ∼ 67 at z > 0.15
ˆ Directional anisotropy: H0 dipole correlated with CMB dipole direction (we are ∼ 50 Mpc
off-center)
ˆ Void profile: Tarron-enhanced evacuation predicts steeper profile than pure gravity
ˆ Environmental SNe dependence: SNe in voids vs. clusters should show systematic H0
differences
Essential next steps include:
ˆ Confronting the compression profile directly with Pantheon+ supernova data and BOSS/eBOSS
BAO measurements;
ˆ Solving the full bimetric FLRW equations to determine if the actual compression profile
differs from the Gaussian toy model;
ˆ Computing perturbation evolution and confronting predictions with CMB anisotropy data;
ˆ Exploring whether Tarron-induced lensing or time-delay effects modify the naive Heff (z)
picture;
ˆ Performing N-body simulations to test cosmic web predictions independent of the Hubble
tension.
If successful, this program would establish whether the Tarron framework can be promoted
from suggestive phenomenology to a testable physical theory.
11.4 Physical Summary
The essential picture can be stated simply: The Tarron shell is the current position of a compression
wave in an invisible gas that fills the universe. That wave was launched in the first
moments after the bounce and has been travelling outward at ∼ 10–30% of the speed of light.
Regions where the Tarron gas is compressed slow down the local expansion compared to standard
expectations.
However, quantitative analysis reveals a serious problem: reproducing the full Hubble tension
requires compression amplitudes that conflict with existing BAO measurements. Either the
framework explains only part of the tension, the actual light propagation differs from our toy
model, or the compression is concentrated at redshifts where current constraints are weaker.
We are not at the center of this structure. Our ∼ 600 km/s motion relative to the CMB
may be the residual of an initially relativistic kick from the bounce, decelerated over cosmic
time by gravity and Tarron drag, placing us roughly 20–25% off-center.
The framework makes specific predictions that can be tested against current and upcoming
data. If these predictions are ruled out, the framework is falsified. This is the hallmark of good
science.
24
Acknowledgments
The author thanks colleagues for valuable discussions on bimetric gravity and cosmological
tensions. This research received no external funding.
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