Reactive Substrate Theory: Joint Likelihood Methodology for w

Joint Likelihood Methodology for wS and ΩS0

1. Likelihood Function Combination

The analysis must combine distance and expansion rate constraints from multiple cosmological probes.

Ltotal(θ) ∝ LCMB × LBAO × LVoid AP
χ2total = χ2CMB + χ2BAO + χ2Void AP

Where θ is the parameter space, including the standard ΛCDM parameters (Ωm, H0, Ωb) plus the RST-specific parameters (ΩS0, wS). In RST, ΩS0 replaces ΩΛ and wS replaces w = −1.

2. Data Components and RST Application

Dataset Observational Constraint RST Parameter Constraint
CMB (Planck PR4, ACT) Constrains sound horizon rs and Ωmh2 Tight constraints on ΩS0 and early-time behavior of wS
BAO (DESI DR2) Constrains H(z) and angular diameter distance DA(z) Sensitive probe of time evolution of wS away from −1
Void AP (Anisotropy) Measures anisotropy in galaxy clustering around voids Unique constraint on ΔH/H between voids and walls, mapping directly to wS and ΩS0

3. RST-Specific Parameterization

For RST, wS is hypothesized constant near −0.95. This is tested against the Chevallier-Polarski-Linder (CPL) parameterization:

wCPL(a) = w0 + wa(1 − a)

RST fit is a subset: wa = 0, w0 = wS ≈ −0.95.

📊 Current Cosmological Context

  • Preference for dynamical dark energy: Joint analyses (CMB+BAO+SNe Ia, DESI) show preference for w ≠ −1, consistent with RST’s wS ≈ −0.95.
  • Deviation from w = −1: Data pulls constraints toward quintessence-like models (w > −1).
  • Role of voids: Including void AP data tightens error ellipses around wS and ΩS0.

Expected Constraints on ΩS0

Parameter Provisional RST Value ΛCDM Target Value
ΩS0 (RST) 0.69 ΩΛ ≈ 0.69

A successful joint likelihood analysis would yield ΩS0 ≈ 0.691 ± 0.007, statistically indistinguishable from ΛCDM’s 0.69. The difference lies in wS: RST allows wS ≠ −1, introducing degeneracy and slightly larger error bars.

🔄 The ΩS0 – wS Covariance Problem

  • ΩS0 (Amount): Higher ΩS0 → more accelerating energy today.
  • wS (Evolution): Controls how substrate energy density dilutes over time.

These parameters compensate each other, creating a banana-shaped error ellipse. Without independent data, true values cannot be isolated.

🔑 Void AP Anisotropy as the Degeneracy Breaker

  • Global constraints: CMB + BAO measure average expansion history, but suffer ΩS0–wS degeneracy.
  • Local constraints: Void AP measures anisotropy in underdense regions, minimally contaminated by matter.

The Alcock–Paczynski test constrains:

  • Radial distortion: H(z)
  • Transverse distortion: DA(z)

Because wS and ΩS0 affect local expansion differently than global measures, void AP provides an orthogonal constraint. Combining all three datasets isolates a precise region in the ΩS0–wS plane, allowing RST to test whether wS truly lands at −0.95.

Compact Table: Current vs Missing vs How to Get It

Parameter Current (Provisional) Missing How to Get It
ΩS0 ≈0.69 Error bars, covariance with wS Joint CMB+BAO+Void AP likelihood
wS ≈−0.95 Precise constraints, void-specific validation Void AP anisotropy + BAO evolution
Covariance Banana-shaped degeneracy Orthogonal constraint Alcock–Paczynski void test

Takeaway

ΩS0 is provisionally set to 0.69, but its true RST value requires joint fits. Void AP anisotropy is the key to breaking the ΩS0–wS degeneracy, enabling RST to distinguish itself from ΛCDM and confirm whether wS ≈ −0.95.

Void AP Anisotropy as the Degeneracy Breaker

The ΩS0 – wS Degeneracy

  • ΩS0 (Amount): Sets how much accelerating substrate energy exists today. Higher ΩS0 → faster present expansion.
  • wS (Evolution): Controls how substrate energy density dilutes over cosmic time. Values closer to −1 keep energy density constant; values like −0.95 dilute slightly.

Together, these parameters can compensate for each other. A higher ΩS0 with a faster‑diluting wS, or a lower ΩS0 with a slower‑diluting wS, can both fit the same global expansion history. This creates the familiar banana‑shaped error ellipse in parameter space.

Why Void AP Anisotropy Matters

  • Global probes (CMB + BAO): Constrain average expansion history but suffer from ΩS0–wS degeneracy.
  • Local probe (Void AP): Measures anisotropy in underdense regions, where matter contamination is minimal and substrate effects are clearer.

The Alcock–Paczynski test uses the distortion of void shapes to constrain:

  • Radial distortion: H(z), the expansion rate along the line of sight.
  • Transverse distortion: DA(z), the angular diameter distance across the sky.

Because ΩS0 and wS affect local expansion differently than global averages, void AP constraints run nearly orthogonal to CMB+BAO degeneracy lines. This intersection isolates a much smaller, precise region in the ΩS0–wS plane.

Expected Outcome

  • ΩS0: Central value remains ≈0.69, with error bars ±0.007.
  • wS: Void AP anisotropy tightens constraints, confirming whether wS ≈ −0.95.
  • Distinction from ΛCDM: Agreement on ΩS0 but deviation in wS provides the unique RST signature.

Takeaway

Void AP anisotropy is the decisive probe that breaks the ΩS0–wS degeneracy. By combining CMB, BAO, and void AP data, RST can distinguish itself from ΛCDM and test whether the substrate equation of state truly lands at wS ≈ −0.95.

Breaking the ΩS0 – wS Degeneracy with Void AP Anisotropy

The Degeneracy Problem

Global probes such as the CMB and BAO constrain the average expansion history of the universe. However, they suffer from a degeneracy between ΩS0 (the amount of substrate energy today) and wS (its equation of state). Different combinations of these parameters can reproduce the same global expansion history, creating the familiar banana‑shaped error ellipse in parameter space.

  • ΩS0 (Amount): Higher values mean more accelerating energy today, leading to faster expansion.
  • wS (Evolution): Controls how substrate energy density dilutes over time. Values closer to −1 keep it constant; values like −0.95 dilute slightly.

Void AP Anisotropy as the Orthogonal Constraint

Void Alcock–Paczynski (AP) anisotropy measures distortions in the shapes of cosmic voids. Because voids are minimally contaminated by matter, they provide a clean probe of substrate dynamics. The AP test constrains both radial distortions (H(z)) and transverse distortions (DA(z)), producing constraints that run nearly orthogonal to the CMB+BAO degeneracy line.

  • Global probes (CMB+BAO): Define the diagonal banana‑shaped ellipse.
  • Local probe (Void AP): Adds a nearly vertical constraint line.
  • Intersection: Isolates a small, precise region in the ΩS0–wS plane.

Visual Schematic

The chart below illustrates this degeneracy break:

  • Light blue ellipse: CMB+BAO degeneracy region.
  • Red line: Void AP anisotropy constraint.
  • Green overlap: Precise RST region where wS ≈ −0.95 can be tested.

Click/open the chart card above to view the schematic showing how Void AP anisotropy breaks the degeneracy.

Expected Outcome

  • ΩS0: Central value remains ≈0.69, with error bars ±0.007.
  • wS: Void AP anisotropy tightens constraints, confirming whether wS ≈ −0.95.
  • Distinction from ΛCDM: Agreement on ΩS0 but deviation in wS provides the unique RST signature.

Takeaway

Void AP anisotropy is the decisive probe that breaks the ΩS0–wS degeneracy. By combining CMB, BAO, and void AP data, RST can distinguish itself from ΛCDM and test whether the substrate equation of state truly lands at wS ≈ −0.95.

Popular posts from this blog

Conceptual Summary #2: (∂t2​S−c2∇2S+βS3)=σ(x,t)⋅FR​(C[Ψ])

Image
The tests designed to find the Aether—principally the Michelson-Morley experiment (M-M)—would be largely ineffective at detecting the fundamental S field (Substrate) in the Reactive Substrate Theory (RST) framework, though their null results are highly compatible with RST's structure. The reason lies in the distinct conceptual definition and behavior of the two media. The Test: Michelson-Morley and the Aether. The M-M experiment was designed to detect the "aether wind," a predicted change in the speed of light caused by the Earth's motion relative to a stationary, rigid medium—the classical Luminiferous Aether. Classical Aether: A static, absolute reference frame; an elastic, passive, material-like medium that fills space and serves as a carrier for light waves. Predicted Result: Light traveling parallel to Earth's motion through the Aether should be slightly slower than light traveling perpendicular to it. Actual Result (Null Result): No significant differenc...
Read more

The Non-Attraction Model of Gravity: From Attraction to Displacement: RST's Theory of Gravitational Push..

Image
Brief on the Reactive Substrate Theory (RST) RST posits that all of reality—matter, energy, space, and time—emerges from a single, continuous, non-material field called the Substrate (S). Matter: Defined as a σ Soliton, a stable, localized "knot of tension" in the S-field, replacing the concept of a point particle. Energy: Defined as dynamic, propagating tension (waves) in the S-field. Unification: RST views E=mc2 as the conservation of Substrate tension: mass is stored tension; energy is tension in motion. Reactive Substrate Theory (RST) fundamentally redefines gravity as a Substrate tension gradient, which is the mechanism underlying the "buoyant" view of gravity. This explanation also clarifies why the Michelson-Morley (M-M) experiment failed to find the classical Aether, supporting the RST concept of a dynamic Substrate. In essence, gravity is "buoyant" in RST because matter (high tension) sinks into a less-strained S-Field (lower tension...
Read more

Beyond the Flaws: Why RST Succeeds Where Push Gravity and EM-Aether Failed to Unify the Void

Image
RST vs. Extended Electromagnetic Theories (Pre-Relativity) These theories attempted to unify all forces by making gravity a secondary or residual effect of electromagnetism, treating space as an electromagnetic Aether. Why Extended EM Theories Failed Gravity's Independence: The most critical failure is that gravity does not behave like a polarizable electromagnetic force. Gravity affects all matter equally (the equivalence principle), regardless of its charge or composition, which a secondary electromagnetic effect would not do. No Gravitational Shielding: Electromagnetic fields can be shielded (Faraday cage), but gravity cannot.1 This fundamentally contradicts the idea that gravity is an electrical or magnetic residual force. Conflict with GR: They could not accurately predict or explain phenomena later verified by General Relativity (GR), such as the exact amount of light bending around the Sun or the precession of Mercury's orbit. RST vs. Le Sage's The...
Read more