Research roadmap: Constraining β with SN, BAO, CMB, and GW datasets

Research roadmap: Constraining β with SN, BAO, CMB, and GW datasets

Overview

Goal: Derive quantitative constraints on the nonlinear elasticity constant β in Reactive Substrate Theory (RST) by integrating Supernova (SN), Baryon Acoustic Oscillation (BAO), Cosmic Microwave Background (CMB), and Gravitational Wave (GW) observations. The plan proceeds from background expansion fits (SN+BAO+CMB) to direct horizon-scale dispersion tests (GW), building a stepwise, defensible case.

Phase 1 — Background expansion (SN Ia)

  • Objective: Establish the baseline expansion history and anchor the effective fluid parameters (wS, Ωm0, ΩS0, H0).
  • Model: H2(z) = H02[Ωm0(1+z)3 + ΩS0(1+z)3(1+wS)].
  • Data: Compiled SN Ia distance moduli (z ≲ 1.5), standardized via light-curve fits.
  • Steps:
    1. Prepare likelihood: Build SN-only likelihood with nuisance parameters (stretch, color, absolute magnitude) marginalized.
    2. Priors: Start with wS ≈ −0.95 ± 0.05; flat priors on Ω fractions consistent with flatness.
    3. Fit: Sample (wS, Ωm0, ΩS0, H0) to obtain posteriors.
    4. Deliverable: Posterior means/credible intervals; derived q0.
  • Outcome for β: Indirect. Tightens the background that β must respect (no dispersion signal here).

Phase 2 — Mid-redshift geometry (BAO)

  • Objective: Sharpen constraints on H(z) and DA(z) to refine wS and Ωm0.
  • Data: BAO peak measurements in radial/transverse directions across z ≈ 0.1–1.1.
  • Steps:
    1. Joint SN+BAO fit: Combine SN likelihood with BAO (DV, DM, H(z)) measurements.
    2. Curvature test: Validate flatness; if needed, allow small curvature to stress-test robustness.
    3. Consistency checks: Ensure inferred Ωm0 matches clustering/lensing priors.
    4. Deliverable: Tighter posteriors on (wS, Ωm0, ΩS0).
  • Outcome for β: Indirect. Defines the allowed background expansion envelope within which β-induced dispersion must fit.

Phase 3 — Early anchors and lensing (compressed CMB)

  • Objective: Ensure RST parameters are consistent with early-universe anchors and lensing amplitudes.
  • Data: Compressed CMB parameters (θ*, Ωbh2, Ωch2, ns) and lensing amplitude constraints.
  • Steps:
    1. Add CMB priors: Integrate compressed likelihoods into SN+BAO fit.
    2. Check growth: Verify consistency of lensing amplitude with inferred Ωm0, σ8 from RST background.
    3. Deliverable: Globally consistent parameter set (wS, Ωm0, ΩS0, H0).
  • Outcome for β: Indirect. Confirms the background against which β’s dispersion will be tested; rules out β values that would disrupt early-time calibration.

Phase 4 — Direct horizon-scale dispersion (GW)

  • Objective: Measure or bound β via frequency- and scale-dependent gravitational-wave propagation.
  • Data: Pulsar Timing Arrays (PTA; nano-Hz), standard sirens (kHz with EM counterparts), and future low-frequency space missions.
  • Steps:
    1. Dispersion model: Derive β-dependent phase/group velocity law on cosmological baselines (k ≈ 10−27 m−1 regime).
    2. PTA analysis: Fit timing residual spectra for scale-dependent phase shifts indicative of horizon-scale dispersion.
    3. Standard siren cross-check: Compare GW-inferred luminosity distances to EM counterparts for redshift- or frequency-dependent offsets.
    4. Forecasts: Use Fisher/Bayesian forecasts to set expected β sensitivity for upcoming datasets (PTA, LISA, next-gen CMB B-modes).
    5. Deliverable: Upper limits or posterior on β; demonstration that any dispersion is horizon-scale-only (no local deviations).
  • Outcome for β: Direct. Provides the first quantitative bounds or detection-level constraints on β.

Integration and decision criteria

  • Joint posterior: Combine SN+BAO+CMB background fits with GW dispersion constraints to obtain a unified posterior over (β, wS, Ωm0, ΩS0, H0).
  • Model comparison: Compute AIC/BIC relative to ΛCDM; assess whether RST’s added predictivity (β-driven dispersion) is supported.
  • Viability thresholds:
    • Evidence for dispersion: Non-zero β with horizon-scale-only signature and null local tests.
    • Distinct wS: Posterior meaningfully different from −1 while maintaining fit quality.
    • Growth compatibility: Lensing/RSD consistent with inferred Ωm0, σ8.

Milestones and deliverables

  • M1: SN+BAO+compressed CMB fit; publish (wS, Ωm0, ΩS0, H0, q0).
  • M2: GW dispersion analysis (PTA + sirens); publish β upper bounds or posteriors.
  • M3: Integrated RST vs. ΛCDM comparison (AIC/BIC) with emphasis on unique horizon-scale predictions.

Practical guidance

  • Priors: Begin with conservative priors (wS = −0.95 ± 0.05; broad β log-prior centered near 10−26 m−2·J−1).
  • Transparency: Publish corner plots and residual diagnostics for each phase.
  • Robustness: Stress-test results under mild curvature and alternate SN/BAO compilations.
  • Clarity: Clearly state that β affects GW propagation only at horizon scales in RST; local constraints remain intact.

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