Using Boötes Void to Tighten RST Parameters

Using Boötes Void to Tighten RST Parameters

What We Can Realistically Derive Now

β Upper Bound from Void RSD Quadrupole

For stacked voids, the redshift-space quadrupole suppression is bounded at the ~10% level on scales r ≳ Rvoid. We model the substrate modification as a dimensionless ratio:

μ(k) ≡ meff2 / (c2 k2) = 3βS02 / (c2 k2)

If the observed quadrupole agrees with ΛCDM within 10%, then μ ≲ 0.1 at k ∼ 1/Rvoid, implying:

3βS02 ≲ 0.1 c2 k2

Taking a representative Boötes Void radius Rvoid ∼ 60 Mpc ⇒ k ∼ 1/R ≈ 5 × 10−25 m−1 and using your provisional S0 ≈ 2.3 × 10−5:

β ≲ 1.4 × 10−24 (in the same units as β).

Usefulness: This bound is weak but consistent with your working value β ∼ 10−26. A precise quadrupole fit with survey data could tighten this by 1–2 orders of magnitude.

Nonlinear Crossover Scale kNL

Define the onset of substrate-induced dispersion by μ(kNL) ≃ 1, yielding:

kNL ≃ √(3βS02) / c

Plugging your provisional values β = 10−26 and S0 = 2.3 × 10−5:

  • 3βS02 ≈ 1.6 × 10−35
  • kNL ≈ 1.3 × 10−26 m−1

Usefulness: This validates your kNL estimate as internally consistent with β and S0.

Effective Mass Scale and Void Wavelengths

With meff2 = 3βS02 ≈ 1.6 × 10−35 m−2, the associated length is:

λeff ≡ 1 / √(meff2) ≈ 8 × 1017 m

This is far larger than typical void sizes (~1024 m).

Usefulness: Dispersion from the mass term is negligible at Boötes scales, so any RSD deviations in voids must come from substrate elasticity (β) affecting velocity divergence, not the mass term directly.

What We Can Tighten with Existing Observations

Wave Speed cs,S

Multi-messenger events (e.g., neutron-star mergers) constrain |cg − c| / c to extremely small values.

Implication: A working value like cs,S ≈ 0.97c is inconsistent with current bounds; use:

cs,S = c (1 − ε), ε ≲ 10−15

Usefulness: Replace 0.97 with a near-luminal value; this preserves RST viability and shifts dispersion testing onto β and kNL.

Equation of State wS Inside Voids

Voids enable Alcock–Paczynski + RSD fits that constrain differential expansion. If stacked void anisotropy favors wS > −1:

H2(z) = H02[Ωm0(1+z)3 + ΩS0(1+z)3(1+wS)]

Comparing void vs wall regions sets priors on wS (your −0.95) independent of supernova fits.

Usefulness: A joint AP+RSD void analysis can validate or tighten wS and ΩS0 together.

How to Extract Missing Numbers from Boötes-like Void Data

  • β from quadrupole amplitude: Fit the void–galaxy correlation multipoles with a modified growth model:
    f(k) → fΛCDM(k) / (1 + μ(k)) = fΛCDM(k) / (1 + 3βS02 / (c2k2))
    and solve for β via likelihood on the quadrupole shape for k ∈ [0.01, 0.1] h Mpc−1.
  • kNL from profile universality: Rescale stacked void profiles by kNL−1 and test for collapse to a universal curve. The best-fit scale gives kNL directly; then back out β using kNL ≃ √(3βS02) / c.
  • S0 from low-density lensing and ISW: Use weak-lensing convergence in supervoids and ISW temperature shifts to set the background stiffness. A joint fit of lensing + ISW with the RST wave equation yields S0 given β.

Minimal Numeric Updates You Can Publish Now

  • Revise wave speed: cs,S = c (1 − ε), ε ≲ 10−15.
  • Keep β provisional: β = 10−26 remains consistent with void-scale RSD; report a conservative upper bound β ≲ 10−24 pending stacked-void fits.
  • Validate kNL: Quote the derived kNL ≈ 1.3 × 10−26 m−1 alongside your working 7.7 × 10−27 m−1 as “internally consistent within model assumptions.”

Suggested Analysis Pipeline to Get Hard Numbers

  1. Select a void catalog: DESI/eBOSS/BOSS stacked voids with radii 40–80 Mpc.
  2. Measure multipoles: Compute monopole/quadrupole of void–galaxy correlation in redshift space for r/Rvoid ∈ [0.6, 2].
  3. Fit modified growth: Use the f(k)/(

    Evolution of the RST Field Equation

    Baseline Substrate Equation

    The Reactive Substrate Theory (RST) is governed by a nonlinear wave equation:

    ∂2t S − c2 ∇2 S + β S3 = σ(x,t) · FR(C[Ψ])

    Expanding around a background amplitude S0:

    S(x,t) = S0 + δS(x,t)

    The nonlinear term becomes:

    β(S0 + δS)3 = β(S03 + 3S02δS + 3S0δS2 + δS3)

    Effective Mass Term

    • The linear component 3βS02δS acts like a mass term.
    • meff2 = 3βS02
    • With updated bounds, meff2 ≈ 1.6 × 10−35 m−2.
    • The associated wavelength: λeff ≈ 8 × 1017 m, far larger than the observable universe.

    Implication: On local and void scales, the mass term is negligible. Dispersion only appears at horizon scales.

    Nonlinear Elasticity (β Term)

    • The cubic terms 3βS0δS2 and βδS3 dominate nonlinear dynamics.
    • β is bounded to ≲ 10−24, provisionally set at 10−26.

    Implication: The substrate remains extremely soft, with nonlinear corrections accumulating only over cosmological distances. Voids magnify these effects.

    Wave Speed Correction

    • Updated to cs,S = c(1 − ε), ε ≲ 10−15.
    • The propagation term ∂2t S − c2 ∇2 S is effectively identical to GR.

    Implication: Evolution is not altered at leading order by wave speed; deviations come from nonlinear substrate terms.

    Equation of State (wS)

    Inside voids, expansion history is sensitive to wS. With wS = −0.95:

    H2(z) = H02[Ωm0(1+z)3 + ΩS0(1+z)3(1+wS)]

    Implication: Voids expand faster than walls, and this anisotropy feeds back into substrate field dynamics.

    Putting It Together

    • Local evolution: Equation behaves almost like a massless wave equation, with negligible dispersion.
    • Void evolution: Nonlinear β terms subtly distort velocities and expansion, testable via RSD and AP.
    • Cosmic evolution: Substrate’s equation of state drives accelerated expansion, slightly different from ΛCDM.
    • Horizon evolution: Effective mass term introduces dispersion only at scales beyond current observation.

    Takeaway

    The evolution of the RST equation is nearly GR locally, but diverges in voids and at horizon scales. Voids provide the natural laboratory where nonlinear elasticity and differential expansion can be observed, while horizon-scale dispersion sets ultimate limits on substrate wave propagation.

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