28-EFT.WP.Propagation.PathRedshift v1.0 | Chapter 6 — Cosmological Redshift (FRW / Low-z Engineering Approximation)

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Chapter 6 — Cosmological Redshift (FRW / Low-z Engineering Approximation)

One-sentence goal: Under the FRW metric and low-redshift engineering approximations, establish a computable convention for cosmological redshift z_cos, orthogonally decomposed from motion/gravity/media terms; provide the companion relationship to the two forms of T_arr^{form1/form2}, with contracts and manifest publication.

I. Scope & Objects

1. Inputs
   • Cosmological parameters & epochs: H_0, Ω_m, Ω_Λ, Ω_k, a(t) or E(z) = H(z)/H_0; reference epochs t_emit / t_obs; RefCond (parameter provenance and version hashes).
   • Path & observation: TX/RX coordinates (comoving/local), observed spectral line / carrier f_obs or a derived z_meas (see Chapter 9).
   • Local terms: station/source local motion (already in z_kin), gravitational terms (Ch. 4), media terms (Ch. 5).
2. Outputs
   • z_cos and its approximations: z_cos(low-z), z_cos(exact); its composition within z_path;
   • Companion arrival-time correction ΔT_cos and attribution for the two-form gap delta_form;
   • Manifest manifest.redshift.cos.* with uncertainties u / U.
3. Boundary
   Default to homogeneous–isotropic FRW; strong lensing / large-scale velocity-field effects (ISW/Rees–Sciama) are handled as path extensions in Chapter 4.

II. Terms & Variables

• FRW & scale factor:
• ds^2 = -c_ref^2 dt^2 + a^2(t)[ dr^2/(1-kr^2) + r^2 dΩ^2 ], a(t), k ∈ {−1,0,+1}
• Cosmological parameters: H_0 (unit = "[1]/[T]"), Ω_m, Ω_Λ, Ω_k (dimensionless),
  E(z) = √(Ω_m(1+z)^3 + Ω_k(1+z)^2 + Ω_Λ).
• Redshift & arrival time: 1+z = a(t_obs)/a(t_emit); T_arr^{form1/form2}, ΔT_cos.
• Dimensions: unit(z) = 1, unit(H_0) = [1]/[T], unit(T_arr) = [T].

III. Postulates P65-6x

• P65-601 (Orthogonal composition): z_cos captures only global expansion and is orthogonal to z_kin / z_grav / z_med; composition follows
• 1+z_path = (1+z_kin)(1+z_grav)(1+z_med)(1+z_cos)(1+z_inst)(1+z_proc).
• P65-602 (Epochs & parameter traceability): RefCond must include parameter provenance (Planck/SH0ES, etc.), epoch/calendar, and coordinate-convention hashes.
• P65-603 (Explicit measures): any integral along redshift/time declares its domain:
  ( ∫_{0}^{z} • dz' ), ( ∫_{t_emit}^{t_obs} • dt ).
• P65-604 (Two-form pairing): publish z_cos / ΔT_cos together with T_arr^{form1/form2} and delta_form, and contractually enforce ≤ tol_Tarr.
• P65-605 (Dimensional compliance): all fields pass check_dim( y − f(x) ); record any log↔linear conversions via scale.note.

IV. Minimal Equations S65-6x

1. FRW exact form & low-z approximation

• S65-601 (Definition):
• 1 + z_cos = a(t_obs)/a(t_emit).
• S65-602 (Low-z z≲0.01–0.1 engineering approximation):
• v_H ≈ c_ref z_cos ≈ H_0 D (D: appropriate distance),

or, in narrow time-window comparisons,

z_cos ≈ H_0 Δt.

• S65-603 (General H(z) form):
• H(z) = H_0 E(z), E(z)=√(Ω_m(1+z)^3+Ω_k(1+z)^2+Ω_Λ).

2. Distance–redshift relations (for engineering conversions and guards)

• S65-604 (Comoving distance):
• χ(z) = ( c_ref / H_0 ) ( ∫_{0}^{z} dz' / E(z') ).
• S65-605 (Luminosity / angular-diameter distances):
• D_L = (1+z) S_k(χ), D_A = D_L/(1+z)^2;
• S_k(χ) = { sin(√|Ω_k| χ)/√|Ω_k|, χ, sinh(√|Ω_k| χ)/√|Ω_k| } for Ω_k>0,=0,<0.

Note: used here only for thresholding and guardband estimation, not for astronomical inference.

3. Removal of local motion (peculiar velocity)

• S65-606 (First-order correction):
• 1+z_cos ≈ (1+z_meas)/(1+z_kin_local),

where z_kin_local is obtained per Chapter 3 (projected local velocities at station/source).

4. Companion arrival-time correction

• S65-607. For narrowband engineering links, the single-pass timing impact of cosmology is typically negligible; for cumulative (deep-space, long-term) effects, provide
• ΔT_cos ≈ ( ∫_{0}^{z} dz' / H(z') ) − ( z / H_0 )

as a first-order difference, and mark it as a long-term metrology term.

5. Composition & small-signal approximation

• S65-608.
• 1+z_cos = a(t_obs)/a(t_emit)

composed multiplicatively with other terms; at low redshift,

z_cos ≈ H_0 D / c_ref,

and it enters linear-sum approximations of z_path.

V. Metrology Pipeline M65-6 (Ready → Modeling → Verification → Persistence)

1. Ready: select parameter sources (Planck / SH0ES / local fit), lock RefCond with version hashes; declare coordinates/epochs.
2. Model / estimate:
   • From z_meas and Chapter 3’s z_kin_local derive z_cos (S65-606);
   • Or compute z_cos from D or t_emit, t_obs with H_0, E(z) (S65-601–605).
3. Verify:
   • check_dim(z) = 1;
   • Compute T_arr^{form1/form2} in parallel and record delta_form ≤ tol_Tarr;
   • Record uncertainties u/U: u(H_0), u(Ω_*), u(z_meas), u(z_kin_local) and propagate (Ch. 13).
4. Persist:

manifest.redshift.cos = { cosmo.hash, params:{ H_0, Ω_m, Ω_Λ, Ω_k },

z_cos, method:{ low-z | FRW }, ΔT_cos,

T_arr_forms, delta_form, u/U, RefCond, contracts.*, signature }

VI. Contracts & Assertions C65-6x (suggested thresholds)

• C65-601 (Two-form gap): delta_form_p95 ≤ tol_Tarr.
• C65-602 (Parameter freshness): age(cosmo.params) ≤ Δt_max; cross-source consistency | H_0^{A} − H_0^{B} | ≤ τ_{H0} when multi-source is used.
• C65-603 (Local-motion removal): computing z_cos must reference Chapter 3’s z_kin_local; if omitted, do not publish.
• C65-604 (Method annotation): method ∈ { low-z, FRW } must be persisted; low-z applicability z ≤ z_max_low is stated.
• C65-605 (Dimensional compliance): unit(z) = 1, unit(H_0) = [1]/[T], unit(ΔT_cos) = [T].

VII. Implementation Bindings I65-6* (interfaces, I/O, invariants)

• I65-61 frw_Hz(H0, Ω_m, Ω_Λ, Ω_k, z) -> { H_z, E_z, meta }
• I65-62 comoving_distance(H0, Ω_*, z) -> { χ, D_L, D_A, meta }
• I65-63 estimate_z_cos_from_meas(z_meas, z_kin_local) -> { z_cos, meta }
• I65-64 estimate_z_cos_from_D_or_t(D|t_emit,t_obs, H0, Ω_*) -> { z_cos, meta }
• I65-65 cosmology_to_Tarr_correction(z_cos, method) -> { ΔT_cos, T_arr_forms }
• I65-66 assert_cosmo_contracts(ds, rules) -> { report, pass }
• I65-67 emit_cosmo_manifest(results, policy) -> manifest.redshift.cos
  Invariants: two_forms_present = true; check_dim(*) passes; parameter sources & epoch hashes are traceable; method and applicability are recorded.

VIII. Cross-References

• Redshift baseline & two forms: Chapter 2; kinematics: Chapter 3 (for local-motion removal); gravity/path terms: Chapter 4; media: Chapter 5.
• Worldlines / path integrals: Chapter 8; observation conventions: Chapter 9; fusion & calibration: Chs. 10 / 11; uncertainty & publication: Ch. 13 and Appendices C / E.

IX. Quality & Risk Control

• SLI / SLO: | z_cos |_p95 (near-Earth links should be minute),
  z_resid_p95 = | z_meas − ( z_kin + z_grav + z_med + z_cos ) |_p95, delta_form_p95, panel_freshness, params_age_p95.
• Fallback: parameter sources inconsistent/expired → revert to previous version and enlarge guardbands; z_resid breach → shorten windows or switch to low-z approximation; two-form excess → standardize to form2 and recheck other components.
• Audit: cosmological parameter sources & hashes, low-z/FRW switch points, near-Earth vs deep-space case evidence, and the manifest.redshift.cos signature chain with replay scripts.

Summary

• Using the FRW and low-redshift approximations as baselines, this chapter provides a computable–verifiable–publishable convention for z_cos, paired with the two-form T_arr.
• With M65-6 / C65-6x / I65-6* and manifest.redshift.cos.*, cosmological redshift can be stably isolated in deep-space and engineering links and consistently composed with motion/gravity/media components into z_path.