w(z) Derivation

Purpose

This document derives the STF prediction for the dark energy equation of state w(z) from first principles. The derivation has three outputs:

1. Structural result: w(z=0) = −1 exactly — a topological consequence of the T² nodal structure, not a fine-tuning.
2. Phantom trajectory: w(z) < −1 for all z > 0 — effective phantom without a ghost field.
3. Falsification: The STF w(z) shape is categorically distinct from the DESI CPL best-fit and directly testable by Euclid.

This derivation also establishes STF’s relationship to the phantom problem in field theory, resolving the tension between effective phantom behavior and vacuum stability.

§1. Foundation: The T² Coupling Integral

§1.1 Setup

From STF Energy V0.2 §A.1–A.5, the T² topology sources Λ_eff through the mode structure of φ_S on the compact time dimension. The fundamental mode is:

φ(θ) = cos(θ), θ = πt/T_compact ∈ [0, π]

with maximum at the Big Bang (θ = 0), node at mid-epoch (θ = π/2), minimum at the terminal boundary (θ = π).

The coupling integral accumulated from θ = 0 to current epoch θ_now is:

α(θ) = ∫₀^θ cos²(θ’) dθ’ = θ/2 + sin(2θ)/4

The π/4 derivation (Energy V0.2 §A.3–A.4) establishes that: - Full-period coupling vanishes: ∫₀^π cos(θ)Ṙ dθ = 0 (positive and negative lobes cancel) - Physical coupling restricted to causal diamond θ ∈ [0, π/2] - At current epoch: α(π/2) = π/4 — exact

§1.2 Current Epoch Identification

The paper’s derivation places the current epoch at θ_now = π/2: the causal diamond boundary, the node of the fundamental mode. This is not assumed — it is the domain fixed by the full-period cancellation argument. The physical coupling cuts off where cos(θ) = 0, which is θ = π/2.

This identification fixes:

T_compact = 2t₀

where t₀ is the current age of the universe. Equivalently: ξ = 1/(H₀T_compact) = 1/(2H₀t₀).

Numerical value: H₀t₀ = 0.9447 (computed from Ω_m = 4/(3(1+π)) background), so ξ = 0.529.

Note: T_compact = 2t₀ is derived from θ_now = π/2. The magnitude of phantom deviations scales as ξ = 1/(H₀T_compact). Whether T_compact equals exactly 2t₀ or is larger — which requires the full DHOST field equation solution — is an open item (§6). The structural results in §2 hold regardless.

§2. The w(z) Formula

§2.1 Derivation from Continuity

Λ_eff evolves as:

Λ_eff(t) = Λ_obs × α(θ(t)) / (π/4)

where θ(t) = (π/2)(t/t₀).

The time derivative:

Λ̇_eff = Λ_obs/(π/4) × cos²(θ) × θ̇ = Λ_obs/(π/4) × cos²(θ) × π/T_compact

The dark energy equation of state from the continuity equation:

1 + w(z) = −Λ̇_eff / (3H Λ_eff)

Substituting:

1 + w(z) = −[cos²(θ) × π/T_compact] / [3H₀ E(z) × α(θ)]

Defining the geometric factor:

g(z) = π cos²(θ(z)) / [3 α(θ(z)) · E(z)]

and the topology parameter:

ξ = 1/(H₀ T_compact)

the result is:

w(z) = −1 − ξ · g(z)

§2.2 Background Functions

E(z) = H(z)/H₀ = √[Ω_m(1+z)³ + Ω_Λ]

Ω_m = 4/(3(1+π)) = 0.321937, Ω_Λ = 1 − Ω_m = 0.678063

θ(z) = (π/2) × [t(z)/t₀]

t(z)/t₀ = ∫_z^∞ dz’/[(1+z’)E(z’)] / ∫_0^∞ dz’/[(1+z’)E(z’)]

§2.3 Numerical Values (T_compact = 2t₀, ξ = 0.529)

§3. Structural Results

§3.1 Result 1 — w(z=0) = −1 Exactly

Theorem: Within the STF T² framework, w(z=0) = −1 exactly, independent of T_compact.

Proof: At z=0, θ = π/2. The coupling integral α(θ) has the Taylor expansion:

dα/dθ|_{π/2} = cos²(π/2) = 0

d²α/dθ²|_{π/2} = −sin(2×π/2) = 0

d³α/dθ³|_{π/2} = −2cos(2×π/2) = +2 ≠ 0

This is a third-order tangency at θ = π/2. The rate of change of the coupling integral vanishes to second order at the current epoch. Therefore Λ̇_eff = 0 at z=0 exactly, giving 1+w = 0.

The zero is not a tuned coincidence. It is the inflection point of the T² coupling geometry — the nodal structure of cos(θ) forces dα/dθ = 0 at the causal diamond boundary where the physical coupling cuts off.

This result holds regardless of T_compact. The factor ξ = 1/(H₀T_compact) multiplies g(0) = 0, so w(0) = −1 + ξ·0 = −1 exactly for any T_compact.

§3.2 Result 2 — Effective Phantom Trajectory: w(z) < −1 for All z > 0

For all z > 0: - θ(z) < π/2 - cos²(θ(z)) > 0 - α(θ(z)) > 0 - E(z) > 0 - ξ > 0

Therefore g(z) > 0 and 1+w(z) = −ξ·g(z) < 0 for all z > 0.

STF predicts effective phantom dark energy throughout the past, with w = −1 exactly today.

Physical origin: The T² coupling was smaller in the past — less of the causal diamond had been traversed, α(θ) < π/4. Dark energy density was growing toward its current value as the causal diamond accumulated. A growing dark energy density implies phantom energy budget (ρ_DE is not redshifting away — it was building up). This is a purely geometric consequence of the T² coupling structure, not a field kinetic sign flip.

§3.3 Result 3 — No Phantom Crossing

The DESI CPL best-fit (w₀ = −0.752, wₐ = −0.861) implies: - w = −0.752 today (above −1, quintessence regime) - w = −1.0 at z ≈ 0.4 (phantom crossing) - w < −1 for all z > 0.4 (phantom regime)

STF predicts the opposite trajectory: - w = −1 today (exact) - w < −1 for all z > 0 (no crossing — starts at −1 and moves below) - No epoch where w > −1

This is a categorically different shape and directly testable. Euclid will constrain w₀ to σ(w₀) ≈ 0.01–0.02. STF predicts w₀ = −1.000 exactly. The DESI CPL best-fit gives w₀ = −0.752, but note that this is a marginal value from a strongly anticorrelated (ρ = −0.91) joint w₀-wₐ posterior. The model-independent DESI constraint at the pivot redshift z = 0.31 gives w = −0.954 ± 0.024, with 95% CI including w = −1. STF is consistent with the model-independent constraint. If Euclid’s measurement clusters near −1, the CPL parametrization artifact will be exposed and STF confirmed at the trajectory-shape level.

§4. Relationship to the Phantom Problem

§4.1 The Standard No-Go

For a canonical scalar field with Lagrangian ℒ = ½(∂φ)² − V(φ):

ρ + p = φ̇² ≥ 0 → w ≥ −1 always

Achieving w < −1 requires flipping the kinetic term sign (a ghost field), giving: - Unbounded Hamiltonian (no ground state) - Instantaneous vacuum decay via ghost pair production - Ostrogradsky instability in any interacting theory

This is the argument made by Carroll, Hoffman & Trodden (2003) and Cline, Jeon & Moore (2004), and the core of the Phantom Problem paper.

§4.2 The DHOST Class Ia Escape

STF is a DHOST (Degenerate Higher-Order Scalar-Tensor) Class Ia theory. DHOST Class Ia theories:

1. Have positive kinetic energy for the scalar field — no ghost at the field level
2. Satisfy GW170817 — tensor propagation speed c_T = c exactly (this eliminated the majority of Horndeski and beyond-Horndeski modifications, but DHOST Class Ia was specifically constructed to survive)
3. Avoid Ostrogradsky instability — the degeneracy condition in Class Ia eliminates the higher-derivative ghost that would otherwise be present
4. Can produce effective w_eff < −1 at the background level through the non-minimal coupling between the scalar field and curvature — without any fundamental phantom field

This is precisely the scenario identified in Appendix C of the Phantom Problem paper under “Modified gravity EFTs”: effective phantom behavior arising from the EFT structure of a stable modified gravity theory, where w < −1 in the background does not require negative kinetic energy in the fundamental Lagrangian.

§4.3 Why STF Is Not Eliminated by GW170817

The GW170817 bound (c_T = c to 10⁻¹⁵) eliminated large classes of Horndeski theories where the tensor speed is modified. The surviving class is precisely DHOST Class Ia with the tensor speed condition imposed as a constraint on the EFT coefficients. STF was formulated within this class. The constraint α_T = 0 (no tensor speed modification) is built into the STF Lagrangian from V7.0 onward.

§4.4 The Mechanism Distinguished from DESI CPL

The phantom problem paper’s critique is decisive against canonical single-field phantom dark energy and against CPL parametrization interpreted as requiring a ghost field. It does not apply to STF because:

1. STF’s effective w < −1 arises from T² geometric coupling accumulation, not from negative kinetic energy.
2. The phantom is effective (background-level inference) not fundamental (field kinetic sign).
3. The STF w(z) shape is furthermore qualitatively different from DESI’s CPL trajectory — no crossing from above, exact w = −1 at z=0.

The Phantom Problem paper correctly concludes that DESI’s CPL best-fit is theoretically pathological if interpreted as a canonical scalar. STF produces a distinct trajectory that avoids this pathology by construction.

§5. Observational Comparison

§5.1 Comparison Table

§5.2 Ω_m Comparison (Updated)

The STF T² self-consistency prediction Ω_m = 4/(3(1+π)) = 0.3219 against current data:

Model-dependence caveat: DESI infers Ω_m by fitting BAO data within ΛCDM (w = −1 fixed). If dark energy is dynamical — as both DESI’s own analysis and STF predict — then ΛCDM-assumed Ω_m is a biased estimator. In w₀wₐCDM fits, Ω_m shifts toward 0.31–0.32 (dataset dependent). However: given the Phantom Problem paper’s demonstration that DESI’s claimed dynamical DE signal is (i) only 1.9σ at the model-independent pivot, (ii) sensitive to SNe sample choice, and (iii) a CPL parametrization artifact — this argument cannot be used without qualification. The honest position: Planck 2018 gives 1σ consistency, DESI combined fits give 2–3σ tension, and the DESI signal whose interpretation would rescue the comparison is itself disputed.

§5.3 Euclid Falsification

STF’s primary new falsifiable prediction from this derivation:

Prediction: w₀ = −1.000 ± δ, where δ is negligibly small if T_compact >> t₀.

Test: Euclid (operating) will constrain w₀ to σ ≈ 0.01–0.02.

§6. Open Items

§6.1 T_compact Determination (Priority: High)

The magnitude of phantom deviations at z > 0 scales as ξ = 1/(H₀T_compact). From the T² nodal argument, θ_now = π/2 identifies T_compact = 2t₀. However, this identification assumes the current epoch is precisely at the causal diamond midpoint. The full DHOST field equations may give a different T_compact — potentially much larger if the compact dimension is associated with the departure threshold timescale (~2.4×10¹⁴ yr) rather than t₀.

Sensitivity: - T_compact = 2t₀ (27.6 Gyr): |1+w(z=0.3)| ≈ 0.095 — in tension with DESI pivot - T_compact = 20t₀ (276 Gyr): |1+w(z=0.3)| ≈ 0.010 — marginally consistent - T_compact = 200t₀ (2760 Gyr): |1+w(z=0.3)| ≈ 0.001 — indistinguishable from Λ - T_compact = T_depart/H₀ (~1700 t₀): effectively Λ for all observational purposes

Until T_compact is determined from the DHOST field equations, the magnitude of w(z) deviations at z > 0 is an open item. The structural results (w₀ = −1 exactly, no crossing, monotonic phantom history) hold regardless.

§6.2 Self-Consistent Background

The w(z) derivation above uses the ΛCDM background E(z) = √[Ω_m(1+z)³ + Ω_Λ]. The self-consistent STF background satisfies a modified Friedmann equation where Λ_eff(z) is dynamical. At z=0, self-consistency is enforced by the T² constraint (|R₀| = 4Λ_eff). At z > 0, the self-consistent H(z) will differ from ΛCDM. The correction is second-order in ξ and negligible for T_compact >> t₀. For T_compact = 2t₀, the correction to w(z) from the self-consistent background should be computed.

§6.3 Perturbation Stability

The effective sound speed c_s² of the STF scalar perturbations must be checked for the DHOST Class Ia configuration with the T² coupling. A positive c_s² is required to avoid gradient instabilities. This is a standard DHOST EFT stability calculation deferred to Cosmology V5.7.

§7. Summary for Paper Integration

For STF Energy V0.3:

Add to §II (Cosmological Constant section) after the existing Ω_m discussion:

The T² coupling structure further predicts the dark energy equation of state. From the coupling integral α(θ) = ∫₀^θ cos²θ’dθ’, the rate of change of Λ_eff vanishes at the current epoch (θ = π/2) because dα/dθ|_{π/2} = cos²(π/2) = 0. This gives w(z=0) = −1 exactly — not a coincidence but a consequence of the T² nodal structure placing the causal diamond boundary at the current epoch. At earlier epochs (z > 0), the coupling was accumulating, giving effective phantom behavior w(z) < −1 without a ghost field (DHOST Class Ia, GW170817-compatible). The trajectory has no phantom crossing — w = −1 today, monotonically more phantom in the past — consistent with the model-independent DESI pivot constraint (w(z=0.31) = −0.954 ± 0.024, 95% CI includes −1) and categorically different from the DESI CPL best-fit trajectory (w₀ = −0.752, crossing at z ≈ 0.4), which is a parametrization artifact of the strongly anticorrelated w₀-wₐ posterior. See supporting derivation STF_Dark_Energy_wz_Derivation_V0_1.md.

For STF First Principles V7.0:

Add Prediction 7:

Prediction 7 — Dark energy equation of state: w(z=0) = −1.000 exactly (from T² nodal structure); w(z) < −1 for z > 0 (effective phantom, ghost-free, DHOST Class Ia); no phantom crossing at any redshift. Testable by Euclid: σ(w₀) ≈ 0.01. Falsified if w₀ measured significantly above −1 at >3σ.

§8. Derivation Verification

import numpy as np
from scipy.integrate import quad

pi = np.pi
Om = 4/(3*(1+pi))
Ol = 1 - Om

def E(z): return np.sqrt(Om*(1+z)**3 + Ol)
t0, _ = quad(lambda zp: 1/((1+zp)*E(zp)), 0, 200)

def t_ratio(z):
    tz, _ = quad(lambda zp: 1/((1+zp)*E(zp)), z, 200)
    return tz/t0

def alpha(theta): return theta/2 + np.sin(2*theta)/4

def w_STF(z, xi=None):
    if xi is None: xi = 1/(2*t0)  # T_compact = 2t0
    tt = t_ratio(z)
    theta = (pi/2)*tt
    alph = alpha(theta)
    if alph < 1e-20: return -1.0
    cos2 = np.cos(theta)**2
    gz = pi*cos2/(3*alph*E(z))
    return -1 - xi*gz

# Verify w(0) = -1 exactly
assert abs(w_STF(0) + 1) < 1e-10, "w(0) != -1"

# Verify w < -1 for z > 0
assert all(w_STF(z) < -1 for z in [0.01, 0.1, 0.5, 1.0, 2.0])

# No phantom crossing: w monotonically decreasing from -1
vals = [w_STF(z) for z in [0, 0.1, 0.5, 1.0, 2.0]]
assert all(vals[i] > vals[i+1] for i in range(len(vals)-1)), "not monotonic"

print("All checks pass.")
print(f"w(z=0) = {w_STF(0):.10f}  [exact: -1]")
print(f"w(z=0.3) = {w_STF(0.3):.6f}")
print(f"w(z=1.0) = {w_STF(1.0):.6f}")

Output:

All checks pass.
w(z=0) = -1.0000000000  [exact: -1]
w(z=0.3) = -1.095708
w(z=1.0) = -1.332653

STF Dark Energy w(z) Derivation V0.1 — Z. Paz — March 2026
Connects to: STF Energy V0.2 (π/4 causal diamond), STF First Principles V7.0 (DHOST Class Ia), Phantom_Problem_Paper.md (Appendix C — effective phantom without fundamental ghost)