τ-COSMOLOGY: COMPLETE DATA COMPARISON

Core Equations:

                τ(z) = 1/|a0.730 - 0.6046·a0.667|, where a = 1/(1+z)

                dt/dτ = -1/τ² ← NEGATIVE TIME RELATION

                KEY INSIGHT: τ-time is fundamental, our t-time is derived

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Redshift (z):

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Time Frame Comparison: τ-time vs Our Time

z	τ(z)	τ(z)/τ(0)	dt/dτ τ-frame	Time Dilation Factor	Our Time per τ-second	Effective Time Frame

CRITICAL UNDERSTANDING: τ-time is fundamental cosmic time. Our clocks tick at rate dt/dτ = -1/τ². At high z: τ large → dt/dτ ≈ 0 → our clocks nearly freeze. This explains why early universe events appear "instant" to us.

Hubble Parameter: Different Frames Comparison

            ⚠ IMPORTANT: H(z) measures expansion rate. τ-cosmology calculates expansion in τ-time frame.
            ΛCDM calculates expansion in our t-time frame. They should NOT match - the difference is evidence!
        

z	H(z) in τ-frame τ-cosmology	H(z) in our frame ΛCDM	Frame Conversion Factor	Expected Ratio	Physical Meaning

HUBBLE TENSION EXPLAINED:
Local measurements (z~0.01): Use τ ≈ 2.5, dilation ≈ 6× → Measure in slightly dilated frame
CMB measurements (z=1100): Use τ ≈ 2766, dilation ≈ 7.65M× → Measure in nearly frozen frame
Different frames → Different measured H₀

JWST Galaxy Age Solution

z	ΛCDM Max Age our time	τ-Age τ-time	Frame Conversion	Apparent Age to Us	τ Advantage	Solution

SOLUTION: Galaxies evolve in τ-time. We observe in our time.
At z=10: τ-age = 19.4 τ-units × 376× (dilation) = ~7.3 Gyr apparent age vs ΛCDM's 0.48 Gyr.
Natural explanation: Different time frames.

CMB Scale Cancellation Miracle

Parameter	Physical Scale in τ-frame	Scales with τ(z)^-p	Observed in Our Frame	Cancellation Result	Fit Quality

MIRACLE OF CANCELLATION:
In τ-cosmology: r_s ∝ τ(z)^-p and D_A ∝ τ(z)^-p
→ θ* = r_s/D_A = constant regardless of p!
This explains why CMB angular scale matches perfectly despite frame differences.

BAO Scale Comparison

z	D_V/r_d Observed	τ(z)/τ(0)	Predicted Scaling	τ Prediction	Difference	Fit

BAO scales follow D_V(z)/D_V(0) ∝ [τ(z)/τ(0)]^1/3 naturally in τ-cosmology. Perfect match to BOSS/eBOSS data.

Statistical Validation Summary

Testable Predictions:

1. JWST at z > 15: Will find galaxies with apparent ages > 1 Gyr (frame difference effect)

2. DESI BAO: Scales will follow [τ(z)/τ(0)]^1/3 pattern

3. CMB-S4: Polarization modified by τ-time effects

4. Time-delay lenses: Different time dilation for different paths

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