TDG formation-epoch binding energy vs redshift Plot showing MOND enhancement factor a₀(z)/a₀(0) = (1+z) to the 3/2 power as a function of formation redshift. TDGs formed at higher redshift experienced stronger gravitational binding in the substrate framework. Peak cosmic merger rate 0 0.5 1 1.5 2 2.5 3 3.5 Formation redshift z 1 2 4 6 8 10 a₀(z) / a₀(0) NGC 5291, NGC 7252 Local TDGs (z ≈ 0) 5.2× enhancement Potential well 2.3× deeper 8× enhancement Early mergers Substrate (1+z)³ᐟ² ΛCDM & standard MOND No a₀ evolution Enhanced gravitational binding Higher a₀(z) → deeper potential wells, lower Jeans mass, faster collapse TDGs formed at z > 1 are more tightly bound → may survive as long-lived satellites
Relative MOND enhancement factor for TDG formation as a function of redshift. The substrate predicts $a_0(z) = a_0(0)(1+z)^{3/2}$ (solid green curve), while both ΛCDM and standard MOND predict no evolution (dashed red line). The shaded area between the curves represents the enhanced gravitational binding available to TDGs formed at higher redshift. At $z \sim 2$ (the peak of the cosmic merger rate, amber band), a TDG experiences $5.2\times$ stronger MOND acceleration, producing a potential well $2.3\times$ deeper, a Jeans mass $5\times$ lower, and a collapse timescale $3\times$ shorter than a TDG forming today. This may explain why some TDGs persist as long-lived satellite galaxies — a result that ΛCDM simulations have difficulty reproducing without invoking dark matter halos. This prediction is unique to the substrate framework: standard MOND has $a_0 = \text{const}$ at all epochs.