Comparison between our model and the GBH parametrization. Top: Density contrast of matter today, δ(r, t₀), as a function of the FLRW comoving coordinate. Our compensated model (solid blue line) satisfies , while the ΛCDM background is not exactly recovered for the GBH models (dashed red and dotted green lines). Middle: Deviations from the background density of matter, , as a function of the FLRW comoving coordinate. The GBH model with r₀ = r_B (red dashed line) largely deviates from at . On the other hand, the choice of r₀ = r_L for the GBH leads to deviations of approximately −0.01 from the background at . The model assumed in this work fully recovers at any . Bottom: Deviations from the background Hubble expansion as a function of the FLRW comoving coordinate. GBH models (dashed red and dotted green lines) do not perfectly match the ΛCDM background expansion history at .