Hubble rate $H(z)=\dot{a}(z)/a(z)$ as a function of redshift. The dotted black line assumes a flat ΛCDM cosmology with Ω_m = 0.6 and Ω_Λ = 0.4. The dashed black line assumes a flat ΛCDM cosmology with Ω_m = 0.3 and Ω_Λ = 0.7. The colored lines represent Hubble rates for a ULA cosmology with fixed relative densities Ω_a = 0.3, Ω_m = 0.3, and Ω_Λ = 0.4. The colored Hubble rates differ only in the ULA mass, as shown by the color bar. This plot demonstrates that ULAs undergo a domain change in the low-mass regime between log₁₀(m_a[eV]) = −32 and log₁₀(m_a[eV]) = −33. On the low-mass end, the Hubble rate approaches that of a flat LCDM cosmology but explains parts of the dark energy component. In the modifications to the pipeline used in G24, we use this accurate Hubble rate extracted from axionCAMB in the distance measure computations.