What does this equation mean?

The European Space Agency’s Planck satellite, launched in 2009 and operating until 2013, extended this to still finer angular resolution and a wider frequency range, cooling its bolometer detectors to a fraction of a degree above absolute zero to suppress instrumental noise. Planck’s 2018 final cosmological parameter release fit the standard six-parameter flat Lambda-CDM model to its temperature and polarization data, reporting the acoustic scale — the characteristic angular size of the sound-wave pattern imprinted on the CMB — to 0.03 percent precision, a cold dark matter density of 0.120 ± 0.001 (in units of the density parameter times the reduced Hubble parameter squared), and a Hubble constant inferred from CMB physics of about 67.4 kilometers per second per megaparsec [ 6 ] . The acoustic scale itself follows from a genuine physical model worth writing down explicitly, because the whole CMB-based measurement chain depends on it: $\theta_* = \frac{r_*}{D_A(z_*)}$. where $r_*$ is the physical size of the sound horizon at recombination — how far a pressure wave in the primordial plasma could travel before the plasma became neutral and transparent — and $D_A(z_*)$ is the angular diameter distance to that same epoch. Planck measures $\theta_*$ directly from the angular spacing of peaks in the CMB power spectrum to extraordinary precision; converting that angle into a present-day expansion rate $H_0$ requires assuming a cosmological model (matter content, radiation content, and the equation of state of dark energy) to relate $r_*$ and $D_A(z_*)$ to $H_0$ . This is the crux of the tension discussed below: Planck’s $H_0$ is a model-dependent inference from an early-universe ruler, not a direct local measurement.