Equation 5 · From Origins to Frontier: A History of Precision Cosmology

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H_0

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a model-dependent inference from an early-universe ruler, not a direct local measurement. Read the equation part by part below; each part has a contextual explanation and a link to its mathematical background.

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H_0

Symbol H_0

a model-dependent inference from an early-universe ruler, not a direct local measurement.

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subscript

The lower label selects a particular version, component, or indexed member of the quantity. For example, x₀ and xₜ can be values at different positions.

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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…

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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.

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