Notes and Useful Info

The Hubble constant is determined through obtaining the angular diameter distance to the last scattering surface. That's not a direct observable; instead it's inferred through trigonometry. We can directly measure the angular scale of the Baryon Acoustic oscillations in the CMB – it's the distance between troughs in the power spectrum. In the standard \(\Lambda\)CDM cosmological model, we also know the physical scale of the BAO feature, known as the sound horizon length. The angular diameter distance is then defined as

\(D_A = \frac{r_s}{\theta_s}\)

where the numerator is the known physical scale and the denominator is the measured angular scale. The angular diameter distance is a well-known function of the Hubble rate, and you can infer the Hubble rate from getting the angular diameter distance (assuming the only pertinent species of particles in the universe are dark matter, baryons, photons, neutrinos, and the cosmological constant). In particular, the equation is

\(\int_0^{z^*}\)

\(D_A = \int_0^{z^*}\)

\( D_A = \int_0^{z^*} \)

\( D_A = \frac{1}{1+z^*} \frac{dz}{H(z)} \)

\( D_A = \frac{1}{1+z^*} \int_0 \frac{dz}{H(z)} \) (works)

$D_A = \frac{1}{1+z^*} \int_0 \frac{dz}{H(z)}$ (works)

$D_A = \frac{1}{1+z^*} \int_0^z \frac{dz}{H(z)}$ (works) $D\_A = \frac{1}{1+z^*} \int_0^z \frac{dz}{H(z)}$

$D_A = \frac{1}{1+z^*} \int_0^{z^1} \frac{dz}{H(z)}$ (works) $D\_A = \frac{1}{1+z^1} \int_0^{z^1} \frac{dz}{H(z)}$

$D_A = \frac{1}{1+z^*} \int_0^{z*} \frac{dz}{H(z)}$ (works) $D\_A = \frac{1}{1+z^\*} \int_0^{z\*} \frac{dz}{H(z)}$

$D_A = \frac{1}{1+z^*} \int_0^{z^*} \frac{dz}{H(z)}$ (works) $D\_A = \frac{1}{1+z^\*} \int_0^{z^\*} \frac{dz}{H(z)}$

$D_A = \frac{1}{1+z^} \int_0^{z^} \frac{dz}{H(z)}$ (does not work) $D\_A = \frac{1}{1+z^*} \int_0^{z^*} \frac{dz}{H(z)}$

\( D_A = \frac{1}{1+z^} \int_0^{z^} \frac{dz}{H(z)} \) (does not work) \\( D\_A = \frac{1}{1+z^*} \int_0^{z^*} \frac{dz}{H(z)} \\)

$H(z) = H_0 \sqrt{\Omega_m (1+z)^3 + \Omega_{rad} (1+z)^4 + \Omega_{\Lambda}}$

with $z^*$ is the redshift of the CMB (~1100), and the $\Omega$ density parameters corresponding to the known total matter density, radiation density, and vacuum energy density today, respectively.

Tags: #CMB #Cosmology #Physics