Distances and the cosmic ladder

• Parallax: $d\,[{\rm pc}] = 1/\pi\,['']$, so 1 mas means 1 kpc. Errors: $\Delta d/d = \Delta\pi/\pi$. Beyond about 20% parallax error, $1/\pi$ becomes biased, which is why HST and Gaia matter (2017 D1 used HST parallaxes of about 2–3 mas).
• Angular size: $D = 2R/\theta$. If you know a physical size, an angle gives the distance. Examples: the Moon's radius from the eclipse geometry (2016 D2), eclipsing binaries with surface brightness (2023 DA1).
• Triangulation in the Solar System: the Sun–Earth–Moon triangle at quarter phase (Aristarchus), Venus transits, radar.

A standard candle has a known $M$, so $m - M$ gives the distance.

• Cepheids: the period–luminosity relation $M = \beta'\log P + C'$, calibrated with parallaxes (2017 D1) and applied to the LMC or other galaxies (2015 P2: IC 342).
• Type Ia supernovae: peak $M\approx-19.3$, corrected for light curve width. Slower decliners are brighter (the Phillips relation; the stretch factor in 2016 D3).
• Novae: the maximum-magnitude–rate-of-decline relation, $M_0$ from $t_2$ or $t_3$ (2019 Nova Del 2013).
• Tip of the red giant branch and RR Lyrae: $M_I\approx-4.0$ and $M_V\approx+0.6$, respectively.

Each candle's calibration error is a systematic error for every distance it gives.

• Baade–Wesselink: integrate a pulsating star's radial velocity to get $\Delta R$, combine with the flux and temperature at two phases to get $R$ and then $d$ (2015 P1).
• Eclipsing binaries: radii in metres from the light and velocity curves, angular diameters from the surface-brightness–colour relation, so $d = 2R/\theta$ (2023 DA1; the method behind the 1% LMC distance of 2019).
• Microlensing parallax: $\theta_E$ and $\pi_E$ together give the lens mass and distance without any luminosity at all (2023 DA2).

For $z\ll1$, $v = cz = H_0d$, so a distance to a galaxy with a known redshift gives $H_0$, and the Hubble time

sets the age scale (2016 D3). Peculiar velocities of a few hundred km/s make nearby galaxies unreliable for this, so use galaxies beyond about 50 Mpc.