Independent measurements of distances and redshifts, z, allow astronomers to constrain cosmological models, since they both define the distance–redshift relation. When it was determined that type Ia supernovae (SNIa) could be standardized and therefore used to measure distances, this led to the discovery of the accelerated expansion of our Universe (Riess et al. 1998; Perlmutter et al. 1999). The combination of SNIa (Betoule et al. 2014), baryonic acoustic oscillations (Eisenstein et al. 2005; Alam et al. 2017), and the cosmic microwave background (CMB; Komatsu et al. 2011; Planck Collaboration et al. 2018) led to the emergence of the concordance Λ cold dark matter (ΛCDM) model, in which the energy density is dominated by dark energy as a cosmological constant Λ. In a flat Friedmann–Lemaître–Robertson–Walker universe, the comoving distance is defined as 
(1)$$\begin{eqnarray*}
D(z) = \int _0^z \frac{c\mathrm{d}z}{H(z)},
\end{eqnarray*}$$ where, in the ΛCDM model, 
(2)$$\begin{eqnarray*}
H(z) = H_0 \sqrt{\Omega _\mathrm{m} (1+z)^3 + 1-\Omega _\mathrm{m}}
\end{eqnarray*}$$is the Hubble parameter, H0 is the Hubble–Lemaître constant, and Ωm is the matter energy density at the current epoch. The luminosity and angular diameter distances are defined as 
(3)$$\begin{eqnarray*}
D_\mathrm{L}(z) = (1+z) D(z)
\end{eqnarray*}$$(4)$$\begin{eqnarray*}
{\mathrm{ and}} \,D_\mathrm{A}(z) = \frac{R}{\theta }= \frac{D(z)}{(1+z)}.
\end{eqnarray*}$$Type Ia supernovae can only be used up to redshift of around 2 (Jones et al. 2013), and there are tensions between direct local measurements of the Hubble–Lemaître constant and model-dependent estimates using CMB observations (Planck Collaboration et al. 2018). Therefore, independent distance measurements to extragalactic objects are desired. We should emphasize here, that model-independent distance indicators can have various important applications in physical cosmology. In particular, to test different aspects of cosmological models and theories of gravity. For instance, we can use these model-independent measurements to test the FLRW metric (Clarkson, Bassett & Lu 2008; Wiltshire 2009; Shafieloo & Clarkson 2010; L’Huillier & Shafieloo 2017; Shafieloo, L’Huillier & Starobinsky 2018; Cao et al. 2019a; Qi et al. 2019a), to test general relativity and some modified gravity models (Cao et al. 2012; Shafieloo, Kim & Linder 2013a; Cao et al. 2015, 2017b; Qi et al. 2017; L’Huillier, Shafieloo & Kim 2018; Shafieloo et al. 2018; Xu et al. 2018; Chen, Sesana & Conselice 2019; L’Huillier et al. 2020), to test natural constants such as the speed of light (Cao et al. 2018), to test cosmic duality relationships, and to measure cosmic curvature (Shafieloo et al. 2013b; Cao et al. 2019b; Qi et al. 2019a, b; Zheng et al. 2020). Using a combination of such model-independent distance indicators can also be used to measure some key cosmological parameters such as the Hubble Constant (Liao et al. 2015; Suyu et al. 2017; Jee et al. 2019; Liao et al. 2019, 2020; Wong et al. 2019). Amongst the most energetic objects in our Universe are active Galactic nuclei (AGNs). AGNs are the nuclei of massive galaxies that sometimes produce relativistic jets of material launched from near a central supermassive black hole (SMBH). When these jets are not aligned close to our line of sight, AGNs are observed as radio galaxies, whereas if the jet is aligned to within a small angle to our line of sight, they are observed as blazars (Urry & Padovani 1995). Blazars and quasars are amongst the most consistently bright objects in our Universe and can be observed at redshifts as high as ∼7 (Mortlock et al. 2011). Attempts have been made to measure distances to AGNs in various ways, with some claiming deviations from the expected cosmology at high redshifts (Risaliti & Lusso 2017, 2019; Turner & Shabala 2019). Very long baseline interferometry (VLBI) has also been used to attempt to measure cosmological distances. The approach pioneered by Gurvits, Kellermann & Frey (1999) attempted to measure cosmological parameters by assuming that AGN could be used as a standardizable rod. Vishwakarma (2001) used this data set and compared it with supernovae data, and found it was not possible to differentiate different cosmological models with the VLBI data of Gurvits et al. (1999). Cao et al. (2015) revisited this technique and investigated the evolution of the standard rod by assuming a Planck cosmology. Cao et al. (2017a, c) then introduced a cosmology independent method for calibrating the standard rod and was able to provide reasonable constraints on cosmological parameters. Our approach differs from this by using the speed of light to normalize the rod. This approach was first attempted by Wiik & Valtaoja (2001), which found that that the apparent angular sizes of AGNs maximized at z ∼ 2. In this paper, we demonstrate the method on the famous nearby radio galaxy 3C 84 (NGC 1275) and discuss possible systematic errors. The source is known to exhibit extremely high energy emission despite not exhibiting strong relativistic effects (Jorstad et al. 2017; Liodakis et al. 2018), and has multiple independent measures of distance (Theureau et al. 2007; Hicken et al. 2009), thus making it an ideal source to test our methodology.
Citation: Cao et al. 2018
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