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tokens: redshifts that is insensitive to assumptions about the cosmological parameters. In the next section, we derive relations between cosmological distances that are independent of cosmological parameters and allow us to construct combinations of that no longer carry a dependence on cosmology. 3. Distance relation in isotropic universes The two-dimensional separation vector between two infinitesimally close light rays follows the geodesic deviation equation where is the affine parameter along the light rays, and is the optical tidal matrix, which is determined by the Ricci and Weyl tensors of the spacetime metric (see, e.g., Chap.3 of Schneider et al. 1992 ; Seitz et al. 1994 ). In an isotropic universe, the tidal part of vanishes, so that , where is the two-dimensional unit matrix and is a scalar function, proportional to the local density. If the two light rays intersect at a vertex at under an angle , we can write , where satisfies the differential equation with initial conditions , , where the latter value depends on the choice of the affine parameter. By definition, is the angular-diameter distance of a source at , seen from an observer at . The second-order differential Eq. ( 7 ) has two independent solutions; we chose them to be and , that is, the angular-diameter distances as measured from observers located at and , respectively. A third solution, , must necessarily be a linear combination of the other two. If we write it in the form where is a constant, then we see that the first initial condition is satisfied, . For the second initial condition, we have where is the Wronskian From Eq.( 7 ), we directly see that , i.e., is constant. We can calculate this constant considering at , which yields yielding . Therefore, Thus, we have expressed the angular-diameter distance as seen from an observer at affine parameter by the corresponding expression for the angular-diameter distance for observers at and . The resulting expression contains the initial conditions at and . If we now specialize Eq.( 9 ) to the case of , with the corresponding angular-diameter distance denoted as , we obtain Combining Eqs.( 9 ) and ( 10 ), we then find The final equality no longer contains the initial conditions. It is valid for any spacetime that is isotropic around the observer through which the radial light bundle passes. Furthermore, this equation does not refer to the Einstein field equation and is therefore independent of the validity of General Relativity, as long a spacetime is characterized by a metric. A more graphical derivation of the result ( 11 ) is given in Schneider (2016) . If, in addition to isotropy, we assume a spatially homogeneous spacetime, which can then be expressed in the form of the Robertson–Walker metric, the initial conditions are , where is the redshift with which sources at are seen by the observer at (or redshift zero). For this, we chose a parametrization of the affine parameter such that it locally coincides with the comoving