Coordinate independent approach to the calculation of the effects of local structure on the luminosity distance

Local structure can have important effects on luminosity distance observations, which could for example affect the local estimation of the Hubble constant based on low red-shift type Ia supernovae. Using a spherically symmetric exact solution of the Eistein's equations and a more accurate expansion of the solution of the geodesic equations, we improve the low red-shift expansion of the monopole of the luminosity distance in terms of the curvature function. Based on this we derive the coordinate independent low red-shift expansion of the monopole of the luminosity distance in terms of the monopole of the density contrast. The advantage of this approach is that it relates the luminosity distance directly to density observations, without any dependency on the radial coordinate choice. We compare our formulae to numerical calculations, and find that the formula in terms of the density contrast is in good agreement with the numerical calculations, in the non linear regime is more accurate than the results obtained using linear perturbation theory, especially in the regime of large Laplacian of the perturbations, and is also more accurate than the formula in terms of the curvature function.