Conditions for low-redshift positive apparent acceleration in smooth inhomogeneous models

It is known that a smooth LTB model cannot have a positive apparent central acceleration. Using a local Taylor expansion method we study the low-redshift conditions to obtain an apparent negative deceleration parameter derived from the luminosity distance for a central observer in a LTB space, confirming that central smoothness implies a positive central deceleration. Since observational data is only available at redshift greater than zero we find the critical values of the parameters defining a centrally smooth LTB model which give a positive apparent acceleration at , providing a graphical representation of the conditions in the plane, which are respectively the zero and first order terms of the central Taylor expansion of . We finally derive a coordinate independent expression for the apparent deceleration parameter based on the expansion of the relevant functions in red-shift rather than in the radial coordinate. We calculate with two different methods to solve the null geodesic equations, one based on a local central expansion of the solution in terms of cosmic time and the other one using the exact analytical solution in terms of generalized conformal time.