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 $q^{app}(z)$
derived from the luminosity distance $D_L(z)$ 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 $z>0$,
providing a graphical representation of the conditions in the
$q_0^{app},q_1^{app}$ plane, which are respectively the zero and first order
terms of the central Taylor expansion of $q^{app}(z)$. 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 $q^{app}(z)$ 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. %The expansion of the solution in terms of
cosmic time is quite useful also for other applications requiring foliation %of
space-time in space-like hyper-surfaces, such as spatial averaging, which is
much more difficult to study using the %analytical solution in terms of the
generalized conformal time coordinate.
