Can smooth LTB models mimicking the cosmological constant for the luminosity distance also satisfy the age constraint?

Understand

The central smoothness of the functions defining a LTB solution plays a crucial role in their ability to mimick the effects of the cosmological constant.

• So far attention has been focused on $C^{1}$ models while in this paper we approach it a more general way, investigating the implications of higher order central smoothness conditions for LTB models reproducing the luminosity distance of a $\Lambda CDM$ Universe.
• Our analysis is based on a low red-shift expansion, and extends previous investigations by including also the constraint coming from the age of the Universe and re-expressing the equations for the solution of the inversion problem in a manifestly dimensionless form which makes evident the freedom to accommodate any value of $H_0$ as well, correcting some wrong claims that the observed value of $H_0$ would be enough to rule out LTB models.
• Higher order smoothness conditions strongly limit the number of possible solutions respect to the first order condition.