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

Abstract

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. Neither a $C^{1}$ or a $C^{i}$ LTB model can both satisfy the age constraint and mimick the cosmological constant for the luminosity distance. One difference is in the case in which the age constraint is not included and the bang function is zero, in which there is a unique solution for $C^1$ models but no solution for the $C^{i}$ case. Another difference is in the case in which the age constraint is not included and the bang function is not zero, in which the solution is undetermined for both $C^1$ and $C^{i}$ models, but the latter ones have much less residual parametric freedom. Our results imply that any LTB model able to fit luminosity distance data and satisfy the age constraint is either not mimicking exactly the $\Lambda CDM$ red-shift space observables theoretical predictions or it is not $C^{\infty}$ smooth.