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

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Abstract

The central smoothness of the functions defining a LTB solution plays a crucial role in their ability to mimic the effects of the cosmological constant. Even if non-smoothness is not physically inconsistent with the theory of general relativity, smoothness is still an important geometrical property characterizing the solution of the Einstein’s equations. So far attention has been focused on \(C^{1}\) models while in this paper we approach it in 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. 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 mimic the cosmological constant for the luminosity distance. This implies that it is not necessary to include any additional observable to distinguish mathematically the theoretical predictions of a smooth LTB model from a \(\Lambda CDM\). 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 theoretical predictions or it is not smooth.

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Acknowledgments

I thank the members of the Dark energy LeCosPa working group, Misao Sasaki and Marco Regis for comments and discussions. I also thank the CERN theoretical division for its support and hospitality. A.E.R. is also supported by the CODI project IN615CE and Sostenibilidad 2013/2014 of UDEA, and the dedicacion exclusiva program of the Vicerectoria de Docencia of UDEA.

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  1. Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan

    Antonio Enea Romano

  2. Instituto de Fisica, Universidad de Antioquia, A.A. 1226, Medellin, Colombia

    Antonio Enea Romano

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Romano, A.E. Can smooth LTB models mimicking the cosmological constant for the luminosity distance also satisfy the age constraint?. Gen Relativ Gravit 45, 2529–2544 (2013). https://doi.org/10.1007/s10714-013-1602-1

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