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A four-dimensional \(\Lambda \)CDM-type cosmological model induced from higher dimensions using a kinematical constraint

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Abstract

A class of cosmological solutions of higher dimensional Einstein field equations with the energy-momentum tensor of a homogeneous, isotropic fluid as the source are considered with an anisotropic metric that includes the direct sum of a 3-dimensional (physical, flat) external space metric and an \(n\)-dimensional (compact, flat) internal space metric. A simple kinematical constraint is postulated that correlates the expansion rates of the external and internal spaces in terms of a real parameter \(\lambda \). A specific solution for which both the external and internal spaces expand at different rates is given analytically for \(n=3\). Assuming that the internal dimensions were at Planck length scales when the external space starts with a Big Bang (\(t=0\)), they expand only 1.49 times and stay at Planck length scales even in the present age of the universe (13.7 Gyr). The effective four dimensional universe would exhibit a behavior consistent with our current understanding of the observed universe. It would start in a stiff fluid dominated phase and evolve through radiation dominated and pressureless matter dominated phases, eventually going into a de Sitter phase at late times.

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Notes

  1. We would like to note that kinematics similar to that we obtained for the external space for \(\lambda \ne 0\) is also noted by Capozziello et al. [17], although with a totally different reasoning in the context of conventional, four dimensional relativistic cosmology.

  2. Stiff fluid is the most promising EoS of matter at ultra-high densities for representing the very early universe (see [15, 16]) and is described with an EoS parameter \(p/\rho =1\), where \(\rho \) and \(p\) are the energy density and pressure, respectively.

  3. If \(c_{1}c_{2}>0\), in the case \(c_{1}\ne c_{2}\) the evolution of the Hubble and deceleration parameters turn out to be exactly the same with the ones in the case \(c_{1}=c_{2}\), but shifted along the time axis.

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Acknowledgments

Özgür Akarsu and Tekin Dereli appreciate the financial support given by the Turkish Academy of Sciences (TÜBA). Ö. Akarsu acknowledges also the financial support he is receiving from Koç University.

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  1. Department of Physics, Koç University, 34450 , Sarıyer, İstanbul, Turkey

    Özgür Akarsu & Tekin Dereli

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  1. Özgür Akarsu
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Correspondence to Özgür Akarsu.

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Akarsu, Ö., Dereli, T. A four-dimensional \(\Lambda \)CDM-type cosmological model induced from higher dimensions using a kinematical constraint. Gen Relativ Gravit 45, 1211–1226 (2013). https://doi.org/10.1007/s10714-013-1521-1

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