Abstract
We discuss a Brans-Dicke model with a cosmological constant, a negative value of the $w$ parameter, and an arbitrary (in general nonvanishing) scale factor at the big bang. The Friedmann equations for a flat universe are considered. The current observational values for Hubble constant ${H}_{0}$ and deceleration parameter ${q}_{0}$ play the role of initial conditions. We follow the approach of Uehara and Kim in order to solve field equations analytically. In K. Uehara and C. W. Kim [Phys. Rev. D 26, 2575 (1982)] only positive values of $w$ were considered; we extend the study to a complete set of possible $w$ values. Our main result is that the scale factor (during its evolution back in time direction) may not vanish, unlike in the standard $\ensuremath{\Lambda}\mathrm{CDM}$ case. In other words, the considered model demonstrates a cosmological bounce instead of the initial singularity. The famous formula (24), that leads to the bounce, is valid only for the dust-filled universe with $p=0$ and, therefore, is not adequate for the early universe hot stage when the bounce happens. So, our results are qualitative in nature and must be used to obtain initial values for the hot stage of the Universe.
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