This came up in discussion around a class I'm taking. For a Universe with $\Lambda$ and matter contributions to energy density (and implicitly curvature, but no radiation), can you have a universe with open geometry ($\Omega_\Lambda + \Omega_m < 1$) that fits the description of a "big bounce" universe?
All the possible descriptions/behaviours of such universe models are summarized in this diagram:
An equivalent way of asking this question is: Does the line separating Big Bang/No Big Bang models approach the $\Omega_m=0$ line asymptotically, or does it meet it at $\Omega_\Lambda=1$? We had a go at sorting this out but couldn't come to any agreement...
My suspicion is that "if open geometry then Big Bang". Follow up question assuming this is the case: Is there an intuitive interpretation as to why open geometries MUST have a Big Bang (given the restrictions of these models of course, e.g. positive matter energy density, no relativistic species)?