Variety of $(d + 1)$ dimensional Cosmological Evolutions with and without bounce in a class of LQC -- inspired Models

Abstract: The bouncing evolution of an universe in Loop Quantum Cosmolgy can be described very well by a set of effective equations, involving a function $sin \; x$. Recently, we have generalised these effective equations to $(d + 1)$ dimensions and to any function $f(x) \;$. Depending on $f(x) \;$ in these models inspired by Loop Quantum Cosmolgy, a variety of cosmological evolutions are possible, singular as well as non singular. In this paper, we study them in detail. Among other things, we find that the scale factor $a(t) \; \propto \; t^{ \frac {2 q} {(2 q - 1) \; (1 + w) d}} \;$ for $f(x) = x^q \;$, and find explicit Kasner--type solutions if $w = 2 q - 1 \;$ also. A result which we find particularly fascinating is that, for $f(x) = \sqrt{x} \;$, the evolution is non singular and the scale factor $a(t)$ grows exponentially at a rate set, not by a constant density, but by a quantum parameter related to the area quantum.