  Space-based missions studying the cosmic microwave background (CMB) have
progressively refined the parameter space in conventional models of inflation
shortly ($\sim 10^{-37}$ seconds) after the big bang. While most inflationary
scenarios proposed thus far in the context of GR have since been ruled out, the
basic idea of inflation may still be tenable, albeit with several unresolved
conundrums, such as conflicting initial conditions and inconsistencies with the
measured CMB power spectrum. In the new slow-roll inflationary picture,
inflation arising in plateau-like potentials requires an initiation beyond the
Planck time. This delay may be consistent with the cutoff, $k_{\rm min}$,
measured recently in the primordial power spectrum. However, the actual value
of $k_{\rm min}$ would imply an initiation time too far beyond the big bang for
inflation to solve the horizon problem. In this paper, we also describe several
other undesirable consequences of this delay, including an absence of well
motivated initial conditions and a significant difficulty providing a viable
mechanism for properly quantizing the primordial fluctuations. Nevertheless,
many of these inconsistencies may still be avoided if one introduces
non-conventional modifications to inflation, such as a brief departure from
slow-roll dynamics, possibly due to a dramatic change in the inflationary
potential, inflation driven by multiple fields, or a non-minimal coupling to
gravity. In addition, some of these difficulties could be mitigated via the use
of alternative cosmologies based, e.g., on loop quantum gravity, which replaces
the initial big-bang singularity with finite conditions at a bounce-like
beginning.
