Second-order Gauge-invariant Cosmological Perturbation Theory: Current
  Status updated in 2019
  The current status of the recent developments of the second-order
gauge-invariant cosmological perturbation theory is reviewed. To show the
essence of this perturbation theory, we concentrate only on the universe filled
with a single scalar field. Through this review, we point out the problems
which should be clarified for the further theoretical sophistication of this
perturbation theory. This review is an extension of the review paper
[K.~Nakamura, "Second-Order Gauge-Invariant Cosmological Perturbation Theory:
Current Status", Advances in Astronomy, vol.2010 (2010), 576273.]. We also
expect that this theoretical sophistication will be also useful to discuss the
future developments in cosmology as a precise science.
