Robertson-Walker (RW) metric, $$ds^2 = - dt^2 +
  a^2(t) \left[
    dx^2
    + {S_K}^2(x) \left(d\theta^2 + \sin^2\theta \phi^2\right)
  \right],
\label{eq3-1}$$ where we employ the unit system with $c=1$, and adopt a notation, $$S_K(x) =
  \left\{
  \begin{array}{ll}
    \displaystyle
    \frac{{\rm sinh}\left(\sqrt{-K} x\right)}{\sqrt{-K}}, &
    (K < 0), \\
    x, & (K = 0), \\
    \displaystyle
    \frac{\sin\left(\sqrt{K} x\right)}{\sqrt{K}}, &
    (K > 0),
  \end{array}
  \right.
\label{eq3-2}$$ The comoving angular diameter distance is related to the last function by $$D_{\rm A}(z) = S_K\left(x(z)\right).
\label{eq3-3}$$ where $z$ is the “RW redshift” in which the peculiar velocity effect is not included, i.e., the redshift of the unperturbed universe. The radial coordinate $x$ is the comoving distance from the observer, and the relation to the RW redshift is given by $$x(z) = \int_0^z  \frac{dz'}{H(z')},
\label{eq3-4}$$ where $H(z) = \dot{a}/a$ is the time-dependent Hubble parameter. We consider the general case that the dark energy has an equation of state $p(z) = w(z) \rho(z)$, in which case, $$H(z) = H_0
   \sqrt{
      (1+z)^3 {{\mit \Omega}}_{\rm M0} - 
      (1+z)^2 {{\mit \Omega}}_{\rm K0} +
      (1+z)^3 \exp\left(3\int_0^z \frac{w dz}{1+z}\right) {{\mit \Omega}}_{\rm Q0}
   },
\label{eq3-5}$$ where ${{\mit \Omega}}_{\rm M0}$ is the density parameter of matter(s), ${{\mit \Omega}}_{\rm Q0}$ is the density parameter of the dark energy, and 