
\magnification=\magstep2



$$
D_A \quad \equiv \quad {{S}\over{\theta}}
$$

$$
\theta(t_e) \quad = \quad {{S}\over{ {\tilde{\omega}} a(t_e)} }
$$

$$
\theta(z) \quad = \quad {{S (1 + z)}\over{ {\tilde{\omega}}(z)}  }
$$


$$
{{a(t_e)}\over{a(\rm now)}} \quad = \quad {{1}\over{1+z}} 
$$

$$
{\tilde{\omega}}(z) \quad = \quad 
    {{2c}\over{H_0}} \thinspace
    {\left( {{1}\over{1+z}} \right)} \thinspace
    {\left( {{1}\over{\Omega_M^2}} \right) } \thinspace
    {\left[ \Omega_M z 
           - (2 - \Omega_M) 
         \left( \sqrt{1 + \Omega_M z} - 1 \right) 
         \right]}
$$

$$
\eqalign{
\theta(z) \quad &= 
 { \quad
  { 
    S (1 + z) 
  }
  \over 
   {
    {{2c}\over{H_0}} \thinspace
    {\left( {{1}\over{1+z}} \right)} \thinspace
    {\left( {{1}\over{\Omega_M^2}} \right) } \thinspace
    {\left[ \Omega_M z 
           - (2 - \Omega_M) 
         \left( \sqrt{1 + \Omega_M z} - 1 \right) 
         \right]}
   }
 }
 \cr
 \cr
 &= \quad
 { {
    ( S H_0 / 2 c ) \thinspace
    {\left( {1+z} \right)^2} \thinspace
    { \Omega_M^2 } \thinspace
   }
   \over
   {
    {\left[ \Omega_M z 
           - (2 - \Omega_M) 
         \left( \sqrt{1 + \Omega_M z} - 1 \right) 
         \right]}
   }
 }
}
$$



\bye
