\magnification=\magstep2



$$
\eqalign{
M_{\rm int} \quad &= \quad {{4 \pi}\over{3}} \rho(t) r(t)^3 \cr
   \cr
                  &= \quad {{4 \pi}\over{3}} \rho_0 r_0^3 \cr
}
$$


$$
{\rm KE + GPE} \quad = \quad {{1}\over{2}} m v^2 - {{G M_{\rm int} m} \over {r}} 
         \quad = \quad 
         - {{1}\over{2}} m k c^2 \tilde{\omega}^2
$$

$$
r(t) \quad \equiv \quad \tilde{\omega} a(t)
$$

$$
\eqalign{
\tilde{\omega} \quad &{\rm comoving\ coordinate\ \ } \quad {\rm meters} \cr
a(t) \quad &{\rm scale\ factor\ \ } \qquad\quad\qquad\thinspace {\rm no\  units} \cr
}
$$

$$
\tilde{\omega}^2 \left( {{da}\over{dt}} \right)^2 - \thinspace
           {{8 \pi}\over{3}} G \rho \tilde{\omega}^2 a^2 
     \quad = \quad 
          - k c^2 \tilde{\omega}^2
$$

$$
 \left( {{da}\over{dt}} \right)^2 - \thinspace
           {{8 \pi}\over{3}} G \rho a^2 
     \quad = \quad 
          - k c^2 
$$

$$
H(t) \quad = \quad {{1}\over{a(t)}} \thinspace {{da}\over{dt}}
$$

$$
 \left[ H(t)^2 - \thinspace
           {{8 \pi}\over{3}} G \rho(t) \right]  a(t)^2 
     \quad = \quad 
          - k c^2 
$$

$$
 \left[ H_0^2 - \thinspace
           {{8 \pi}\over{3}} G \rho_0 \right]  a_0^2 
     \quad = \quad 
          - k c^2 
$$

$$
\Omega_M \quad \equiv \quad {{\rho_0}\over{{\rho_{\rm crit}}}}
$$

$$
\left( {{da}\over{dt}} \right)^2 - \thinspace
           {{8 \pi}\over{3}} G \rho(t) a(t)^2 
     \quad = \quad 
          - k c^2 
$$

$$
\eqalign{
\rho_0 {a_0}^3 \quad &= \quad \rho(t) {a(t)}^3 \cr
\rho_0 {1}^3 \quad        &= \quad \rho(t) {a(t)}^3 \cr
{{\rho_0}\over{a(t)^2}}  \quad       &= \quad \rho(t) {a(t)}^2 \cr
}
$$

$$
\left( {{da}\over{dt}} \right)^2 - \thinspace
           \left( {{8 \pi}\over{3}} G \rho_0 \right) {{1}\over{a(t)}}
     \quad = \quad 
          - k c^2 
$$

$$
a(x) \quad = \quad {{1}\over{2}} {{\Omega_M}\over{\Omega_M - 1}} 
        \left( 1 - \cos(x) \right) 
$$


$$
t(x) \quad = \quad {{1}\over{2 H_0}} {{\Omega_M}\over{(\Omega_M - 1)^{3/2}}} 
        \left( x - \sin(x) \right) 
$$


$$
a(t) \quad = \quad \left( 6 \pi G \rho_0 \right)^{1/3} \thinspace t^{2/3}
$$

$$
a(x) \quad = \quad {{1}\over{2}} {{\Omega_M}\over{1 - \Omega_M}} 
        \left( \cosh(x) - 1 \right) 
$$

$$
t(x) \quad = \quad {{1}\over{2 H_0}} {{\Omega_M}\over{(1 - \Omega_M)^{3/2}}} 
        \left( \sinh(x) - x \right) 
$$

$$
z \quad = \quad {{\lambda_{\rm obs} - \lambda_{\rm emit}}\over{\lambda_{\rm emit}}}
             \quad = \quad
                {{a_{\rm obs} - a_{\rm emit}}\over{a_{\rm emit}}}
$$


$$ 
q(t) \quad \equiv \quad {{ a(t) \left( d^2a/dt^2 \right) } 
                             \over
                           { {(da/dt)}^2 } }
$$


$$
q(t) \quad = \quad {{1}\over{2}} \thinspace \Omega_M(t)
$$

$$
(ds)^2 \quad = \quad (c \thinspace dt)^2 -
        a(t)^2 \left(   {  { {d \tilde{\omega}}^2 } \over {1 - k \tilde{\omega}^2} }  \right)
$$

$$
c^2 \thinspace dt^2 \quad = \quad
        a(t)^2 \left(   {  { {d \tilde{\omega}}^2 } \over {1 - k \tilde{\omega}^2} }  \right)
$$

$$
c \thinspace {{dt}\over{a(t)}}  \quad = \quad
       {  { {d \tilde{\omega}} } \over \sqrt{1 - k \tilde{\omega}^2} } 
$$

$$
c \thinspace \int_{0}^{t_{\rm obs}} { {{dt}\over{a(t)}} } 
                 \quad = \quad
       \int_{0}^{\tilde{\omega}_e} {  { {d \tilde{\omega}} } \over \sqrt{1 - k \tilde{\omega}^2} } 
$$

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

$$
f \quad {\buildrel{?}\over{=}} \quad  {  {L}\over{4 \pi \tilde{\omega}^2} }
$$

$$
f_{\rm photon} \quad = \quad {  {L}\over{4 \pi \tilde{\omega}^2 (1 + z)} }
$$

$$
D_{L{\rm \ photon}} \quad = \quad \sqrt{{{L}\over{4 \pi f_{\rm photon}}}} \quad = \quad {\tilde{\omega}} \sqrt{1 + z}
$$


$$
f_{\rm energy} \quad = \quad {  {L}\over{4 \pi \tilde{\omega}^2 (1 + z)(1 + z) } }
$$

$$
D_{L{\rm \ energy}} \quad = \quad \sqrt{{{L}\over{4 \pi f_{\rm energy}}}} \quad = \quad {\tilde{\omega}} (1 + z)
$$


\bye