If the universe has 
consists of protons and electrons. If the universe were homogeneous, fully ionized, and contained no He, the optical depth due to Thomson scattering along the line of sight to relativistic redshift is
where
is the Thomson cross section. The proper length is
Now, since the electrons have constant co-moving density,
so
If
, then this simplifies to
consists of protons and electrons. If the universe were homogeneous, fully ionized, and contained no He, the optical depth due to Thomson scattering along the line of sight to relativistic redshift is(1)
where
is the Thomson cross section. The proper length is (2)
Now, since the electrons have constant co-moving density,
(3)
so
(4)
If
, then this simplifies to (5)
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