Counting the number N of astronomical objects out to a distance r by luminosity gives
but the flux satisfies
so
For magnitude,
where a is the expansion parameter and
is the luminosity distance.
(1)
but the flux satisfies
(2)
so
(3)
For magnitude,
(4)
(5)
where a is the expansion parameter and
is the luminosity distance.(6)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
(17)
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