The relativistic redshift for a particle is given by
where
is the rest wavelength. From the relativistic Doppler shift
So, for
Since, to first-order approximation
using
gives
The time derivative of z is
where a is the expansion parameter and H is Hubble constant. In the Einstein-de Sitter cosmological model, the derivative is
In terms of the expansion parameter,
But
so
(1)
where
is the rest wavelength. From the relativistic Doppler shift(2)
So, for
(3)
Since, to first-order approximation
(4)
using
(5)
gives
(6)
(7)
(8)
The time derivative of z is
(9)
where a is the expansion parameter and H is Hubble constant. In the Einstein-de Sitter cosmological model, the derivative is
(10)
(11)
(12)
In terms of the expansion parameter,
(13)
(14)
(15)
But
(16)
so
(17)
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