Redshift and conformal time What is the relationship between redshift and conformal time ? 
For example in a paper i found: 
taking $z_e = 3234 $ at the time of radiaton-matter equality yields the conformal time $\frac{\eta_e}{\eta_0} = 0.007$ and taking $z_E=0.39$ at matter-$\Lambda$ equality yields $\frac{\eta_E}{\eta_0}=0.894$ and setting redshift at decoupling $z_d=1089$ yields $\frac{\eta_d}{\eta_0} = 0.0195$  where $\eta_0$ is the present decoupling time.
Further some cosmological parameters are given as : 
$\Omega_r = 8.36 \times 10^{-5},
\Omega_m = \Omega_b + \Omega_{dm} = 0.044 +0.226,
\Omega_\Lambda = 0.73, 
H_0=0.72$
Now how can i calculate all those $\eta_e, \eta_E, \eta_d, \eta_0$ from given redshift values and/or above parameters ? I searched whole of my text books trying to find an explicit relation for conformal time $\eta$ but all i got was $ \mathrm{d}t=a(t)\mathrm{d}\eta$. Any help would be very helpful. 
 A: As you state, conformal time is defined as
$$
\eta(t) = \int_0^t\frac{\text{d}t'}{a(t')}.
$$
Using
$$
\dot{a} = \frac{\text{d}a}{\text{d}t},
$$
this can be written in the form
$$
\eta(a) = \int_0^a\frac{\text{d}a}{a\dot{a}} = \int_0^a\frac{\text{d}a}{a^2H(a)},
$$
with
$$
H(a) = \frac{\dot{a}}{a} = H_0\sqrt{\Omega_{R,0}\,a^{-4} + \Omega_{M,0}\,a^{-3} + \Omega_{K,0}\,a^{-2} + \Omega_{\Lambda,0}}.
$$
The scale factor $a$ is related to the redshift as
$$
1 + z = \frac{1}{a},
$$
so that
$$
\eta(z) = \frac{1}{H_0}\int_0^{1/(1+z)}\frac{\text{d}a}{\sqrt{\Omega_{R,0} + \Omega_{M,0}\,a + \Omega_{K,0}\,a^2 + \Omega_{\Lambda,0}\,a^4}}.
$$
Basically, the conformal time is equal to the distance of the particle horizon, divided by $c$ (see this post for more info). $\eta_0$ refers to the current conformal age of the universe.

edit
I just checked the values that you posted with my own cosmology calculator. I get
$$
\eta_0 = 45.93\;\text{Gigayears}
$$
for the current conformal age of the universe, and
$$
\begin{align}
z_e &= 3234,& a_e &= 0.000309,& t_e &= 5.54\times 10^{-5}\;\text{Gy},&\eta_e &= 0.3804\;\text{Gy},\\
z_E &= 0.39,& a_E &= 0.719,& t_E &= 9.359\;\text{Gy},& \eta_E &= 41.08\;\text{Gy},\\
z_d &= 1089,& a_d &= 0000917,& t_d &= 0.00037\;\text{Gy},& \eta_d &= 0.911\;\text{Gy},
\end{align}
$$
so that
$$
\frac{\eta_e}{\eta_0} = 0.00828,\quad \frac{\eta_E}{\eta_0} = 0.894,\quad \frac{\eta_d}{\eta_0} = 0.0198.
$$
So my results are almost the same, but there's a small discrepancy. Apparently there's a small numerical error somewhere.
