An important consideration in the post-recombination epoch is the issue of the optical depth $\tau$ of the Universe due to Compton scattering. This is a dimensionless quantity such that $exp(−\tau)$ (often called the visibility) describes the attenuation of the photon flux as it traverses a certain length. The probability $dP$ that a photon has suffered a scattering event from an electron while travelling a distance $cdt$ is given by
$$dP=-\frac{dN_\gamma}{N_\gamma}=-\frac{dI}{I}=\frac{dt}{\tau_{\gamma e}}=n_e\sigma_Tcdt=-\frac{x\rho_m}{m_p}\sigma_T c\frac{dt}{dz}dz=-d\tau\hspace{1cm} (9.4.2)$$
where $N_\gamma$ is the photon flux, so that
$$I(t_0,z)=I(t)exp\left(-\int_0^z\frac{x\rho_m}{m_p}\sigma_T c\frac{dt}{dz}dz\right)=I(t)exp[-\tau(z)];\hspace{2cm} (9.4.3)$$
$I(t_0,z)$ is the intensity of the background radiation reaching the observer at time $t_0$ with a redshift $z$ if it is incident on a region at a redshift $z$ with intensity $I[t(z)]$; $\tau(z)$ is called the optical depth of such a region. The probability that a photon, which arrives at the observer at the present epoch, suffered its last scattering event between $z$ and $z-dz$ is
$$-\frac{d}{dz}\{1-exp[-\tau(z)]\}dz=exp[-\tau(z)]d\tau=g(z)dz\hspace{4cm}{(9.4.4)}$$
The quantity $g(z)$ is called the differential visibility or effective width of the surface of last scattering;
This text was taken from "Cosmology. The origin and evolution of cosmic structure" by P.Coles, F.Lucchin.
I have some doubt about $(9.4.2)$ and $(9.4.4)$.
I would write for $(9.4.2)$:
$$dP=-\frac{dN_\gamma}{N_\gamma}=-\frac{dI}{I}=...=...=-\frac{x\rho_m}{m_p}\sigma_T c\left|\frac{dt}{dz}\right|dz=-d\tau$$
because $z$ decreases from the Big Bang up to us and $t$ increases, so $\frac{dt}{dz}<0$, $dz<0$ and $dt>0$. (The same correction obviously inside the integral in $(9.4.3)$).
About $(9.4.4)$, what is not clear is the way in which "The probability that a photon, which arrives at the observer at the present epoch, suffered its last scattering event between $z$ and $z-dz$", was derived. How do I get the first expression of $(9.4.4)$?
Someone has any suggestion?
Thank you very much.
\exp
(with a leading backslash) sets that operator in upright type. $\endgroup$