readily calculable from basic principles and is given by [@2004ApJ...608..622Z their § 2] $$\begin{array}{rl}
\label{eqn_intro_temp} \delta T_{21}(\vec{\theta}, z) \approx~&
23~(1+\delta)~x_{HI} \left ( 1 - \frac{T_\gamma}{T_S} \right ) \\
& \times \left ( \frac{\Omega_b~h^2}{0.02} \right ) \left [ \left (
\frac{0.15}{\Omega_m~h^2} \right ) \left ( \frac{1+z}{10} \right )
\right ]^{1/2} \mbox{mK},
\end{array}$$ where $\delta(\vec{\theta},z)$ is the local matter over-density, $x_{HI}(\vec{\theta},z)$ is the neutral fraction of hydrogen in the IGM, $T_\gamma(z) = 2.73~(1+z)$ K is the temperature of CMB at the redshift of interest, $T_S(\vec{\theta},z)$ is the spin temperature that describes the relative population of the ground and excited states of the hyperfine transition, and $\Omega_b$ is the baryon density relative to the critical density, $\Omega_m$ is the total matter density, and $h$ specifies the Hubble constant according to $H_0=100~ h$ km s$^{-1}$ Mpc$^{-1}$. From Equation \[eqn\_intro\_temp\], we see that perturbations in the local density, spin temperature, and neutral fraction of hydrogen in the IGM would all be revealed as fluctuations in the brightness temperature of the observed redshifted 21 cm line.

The differential brightness temperature of the redshifted 21 cm line is very sensitive to the spin temperature. When the spin temperature is greater than the CMB temperature, the line is 