I have integrated a distribution function (for sterile neutrinos), of a form similar to:
$$\left( \frac{\partial f_s}{\partial T}\right)_{E/T}= ..... \tag{1}$$
Meaning that throughout the integration $E/T$ should be treated as a constant. Where $E$ signifies momentum and $T$, temperature.
At the end of the integration my answer is of the form:
$$f_s = \frac{E}{T} x $$
where $x$ is some variable.
I am trying to plot this in terms of $T$ and $x$ but I am not sure of what to do because I have the following doubt:
Is $E/T$ still seen as a constant even after the integration has been done?
If the ratio is still a constant, can I consider $T$ to be a variable and to therefore plot $f_s /T$?
$T$ is not a constant, but $E/T$ is. So you have $f_s = kx,$ where $k=E/T.$ This gives $f_s/T = kx/T.$