A Convention of units for densities in cosmology

AI Thread Summary
The discussion centers on the correct interpretation of the units for galaxy density in cosmology, specifically whether to use ##\text{d}N/\text{d}\Omega\text{d}z## or ##\text{d}N/\text{d}\Omega/\text{d}z##. The user presents two calculation cases for the expected number density of galaxies, leading to significantly different results based on the interpretation of the units. There is uncertainty about whether to multiply or divide by the redshift interval, ##\Delta z##, which affects the calculations. Clarification on the convention for these units is sought to resolve the confusion. Understanding the correct unit convention is crucial for accurate density calculations in cosmological surveys.
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I would like to know the Convention of units for densities in cosmology : I wonder if consistent units are used by multiplying or dividing with Delta_z
I have a table of densities of galaxies :

Expected number density of galaxies for photometric survey per unit area and redshift intervals, ##\mathrm{d} N / \mathrm{d} \Omega \mathrm{d} z\left[\mathrm{sr}^{-1}\right]## and the corresponding density of galaxies per ##\operatorname{arcmin}^2## for each redshift

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I wonder if the second row values are correct : indeed, I hesitate between both calculus, for example for the bin :

- case 1

3 / 11818102.860 * 0.119 = 4219062.72 (rounded to 4219063) in units ##\text{d}N/\text{d}\Omega\text{d}z##

OR should I set rather :

- case 2

3 / 11818102.860 / 0.119 = 297935366.218 (rounded to 297935366) in units ##\text{d}N/\text{d}\Omega\text{d}z##

One of both is wrong since I don't know if the units are ##\text{d}N/\text{d}\Omega\text{d}z## or ##\text{d}N/\text{d}\Omega/\text{d}z##.

Could anyone help me what is the convention for the units of the writing ##\text{d}N/\text{d}\Omega\text{d}z## that causes some confusions ( we don't know if we have to multiply or divide by ##\Delta z## ?
 
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