The hardness of fresh water is primarily determined by the presence of certain metal ions, specifically calcium (Ca^{2+}) and magnesium (Mg^{2+}) ions. The other options provided do not contribute to the hardness index of fresh water. Here is the explanation for each option:

1. Total concentration of the Ca^{2+} and Mg^{2+} ions in the fresh water: This is the correct answer. The hardness of water is generally defined by the concentration of dissolved calcium and magnesium ions. The more Ca^{2+} and Mg^{2+} ions present, the harder the water.
2. Difference of the pH value of the fresh water with respect to sulphuric acid: The pH value of water is a measure of its acidity or alkalinity, not hardness. Hence, this option is incorrect.
3. The pH value of fresh water measured at room temperature and at one atmospheric pressure: Like the previous option, the pH value measures acidity/alkalinity and does not determine hardness. This option is also incorrect.
4. The molar ratio of D_2O to H_2O in fresh water: This ratio refers to the presence of deuterium oxide (heavy water) compared to normal water, which does not impact water hardness. Thus, this is incorrect.

Therefore, the hardness index of fresh water is determined by the total concentration of the Ca^{2+} and Mg^{2+} ions present in it.