# Why the $\delta \, ^{18}\text{O}$ in foraminifera shells decrease with temperature even if the oceanic $\delta \, ^{18}\text{O}$ stay constant?

I understand why foram shells contain more $^{18}\text{O}$ when there are ice sheets present, since $^{16}\text{O}$ evaporates more readily and gets trapped in the ice, increasing the relative abundance of $^{18}\text{O}$ in seawater. However, in the absence of ice sheets: What is the reason for there being more $^{18}\text{O}$ in the foram shells in colder climates? Since I assume the evaporated $^{16}\text{O}$ would just get returned to the sea by rain and rivers if it's not being trapped in ice.

PS: This is an adaptation of a question asked in the private discussion board of a first year university course on Earth Sciences, that I'm copying here so a wider community can benefit from it and the corresponding answer. As suggested by the moderators, I'll donate any reputation received from this question in the form of a bounty awarded to exceptional answers (in particular by new or low-rep users).

That is because the chemical reactivity of $^{18}\text{O}$ is slightly higher than that of $^{16}\text{O}$. For that reason, the biochemical reactions that produce calcium carbonate in foraminifera prefer $^{18}\text{O}$ over $^{16}\text{O}$.
However, the temperature dependence arise from the fact that the difference in reactivity is smaller when temperature increase. Therefore, in warmer temperatures the $\delta ^{18}\text{O}$ will be more similar to seawater (no preference for any isotope), but at colder temperatures the chemical preference for $^{18}\text{O}$ will be stronger and, therefore, the $\delta ^{18}\text{O}$ will increase.
This correlation with temperature is also known as the "paleotemperature equation", but as it is an empiric equation there are many slightly different versions, however all of them are pretty consistent. Several ones are summarized in figure 5 of the paper Oxygen isotopes in foraminifera: Overview and historical review (Pearson 2012) copied below, showing how the excess of $\delta ^{18}\text{O}$ in shells ($\delta ^{18}\text{O}_{\text{cc}}$) relative to the $\delta ^{18}\text{O}$ in seawater ($\delta ^{18}\text{O}_{\text{sw}}$) correlates with water temperature.
The deeper reason that explain the chemical preference for $^{18}\text{O}$ come from a process named "equilibrium isotope fractionation", associated to the vibrational characteristics of the light and heavy isotopes related to covalent atomic bonding. As explained by Gussone et. al (2003):