Hydrogen is the most abundant element in the solar system and has been considered one of the main light elements in the Earth's core. The hydrogen content in the Earth's core is determined normally by matching the volume expansion caused by the incorporation of hydrogen into FeHx to the Earth's core density deficit. The magnitude of this volume expansion at the pressure (P) and temperature (T) conditions of the Earth's core is still unknown, and the effect of spin transition in FeHx at high pressure is usually ignored. In this study, we simulate the Fe spin transition, equation of state, and hydrogen-induced volume expansion in Fe-H binaries at high P-T conditions using density functional theory (DFT) calculations. Our results indicate that hydrogen could stabilize the magnetic properties of fcc Fe from similar to 10 to similar to 40 GPa. A volume expansion induced by hydrogen is linear with pressure except at the Fe spin transition pressure, where it collapses significantly (similar to 30%). The fcc FeH lattice is predicted to expand at an average rate of similar to 1.38 and 1.07 angstrom(3) per hydrogen atom under the Earth's outer and inner core P-T conditions, where the hydrogen content is estimated to be similar to 0.54-1.10 wt% and similar to 0.10-0.22 wt%, respectively. These results suggest that the Earth's core may be a potentially large reservoir of water, with up to similar to 98 times as much as oceans of water being brought to the Earth's interior during its formation. Based on our predicted hydrogen content in the Earth's core, we propose that the presence of hydrogen would induce a relatively lower core temperature, similar to 300-500 K colder than it has been previously speculated.

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