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A key question regarding the unconventional superconductivity ofSr$_2$RuO$_4$ remains whether the order parameter is single- or two-component.Under a hypothesis of two-component superconductivity, uniaxial pressure isexpected to lift their degeneracy, resulting in a split transition. The mostdirect and fundamental probe of a split transition is heat capacity. Here, wereport development of new high-frequency methodology for measurement of heatcapacity of samples subject to large and highly homogeneous uniaxial pressure.We place an upper limit on the heat capacity signature of any second transitionof a few per cent of the primary superconducting transition. The normalizedjump in heat capacity, $\Delta C/C$, grows smoothly as a function of uniaxialpressure, but we find no qualitative evidence of a pressure-induced orderparameter transition. Thanks to the high precision of our measurements, thesefindings place stringent constraints on theories of the superconductivity ofSr$_2$RuO$_4$.

High sensitivity heat capacity measurements on Sr2RuO4 under uniaxial pressure