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Lessons from $T^μ_{~ μ}$ on inflation models: two-scalar theory and Yukawa theory

We demonstrate two properties of the trace of the energy-momentum tensor $T^μ_{~ μ}$ in the flat spacetime. One is the decoupling of heavy degrees of freedom; i.e., heavy degrees of freedom leave no effect for low-energy $T^μ_{~ μ}$-inserted amplitudes. This is intuitively apparent from the effective field theory point of view, but one has to take into account the so-called trace anomaly to explicitly demonstrate the decoupling. As a result, for example, in the $R^{2}$ inflation model, scalaron decay is insensitive to heavy degrees of freedom when a matter sector ${\it minimally}$ couples to gravity (up to a non-minimal coupling of a matter scalar field other than the scalaron). The other property is a quantum contribution to a non-minimal coupling of a scalar field. The non-minimal coupling disappears from the action in the flat spacetime, but leaves the so-called improvement term in $T^μ_{~ μ}$. We study the renormalization group equation of the non-minimal coupling to discuss its quantum-induced value and implications for inflation dynamics. We work it out in the two-scalar theory and Yukawa theory.