Unimodular Theory of Gravity and Inflation

We investigate inflation and its scalar perturbation driven by a massive scalar field in the unimodular theory of gravity. We introduce a parameter $\xi$ with which the theory is invariant under general unimodular coordinate transformations. When the unimodular parameter is $\xi=6$, the classical picture of inflation is reproduced in the unimodular theory because it recovers the background equations of the standard theory of general relativity. We show that for $\xi=6$, the theory is equivalent to the standard theory of general relativity at the perturbation level. Unimodular gravity constrains the gauge degree of freedom in the scalar perturbation, but the perturbation equations are similar to those in general relativity. For $\xi \neq 6$, we derive the power spectrum and the spectral index, and obtain the unimodular correction to the tensor-to-scalar ratio. Depending on the value of $\xi$, the correction can either raise or lower the value of the tensor-to-scalar ratio.