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Cosmological perturbation of Unimodular Gravity and General Relativity are identical

Unimodular Gravity (UG) is a restricted version of General Relativity (GR) in which the determinant of the metric is a fixed function and the field equations are given by the trace-free part of the full Einstein equations. The background equations in UG and GR are identical. It was recently claimed that, the first order contribution in the temperature fluctuation of the Cosmic Microwave Background (CMB) in UG is different from GR. In this work, we calculate the first order perturbation equations in UG and show that the Sachs-Wolfe effect in UG, in terms of gauge invariant variables, is identical to GR. We also show that the second order perturbation equation of Mukhnanov-Sasaki variable in UG, is identical to GR. The only difference comes from the gauge choices due the constraint on the metric determinant. Hence, UG and GR are identical and indistinguishable in CMB data on large scales.