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The spatial gauge-dependence of single-field inflationary bispectra

In single-field inflationary models the bispectra are usually given in the $ζ$-gauge, because its temporal part leads to the super-horizon conservation of fluctuations. However, this property is independent of the choice of {\it spatial} gauge, so in this letter we explore this freedom. We compute the variation of the bispectra under the most general spatial gauge transformation that is globally defined and privileges no point, direction or scale. In the squeezed configuration we then obtain a generalization of the classic $ζ$-gauge consistency relations, which we also derive through the `large diffeomorphism' approach for all four bispectra. At leading order in the long wave-number the transformation only affects the case where the long mode is a scalar. The first effect is a shift of the tilt factor, so that one can significantly reduce the amplitude of that contribution. Secondly, there is now an extra term depending on the triangle shape, the same as in solid inflation, which is due to the fact that the 3-metric has a scalar anisotropy in generic spatial gauge. At next-to-leading order there is no variation, so the conformal consistency relations of the $ζ$-gauge are preserved.