Kavli Affiliate: Wayne Hu

| First 5 Authors: Keisuke Inomata, Hayden Lee, Wayne Hu, ,

| Summary:

We study the $N$-point function of the density contrast to quadratic order in

the squeezed limit during the matter-dominated (MD) and radiation-dominated

(RD) eras in synchronous gauge. Since synchronous gauge follows the free-fall

frame of observers, the equivalence principle dictates that in the gradient

approximation for the long-wavelength mode there is only a single, manifestly

time-independent consistency relation for the $N$-point function. This simple

form is dictated by the initial mapping between synchronous and local

coordinates, unlike Newtonian gauge and its correspondingly separate dilation

and Newtonian consistency relations. Dynamical effects only appear at quadratic

order in the squeezed limit and are again characterized by a change in the

local background, also known as the separate universe approach. We show that

for the 3-point function the compatibility between these squeezed-limit

relations and second-order perturbation theory requires both the initial and

dynamical contributions to match, as they do in single-field inflation. This

clarifies the role of evolution or late-time projection effects in establishing

the consistency relation for observable bispectra, which is especially

important for radiation acoustic oscillations and for establishing consistency

below the matter-radiation equality scale in the MD era. Defining an

appropriate angle and time average of these oscillations is also important for

making separate universe predictions of spatially varying local observables

during the RD era, which can be useful for a wider range of cosmological

predictions beyond $N$-point functions.

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