The Observed squeezed limit of cosmological three-point functions

Abstract

The squeezed limit of the three-point function of cosmological perturbations is a powerful discriminant of different models of the early Universe. We present a conceptually simple and complete framework to relate any primordial bispectrum in this limit to late time observables, such as the CMB temperature bispectrum and the scale-dependent halo bias. We employ a series of convenient coordinate transformations to capture the leading non-linear effects of cosmological perturbation theory on these observables. This makes crucial use of Fermi Normal Coordinates and their conformal generalization, which we introduce here and discuss in detail. As an example, we apply our formalism to standard slow-roll single-field inflation. We show explicitly that Maldacena's results for the squeezed limits of the scalar bispectrum [proportional to (ns − 1) in comoving gauge] and the tensor-scalar-scalar bispectrum lead to no deviations from a Gaussian universe, except for projection effects. In particular, the primordial contributions to the squeezed CMB bispectrum and scale dependent halo bias vanish, and there are no primordial "fossil" correlations between long-wavelength tensor perturbations and small-scale perturbations. The contributions to observed correlations are then only due to projection effects such as gravitational lensing and redshift perturbations. Generation comoving coordinates kS < aH −−−−−→ Eq. (12) Evolution FNC τem −−−−−→ Eq. (38) Emission FNC(emitter) τ obs − −−−−−−−−−− → cosmic rulers [6]Observation FNC(observer) FIG. 1: Sequence of coordinates employed in the computation of observables predicted by a primordial bispectrum in the squeezed limit. The arrows represent the change from one set of coordinates to the next, indicating when this transformation is most conveniently performed and with a reference to the relevant equations. τem and τ obs denote the conformal times at which the observed photons were emitted and observed, respectively.

On separate universes

“…While consistency relations are a statement about transformation properties under diffeomorphisms, in a local observer's frame the diffeomorphism freedom is completely fixed by the observer's proper time and distance standards. As shown in [10] (see also [11][12][13][14]), this implies that squeezed-limit consistency relations drop out of locally measurable quantities. Non-single-field models which violate the consistency relation will on the other hand contribute to physical observables and can be measured in LSS surveys through the bispectrum or scale-dependent bias b(k) ∝ k −2 [15].…”

“…To solve small-scale clustering in this modified background, one can take a Lagrangian perspective -if nongravitational forces are negligible at large distances, one can follow inertial observers in the coarse-grained universe and their vicinity. As initially proposed in [10] and rigorously proven in [21], one can locally construct a frame, Conformal Fermi Coordinates (CFC), valid across a comoving distance of order Λ −1 at all times, which realizes this physical picture. In CFC, the observer interprets the small-scale structure around her as evolving in an FLRW universe that is modified due to long-wavelength modes with comoving wavenumber k L < Λ, with corrections quadratic in the spatial distance to the observer…”

“…A detailed treatment of projection effects is beyond the scope of this paper. For recent literature on this subject, see [10,[22][23][24].…”

“…[10] showed that the consistency relation for single-clock inflation, transformed into CFC, states that there is no primordial correlation between long-and short-wavelength potential perturbations. This provides the trivial initial conditions for the small-scale fluctuations in CFC.…”

“…We first briefly review the basic concept of Conformal Fermi Coordinates (CFC), which was first introduced in [10] and defined rigorously in [21]. Consider an observer free-falling in some spacetime.…”