Conformal Fermi Coordinates

TL;DR

Conformal Fermi Coordinates (CFC) generalize Fermi Normal Coordinates to describe a local FLRW spacetime around a free-falling observer, valid on scales beyond the horizon. By performing a controlled coarse-graining and mapping to observations, the authors show that long-wavelength perturbations enter local dynamics mainly through the background expansion $a_F$ and tidal terms via the conformal Riemann tensor, while short-scale Einstein equations remain linear in the subhorizon regime. The formalism clarifies the physical content of long-short mode couplings, recovers known results for tensor-scalar interactions, and provides a transparent framework separating locally measurable effects from projection effects. The approach thus offers a practical, gauge-invariant method to connect early-universe perturbations to late-time observables across all relevant cosmological scales, with clear prescriptions for translating to distant observations.

Abstract

Fermi Normal Coordinates (FNC) are a useful frame for isolating the locally observable, physical effects of a long-wavelength spacetime perturbation. Their cosmological application, however, is hampered by the fact that they are only valid on scales much smaller than the horizon. We introduce a generalization that we call Conformal Fermi Coordinates (CFC). CFC preserve all the advantages of FNC, but in addition are valid outside the horizon. They allow us to calculate the coupling of long- and short-wavelength modes on all scales larger than the sound horizon of the cosmological fluid, starting from the epoch of inflation until today, by removing the complications of the second order Einstein equations to a large extent, and eliminating all gauge ambiguities. As an application, we present a calculation of the effect of long-wavelength tensor modes on small scale density fluctuations. We recover previous results, but clarify the physical content of the individual contributions in terms of locally measurable effects and "projection" terms.

Figures (2)

• Figure 1: Construction of the CFC. At point $P$, the observer's geodesic $G$ intersects a spatial hypersurface $\Sigma$, having constant conformal time $\tau$ and scale factor $a(\tau)$ in some global coordinate system. The spatial hyper-surface $\Sigma_F$, having constant CFC conformal time $\tau_F$ and CFC scale factor $a_F(\tau_F)$, also intersects $G$ at $P$, but does not coincide with $\Sigma$ in general. Another point $Q$ on $\Sigma_F$ is connected to $P$ by a conformal geodesic, and is parameterized by the same $\tau_F$ but nonzero $x^i_F$. The CFC coordinates are valid within a tubelike region bounded by hyper-surface $B$ surrounding $G$.
• Figure 2: Illustration of the mapping between nearby CFC frames $\{x_F\}$, $\{ x_{F'} \}$ (tetrads indicated as arrows) constructed for the geodesics $G$, $G'$, respectively (thick solid and dashed). The initial separation at $\tau_F = 0$ of the two geodesics in the $\{ x_F \}$ frame is given by $\mathbf{q}$ (dotted arrow), which at late times is modified to $\mathbf{q}+\mathbf{s}$ due to the local geodesic deviation [Eq. (\ref{['eq:shift']})].