Large-Scale Structure and Gravitational Waves III: Tidal Effects

TL;DR

This paper shows that long-wavelength metric perturbations imprint a local tidal field that modulates small-scale density fluctuations at second order, with a fossil-like persistence for tensor and vector tides once the modes enter the horizon. Using conformal Fermi Normal Coordinates and Lagrangian perturbation theory, the authors derive a general form for the second-order density response $\delta_{2,t}$ in terms of coefficient functions $\alpha(\tau)$ and $\beta(\tau)$, recover the standard scalar $F_2$ result, and reveal a non-decaying tensor/tidal imprint even after horizon entry. They extend the analysis to include radiation and $\Lambda$, perform projection to observed quantities, and explore implications for intrinsic alignments and B-mode cosmic shear, finding potentially detectable tensor-induced signals at low redshift. The results provide a framework to connect primordial gravitational waves and large-scale tidal fields to observable small-scale statistics, including a concrete forecast for tensor-induced B-modes and intrinsic alignments that could inform future surveys.

Abstract

The leading locally observable effect of a long-wavelength metric perturbation corresponds to a tidal field. We derive the tidal field induced by scalar, vector, and tensor perturbations, and use second order perturbation theory to calculate the effect on the locally measured small-scale density fluctuations. For sub-horizon scalar perturbations, we recover the standard perturbation theory result ($F_2$ kernel). For tensor modes of wavenumber $k_L$, we find that effects persist for $k_Lτ\gg 1$, i.e. even long after the gravitational wave has entered the horizon and redshifted away, i.e. it is a "fossil" effect. We then use these results, combined with the "ruler perturbations" of arXiv:1204.3625, to predict the observed distortion of the small-scale matter correlation function induced by a long-wavelength tensor mode. We also estimate the observed signal in the B mode of the cosmic shear from a gravitational wave background, including both tidal (intrinsic alignment) and projection (lensing) effects. The non-vanishing tidal effect in the $k_Lτ\gg 1$ limit significantly increases the intrinsic alignment contribution to shear B modes, especially at low redshifts $z \lesssim 2$.

Figures (4)

• Figure 1: Top panel: the function $F(\tau)$ [Eq. (\ref{['eq:FVtensor']})], corresponding to $\beta(\tau)$ in Eq. (\ref{['eq:d2ttensor']}), for tensor modes of various wavenumbers $k_L$ as function of scale factor. Bottom panel: coefficient $\alpha(\tau)$ in Eq. (\ref{['eq:d2ttensor']}), as a function of $a$ for tensor modes with the same wavenumbers $k_L$ as in the upper panel. For wavenumbers entering the horizon during matter domination, this reduces to Eq. (\ref{['eq:alphaMD']}). All results were obtained by numerical integration of the linear and second order equations (see App. \ref{['app:numerics']}) for a flat $\Lambda$CDM cosmology.
• Figure 2: Coefficient functions $\alpha(k_L,\tau),\,\beta(k_L,\tau)$ in Eq. (\ref{['eq:d2ttensor']}) as function of $k_L$ for fixed scale factors $a(\tau) = 1$ and $a(\tau) = 1/3$. The same $\Lambda$CDM cosmology as in Fig. \ref{['fig:FDsigma']} was assumed.
• Figure 3: Coefficient function $\alpha(k_L,\tau_0)$ as in Fig. \ref{['fig:d2t']} (green dashed), and for a flat matter dominated (Einstein-de Sitter, EdS) Universe (blue solid). The black dotted line shows the EdS result plotted at the same $k_L\tau$ as $\Lambda$CDM (see text). All curves are for $a(\tau_0)=1$.
• Figure 4: Tensor mode contribution to the $B$ mode angular shear power spectrum from a gravitational wave background with tensor-to-scalar ratio $r=0.1$. The blue solid lines show the result using the matching of the second order density [Eq. (\ref{['eq:gammaIAt']})], green dashed lines show the result when using the instantaneous tensor tidal field as adopted in GWshear, and black dotted lines show the result in the absence of intrinsic alignment, i.e. only including the lensing contribution. In all cases, thick lines are for a source redshift of $z=2$, while thin lines are for $z=0.8$.