Anisotropic imprint of long-wavelength tensor perturbations on cosmic structure

TL;DR

This paper shows that long-wavelength tensor perturbations induce a quadrupolar modulation of the local matter power spectrum, but in single-field slow-roll inflation the observable effect is infrared-safe and suppressed by horizon-scale dynamics and projection effects. The authors carefully separate primordial, post-inflation nonlinear evolution, and projection contributions, demonstrating that the naive $K\to0$ divergence cancels while a finite, small residual quadrupole remains. They quantify the quadrupole in the observed galaxy power spectrum using a detailed treatment of projection and nonlinear mode coupling, and they show that the resulting rms quadrupole is of order $\sim 1.2\sqrt{\Delta^2_{\gamma}}$ at cosmological redshifts, typically far below current detectability. Consequently, detecting a larger quadrupole would falsify SFSR inflation or indicate a violation of the squeezed-limit consistency relation, while null results place constraints on such alternative scenarios. The framework also provides a clear pathway to interpret potential signals in future 21-cm and galaxy surveys as probes of the primordial tensor background.

Abstract

Inflationary models predict a correlation between primordial density perturbations (scalar metric perturbations) and gravitational waves (tensor metric perturbations) in the form of a scalar-scalar-tensor three-point correlation, or bispectrum in Fourier space. The squeezed limit of this bispectrum implies a quadrupolar asymmetry in the observed local power spectrum for matter and galaxies. Here we show (like others before) that an infrared divergence in the amplitude of this power asymmetry predicted in single-field slow-roll models is canceled by projection effects when considering the observed power spectrum. We then further evaluate the nonzero, but finite, residual quadrupolar power asymmetry that remains after the divergences are canceled. While the quadrupolar power asymmetry is small, it is conceptually important. Our calculation moreover clarifies how the predictions for this power asymmetry may change in models with different scalar-scalar-tensor bispectra, and shows that convincing detection of the quadrupolar power asymmetry would rule out the single-field slow-roll models of inflation.

Figures (3)

• Figure 1: The mode-coupling kernel $\mathcal{S}(K)$ is plotted assuming matter domination. It is only a function of the combination $K\eta$. The vertical line marks the comoving Hubble scale.
• Figure 2: The contribution to $\overline{\mathcal{Q}^2}$ from per logarithmic interval of $K$ for source redshift $z=2$, normalized to $\Delta^2_{\gamma}$. The dash-dotted line (P) is the prediction from SFSR scalar-scalar-tensor bispectrum alone. (first term of Eq. (\ref{['eq:quadrupole-tensor']})). The dashed line (P+NL) includes nonlinear mode couplings (second term of Eq. (\ref{['eq:quadrupole-tensor']})). The solid line (P+NL+proj) is the full result with the projection effects (last term of Eq. (\ref{['eq:quadrupole-tensor']})). The vertical line marks the horizon scale at present time.
• Figure 3: The variance of the galaxy power quadrupole as a function of the source redshift $z$, normalized to $(\Delta^2_{\gamma})^{1/2}$. We compare results for different choices of the small-scale cutoff $K_{\mathrm{max}}$ around the scale $k_{\mathrm{eq}}$.