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Novel Quantity for Probing Matter Perturbations Below the Fresnel Scale in Gravitational Lensing of Gravitational Waves

So Tanaka, Teruaki Suyama

TL;DR

This work analyzes gravitational-wave lensing in the wave-optics regime and identifies a Fresnel-scale limitation in probing small-scale structure. By introducing the quantity $I(\omega)$ and its frequency correlation $\Delta_I(\Omega)$, the authors show that one can access perturbations at arbitrarily small scales through an effective Fresnel scale $r_F(\Omega,\chi)$, which becomes increasingly suppressed as $\Omega$ grows. The approach remains viable under finite observation time $T$ (with $r_{F,\rm min} \sim r_F/\sqrt{\omega T}$), enabling parsec-scale sensitivity for mHz GWs in year-long observations. Observability estimates suggest $\sqrt{\langle |I^{\rm 1h}(\omega)|^2\rangle}$ of order $10^{-2}$ can be achieved, surpassing anticipated LISA noise levels, and providing a new avenue to probe the small-scale matter power spectrum and dark-matter physics using GW lensing. Overall, the paper proposes a practical, model-independent pathway to study fine-grained cosmic structure via GW wave optics.

Abstract

Gravitational lensing of gravitational waves provides a powerful probe of the mass density distribution in the universe. Wave optics effects, such as diffraction, make the lensing effect sensitive to the structure around the Fresnel scale, which depends on the gravitational wave frequency and is typically sub-Galactic for realistic observations. Contrary to this common lore, we show that wave optics can, in principle, probe matter perturbations even below the Fresnel scale. This is achieved by introducing a new quantity derived from the amplification factor, which characterizes the lensing effect, and analyzing its correlation function. Our results demonstrate that this quantity defines an effective Fresnel scale: a characteristic scale that can be arbitrarily small, even when observational frequencies are bounded. In practice, the effective Fresnel scale is constrained by the observation time $T$ and is suppressed by a factor of $1/\sqrt{fT}$ relative to the standard Fresnel scale at frequency $f$. Nevertheless, it remains significantly smaller than the conventional Fresnel scale for $fT \gg 1$; for instance, in one-year observations of mHz GWs, the effective Fresnel scale can be as small as 1 pc. This approach opens new avenues for probing the fine-scale structure of the universe and the nature of dark matter.

Novel Quantity for Probing Matter Perturbations Below the Fresnel Scale in Gravitational Lensing of Gravitational Waves

TL;DR

This work analyzes gravitational-wave lensing in the wave-optics regime and identifies a Fresnel-scale limitation in probing small-scale structure. By introducing the quantity and its frequency correlation , the authors show that one can access perturbations at arbitrarily small scales through an effective Fresnel scale , which becomes increasingly suppressed as grows. The approach remains viable under finite observation time (with ), enabling parsec-scale sensitivity for mHz GWs in year-long observations. Observability estimates suggest of order can be achieved, surpassing anticipated LISA noise levels, and providing a new avenue to probe the small-scale matter power spectrum and dark-matter physics using GW lensing. Overall, the paper proposes a practical, model-independent pathway to study fine-grained cosmic structure via GW wave optics.

Abstract

Gravitational lensing of gravitational waves provides a powerful probe of the mass density distribution in the universe. Wave optics effects, such as diffraction, make the lensing effect sensitive to the structure around the Fresnel scale, which depends on the gravitational wave frequency and is typically sub-Galactic for realistic observations. Contrary to this common lore, we show that wave optics can, in principle, probe matter perturbations even below the Fresnel scale. This is achieved by introducing a new quantity derived from the amplification factor, which characterizes the lensing effect, and analyzing its correlation function. Our results demonstrate that this quantity defines an effective Fresnel scale: a characteristic scale that can be arbitrarily small, even when observational frequencies are bounded. In practice, the effective Fresnel scale is constrained by the observation time and is suppressed by a factor of relative to the standard Fresnel scale at frequency . Nevertheless, it remains significantly smaller than the conventional Fresnel scale for ; for instance, in one-year observations of mHz GWs, the effective Fresnel scale can be as small as 1 pc. This approach opens new avenues for probing the fine-scale structure of the universe and the nature of dark matter.

Paper Structure

This paper contains 8 sections, 58 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Amplification Factor in the Born Approximation
  3. A Novel Quantity for Probing Matter Perturbations Below the Fresnel Scale
  4. Formulation
  5. Observability
  6. Conclusion
  7. Relation Between $I$ and the Convergence Field
  8. Estimation of $\langle |I|^2\rangle$ in the Halo Model

Figures (1)

  • Figure 1: Schematic picture of gravitational lensing. $\chi$-axis is directed from the observer to the wave source, and the source is at $\chi=\chi_s.$$\bm{r}_\perp{}$ is the position vector perpendicular to the $\chi$-axis.