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Probing Gravitational-Wave Four-Point Correlators

Martina Ciprini, Maria Lucia Marcelli, Gianmassimo Tasinato

Abstract

Stochastic gravitational-wave backgrounds (SGWBs) of primordial origin offer a powerful probe of early-Universe physics and possible dark-sector dynamics. While most searches focus on the GW power spectrum, additional information is encoded in higher-order correlators that characterize the statistical properties of the signal. In this work we study non-Gaussian features of a cosmological SGWB generated at second order by vector fluctuations, a class of sources well motivated in early-Universe scenarios. Within this framework we develop tools to characterize higher-order GW correlators and compute representative four-point functions that generate a connected contribution to the GW trispectrum. We show that the trispectrum amplitude scales as the square of the GW power spectrum and peaks in characteristic folded momentum configurations, reflecting the structure of the nonlinear source. We then explore the observational implications. First, we demonstrate that the connected trispectrum contributes to the variance of two-point overlap reduction functions, including the Hellings-Downs curve relevant for pulsar timing arrays. We then construct the optimal estimator to measure the connected trispectrum with ground-based interferometers. Our results highlight how non-Gaussian SGWB statistics provide a complementary observable to probe the origin of GW backgrounds and to distinguish cosmological from astrophysical sources.

Probing Gravitational-Wave Four-Point Correlators

Abstract

Stochastic gravitational-wave backgrounds (SGWBs) of primordial origin offer a powerful probe of early-Universe physics and possible dark-sector dynamics. While most searches focus on the GW power spectrum, additional information is encoded in higher-order correlators that characterize the statistical properties of the signal. In this work we study non-Gaussian features of a cosmological SGWB generated at second order by vector fluctuations, a class of sources well motivated in early-Universe scenarios. Within this framework we develop tools to characterize higher-order GW correlators and compute representative four-point functions that generate a connected contribution to the GW trispectrum. We show that the trispectrum amplitude scales as the square of the GW power spectrum and peaks in characteristic folded momentum configurations, reflecting the structure of the nonlinear source. We then explore the observational implications. First, we demonstrate that the connected trispectrum contributes to the variance of two-point overlap reduction functions, including the Hellings-Downs curve relevant for pulsar timing arrays. We then construct the optimal estimator to measure the connected trispectrum with ground-based interferometers. Our results highlight how non-Gaussian SGWB statistics provide a complementary observable to probe the origin of GW backgrounds and to distinguish cosmological from astrophysical sources.
Paper Structure (19 sections, 119 equations, 5 figures)

This paper contains 19 sections, 119 equations, 5 figures.

Table of Contents

  1. Introduction and Conclusions
  2. Our set-up, and the SGWB two-point function
  3. An example of vector field models from the early universe.
  4. Computation of the induced SGWB spectrum
  5. Explicit examples.
  6. Computation of induced GW non-Gaussianity
  7. Definitions and motivations
  8. The time integral in Eq. \ref{['def_tintA']}.
  9. The momentum integral of Eq. \ref{['def_tintB']}.
  10. An explicit, simple example.
  11. Summary of this Section
  12. Consequences for GW experiments
  13. Motivations
  14. The variance of the ORF.
  15. The ORF for the four-point function of GW signals.
  16. ...and 4 more sections

Figures (5)

  • Figure 1: The GW energy spectrum $\Omega_{\rm GW}$ associated a monochromatic (delta-function) magnetic power spectrum, plotted as a function of $f/f_\star$ for different values of $k_\star|\tau_R|$. See Eq. \ref{['Omega_deltafunction']}. Notice that the the spectrum drops to zero for $f/f_\star\ge2$.
  • Figure 2: Profiles of $\Omega_{\rm GW}$ as function of GW frequency over a characteristic scale $f_\star$. Left panel: GW spectrum induced by a broken power law profile for the magnetic field, \ref{['eq_bplp']}, choosing $n_1=4$, $n_2=-2$, $\sigma=1$. Right panel: GW spectrum associated to the log-normal profile of Eq. \ref{['eq_logp']}, with $\Delta=0.8$. In both cases, $P_0=10^2$, and fix the quantity $|k_\star \tau_R|=10^{-2}$ in Eq.\ref{['exp_ph3']}.
  • Figure 3: Graphical representation of the shapes of closed quadrilatera associated with the GW trispectrum. Left panel: a generic closed shape. Right panel: a flattened trispectrum shape, as the one we reduce to in our set-up. See main text for details.
  • Figure 4: A particular lay-out for the four building blocks of magnetic fields leading to connected GW four-point functions, located on the vertexes of a square and linked by vector-field correlations. See main text for details.
  • Figure 5: Representation of the Hellings-Down function (black line); its total variance associated with disconnected contributions to the signal four-point function (brown band); finally, the total variance accounting for non-vanishing contributions of the connected GW trispectrum (green band). See main text for details.