Nonlinear Gravitational Wave Memory : Universal Low-Frequency Background

TL;DR

This work establishes a universal low-frequency contribution to the stochastic gravitational-wave background from nonlinear memory, showing it is sourced by the energy in emitted GWs and yields distinct infrared scalings that depend on the cosmic equation of state. By combining a peaked linear-spectrum ansatz with Green-function methods for a generic EOS, the authors derive analytic expressions for the nonlinear memory background and demonstrate how the memory component can dominate in the infrared, following a slope $\Omega_{\rm GW,mem} \propto f^{3-2\left|\frac{3w-1}{3w+1}\right|}$ (with a $\ln^2 f$ enhancement in radiation domination, $w=1/3$). They apply the framework to cosmological and astrophysical SGWBs, producing complete frequency spectra and highlighting observational prospects across future multi-band detectors, which could also probe the thermal history of the universe via memory features. The results enable separation of memory from linear signals through spectral and polarization characteristics, offering a robust handle on gravity and high-energy physics through SGWB observations.

Abstract

A universal contribution exists in the infrared (low frequency) regime of all gravitational waves, which results from nonlinear memory. Nonlinear memory is sourced by linear order gravitational waves and exists for any gravitational-wave background. We calculate the stochastic nonlinear memory signal of various stochastic backgrounds of cosmological (scalar induced, reheating, phase transition, topological defect, turbulence) and astrophysical (binary mergers of stellar-mass, intermediate mass, supermassive, and primordial black holes) origins. These results allow us to derive the complete frequency spectrum of cosmological and astrophysical SGWB. We calculate how to probe the thermal state of the universe, i.e. the equation of the state, via the memory spectrum's slope and also discuss the detection prospects at various frequency bands with future experiments.

Figures (5)

• Figure 1: Decomposition of GW signal : Linear (black), nonlinear memory (red) and total (blue). Left: Character of a cosmic SGWB produced in radiation domination, where peak frequency $f_{*}=k_*/2\pi$ can be anywhere from nHz (MeV) to THz ($10^{18}$ GeV). We choose two sets of example signals to compare the effects of amplitude of linear power spectrum, $\Delta_h^2$, and momentum to Hubble ratio, $k_*/{\cal H}_*$: i)Top one has $k_*/{\cal H}_*\sim 1$ and $\Delta_h^2 (k_*) = 1/4$; ii)The bottom one has $k_*/{\cal H}_*=30$ and $\Delta_h^2 (k_*) = 10^{-8}$. The memory surpasses linear signal at $f^{\rm cosmo}_{\rm mem. dom.}$ according to Eq. \ref{['eqcosmomemdom']} or Eq. \ref{['eqgenericcosmomemdom']} depending on the equation of state. Right: Character of astrophysical SGWBs, where peak frequency $f_{*}$ can be $10^3, \, 10^{-1}, 10^{-7}\, {\rm Hz}$ for the SGWB of stellar-mass, intermediate and supermassive BBH mergers. Ratio of linear signal to nonlinear memory is nearly the same for BBH mergers for different masses and event rates, i.e. $\Omega_{\rm GW,mem} / \Omega_{\rm GW,lin} \approx 2-3 \cdot 10^{-4}$. Below peak frequency, linear signal scales with $f^{2/3}$ until the turn-around, and with $f^3$ below turn-around; while memory scales with $f$. The memory starts dominating the linear signal at $f^{\rm astro}_{\rm mem. dom.}$ according to Eq. \ref{['eqastromemdom']}.
• Figure 2: Solid (dashed) lines denote the total (memory) signals. Left: SGWB from cosmic sources with $\Delta_h^2=0.01$ and $k_* \simeq {\cal H}_*$. Memory dominates the linear signal, with a gentle logarithmic dependence for (I) in radiation domination, and with quadratic power for (III) in kination domination. Memory is subdominant for (II) in matter domination. Three example phenomena have peaks at distinct frequencies : (I) nHz band (PTA NANOGrav:2023gorNANOGrav:2023hvmEPTA:2023fykEPTA:2023xxkReardon:2023gzhXu:2023wog and SKA Janssen:2014dkaWeltman:2018zrl), (II) mHz (LISA LISA:2022yaoLISACosmologyWorkingGroup:2022jokLISA:2022kgy, TianQIN TianQin:2015yph, Taiji Hu:2017mdeRuan:2018tsw, DECIGO-BBO Yagi:2011wg, $\mu$ARES Sesana:2019vho and APTA Alves:2024ulc) and Hz band (ET Punturo:2010zz and CE Evans:2021gyd), (III) MHz and higher band. Right: SGWB from stellar-mass BH, IMBH, SMBH and primordial BH (PBH) mergers. For SMBHs the estimated event rate density is $10^{-5}-10^{-2}~{\rm Gpc}^{-3}{\rm yr}^{-1}$2021MNRAS.502L..99M2024AA...685A..94E, for StMBHs it is $10-100~{\rm Gpc}^{-3}{\rm yr}^{-1}$theligoscientific. For IMBHs it is very uncertain and can be between $10^{-2}$ to 10 ${\rm Gpc}^{-3}{\rm yr}^{-1}$2022ApJ...933..170F. For PBHs, we take $10^{-4} < \rho_{\rm PBH} / \rho_{\rm DM} < 10^{-3}$ and $M_{\rm PBH} \sim 10^{-6} {\rm M}_\odot$ as a fiducial value.
• Figure 3: Comparing exact time integral and our approximation in Eq. \ref{['eq:timeintegapproximation']} which leads to final expression \ref{['eq:genericformemory']}
• Figure 4: Memory slope with distinct equation of states ($n=\frac{2}{3w+1}$) for $k_*\sim {\cal H}_*$. Red curves ($n<1$) due to persistent source and blue curves ($n>1$) due to causal slope. $n=0.5, \, 0.75, \, 1,\, 1.75,\, 2,\, 2.25, \, 2.5$ correspond to $w = 1,\, 5/9, \, 1/3, \, 1/21,/, 0, \, -1/27, \, -1/15$, respectively.
• Figure 5: Comparison of the exact and approximate solutions for $n={0.5, \, 1.5, \, 2, \, 2.5, \, 3}$ corresponding to equations of states $w={1, \, 1/9, \, 0, \, 1/15, \, -1/9}$, respectively.