Stochastic Limit of Growing Gravitational Wave Memory from Sources in the Early Universe and Astrophysical Sources

TL;DR

The paper identifies a growing gravitational-wave memory mechanism in early-universe and astrophysical contexts, showing that the resulting stochastic memory converges to fractional Brownian motion with a Hurst parameter $H$ in $(\tfrac{1}{2},1)$. By modeling memory events as Gaussian long-range dependent series with power-law growth $|t|^{1-\beta}$ or $|t|^{1-2\beta}$, the authors derive explicit conditions under which the memory background becomes $B_H(t)$ and characterize its covariance. They treat both asymptotically-flat and cosmological spacetimes, deriving how environmental fall-off and expansion affect the memory amplitude via redshift and lensing, while preserving the fBM scaling. The results offer a practical pathway to detect and characterize memory signals in PTA data and provide a theoretical framework linking GW memory to long-range dependence, potentially opening a window into the early universe's conditions and the behavior of primordial black holes. Overall, the work extends stochastic memory theory to gravitational-wave memories, providing quantitative growth laws, cross-source composition, and implications for data analysis and cosmology.

Abstract

We show that the stochastic background of gravitational wave memory of growing type leads to a fractional Brownian motion increasing at the order of $t^{H}$ for large $t$ where $\frac{1}{2} < H <1$. This beats the scaling law of Brownian motion. In this article we investigate sources of gravitational waves in the early universe as well as in astrophysical settings. Cosmological sources may include primordial black holes or other sources immediately after the Big Bang when there were pockets of hot material, and large density fluctuations. Gravitational waves from mergers of primordial black holes produce memory. We show that due to the conditions in which these are taking place the gravitational wave memory will be increasing in time following a certain power law. Corresponding results hold for any gravitational wave memory from a cosmological source where the surrounding conditions are similar. The stochastic limit of these memories is a stochastic process growing in time faster than the $\sqrt{t}$ scaling law of Brownian motion. The latter is also typical for noise and for the limit of memory events as they have been mostly considered in the literature. In an expanding universe, the memory is enhanced by the expansion itself. Our results provide a tool to extract gravitational wave sources of this type from data using this memory signature. This would be particularly useful for the PTA data that has been already observed, answering the long-standing question on how to extract memory signals from the data. Further, the new results open up a new door to explore the conditions right after the Big Bang using the long-range dependence and further probability analysis.