Physical Effects of Gravitational Waves at Second Order
Guillem Domènech, Shi Pi, Ao Wang
TL;DR
The paper analyzes gauge ambiguities in second-order induced gravitational waves (GWs) from primordial fluctuations. By modeling geodesic clocks exchanging electromagnetic signals, it derives the second-order time delay and redshift, showing the observable GW effect corresponds to the gauge-invariant quadrupole h^{N(2)}_{ij}, which matches the transverse-traceless part in the Newton gauge. The main result is an explicit, gauge-invariant expression for the second-order GW contribution to the time delay, establishing the Newton gauge TT component as the physical strain and clarifying boundary terms from observer motion. It discusses implications for pulsar timing arrays and GW memory, noting constant, non-propagating modes do not contribute to redshift. Overall, the work resolves the gauge issue for induced GWs and provides a robust framework for predicting second-order GW signals.
Abstract
There is currently no rigorous definition of gravitational wave strain at second order in cosmological perturbation theory. The usual association of gravitational waves with transverse and traceless fluctuations of the metric on spatial hypersurfaces becomes ambiguous at second order, as it inherently depends on the spacetime slicing. While this poses no practical issues in linearized gravity, it presents a fundamental problem for secondary gravitational waves, especially notorious for gravitational waves induced by primordial fluctuations. We compute, for the first time, the physical effects of gravitational waves at second order, as measured by geodesic observers that emit and receive electromagnetic signals, thereby settling the debate on gauge ambiguities. We find that the measured gravitational wave strain coincides with the transverse-traceless components in the Newton gauge.