Approximate gauge independence of the induced gravitational wave spectrum

Abstract

Gravitational waves (GWs) induced by scalar curvature fluctuations are an important source of the cosmological GW background and a crucial counterpart of the primordial black hole scenario. However, doubts have been cast on the theoretically predicted induced GW spectrum due to its seeming gauge dependence. In this paper, we shed light on the gauge dependence issue of the induced GW spectrum in general cosmological backgrounds. First, inspired by the Hamiltonian formalism we provide very simple formulas for the tensor modes at second order in cosmological perturbation theory. We also emphasize the difference between observable and gauge invariant variables. Second, we argue that the Newton (or shear-free) gauge is suitable for both the calculation of induced GWs and the physical interpretation. We then show that, most notably, the induced GW spectrum is invariant under a set of reasonable gauge transformations, i.e. physically well behaved on small scales, once the source term has become inactive. This includes the commonly used flat, constant Hubble and synchronous gauges but excludes the comoving slicing gauge. We also show that a particular solution of the GW equation in a dust dominated universe while the source term is active can be gauged away by a small change of gauge.

Gauge-Invariant Tensor Perturbations Induced from Baryon-CDM Relative Velocity and the B-mode Polarization of the CMB

“…In practice, it may not be feasible to find the analytic form of χ h , the gauge condition, but the choice of χ h (the zero-IGW gauge) satisfying the above relation makes the IGW h ij in that gauge vanishes. This does not imply that the IGW can be "gauged away" as mentioned in [22]; h ijχ is the same as h ij in the zero-shear gauge and indicates a gauge-invariant combination, thus cannot be gauged away. It merely reflects that the IGW is gauge-dependent, and therefore we can find the gauge condition where the IGW vanishes.…”

“…Most recently, Ref. [22] preferred the Poisson gauge based on the coincidence among some gauges far inside the horizon for nonvanishing pressure. However, this mathematical observation alone cannot replace physical justification for a particular gauge choice.…”

“…Ref. [22] also claimed that the IGW in the dust era could be "gauged away". While it is true that the gauge dependent IGW can be eliminated by a gauge transformation, this is not a unique property of the dust dominated universe and does not imply that the IGW is physically irrelevant.…”

Probing primordial–black-hole dark matter with scalar induced gravitational waves

“…They found the new gauge invariant second-order GWs converged and were identical with the one in Newton gauge [179]. The idea of constructing new observable quantities was also tried in [180], where the authors found, for a series of gauges where the source term decay at a late time, the energy density of SIGWs will be identical with that in Newton gauge.…”

NANOGrav Signal from double-inflection-point inflation

“…It is well known that at first order in perturbation theory the scalar and tensor perturbations are decoupled, however, at second order the free wave equation of tensor perturbations gets a source term of scalar perturbations. Thus, when the scalar perturbations reenter the Hubble radius in the radiation-dominated(RD) era, it can leads to the production of secondorder GWs [8,9], and if the power spectrum of scalar perturbations is enhanced at small scales, the induced GWs can be sizable to be detected by experiments in near future [10][11][12][13][14][15][16][17][18][19][20][21][22].…”